Damping of hot giant dipole resonance due to complex configuration mixing

Transcription

1 31 December 1998 Physics Letters B Damping of hot giant dipole resonance due to complex configuration mixing Nguyen Dinh Dang a,b,1,, Kosai Tanabe a, Akito Arima b,c a Department of Physics, Saitama uniõersity, 55 Shimo-Okubo, Urawa, Saitama, Japan b Cyclotron Laboratory, RIKEN, -1 Hirosawa, Wako, Saitama , Japan c Ministry of Education, Chiyoda-ku, Tokyo, Japan Received 15 June 1998; revised 10 November 1998 Editor: W Haxton Abstract An approach is presented to study the width of the giant dipole resonance Ž GDR at non-zero temperature T, which includes all forward-going processes up to two-phonon ones Calculations are performed in 10 Sn and 08 Pb An overall agreement between theory and experiment is found for the GDR width and shape The total width of the GDR due to coupling of the GDR phonon to all ph, pp and hh configurations increases sharply as T increases up to T;3 MeV and saturates at T; 4 6 MeV The quantal width GQ due to coupling to ph configurations decreases with increasing T Itis almost independent of T if the contribution of two-phonon processes at T/ 0 is omitted q 1998 Elsevier Science BV All rights reserved PACS: 110Pc; 410Pa; 430Cz Keywords: Single-particle levels; Thermal and statistical models; Giant resonances The giant dipole resonance Ž GDR built on compound nuclear states Ž hot GDR, has been the subject of a considerable number of experimental and theoretical studies during the last fifteen years ŽSee wx 1 for the reviews The width of the GDR built on the ground state Ž gs GDR is composed mainly of Landau splitting, the escape width G, and the spreading width G x The Landau splitting represents the distribution of the GDR over a number of harmonic oscillators Ž phonons with frequencies around the GDR energy It can be well described within the random-phase approximation Ž RPA and amounts to only a small fraction of the total width The escape width G arises from particle emission It can be studied via coupling to the continuum and x is also small few hundreds kev The main part of the total GDR width comes from the spreading width G 1 Electronic address: dang@rikaxprikengojp On leave of absence from the Institute of Nuclear Science and Technique, VAEC, Hanoi Vietnam r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

2 ( ) N Dinh Dang et alrphysics Letters B due to coupling to complicated configurations such as ph and even more complex ones The extension of the microscopic descriptions of the width of the gs GDR such as those in w,3x to non-zero temperature Ž T/ 0 has shown that all the three components of the GDR width depend weakly on temperature w4 7 x This is in contradictory to the experimental systematic of the width of hot GDR, which increases rapidly at low excitation energies E ) Ž or temperature T At higher excitation energies the observed width increases slowly and even ) saturates at E G 130 MeV in tin isotopes w8 14 x The increase of the width at T/ 0 can be described by the thermal fluctuations of nuclear shapes w15 17 x They were also included in the recent adiabatic coupling model Ž ACM w18 x, whose results agree well with the data from the inelastic a-scattering experiments in w14x for the GDR width in 10 Sn and 08 Pb within 1 MeV -TF 3 MeV The width of the GDR may depend noticeably on the angular momentum J if the latter reaches a rather high value JG 35 " at T,15 18 MeV in a lighter 106 nucleus Sn w19 x Recently we have shown that the coupling of the collective dipole vibration Ž GDR phonon to the incoherent particle-particle Ž pp and hole-hole Ž hh configurations appearing at T/ 0 Ž the thermal damping, which is de-facto taking shape fluctuations into account, is decisively important for an adequate description of the width s increase and its saturation w0 x It has also been concluded that the quantal damping due to coupling to only ph configurations decreases slowly as T increases The application of this approach in a systematic study of the hot GDR in 90 Zr, 10 Sn, and 08 Pb has shown a reasonable agreement with the experimental data w0 x The higher-order graphs such as 1 p1hmphonon-, 1 p1 pmphonon-, 1h1hmphonon- orrand two-phonon ones have not been included explicitly in w0 x, but rather effectively in the parameters of the model The aim of the present letter is to study the contribution of these higher-order processes to the width of the hot GDR For this purpose a complete set of approximate equations will be derived including all the forward-going processes up to two-phonon ones at T/ 0 and applied in numerical calculations in 10 Sn and 08 Pb As the present paper is a further development of the approach in w0 x, the latter will be frequently quoted as I hereafter We consider the same model Hamiltonian as in I for the description of the coupling of collective oscillations Ž phonons to the field of ph, pp and hh pairs This Hamiltonian is composed of three terms: Hs Ea a q v Q Q q F Ž q a a X Q X qq Ž 1 Ý Ý Ý s s s q q q ss s s q q X s q ss q Ž The first term in the rhs of Eq 1 is the field of independent single particles, where as and as are creation and destruction operators of a particle or hole state with energy EssesyeF with es being the single-particle energy and ef - the Fermi surface s energy We will simply call the energy Es as the single-particle energy The second term stands for the phonon field, where Q q and Qq are the creation and destruction operators of a phonon with energy v The last term describes the coupling between the first two terms The indices s and s X q denote particle Ž p, E ) 0 or hole Ž h, E - 0, while the index q is reserved for the phonon state qsl,i4 p h with multipolarity l Ž the projection m of l in the phonon index is omitted in the writing for simplicity The sums in the last two terms in the rhs of Eq Ž 1 are carried over lg 1 In order to include all the propagators up to two-phonon ones let us introduce the double-time Green functions in the standard notation w3 x, which describe the following processes: a The propagation of a free phonon: G Ž X ²² Ž X :: Ž X X tyt s Q X t ;Q t,b The transition between a nucleon pair and a phonon: G X tyt q ;q q q ss ;q s²² a Ž t a Ž Ž X X t ;Q t ::,c s s q The transition between a nucleon pair m phonon configuration and a phonon: y,q G Ž X ²² Ž Ž Ž Ž X X X t y t s a t a X t Q X t ;Q t ::, and d ss q ;q s s q q The transition between two- and one-phonon configura- yy,q tions: G Žtyt X s²² Q Ž t Q Ž t ;Q Žt X :: q q q q q The effect of single-particle damping on the GDR width has 1 1 been found in I to be rather small up to high T Therefore we will not consider it here again The q,q Ž X ²² Ž Ž X :: q,q backward-going processes G Ž X ²² Ž Ž Ž Ž X X t y t s Q X t ;Q t, G X X t y t s a t a X t Q X t ;Q t :: q ;q q q ss q ;q s s q q,, and qq,q G Žty t X s²² Q Ž t Q Ž t ;Q Žt X :: q q q q q will be omitted as their effects on the damping of the GDR are 1 1 expected to be negligible Applying now the standard method of the equation of motion for the double-time Green function w3 x, we obtain a set of coupled equations for an hierarchy of Green functions Employing the decoupling approximation discussed previously in I, we can close this set to the functions Ž a Ž d Making then

3 ( ) N Dinh Dang et alrphysics Letters B the Fourier transformation to the energy plane E, we obtain a set of four equations for the Fourier transforms of the Green functions Ž a Ž d in the form: 1 Ž q Eyv G Ž E y F Ẋ G X Ž E s, Ý q q;q s1s1 s1s 1;q ss X p 1 1 Ý X X Ž q y,q Ž q y,q Ž EyE XqE G X E y F G X E yf G X X s s ss ;q sx s ss q ;q s s s s q ;qž E s0, Ž X sq 1 Ý yy,q Ž q 1 y,q Ž q y,q Eyv yv G E y F G X E qf G X q q q q ;q ssx ss q ;q ssx ss q ;q E s X ss Ý Ý y,q EyE XqE yv X G X X Ž E y 1yn Xqn X F Ž qx X G Ž E q Ž n qn X F Ž qx G X Ž E s s q ss q ;q s q s s1 ss 1;q s q s1s s1s ;q s1 s1 Ý yž n yn X F Ž q1 X G yy,q X s s s s q q ;q Ž E s0, Ž 5 1 q 1 The first three equations in this set are exact, while the last Eq Ž 5 is approximated due to the decoupling scheme mentioned above The latter leads to the single-particle occupation number ns and phonon occupation number n, whose explicit expressions have been derived in I Eliminating G X Ž E from Eqs Ž and Ž q ss ;q 3, we y,q obtain one exact equation, which relates G E to G X Ž E, in the form q;q ss,q X X Ž q Žq y,q Ž q y,q F F X G X X E yf G X X ssx ss ss q;q s s s s q ;qž E Eyvq Gq;q Ž E y s Ž 6 X X EyE XqE p Ý ss s 1 q s yy,q y,q Eliminating now G E by expressing it in terms of G Ž E using Eq Ž 4 q q ;q and inserting the result into Eq 1 Ž y,q 5, we come to the an approximate equation, which relates functions G Ž E to G X Ž E ss,q Making again the decoupling for all the Green functions under the sums in this equation, which truncates the chain at the second wž Ž q X x Ž q X y,q order O F X of the interaction strength F X, we can express G X X Ž E in terms of G X Ž E as ss ss ss q ;q q,q Ý y,q EyE XqE yv X G X X Ž E s M q 1q X X Ž E G Ž E Ž 7 s s q ss q ;q ss q 1;q q 1 y,q Inserting G X X Ž E from the approximate Eq Ž 7 into the exact Eq Ž ss q ;q 6, we end up with the final equation for the propagation of a single phonon Ž q sq in the following form Gq,qŽ E s Ž 8 p EyvqyPqŽ E q 1 q X Ž Ž The polarization operator P E and the vertex function M E in Eqs 7 and 8 are q ss X Ý X X X X Ž q Ž q qq Ž q qq F F X M E F M X ssx s s ss s s s sž E Pq Ž E s y, Ž 9 X X EyE XqE EyE qe yv X EyE XqE yv X ss s1q s s s1 s q s s1 q ½ Ž 1yn Xqn X Ž n yn Ž n qn X Ž n yn X X s q s s X s q s s X qq Ž q Ž q Ž q Ž q X Ý X X ss ss ss ss ss X s EyEs qes EyEs qe s X Ž q Ž q Ž q1 Ž q1 F X F F X ss s s ss F ss X X s s s qqý 5 Eyv yv X Eyv yv X q q q1 q1 q M E s F F y F F s qn Ž n yn qd Ž 10

4 4 ( ) N Dinh Dang et alrphysics Letters B The damping width GGDR of the hot GDR located at energy vsvgdr is defined via the imaginary part of the analytical continuation of the polarization operator P Ž E into complex energy plane E s v" i as q G sg Ž v < simp < Ž v"i < Ž 11 GDR q vsvgd R q vsv GDR X X The polarization operator P Ž E includes all 1s1s and 1s1s mphonon processes ŽŽs,s X sž p,h, Žp, p X q and Žh,h X and two-phonon ones at the same second order in the interaction strength In the limit of high T it is qq easy to verify that the vertex function M X Ž y1 X E in Eqs 9 and 10 decreases as OT ss with increasing T y1 because of the factor T in front of two-phonon terms on the rhs of Eq Ž 10 Neglecting these two-phonon terms would lead to a constant width at high temperature because the first two terms on the rhs of Eq Ž 10 are independent of T under the assumption that the temperature dependence of the interaction, of the phonon energy, and of the difference E X s y Es is negligible The decrease of the quantal width GQ of the GDR as OT Ž y1 at high T has also been estimated analytically in the SRPA in w4 x Although we proceeded the calculations up to TF 6 MeV, the high T-limit indicates the decreasing trend of GQ with increasing T as will be seen below The calculations of the GDR width G from Eq Ž 11 GDR have been carried out in Sn and Pb within the interval 0FTF 6 MeV The schematic model in I is employed, according to which the gs GDR is generated by a single collective and structureless phonon width energy vq closed to the energy EGDR of the gs GDR The realistic single-particle energies, calculated in the Woods-Saxon potentials at Ts0 for 10 Sn and 08 Pb, were used in calculations The calculations in w18x have shown that the major contribution of the shape fluctuations in the increase of the GDR width at T/ 0 comes from the quadrupole shape fluctuations Therefore, we retain here only dipole and quadrupole phonons in the two-phonon configuration mixing for simplicity Consequently, X from the sums on the rhs of Eqs Ž 9 and Ž 10 there remain only one dipole phonon with qsq, which q corresponds to the GDR ls 1 and one quadrupole phonon with q 1, which corresponds to the energy of 1 Ž Ž1 state ls The parameters of the model have been selected as follows We first set the ratio rsfi rfi Ž1 is 1, and choose v in Eq 10 to be close to E q q The values of vq in Eq 10 and of F1 are then 1 1 selected in such a way that the solution v of the equation for the pole of the Green function in Eq Ž 8 vyv yp Ž v s0 is equal to the GDR energy vse while the total width G Ž v from Eq Ž 11 q q GDR GDR Ž1 reproduces the empirical width of the GDR at Ts 0 The value of F is chosen so that v is stable while T is varied For 10 Sn the set of parameters has been found as: vq s170 MeV, F1 Ž1 s 661=10 y3 MeV, F Ž1 s 1845=10 y1 MeV, and rs 8603=10 y For 08 Pb the values vq s138 MeV, F1 Ž1 s 7=10 y3 MeV, F Ž1 s 995=10 y MeV, and rs 88=10 y1 have been selected These parameters are kept unchanged with changing T in the present paper This ensures that all thermal effects come from the microscopic configuration mixings, but not from varying parameters In fact, as has been shown, eg in wx 5, the first quadrupole state within the RPA has a considerable fragmentation with increasing T This offers some possibility to finely tune the parameters of the model, which we will not consider in this stage The inclusion of higher multipolarities such as octupole phonons in 08 Pb, etc is also expected to improve the results However it would certainly make the calculations more complicate The calculations have used a value equal to 05 MeV for the smearing parameter in Eq Ž 11 The results have been checked to be stable against varying within the interval 0 MeVF F 10 MeV The total widths G, calculated in Sn and Pb as a function of T, are shown in Fig 1 Ž solid GDR An overall agreement is found between theory and recent data from the inelastic a-scattering w14x and heavy-ion 10 fusion experiments w8,10,11 x In both nuclei the region of width s saturation is at around T;4 6 MeV In Sn 08 the saturated value of the width is about 1 MeV in agreement with the data from w8 13 x In Pb it is around 105 MeV As has been demonstrated in I, the total width GGDR is composed of the quantal width GQ due to coupling of the GDR phonon to ph configurations and the thermal width GT due to coupling to pp and hh configurations at T/ 0 The main conclusion of I is that the behavior of the total width at high temperatures is mostly driven by the thermal width GT since the quantal width GQ decreases as temperature increases In order to see whether this conclusion still holds within the present more refined approximation we have also switched

5 ( ) N Dinh Dang et alrphysics Letters B Fig 1 GDR width as a function of temperature for Sn Ž a and Pb Ž b Solid: total width G GDR; dashed: quantal width G Q; dash-dotted: quantal width G when the contribution of the two-phonon graphs at T / 0 is omitted Diamonds: data of w14 x Q ; squares and triangles: data by Enders et al and Hofmann et al w11 x, respectively; cross and asterisk: data from w8x and w10 x, respectively A level ) density parameter ; Ar8 has been used to translate E to T in fusion data of w8,10x off the coupling to pp and hh configurations in the sums on the rhs of Eqs Ž 9 and Ž 10 The results obtained are shown by the dashed curves in Fig 1 They restore perfectly the results for quantal width GQ in I, which show a clear decrease as T increases Switching off the two-phonon terms at T/ 0 from Eq Ž 10 in calculating the width GQ has resulted in a quantal width, which is practically independent of T as shown by the dash-dotted curves in Fig 1 in agreement with the conclusion within the NFT at T/ 0 w6,5 x These results show that the NFT indeed includes the graphs, which are most important for an adequate description of the quantal width GQ at Ts 0, namely the 1 p1h and 1 p1hmphonon ones However, in calculating G Q, the NFT neglects entirely the coupling to two-phonon configurations which enter in the same second order of the Ž interaction strength F q X ss These two-phonon processes at T/ 0 indeed lead to the decrease in the quantal width with increasing T as has been mentioned above Within the NFT, where the phonon operator has a ph nature, the exclusion of two-phonon terms was necessary to avoid double counting We also notice that the NFT solved an RPA equation at T/ 0 with the sum carried over mainly n yn as has been shown by Eq Ž 4 of wx h p 6 The self-energy graphs within the NFT at T s 0 described just the damping with ph-vibration doorways as has been shown by Fig 6 and Eqs Ž B8 and Ž B9 of wx 3 The deformation of Fermi surface at T/ 0 within NFT leaded to some ground-state correlation graph and single-particle renormalization, whose effect on the GDR damping are known to be small ŽFigs 5, 6, and Eqs Ž 11 Ž 14 of wx 6 Shown in Fig are the GDR shapes calculated within our model ŽŽ a and Ž b using the GDR strength GDR y1 w Ž function S v s p g v r v y v q g v x and those obtained within the ACM w18x ŽŽ c and Ž d q q q The areas of experimental divided spectra as well as the results from the ACM have been taken from w6 x The experimental spectra at low Ž MeV and high Ž MeV excitation energies are put in a one-to-one

6 6 ( ) N Dinh Dang et alrphysics Letters B Fig GDR shapes in Sn at low Ž MeV and high Ž MeV excitation energies In each panel the vertical bars denote the area of experimental divided spectra Results of the present work are plotted in Ž a and Ž b, where dashed curves show the shapes calculated without an additional parameter Dg Ž see the text while solid curves denote those obtained by adding Dg to g Ž v ŽDgs 05 MeV in Ž a q and MeV in Ž b Results of ACM using a strength parameter Ss 1 Ž dashed and 08 Ž solid are plotted in Ž c and Ž d correspondence with the data of w14x at T s 14 MeV and 31 MeV, respectively The calculated shape Ždashed curves in Figs Ž a and Ž b is slightly narrower than the experimental one Among the possible reasons there are the effects of mixing with more complex configurations, other multipolarities, angular momentum w19 x, evaporation width w7x at high T, and coupling to continuum, which are not considered in the present scheme An effective way to take into account these contributions is to add a smearing parameter Dg to the damping g Ž v q to recover the value of the total width at high T obtained in I By doing so the a better agreement between theory and experiment is achieved Žsolid curves in Figs Ž a and Ž b It is seen in both cases that our approach offers a reasonable agreement with the observed GDR shape in both regions of low and high excitation energies while the calculated strength function in the ACM does not follow a Lorentzian-like shape at high E ) Reducing the value of the GDR energy weighted sum rule value by 0% helped the ACM to improve the agreement with data at high E ) but worsened it at low excitation E ) In summary we have developed an approach to study the width of the GDR as a function of temperature, which includes all the forward-going processes up to two-phonon ones in the second-order of the interaction strength The present paper show that: Ž 1 the total width GGDR of the hot GDR arises mainly from the coupling of the GDR vibration to all ph, pp and hh configurations It increases sharply as temperature increases up to T;3 MeV At higher temperatures the width increase is slowed down to reach a saturated value of around MeV in Sn and around 105 MeV in Pb at T;4 6 MeV; Ž the quantal width GQ of the GDR due to coupling of GDR vibration to only ph configurations decreases as T increases Neglecting the two-phonon processes in the expansion to higher-order propagators results in a quantal width, which is almost independent of T; Ž 3 the calculated GDR shape in our model agrees reasonably well with the data The numerical calculations are limited within a schematic case whose parameters were selected to reproduce the empirical values of the GDR width and its energy at Ts 0 More refined microscopic studies in this direction are highly

7 ( ) N Dinh Dang et alrphysics Letters B desirable since the actual complexity of the wave functions can reveal the detail relationship between the eigenstates of the model Hamiltonian and the representation basis, and, therefore, may shed more light on the physics of the problem Other ingredients such as the temperature dependence of the single-particle energies, the contribution of the evaporation width, the effects of high values of the angular momentum have been also left out in the present study While there have been some indications on that these effects may not be crucial within the temperature region under consideration andror in nuclei with mass number A G 10 w11,14,19,6 x, they certainly deserve thorough studies in the future Acknowledgements NDD acknowledges the financial support of the Japanese Society for the Promotion of Science References wx 1 K Snover, Annu Rev Nucl Part Sci 36 Ž ; JJ Gaardhøje, Annu Rev Nucl Part Sci 4 Ž wx VG Soloviev, Theory of Atomic Nuclei Quasiparticles and Phonons, IOP, Bristol, 199 wx 3 GF Bertsch, PF Bortignon, RA Broglia, Rev Mod Phys 55 Ž wx 4 H Sagawa, GF Bertsch, Phys Lett B 146 Ž wx 5 N Dinh Dang, J Phys G 11 Ž 1985 L15 wx 6 PF Bortignon et al, Nucl Phys A 460 Ž wx 7 N Dinh Dang, Nucl Phys A 504 Ž wx 8 JJ Gaardhøje et al, Phys Rev Lett 53 Ž ; 56 Ž ; 59 Ž wx 9 DR Chakrabarty et al, Phys Rev C 36 Ž w10x A Bracco et al, Phys Rev Lett 6 Ž ; 74 Ž w11x G Enders et al, Phys Rev Lett 69 Ž ; HJ Hofmann et al, Nucl Phys A 571 Ž w1x JH Le Faou et al, Phys Rev Lett 7 Ž w13x D Perroutsakou et al, Nucl Phys A 600 Ž w14x E Ramakrishnan et al, Phys Rev Lett 76 Ž ; T Baumann et al, Nucl Phys A 635 Ž w15x M Gallardo et al, Nucl Phys A 443 Ž w16x N Dinh Dang, J Phys G 16 Ž w17x Y Alhassid, B Bush, S Levit, Phys Rev Lett 61 Ž ; Y Alhassid, B Bush, Nucl Phys A 509 Ž ; A 531 Ž , 39; B Bush, Y Alhassid, Nucl Phys A 531 Ž w18x WE Ormand, PF Bortignon, RA Broglia, Phys Rev Lett 77 Ž ; WE Ormand et al, Nucl Phys A 614 Ž w19x M Matiuzzi et al, Nucl Phys A 61 Ž w0x N Dinh Dang, A Arima, Phys Rev Lett 80 Ž ; Nucl Phys A 636 Ž w1x N Dinh Dang, F Sakata, Phys Rev C 57 Ž ; C 55 Ž wx N Dinh Dang, Phys Rep 64 Ž w3x DN Zubarev, Sov Phys Uspekhi 3 Ž w4x K Tanabe, Nucl Phys A 569 Ž c w5x P Donati et al, Phys Lett B 383 Ž w6x G Gervais, M Thoennessen, WE Ormand, Phys Rev C 58 Ž R w7x Ph Chomaz, Phys Lett B 347 Ž

8 31 December 1998 Physics Letters B Strange r anti-strange asymmetry in the nucleon sea HR Christiansen 1, J Magnin Centro Brasileiro de Pesquisas Fısicas, CBPF - DCP, Rua Dr XaÕier Sigaud 150, Rio de Janeiro, Brazil Received 19 January 1998; revised 6 October 1998 Editor: J-P Blaizot Abstract We analyze the non-perturbative structure of the strange sea of the nucleon within a meson cloud picture In a low Q approach in which the nucleon is viewed as a three valon bound state, we evaluate the probability distribution of an in-nucleon Kaon-Hyperon pair in terms of splitting functions and recombination The resulting kaon and hyperon probability densities are convoluted with suitable strange distributions inside the meson and baryon in order to obtain non-perturbative contributions to the strange sea of the nucleon We find a structured strangeranti-strange asymmetry, displaying a clear excess of quarks Ž anti-quarks for large Ž small momentum fractions q 1998 Elsevier Science BV All rights reserved 1 Motivation According to the Quark Parton Model, with its subsequent improvements coming from Quantum Chromodynamics, it is known that hadrons are built up from a fixed number of valence quarks plus a fluctuating number of gluons and sea quark anti-quark pairs As the momentum scale rises up, the nucleon s sea is mainly perturbatively generated and consequently quarks and anti-quarks have the same probability and momentum distributions However, a small fraction of the sea may be associated with non-perturbative processes raising the possibility of generating unequal quark and anti-quark distributions For instance, a non-perturbative q-q asymmetry in the charmed sea of the nucleon could play an important role in explaining the excess of events at large x and 1 hugo@catcbpfbr jmagnin@lafexcbpfbr Q in e q p neutral and charged current deep inelastic scattering wx 1, as seen at HERA by H1 wx and ZEUS wx 3 Regarding the strange nucleon s sea, the present status of the experimental data does not exclude the possibility of asymmetric s and s distributions Actually, it is an important question related to current experimental research Recently, the CCFR Collaboration has analyzed the strange quark distribution in nucleons allowing explicitly for sž x / sž x distriwxž 4 see also the CTEQ analyses wx 5 Al- butions though the analysis of CCFR seems to indicate a rather small asymmetry, the error bars are still too large to be conclusive, leaving enough place to conjecture about sizeable unequal s and s distributions Žsee discussion in Refs w6,7 x Most recently, the E866 Collaboration wx 8 has measured a large u-d asymmetry in the nucleon which may be succesfully accomodated in the framework of a pion cloud model The question of a non-perturbative asymmetry in the nucleon sea has been addressed by a number of r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

9 ( ) HR Christiansen, J MagninrPhysics Letters B authors The most significant framework to discuss this issue appears to be the Meson Cloud Model Ž MCM In this context, it is assumed that the nucleon can fluctuate to a meson-baryon bound state with a small probability Non-perturbative sea-quark distributions are then associated with the valence quark and anti-quark distributions inside the baryon and the meson respectively A first attempt in this direction was carried out in Ref wx 6 where the nucleon-meson-baryon vertex is parameterized with adwx 7, intrinsic sea quark and hoc form factors In Ref anti-quarks are rearranged with valence quarks using simple two-body wave functions for describing meson-baryon bound states Finally, let us also mention a different approach to the problem, based on a generalized chiral Gross-Neveu model at large N c wx 9 In this letter, we employ both effective and perturbative degrees of freedom and show that a wellknown scheme related to hadron production in hadronic collisions, viz recombination, may be succesfully applied for describing meson-baryon fluctuations yielding non-perturbative sea quark-antiquark asymmetries Our description of the nucleon-mesonbaryon vertex starts from the perturbative production of a qq pair out of a dressed valence quark Such perturbative sea quarks subsequently rearrange with the remaining valence quarks Õia recombination, in order to give rise to a hadronic nucleon fluctuation As we will see, this provides a phenomenologically motivated one-parameter dependent description of the low Q scale meson and baryon probability distributions inside the nucleon The non-perturbative contributions to the strange sea of the nucleon are then calculated by means of widely used convolution forms We obtain a structured asymmetry indicating a clear excess of strange quarks Žanti- quarks for large Ž small momentum fraction, becoming negligible beyond x; 08 Strange meson-baryon fluctuation and s-s asymmetry in the nucleon We start by considering a simple picture of the nucleon in the infinite momentum frame as being formed by three dressed valence quarks Õalons which carry all of its momentum The valon distribution in the nucleon has been calculated by Hwa in Ref w10 x, and it has been found to be 105 ÕŽ x s ' x Ž 1yx Ž 1 16 For the sake of simplicity, we will not distinguish between u and d valon densities, so identical momentum distribution will be assumed for both flavors In the framework of the MCM, the nucleon can fluctuate to a meson-hyperon bound state carrying zero net strangeness As a first step in such a process, we may consider that each valon can emit a gluon which, before interacting, decays perturbatively into a ss pair The probability of having such a perturbative qq pair can then be computed in terms of Altarelli-Parisi splitting functions w11x 41qŽ 1yz PgqŽ z s, 3 z 1 PqgŽ z s z q Ž 1yz These functions have a physical interpretation as the probability of gluon emission and qq creation with momentum fraction z from a parent quark or gluon respectively Hence, 1 / ast Q dy x qž x,q sqž x,q sn H P qg Ž p y ž x y H gqž / 1 = dz P y ÕŽ z Ž 3 z z y is the joint probability density of obtaining a quark or anti-quark coming from subsequent decays Õ Õ qg and g qqq at some fixed low Q As the valon distribution does not depend on Q w10 x, the scale dependence in Eq Ž 3 only exhibits through the strong coupling constant a st The range of values of Q at which the process of virtual pair creation occurs in our approach is typically about 1 GeV,as dictated by the valon model of the nucleon For definiteness, we will use Q s 08 GeV as in Ref w x 10, for which a st;03 is still sufficiently small to allow for a perturbative evaluation of the qq pair production Since the scale must be consistent with the valon picture, the value of Q is not really free and cannot be used to control the flavor produced at

10 10 ( ) HR Christiansen, J MagninrPhysics Letters B the gqq vertex Instead, this role can be ascribed to the normalization constant N, which must be such that to a heavier quark corresponds a lower value of N For instance, it could be fixed by comparing the output of Eq Ž 3 with fit analyses of experimental data of sea quark distributions in the nucleon at low Q However, we will for simplicity include the whole factor appearing in Eq Ž 3 in the global normalization of the < MB: Fock state Once a ss pair is produced, it can rearrange itself with the remaining valons so as to form a most energetically favored meson-baryon bound state To obtain the meson and baryon probability densities inside the nucleon, one has to employ effective techniques in order to deal with the non-perturbative QCD processes inherent to the dressing of quarks into hadrons Although in-nucleon meson and baryon are virtual states, one may assume that the mechanisms involved in their formation are similar to those at work in the production of real hadrons in hadronic collisions Therefore we shall proceed to describe the N MB fluctuation by means of a well-known rew1 x Notice that as the nucleon fluctuates into a me- combination model approach son-baryon bound state, the meson and baryon distributions inside the nucleon are not independent Actually, to ensure the zero net strangeness of the nucleon and momentum conservation, the in-nucleon meson and baryon distributions must fulfill two basic constraints: H B M 0 1 dx P Ž x yp Ž x s0, H B M 0 1 dx xp Ž x qxp Ž x s1, Ž 4 which can be both guaranteed by choosing P Ž x sp Ž 1yx Ž 5 M B for all momentum fractions x 3 In addition to satisfying conservation laws, condition Ž 5 enables us to calculate just one of the two distributions Therefore, we can proceed to compute the strange meson distri- 3 Although condition Ž 5 has been generally employed Žsee eg Refs w1,13 x, it is not a unique choice since Eqs Ž 4 relate the integrals of the meson and baryon probability densities and not the distributions themselves bution P Ž x along the lines of Ref w1x M and then relate it to the hyperon probability as indicated in Eq Ž 5 Using the recombination model, the probability density of having a meson out of two quarks is given by xdy xyy dz PM Ž x sh H FŽ y, z RŽ x, y, z, Ž 6 y z where F y, z 0 0 is FŽ y, z sb yõž y zqž zž 1yyyz Ž 7 and RŽ x, y, z is the recombination function associated with the meson formation yz yqz RŽ y, z sa d 1y Ž 8 ž / x x The normalization of the meson distribution is not given by the recombination model We shall thus fix the overall normalization as given by the experimental probability that the strange hadronic fluctuation occurs; this value is phenomenologically estimated to be about 4 10% Žsee eg Refs w7,13 x In Fig 1 we show our calculated meson and baryon probability densities inside the nucleon The non-perturbative strange and anti-strange sea distributions can be now computed by means of the two-level convolution formulas 1 dy NP s Ž x sh PBŽ y sbž xry, y x 1 dy NP s Ž x sh PMŽ y smž xry, Ž 9 y x where the sources s Ž x and s Ž x B M are primarily the probability densities of the strange valence quark and anti-quark in baryon and meson respectively, evaluated at the hadronic scale Q wx 6 In principle, to obtain the non-perturbative distributions given by Eqs Ž 9, one should sum over all the meson-hyperon fluctuations of the nucleon but, since such hadronic Fock states are necessarily off-shell, the most likely configurations are those closest to the nucleon energy-shell, namely L 0 K q, S q K 0 and S 0 K q, for a proton state However, as we are using the same distribution for u and d valons, the three configurations above contribute in the same way to the strange non-perturbative structure of the nucleon Other fluctuations involving heavier in-nucleon mesons and

11 ( ) HR Christiansen, J MagninrPhysics Letters B Fig 1 Meson and baryon probability densities in the nucleon, normalized to 4% Full line is for the strange meson distribution in the nucleon Dashed line shows the strange baryon distribution P Ž x sp Ž 1yx B M baryons should be strongly suppressed due to their high virtuality 4 As long as experimental measurements are lacking at present, several choices are possible for the s M and sb distributions in K-ons and L or S baryons respectively In this respect, it has been a common practice to employ modified light valence quark distributions of pions and protons w6,13 x However, we think that one should better use simple forms reflecting the fact that strange valence quarks should carry a rather large amount of momentum in Ss"1 hadron states at low Q scales Indeed, the momenw14x in the tum distribution found by Shigetani et al framework of a Nambu-Jona Lasinio model at low 4 For instance, a loosely bound pf configuration can be neglected since it is rather heavier than a LK state and Zweig s rule suppressed Q, adequately reflects this feature see also Ref w15 x A similar form for the strange distribution in K-ons has been found very recently in Ref w16 x, using a Monte Carlo based program for generating valence distributions in hadrons We thus put forward the following simple form Ž xsm Ž x s6 x Ž 1yx, Ž 10 which is an excellent approximation to these phenomenological data For similar reasons, in L and S baryons we expect a sb momentum distribution peaked around 1r, which after normalization can be approximated by B xs x s1 x 1yx 11 For comparison, we have also considered other forms for sm and sb in order to display their effect on the shape of the non-perturbative distributions of Eqs Ž 9 As suggested in Refs w6,13 x, we have used

12 1 ( ) HR Christiansen, J MagninrPhysics Letters B NP NP Fig Upper: The non-perturbative strange distributions in the nucleon sea Full lines are for s thin and s thick, as obtained by using Eqs Ž 10 and Ž 11 Point lines are for the non-perturbative distributions coming from the choice s su r Ž thin and s sž B N M 1y 018 p x q Ž thick Lower: The sž x ysž x asymmetry in the nucleon sea The full line is the asymmetry obtained by using sb and sm as given by Eqs Ž 10 and Ž 11, whereas the point line is the asymmetry coming from a sbsunr distribution in the hyperon and a s sž 1yx 018 q p in the kaon M s M s 1yx 018 q p, where q p is the valence distribution in pions, and sbsunr, with un the up valence distribution in the nucleon; q p and un are w x NP those given in Ref 16 In Fig we show the s NP and s distributions as well as the sys asymmetry predicted within our model 3 Summary and discussion By means of an approach involving both effective and perturbative degrees of freedom, we have obtained s and s non-perturbative distributions in the nucleon s sea We based our approach on a valon description of the initial state of the nucleon, which perturbatively produces sea quarkranti-quark pairs Thereafter these quarks and anti-quarks give rise to a hadronic bound state by means of recombination with the remaining valons An interesting feature of the model is that it allows a full representation of the non-perturbative processes inside the nucleon in terms of a well-known effective scheme of hadron physics w1 x In this way, a specific connection between the physics of hadronic reactions and that of hadron fluctuations is established in the approach This is a pleasant aspect of the approach since the same physical principles, related to hadronization, should be involved in both situations The present model also exhibits a remarkable economy of parameters when compared to other approaches in the literature Notice for instance that neither nucleon-meson-hyperon coupling constants nor vertex functions are used to recognize the extended nature of the nucleon-meson-hyperon vertex,

13 ( ) HR Christiansen, J MagninrPhysics Letters B avoiding the need of such parameters, whose values are still controversial As can be seen in Fig, the model predicts a definite structured asymmetry in the strange sea of the nucleon As we have discussed in the introduction, our results are compatible with the present status of experimental data coming from n N and n N scattering, due to the large error bars allowed for such distributions wx 4 One should however notice NP NP that the predicted non-perturbative s and s distributions depend on the form of the strange and anti-strange distributions in the in-nucleon baryon and meson respectively So, although the model is reliable in predicting different distributions for sea quarks and anti-quarks, their exact shapes cannot be determined until we have more confident results for quark densities inside strange hadrons Nevertheless, the model clearly predicts an excess of sea quarks over anti-quarks carrying a large fraction of the nucleon s momentum Ž see Fig and this appears to be independent of the exact form of the strange and anti-strange distributions inside the in-nucleon baryon and meson This result coincides with the predictions of Refs w6,7,9x and strongly differs with the outcome w x NP NP of Ref 13 Our s and s distributions, as well as the difference between them, are similar in shape and magnitude to those found in a recent analysis with a different approach based on two-body wave functions with a number of parameters wx 7 Another point deserving a comment concerns the probability of the strange meson-baryon fluctuation As far as we know, direct experimental information is still not available and one then has to rely on phenomenological estimates that it should range between 4 and 10% It has been recently shown that double polarization observables in f meson photoproduction off protons w17x are very sensitive to the strange quark content of the proton itself, indicating an interesting source of experimental information for giving further support to the current discussion The present analysis can be easily extended to study non-perturbative contributions to other flavors A pion cloud version of the model should also predict a noticeable u-d asymmetry in the sea of the nucleon, as announced by the recent data of the E866 Collaboration wx 8 In this sense, it is worth to mention that the data analysis resulting from the above mentioned experiment favors an explanation in terms of the MCM rather than some alternative approaches Acknowledgements We acknowledge useful discussions with M Malheiro We also thank Centro Brasileiro de Pesquisas Fısicas Ž CBPF for the warm hospitality extended to us during this work HRC was partially supported by Centro Latino Americano de Fisica Ž CLAF JM and HRC are supported by Fundaçao de Amparo a` Pesquisa do Estado de Rio de Janeiro Ž FAPERJ References wx 1 W Melnitchouk, AW Thomas, Phys Lett B 414 Ž wx C Adloff et al Ž H1 Colaboration, Z Phys C 74 Ž wx 3 J Breitweg et al, ZEUS Collaboration, Z Phys C 74 Ž wx 4 AO Bazarko et al, CCFR Collaboration, Z Phys C 65 Ž wx 5 J Botts et al, CTEQ Collaboration, Phys Lett B 304 Ž ; H Lai et al, Phys Rev D 51 Ž wx 6 A Signal, AW Thomas, Phys Lett B 191 Ž wx 7 SJ Brodsky, BQ Ma, Phys Lett B 381 Ž ; BQ Ma, SJ Brodsky, SLAC-PUB-7501, hep-phr wx 8 EA Hawker et al, E866rNuSea Collaboration, Phys Rev Lett 80 Ž ; JC Peng et al, E866rNuSea Collaboration Phys Rev D 58 Ž , hep-phr wx 9 M Burkardt, BJ Warr, Phys Rev D 45 Ž w10x RC Hwa, Phys Rev D Ž , 1593 w11x G Altarelli, G Parisi, Nucl Phys B 16 Ž w1x KP Das, RC Hwa, Phys Lett B 68 Ž w13x W Melnitchouk, M Malheiro, Phys Rev C 55 Ž w14x T Shigetani, K Suzuki, H Toki, Phys Lett B 308 Ž w15x JT Londergan, GQ Liu, AW Thomas, Phys Lett B 380 Ž w16x A Edin, G Ingelman, Phys Lett B 43 Ž w17x AI Titov, Y Oh, SN Yang, T Morii, Phys Rev C 58 Ž nucl-thr

14 31 December 1998 Physics Letters B Fragmentation of tensor polarized deuterons into cumulative pions S Afanasiev a, V Arkhipov a, V Bondarev a,b, M Ehara c, G Filipov a,d, S Fukui c, S Hasegawa c, T Hasegawa e, N Horikawa c, A Isupov a, T Iwata c, V Kashirin a, M Kawano e, T Kageya c,1, A Khrenov a, V Kolesnikov a, V Ladygin a, A Litvinenko a, A Malakhov a, T Matsuda e, H Nakayama c, A Nikiforov a, E Osada e, Yu Pilipenko a, S Reznikov a, P Rukoyatkin a, A Semenov a, I Semenova a, N Takabayashi c, A Wakai c, L Zolin a, a Joint Institute for Nuclear Research, Dubna , Moscow region, Russia b St-Petersburg State UniÕersity, St-Petersburg, , Russia c Nagoya UniÕersity, Nagoya , Japan d INPNE, 1784 Sofia, Bulgaria e Miyazaki UniÕersity, Miyazaki 889-1, Japan Received 11 May 1998; revised 4 November 1998 Editor: JP Schiffer Abstract The tensor analyzing power T of the reaction dqa p y Ž 08 0 qx has been measured in the fragmentation of 9 GeV tensor polarized deuterons into pions with momenta from 35 to 53 GeVrc on hydrogen, beryllium and carbon targets This kinematic range corresponds to the region of cumulative hadron production with the cumulative variable xc from 108 to 176 The values of T0 have been found to be small and consistent with positive values This contradicts the predictions based on a direct mechanism assuming NN collision between a high momentum nucleon in the deuteron and a target nucleon Ž NN NNp q 1998 Elsevier Science BV All rights reserved PACS: 160-n; 470qs; 485qp; 510qs Keywords: Deuteron; Tensor analyzing power; Pion; Cumulative production To study the deuteron core structure, a number of experiments in a GeV region including the polariza- 1 Present address: Univ of Michigan, Ann Arbor MI address: zolin@moonhejinrru tion ones, have been performed in the last two decades The nucleon momentum distributions in a deuteron, extracted from the deuteron breakup wx 1 and ed- inelastic scattering wx, are in good agreement with each other confirming that both hadron and electron probes give reliable information on the deuteron structure These data were reproduced rather r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

15 ( ) S AfanasieÕ et alrphysics Letters B well by the relativistic impulse approximation Ž IA calculations using conventional deuteron wave functions Ž DWF w3,4x at internal momenta up to k s 05 GeVrc 3 In the higher momentum region, however, a marked deviation from the calculations was observed Taking into account additional mechanisms to IA w6,7x gives a satisfactory description of the momentum distribution data A deviation from IA was also observed in the data on the tensor analyzing power T0 for deuteron inclusive breakup with the emission of the proton at a zero angle w8 10x and for dp-backward elastic scattering w11,1x as shown in Fig 1 The same values of T0 are assumed in both reactions since one nucleon exchange mechanism is to be dominant in both of them However, the data for these reactions disagree in a high momentum region Furthermore, the dp-backward data demonstrate a strong oscillation structure in the region k) 03 GeVrc The experimental data over all internal momentum region cannot be described by The calculations takw6,13x or ing account of additional mechanisms to IA considering additional components to the DWF due to relativistic effects w14 17 x On the other hand, the behavior of T0 at high nucleon momenta is explained by the models incorporating the nonw18,19 x These theoretical investigations give different expla- nucleonic degrees of freedom in the deuteron nations to the data on different reactions, in particular in a high momentum region, based on different pictures of the deuteron core structure and nucleon correlations at short distances It is desirable to have a global picture which could explain the data The deuteron fragmentation into cumulative hadrons is one of the suitable reactions to study the deuteron core structure Cumulative hadrons are those produced in a nuclear reaction beyond the kinematic limit of free nucleon-nucleon collisions w0,1 x The simplest model used to describe their production is a 3 At high energy deuteron fragmentation it is preferable to use the definition of the internal momenta in the infinite momentum frame IMF ks( Ž mn q ph rž 4a Ž 1y a y m N, where a s Ž E q p rž E q p sž E q q N N<< d d q << rmd is the light cone variable; E N, pn<< and E q,q<< are the nucleon energies and momenta parallel to the incident deuteron momentum pd in the laboratory and in the deuteron frames, respectively; ph is the transverse component of the nucleon momentum Žfor references see wx 5 Fig 1 The experimental data for the reactions induced by tensor polarized deuterons in GeV region: the T0 data for deuteron breakup da px on the hydrogen and carbon targets w8 10 x; and the T data for backward elastic dp pd w11,1 x 0 The solid curve is the IA calculation with the Paris DWF wx 3 for both breakup and backward elastic scattering, and the dashed curve is the calculation wx 6 taking account of final state interactions for breakup at 9 GeVrc The direct mechanism IA-prediction for dc p Ž 08 X w3x is also shown Ž dot-dashed curve direct mechanism, where a nucleon with a high momentum in the projectile, for instance in the deuteron, collides against a nucleon in the target nucleus and produces a cumulative hadron Žeg NN NNp w x The features of cumulative reactions reflect the nuclear structure at small distances where short range nucleon correlations take place In case of the deuteron, spin observables in the fragmentation of deuterons into cumulative hadrons allow one to extract information on the deuteron core spin structure In order to compare the data on cumulative hadrons obtained under different kinematic conditions, the cumulative variable x Ž c cumulative num- ber w0,3x is often used in the literature For the reaction of cumulative pion production, AqB p qx, it is expressed as Ž pb pp y Ž m pr x c s, Ž 1 Ž p p ym y Ž p p A B N A p where p A, pbare the four-momenta per nucleon for projectile and target nuclei, respectively, ph is the four-momentum of the produced pion, m and m N p

16 16 ( ) S AfanasieÕ et alrphysics Letters B are the nucleon and pion masses In its meaning, the cumulative number, x c, is a minimum target mass Ž measured in nucleon mass units in the rest frame of a fragmenting nucleus needed to obey the 4-momentum conservation law for particle production in nucleon-nucleus collision Hence, its range is up to A, the mass number of the fragmenting nucleus When the hadron with x c larger than 1 is produced, it is assigned to a cumulative hadron The backward elastic scattering dp pd is the special case of cumulative reactions when x c s In the framework of the direct mechanism, the cumulative hadron is produced by a nucleon with a high internal momentum k in the fragmenting nucleus Hence, the minimum momentum of the nucleon, k min, which is wanted for cumulative hadron production can be related to x c: kmin increases from 0 as x c increases from 1 In this paper, the data on the tensor analyzing power T at the fragmentation of deuterons into cumulative pions are presented vs the variables x c and k min The first measurement of T in cumulative pion production in the forward direction dqa p " Ž 08 qx has been performed in our previous experiment w3x using a carbon target The values of < T < 0 were found to be about one order smaller than those predicted by the IA calculations However, it might be due to the nuclear effect of carbon target Moreover, it was difficult to achieve definite conclusion concerning N ' q ž y1 y N / 0 N q y q pzzypzz N y T s Ž 3 the sign of T0 due to large errors Here, we report new data on T0 obtained in the fragmentation of deuterons into negative pions on hydrogen, beryllium and carbon targets Ž A s 1,9,1 at a zero emission angle over the range of the cumulative variable x c from 108 to 176 The val- ues of T0 were determined by measuring the ratio Ž q of pion yields N rn y emitted at forward angles Ž u F 15 mrad at different tensor polarizations Ž p " p zz of the deuteron beam: The measurements were carried out with a 9 GeVrc polarized deuteron beam slowly extracted from the Dubna 10 GeV accelerator ŽSynchro- 0 0 phasotron The ion source POLARIS w4 x, operating in the tensor polarization mode, provided three tensor polarization states q,y,0 Ž 0 means the absence of polarization A series of polarization states was cyclically repeated every three spills The beam polarization was measured with the two-arm polarimeter ALPHA w5x by detecting dp-elastic scattering at a deuteron momentum of 3 GeVrc w6 x The values of tensor polarization for the q and q states were p s 0643 " 0033Ž stat zz " y 006 syst and p sy079 " 004Ž stat zz " 009Ž syst To study beam polarization stability for a 5-day run, the beam line polarity was switched periodically to register protons and the T0 p of the deuteron breakup reaction dqc pž 08 qx was p measured 4 times in all The value of T0 at k;03 GeVrc averaged over the run is shown by a star in Fig 1 It is consistent with the data obtained w x p in other experiments 8,9 The analysis of T0 fluctuations during the run showed that the values of polarization remained stable within "% w7 x The deuteron beam intensity was 15 to 5P10 9 drspill The duration of the beam spill was 05 s The deuterons with a fixed momentum of 9 GeVrc were led to the target T located at focus F5, the starting point of secondary beam line 4V as shown in Fig The position of the beam and its profile Ž s s4 mm, s s9 mm at the target point were x y Fig The layout of the setup of the 4V beam line in the Synchrophasotron experimental hall M i; bending magnets, L i; quadrupole magnets, F i,c i; scintillation and Cherenkov counters, HT, H0 XY, H0UV; scintillation hodoscopes, T; target, IC; ionization chamber

17 ( ) S AfanasieÕ et alrphysics Letters B monitored with the multiwire chambers The beam intensity was monitored with the ionization chamber IC placed 1 m upstream the target The magnetic optics of line 4V, consisting of four magnetic dipoles M0 M3 and three quadrupole doublets L1 L3, provided a momentum resolution of sprp s"3% and an angular acceptance of DVs1P10 y4 sr The particles produced in the target were led to final focus F6 located 4 m downstream the target The scintillation hodoscopes H0XY and H0UV were used to control the beam size at F6 The detecting system consists of two counter sets as shown in Fig One set, a scintillation hodoscope HT and counters F 56 1y 4, was placed behind the collimator at the intermediate beam focus, and the other, counters F 6 1y 3, was 7 m downstream at focus F6 Pions were selected by time of flight Ž TOF between the intermediate focus and the focus F6 with a time resolution of stof s 0 ns To reject background due to the pile-up effect in F 56 1y 4, a correlation of two TOF pairs was required Although events corresponding to pions are only seen in the TOF spectrum, muons from the pions decay should contribute to the peak Their number was estimated to be no more than 8% w3 x It was confirmed by GEANT simulation w8x that they originate mainly from pions decaying in the final section of beam line 4V For this reason we did not exclude muon events when we calculate the N q rn y -ratio to extract T0 according to Ž 3 The pion momentum pp varied from 35 to 53 GeVrc at a fixed deuteron beam momentum of pds90 GeVrc to range x c from 108 to 176 The measurements at pp s 35, 40, 45 and 50 GeVrc were carried out with the hydrogen target Ž7 grcm To study the target dependence, the measurement at pp s50 GeVrc was performed also with the beryl- lium target Ž36 grcm, and the highest x c point, x s 176, was taken with the carbon target Ž c 55 grcm Such relatively thick targets do not affect on the T According to Ž 3 the T depends on the 0 0 N q rn y -ratio only at fixed p " It means that the zz results of T -measurements are not affected by the 0 distortion of the cumulative pion spectra due to secondary interactions in a target and beamline materials as the distortion factor is the same for both polarization states of the primary deuteron beam Monte Carlo simulation shows that pion rescattering in the carbon target Ž55 grcm increases the angular acceptance by 10% with respect to a thin target; an effective value of x c increases by, % due to energy losses and p A-interactions in the target Experience in studying T0 in deuteron breakup confirmed its independence of target thickness w9,10 x The background pions not from the target, were measured to be about 3% for a 36 grcm target and 8% for a 7 grcm target They cannot change radically the N q rn y -ratio To estimate an effect of pions originated not from the reaction Ž which can affect on N q rn y, we estimated a contribution from the two step process where high momentum nucleons from a deuteron breakup produce high momentum pions in secondary collisions in a target: da NX, NA p X Our numerical estimations for a thick target Ž55 grcm, based on the known approximation of the cross sections near the kinematic limit for NA p X w9 x, showed that the contamination of pions from this process is no more than 1% Table 1 presents values on T0 obtained in this experiment for p y production as a function of x c and k min At Fig 3 the new T0 data are plotted together with the p " data obtained in the previous experiment on carbon w3 x Comparing the values of T0 obtained on hydrogen, beryllium and carbon targets, T0 in pion production was found to be independent of the mass number of the target Although a weak target dependence was seen in the inclusive cross sections for cumulative hadron production w30,31 x, it is evident that T0 is predominantly ruled by the internal structure of the fragmenting nucleus, in this case the deuteron In the framework of the direct mechanism, the increasing x c corresponds to the increasing k min, thus decreasing internucleonic distances r NN Using the momentum-space transformation, rnn is evaluated to be r NN,04 fm at x cs17 This internucleonic distance corresponds to a large D-wave contribution to the DWF, and therefore a large value of the tensor analyzing power could be expected The rew3x performed in the frame- sults of the calculations work of the direct mechanism using the Paris DWF wx 3 are shown in Fig 3 It is notable that the measured values of T0 have a positive sign opposite to the sign predicted by the calculations The value of T is nearly zero at x s10 and increases up to 0 c

18 18 ( ) S AfanasieÕ et alrphysics Letters B Table 1 T as a function of x and k ; d x is the width of x -bin Ž 1s 0 c min c c due to the beam momentum resolution and energy losses in the target Ž 3 with the indicated thickness DT grcm All the momenta p p,kmin are in GeVrc The invariant cross section sinvsep dsrd pp is in mbpgev y Pc 3 Psr y1 rnucleus, the uncertainty of sinv is estimated to be equal to 0% The systematic error of T0 is caused by the uncertainty of beam polarization p Target D x d x k s x T " stat" sys p T c c min inv c 0 35 H " " 0059" H " " 0047" H " " 0058" H " " 0098" Be " " 0056" C " " 0118" 0010, 015 at x c s 17 not showing an extremum at k,03 GeVrc peculiar to IA predictions Ž Fig 1 Taking account of relativistic effects w3x or processes additional to the direct mechanism like p-rew33x cannot explain such a behaviour of scattering T Ž k 0 The deuteron spin structure is also studied in the experiments with electron probes w35,36x although the data are limited within a lower internal momentum region Based on the hypothesis of NN-core in the deuteron, oscillating behavior of the tensor analyzing power due to variation of the D-wave contribution was predicted for ed- elastic scattering and also for cumulative pion production w34 x The availw35,36x 0 con- able T data for ed- elastic scattering firmed this prediction Hence, we can conclude that IA calculations are successful in all previously studied reactions induced by deuterons Žthe deuteron breakup and dp-, ed-scattering at low internal momenta; on the contrary, a marked deviation from IA is observed at higher k Ž )03 GeVrc It is different in form in different reactions as shown in Fig 1 The reasons of these deviations were discussed by many authors: in particular, final state interactions Ž ) and non-nucleonic contributions NN, DD or 6 q- configurations in the DWF were considered In the case of cumulative pion production, we met a more serious problem: the sign of T Ž k 0 was found to be opposite to the IA calculation over the whole region of internal momenta from 0 to 05 GeVrc studied in the experiment This contradiction seems not to be solved by any current models, based on the traditional picture of the deuteron as a bound state of a neutron and a proton It may be more successfully the T pion data can be explain by 0 introducing quark scenarios for cumulative meson production w37,38 x, where the deuteron core is considered as a multiquark configuration and a quarkspectator with a high momentum is softly hadronized into a cumulative particle To explain the behaviour of spin observables in cumulative hadron production the similar quark approaches introducing in consideration of spin degrees of freedom in nuclei should be developed The results of the present work are the following: The tensor analyzing power T0 of the reaction dqhž Be,C p y Ž 08 qx at a deuteron momentum of 9 GeVrc and pion momenta from 30 to 53 GeVrc has been measured In the framework of the Fig 3 T vs x and k Ž MeVrc 0 c min for the reaction dq A p y Ž 08 q X at the fragmentation of 9 GeV deuterons on the hydrogen, beryllium and carbon targets obtained in this experiw3x obtained with the carbon ment along with the previous data target for p y Ž open squares and p q Ž open cross The dashed Ž line is a linear fit of the data x rdofs05 The solid curve is our calculation based on the Paris DWF w3 x

19 ( ) S AfanasieÕ et alrphysics Letters B direct mechanism, this kinematic region corresponds to internal momenta of the nucleon in the deuteron up to k,500 MeVrc It has been observed that the behavior of T0 is independent of the atomic number of the target This fact and a weak A-dependence of the cross section shape of deuteron fragmentation into cumulative piw30,31x evidently confirm that T is ons predomi- nantly determined by the internal structure of the deuteron The sign of the measured T0 is positive and opposite to that obtained by the calculations using standard DWFs assuming the direct mechanism The T0 data cannot be reproduced by relativistic corrections or taking pion rescattering into account either Attention should be given to more sophisticated approaches taking account of non-nucleonic degrees of freedom in nuclei Acknowledgements The authors are grateful to the LHE accelerator staff and to the cryogenic division staff supporting POLARIS and the liquid hydrogen target operation, especially to A Kovalenko, A Kirillov, L Golovanov, V Fimushkin, V Ershov, Yu Borzunov, A Tsvinev We would like to thank L Strunov, V Sharov, and A Nomofilov for the primary deuteron polarization measurements by the ALPHA polarimeter This research was supported in part by Russian Foundation for Fundamental Research under grant No and by Japan Society for the Promotion of ScienceŽ JSPS, Daikou Foundation and Toyota Foundation References wx 1 VG Ableev et al, Nucl Phys A 393 Ž ; A 411 Ž E wx P Bosted et al, Phys Rev Lett 49 Ž wx 3 M Lacombe et al, Phys Lett B 101 Ž wx 4 RV Reid, Ann Phys 50 Ž wx 5 LL Frankfurt, M Strikman, Phys Rep 76 Ž wx 6 GI Lykasov, Phys Part Nucl 4 Ž wx 7 C Ciofi degli Atti et al, Phys Rev C 36 Ž wx 8 CF Perdrisat et al, Phys Rev Lett 59 Ž ; V Punjabi et al, Phys Rev C 39 Ž wx 9 T Aono et al, Phys Rev Lett 74 Ž w10x LS Azhgirey et al, Phys Lett B 387 Ž w11x V Punjabi et al, Phys Lett B 350 Ž w1x LS Azhgirey et al, Phys Lett B 391 Ž 1997 w13x A Nakamura, L Satta, Nucl Phys A 445 Ž w14x WW Buck, F Gross, Phys Rev D 0 Ž w15x MA Braun, MV Tokarev, Part and Nucl Ž w16x L Kaptari et al, Phys Lett B 351 Ž w17x BD Keister, JA Tjon, Phys Rev C 6 Ž ; L Kaptari et al, Phys Rev C 57 Ž w18x AP Kobushkin, J Phys G: Nucl Phys 19 Ž w19x AP Kobushkin, Phys Lett B 41 Ž w0x AM Baldin, Nucl Phys A 434 Ž c w1x VB Kopeliovich, Phys Rep 139 Ž wx IA Schmidt, R Blankenbecler, Phys Rev D 15 Ž ; Ch-Y Wong, R Blankenbecler, Phys Rev C Ž w3x S Afanasiev et al, Nucl Phys A 65 Ž w4x VP Ershov et al, in: Proc Intern Workshop on Polarized Beams and Polarized Gas Targets, Cologne, Germany, June 6 9, 1995, p 193 w5x VG Ableev et al, Nucl Instr and Meth A 306 Ž w6x V Ghazikhanian et al, Phys Rev C 43 Ž w7x LS Zolin et al, JINR Rapid Comm w88x-98 Ž w8x R Brun et al, GEANT 3, CERN report DDrEEr84-1, 1989 w9x VS Barashenkov, NV Slavin, Phys Part Nucl 15 Ž w30x E Moeller et al, Phys Rev C 8 Ž w31x YuS Anisimov et al, Sov J Nucl Phys 60 Ž w3x MV Tokarev, in: Proc Int Workshop Dubna Deuteron- 91, Dubna, 1991, Ž JINR, E-9-5, Dubna, 199 p 84 w33x A Wakai, Talk at Intern Symposium Dubna Deuteron-97, Dubna, July 5, 1997 Žto be published in Proc of the Symposium w34x LL Frankfurt, M Strikman, Nucl Phys A 405 Ž w35x CE Jones et al, in: Proc of the 14th Intern European Conference on Few Body Problems in Physics, Amsterdam, August 3 7, 1993, p 11 w36x M Garcon et al, Phys Rev C 49 Ž w37x VV Burov et al, Phys Lett B 67 Ž w38x VK Lukyanov, AI Titov, Phys Part Nucl 10 Ž

20 31 December 1998 Physics Letters B Measurement of g p K q L and g p K q S 0 at photon energies up to GeV 1 SAPHIR Collaboration MQ Tran a,, J Barth a, C Bennhold e,3, M Bockhorst a,, W Braun a, A Budzanowski c, G Burbach a,, R Burgwinkel a,, J Ernst b, KH Glander a, S Goers a, B Guse a,, K-M Haas a,, J Hannappel a, K Heinloth a,, K Honscheid a,, T Jahnen a,, N Jopen a, H Jungst b,, H Kalinowsky b, U Kirch a,, FJ Klein a,, F Klein a, E Klempt b, D Kostrewa a,, A Kozela c, R Lawall a, L Lindemann a,, J Link b, J Manns a,, T Mart f,e, D Menze a, H Merkel a,, R Merkel a,, W Neuerburg a, M Paganetti a,, E Paul a, R Plotzke b, U Schenk a,, J Scholmann a,, I Schulday a, M Schumacher d, P Schutz a,, WJ Schwille a,4, F Smend d, J Smyrski c, W Speth a,, HN Tran d, H van Pee b, W Vogl a,, R Wedemeyer a, F Wehnes a, B Wiegers a, FW Wieland a, J Wißkirchen a, A Wolf a, a Physikalisches Institut der UniÕersitat Bonn, Bonn, Germany b Institut fur Strahlen- und Kernphysik, Bonn, Germany c Jaggellonian UniÕersity, Krakow, Poland d II Physikalisches Institut, Gottingen, Germany e Dep of Physics, George Washington UniÕersity, Washington, DC, USA f Jurusan Fisika, FMIPA, UniÕersitas Indonesia, Indonesia Received 8 July 1998; revised 1 October 1998 Editor: JP Schiffer Abstract Associated strangeness production in the reactions g p K q L and g p K q S 0 was measured with the SAPHIR detector at the electron stretcher ring ELSA at Bonn Data on total and differential cross sections and on hyperon 1 This work is supported in part by the Bundesminister fur Bildung, Wissenschaft, Forschung und Technologie Ž BMBFŽ ProjNr06BN663 and by the Deutsche Forschungsgemeinschaft Ž DFG Ž SPP No longer working at this experiment 3 Supported by DOE grant DE-FG0-95-ER schwille@physikuni-bonnde r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

21 ( ) MQ Tran et alrphysics Letters B polarizations are presented The total cross section for L production shows a strong threshold enhancement whereas the S 0 data have a maximum at about Egs145 GeV Along with the angular decomposition of the differential cross section into polynomials, this suggests resonance production However, the angular distributions of both hyperon polarizations vary only slightly with the photon energy L and S 0 polarizations show opposite signs and change sign over the angular range q 1998 Elsevier Science BV All rights reserved Keywords: Photoproduction; Strangeness; Hyperons; Cross sections; Polarizations The production of open strangeness in photo-induced reactions at intermediate energies allows studies of the transition from the conventional hadron dynamics to the underlying dynamics of quarks, since a strange quark and antiquark must be created A measurement of total and differential cross sections and hyperon polarizations should provide for a deeper insight into the leading contributions to the process The most interesting questions are: Do quark degrees of freedom control the open strangeness production? Does chiral symmetry govern the threshw1, x? Is the old region and up to which energy Feynman graph technique through isobaric models sufficient for an adequate description w3 5x or are different concepts like Regge exchange wx 6 more appropriate to understand associated strangeness production? Obviously data of sufficient accuracy are needed before these questions can be answered As a first step towards this goal, data on the total and differential cross sections and the hyperon polarq ization of the reactions g p K LL pp Ž y and q 0 Ž 0 g p K S S Lg ; L pp y for the full angular range between threshold and a photon energy of GeV are presented The Data are available through the Internet Ž q bonnde First results of the reactions g p K L and g p K q S 0 were published from a data set taken with SAPHIR in 199 wx 7 The data were taken with the multiparticle detecwx 8 at the 35 GeV electron stretcher ring tor SAPHIR ELSA wx 9 An extracted electron beam of 16, 17, and GeV, respectively, provided tagged photons in the energy range Ž 053y09 PE e The trigger was defined by a coincidence of signals from the scattered electron in the tagging system, at least two charged particles in the scintillator hodoscopes of SAPHIR, and no signal from the beam veto counter The data are based on 3P10 7 triggers including the reanalysed data from wx 7 In contrast to previous data from other experiments w10 1x this experiment provides the full angular range for differential cross sections and hyperon polarizations From the events on tape those with one negative and two positive tracks were selected in a first step and the reconstruction of the L pp y decay vertex with < m ym y< L pp - 15 MeV was required Then the reconstructed L and the remaining positive K q candidate were combined to the primary vertex The final event samples of both reactions were selected by means of kinematic fits at the level of the single Ž 0 vertices L and S decay vertex and primary vertex and by a combined overall fit for the complete reactions g p K q L and g p K q S 0 Details on software algorithms and performance criteria are given elsewhere w7,13 x The final event samples used for this analysis comprise 7591 events of g p K q L and 5869 events of g p K q S 0 The kinematical overlap between both reactions and the possible background from competing reactions were estimated from Monte-Carlo generated events, taking into account the full experimental setup and the standard reconstruction and analysis software The results are given in Table 1 They indicate that all false fit decisions are on a level of a few percent or less and can be neglected for the present analysis That the two reactions are identified without noticeable background in our data can also Table 1 Contribution of several backgound reactions Estimated background for q q 0 Reaction g p K L g p K S q y g p pp p 0006% 0006% q y 0 g p pp p p 01% 0% 0 q g p K p L - 005% 01% L q 0 g p K pl - 005% - 005% q g p K L % q 0 g p K S 18%

22 ( ) MQ Tran et alrphysics Letters B Fig 1 Measured L life time be seen from the L decay distribution in Fig 1 which is fitted with one exponential function without background term The fitted L life time of tsž70 y10 "007 P10 s agrees well with the accepted value w14 x The separation of the reactions from each other is shown by the distribution of the missing mass q M X for g p K X in Fig The missing mass was calculated from the measured momenta for those events passing the kinematics for these reactions The acceptance was determined by means of Monte-Carlo events, where the created particles were tracked through the SAPHIR-detector, taking into account all particle decays, energy losses and multiple scattering The calculation comprises the accep- Fig Missing mass distribution

23 tance on the trigger level, the event reconstruction efficiencies based on the measurement in the central chamber of SAPHIR and the event selection For both reactions the acceptance wx 7 varies slowly within the range of the photon energy Ž E - 0 GeV g and Ž Ž CMS the angular range y1fcos Q q Fq1 K, where the latter is the production angle of the K q with respect to the photon beam direction in the overall CMS Moreover the acceptance is flat for the L decay angular distribution which is relevant for the measurement of the L and S 0 polarization The L polarization is determined from a fit to the angular distribution of the decay proton in the L rest system, whereas the S 0 polarization is calculated from the polarization of its decay L w15 x Total cross sections for both reactions as a function of the photon energy are shown in Fig 3 The error bars at the data points are statistical only The systematical uncertainties were estimated and are shown at the bottom of the graphs They take into account uncertainties of the photon flux determination and of the overall acceptance The latter uncertainty was determined by comparing previously mea- ( ) MQ Tran et alrphysics Letters B Fig 4 Differential cross sections of g p K q L for various energy bins The full bands represent the systematic errors q y sured pp p cross sections w16x with the corresponding data from our experiment The total cross section of g p K q L shows a steep rise from threshold up to Eg f11 GeV This photon energy corresponds to a g p invariant mass of 17 GeV which coincides with the masses of some ) N resonances ŽS Ž 1650, P Ž 1710, P Ž that are known to decay into K q L w14 x A polynomial of the frequently used form dsrdvs qrk CMS a qa cosž Q q 0 1 K qa cos Q q a cos Q CMS 3 CMS q q K 3 K 5 was fitted to the differential cross sections Fig 4 The phase space factor qrk is given by the ratio of the meson and photon momenta in the CMS The Fig 3 Total reaction cross sections for the reactions g p K q L q top and g p K S 0 Ž bottom The systematic errors Ž see text are indicated by the dashed lines 5 Fits with Legendre polynomials yield corresponding results, the coefficients are available via Internet

24 4 ( ) MQ Tran et alrphysics Letters B resulting coefficients are shown in Fig 6 The steep rise of the cross section at threshold is mainly described by the coefficient a 0, indicating, together with the flat angular distribution, a large s-wave contribution Furthermore, the steep decrease of a 0 immediately behind the K q S 0 threshold indicates a cusp due to the opening of this channel This cusp is well reproduced in a coupled-channel analysis wx The steady Ž but slower increase of the a1coefficient is suggestive of non-negligible contributions from an s-p-wave interference This is supported by the polarization Ž Fig 7 which is well described by a Ž CMS Ž Ž CMS sin Q q aqb cos Q q K K fit that takes into account l s 0,1 contributions only From these facts we are led to the conjecture that the photoproduction of the K q L system has significant contributions from isobar excitation in the threshold region This interpretation is in accordance with a recent coupled-channels analysis of hadronic and electromagnetic K L production wx 4 which found appreciable branching ratios of 10-0% of the S Ž 1650 and P Ž states into the K L channel Investigations using tree-level effective Lagrangian approaches came to similar conclusions wx 3 On the Fig 5 Differential cross sections of g p K q S 0 for various energy bins The full bands represent the systematic errors Fig 6 Fit coefficients see text for the differential cross sections Top: g p K q L Bottom: g p K q S 0 other hand, it will be difficult to explain the polarization throughout the full energy range with this assumption since its energy dependence is much weaker than the energy dependence of the coefficients found to describe the differential cross sections It should be kept in mind, though, that K L photoproduction receives significant contributions from the Born terms due to the magnitude of the gk L N coupling constant It is also known that at higher energies the K ) t-channel exchange becomes important w3,5 x The total cross section of g p K q S 0 rises smoothly from threshold up to its peak at about 145 GeV which corresponds to an g p invariant mass of 19 GeV At this energy a series of D states exist, but for none of those a decay fraction into K q S 0 is experimentally known w14 x Isobar models w3x have found the S Ž 1900 and P Ž states to play important roles in the description of K S production Due to the small gk S N coupling constant Born terms are expected to be small and the resonance behavior is thus more pronounced A polynomial fit to the angular distributions of the differential cross sections Ž Fig 5 indicates contributions from s-, p- and d-waves of approximately equal importance and growing with energy from threshold to 15 GeV Even contributions from f-waves may be present at the peak cross section; this suggests that at least the F Ž 1905 or F Ž 1950 state is produced, interfering with lower partial waves The S 0 polarization is shown in Fig 7, with a fit up to ls1, as in the L case Comparing the L and the S 0 polarizations two features are obvious: the

25 ( ) MQ Tran et alrphysics Letters B Fig 7 Hyperon polarizations for three energy bins L Ž top and S 0 Ž bottom The full line corresponds to a fit up to ls1 angular dependences are generally opposite in sign, as expected from SUŽ 6 wave functions w17x if the same production mechanism for the s-quark is presumed in both reactions and vary only slightly with energy In summary, total and differential cross sections as well as the hyperon polarizations were measured for the reactions g p K q L and g p K q S 0 over the full angular range from threshold up to a photon energy of GeV Total and differential cross sections show resonant contributions, for L production near threshold and around 15 GeV for the S 0 channel The shape of the hyperon polarizations show a forward-backward asymmetry, along with no significant energy variation Acknowledgements We gratefully acknowledge the help of the technical staff of the Physics Institute in Bonn and the engagement of the accelerator group at ELSA References wx 1 S Steininger et al, Phys Lett B 391 Ž wx N Kaiser et al, Nucl Phys A 61 Ž wx 3 T Mart, C Bennhold, CE Hyde-Wright, Phys Rev C 51 Ž 1995 R1074; T Mart, C Bennhold, Few-Body Syst Suppl 9 Ž ; C Bennhold, T Mart, D Kusno, in: T-SH Lee, W Roberts Ž Eds, Proc 4th CEBAFrINT Workshop on N ) Physics, World Scientific, Singapore, 1997, p 166 wx 4 T Feuster, U Mosel, Phys Rev C 58 Ž ; nuclthr , 1998 wx 5 JC David et al, Phys Rev C 53 Ž wx 6 M Guidal, JM Laget, M Vanderhagen, Nucl Phys A 67 Ž ; M Vanderhagen, M Guidal, JM Laget, Phys Rev C 57 Ž ; M Guidal, These ŽDAPNIArSPhN T, 1r1997, unpublished wx 7 SAPHIR-Collaboration, M Bockhorst et al, Z Phys C 63 Ž ; L Lindemann, PhD Thesis, Bonn preprint, BONN-IR-93-6, 1993 wx 8 WJ Schwille et al, Nucl Instr Meth A 344 Ž wx 9 D Husmann, WJ Schwille, Phys Bl 44 Ž w10x Aachen-Berlin-Bonn-Hamburg-Heidelberg-Munchen Collaboration, Phys Rev 188 Ž w x q 11 Data from other experiments for the reaction g p K L: RL Anderson et al, Phys Rev Letters 9 Ž CW Peck, Phys Rev 135 Ž ; AJ Sadoff et al, Bull Am Phys Soc 9 Ž RL Anderson et al, in: Proc Int Symp on electron and photon interactions, Hamburg, 1965, p 03; S Mori, PhD thesis, Cornell University, 1966; H Thom, a compilation, Phys Rev 151 Ž ; DE Groom, JH Marshall, Phys Rev 159 Ž ; A Bleckmann et al, Z Phys 39 Ž ; G Goeing, W Schorch, J Tietge, W Weilenbock, Nucl Phys B 6 Ž ; T Fujii et al, Preprint Ž INS, Univ of Tokyo, 1970 ; P Feller et al, Nucl Phys B 39 Ž ; M Grilli et al, Nuovo Cim 38 Ž ; R Haas et al, Nucl Phys B 137 Ž ; D Decamp et al, LAL Ž ; RL Anderson et al, Phys Rev D 14 Ž

26 6 ( ) MQ Tran et alrphysics Letters B w x q 0 1 Data from other experiments for the reaction g p K S : P Feller et al, Nucl Phys B 39 Ž ; D Decamp et al, LAL Ž ; RL Anderson et al, Phys Rev D 14 Ž ; BD McDaniel et al, Phys Rev 115 Ž ; HM Brody et al, Phys Rev 119 Ž ; RL Anderson, Phys Rev 13 Ž ; RL Anderson, Phys Rev Lett 9 Ž ; E Gabathuler et al, Bull Am Phys Soc 9 Ž ; H Goeing et al, in: Proc Int Symp on electron and photon interactions, Daresbury, 1969, p 30; T Fujii et al, Preprint Ž INS, Univ of Tokyo, 1970 w13x MQ Tran, PhD Thesis, Bonn preprint, BONN-IR-97-11, 1997 w14x Particle Data Group, Phys Rev D 54 Ž 1996 w15x TD Lee, CN Yang, Phys Rev 106 Ž ; R Gatto, Phys Rev 109 Ž w x q y 16 Data from other experiments for the reaction g p pp p : ABBHHM Collaboration, Phys Rev 175 Ž ; AHHM Collaboration, Nucl Phys B 108 Ž w17x See eg W Thirring, Electromagnetic Properties of Hadrons in the Static SU6 Modell, Acta Physica Austriaca, Sup II, 1966

27 31 December 1998 Physics Letters B Superradiance and the statistical-mechanical entropy of rotating BTZ black holes Jeongwon Ho 1, Gungwon Kang Department of Physics and Basic Science Research Institute, Sogang UniÕersity, CPO Box 114, Seoul , South Korea Received 6 July 1998; revised November 1998 Editor: M Cvetič Abstract We have considered the divergence structure in the brick-wall model for the statistical mechanical entropy of a quantum field in thermal equilibrium with a black hole which rotates Especially, the contribution to entropy from superradiant modes is carefully incorporated, leading to a result for this contribution which corrects some previous errors in the literature It turns out that the previous errors were due to an incorrect quantization of the superradiant modes Some of main results for the case of rotating BTZ black holes are that the entropy contribution from superradiant modes is positive rather than negative and also has a leading order divergence as that from nonsuperradiant modes The total entropy, however, can still be identified with the Bekenstein-Hawking entropy of the rotating black hole by introducing a universal brick-wall cutoff Our correct treatment of superradiant modes in the angular-momentum modified canonical ensemble also removes unnecessary introductions of regulating cutoff numbers as well as ill-defined expressions in the literature q 1998 Elsevier Science BV All rights reserved PACS: 0470Dy; 046qv; 0470-s Keywords: Black hole entropy; Rotating BTZ black holes; Superradiance; Brick-wall model Since Bekenstein wx 1 suggested that black holes carry an intrinsic entropy proportional to the surface area of the event horizon, and Hawking wx provided a physical basis for this idea by considering quantum effect, there have been various approaches to understanding the black hole entropy One of them is the so-called brick-wall model introduced by t Hooft wx 3 He has considered a quantum gas of scalar particles propagating just outside the event horizon of the Schwarzschild black hole The entropy obtained just by applying the usual statistical mechanical method to this system turns out to be divergent due to the infinite blue shift of waves at the horizon t Hooft, however, has shown that the leading order term on the entropy has the same form as the Bekenstein-Hawking formula for the black hole entropy by introducing a brick-wall cutoff which is a property 1 jwho@physics3sogangackr kang@theoryyonseiackr r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

28 8 ( ) J Ho, G KangrPhysics Letters B of the horizon only and is the order of the Planck length The appearance of this divergence w4,5x and relationships of this statistical-mechanical entropy of quantum fields near a black hole with its entanglement entropy wx 6 and quantum excitations of the black hole wx 7 have been studied, leading a great deal of interest recently wx 8 The brick-wall model originally applied to the four dimensional Schwarzschild black hole wx 3 has been extended to various situations The application to the case of rotating black holes has also been done for scalar fields in BTZ black holes in three-dimensions w9,10x and in Kerr-Newman and other rotating black holes in four-dimensions w11,1 x In a background spacetime of rotating black holes, it is well known that scalar fields have a special class of mode solutions, giving superradiance It is claimed in Ref w10x that the statistical-mechanical entropy of a scalar matter is not proportional to the area Žie, the circumference in the three-dimensional case of the horizon of a rotating BTZ black hole and that the divergent parts are not necessarily due to the existence of the horizon Contrary to it, in Ref wx 9, the leading divergent term on the entropy is proportional to the area of the horizon, and it is possible to introduce a universal brick-wall cutoff which makes the entropy equivalent to the black hole entropy Moreover, it is claimed in Ref wx 9 that the contribution from superradiant modes to entropy is negatiõe and its divergence is in a subleading order compared to that from nonsuperradiant modes On the other hand, for the case of Kerr black holes in Ref w1 x, the divergence is in the leading order but the entropy contribution is still negative One may expect that the leading contribution to the entropy comes from the region very near the horizon as in the case of nonrotating black holes Since the vicinity of a rotating horizon can also be approximated by the Rindler metric, it is seemingly that the essential feature of the leading contribution will be same as that in nonrotating cases Our study in detail shows this naive expectation is indeed true That is, we point out that previous erroneous results appeared in the literature were mainly related to an incorrect quantization of superradiant modes For the case of rotating BTZ black holes, we have explicitly shown that superradiant modes also give leading order divergence to the entropy as nonsuperradiant ones Moreover, its entropy contribution is positiõe rather than negative found in Refs w9,1 x However, the total entropy of quantum field can still be identified with the Bekenstein-Hawking entropy by introducing a universal brick-wall cutoff It also has been shown that the correct quantization of superradiant modes in the calculation of the angular-momentum modified canonical ensemble removes various unnecessary regulating cutoff numbers as well as ill-defined expressions in the literature Let us consider a quantum gas of scalar particles confined in a box near the horizon of a stationary rotating black hole The free scalar field satisfies the Klein-Gordon equation given by ŽI q m fs 0 with periodic boundary conditions f Ž rqqh sf Ž L Ž 1 Here, r q, rqqh, and L are the radial coordinates of the horizon and the inner and outer walls of a spherical box, respectively Suppose that this boson gas is in a thermal equilibrium state at temperature b y1 Due to the existence of an ergoregion just outside the event horizon, any thermal system sitting in this region must rotate with respect to an observer at infinity Accordingly, in order to obtain the appropriate grand canonical ensemble for this rotating thermal system, one should introduce an angular momentum reservoir as well in addition to a heat bathrparticle reservoir characterized by temperature Tsb y1 and angular speed V with respect to an observer at infinity w13 x All thermodynamic quantities can be derived by the partition function ZŽ b, V given by yb :Ž HyV ˆ J ˆ: ZŽ b,v str e, where : H: ˆ and : J: ˆ are the normal ordered Hamiltonian and angular momentum operators of the quantized field, respectively w14,13,8 x Here, we assume that particle number of the system is indefinite

29 ( ) J Ho, G KangrPhysics Letters B As usual, by using the single-particle spectrum, one can obtain the free energy FŽ b,v of the system in the following form Ý Ý yb Ž yv j k bfsylnzsy ln we l l x Ž 3 l k yb Ž lyv jl yý lnw1qe x for fermions, ~ l s Ž 4 yb Ž lyv jl Ý lnw1ye x for bosons with lyv j l )0 l where l denotes the single-particle states for the free gas in the system s² 1 < :H: ˆ < 1 : and j s² 1 < :J: ˆ < 1: l l l l l l are normal ordered energy and angular momentum associated with single-particle states l, respectively The occupation number ks0,1,, PPP for bosonic fields and ks0,1 for fermionic fields Note that, if there exists a bosonic single-particle state with its energy l and angular momentum jl such that ly V jl- 0, the expression in Eq Ž 3 becomes divergent and so is ill-defined In order to compute the free energy in Eq Ž 3, one must know all single-particle states and their corresponding values of l and jl for a given system As pointed out in Refs w15,16 x, the quantization of matter fields on a stationary rotating axisymmetric black hole background is somewhat unusual due to superradiant modes which occur in the presence of an ergoregion yi v tqimw The mode solutions will be of the form, f x ; e, because this background spacetime possesses two Killing vector fields denoted by E and E Since the partition function in Eq Ž t w is defined with respect to an observer at infinity, the vacuum state to be defined by the standard quantization procedure should be natural to that observer at infinity in the far future Thus, we expand the neutral scalar field in terms of a complete set of mode solutions as follows: Ý H ` ` out out out ) out in in in ) in v m v m v m v m Ý v m v m v m v m 0 m m V H H f Ž x s dv b u qb u q dv b u qb u Ž 5 m mv ÝH H in in in ) in yvym yvym yvym yvym m 0 q dv b u qb u Ž 6 out q Here u x describes unit outgoing flux to the future null infinity T and zero ingoing flux to the horizon q in q q H while u x describes unit ingoing flux to H and zero outgoing flux to T These mode solutions are orthonormal ² out out : ² in in : ² in in X u, u X X s u, u X X s u, u X X: sdž vyv d X Ž 7 v m v m v m v m yvym yv ym mm with respect to the Klein-Gordon inner product i ² f, f : s f E f ds Ž 8 H l ) m 1 1 m tsconst yi v tqimw Note that uv m x ;e and we suppressed other quantum numbers Modes with vsvyvh m-0 exhibit the so-called superradiance Here VH is the angular speed of the horizon with respect to an observer at infinity An observer at infinity would measure positive frequency for all modes u out v m and u in v m with v)0, but in measure negative frequency for u with v- 0 A ZAMO w17x yvym near the horizon, however, would see positive frequency waves for uyvym in with v-0 as well as for u in v m with v)0 Hence, in the terminology of w x out in Ref 15, we adopt the distant-observer viewpoint for u x and the near-horizon viewpoint for u Ž x And the conventions are chosen so that they agree with viewpoints

30 30 ( ) J Ho, G KangrPhysics Letters B Now the mode solutions for particles confined in the near-horizon box would be constructed by linearly superpose u in and u out above as follows: ½ u out v m qav mu in v m for v)0, fv mž x ; Ž 9 u out qa u ) in for v-0, v m v m yvym with appropriate normalization factor av m is chosen so that the modes satisfy the periodic boundary condition Ž 3 in Eq 1 Thus, only some discrete real values of v will be allowed f Ž x v m are understood to be cut off yi v tqimw everywhere outside the box Note that fv m x ;e for all v The inner product of these modes becomes H L y1 X X X X v m v m vv mm 0 v m rqqh ² f, f : sd d Ž vyv m < f < N ds, Ž 10 m m m y1 where we have used ds sn ds and the unit normal to a tsconst surface n sn Ž E tqv 0E w m Here V Ž r is the angular speed of ZAMO s w17 x Since V Ž r FV sv Ž rsr 0 0 H 0 q, the norm of a mode solution with v)0 is positive if vsvyvh m)0 When v-0, the norm could be either positive or negative depending on the radial behavior of the solution If the norm of f Ž x v m is negative, we can easily see that i v tyimw fyvym x ;e has the positive norm Let us define a set SR consisting of mode solutions fv m with v) 0 whose norms are negative Then, the quantized field inside the box can be expanded in terms of orthonormal mode solutions as Ý Ý ) ) f x s avmfv m x q av mfv m x q ayvymfyvym x q ayvymfyvym x, 11 lgu SR lgsr where the single-particle states are labeled by lsž v,m The Hamiltonian operator in the reference frame of a distant observer at infinity becomes then Hs Ý v Ž avm a v m qa v m av m q Ý Ž yv Ž ayvym a yvym qa yvym ayvym lgu SR lgsr Ý Ý s 1 1 v Nv m q q yv Nyvym q, 1 lgsr lgu SR where Nvm sa v m av m and Nyvym sa yvym ayvym are number operators Now, by following the standard procedure for defining a vacuum state and single-particle states w16,15 x, we can easily see that Ž, j sž v, m l l for single-particle states l sž v, m gu SR while Ž, j sž yv,y m, instead of Ž v, m l l, for single-particle states lsž v,m gsr In Refs w9,10,1 x, however, Ž, j sž v,m l l for lgsr have been used, and our study shows that this error comes from the incorrect quantization of superradiant modes This important difference is a peculiar feature of the quantization of matter fields in the presence of an ergoregion and turns out to make our angular-momentum modified canonical ensemble in Eq Ž being well-defined as shall be shown below in detail It also makes somewhat unphysical treatment of superradiant modes and introduction of various cutoff numbers unnecessary in the calculation of statistical-mechanical entropy in the literature For example, in Ref w10 x, a cutoff in the occupation number k was introduced to avoid the divergent sum for superradiant modes in the log in Eq Ž 3 3 w x We assume that no complex frequency mode occurs at the present situation As shown in Ref 18, in the presence of an ergoregion only without horizon inside, complex frequency modes may occur basically due to a repeated scattering of a negative energy wave packet in the ergoregion after passing through the center which leads to an exponential amplification of energy accordingly If there exists a horizon, however, this wave packet never comes back to the ergoregion once it passes the horizon In addition, the inner brick wall should be regarded as simply representing a regularizing parameter in the field theory calculation

31 ( ) J Ho, G KangrPhysics Letters B In general, v is discrete due to the finite size of the box, but the gap between adjacent values goes small as the size of the thermal box becomes large In this continuous limit, one may introduce the density function defined by gž v,m se nž v,m rev where nž v,m is the number of mode solutions whose frequency or energy is below v for a given value of angular momentum m Thus, gž v, mdv represents the number of single-particle states whose energy lies between v and vqdv, and whose angular momentum is m Using this density function, the free energy in Eq Ž 3 can be re-expressed as Ý H Ý yb Ž yv j k bfsy dv g v,m ln we l l x 13 m k The angular speed V in Eq 13 is a thermodynamic parameter defined, in principle, by its appearance in the thermodynamic first law for the reservoir, namely TdSsdEyV djq PPP Since a particle cannot move faster than the speed of light, its angular velocity with respect to an observer at rest at infinity should be restricted The possible maximum and minimum angular speeds are ( " 0 t w w w t t w w V Ž r sv Ž r " E PE re PE ye PE re PE, Ž 14 respectively We see that, as r r, the range of angular velocities a particle can take on narrows down Ž q ie, V Ž r V " H, and so the angular speed of particles near the horizon will be V H For a rotating body in flat spacetime, one knows that all subsystems must rotate uniformly when the body is in a thermal equilibrium state w13 x In fact, it is a part of the thermodynamic zeroth law In curved spacetimes, the uniform rotation of all subsystems in thermal equilibrium may not be true to hold any more However, since we will be finally interested in the quantum gas only in the vicinity of the horizon, we shall assume VsVH below Now, one can see that the sum in the log in Eq Ž 13 is defined well for states belonging to SR since y V j syž v y V m l l H ) 0 by the definition of the set SR For some states with v y VH m- 0 not belonging to SR, however, the sum becomes divergent As will be shown below explicitly, however, the main contribution to the entropy of the system comes from the infinite piling up of waves at the horizon For such localized solutions near the horizon, the signature of norms in Eq Ž 10 will be determined by that of vyv H m since V Ž r 0, VH for the range of integration giving dominant contributions Therefore, we assume that all states with v y V m- 0 belong to the set SR Then, the free energy in Eq Ž 13 H can be written as FsFNS qf SR Here Ý H yb Ž vyv w H m bf s dv g Ž v,m ln 1ye x, Ž 15 NS lgu SR Ý H bf s dv g Ž v,m lnw1ye Ž 16 SR lgsr bž vyv H m x In general cases, it is highly nontrivial to compute gž v, m exactly except for some cases in two-dimensional black holes w19 x For suitable conditions, however, one can approximately obtain gž v, m by using the WKB method as in the brick-wall model wx 3 For simplicity, let us consider a scalar field in a rotating BTZ black hole in 3-dimensions w0 x For the case of Kerr black holes in 4-dimensions, although the essential result is the same as that in the case of BTZ black holes, it requires some modified formulation basically due to the fact that the geometrical property near the horizon changes along the polar angle w1 x The metric of a rotating BTZ black hole is given by y Ž 0 ds syn dt qn dr qr dwyv dt, 17 where N sr rl ymqj r4r s r yr r yr rr l, Ž 18 Ž qž y

32 3 ( ) J Ho, G KangrPhysics Letters B and the angular speed of ZAMO s is V 0 sjrr Here r " denote the outer and inner horizons, respectively yi v tqimw Note that VH sv 0 rsrq sryrrql Then, mode solutions are fv m x sfv m r e Here the radial part f Ž r v m satisfies d d 4 rn rn fv mž r qr N k Ž r;v,m fv mž r s0, Ž 19 dr dr where y4 k r;v,m sn vyvqm vyvym ym N 0 Here V Ž r sv Ž r "Nrr for a rotating BTZ black hole in Eq Ž 14 " 0 In the WKB approximation, the discrete value of energy v in Eq Ž 19 is related to nž v,m as follows H L p nž v,m s dr k Ž r ;v,m, Ž 1 rqqh where k Ž r;v, m is set to be zero if k Ž r;v, m becomes negative for given Ž v, m wx 3 Since k y r;v, m, N and NŽ r 0 as one approaches to the horizon, we can easily see that the dominant contribution in Eq Ž 1 comes from the integration in the vicinity of the horizon as the inner brick-wall approaches to it Ž ie, h 0 Now, Eq Ž 15 becomes E 1 L bfns s Ý Hdv H dr k Ž r ;v,m lnw1ye Ev yb Ž vyv H m lgu p SR r q qh b L kž r;v,m 1 L yb Ž vyv m v maxž m w H H ÝH Ý x b Ž vyv m H v minž m p H r qh e y1 p q m rqqh m sy dr dv q dr k r ;v,m ln 1ye by using the integration by parts in v For convenience, one can divide F where x NS into two parts F sf Ž m) 0 qf Ž m- 0, Ž 3 NS NS NS 1 L Ž vyvqmž vyvym Ž m) 0 y FNS sy H dr N H dmh dv Ž 4 b Ž vyv m p e H y1 ` ` ( rqqh 0 Vqm from states with positive angular momenta and / 1 rerg 0 ` L 0 ` Ž m- 0 y y NS ž H H H H H H p r q qh y` 0 r erg y` V y m F sy dr N dm dv q dr N dm dv ( Ž vyv m Ž vyv m q y r = 1 erg 0 y bvh m y dr N dm( V V m lnž 1ye Ž 5 b Ž vyv m H H q y e H y1 pb r qh y` ' from states with negative angular momenta rerg s Ml is the radius of the outer boundary of the ergoregion where V Ž r s r y erg s 0 Here we considered a massless scalar field for simplicity Similarly, from states belonging to SR, Eq Ž 16 becomes 1 erg V m Ž vyvqmž vyvym y y FSR sy H dr N H dmh dv yb Ž vyv H m p e y1 r ` ( rqqh rerg ` y ybv H m H H ( q y pb r q qh 0 q q dr N dm V V m lnž 1ye Ž 6

33 ( ) J Ho, G KangrPhysics Letters B Note that gsyenrev for lgsr, and that the boundary term in FSR exactly cancels that in FNS Ž m- 0 From Eqs Ž 4 Ž 6, we can obtain leading order dependence on the brick-wall cutoff h for the free energy as follows; 3 (r yr z Ž 3 rql q y rq ry r Ž m) 0 q NS 3 b Ž ' h p ž h / rqyr y 3 z Ž 3 rql rqyry r Ž m- 0 q Ž ' NS 3 ž / b r yr p ry h q y Ž ' F sy q ln qq h, F sy ln qq h, ( ( 3 z Ž 3 rql (rqyr y rq ry rq rqyry rq F sy y ln y ln q q Ž ' h Ž 7 b r yr ' h p h p ry h Ž q y SR 3 ž / ž / The entropy of this boson gas which is assumed to be in thermal equilibrium with the rotating black hole can be obtained from the free energy by using the thermodynamic relation, Ssb E FrEb < sy3bf < ; ( ( (rqyr y q y q q y q ' 3z 3 r r r r yr r S s q ln q ln qq Ž h, 4p l ' h p h p ry h NS ž / ž / (rqyr y q y q q y q ' bsbh bsb H 3z 3 r r r r yr r S s y ln y ln q q Ž h, Ž 8 4p l ' h p h p ry h SR ž / ž / where the temperature of a rotating BTZ black hole w0x is b y1 s r yr rp r l Ž 9 H q y q Now the total entropy of the system becomes ( (rqyr y rq ' 3z 3 Ss qq Ž h Ž 30 ' 4 p l h In Ref wx 9, it is claimed that the contribution from superradiant modes is a subleading order compared with that from nonsuperradiant modes In our results above, however, we find that superradiant modes also give a leading order contribution which is in fact exactly same as that from nonsuperradiant modes in the leading order of ( r q rh It should be pointed out that the entropy associated with superradiant modes is positiõe in our result whereas it is negatiõe in Refs w9,1 x In addition, since the log terms in Eq Ž 8 are exactly cancelled, our result for the entropy of quantum field smoothly reproduces the correct result in the non-rotating limit Žie, J 0orr 0 whereas the entropy obtained in Refs w9,1x y becomes divergent in that limit rqqh If we rewrite the entropy in terms of the brick-wall cutoff in proper length defined as hshr ( grr dr, Eq q Ž 30 becomes 3z Ž 3 r8p 3 Ss Cqq Ž h, Ž 31 h where Cs p rq is the circumference of the horizon Thus, by recovering the dimension and introducing an appropriate brick-wall cutoff 3z Ž 3 y3 hs l P,73=10 lp Ž 3 16p 3

34 34 ( ) J Ho, G KangrPhysics Letters B which is a universal constant, one can make the entropy of quantum field finite and being equivalent to the Bekenstein-Hawking entropy of a rotating BTZ black hole w0x Ss4p rqrlp ssbh Ž 33 in leading order Here lp is the Planck length For a fermionic field, although modes with v-0 do not reveal superradiance, it turns out that only the overall numerical factor in Eq Ž 30 is different As mentioned before, the extension of our study to the case of Kerr black holes in four-dimensions is straightforward, but requires some modifications mainly due to the polar angle dependence of the near horizon geometry A calculation in the phase space shows that the essential feature of the leading order divergence in the entropy of quantum fields is the same as that of the present case w1 x Other thermodynamic quantities of quantum field such as the angular momentum and internal energy can also be obtained as follows; 3 E F 3z Ž 3 r16p rqry Jmattersy s qq Ž lnh Ž 34 EV bsb,vsv h l H H Here the derivative with respect to V has been taken for Eqs Ž 4 Ž 6 If we put the cutoff value in Eq Ž 3, we have rqry Jmatters sj BH Ž 35 l The internal energy of the system with respect to an observer at infinity is 1 3 rqq ry E 3z Ž 3 r16p 4 4 ry Es Ž bf qvh Jmatters qq Ž lnh s MBHy, Eb bsb,vsv h 3 l 3 3 l H H Ž 36 4 where the black hole mass is M smsž r qr BH q y rl One can easily see that Jmatter 0 and E 3 MBH in the limit of non-rotating black holes Ž eg, J sj 0 BH Therefore, we find that the entropy and angular momentum of quantum field can be identified with those of the rotating black hole by introducing a universal brick-wall cutoff although the internal energy is not proportional to the black hole mass What kind of relationships could be held among parameters characterizing a rotating black hole and thermodynamic quantities of the system of quantum fields in equilibrium with the black hole? To see this, let us 4 y3 consider a system whose free energy depends on the temperature such that F b,v, M, J sb fž V, M, J Suppose the entropy of the system is identified with that of the black hole after an appropriate regularization, ž / E F Ssb ss s4p r Ž 37 Eb V bsb H,VsV H BH The internal energy of the system with respect to a corotating observer is q E S 4 rqyr X y E s Ž bf s s Ž 38 Eb 3 b 3 l V bsb H,VsV H H 4 One can show that in general this form of the free energy holds for a massless field by introducing new integration variables defined as xs bv and ys b m in Eq 13, or, more explicitly, in Eqs 4 6

35 ( ) J Ho, G KangrPhysics Letters B Now suppose the angular momentum of the system is proportional to that of the rotating black hole, E F rqry Jmattersyž sajsa Ž 39 EV l / b bsb H,VsV H The internal energy of the system with respect to an observer at infinity is then 4 rq qž 3ary1 ry X EsE qvh Jmatters, Ž 40 3 l Ž which becomes proportional to the mass of the black hole, Ms rqqr y rl, only if as4r3 Therefore, we expect the relationships are probably 4 4 Jmatters 3J, Es 3M Ž 41 If we apply the same argument to the case of Kerr black holes in 4-dimensions, we obtain 3 3 Jmatters 4J, Es 8M Ž 4 of which the second relationship has been explicitly shown for the Schwarzschild black hole in the brick-wall model by t Hooft wx 3 We have not obtained the relationships in Eq Ž 41 at the present letter The reason for these discrepancies is not understood at the present It will be very interesting to see how the Pauli-Villars regularization method, which does not require the presence of a brick-wall as shown in Ref wx 5 for the case of a charged non-rotating black hole, works for the case of rotating black holes Acknowledgements Authors would like to thank, for useful discussions, WT Kim, YJ Park, and HJ Shin Especially, GK would like to thank J Samuel, T Jacobson, and HC Kim for many helpful discussions and suggestions GK was supported by Korea Research Foundation References wx 1 JD Bekenstein, Lett Nuovo Cim 4 Ž ; Phys Rev D 7 Ž ; D 9 Ž wx SW Hawking, Commun Math Phys 43 Ž wx 3 G t Hooft, Nucl Phys B 56 Ž wx 4 L Susskind, J Uglum, Phys Rev D 50 Ž ; T Jacobson, Phys Rev D 50 Ž ; Black Hole Entropy and Induced Gravity, gr-qcr wx 5 J-G Demers, R Lafrance, RC Meyers, Phys Rev D 5 Ž wx 6 V Frolov, I Novikov, Phys Rev D 48 Ž ; D Kabat, MJ Strassler, Phys Lett B 39 Ž ; C Callan, F Wilczek, Phys Lett B 333 Ž wx 7 AO Barvinsky, VP Frolov, AI Zelnikov, Phys Rev D 51 Ž wx 8 VP Frolov, DV Fursaev, Class Quantum Grav 15 Ž wx 9 SW Kim, WT Kim, YJ Park, H Shin, Phys Lett B 39 Ž w10x I Ichinose, Y Satoh, Nucl Phys B 447 Ž ; see also RB Mann, SN Solodukhin, Phys Rev D 55 Ž , where conical deficit methods were used to compute the one-loop corrections to the black hole entropy w11x MH Lee, JK Kim, Phys Rev D 54 Ž w1x J Ho, WT Kim, YJ Park, H Shin, Class Quantum Grav 14 Ž w13x EM Lifshitz, LPP Pitaevskii, Statistical Physics, Pergamon Press Ltd, New York, 1980; RM Wald, Phys Rev D 56 Ž w14x A Vilenkin, Phys Rev D 1 Ž ; B Allen, Phys Rev D 33 Ž w15x VP Frolov, KS Thorne, Phys Rev D 39 Ž w16x AL Matacz, PCW Davies, AC Ottewill, Phys Rev D 47 Ž ; LH Ford, Phys Rev D 1 Ž w17x ZAMO s are locally nonrotating observers whose trajectories are tangent to E se q V E where V Ž r t t 0 w 0 syetpewre wpe w w18x G Kang, Phys Rev D 55 Ž w19x Y Shen, D Chen, T Zhang, Phys Rev D 56 Ž w0x M Banados, C Teitelboim, J Zanelli, Phys Rev Lett 69 Ž w1x J Ho, G Kang, in preparation

36 31 December 1998 Physics Letters B Emission of axions in strongly magnetized stars Deog Ki Hong 1 Department of Physics, Pusan National UniÕersity, Pusan , South Korea Institute of Fundamental Theory, UniÕersity of Florida, GainesÕille, FL 3611, USA Received 4 August 1998; revised 5 November 1998 Editor: M Cvetič Abstract We show that the axion decay constant does not receive any correction at any order from external magnetic fields On the other hand, in the context of the Wilsonian effective action under external magnetic fields, the axial currents get a finite correction We then calculate the effect of strong magnetic fields B)10 18 G on the axion-nucleon coupling and find that magnetic fields enhance the axion emission rate If BR10 0 G in strongly magnetized neutron or white dwarf stars, the effective four-quark interaction may become strong at low energy and enhance the emission rate by an order of magnitude q 1998 Elsevier Science BV All rights reserved PACS: 1480Mz; 165qf; 9530Cq; 6760Jd Keywords: Axion; Magnetic field; Neutron star; Magnetic catalysis Stellar evolution tightly constrains the emission of weakly interacting, low-mass particles such as neuwx 1 whose properties are otherwise trinos or axions hard to probe wx Recently, the emission rate of such particles from stars has been calculated more accuwx 3, the effect rately by including many-body effects of strong magnetic fields w4,5 x, and those of thermal pions and kaons w6,7 x The many-body effects in the dense core of stars are found to suppress the axion or neutrino emission rate by a factor of about, while the thermal pions and kaons increase the axion emis- 1 address: dkhong@hyowonccpusanackr Permanent address Address from Aug 1, 1998 till July 31, 1999: Phys Dept, Boston Univ Boston, MA 015, USA sion rate by a factor of 3 4 within a perturbative treatment Under a strong magnetic field B the cyclotron emission of neutrinos and axions offers a new cooling mechanism for magnetized white dwarfs and neutron stars It puts a bound gaeq10 y10 on the axion electron coupling constant Similarly from magnetic white dwarf Ž WD stars gaeq 9 = y13 Ž 7 5r4 Ž Tr10 K Br10 G y, where T is the internal temperature of the white dwarf Large magnetic fields B; G have been estimated at the surface of neutron stars from the synchrotron radiation spectrum wx 8 Though the core of neutron stars or white dwarfs is poorly understood, magnetic fields as strong as B;10 18 G may be assumed in the core by the scalar virial theorem wx 9 In this letter, we calculate the effect of such strong magnetic fields on the axion-nucleon cou r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S X

37 ( ) DK HongrPhysics Letters B pling As we will see later, the coupling gets enhanced and becomes quite strong for fields B R 0 10 G Ž or ( qb R4L,800 MeV QCD We then estimate the corrections to the emission rate of axions by nucleons in strongly magnetized neutron or WD stars whose core magnetic field is BR10 18 G 3 Let us first consider how the axion decay constant is renormalized in QED under a strong magnetic field The calculation in QCD will parallel this one Under a constant magnetic field oriented along the z axis, B s Bz, ˆ the spectrum of charged particles of charge q and mass m is quantized by the Landau level ( EAsa m qk zq qb n, 1 where a s" is the sign of the energy and the quantum number n labeling the Landau level nsn q1q< m < ysgnž qbž m qb r L L 1 with bs" the spin component along the magnetic field Here nr is the number of nodes of the radial eigenfunction and ml is the angular momentum ŽNote that there will be corrections to the energy eigenvalue Eq Ž 1 due to QED loop effects like the anomalous magnetic moment, but they are irrelevant since the corrections are smaller than the Landau gap At an energy lower than the Landau gap, E -( qb, the relevant degrees of freedom in QED are photons and the lowest Landau level Ž LLL fermions Their interaction can be described by an effective action derived by integrating out the modes at the higher Landau levels Ž n)0 w11 x The effective action contains a Ž marginal four-fermion interaction, which leads to chiral symmetry breaking even at weak attraction w1 14 x Using this effective action, we calculate the axion decay amplitude to two 3 Magnetic fields higher than G may be consistently possible in the core of neutron stars due to other binding forces like the strong interactions or due to the effect of general relativity at high matter density Even though there is no proof of such high magnetic fields, the effect of the high magnetic field B;10 0 G in the core of neutron stars is studied in Ref w10 x, based on the wx 18 fact that in Ref 9, which arrives at BF10 G, only the Newtonian limit of the general relativity was incorporated in the analysis of the virial theorem photons The Wilsonian effective action of QED is given as w11x 1 Leff sy4 1qa1 Fmnq 1qb1 c idu ym c ie l :m 1 m I yž 1qc c Aug PE Au c < eb< ie l m y c Aug m E H Au c < eb < g1 5 q cc q c ig c q PPP, Ž 3 < eb< where the coefficients a 1,b 1, and c1 are of order e while g1 is of order e 4, determined by the one-loop matching conditions, and the ellipses are the higher order operators in the momentum expansion The components E Ž E are perpendicular Ž parallel H I to the magnetic field and ig m g 1 g m ln sgnž eb if ms0,3 g s ½ Ž 4 m g if ms1, Now, we consider the axion coupling to electrons: 1 m Lints E acgm g5c, Ž 5 f PQ where a is the axion field and fpq is the axion decay constant We integrate out all the modes Ž C except the LLL electron Ž c n/ 0 to get the low energy effective interaction for a and c, which is given as 1 e eff m Lint s E acgm g5 cy f < eb< f PQ l PQ a m b = cg g E ag i E g ca q PPP, Ž 6 5 a m b where the ellipses denote the effective interaction terms containing the higher powers of axions and momenta, generated by the exchange interaction of C n/ 0 In leading order, there are three diagrams that contribute to the axion decay amplitude see Fig 1 Fig 1 Axion decay amplitude Wiggly lines denote photons; solid lines, the LLL fermions; and broken lines, axions

38 38 ( ) DK HongrPhysics Letters B We find that the first two diagrams vanish but the third diagram gives ie 4 4 A1 sy Ž p d Ž kyp eb f PQ H n a m 1 b q = Ž qqk k Ž p Ž p Tr g S Ž qqp g g g g S Ž q Ž 7 m a n b = 1 5 e 4 mnab sy d Ž kyp e p Ž p 1m n 1 4p f PQ =p a bž p, Ž 8 where Psp qp, Ž p 1 m is the photon wave function, and the LLL electron propagator is i yl H r eb y I SŽ l s P e Ž 9 lu ym 1 with P s 1 y ig g sgnž eb y The effective La- grangian for the axion decay amplitude is then e a eff L s FF agg Ž 10 16p f PQ We find that the axion decay constant is not renormalized in leading order under an external magnetic field, f Ž B s f Ž 0 PQ PQ, in contrast with the pion decay constant that increases under an external magw15 x In fact, as is shown in appendix A, netic field the non-renormalization of the axion decay constant under external magnetic fields is exact simply because the axion couples to E P B not to magnetic field alone Similarly, we calculate the decay amplitude of the axion into one photon Ž Fig, using the effective Lagrangians Eqs Ž 3 and Ž 6, to get ie 4 4 a b A s p d kyp p k f PQ a b = H 5 q Tr g SŽ q g g S Ž qyk Ž 11 e 4 4 1ab s Ž p d Ž kyp e 4p f PQ = Ž p k B Ž 1 a b Fig Axion conversion into a photon Solid lines denote electrons This result agrees with the direct calculation done with the effective Lagrangian Eq Ž 10, where one of two photons is replaced by an external photon, assuring that fpq is not renormalized We see that the external magnetic field does not change the axionphoton interaction Therefore, the energy-loss processes involving only axions and photons such as axion-photon conversion or the Primakoff process are not affected by the dimensional reduction Now we consider the processes in which electrons or quarks emit axions For an extremely strong magnetic field B)B, all the charged particles are in the c lowest Landau level and thus Bremsstrahlung emission is more relevant than cyclotron emission, unless the interior temperature of stars is comparable to the Landau gap The one-loop correction to the axion Bremsstrahlung amplitude is see Fig 3 i g 1 X A s ² p,k< BR H ceuag5 cž y f eb x, y = PQ Ž 5 cc Ž x q c ig c < p : Ž 13 yig X s Ž p d Ž pyp yk 4p f PQ ž / 4 k X PuŽ p kug 5uŽ p 1q qppp Ž 14 3 m We see that the Bremsstrahlung process has an extra contribution coming from the effective four-fermion Ž interaction Note that here we assume k <m But if m 0, the contribution goes to zero In QED, 4 g A e and is therefore quite small w11 x 1, but in 19 QCD, for a strong magnetic field B) 10 G Žor ( < qb < )L QCD such that the effective action for QCD is valid, g s 1 R as and is no longer negligible Ž see appendix B for details In particular, for quarks under a strong external magnetic field BR10 0 G, g1 s will be quite large Thus the four-quark interaction will be stronger than the one-gluon exchange

39 ( ) DK HongrPhysics Letters B Fig 3 Axion Bremsstrahlung emission Solid lines denote electrons or quarks interaction and will break the chiral symmetry of QCD at an energy above L QCD The axion emission process in neutron stars was first calculated by Iwamoto, and it was found that axion bremsstrahlung from nucleon-nucleon collision is the dominant energy-loss process in neutron stars w16 x Since the emission rate is proportional to the square of the axion-nucleon coupling gan, any cor- rection to this coupling will affect the emission rate What we calculated above can be thought of as a finite correction to the axial singlet current of quarks and equally valid to the axial octet quark currents a Namely, both qgm T g5 q and qgm g5 q obtain the same Ž s correction 1qg ržp 1 As was discussed in Ref w17 x, the derivative couplings of the axion to the axial vector baryon current is E m a a m a tr QT FtrBg gwt, Bx f PQ ½ Ž A 5 m a qdtrbg g5t, B 4 S m A 5 5 q trq trbg g B, Ž 15 3 where B is the baryon octet matrix and QA the axion charge According to current algebra, F, D, and S are defined to be proportional to the nucleon matrix elements of the axial quark currents w18 x Therefore we find that the axion-nucleon coupling gets a correction g s 1 d gans g an Ž 16 p Besides the correction due to the four-fermion interaction, the emission rate gets corrections since the axion-nucleon coupling and the nucleon mass will change due to the strong magnetic field As a good approximation, we can take the nucleon mass to be m A² qq : 1r3 Ž 17 n In chiral perturbation theory, the quark condensate is given as w19x E vac m q, B B ² qq: sy < m s0, Ž 18 q E m q where Evac is the vacuum energy density The one- loop vacuum energy under a magnetic field was calculated by Schwinger w0x as 1 ` ds ymp s vac vac H 3 16p 0 s E B se 0 y e = ebs sinhž ebs y1 Ž 19 Therefore, the condensate under an external magnetic field is, using the Gell-Mann-Oakes-Renner relation F M s² qq:ž m q m, ž / p p u d eb ln B Bs0 ² qq: s² qq: 1 q qppp, Ž 0 16p F p where the ellipses denote the higher loop corrections w15 x Since the axion-nucleon coupling is proportional to the nucleon mass Ž g Am rf an n PQ and the axion decay constant does not change under magnetic fields, ž / eb ln ganž B sganž 0 1q qppp Ž 1 48p F p The energy-loss rate per unit volume due to the nucleon-nucleon axion bremsstrahlung is calculated in Refs w16,1,7x and can be written as ž / 4 f Q A m g T, a n an M p where f is the pion-nucleon coupling and T is the temperature Therefore, we find the correction due to the strong magnetic field is ž / s g a a Q B sq 0 1q qppp p = ž / 5r3 eb ln 1q qppp, Ž 3 16p F p where we have used the Goldberg-Treiman relation mn s ffp together with the Gell-Mann-Oakes-Renner relation Similarly, the lowest-order energy emission

40 40 ( ) DK HongrPhysics Letters B rate per unit volume by the pion-axion conversion y p qp nqa wx 6 has corrections due to the strong magnetic field ž / g s y y p p 1 a a Q B sq 0 1q qppp p = ž / y1r3 eb ln 1q qppp, Ž 4 16p F p Note that the above derivation of the change in the quark condensate relies on chiral perturbation theory and the one loop vacuum energy calculation, which will presumably not be valid for an extremely strong magnetic field such that the Landau gap becomes bigger than the r meson mass ( eb )m r But the condensate will still increase as B even for such a strong magnetic field because the increment of vacuum energy due to the external field will be balanced by the formation of larger condensates The correction to the emission rate due to the external magnetic field comes from two sources The first source is the finite correction to the axial current due to the effective four-quark operator and the second is the change of the quark condensate under the external field For Bs10 18 ;10 19 G, the second correction is less than a few percent while the first correction is of order one For Bs10 0 G or ( eb s 077 GeV, the second correction is about 4% for the Bremsstrahlung and about y10% for the pion-axion conversion, while the first correction is about 006as for both processes, because the four-quark operator will evolve sufficiently For B R 10 0 G the Landau gap is large enough, compared to L QCD, and thus the coupling g1 s will run long enough to become very large at low energy Though the one-loop correction is no longer valid in this case, we expect that the corrections due to the effective four-quark operator will be quite large But one has to rely on a nonperturbative method to determine the exact size of the corrections to the emission rate if BR10 0 G in stars In conclusion, we have shown that the axion decay constant does not obtain corrections from external magnetic fields and we have calculated the correction to the axion-nucleon coupling due to external magnetic fields We find that the magnetic field enhances the axion emission rate of strongly magnetized neutron or white dwarf stars and, if BR10 0 G in the stars, there is a strong indication that they may emit axions too quickly and may not survive to the present If such a strongly magnetized star is observed, the nonpertubative effect of the strong four-quark interaction on the axion-nucleon coupling has to be understood before one can estimate an accurate bound on the axion-nucleon coupling Acknowledgements The author is grateful to S Chang, John Ellis, Pierre Ramond, and Pierre Sikivie for discussions and also to the members of the IFT of University of Florida for their hospitality during his stay at IFT The author wishes to acknowledge the financial support of Korea Research Foundation made in the program year of 1997 Appendix A QED vacuum energy The low energy effective action of QED, integrating out fermions and scalars, has been calculated for various external fields w3 x Here, we calculate the QED vacuum energy, integrating out the axion fields: yh E Ž E, B 1 1 vac H w x Hž x 4 Ž m x a ie aff 0 ž / / fpq 16p fpq e s d a exp y F q E a yv cos q, Ž A1 where we neglect other matter fields for simplicity Their inclusion does not change our argument The vacuum energy density is then 1 a Evac E, B s E qb qevac E, B, A where the second term is due to axions and is obviously a function of FF or EPB If EPBs0, the vacuum energy does not change and neither does the axion decay constant But if EPB/0, the vacuum energy increases by the axion loop contribution because the axion-photon coupling in Euclidean space

41 ( ) DK HongrPhysics Letters B is imaginary Using the Schwarz inequality, one can easily show that E a Ž E, B vac G0 Thus under the ex- ternal fields the condensate has to increase to balance the vacuum energy and the axion decay constant gets bigger For general electromagnetic fields, we get f Ž E, B sf Ž EPB Gf Ž 0 PQ PQ PQ If we keep only the axion mass term in the potential in Eq Ž A1 and integrate over a, we get for slowly varying fields e EPB a vac ž / ž 16p L QCD / E Ž E, B,8, Ž A3 where we used the axion mass m sl rf a QCD PQ Appendix B QCD under a constant external magnetic field Since quarks have electric charges, magnetic catalysis will operate for quarks as well when the external magnetic field is sufficiently strong, B) G or ( < qb < ) L QCD, 00 MeV Using renormalization group analysis, we will see that if the magnetic field is stronger than 10 0 G, the four-quark interaction is stronger than the one-gluon exchange interaction and thus the chiral symmetry will break at a scale higher than L QCD The derivation of the low energy effective Lagrangian will be the same as in QED except for group theoretical factors At low energy Ž but above L the relevant degrees of QCD freedom are LLL quarks, gluons, and photons Since the gluons couple to quarks more strongly than photons, we will neglect the photon interactions As in QED, the exchange of quarks at the higher Landau levels will generate tree-level interactions among quarks and gluons, which will induce a four-quark interaction at one-loop matching Ž see Fig 4 ; s g 1 1 eff L QQ q Qig Q, Ž B1 4 qb where Q0 is the LLL quark and q is the electric charge of the quark The one-loop RG equation for the four-quark interaction is given as d s s m g sy a Ž ln, Ž B dm where as is the coupling constant of the strong interactions Unlike in QED, the strong coupling constant runs at scales below the Landau gap L L s ( qb as wx m s q ln Ž B3 a Ž m a Ž L p L s s L L Note that quarks do not contribute to the running coupling since the quark loop is finite due to dimensional reduction Combining the above RG Eqs Fig 4 Matching conditions Double lines denote quarks at the higher Landau levels Ž n) 0 ; single lines, LLL quarks; and curled lines, gluons Ž a tree level, Ž b one loop level

42 4 ( ) DK HongrPhysics Letters B B and B3, we find how the four-quark coupling changes with scale; g1 s Ž m s1144 asž m yasž LL qg1 s Ž L L, Ž B4 where g s 1 LL is of the order of as L L We see that the four-quark coupling becomes stronger than the strong coupling constant at a scale larger than the QCD scale m)lqcd if LL R4LQCD or BR10 0 G For such a strong magnetic field, the chiral symmetry of QCD will be broken at a scale larger than L QCD, the usual chiral symmetry breaking scale of QCD References wx 1 For a review, see MS Turner, Phys Rep 197 Ž ; G Raffelt, Phys Rep 198 Ž ; Stars as Laboratories for Fundamental Physics, University of Chicago Press, Chicago, 1996 wx For the axion search in laboratories, see C Hagman et al, Nucl Phys Ž Proc Suppl 51B Ž ; I Ogawa, S Matsuki, K Yamamoto, Phys Rev D 53 Ž 1996 R1740 wx 3 G Raffelt, D Seckel, Phys Rev Lett 67 Ž ; Phys Rev D 5 Ž ; H-Th Janka, W Keil, G Raffelt, D Seckel, Phys Rev Lett 76 Ž wx 4 AV Borisov, VYu Grishina, JEPT 79 Ž wx 5 M Kachelrieß, C Wilke, G Wunner, Phys Rev D 56 Ž wx 6 R Mayle, DN Schramm, MS Turner, JR Wilson, Phys Lett B 317 Ž ; MS Turner, Phys Rev D 45 Ž wx 7 W Keil et al, Phys Rev D 56 Ž wx 8 C Woltjer, Astrophys J 140 Ž ; TA Mihara et al, Nature 346 Ž ; G Ghanmugan, Annu Rev Astron Astrophys 30 Ž wx 9 D Lai, SL Shapiro, Astrophys J 383 Ž , and references therein w10x SL Shapiro, SA Teukolsky, Black Holes, White Dwarfs and Neutron Stars, Willey, New York, 1993; S Chakrabarty, D Bandyopadhyay, S Pal, Phys Rev Lett 78 Ž ; D Bandyopadhyay, S Chakrabarty, S Pal, Phys Rev Lett 79 Ž w11x DK Hong, Phys Rev D 57 Ž hep-phr w1x VP Gusynin, VA Miransky, IA Shovkovy, Phys Rev Lett 73 Ž ; Phys Rev D 5 Ž ; Nucl Phys B 46 Ž w13x CN Leung, YJ Ng, AW Ackley, Phys Rev D 54 Ž ; DS Lee, CN Leung, YJ Ng, Phys Rev D 55 Ž w14x DK Hong, Y Kim, S Sin, Phys Rev D 54 Ž w15x IA Shushpanov, AV Smilga, Phys Lett B 40 Ž w16x N Iwamoto, Phys Rev Lett 53 Ž w17x D Kaplan, Nucl Phys B 60 Ž ; M Srednicki, Nucl Phys B 60 Ž ; H Georgi, D Kaplan, L Randall, Phys Lett B 169 Ž w18x R Shrok, L-L Wang, Phys Rev Lett 41 Ž ; R Jaffe, A Manohar, Nucl Phys B 337 Ž w19x J Gasser, H Leutwyler, Nucl Phys B 50 Ž w0x J Schwinger, Phys Rev 8 Ž ; Particles, Sources, and Fields, Addison-Wesley, 1973 w1x RP Brinkmann, MS Turner, Phys Rev D 38 Ž wx H Arason et al, Phys Rev D 46 Ž w3x See, for example, JS Heyl, L Hernquist, Phys Rev D 55 Ž

43 31 December 1998 Physics Letters B Limits on cosmic chiral vortons CJAP Martins 1, EPS Shellard Department of Applied Mathematics and Theoretical Physics, UniÕersity of Cambridge, SilÕer Street, Cambridge CB3 9EW, UK Received 5 June 1998; revised 1 October 1998 Editor: PV Landshoff Abstract We study chiral vorton production for Witten-type superconducting string models in the context of a recently developed analytic formalism We delineate three distinct scenarios: First, a low energy regime Ž including the electroweak scale where vortons can be a source of dark matter Secondly, an intermediate energy regime where the vorton density is too high to be compatible with the standard cosmology Ž thereby excluding these models Finally, a high energy regime Žincluding the GUT scale in which no vortons are expected to form The vorton density is most sensitive to the order of the string-forming phase transition and relatively insensitive to the current-forming transition For a second-order string transition, vorton y8 y10 Ž 5 14 production is cosmologically disastrous for the range 10 QGmQ10 10 GeVQT Q10 GeV c, while for the y0 y14 Ž 9 1 first-order case we can only exclude 10 QGmQ10 10 GeVQT Q10 GeV c We provide a fitting formula which summarises our results q 1998 Elsevier Science BV All rights reserved PACS: 9880Cq; 117qd Keywords: Cosmology; Cosmic strings; Vortons 1 Introduction It is well known that cosmic strings can Žand are perhaps likely to behave as superconducting wires carrying large currents and charges wx 1 The charge carriers can be either bosons or fermions Žsee wx for a review String superconductivity has a number of important cosmological consequences, the most important of which is the possibility of forming vortons wx 3 These are stationary loops Žstabilised by angular momentum that do not radiate classically At large 1 Also at CAUP, Rua do Campo Alegre 83, 4150 Porto, Portugal Electronic address: CJAPMartins@damtpcamacuk Electronic address: EPSShellard@damtpcamacuk distances they look like point particles with quantised charge and angular momentum The reason why they are important w3 5x comes from the fact that they redshift like matter Žas opposed to radiation, and they can be a dark matter candidate However, it is also possible that so many of them are produced that the universe becomes dominated by matter much earlier than in the standard scenario Žin disagreement with observations In this case, this can provide rather strict constraints on allowed particle physics models In the past few years, we have developed a forw6,7,5x which allows one to calculate vorton malism densities in any chosen superconducting string model Žsee wx 8 and references therein for some theoretical r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S X

44 44 ( ) CJAP Martins, EPS ShellardrPhysics Letters B possibilities subject to some approximations we refer the reader to wx 5 for a detailed discussion of the formalism and some basic results Recently, vorton studies have been mostly done in the context of macroscopic so-called elastic models, and mostly concerned themselves with studying the stability of given vorton solutions see wx 9 for some examples Here we go back to basics, that is to the microscopic Lagrangian density, and study the classical evolution of relevant loop configurations wx 5, trying to determine which ones will reach the vorton state ŽWe focus on the Witten action in this letter, but the formalism would apply equally well to any other microscopic action We note, however, that connections can be made between the macroscopic and microscopic approaches to the problem this was discussed in wx 5 Very briefly, vorton density calculations are a two stage process First, one studies the evolution of a number of relevant loop solutions to the required microscopic equations of motion, and from the results of this analysis one introduces parameters characterising the loop s ability to evolve into a vorton state Secondly, one uses a phenomenological model for the evolution of the superconducting currents on the long cosmic string network w10 x, to estimate the currents carried by string loops formed at all relevant times, and thus decide if these can become vortons and calculate the corresponding density In a previous paper wx 5 we have applied this formalism to two extreme cases, showing that GUTscale superconducting string networks do not produce any vortons, while a network formed at the electroweak scale can produce vortons that make up to 6% of the critical density of the universe We have also conjectured that more generally there will be three different regimes: high-energy strings produce no vortons, intermediate-energy ones produce too many of them Žand are therefore observationally ruled out, and low-energy ones produce an acceptable amount of them Žwhich might be an important part of the dark matter in the universe Here we extend this work by numerically analysing vorton production at all relevant energy scales Žfrom Planck to electroweak and initial conditions We are therefore able to confirm the above conjecture and to provide a quantitative measure of the range of energies and initial conditions in which each of the regimes holds Throughout this paper we will use fundamental units in which "scsk B sgm Pls1, and all logarithms will be base ten Witten s superconducting model The idea that a cosmic string can be superconwx 1, who was also ducting is originally due to Witten the first to derive a low-energy effective action for them One finds that they are describable by the following Lagrangian density 1,a m Lsym0 q f,af yam J 1 Here, the three terms are respectively the usual Goto-Nambu term, the inertia of the charge carriers and the current coupling to the electromagnetic potential The microscopic string equations of motion can then be obtained by in the usual Ž variational way Numerical simulations of contracting string loops at fixed charge and winding number have shown w11x that a chiral state with equal charge and current densities is approached as the loop contracts For this reason, we will restrict ourselves to the chiral case for the remainder of this letter In this limit, introducing the simplifying function F defined as f Fs, m a Ž 1yx 0 and choosing the standard temporal transverse gauge the string equations of motion in an FRW Ž background with the line element ds s a dt y dx have the form e ȧ X e Ž 1qF q x sf y ef, 3 l a d Žwith dots and primes respectively denoting derivatives with respect to the time-like and space-like coordinates on the worldsheet as usual and e x X X ež 1qF xq Ž 1yx xs Ž 1yF l e d ž / ȧ q Fq F x a qf x X 4 X

45 ( ) CJAP Martins, EPS ShellardrPhysics Letters B For simplicity we have introduced the damping length ž / f 1 1 sa Hq 5 l l d Notice that this includes a frictional term due to particle scattering on strings Žsee wx 6 for a discussion of the inclusion of this term As shown in w10 x, plasma effects are subdominant, except possibly in the presence of background magnetic fields either of primordial origin or generated Žtypically by a dynamo mechanism once proto-galaxies have formed Hence one expects Aharonov-Bohm scattering to be the dominant effect 3 Calculating vorton densities The calculation of vorton densities is a two-stage process This has been detailed in wx 5, so in the present letter we will restrict ourselves to a brief summary of the ideas involved The first stage consists of studying the microphysics of the particular model that one is interested in, in order to derive its microscopic equations of motion and to numerically study some of its solutions From this one can establish criteria to decide which loop configurations will produce vortons Pending a detailed study of the quantum-mechanical stability of these objects, our criterion should be that loops whose velocity is always small will become vortons, while those which are Ž sometime during their evolution relativistic will suffer significant charge losses, so that one cannot realistically expect them to become vortons In this letter we take relativistic to mean a velocity larger than Õ s 05 Žsee wx vor 5 for a discussion of the effect of varying this cutoff The outcome of this task is to pick out the region of the tri-dimensional parameter space formed by the epoch of loop formation t i, its initial size l i and its winding number W where vortons can form The second stage is to determine the actual properties of the loops produced by the relevant network The evolution of the string network Žincluding the mechanism of loop production is described via the velocity-dependent one-scale model w6,7 x This is a quantitative model which has two key macroscopic Ž or thermodynamic quantities: the long-string correlation length r ' mrl Ž ` where m is the string mass per unit length and the string RMS velocity, Õ ' ² x : The evolution equations for these can be derived from the microscopic string equations of motion w6,7 x, at the expense of introducing two phenomenological parameters, a loop chopping efficiency c and a small-scale structure parameter k In addition, we use a simple model for the evolution of the superconducting currents on the long strings w10,5 x Assuming that there is a superconducting correlation length, denoted j, which measures the scale over which one has coherent current and charge densities on the strings, we can define N L to be the number of uncorrelated current regions per long-string correlation length, that is N s Lrj By L considering how the dynamics of the string network affects N L, one can and obtain an evolution equation for it Again, one needs to introduce a further phe- nomenological parameter Ždenoted f in w10,5 x, which quantifies the importance of equilibration between neighbouring current regions Now, the winding number can be written as ps 1r sn Ž where S is a model dependent constant L so that the outcomes of these two steps can be easily related: knowing the properties of the loops produced by the string network of interest and having decided which of these will produce vortons, it is a simple matter to determine the corresponding density We note that there are some uncertainties in the values of the phenomenological parameters mentioned above, but these do not significantly affect the numerical results Although in our previous work wx 5 we restricted ourselves to calculating vorton densities for GUT and weak-scale strings, a number of general points about the vorton problem became apparent The most important one is that since loops must be Žloosely speaking non-relativistic if they are to form vortons, the efficiency of vorton production of a given string network will vary with time For the purposes of the present discussion, this efficiency can be defined as the fraction of all the loops produced by a string network in a given Hubble time that will eventually become vortons In fact, the efficiency will be largest just after the string-forming and superconducting phase transitions Žassumed to occur at about the

46 46 ( ) CJAP Martins, EPS ShellardrPhysics Letters B same temperature, since this is the most favourable case see wx 5, and decreases afterwards This is because the relative importance of the frictional force on the string dynamics decreases with time In particular, no vortons will be formed once the network has ceased to be friction-dominated Since the duration of this friction-dominated phase increases as the string mass parameter Gm decreases, we expect that overall GUT-scale networks will be the less efficient ones, while electroweak-scale networks will be the most efficient Indeed, we have shown in wx 5 that no GUT-scale vortons are expected to form, while most electroweak-scale loops become vortons The second general point to be made is that relatively high currents are needed on the string loops to prevent them from becoming relativistic Ž and hence liable to charge losses Indeed, as was first pointed out in wx 5, unless there are outside sources Ž such as primordial magnetic fields, the required currents are much harder to get than was previously expected Given this need to have very large currents, it turns out that the order of the string-forming and superconducting phase transitions is the crucial parameter determining the efficiency of vorton formation for a string network of a given energy scale In particular, this is much more important than any of the phenomenological parameters used in the analytic modelling of these networks w6,7,5 x As one could intuitively have guessed, it is comparatively much harder to form vortons if the phase transitions are of first order, since in this case the early stages of evolution of the network occur in a stretching regime wx 7 where only a few relatively large loops are produced with quite small currents Furthermore, the order of the string-forming transition is the most crucial of the two the effects of the superconducting phase transition are quickly forgotten by the string network 4 Vorton densities for all energy scales In this letter, we extend our previous work by performing the calculation of the vorton density for typical cosmic string networks of all relevant masses, from the GUT to the electroweak scale Since the detailed algorithm for the calculation has been presented elsewhere wxž 5 and summarised above, we now simply proceed to present and discuss the results We will be discussing the two extreme cases where the string-forming and superconducting phase transitions are both of first or second order Before discussing what we believe is the right result, however, let us discuss a naive maximal argument that has appeared a number of times in the literature We start by analysing the hypothetical case where all loops produced by the string network become vortons The results in such a case are summarised in Fig 1 This shows, for different values of the string mass parameter Gm Žranging from Planck-scale strings with Gm; 1 to elecy34 troweak-scale strings with Gm s 10 the epoch at which each network would form, in orders of magnitude after the Planck time Žthe dotted line marked t c, and the epoch at which a universe with such a string network would become matter-dominated for initial conditions typical of first Ž dashed curve and second Ž solid curve order phase transitions Also shown are the epochs of equal matter and radiation densities in the standard cosmological scenario and the present epoch Ždotted lines marked teq and t 0, respectively Note the strong dependence on initial conditions, as claimed If the string-forming and superconducting phase transitions are of second order, then all vorton-forming strings above the electroweak scale would be excluded, as vorton production would make the universe matter-dominated too early to be consistent with observations On the other hand, if the phase transitions were of strongly first order, then only strings heavier than Gms10 y0 Ž 9 that is, those formed above T ;10 GeV c would be excluded However, we emphasise that Fig 1 is incorrect since, as we claimed above Žand have shown in wx 5 this perfect efficiency is unrealistic There is a considerable region of the parameter space of possible loop initial conditions for which loops are not prevented from becoming relativistic and hence no vortons form This is largest for GUT-scale strings and its importance in the total available phase space decreases with the string mass parameter Gm On the other hand, there is a minimum size for loops, below which they will simply disintegrate The reason for us to present Fig 1 is to emphasise that in this type of cosmological problem choosing assumptions that

47 ( ) CJAP Martins, EPS ShellardrPhysics Letters B Fig 1 Cosmological properties of maximal vorton-producing string networks This shows, for all allowed values Gm, the epoch at which each network would form, in orders of magnitude after the Planck time Ž t line c, and the epoch of onset of matter-domination in such a universe for initial conditions typical of first Ž dark grey region and second Ž extended light gray order phase transitions Also shown are the epochs of equal matter and radiation densities in the standard cosmological scenario and the present epoch Ž teq and t lines, respectively 0 Note that this plot is an incorrect overestimate of vorton production Ž See text for explanation Fig Same as Fig 1, but for realistic Witten-type superconducting strings Ž ie, this is the correct plot The string network forms at t c, and the onset of matter domination is brought forward to unacceptably early times for an intermediate Gm range, which depends on the phase transition order

48 48 ( ) CJAP Martins, EPS ShellardrPhysics Letters B are too optimistic Žas was done by a number of authors in the past w3,4x can have rather dramatic consequences Having made this point clear, let us now discuss what we believe is the correct answer to the vorton problem, or at least the best one that can be presently given, pending a detailed study of the question of quantum mechanical stability of these objects ŽFig As we already mentioned, the problem with Fig 1 is at the high-mass end: in this region, the frictiondominated epoch is very short and there is not time for loops to build up high enough currents to prevent them from becoming relativistic Hence in this region we do not expect any vortons to form, so that no cosmological constraints on these superconducting models can be set As was already pointed out in wx 5, this region includes GUT-scale string When vortons start to be produced, however, their high mass means that even a very small number of them is enough to start matter domination Hence there is an intermediate Gm range where we can rule out Witten-type superconducting models It should be noted, however, that the excluded region is strongly dependent on initial conditions in particular, on the order of the string-forming phase transition Ž Fig 3 For a first-order phase transition, the exclusion region corresponds to energy scales T c ; GeV, while for a second-order one it is much larger, T c ; GeV Finally, for lower-mass superconducting strings, vortons can be a source of dark matter Still, note again the importance of the initial conditions At the electroweak scale Žwhere essentially all but the smallest loops become vortons, the present density of the universe in the form of vortons can be up to V Õ ;006 if the phase transitions were of second order, but it will only be V Õ ;10 y16 in the most unfavourable case of first-order phase transitions Finally, we point out that a superconducting cosmic string network formed 4 8 at an energy scale T ;10 10 GeV can Ž c depending on initial conditions, provide a rather simple and elegant solution to the dark matter problem of the universe This being so, the recently popular low-energy super-symmetry-breaking models which admit superconducting cosmic strings Žsee for example w1x might well prove useful Finally, we provide a simple fitting formula which approximately reproduces our results Ž see Fig 4 Choose a string-forming energy scale T, or equivac lently T ž c m Pl / Gm; Ž 41 Then the initial correlation length of the string network must obey Žsee wx 6 l QL Ql, Ž 4 f c H where l st is the horizon length and l ; Ž Gm 1r H f is the friction length One expects strings formed in strongly first order phase transitions to be close to the above upper limit, while those formed in second order transitions should be close to the lower one We therefore introduce a parameter s characterising the strength of the phase transition and defined as follows logž Lcrtc ss1y Ž 43 log l rt f c This will therefore vary between zero Žsecond order transition, L ; Ž Gm 1r t and unity Ž c c strong first order transition, L ; t c c Given these, our results indicate that no vortons form if logž Gm Ry4sy 10 On the other hand, the region y133s q13sy8qlogž Gm Qy4sy10 Ž 44 is excluded, since vortons would become the dominant component of the universe, leaving a fraction of the critical density in baryons that is incompatible with nucleosynthesis Finally, for low-mass string networks, log Gm Qy133s q13sy8, our results for the present density in vortons can be reproduced by the following expression t,10 a 1ŽlogGm qa loggmqa 3 V Ž 0, Ž 45 where a1 s0011sy009, Ž 46 a s181sy16, Ž 47 a s3363sy9 48 3

49 ( ) CJAP Martins, EPS ShellardrPhysics Letters B Fig 3 The present contributions of chiral vortons to the density of the universe, for Witten-type superconducting strings produced at first Ž top and second Ž bottom order phase transitions Note that in each case there are three different regions depending on the string mass Žsee text However, a word of caution is needed here This fit is accurate in the limits ss0 and ss1, but less accurate in intermediate cases In this case, a scan of the parameter space of possible initial conditions shows that the above expression is accurate at the order of magnitude level, but not more than that Hence it should be used with some care, in particular when one is interested in cases that lie close to the

50 50 ( ) CJAP Martins, EPS ShellardrPhysics Letters B Fig 4 The expected present vorton density, according to the fitting formula Ž 45, as a function of the string energy scale and the strength of the phase transition that produced them Ž see text Several relevant contours are marked border between the excluded and dark matter producing regions Žin which case a detailed calculation should be required Finally, we emphasise that our results apply only to Witten-type superconducting string models As was pointed out elsewhere wx 5, strong model dependence is expected 5 Conclusions In this letter we have used a recently developed wx 5 semi-analytic model for vorton formation to study the mechanism of chiral vorton production in Witten-type superconducting string models of all relevant masses We have numerically confirmed the claim wx 5 that in addition to a low-energy regime where vortons can be a source of dark matter and an intermediate regime where vorton-producing networks are observationally excluded there is also a high-energy regime where no vortons are produced We have emphasised that there is a crucial dependence of the results on the initial conditions of the formation of the string networks Nevertheless, we are able to rule out any vorton-forming cosmic strings y0 y14 Ž 9 in the range 10 QGmQ10 that is 10 GeV 1 Q T Q 10 GeV c, irrespective of initial conditions For the particular case of a second-order string transition Ž producing a very high-density string network, vorton production will be cosmologically disastrous y8 y10 Ž 5 for the range 10 QGmQ10 or 10 GeVQT c Q10 GeV These bounds assume that the energy scales of the string and current-forming phase transitions are not too widely separated On the other hand, cosmic string networks formed slightly below the lower limit of the excluded energy region can provide a significant contribution to the dark matter of the universe We therefore put forward the claim that cosmic strings produced in lowenergy super-symmetry-breaking models are good candidates for solving the dark matter problem of the universe We have also provided a simple fitting formula which approximately summarises our results To conclude, let us emphasise that there is a crucial aspect of the vorton phenomenology that still remains to be considered namely, that of their quantum mechanical stability Obviously, this is a very difficult and delicate problem, which is the 14

51 ( ) CJAP Martins, EPS ShellardrPhysics Letters B reason why it has been largely untouched for nearly a decade since the first references to vortons apwx 3 However, it is clear that any further peared attempt of a more detailed study of vorton cosmology must necessarily start in that direction, and we hope to concentrate our own efforts on this important task Acknowledgements CM is funded by FCT Ž Portugal under Programa PRAXIS XXI Žgrant no PRAXIS XXIrBPDr11769r97 EPS is funded by PPARC and we both acknowledge the support of PPARC and the EPSRC, in particular the Cambridge Relativity rolling grant Ž GRrH71550 and a Computational Science Initiative grant Ž GRrH6765 References wx 1 E Witten, Nucl Phys B 49 Ž wx A Vilenkin, EPS Shellard, Cosmic Strings and other Topological Defects, Cambridge University Press, 1994 wx 3 RL Davis, EPS Shellard, Nucl Phys B 49 Ž wx 4 RH Brandenberger et al, Phys Rev D 54 Ž wx 5 CJAP Martins, EPS Shellard, Phys Rev D 57 Ž wx 6 CJAP Martins, EPS Shellard, Phys Rev D 53 Ž ; D 54 Ž wx 7 CJAP Martins, EPS Shellard, Phys Rev B 56 Ž wx 8 B Carter, Brane dynamics for treatment of cosmic strings and vortons, hep-thr970517, 1997 wx 9 X Martin, P Peter, Phys Rev D 51 Ž ; B Carter, P Peter, A Gangui, Phys Rev D 55 Ž w10x CJAP Martins, EPS Shellard, Phys Lett B 43 Ž w11x RL Davis, EPS Shellard, Phys Lett B 09 Ž w1x A Riotto, Phys Lett B 413 Ž 1997

52 31 December 1998 Physics Letters B Supermultiplets of AdS black holes in q 1 dimensions Sharmanthie Fernando 1, Freydoon Mansouri Physics Department, UniÕersity of Cincinnati, Cincinnati, OH 451, USA Received 18 September 1998 Editor: M Cvetič Abstract We construct super AdS black holes in q1 dimensions in terms of Chern Simons gauge theory of NsŽ 1,1 super AdS group coupled to a Ž super source We take the source to be a super AdS state specified by its Casimir invariants We show that the corresponding space-time is a supermultiplet of AdS space-times related to each other by supersymmetry transformations We give explicit expressions for the masses and the angular momenta of the black holes in a supermultiplet With one exception, for NsŽ 1,1 one pair of extremal black holes can be accommodated in such all-black hole supermultiplets The requirement that the source be a unitary representation leads to a discrete tower of excited states which provide a microscopic model for the super black hole q 1998 Elsevier Science BV All rights reserved A conventional method of searching for signs of supersymmetry in black hole solutions is to look for Killing spinors Many works along these lines already exist in the literature We cite a representative few here w1 9 x, from which more references can be traced One way to see whether a given black hole solution admits Killing spinors is identify it with the bosonic part of an appropriate supergravity theory w1 9 x Then by requiring that the fermions as well as their variations vanish, one arrives at Killing spinor equationž s The asymptotic supersymmetries depend on the number of non-trivial solutions of these equations consistent with the black hole topology For example, in asymptotically flat space-times, a typical supermultiplet consists of a black hole and a number of ordinary particles all with the same mass In 1 address: fernando@physungphyucedu address: Mansouri@ucedu contrast to the familiar situation in particle physics, where we have Supermultiplets consisting of particles only, in this approach there is no systematic way of looking for supermultiplets consisting of black holes only The main purpose of the present work is to show that in q1 dimensions it is possible to construct a theory which permits macroscopic solutions consisting of all AdS black hole supermultiw10 x It involves the Chern Simons gauge theory plets of the Ž 1,1 super AdS group w11,1x coupled to a super AdS state Ž source As we shall see below, to be able to accommodate the structure of the solution which emerges from such a theory it becomes necessary to broaden the standard notions of classical geometries to include some quantum mechanical elements The theory which we will describe below is the supersymmetric generalization of a theory w13x which has been used recently to provide both a microscopic and a macroscopic description of the BTZ black hole r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

53 ( ) S Fernando, F MansourirPhysics Letters B w14 x Since the concepts used in Ref w13 x, in which the gauge group is the AdS group, are essential for the understanding of the present work, we begin with a brief summary of the results of that work This will also allow us to establish our notation and to describe our unconventional use of Chern Simons thew13 x, we take the ory In the same sense as in Ref Chern Simons theory to be defined on a manifold M with topology R=S, where S is a two dimensional space Moreover, we consider the theory to be an explicit realization of the Mach Principle, so that in the absence of sources the field strengths vanish and the topology is trivial Ž no punctures In this way, we associate non-trivial topologies to the presence of sources w15,16 x We then show that the information encoded in such a theory identifies the physical Ž metrical space-time as the space Mq of 0 q 1 dimensional fields which represent the source Taking the source to be a super AdS state, we find that the emerging physical space-time has a multilayered structure, distinct from the manifold M over which the gauge theory is defined The layers are connected to each other by supersymmetry transformations As a result, for N sž 1,1 and appropriate ranges of Casimir invariants of the source state, the physical space-time becomes a super AdS Ž super multiplet consisting of four AdS black holes Let us now consider the details It will be recalled that the anti-de Sitter space in q1 dimensions can be viewed as a subspace of a flat 4-dimensional space with the line element ds sdxadx A sdx0 ydx1 ydx qdx3 Ž 1 It is determined by the constraint Ž X0 y Ž X1 y Ž X q Ž X3 sl where l is a real constant The set of transformations which leave the line element invariant form the anti-de Sitter group SOŽ, which is locally isomorphic to SLŽ, R =SLŽ, R From here on by anti-de Sitter group we shall mean its universal covering group With as0,1,, we can write the AdS algebra in two convenient forms w13 x: JasJ q a qj y a, lp a sj qa yj ya Ž 3 Setting e 01 s1; h ab s Ž 1,y1,y1 Ž 4 the commutation relations in, say, J a " take the form, basis will " q c " q y J a, Jb syieabj c ; J a, Jb s0 5 The Casimir operators are then given by j" sh ab J a " J b " Ž 6 Alternatively, we can take a combination of these with eigenvalues corresponding to the parameters of the BTZ solution: Msl P a P ql y J a J s j qj, Ž a a Ž q y a Ž q y JrlslP J a s j yj Ž 7 Unless stated otherwise, we will use the same symbols for operators and their eigenvalues As exw13 x, an irreducible representation plained elsewhere associated with an AdS black hole can be labeled either with the pair Ž M, J or the pair Ž j, j q y Itis also possible to label these representations by eigenvalues which are proportional to the Horizon radii r of the AdS black hole w13 x " To write down the Chern Simons action, we begin by expressing the connection in SLŽ, R =SLŽ, R basis Am svm a Ja qem a Pa sa qa m J q a qa ya m J y a Ž 8 where A " m a svm a "l y1 em a Ž 9 Eq Ž 9 should be viewed as definitions of e and v in terms of the two SLŽ, R connections The covariant derivative will have the form Dm semyiamsemyia qa m J q a yia ya m J y a Ž 10 Then the components of the field strength are given by qa q ya y D m, Dn syifmn Ja yifmn Ja syif q w A q x yif y w A y x Ž 11 mn For a simple or a semi-simple group, the Chern Simons action has the form 1 Ics s TrH AnŽ daq 3 AnA Ž 1 4p M where Tr stands for trace and AsA dx m sa q qa y 13 m mn

54 54 ( ) S Fernando, F MansourirPhysics Letters B So, The Chern Simons action with SLŽ, R = SLŽ, R gauge group will take the form 1 1 Ics s TrH A n da q 3 A n A 4p a M q q q q q 1 y y y y q A n da q A n A Ž 14 a y 3 Here the quantities a" are, in general, arbitrary coefficients, reflecting the semisimplicity of the gauge group Up to an overall normalization, only their ratio is significant In the presence of a source Ž or of sources in M, any a priori choice of the coefficients a" reduces the class of allowed holonomies w13 x We will, therefore, keep the coefficients a" as free parameters in the sequel As stated above, the manifold M has the topology R=S with R representing x 0 Then subject to the constraints F w A xs h e E A ye A " " 1 ij " b " b a ab i j j i Ž qe b A " c A " d s0 Ž 15 cd i j the Chern Simons action for SOŽ, will take the form 1 0 p I s dx d x cs H H aq R S Ž ab i 0 j 0 a = ye ij h A qa E A qb qa qa F q 1 0 q dx d x H H ay R S Ž ab i 0 j 0 a = ye ij h A ya E A yb qa ya F y Ž 16 where i, js1, To introduce interactions, we follow an approach which has been successful in coupling sources to Poincare and super Poincare Chern Simons theories w15,16x and take a source for the present problem to be an irreducible representation of anti-de-sitter group characterized by Casimir invariants M and J Within the representation, the states are further specified by the phase space variables of the source P A and q A, As0,1,,3, subject to anti-de Sitter constraints The relevant irreducible representations of the AdS group have been discussed in Ref w13 x Here we note that to allow for the possibility of quantizing the Chern Simons theory consistently, we must require that our sources be represented by unitary representations of AdS group Since the AdS group in q1 dimensions can be represented in the SLŽ, R =SLŽ, R form, the unitary representations of SOŽ, can be constructed from those of SLŽ, R The latter group has four series of unitary representaw17 x Of tions all of which are infinite dimensional these, the relevant representations for our purposes turn out to be the discrete series bounded from below w13 x For this series, the states in an irreducible representation of SLŽ, R are specified by the eigenvalues of its Casimir operator j Žsee Eq Ž 6 and, eg, the element J 0, where we have suppressed the superscripts " distinguishing our two SLŽ, R s Thus, we have j < F,m: sfž Fy1 < F,m :, J < F,m: s Ž Fqm < F,m: 0 In these expressions Fsreal numberg0; ms0,1,, Ž 17 So, for this series, the eigenvalues of the Casimir invariants of SLŽ, R =SLŽ, R can be written as, j" sf" yf" Ž 18 It follows that the infinite set of states can, in a somewhat redundant notation, be specified as < j,f qm :; m s0,1,, Ž 19 " " " " Clearly, the integers m" are not necessarily equal Using these states, we can construct the discrete series of the unitary representations of SOŽ, A typical state will have the following labels: < : < M, J s j, j,f qm,f qm : q y q q y y Ž 0 To be able to identify the labels M and J with the corresponding labels in the AdS black hole, we must require that F G1 w13 x " It would then follow that < Jrl< F< M <, as required for having a black hole solution With this background, let us now consider the coupling of sources to the AdS Chern Simons theory It is given by H I s dt P E q y Ž A J qa J A qa q ya y s A t a a C H A qa q Ž A qž a q C ql q q yl q dt l J J yl j ya y qly J Ja yl jy 1

55 ( ) S Fernando, F MansourirPhysics Letters B In this expression, C is a path in M, t is a parameter along C, and J a " play the role of c-number general- ized angular momenta which transform in the same way as the corresponding generators which label the source The quantities l and l" are Lagrange multipliers The first constraint in this action ensures that q A Ž t satisfy the AdS constraint As explained in previous occasions w13 15 x, it is not the manifold M over which the gauge theory is defined but the space of q A s which give rise to the classical space-time The last two constraints identify the source being coupled to the Chern Simons theory as an anti-de Sitter state with invariants jq and j y These con- straints are crucial in relating the invariants of the source to the asymptotic observables of the coupled theory via Wilson loops The total action for the theory is given by: IsI qi cs s It is easy to check that in this theory the components of the field strength still vanish everywhere except at the location of the sources So, the analog of Eq Ž 15 become e ij F ij " a sp a" J " a d Ž x, x0 Ž 3 In particular, fixing the gauge so that SOŽ, symmetry reduces to SOŽ =SOŽ, we get e ij F ij " 0 sp a" F" d Ž x, x 0 Ž 4 where F" are the invariant labels of the state as in Eq Ž 18 All other components of the field strength vanish We thus see that because of the constraints appearing in the action given by Eq Ž, the strength of the sources corresponding to the maximal compact subgroup of the gauge group become related to their Casimir invariants These invariants, in turn, determine the asymptotic observables of the theory Since such observables must be gauge invariant, they are expressible in terms of Wilson loops, and a Wilson loop about our source can only depend on, eg, jq and j y From the data on the manifold M given above, one can determine the properties of the emerging space-time by solving Eq Ž 4 The only non-vanishing components of the gauge potential are given by w13x A " u 0 s a" F" Ž 5 where u is an angular variable In particular, using Ž w x Eq 9, we can write 13 vu 0 rl s Ž jqyj y sryrl Ž 6 e 0 u s Ž jqqj y srqrl Ž 7 Although these are components of a connection which is a pure gauge, they give rise to non-trivial holonomies around the source More explicitly, we have ž E u 0 g / ž E u 0 g / 0 Wwexsexp e P Ž 8 0 W wv xsexp v J Ž 9 Here, g is a loop around the source The reduction from Eq Ž 3 to Eq Ž 4 with SOŽ =SOŽ as left over symmetry relies on diagonalizing compact subgroup generators Although this can work for black hole solutions as described in Ref w13 x, the natural left over symmetry from the point of view of black holes is SOŽ 1,1 =SOŽ 1,1 Starting from Eq Ž 3, one can carry out the gauge fixing such that the analog of Eq Ž 4 will have this non-compact symmetry w18 x The analysis of the holonomies will go through as in the case of the compact subgroup The main advantage in this case is that it is no longer necessary to carry out a Wick rotation to make contact with the black hole solution w13 x Irrespectively of whether the left over symmetry were compact or non-compact, it was shown in Ref w13x how these holonomies lead to a discrete identification subgroup of SOŽ,, which shows that the manifold Mq of the 0q1 dimensional fields q A has all the relevant features of the macroscopic AdS black hole solution The approach used in this reference was to improve and make precise some of the previous attempts to obtain this kind of identificaw19,15,16,0 x As we shall see below, the same tions holonomies, suitably interpreted, will play a crucial role in establishing the space-time structure of the supersymmetric theory discussed below To actually obtain the BTZ line element, one must determine a parametrization of q A s consistent with the above

56 56 ( ) S Fernando, F MansourirPhysics Letters B holonomy properties This was carried out in Ref w13 x, a typical parametrization for r)r being ž / ž / rq ryt ž l l / rq ryt ž l l / 1 q sfcos ry rqt fy l l q sfsin ry rqt fy l l 0 ( q s f ql cos fy 3 ( q s f ql sin fy Ž 30 where f r yrq s ; r)r Ž 31 q l r yr q y the important point to note here is that the quantities A q carry the Casimir invariants Ž r,r q y of the source state We now turn to the supersymmetric generalization of the Chern Simons theory described above and show that the corresponding macroscopic theory consists of a supermultiplet of ordinary space-times and, as a special case, a supermultiplet consisting of black holes only The simplest way of obtaining the supersymmetric extension of the anti-de Sitter group is to begin with the AdS group in its SLŽ, R = SLŽ, R basis The Ns1 supersymmetric form of each SLŽ, R factor is the supergroup OSpŽ 1;R < Thus, one arrives at the Ž 1,1 form of the N s super AdS group Its algebra is given by " " c " " " a b " J a, Jb syieab J c ; J a,qa sysa Q a ; 4 4 Q ",Q " sys a J " ; Q q,q y s0; w q a b ab a a b J, J s0 Ž 3 y x The Casimir invariants are given by C" sj" qe ab Q " a Q " b Ž 33 The spinor indices are raised and lowered by antisymmetric metric e ab defined by e 1 sye1 s1 The matrices Žs a b, Ž a s 0,1, a, form a representa- tion of SLŽ, R and satisfy the Clifford algebra 4 a b 1 s,s s h ab 34 q More explicitly, we can take them to be: ž / ž / 0 1 y1 0/ i s s ; s s ; 0 y1 i 0 1 s s ž 35 It is important to note that the supersymmetry generators of OSpŽ 1,R < do not commute with the Casimir invariant of its SLŽ, R subgroup That is, j ",Q a /0 36 Since super AdS group is semi-simple, we can construct its irreducible representations by first constructing the irreducible representations of OSpŽ 1,R < Depending on which OSpŽ 1,R < we are considering, the states within any such supermultiplet are the corresponding irreducible representations of SLŽ, R Characterized by the Casimir invariants jq and j y, respectively Based on the rationale given for the non-supersymmetric case, the irreducible representations of interest for the present case are those which can be obtained from the unitary discrete series of SLŽ, R and which are bounded from below To construct the supermultiplet corresponding to, say, the plus generators in Eq Ž 3, we can take the Clifford vacuum state < V q : to be the q SL, R state with the lowest eigenvalue of J o In the notation of Eq Ž 19, this corresponds to an ms0 state: < : < : < : F,m s F,ms0 s F q q q Then, the superpartner of this state, again with m s 0, is the state < F q 1r: q obtained by the application of one of the Q s The corresponding values of jq are F Ž F y1 and Ž F q1rž F y1r q q q q, respec- tively The supermultiplet for the second OSpŽ 1<,R can be constructed in a similar way We are now in a position to construct the Ž 1,1 super AdS supermultiplet as a direct product of the two OSpŽ 1,R < doublets Altogether, there will be four states in the supermultiplet They will have the following labels: < : < : < : F,F ; F q1r,f ; F,F q1r ; q y q y q y < F q1r,f q1r: Ž 37 q y

57 ( ) S Fernando, F MansourirPhysics Letters B From these, we can also obtain the expressions for the eigenvalues Ž M, J of various states within the supermultiplet: < M, J : s< M, J : 1 1 < M, J : s< Mq F y1r,jqf y1r: q q < M, J : s< Mq F y1r,jyf q1r: 3 3 y y < M, J : s< 4 4 MqŽ FqqF y y1,j qž F yf : q y Ž 38 these states transform into one another under supersymmetry transformations The Chern Simons action for simple and semisimple supergroups has the same structure as that for Lie groups The only difference is that the trace operation is replaced by super trace Ž Str operation So, in the OSpŽ 1,R < = OSpŽ 1,R < basis the Chern Simons action for the super AdS group has the same form as that given by Eq Ž 14 But now the expression for connection is given by " " a " "a " m A s A J qx Q dx Ž 39 m a m a Just as in the non-supersymmetric case, to have a nontrivial theory, we must couple sources to the Chern Simons action To do this in a gauge invariant and locally supersymmetric fashion, we must take a source to be an irreducible representation of the super AdS group As we saw above, such a supermultiplet consists of four AdS states To couple it to the gauge fields, we must first generalize the canonical variables we used in the AdS theory to their supersymmetric forms: PA Ž P A,Pa q A Ž q A,qa Ž 40 Then, the source coupling can be written as H A a q y Iss PAdq qpa dq q Ž A qa C qconstraints Ž 41 where again C is a path in M The constraints here include those discussed for the AdS group and, in addition, those which relate the AdS labels of the Clifford vacuum to the Casimir eigenvalues of the super AdS group The combined action IsIcs qis Ž 4 leads to the constraint equations e ij F " a sp a J " d Ž x, x 0 ; ij " e ij F " a sp a Q " a d Ž x, x 0 Ž 43 ij " Up to this point, everything proceeds in parallel with the non-supersymmetric case Differences begin to show up when one attempts to solve these equations by choosing a gauge again so that the gauge symmetry is reduced to SOŽ =SOŽ : e ij F " 0 sp a" J 0 " d Ž x, x 0 Ž 44 Although this equation is identical in form to Eq Ž 4 for the non-supersymmetric case, there is an essential difference in the underlying physics In the supersymmetric case, the supermultiplet which we couple to the Chern Simons action consist of four SOŽ, states with different values of F " As a result, in the parallel transport of q A around a close path analogous to the non-supersymmetric case, there will be four sets of holonomies with different values of Ž j, j or, equivalently, Ž r, r q y q y Moreover, in the non-supersymmetric case, a single source with Casimir invariants Ž r, r or, equivalently, Ž M, J q y will give rise to an AdS black hole w14x for which the line element is characterized with the corresponding values of M and J: ž / r J dr ds sy ymq dt q l 4l r J ymq l 4r ž / J ž r / qr dfy dt Ž 45 In the supersymmetric case, the source is a supermultiplet in which there are four states of differing Ž M, J values Moreover, recall that the explicit A parametrization of q given by Eq Ž 30, depends on the Casimir invariants Ž r, r q y or, equivalently, on Ž M, J Then, depending on which set Ž M, J that we choose, we will get a different BTZ solution Since M and J are not invariant under supersymmetry transformations, these solutions are transformed into each other under supersymmetry This makes it impossible for a single c-number line element of the type given by Eq Ž 45 to correspond to all the AdS states of a supermultiplet The situation here runs parallel to what was encountered in connection with super Poincare Chern Simons theory w16 x There it was pointed out that standard classical geometries were not capable describing these structures and that one must make use

58 58 ( ) S Fernando, F MansourirPhysics Letters B of nonclassical geometries Such geometries can be based on three elements: 1 An algebra such as a Lie algebra or a Lie superalgebra A line element operator with values in this algebra 3 A Hilbert space on which the algebra acts linearly For the problem at hand, the algebra of interest is the N sž 1,1 super AdS algebra in q 1 dimensions The corresponding Hilbert space is the representation space of the superalgebra given by Eq Ž 38 Then, instead of the BTZ line element given above, we begin with a line element operator with values in the N sž 1,1 superalgebra and assume that its diagonal elements depend on the algebra only through the Casimir operators Ž M, ˆ Jˆ of its SOŽ, subalgebra The hats on top of M and J are meant to distinguish the operators from the corresponding eigenvalues Thus, we have ds sds M, ˆ Jˆ The matrix element of this operator for each state of the supermultiplet will produce a c-number line element: ² < ˆ ˆ < : M, J ds Ž M, J M, J sds Ž M, J Ž 46 k k k k k k In other words, for each state of the supermultiplet, the nonclassical geometry produces a layer of classical space-time The number of the layers is equal to the dimension of the supermultiplet Supersymmetry transformations act as messengers linking different layers of this multilayered space-time For consistency, we must also interpret the quantities J 0 " in Eq Ž 44 as operators Acting on different states of the supermultiplet, they will give the corresponding F" eigenvalues There will therefore be not one set but four sets of holonomies Wwx e and Wwv x Each set will produce the discrete identification subgroup in the corresponding layer of space-time Consider, next, the conditions under which every layer of the supermultiplet corresponds to an AdS black hole For this to be true, we must have M G0; < J < FlM k k k This, in turn implies that FqGF yg1 In the notation of Eq Ž 38, for < J < slm, two layers of the supermultiplet become extreme AdS black holes The only exception is in the limiting case when MsJs0, in which case there will be three extremal black holes in the supermultiplet It is also interesting to note that for an appropriate choice of M and J or, equivalently, Fq and F y, the same supersource which generates a black hole in one layer can generate a naked singularity in another Acknowledgements This work was supported, in part by the Department of Energy under the contract number DOE- FGO-84ER40153 The hospitality of Aspen Center for Physics, where part of this work was carried out, is gratefully acknowledged References wx 1 PC Aichelburg, R Guven, Phys Rev D 4 Ž wx R Kallosh, A Linde, T Ortin, A Peet, A Van Proyen, Phys Rev D 46 Ž wx 3 O Coussaert, M Henneaux, eprint No hep-thr ; Phys Rev Lett 7 Ž wx 4 JM Izquierdo, PK Townsend, eprint No gr-qcr ; Clas Quan Grav 1 Ž wx 5 R Kallosh, D Kaster, T Ortin, T Torma, Phys Rev D 50 Ž wx 6 MJ Duff, J Rahmfeld, Phys Lett B 345 Ž ; eprint No hep-thr ; MJ Duff, H Lu, CN Pope, Phys Lett B 409 Ž wx 7 AR Steif, Phys Rev Lett 69 Ž wx 8 R Kallosh, eprint No hep-thr950309; Phys Rev D 5 Ž wx 9 T Ortin, eprint No hep-thr w10x A preliminary version of this work was reported at the Twenty sixth Global Foundation Conference on High Energy Physics and Cosmology, Miami, FL, December 1 15, 1997; S Fernando, F Mansouri, University of Cincinnati preprint UCTP10198, to appear in the proceedings w11x A Achucarro, P Townsend, Phys Lett B 180 Ž w1x E Witten, Nucl Phys B 311 Ž ; B 33 Ž w13x S Fernando, F Mansouri, eprint No hep-thr , Int Jour Mod Phys A, in press w14x M Banados, C Teitelboim, J Zanelli, Phys Rev Lett 69 Ž ; M Banados, M Henneaux, C Teitelboim, J Zanelli, Phys Rev D 48 Ž w15x F Mansouri, MK Falbo-Kenkel, Mod Phys Lett A 8 Ž ; F Mansouri, in: Z Horwath Ž Ed, Proceedings of XIIth Johns Hopkins Workshop, World Scientific, 1994

59 ( ) S Fernando, F MansourirPhysics Letters B w16x Sunme Kim, F Mansouri, Phys Lett B 397 Ž ; F Ardalan, S Kim, F Mansouri, Int Jour Mod Phys A 1 Ž w17x BG Wybourne, Classical Groups, Wiley, 1974; AO Barut, C Fronsdal, Proc Roy Soc Ž London A 87 Ž w18x S Fernando, F Mansouri, in: L Brink, R Marnelius Ž Eds, Proceedings of XXII Johns Hopkins Workshop, Goteborg, Sweden, August 0, 1998 w19x K Koehler, F Mansouri, C Vaz, L Witten, Nucl Phys B 348 Ž w0x C Vaz, L Witten, Phys Lett B 37 Ž

60 31 December 1998 Physics Letters B Quaternionic Taub-NUT from the harmonic space approach Evgeny Ivanov a,1, Galliano Valent b, a BogoliuboÕ Laboratory of Theoretical Physics, JINR, Dubna, Moscow region, Russia b Laboratoire de Physique Theorique et des Hautes Energies, Unite associee au CNRS URA 80, UniÕersite Paris 7, Place Jussieu, 7551 Paris Cedex 05, France Received 18 September 1998 Editor: L Alvarez-Gaumé Abstract We use the harmonic space technique to construct explicitly a quaternionic extension of the Taub-NUT metric It depends on two parameters, the first being the Taub-NUT mass and the second one the cosmological constant q 1998 Elsevier Science BV All rights reserved 1 An efficient way to explicitly construct hyper-kahler and quaternionic-kahler metrics is provided by the harmonic Ž super space method w1 4 x It was firstly introduced in the context of Ns supersymmetry wx 1 The basic idea was to extend the " i qi y standard N s superspace by a set of internal harmonic variables u,u ui s 1, parametrizing the automorphism group SUŽ of N s superalgebra It was shown in wx 1 that all N s theories admit a manifestly supersymmetric off-shell description in terms of unconstrained superfields given on an analytic subspace of the N s harmonic superspace, harmonic analytic superfields It was soon realized that the harmonics are also relevant to some purely bosonic geometric problems As is shown in wx 3, the constraints defining the hyper-kahler Ž HK geometry can be given an interpretation of the integrability conditions for the existence of analytic fields in a SUŽ harmonic extension of the original im 4n-dimensional HK manifold x 4, Ž is1,;ms1,, n This time, the SU to be harmonized is an extra SUŽ rotating three complex structures of the HK manifold The analytic subspace is spanned by the harmonic variables u " i and half of the initial x-coordinates, x qm The constraints of HK geometry can be solved via an q4 unconstrained analytic HK potential L Ž x qm,u " i It encodes Ž at least, locally all the information about the associated metric Remarkably, it allows one to explicitly construct the HK metrics by simple rules wx eivanov@thsun1jinrru valent@lpthejussieufr r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

61 ( ) E IÕanoÕ, G ValentrPhysics Letters B In wx 4, a generalization of this approach to the quaternionic-kahler Ž QK manifolds was given These manifolds generalize the HK ones in that the extra SUŽ which transforms complex structures becomes an essential part of the holonomy group It was shown in wx 4, that the QK geometry constraints can be also solved q4 in terms of some unconstrained potential L living on the analytic subspace parametrized by SUŽ harmonics and half of the original coordinates The specificity of the QK case is the presence of a non-zero Ž q4 constant Sp 1 curvature on all steps of the way from L to the related metric It is interesting to consider some examples in order to see in detail how the machinery proposed in wx 4 works Only the simplest case of the Ž q4 homogeneous QK manifold Sp nq1 rsp 1 =Sp n corresponding to L s0 was considered in Ref wx 4 The aim of this paper is to demonstrate the power of the harmonic geometric approach on the example of less trivial QK metric, a quaternionic generalization of the well-known four-dimensional Taub-NUT Ž TN metric wx 5 Like in the HK case wx, the computations are greatly simplified due to the UŽ 1 isometry of the quaternionic TN metric The metric depends on two parameters, the TN mass parameter and the constant SUŽ curvature parameter which can be interpreted as the inverse radius of the corresponding flat QK background ;SpŽ rspž 1 =SpŽ 1 We end up by performing the identification of the metric with other forms given in the literature We first recall some salient features of the construction of wx 4 One starts with a 4n-dimensional Riemann mm manifold parametrized by local coordinates x 4,ms1,,, n; ms1,, and uses a vielbein formalism The QK geometry can be defined as a restriction of the general Riemannian geometry in 4 n-dimensions, such that the holonomy group of the corresponding manifold is a subgroup of SpŽ 1 =SpŽ n 3 Thus one can choose the tangent space group from the very beginning to be SpŽ 1 =SpŽ n and define the QK geometry via appropriate restrictions on the curvature tensor lifted to the tangent space Žtaking into account that the holonomy group is generated by this tensor As explained in wx 4, for the QK manifold of generic dimension the defining constraints can be written as a restriction on the commutator of two covariant derivatives Here D syv RG Ž 1 a Ž i, Db k ab Žik E mm m Da is ea i Ž x Dm s ea i Ž x q SpŽ 1 = SpŽ n y connections, Ž m E x mm e Ž x a i being the 4n=4n vielbein with the indices as1,,, n and is1, rotated, respectively, by the tangent local SpŽ n and SpŽ 1 groups, V is the SpŽ n ab -invariant skew-symmetric tensor serving to raise and Ž bg lower the Sp n indices V V sd g, G are the SpŽ 1 ab a Žik generators, and R is a constant, remnant of the SpŽ 1 component of the Riemann tensor Žits constancy is a consequence of the QK geometry constraint and Bianchi identities The scalar curvature coincides with R up to a positive numerical coefficient, so the cases R)0 and R-0 correspond to compact and non-compact manifolds, respectively In the limit Rs0 Eq Ž 1 is reduced to the constraint defining the HK geometry wx 3, in accord with the interpretation of HK manifolds as a degenerate subclass of the QK ones Like in the HK case wx 3, in order to explicitly figure out which kind of restrictions is imposed by Ž 1 on the mm vielbein e Ž x and, hence, on the metric a i g mm n s sea mm i e n s a i, gm n s sem a i en a s i, Ž 3 Ž one should solve the constraints 1 by regarding them as integrability conditions along some complex directions in a harmonic extension of the original manifold 3 Ž For the 4-dimensional case this definition has to be replaced by the requirement that the totally symmetric part of the Sp 1 component of the curvature tensor lifted to the tangent space is vanishing

62 6 ( ) E IÕanoÕ, G ValentrPhysics Letters B Due to the non-vanishing rhs in Ž 1, the road to such an interpretation in the QK case is more tricky mm Modulo these peculiarities, the basic step still consists in extending x 4 by a set of some harmonic variables, mm 4 mm " 4 qi y x x,w, w w s 1 Then, following the general strategy w3,4 x, one passes to a new Ž analytic i i mm basis in x,w " i x mm,w " x qm, x ym,w " i Ž 4 i A A A x A " m sx m i w i " qõ " m Ž x,w, wa qi sw qi yrõ qq Ž x,w w yi, wa yi sw yi, Ž 5 " m qq q where the bridges Õ x,w,õ x,w are chosen so as to make the w -projection of Da i in this basis to be proportional to the partial derivative with respect to x ym E q qi qi m m m q D a ;w Da isw ea i Ž x Em q sea Ž x,w se Ž x,w E Ž 6 ym a m E x A Žsimultaneously, one performs an appropriate SpŽ n rotation of the tangent space index a by a matrix q Sp n - bridge The possibility to reduce Da to this short form amounts to the possibility to define analytic qm " fields living on the analytic subspace x,w i 4 A A The original QK geometry constraints prove to be equivalent to the existence of such analytic fields and subspace wx 4 An essential difference of the QK case from the HK wx qq case 3 is the necessity to shift the harmonic variables with the new bridge Õ q mn Besides the opportunity to make D short, the passing to the harmonic extension of x 4 a and further to the analytic basis and frame Ž the l-world allows one to exhibit the fundamental unconstrained objects of the QK geometry, the QK potential While in the original formulation Ž the t-world the basic geometric objects are the mm vielbeins e Ž x properly constrained by Eq Ž 1 a k, in the analytic basis such objects are the harmonic vielbeins covariantizing the derivatives with respect to the harmonic variables In the original basis the harmonic "" "" " i i 0 0 qi qi yi yi w qq yy x 0 derivatives are D s Ew s w ErE w, D s Ew s w ErE w y w ErE w, E w,ew s E, ie they mn mm contain no partial derivatives with respect to the variables x, because the harmonic space x,w " 4 i has the mm structure of the direct product x 4m w " 4 After passing to the analytic basis by Eq Ž i 5, the derivatives D "" acquire terms proportional to E m " 'ErE x m Besides, in D qq there emerges a term proportional to Ew yy A These new terms appear with the appropriate vielbein components H q3 m, H yy" m, H q4 which are related to the bridges as follows E qq qrõ qq x qm sh q3 m, Ž 7 w A E qq qrõ qq Õ qq syh q4, Ž 8 w 1yRE 1 yy " m yy" m Ew x A sh 9 yy qq w Õ Note that x ym is determined in terms of x qm by the equation E qq yrõ qq x ym sx qm Ž 10 w A A The original QK geometry constraints require H q3 m, H q4 to be analytic E q H q3 m se q H q4 s0 H q3 m sh q3 m x q,w, H q4 sh q4 x q,w, Ž 11 m m A A A A and express H q3 m in terms of H q4 Basically, the analytic harmonic vielbein H q4 is just the unconstrained QK q4 wx q4 potential To be more precise, the QK potential L, as it was defined in 4, is related to H as Žafter properly fixing the l-world gauge freedom H q4 x q,w sl q4 x q,w qx q H q3 m x q,w, x q 'V x qn, Ž 1 A A A A m A A m mn

63 ( ) E IÕanoÕ, G ValentrPhysics Letters B and ˆ ˆ q3n 1 nm y q4 y y q yy H s V Em L, E m ' Em q Rxm E A 13 It can be shown that the only constraint to be satisfied by L q4 is its analyticity, so this object encodes all the information about the relevant QK geometry and metrics, whence its name QK potential Choosing one or q4 " m qq another explicit L, and substituting 1, 13 into Eqs 7 10, one can solve the latter for x and Õ as functions of harmonics and the t-world coordinates x mm Having at hand the explicit form of the variable Ž q4 change 5, it remains to find the appropriate expression of the l-world vielbeins in terms of L in order to be able to restore the t-world vielbein and hence the QK metric itself Skipping intermediate steps Žthey can be found in wx 4, the non-vanishing components of the l-world inverse QK metric are given by the following expressions g sg sv E H ˆ, g sy V E Hˆ E Hˆ E H ˆ, Ž 14 y1 n y1 v y1 mqny nymq mr myny rs Ž m q y3n Ž l Žl r Ž l s r v where m 1 ˆ q ˆyyqm ˆyy" m yy" m q yyq m Ž E H n 'E n H, H ' H, xph'x m H Ž 15 1yRŽ xph Ž Then the t-world metric can be obtained via the change of variables inverse to 4 ž / mm n s vysy q m q n s vqsy g s g ˆy m q n s ˆy Ž l Ev x Es x q gž l Ev x Es x q Ev x n m E q s x m s 16 In the case of 4-dimensional QK manifolds we will deal with in the sequel Ž m,ns1, the t-basis metric Ž 16, after some algebra, can be put in the form 1 1 mm n s m n s g s G, Ž 17 yy qq detž E Hˆ 1yRŽ xph Ž 1yREw Õ Here mm n s lr yy q mm qn s G se Ew Xl Xr q mmln s 18 X qm m 'E q x mm Ž 19 r r are solutions of the system of algebraic equations Xr qm m = m x yn se q r x yn sdr n, Xr qm m = mm x qn se q r x qn s0, Ž 0 R qq yy = mm ' Em q Ž E Õ E Ž 1 yy qq m w 1yRE Õ w As we see, the problem of calculating the QK metric Ž 17, Ž 18 is reduced to solving the differential Ž Ž q4 Ž qm " i " m qq equations 7, 10, 8 which define, by the known L u x,w A, x and Õ as functions of the t-basis coordinates x mm and w " i In general, it is a difficult task However, it is simplified for the QK metrics with isometries, like in the HK case wx We will demonstrate this on the example of the QK analog of the Taub-NUT metric q4 3 The QK counterpart of the TN manifold is characterized by the same L 6 q4 q q qq L sž il x x ' Ž f wx

64 64 ( ) E IÕanoÕ, G ValentrPhysics Letters B Here we introduced the notation 4 q1 q q q q q q q Ž x, x s Ž x,y x, x sž x, Ž x syx Ž 3 We also assume qq qq lsl f sf 4 ŽŽ The basic equations 8, 7, 10 for the given case take the form qq qq qq qq E Õ q RŽ Õ s Ž f, Ž 5 Ž E qq qrõ qq x q sil x q f qq, Ž 6 Ž E qq yrõ qq x y sx q Ž 7 Ž together with their conjugates These equations are covariant under two rigid symmetries preserving the q q " i analytic subspace x, x,w 4: UŽ 1 Pauli-Gursey Ž PG symmetry A qx i a q qx yi a q x se x, x se x, Ž 8 Ž Ž " qq and SU symmetry which uniformly rotates the doublet indices of the harmonic variables x and Õ are scalars with respect to this SUŽ They constitute the UŽ isometry group of the QK TN metric We will firstly solve Eq Ž 5 Defining ˆ ˆ qq qq R Õ q q qq q q qq Õ se Õ, v'e, x 'v x, f sil x x sv f, 9 we rewrite 5, 6 as Ž ˆf qq qq Ž E vsr, Ž 30 v 3 f ˆqq qq q q q qq E xˆ sil x 'il x k Ž 31 v From Eq Ž 31 and the definition of f ˆqq one immediately finds qq E f ˆqq s0 f ˆqq sfˆ ik x w q i w q k 3 ˆ ˆ We observe that Eq Ž 30 coincides with the pure harmonic part of the equation defining the Eguchi-Hanson metric in the harmonic superspace approach wx 8 Its general solution was given in wx 8, it depends on four arbitrary integration constants, that is, in our case, on four arbitrary functions of x mi However, these harmonic constants turn out to be unessential due to four hidden gauge symmetries of the set of equations Ž 5 - Ž 7 One X of them is the scale invariance Õ sõqbž x, while three remaining ones form an extra local SU symmetry wx 7 Using this gauge freedom one can gauge away four integration constants in v and write a solution to Eq Ž 30 in the following simple form ( 1 ˆ ˆ vs 1qRf Õs lnž 1qRf, Ž 33 R ff ˆˆqq qq qq ˆ ˆŽik q y Õ se Õs, f'f Ž x wi w k Ž 34 ˆ 1qRf One can restore the general form of the solution as it was given in wx 8, acting on Ž 33 by a finite form of the aforementioned hidden symmetry transformations In wx 7 we demonstrate that the whole effect of the full gauge 4 We adopt the convention e sye 1 s1 The complex conjugation is always understood as a generalized one, ie the product of the 1 wx ordinary conjugation and Weyl reflection of harmonics 1

65 ( ) E IÕanoÕ, G ValentrPhysics Letters B SUŽ transformation is reduced to the rotation of the t-world metric corresponding to the fixed-gauge solution Ž 34 by some harmonic-independent non-singular matrix which becomes identity upon restriction to x- independent SUŽ transformations Thus in what follows we can stick to this solution Now we are prepared to solve Eq Ž 6 Žor Ž 31 This can be done in a full analogy with the hyper-kahler TN case wx, based essentially upon the PG invariance Ž 8 Using Ž 34, we obtain k qq se qq k, 1 1 Ž 1 R)0,ks arctan' R f ˆ ; R-0,ks arctanh'< R< f ˆ Ž 35 ' R '< R< Ž For definiteness, in what follows we will choose the solution 1 in 35 Then, making the redefinition ˆ 4 ˆ 4 q q q q x sexp ik x, x sexp yik x, 36 we reduce 31 to E qq x q s0 ˆ q i q q qi i q Ži k q y x sx w i, x sxiw syxw i, fsyilx x wi w k, 37 where, in expressing f, ˆ we essentially made use of the PG symmetry Ž 8 q Combining Eqs 9, 33, 36 and 37 we can now write the expressions for x, x q in the following form 1 1 q i q q i q x s expik 4 xw i, x sy expyik 4 xw i, Ž 38 ˆ ˆ 1qRf 1qRf ( ( ˆ i i i i where k and f are expressed through x, x according to Eqs 35 and 37 Comparing 38 with the general definition of the x -bridges Ž 5, we can identify x, x with the t- world coordinates, ie with the coordinates of the initial 4-dimensional QK manifold y y i i " We still need to find x, x as functions of x, x and harmonics w by solving Eq Ž 7 i and its conjugate Dropping intermediate technical steps Ž they involve a number of redefinitions, its general solution can be presented in the following form Here ( ˆ 1 1qRf y y ik Ž i l s ik Ž f ˆ y x s e ye x, l Ž ls yifˆ ( ˆ 1 1qRf y y ik Ž i l s y ik Ž f ˆ y x s e y e x Ž 39 l Ž ls qifˆ il y i y y i y i x sx w, x syxw, ssx x, k ils 'k s arctanh' i i i 0 R Ž ls Ž 40 ' R For what follows it will be convenient to define AŽ s '1yRl s, BŽ s '1q4l sqrl s, CŽ s '1qRsqRl s Ž 41

66 66 ( ) E IÕanoÕ, G ValentrPhysics Letters B Now we are ready to find explicit expressions for the two important quantities entering the general expression for the t-metric Ž 17, Ž 18 : 1yRfˆ C 1qRfˆ yy qq 1yRE Õ sa, 1yRŽ xph s Ž 4 ˆ ˆ 1qRf A 1yRf As a next step towards the QK Taub-NUT metric, one needs to find the entries of the matrix Xn qm i 'E q n x mi by solving the set of algebraic equations Ž 0 In the complex notation, this set is divided into the two mutually conjugated ones, each consisting of four equations It is clearly enough to consider one such set, eg qr k y qrk q qrk " X = rk x s1, X = r k x s0, X = r k x s0, 43 qr k qrk qrk qrk where X 'X 1, X 'yx It is convenient to work with ˆ ( ˆ qr k RÕ qrk qrk X se X s 1qRf X 44 " " It remains to calculate the transition matrix elements = x,= x entering Eq Ž 0 rk r k This can be done straightforwardly, the corresponding expressions look rather involved and by this reason we do not quote them here explicitly Žmore details are given in wx 7 Surprisingly, the expressions for X ˆqr k prove to be much simpler: q1 k 1 kl Ž k l q ik ˆX s 3 AqB e y4l AqC x x w e 0 4, l q k q k l q ik Xˆ s ŽE ˆ xsl Ž AqC Ž x x w e 0 l Ž 45 Ž the remaining components can be obtained by conjugation It will be convenient to rewrite the metric Ž 17, Ž 18 through X ˆqr i m 1 ri,lk ˆ r i,lk g s G, Ž 46 CdetŽ E Hˆ v b ˆ ˆ ˆ ˆ ri,lk r i,lk vb yy q r i qlk G s 1qRf G se E X X q rillk 47 As the last step, one should compute detž E H ˆ After some algebra, it can be represented in the following concise form 1 A ˆ ˆ ab yy ˆq r k yy ˆq ll qn qm detž E H sy Ž 1 y Rf e E Xa E X b = rk x = ll x e nm Ž 48 C 3 As a result of rather cumbersome, though straightforward computation one eventually gets the simple expression for detž E Hˆ B 1ql sql sž qsr 4ik 0 4ik detž E Hˆ sa e s Ž 1yRl s e 0 Ž C Ž 1qRsqRl s The harmonic dependence disappeared in detž E H ˆ, as it should be The calculation of this determinant is the most long part of the whole story Once this has been done, the computation of the t basis inverse metric amounts to the computation of entries of the matrix G ˆ ri,ll The final answer for the metric tensor is as follows D g1k,1ts Ž xkx t, As1yRl s, C B D ~ g k, ts Ž xkx t, Bs1q4l sqrl s, Ž 50 C B 1 g 1k, t s B e kt qdž x k x t, Cs1qRsqRl s C B

67 ( ) E IÕanoÕ, G ValentrPhysics Letters B Here D'l Ž AqCŽ AqB sl Ž qrsž1ql s One should observe that this final expression is valid for any sign of the parameter R even if, in the intermediate steps Žsee relation Ž 35 for instance, the sign choice plays a significant role 4 To compare to the results in the literature wx 5 one has to use wx 9 ž / ž / ds s3 syis i i i 1 1 dx sx qi yx, dsis eijk sjn s k Ž 51 s i Using the notation A P B' A B relation Ž 51 implies i ds s3 ds s yxpdxs qis, dxpdxs q Ž s1 qs qs 3, 4s 4 ssxpx Ž 5 The metric given by Ž 50 becomes 1 B sb sa ds q s q s q s Ž 53 Ž 1 3 sc C C B The most general Bianchi IX euclidean Einstein metrics can be deduced from Carter s results w10 x A convenient standardization w11x is the following with ½ 5 r y1 DŽ r dt sl Ž dr q4 s q Ž r y1 s qs, Ž 54 3 Ž 1 DŽ r r y1 yll 4 DŽ r s r q Ž 1q Ll r y Mrq1qLl Ž 55 3 Ž These metrics are Einstein, with Einstein constant L and isometry group U If we take Ms4r3Ll q1 the metric simplifies to rq1 Ž dr ry1 ½ 3 Ž 1 ry1 S Ž r rq1 5 dt Ž Q sl q4 S Ž r s q Ž r y1 s qs, Ž 56 where now L l S Ž r s1y Ž ry1ž rq3 Ž 57 3 The identifications ry1 s 4 R s Ž 4l yr, Ll s, Ž 58 1qRsqRl s 3 4l yr give the relation B sb sa dt Ž Q 44l Ž yr ds q s qs q s s Ž 59 Ž 1 3 sc C C B l

68 68 ( ) E IÕanoÕ, G ValentrPhysics Letters B The quaternionic metric Ž 56 is complete for L- 0 and is asymptotically Anti de Sitter It has been considered recently in w13x under the name Taub-NUT-AdS metric and reveals itself a useful background for computing black-holes entropy 5 In this paper we made the first practical use of the harmonic space formulation of the QK geometry wx 4 to compute a non-trivial QK metric, the four-dimensional quaternionic Taub-NUT metric As we were convinced, the harmonic space techniques, like in the HK case w,3,8 x, allows one to get the explicit form of the QK metric starting from a given QK potential and following a generic set of rules It would be interesting to apply this approach to find the QK analogs of some other interesting 4- and higher-dimensional HK metrics, in particular, the quaternionic Eguchi-Hanson metric and the quaternionic generalization of the multicenter metrics of Gibbons and Hawking w1 x Finally, we note that the HK Taub-NUT metric plays an important role in the modern p-branes realm, yielding an essential part of one of the fundamental brane-like classical solutions of D s 11 supergravity, the so-called Kaluza-Klein monopole Žsee, eg w14 x It would be of interest to reveal possible brane implications of the QK Taub-NUT metric constructed here Acknowledgements EI thanks the Directorate of LPTHE, Universite Paris 7, for the hospitality extended to him during the course of this work His work was partly supported by the grant of Russian Foundation of Basic Research RFBR and by INTAS grants INTAS ext, INTAS References wx 1 A Galperin, E Ivanov, S Kalitzin, V Ogievetsky, E Sokatchev, Class Quantum Grav 1 Ž wx A Galperin, E Ivanov, V Ogievetsky, E Sokatchev, Commun Math Phys 103 Ž wx 3 A Galperin, E Ivanov, V Ogievetsky, E Sokatchev, Ann Phys Ž NY 185 Ž 1988 wx 4 A Galperin, E Ivanov, O Ogievetsky, Ann Phys Ž NY 30 Ž wx 5 T Eguchi, B Gilkey, J Hanson, Physics Reports 66 Ž wx 6 JA Bagger, AS Galperin, EA Ivanov, VI Ogievetsky, Nucl Phys B 303 Ž wx 7 E Ivanov, G Valent, Harmonic Space Construction of the Quaternionic Taub-NUT metric, in preparation wx 8 A Galperin, E Ivanov, V Ogievetsky, PK Townsend, Class Quantum Grav 7 Ž wx 9 F Delduc, G Valent, Class Quantum Grav 10 Ž w10x B Carter, Commun Math Phys 10 Ž w11x T Chave, G Valent, Class Quantum Grav 13 Ž w1x G Gibbons, SW Hawking, Phys Lett B 78 Ž w13x SW Hawking, GC Hunter, DN Page, Nut Charge, Anti-de Sitter Space and Entropy, hep-thr w14x KS Stelle, BPS Branes in Supergravity, CERN-THr98-80, ImperialrTPr97-98r30; hep-thr

69 31 December 1998 Physics Letters B Octonionic gravitational instantons I Bakas a,1, EG Floratos b,c,, A Kehagias d,3 a Department of Physics, UniÕersity of Patras, GR-6500, Greece b Institute of Nuclear Physics, NRCS Democritos, Athens, Greece c Physics Department, UniÕersity of Crete, Iraklion, Greece d Theory DiÕision, CERN, 111 GeneÕa 3, Switzerland Received 0 October 1998 Editor: L Alvarez-Gaumé Abstract We construct eight-dimensional gravitational instantons by solving appropriate self-duality equations for the spin-connection The particular gravitational instanton we present has SpinŽ 7 holonomy and, in a sense, it is the eight-dimensional analog of the Eguchi-Hanson 4D space It has a removable bolt singularity which is topologically S 4 and its boundary at infinity is the squashed S 7 We also lift our solutions to ten and eleven dimensions and construct fundamental string and membrane configurations that preserve 1r16 of the original supersymmetries q 1998 Elsevier Science BV All rights reserved 1 Introduction The main purpose of the present work is to construct an eight-dimensional analogue of the noncompact gravitational instantons that arise in four dimensions w1 3 x, the simplest example being the Eguchi-Hanson space wx 4 It is well known that in four dimensions the non-trivial structure of such gravitational instantons is encoded into the asymptotic form of the metric whose boundary structure at infinity is given by the lens spaces of S 3 Since S 3 is the Hopf fibration of S 1 over S it is quite natural to employ the topological structure of the seven-sphere, 7 3 Ž 4 S, as Hopf fibration of S ssu over S in our search of eight-dimensional gravitational instantons 1 bakas@nxth04cernch, bakas@ajaxphysicsupatrasgr manolis@timaiosnrcpsariadne-tgr 3 alexandroskehagias@cernch In the context of gauge fields the first Hopf fibration arises in the construction of the UŽ 1 Dirac monopole with unit magnetic charge, while the second Hopf fibration arises in the construction of the SUŽ Yang-Mills instanton with unit topological number wx 5 Because of this analogy we expect that self-dual SUŽ Yang-Mills configurations in four dimensions will play an interesting role in future investigations of eight-dimensional gravitational instantons, for instance in generalizing the notion of nut potential in eight dimensions and classifying the fixed point sets of their isometry groups We will make a few remarks in this direction at the very end of the paper In any event the solution we present here could be viewed as the simplest non-trivial example of a more general class of eight-dimensional gravitational instantons, but further results will be reported elsewhere The metric we construct in the sequel has the squashed seven-sphere Žas opposed to the round seven-sphere as its boundary space at infinity, which r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

70 70 ( ) I Bakas et alrphysics Letters B is essential for establishing the non-triviality of our solution in eight dimensions Alternatively we could have arrived at the same result by deforming the squashed seven-sphere in the interior of an eight dimensional space using appropriately chosen coefficient functions, but we decided to emphasize the aspects of Hopf fibration in our presentation in view of other future generalizations As we have shown in wx 6, eight-dimensional manifolds which satisfy an appropriate self-duality condition for the spin connection have holonomy in SpinŽ 7 This provides us with a systematic way of constructing SpinŽ 7 holonomy manifolds by solving appropriate first-order equations Non-compact such manifolds have first been constructed in wx 7 and further discussed in wx 8, while compact ones were 8 constructed as T orbifolds in wx 9 after appropriate resolution of the singularities Here, by solving the eight-dimensional self-duality condition we find a space with SpinŽ 7 holonomy and isometry group SOŽ 5 = SUŽ It has a bolt singularity which is topologically S 4 and its boundary at infinity is the squashed S 7 Finally, we lift our solution in ten and eleven dimensions and we describe fundamental string and membrane configurations, which we name octonionic, and which preserve 1r16 of the original supersymmetries Octonionic algebra We will recall here some properties of the octonionic Ž Cayley algebra O and its relation to SOŽ 8 following mainly w10,11 x The octonionic algebra is a division algebra, which means it has a non-degenerate quadratic form Q that satisfies QŽ xy s QŽ x QŽ y and in addition QŽ x s0 implies xs0 The other division algebras are the real R, complex C and quaternionic H algebras R, C are commutative, H is non-commutative but associative, while O is neither commutative nor associative A basis for O is provided by the eight elements 1, e a, as1,,7, Ž 1 which satisfy the relation eesc a b abcecyd ab The tensor c c abc sq1 abc is totally antisymmetric with for abcs13, 516, 64, 435, 471, 673, 57 Ž 3 We may also define its dual c abcd 1 fgh cabcds eabcdfghc, Ž 4 3! so that cabcdsq1 for abcd s 145, 671, 356, 473, 5764, 6431, 7531 Ž 5 They satisfy the relations c abc cdhcsdd a dh b ydh a dd b ycdh ab, c abcd c s4 d a d b yd a d b yc ab, as ehcd e h h e eh c abc c sy4c a Ž 6 debc de Let us note here that the relation Ž is similar to the one obeyed by the quaternions Ž Pauli matrices However, the latter satisfy the Jacobi identity as a result of the associativity of the quaternionic algebra, while the octonions ea are not associative and hence do not satisfy the Jacobi identity The tensor c can be assigned to an SOŽ abc 8 representation C abgd, where the Greek indices range from 1 to 8 and also set the notation asž a,8 We have Cabg 8 sc abc, Cabgdsc abcd, Ž 7 which is self-dual 1 zhuk Cabgds eabgdzhukc, Ž 8 4! and belongs to one of the three different 35 s of SOŽ 8 35, 35 Ž related by triality Õ " It satisfies the fundamental identity qž d u d z yd z d u qž d h d u yd u d h C C zhud s d z d h yd z d h d u abgd a b b a g a b b a dg h d h qc zh d u a b b a g ab g qc uz d h qc hu d z qc zh d u ab g ab g ga b qc uz d h qc hu d z qc zh d u ga b ga b bg a qc uz d h qc hu d z 9 bg a bg a

71 ( ) I Bakas et alrphysics Letters B We may use the octonions ea to construct a representation of the SOŽ 7 g-matrices according to Ž ga bcsic abc, Ž ga b8 sid ab, g a,g b4 sd ab, Ž 10 w x Ž We may also form the SOŽ 8 g-matrices G a s Ž G,G,as1,,8 ab 1 so that g s g a,g b are the SO 7 generators a 8 ž a / 0 ig a 0 1 G s, G s, yig a 8 ž / G,G 4sd, Ž 11 a b ab that correspond to the standard embedding of SOŽ 7 Õ in SOŽ 8 The latter is defined as the stability subgroup SOŽ 7 ; SOŽ 8 of the vector representation according to which we have the decomposition 8 Õ s7q1, 8 " s8, Ž 1 where 8 Õ, 8 " are the vector and the two spinorial representations of SOŽ 8 The SOŽ 8 generators ab 1 w a b G s G, G x satisfy the relations G ab sc abc G 9 G 8c, cabcdg ab syž 4q G 9 G cd, Ž 13 where, as usual, G sž 0 y1/ is the chirality matrix It is not then difficult to verify that the generators ab 3 ab 1 ab gd G s 8 G q 6Cgd G, 14 leave the right-handed spinor h q invariant G ab hqs0 Ž 15 The stability group of h is again another SOŽ 7, which we will denote by SOŽ 7 q, according to which 8 Õ s8, 8qs7q1, 8ys8 Ž 16 The singlet in the decomposition of 8q corresponds ab to the null eigenspinor h of G in Eq Ž 15 This can be seen by considering the Casimir G ab G ab which has a zero eigenvalue for the 8 q If we had used in Eq Ž 14 the second 35 of SOŽ 8, the antiself-dual one, then G ab would annihilate the left-handed spinor h y In this case, the stability group of the latter is a third SOŽ 7 y subgroup of SOŽ 8 defined by 8 Õ s8, 8qs8, 8ys7q1 Ž 17 Again, G ab constructed with the antiself-dual 35 has a zero eigenvalue which corresponds to the singlet in 8 The three different SOŽ 7 subgroups of SOŽ y 8, Ž Ž " SO 7 Õ,SO 7 are related by triality () 3 Spin 7 -holonomy spaces Let us now assume that the eight-dimensional spin manifold X admits a spin connection which satisfies the self-duality condition 1 gd vabms Cabgd v m, 18 where C is defined in Eq Ž abgd 7 It is not difficult to verify that a relation of the form vabms lcabgd vm gd is consistent for ls1r,y1r6 as can be verified by multiplying both sides by C abzh and using Eq Ž 9 Here we consider only the case ls 1r, while the other possibility has not been investigated and left to future work Eq Ž 18 is an eight-dimensional analog of the standard self-duality condition in four dimensions where the totally antisymmetric symbol eabcd is replaced by C abgd Eight-di- mensional manifolds with a connection satisfying Eq Ž 18 in the Euclidean regime will be called gravitational instantons Note that the octonions have also appeared in the construction of an eight-dimenw1 x The holonomy group of such sional Kerr metric manifolds is in SpinŽ 7 as can be seen as follows Recall first that a manifold is of SpinŽ 7 holonomy if and only if the Cayley four-form w13x Cse 1 ne ne 3 ne 8 qe 5 ne 1 ne 6 ne 8 qe 6 ne ne 4 ne 8 qe 4 ne 3 ne 5 ne 8 qe 4 ne 7 ne 1 ne 8 qe 6 ne 7 ne 3 ne 8 qe 5 ne 7 ne ne 8 qe 4 ne 5 ne 6 ne 7 qe ne 3 ne 7 ne 4 qe 1 ne 3 ne 5 ne 7 qe 1 ne 3 ne 5 ne 7 qe 1 ne ne 7 ne 6 qe ne 3 ne 5 ne 6 qe 1 ne ne 4 ne 5 qe 1 ne 3 ne 4 ne 6, Ž 19

72 7 ( ) I Bakas et alrphysics Letters B a where e, as1,,8 is an orthonormal frame the i i metric is Ý e me in this frame i, is torsion free, ie, if it is closed dcs0 Ž 0 In this case, the manifold is Ricci flat The Cayley form C is self-dual and, in addition, it is invariant under SpinŽ 7 It is the singlet in the decomposition of the 35 in the non-standard embedding of SOŽ q 7 in SOŽ 8 given in Eq Ž 16 according to which 35Õ s35, 35qs1q7q7, 35ys35 Ž 1 In order to prove now that manifolds which satisfy Eq Ž 18 have SpinŽ 7 holonomy, let us observe that the Cayley four-form C can actually be written as 1 a b g d Cs Cabgd e ne ne ne 4! It is not difficult to verify using Eq Ž 9 and the structure equations de a qv a e b s0, v a sv a dx m b b bm that if the spin connection satisfies Eq Ž 18 the Cayley form C will be closed Thus, manifolds with connection satisfying the self-duality condition Ž 18 are SpinŽ 7 Ricci flat manifolds In addition, such manifolds admit covariantly constant spinors Indeed, for an SOŽ 8 spinor h with 1 ab Dm hs Emy 4 vabmg h, 3 Ž where a,b, s1,,8 are world indices, m,n,s1,,8 are curved space indices and v ab satisfy Eq Ž 18, one may easily verify that h is necessarily right-handed and A 1 ab A Dm hsemh y vabmg h, 4 ab where G has been defined in Eq Ž 14 Thus, h is covariantly constant Dm hs0 if it is a constant spinor and satisfies Eq Ž 15 Note also that the integrability condition of Eq Ž 3 is Rabmn G ab h A s0, Ž 5 from which follows, after multiplication with G m, that spaces of SpinŽ 7 holonomy are indeed Ricci flat 4 The octonionic gravitational instanton We have just seen that a systematic way to construct manifolds of SpinŽ 7 holonomy is provided by solving Eq Ž 18 Selecting one direction, say the eighth, Eq Ž 18 is written as 8 1 pq vr sycrpqv 6 The rest of the equations, namely, p 1 p rs p r8 vq s cqrsv ycqrv, 7 are automatically satisfied if Eq Ž 6 holds The self-duality conditions are then explicitly written as v 81 syž v 3qv65qv 47, v 8 syž v 31qv46qv 57, v 83 syž v1qv54qv 67, v 84 syž v6qv35qv 71, v 85 syž v16qv43qv 7, v 86 syž v51qv4qv 73, v 87 syž v14qv36qv 5 Ž 8 We will assume that the metric of the SpinŽ 7 -holonomy manifold X is of the form 1 4 i ds sf r dr qg r dm q sin ms Ž i i qh r s ya, 9 where S and s, i s 1,,3 are SUŽ i i left-invariant one-forms satisfying 1 1 dsisyeijksjns k, dsisyeijksjns k, and m Ž 30 Aiscos S i Ž 31 In terms of the angular coordinates a, b,g,u,f,c with 0-a,uFp, 0-b,fFp, 0-g,cF4p, the SUŽ left-invariant one-forms S i,si are explic- itly given by S1 scosg daqsing sina db, S sysing daqcosg sina db, S3 sdgqcosa db, s1 scosc duqsinc sinu df, s sysinc duqcosc sinu df, s3 sdcqcosu df,

73 ( ) I Bakas et alrphysics Letters B The surfaces r s const have the topology of Ž 4 SU -bundle over S with A the k s 1 SUŽ i instanton on S 4 The isometry of X for generic gž r,hž r are SOŽ 5 =SUŽ In particular, if gž r s hž r the isometry group is enlarged to SOŽ 8 since in this case the surfaces rsconst are round seven-spheres S 7 written in their Hopf-fibered form It should be noted that the ansantz for the metric Ž 9 is similar to the one for the Eguchi-Hanson 4d gravitational instanton where the Hopf-fibration of the S 7 is replaced with the corresponding fibration of S 3 Employing the orthonormal base ˆ m i 1 i e s gž r sinm S i, e shž r ž siycos S i/, e 7 sfž r dr, e 8 sgž r dm, Ž 3 where is1,,3 and is4,5,6s1,,3, ˆ ˆ ˆ ˆ we find that the spin connection is 1 hž r k kˆ vijsy e q e e ijk, gž r sinm gž r ž / 1 cos mr ˆk k ž hž r gž r sinm / v sy e q e e, ij ˆˆ hž r hž r k 8 vijˆs eijke q dije, gž r gž r 1 hž r i iˆ v 8isy cotm e y e, gž r gž r hž r gž r i 8 v 8iˆsy e, v 87s e, gž r gž r fž r X gž r hž r i iˆ vi7 s e, vi7 ˆ s e, Ž 33 gž r fž r hž r fž r with X sdrdr Then, using the components of the connection, we find that Eq Ž 8 is satisfied if hr, gž r obey the differential equations X hž r 1 hž r q y s0, hž r fž r hž r gž r X gž r 3hŽ r q s0 Ž 34 gž r fž r gž r X X ijk We may use the reparametrization invariance of the X metric under r r sr X Ž r to fix one of the three functions hr, gž r and fž r In particular, choosing 9 0 g r s r, 35 we find after solving Eq 34 that m 10r3 1 5 ž ž r / / hž r s gž r 1y, 1 fž r s, Ž 36 10r3 m 1yž r / where m is an integration constant taken as the moduli Thus, the full metric Ž 9 turns out to be dr 9 1 ds s q r Ž dm q sin ms m 10r3 0 4 i 1yž r / ž / / ž / m 10r3 m 9 q 100 r 1yž siycos S i r Ž 37 It resembles the Eguchi-Hanson metric with moduli m, but in the present case we have a radial fall off rate r y10r3, instead of r y4 in Eguchi-Hanson, because of the eight-dimensional nature of our solution One may easily verify that the apparent singularity at rsm is a removable bolt singularity Indeed, near r s m, the metric Ž 37 at m, a, b,g s const, takes the form 10r3 Ž 9 1 ds ; 5 d r q 4 rs i, 38 with r s r 1 y mrr which is the metric on R 4 in polar coordinates Thus, the topology of X is locally of the form R 4 =S 4 and the singularity at r s m is just a coordinate singularity Asymptotically the metric 37 takes the form ds sdr q r 9 0 = ž m i 5ž i i / / dm q sin msq s ycos S, Ž 39 and thus the boundary at infinity is the squashed S 7 w14 x

74 74 ( ) I Bakas et alrphysics Letters B Finally we note for completeness that the bolt structure and the asymptotic behaviour of the metric are insensitive to the value of the free parameter m This should be compared to the similar behaviour of the Eguchi-Hanson metric in four dimensions, which for ms0 reduces to the flat space metric in polar coordinates modulo global issues that arise from identifications in the allowed range of the Euler angles resulting in S 3 rz at infinity 5 Brane solutions The solution Ž 37 we found above can be lifted to appropriate brane solutions in ten- and eleven-dimensional supergravity In particular, the octonionic gravitational instanton may be taken as the transverse space of a fundamental string in string theory or of a fundamental membrane in M-theory It can also be considered as a domain-wall solution Žeight- brane in massive IIA supergravity In all cases, these brane solutions preserve the minimum amount of supersymmetry since there exists only one covariant constant spinor on the octonionic gravitational instanton Thus, the brane solutions we will describe in the following are 1r16 BPS states 51 Type II octonionic fundamental string We consider ten-dimensional backgrounds of the form M =N 8 with metric ds se A Ž ydt qdx qe B gmndy m dy n, Ž 40 where y m,ms1,,8 are coordinates on the trans- 8 verse space N, AsAŽ y, BsBŽ y and gmn is the metric on N 8 Similarly, we assume that the dilaton f s fž y and the only non-vanishing component of the antisymmetric NS-NS two-form BMN is Btxs ye CŽ y Then, we find that the fermionic shifts equal to 1 yf r NPQ N PQ dcmsdmeq 96 e GM y9dmg HNPQ e 1 1 m yf r MNP dlsy G Emfeq e G HMNP e, ' 4' Ž 41 vanish if 3f f As, Bsy, 4 4 Csf, Ž 4 and the transverse space N 8 admits a covariant constant spinor Finally, from the field equations, we obtain that e y f is harmonic in N 8, ie, = e y f s0 Ž 43 Recall at this point that our octonionic gravitational instanton admits a covariant constant spinor and so it can be taken as a solution for the transverse 8 space N In this case Eq Ž 43 reduces to 10r3 7 yf r r ž r / 1 m E r 1y E e s0 Ž 44 r 7 ž / The solution is described in terms of the Gauss hypergeometric function F Ž a, b;c, z as 1 L y f Ž r yf` r3 e se y F1Ž 1, 5; 5, Ž mrr, r 6 Ž 45 where L is a constant proportional to the string tension T, using the integral representation of F Ž a,b;c, z 1 F1Ž a,b;c, z G Ž c s G Ž b G Ž cyb H t Ž 1yt Ž 1yzt dt, 0 Re c)re b)0 Ž 46 = 1 by1 cyby1 ya To compare it with other results in the literature, we note that a heterotic octonionic string was conw15x with Ž 0,1 supersymmetry on its structed in world-volume In that case, however, the Bianchi identity for HMNP was solved by introducing appropriate self-dual eight-dimensional gauge fields; for the relevant technical details we refer the reader to the literature w16 x Hence, the solution we describe in this paper is different 5 Octonionic fundamental membrane We may also lift the octonionic gravitational instanton in the eleven-dimensional supergravity Here, the bosonic field in the graviton multiplet is the graviton and the antisymmetric three-form A MNP

75 ( ) I Bakas et alrphysics Letters B We will thus consider an eleven-dimensional background of the form M 3 =N 8 with metric ds se A Ž ydt qdx1 qdx qe B gmndy m dy n, Ž 47 and non-vanishing components for A MNP, namely yc A se, where AsAŽ y, BsBŽ y tx x and Cs 1 CŽ y as before In this case, the gravitino shift 1 NPQR N PQR dcmsdmeq 88 GM y8dmg FNPQR e Ž 48 vanish if there exists a covariant constant spinor on N 8 and 1 As C, Bsy C Ž Thus, we may choose for N 8 the octonionic gravitational instanton as before In addition, the field equations are satisfied if e yc is a harmonic function on N 8 In this case we find L yc Ž r 3 yc` r3 e se y F1Ž 1, 5; 5, Ž mrr, r 6 Ž 50 where now L3 is proportional to the membrane tension T 3 The fundamental octonionic membrane we have just constructed is completely different from the octonionic membrane constructed in w17 x We only note that the latter is not supersymmetric and furthermore it has a non-trivial Fmnpq proportional to the self-dual tensor in Eq Ž 7 Hence our result yields a new configuration in eleven-dimensional supergravity 6 Discussion Summarizing, we have shown that eight-dimensional gravitational instantons can be constructed as solutions of self-duality equations for the spin connection The octonions, which are integral part of our method, yield in this fashion Ricci flat manifolds with holonomy in SpinŽ 7 We have also presented an explicit solution with isometry group SOŽ 5 = SUŽ having a bolt singularity and the asymptotic structure of the squashed seven-sphere; it can be regarded as a higher dimensional analogue of the Eguchi-Hanson space Also, because of supersymmetry, we can elevate our solution to ten and eleven dimensions in order to obtain octonionic fundamental string and membrane configurations respectively which are 1r16 BPS states The present framework can be developed further by considering eight-dimensional solutions with SUŽ isometry and attempt a classification of the corresponding gravitational instanton symmetries in analogy with four dimensions wx In the four dimensional case one considers the action of an oneparameter isometry group UŽ 1 and classifies its fixed points into two different types: isolated points called nuts and -surfaces Ž spheres called bolts Then, performing a dimensional reduction of Einstein s equations with respect to the orbits of the UŽ 1 isometry group to three dimensions, one exhibits a duality between the electric and the magnetic aspects of gravity Žcorresponding to bolts and nuts respectively via the Ehlers transform Of course, certain solutions might exhibit more isometries which are not of general value; for instance the Eguchi- Hanson space has an additional SOŽ 3 isometry apart from the UŽ 1 In eight dimensions, although the particular solution we found has an additional SOŽ 5 isometry apart from SUŽ, we think that focusing on more general classes of solutions with SUŽ isometry are worth studying further Selecting gravitational instantons with SUŽ isometry in eight dimensions is also quite natural from the point of view of Hopf fibrations, as is the selection of gravitational instantons with UŽ 1 isometry in four dimensions In the reduction of an eight dimensional theory to five dimensions using the orbits of the corresponding isometry group, S 3, we are naturally led to the notion of nut potential as an antisymmetric -form field that dualizes the SUŽ gauge field connection of the metric The reasoning is the same one that yields an axion-like scalar function as nut potential in four-dimensional spaces with UŽ 1 isometry It will be interesting to carry out this line of investigation in the present case and see whether M and M5 branes, when appropriately taken in eleven-dimensional supergravity, provide the analogue of bolts and nuts in establishing various electric and magnetic aspects of eight dimensional gravitational instantons We hope that in this way, apart for improving our conceptual understanding of the general problem, we will also be able to find more explicit instanton solutions as in

76 76 ( ) I Bakas et alrphysics Letters B four dimensions We plan to report on these issues elsewhere Note added We have been informed by B Acharya Žprivate communication that he has also employed the 7 squashed S in searching for SpinŽ 7 -holonomy manifolds Ž unpublished work We also became aware after circulating the first draft of our paper that the particular metric Ž 37 was constructed some time ago by solving directly Einstein s equations in eight dimensions w18 x; we thank G Gibbons and C Pope for bringing their paper to our attention Their study was based on second order differential equations rather than the first order self-duality Eqs Ž 18, and Ž in a sense made explicit earlier mathematical results on the construction of complete metrics with SpinŽ 7 holonomy w19 x We thank D Morrison for also pointing out the relevance of Ref w19x to the general framework we have adopted here References wx 1 SW Hawking, Phys Lett A 60 Ž wx GW Gibbons, SW Hawking, Comm Math Phys 66 Ž wx 3 T Eguchi, PB Gilkey, AJ Hanson, Rhys Rep 66 Ž wx 4 T Eguchi, AJ Hanson, Phys Lett B 74 Ž ; GW Gibbons, SW Hawking, Phys Lett B 78 Ž wx 5 A Trautman, Int J Phys 16 Ž wx 6 E Floratos, A Kehagias, Phys Lett B 47 Ž hep-thr wx 7 R Bryant, Ann Math 16 Ž wx 8 BS Acharya, M O Loughlin, Phys Rev D 55 Ž hep-thr96118 wx 9 D Joyce, Invent Math 13 Ž w10x M Gunaydin, F Gursey, J Math Phys 14 Ž ; R Dundarer, F Gursey, CH Tze, J Math Phys 5 Ž w11x B de Wit, H Nicolai, Nucl Phys B 31 Ž w1x A Chakrabarti, Phys Lett B 17 Ž w13x SM Salamon, Riemannian geometry and holonomy groups, Pitman Res Notes in Math 01, Longman, 1989 w14x MA Awada, MJ Duff, CN Pope, Phys Rev Lett 50 Ž ; see also, MJ Duff, BEW Nillson, CN Pope, Phys Rep 130 Ž w15x JA Harvey, A Strominger, Phys Rev Lett 66 Ž w16x DB Fairlie, J Nuyts, J Phys A 17 Ž ; S Fubini, H Nicolai, Phys Lett B 155 Ž ; B Grossman, T Kephart, JD Stasheff, Comm Math Phys 96 Ž ; M Gunaydin, H Nicolai, Phys Lett B 351 Ž ; ibid B 376 Ž w17x MJ Duff, JM Evans, RR Khuri, JX Lu, R Minasian, Phys Lett B 41 Ž hep-thr w18x GW Gibbons, DN Page, CN Pope, Commun Math Phys 17 Ž w19x RB Bryant, SM Salamon, Duke Math J 58 Ž

77 31 December 1998 Physics Letters B Equivalence principle, Planck length and quantum Hamilton Jacobi equation Alon E Faraggi a,1, Marco Matone b, a Department of Physics, UniÕersity of Minnesota, Minneapolis, MN 55455, USA b Department of Physics G Galilei, Istituto Nazionale di Fisica Nucleare, UniÕersity of PadoÕa, Via Marzolo 8, PadoÕa, Italy Received 6 October 1998 Editor: M Cvetič Abstract The Quantum Stationary HJ Equation Ž QSHJE that we derived from the equivalence principle, gives rise to initial conditions which cannot be seen in the Schrodinger equation Existence of the classical limit leads to a dependence of the integration constant ls l 1qi l on the Planck length Solutions of the QSHJE provide a trajectory representation of quantum mechanics which, unlike Bohm s theory, has a non-trivial action even for bound states and no wave guide is present The quantum potential turns out to be an intrinsic potential energy of the particle which, similarly to the relativistic rest energy, is never vanishing q 1998 Elsevier Science BV All rights reserved PACS: 03; 0365-w; 0460-m Keywords: Equivalence principle; Quantum potential; Trajectories; Mobius symmetry; Planck length Let us consider a one-dimensional stationary system of energy E and potential V and set W'VŽ q y E In wx 1 the following equivalence principle has been formulated For each pair W a,w b, there is a transformation a b q q sõ( q a ), such that W a Ž q a W aõ Ž q b sw b Ž q b Ž faraggi@mnhepohepumnedu matone@padovainfnit Implementation of this principle uniquely leads to the Quantum Stationary HJ Equation Ž QSHJE wx 1 ž / 1 E S0Ž q " qvž q yeq S 0,q4 s0, m E q 4m where S0 is the Hamilton s characteristic function also called reduced action In this equation the Planck constant plays the role of covariantizing parameter The fact that a fundamental constant follows from the equivalence principle suggests that other fundamental constants as well may be related to such a principle r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

78 78 ( ) AE Faraggi, M MatonerPhysics Letters B We have seen in wx 1 that the implementation of the equivalence principle implied a cocycle condition which in turn determines the structure of the quantum potential A property of the formulation is that unlike in Bohm s theory w,3 x, the quantum potential " QŽ q s S 0,q 4, Ž 3 4m like S 0, is never trivial This reflects in the fact that a general solution of the Schrodinger equation will have the form i i 1 y S0 S " 0 " ( 0 cs Ae qbe Ž 4 S X ž / Ž D If c,c is a pair of real linearly independent solutions of the Schrodinger equation, then we have i S " 0 i a wqi l e se, Ž 5 wyi l where wsc D rc, and Re l/0 We note that l, on which the dynamics depends, does not appear in the conventional formulation of quantum mechanics In this Letter we show that non-triviality of the quantum potential of the free particle with vanishing energy is at the heart of the existence of a length scale Actually, we will show the appearance of the Planck length in the complex integration constant l, indicating that gravity is intrinsically and deeply connected with quantum mechanics Therefore, there is trace of gravity in the constant l whose role is that of initial condition for the dynamical Eq Ž In this context we stress that l plays the role of hidden constant as it does not appear in the Schrodinger equation This is a consequence of the fact that whereas the Schrodinger equation is a second-order linear differential equation, the QSHJE is a third-order non-linear one, with the associated dynamics being deeply connected to the Mobius symmetry of the Schwarzian derivative Before going further it is worth stressing that the basic difference between our S0 and the one in Bohm s theory w,3x arises for bound states In this case the wave function c is proportional to a real function This implies that with Bohm s identification c Ž q srž qexp Ž i S r" 0, one would have S0 scnst Therefore, all bound states, like in the case of the harmonic oscillator, would have S0s cnst Ž for which the Schwarzian derivative is not defined Besides the difficulties in getting a non-trivial classical limit for pseq S 0, this seems an unsatisfactory feature of Bohm s theory which completely disappears if one uses Eq Ž 4 This solution directly follows from the QSHJE: reality of c simply implies that < A< s< B< and there is no trace of the solution S0s cnst of Bohm theory Furthermore, we would like to remark that in Bohm s approach some interpretational aspects are related to the concept of a pilot-wave guide There is no need for this in the present formulation This aspect and related ones have been investigated also by Floyd wx 4 Nevertheless, there are some similarities between the approach and Bohm s interpretation of quantum mechanics In particular, solutions of the QSHJE provide a trajectory representation of quantum mechanics That is for a given quantum mechanical system, by solving the QSHJE for S Ž q 0, we can evaluate pse S Ž q q 0 as a function of the initial conditions Thus, for a given set of initial conditions we have a predetermined orbit in phase space The solutions to the third-order non-linear QSHJE are obtained by utilizing the two linearly independent solutions of the corresponding Schrodinger equation In the case of the free particle with vanishing energy, ie with W Ž q sw 0 Žq 0 '0, we have that two real linearly independent solutions of the Schrodinger equation are c D 0 sq 0 and c 0 s1 As the different dimensional properties of p and q led to introduce the Planck constant in the QSHJE wx 1, in the case of c D 0 and c 0 we have to introduce a length constant Let us derive the quantum potential in the case of 0 the state W By Ž 3 and Ž 5 we have p0 " ž l 0ql 0/ 0 Q sy sy, m 8m< q yi l 0 < where p s E 0 Ž 0 0 S q 0 q 0 Therefore, we see that the quantum potential is an intrinsic property of the particle as even in the case of the free particle of vanishing energy one has Q 'u 0 This is strictly 0 related to the local homeomorphicity properties that the ratio of any real pair of linearly independent solutions of the Schrodinger equation should satisfy,

79 ( ) AE Faraggi, M MatonerPhysics Letters B a consequence of the Mobius symmetry of the Schwarzian derivative wx 5 In this context let us recall that S, q4 0 is not defined for S0s cnst More generally, the QSHJE is well defined if and only if the corresponding w/ cnst is of class C Ž Rˆ with E w differentiable on the extended real line RsR ˆ q j ` 4 wx 5 We stress that Q does not correspond to the Bohm s quantum potential which, in a different conwx 6 Here text, was considered as internal potential in we have the basic fact that the quantum potential is never trivial, an aspect which is strictly related to p q duality and to the existence of the Legendre transformation of S for any W w1,5 x 0 Besides the different dimensionality of c D 0 and c 0, we also note that similarity between the equivalence principle we formulated and the one at the heart of general relativity would suggest the appearance of some other fundamental constants besides the Planck constant We now show that non-triviality of Q 0 or, equivalently, of p, has an important 0 consequence in considering the classical and E 0 limits in the case of the free particle In doing this, we will see the appearance of the Planck length in the complex integration constant l of the QSHJE Let us consider the conjugate momentum in the wx case of the free particle of energy E We have 5 ž / " l Eql E pe s", Ž 7 < y1 k sinkqyi l coskq< E ' where ks me r" The first condition is that in the " 0 limit the conjugate momentum reduces to the classical one ' lim p s" me Ž 8 " 0 E On the other hand, we should also have ž / " l 0ql 0 lim pe sp0 s" Ž 9 E 0 < qyi l 0 < Let us first consider the limit Ž 8 By Ž 7 we see that in order to reach the classical value ' me in the " 0 limit, the quantity l E should depend on E Let us set l sk y1 fž E," ql, Ž 10 E E where f is dimensionless Since le is still arbitrary, we can choose f to be real By Ž 7 we have ' me f Ž E," qmež l E ql E r" pe s" ikq e q Ž fž E," y1qle k coskq Ž 11 Observe that if one ignores le and sets les0, then by Ž 8 we have lim fž E," s1 Ž 1 " 0 We now consider the properties that le and f should have in order that Ž 1 be satisfied in the physical case in which le is arbitrary but for the condition Re l E / 0, as required by the existence of the QSHJE First of all note that cancellation of the divergent term E y1r in y1r " Ž me f Ž E," q" Ž leqle E y1r p ; ", E 0 < qyi" Ž me f Ž E," yil < E Ž 13 yields lim E y1r fž E," s0 Ž 14 E 0 The limit Ž 9 can be seen as the limit in which the trivializing map reduces to the identity Actually, the trivializing map, introduced in wx 1 and further investiwx 5, which connects the state WsyE with gated in the state W 0, reduces to the identity map in the E 0 limit In the above investigation we considered q as independent variable, however one can Ž 0 0 also consider qe q so that lim E 0 qe sq and in the above formulas one can replace q with q E We know from Ž 14 that k must enter in the expression of fž E," Since f is a dimensionless constant, we need at least one more constant with the dimension of a length Two fundamental lengths one can consider are the Compton length " lc s, Ž 15 mc and the Planck length ( "G lp s Ž 16 3 c

80 80 ( ) AE Faraggi, M MatonerPhysics Letters B ( Two dimensionless quantities depending on E are E x csklcs mc, Ž 17 and ( meg x psklps Ž 18 3 "c On the other hand, concerning x c we see that it does not depend on " so that it cannot be used to satify Ž 1 Therefore, we see that a natural expression for f is a function of the Planck length times k Let us set f E," se ya Ž xy1 p, Ž 19 where Ý a x y1 s a x yk Ž 0 p k p kg1 The conditions Ž 1, Ž 14 correspond to conditions on the coefficients a k For example, in the case in which one considers a to be the function a x y1 sa x y1, Ž 1 p 1 p then by 14 we have a 1 )0 In order to consider the structure of l E, we note that although e ya Ž xy1 p cancelled the E y1r divergent term, we still have some conditions to be satisfied To see this note that ya Ž x y1 ' me e p qme Ž l E ql E r" pe s", y1 ikq ya Ž x p e q e y1qkl coskq Ž E Ž so that the condition 8 implies l E Ž 3 lim s0 Ž 4 " 0 " To discuss this limit, we first note that y1 y1 ya Ž x p "k e q" Ž leq le pe s" Ž 5 y1 y1 ya x k sinkqyi k e Ž y1 p q l coskq Ž E So that, since lim k y1 e ya Ž xy1 p s0, Ž 6 E 0 Ž we have by 9 and 5 that l s lim l s lim l sl Ž 7 0 E E 0 E 0 E 0 Let us now consider the limit lim p s0 Ž 8 " 0 0 First of all note that, since ž / " l 0ql 0 p0 s", Ž 9 < 0 q yi l 0 < we have that the effect on p0 of a shift of Im l 0 is equivalent to a shift of the coordinate Therefore, in considering the limit Ž 8 we can set Im l 0 s 0 and distinguish the cases q 0 /0 and q 0 s0 Observe that as we always have Re l 0 /0, it follows that the denominator in the right hand side of Ž 9 is never vanishing Let us define g by Re l ; " g 0 Ž 30 " 0 We have ½ 0 1yg " gq1, q 0 /0, p ; Ž 31 " 0 ", q s0, 0 and by Ž 8 y1-g-1 Ž 3 A constant length having powers of " can be constructed by means of lc and l p We also note that a constant length independent from " is provided by le se rmc where e is the electric charge Then l 0 can be considered as a suitable function of l c, lp and l satisfying the constraint Ž 3 e The above investigation indicates that a natural way to express l is given by E le se yb Ž x p l 0, Ž 33 where Ý b Ž x s b x k Ž 34 p k p kg1 Any possible choice of bž x p should satisfy the conditions Ž 4 and Ž 7 For example, for the modulus l built with bž x E p s b1 x p, one should have b 1 )0

81 ( ) AE Faraggi, M MatonerPhysics Letters B Summarizing, by Ž 10, Ž 19, Ž 7 and Ž 33 we have l sk y1 e ya Ž xy1 p qe yb Ž x p E l 0, Ž 35 where l sl Ž l,l,l 0 0 c p e, and for the conjugate mo- mentum of the state WsyE we have y1 y1 ya Ž x p yb Ž x p k "e q"e Ž l 0q l 0 pe s" y1 y1 y1 ya Ž x p yb Ž x p k sinkqyi k e q e l coskq Ž 0 Ž 36 It is worth recalling that the conjugate momentum does not correspond to the mechanical momentum mq w4,5 x However, the velocity itself is strictly related to it In particular, following the suggestion by Floyd wx 4 of using Jacobi s theorem to define time parametrization, we have that velocity and conjugate momentum satisfy the relation wx 5 1 qs, Ž 37 E p E which holds also classically We stress that the appearance of the Planck length is strictly related to p q duality and to the existence of the Legendre transformation of S0 for any state This p q duality has a counterpart in the c D c duality w1,5x which sets a length scale which already appears in considering linear combinations of c D 0 s q 0 and c 0 s1 This aspect is related to the fact that we always have S 0/cnst and S0Au qqcnst, so that also for the states W 0 and WsyE one has a non-constant conjugate momentum In particular, the Planck length naturally emerges in considering lim E 0 pes p 0, together with the analysis of the " 0 limit of both pe and p 0 Acknowledgements Work supported in part by DOE Grant No DE FG 087ER4038 Ž AEF and by the European Commission TMR programme ERBFMRX CT Ž MM References wx 1 AE Faraggi, M Matone, hep-thr ; Phys Lett B 437 Ž , hep-thr971108; Phys Lett A 49 Ž , hep-thr wx D Bohm, Phys Rev 85 Ž ; 180 wx 3 PR Holland, The Quantum Theory of Motion, Cambridge Univ Press, Cambridge, 1993 wx 4 ER Floyd, Phys Rev D 5 Ž ; D 6 Ž ; D 9 Ž ; D 34 Ž ; Phys Lett A 14 Ž ; Found Phys Lett 9 Ž , quantphr ; quant-phr ; quant-phr wx 5 AE Faraggi, M Matone, hep-thr980916, to appear in Phys Lett B; hep-thr wx 6 JG Muga, R Sala, RF Snider, Phys Scripta 47 Ž

82 31 December 1998 Physics Letters B ž / Anomalous U 1, gaugino condensation and supergravity T Barreiro a,1, B de Carlos a,,3, JA Casas b,4, JM Moreno b,5 a Centre for Theoretical Physics, UniÕersity of Sussex, Falmer, Brighton BN1 9QH, UK b Instituto de Estructura de la Materia, Consejo Superior de InÕestigaciones Cientıficas, Serrano 13, 8006 Madrid, Spain Received September 1998 Editor: R Gatto Abstract The interplay between gaugino condensation and an anomalous Fayet-Iliopoulos term in string theories is not trivial and has important consequences concerning the size and type of the soft SUSY breaking terms In this paper we examine this issue, generalizing previous work to the supergravity context This allows, in particular, to properly implement the cancellation of the cosmological constant, which is crucial for a correct treatment of the soft breaking terms We obtain that the D-term contribution to the soft masses is expected to be larger than the F-term one Moreover gaugino masses must be much smaller than scalar masses We illustrate these results with explicit examples All this has relevant phenomenological consequences, amending previous results in the literature q 1998 Elsevier Science BV All rights reserved 1 Introduction Superstring theories have been, for now a number of years, the most promising candidates for physics beyond the Standard Model Ž SM Two major problems, however, have impeded extracting definite phenomenological predictions from these constructions: the large vacuum degeneracy and the issue of supersymmetry Ž SUSY breaking A large amount of work has been devoted to studying both questions, which has led to the proposal of several mechanisms and models in order to solve them Among those, gaugino condensation wx 1 is the most promising one, and it has been implemented, more or less successfully, in superstring inspired scenarios This is a non-perturbative effect which provides the typically flat stringy fields, the dilaton and the moduli, with a non-trivial potential which could eventually lead to their stabilization at realistic values It can Ž Ž also give rise to SUSY breaking at the so-called condensation scale L ; M exp yarg c P string where MP is the reduced Planck mass and g is the string coupling constant string, relating therefore both major problems of superstring phenomenology So far all this is happening in the hidden sector of the theory, governed by a 1 tiagob@pcssmapssusxacuk bde-carlos@sussexacuk 3 Present address: CERN, TH Division, CH-111 Geneva 3, Switzerland 4 casas@cccsices 5 jmoreno@pinar1csices r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

83 ( ) T Barreiro et alrphysics Letters B strong-type interaction, and, in this simple picture, gravity is responsible for transmitting the breakdown of SUSY to the observable sector Ž where the SM particles and SUSY partners live, parameterized in terms of the gravitino mass m 3r ;L 3 crmp which sets the scale of the soft breaking terms On the other hand, it is typical of many superstring constructions to have anomalous UŽ 1 X symmetries whose anomaly cancellation is implemented by a Green-Schwarz mechanism w 4 x, where the dilaton plays a crucial role This anomalous symmetry induces a Fayet-Iliopoulos Ž F-I D-term in the scalar potential, and this will generate extra contributions to the soft terms Therefore one might expect an interesting interplay between SUSY breaking through gaugino condensation and the presence of the UŽ 1 X Moreover, there is now a new scale in the theory, that of the UŽ 1 X breakdown, j Given that both hidden and observable sector fields are charged under this symmetry, the F-I term will act effectively as an extra source of transmission of the SUSY breaking between sectors In fact, given that in general j-m P, one expects that the F-I will set the scale of the soft breaking terms All these issues have been recently discussed in a number of papers w5 15 x, but always in a global SUSY limit In Ref wx 5 it was pointed out that the contribution from the D-terms to the soft breaking terms is the dominant one, which was claimed to have phenomenological merits In particular, this could be useful to get naturally universal soft masses, thus avoiding dangerous flavour changing neutral current effects On the other hand, in Ref wx 6 it was shown that that discussion had ignored the crucial role played by the dilaton in the analysis Furthermore, it was claimed Ž based on particular ansatzs for the Kahler potential that the situation is in fact the opposite, namely the contribution from the F-terms to the soft breaking terms is the dominant one Here, we present the generalization of the results obtained in Ref wx 6 to the supergravity Ž SUGRA case, which is the correct framework in which to deal with effective theories coming from strings Whereas we have checked that the SUGRA corrections do not affect significantly the minimization of the potential, and thus the vacuum structure of the theory, we have seen that they are crucial for a correct treatment of the soft breaking terms This is due to the fact that the analysis of the soft terms relies heavily on the cancellation of the cosmological constant, an issue which can only be properly addressed in the context of SUGRA This single extra requirement will introduce very powerful constraints on the structure of the soft terms, which entirely contradict previous results The structure of the paper is as follows: in Section we first extend the formulation of the F-I term made in Ref wx 6 to SUGRA, as a previous stage to analyze the vacuum structure of a typical model first presented in Ref wx 5 In Section 3 we study the cancellation of the cosmological constant, and we apply the constraints we get from it to the calculation of the soft breaking terms, obtaining a definite hierarchy between the different contributions to the scalar masses and the gaugino mass In Section 4 we illustrate our general results with a particular example of a gaugino condensation mechanism that stabilizes the dilaton, namely one condensate with non-perturbative corrections to the Kahler potential Finally, in Section 5 we present our conclusions () Anomalous U 1, gaugino condensation and the SUGRA potential Before studying the interplay between gaugino condensation and the presence of a Fayet-Iliopoulos D-term in string theories, let us briefly introduce the general SUGRA formulation of such a F-I term As has been stressed by Arkani-Hamed et al wx 6 the dilaton field, S plays a crucial role in this task Under an anomalous UŽ 1 X transformation with gauge parameter a, S transforms as S SqiadGSr, where the Green-Schwarz coefficient, d, is proportional to the apparent UŽ 1 anomaly GS X 1 dgs s Ý q, Ž 1 i 19p i Ž Ž < < Ž y1 with q the U 1 charges of the matter fields, f typically d QO 10 In order to be gauge invariant i X i GS

84 84 ( ) T Barreiro et alrphysics Letters B GS X the Kahler potential must be a function of SqSyd X, where X is the vector superfield of UŽ 1 wx 3 Therefore, the UŽ 1 invariant D-part of the SUGRA action reads with H X 4 yk r3 L s d uwy3e x, D ) q X KsK fi e i,f i; SqSydGS X 3 Extracting the piece proportional to D in Eq Ž X, and eliminating DX through its equation of motion we find yk r3 ) X 6 DXsg Xe ykiqf i i qk dgsr, where g X is the gauge coupling, the prime indicates a derivative with ) respect to the dilaton S and K i indicates derivatives of the Kahler potential with respect to f i Consequently, the contribution to the potential is with 1 1 ) D X XŽ i i V s D s g K q f q j, Ž 4 g X K X d GS j sy Ž 5 Ž y K r3 K r3 In writing Eq 4 we have absorbed the e factor into the redefinition of the vierbein e e e, asitis usually done in SUGRA theories to get a standard gravity action Eqs ŽŽ 4, 5 were obtained in Ref wx 6 in the global SUSY picture The expression of j remains as given in Ref wx 6 Let us now turn to the gaugino condensation effects, and how are they affected by the presence of the F-I potential In order to discuss this issue, let us consider in detail the model presented initially by Binetruy and Dudas wx 5 and recently reanalyzed by Arkani-Hamed et al wx 6 The starting point is a scenario with gaugino condensation in the hidden sector of the theory, originated by a SUŽ N strong-like interaction Following wx c 6 we take the number of flavours Nfs 1, which corresponds to chiral superfields Q, Q that transform under SUŽ N =UŽ 1 as Ž N,q and Ž N,q respectively The spectrum in this sector is completed by a SUŽ N c X c c c Ž singlet, w, which has charge y1 under the U 1 X It is also assumed that j has positive sign The superpotential of this model is given by ž / ž / P qqq 3Ncy1 1 t w L Ncy1 Wsm q Ž Nc y1, Ž 6 M t where ts( QQ is the meson superfield and L is the condensation scale, which is related to the dilaton by 3 Ncy1 L yž qqqsrd se GS Ž 7 ž / M P As for the Kahler potential, K, it was assumed in Refs w5,6x that it consists of a dilaton dependent part plus canonical terms for t and w The scalar potential for such a theory in the framework of SUGRA is given by ½ 5 1 W W W W 1 K r M X X P Vse XX W qk q W qk q W qk y3 q D, Ž 8 w w t t X K M M M M g P P P P X where, as before, a prime indicates a derivative with respect to S, and the subindices indicate derivatives of the superpotential and Kahler potential with respect to the corresponding fields The terms generated by the SUGRA 6 wx The convention for the sign of D is as in Ref 6 X

85 ( ) T Barreiro et alrphysics Letters B corrections are indicated explicitly by the presence of inverse powers of the Planck mass, so that in the limit M ` we recover the global SUSY case studied by the authors of Refs wx 5 and wx P 6 The D-part of the potential reads 1 qqq 1 D s g < t< y< w < qj Ž 9 X X g X This model leads to SUSY breaking by gaugino condensation Žprovided that S is stabilized at a non-trivial value In order to study the phenomenological implications of the presence of the F-I term, we have to compute the soft breaking terms, separating the F-I contribution The first step in this task is to minimize the potential Following Ref wx 5 we define the parameters ž / ž / qqq 3 Ncy1 1 j L N j c Ncy1 ˆ M j ž mˆ / P msm, es, Ž 10 w x ² : with e < 1 We have checked that the expansion in e presented by Refs 5,6, around w s j and ² t : s0, is still correct in the SUGRA case To be more precise, we will parameterize w sj 1qe Ž qqq qa e q, t sj e1qb eq Ž 11 ² : ² : w x It is straightforward to check that the minimization of V with respect to the w, t fields at lowest order in e imposes the form of the lowest order terms in Eq Ž 11 In order to evaluate the next to leading order coefficients in this expansion, a and b, we have to solve the minimization conditions to the next order in e For this matter, and future convenience, the following Ž lowest order in e expressions are useful Wsmj ˆ e N, Ž 1 c Ž DX sg je ay qqq b, 13 c E W qqq N 3r smje ˆ ' qb, Ž 14 E t N y1 c E W smje ˆ Ž qqq, Ž 15 Ew E W Ž qqq symj ˆ e Ž 16 ES d GS Now, from the minimization condition, EtVs0, we get X X XX X XX Ž Nc y1 qqq KŽ K ydgs K K Ž qqqq dgs K bsb0 q XX q K ž N, Ž 17 where b 0 / c is the result obtained in Ref wx 6 in the global SUSY case ž / b s Ž 18 XX X Ncy1 qqq 1y Nc dgs K y K 0 XX K Nc dgs From the second minimization condition, E Vs0, we get w ž / / K X e mˆ Ž qqq qqq K as Ž qqq by 1y 1y qd, Ž 19 X XX ž d K N d K SUGRA g GS c GS

86 86 ( ) T Barreiro et alrphysics Letters B where ž / K X N d K X c GS DSUGRAsK dgsy XX y XX yž qqq K q Ž Nc qqqq dgs K K Ž qqq 4K X X XX N 3d K X c GS y, Ž 0 qqq to be compared to the global case ž // X 1 mˆ Ž qqq qqq K a0s Ž qqq b0y X 1y 1y XX Ž 1 ž d K N d K g GS c GS From the previous expressions we can write the final form of the D-term X qqq K K DX se mˆ e qqq 1y 1y XX qd SUGRA, ž ž / / N c d GS K where D was given in Eq Ž 0 SUGRA Finally, the third minimization condition, ESV s 0 translates into an equation for the value of the third derivative of the Kahler potential, ž / ž / Ž X ž X K / XXX XX XX X X XX GS c GS c GS X sy X 1y X q X X c GS Ž Nc dgs K yž qqq K qqq K d K N d K N d K qqqq K K N d K K K X X XX 3 GS c GS c GS d N d K N d K q6 qqq y4 qqq K y Ž 3 Ž Nc dgs K yž qqq XXX y For phenomenological consistency we will assume that K satisfies 3 at ReS, g GUT, Notice that, since d is small, one expects GS K XXX 4K XX, Ž 4 wx X XX as was already pointed out in Ref 6 In that reference it was also claimed that one expects K 4K, based on a particular ansatz for K This was crucial to get the result that the F-term contribution Ž in particular the F one S to the soft terms dominates over the D-term one However, as we shall shortly see, this is not a consequence of the minimization and, in fact, general arguments indicate that the most likely case is precisely the opposite 3 Cancellation of the cosmological constant and size of the soft terms In order to get quantitative results for the soft terms we need, beside the above minimization conditions, an additional condition, which is provided by the requirement of a vanishing cosmological constant Notice from Eqs Ž 1 Ž 14 that, for this matter, the D-part of the potential and the W qk W term in Eq Ž t t 8, being of order OŽe 4 and OŽe 3 respectively, are irrelevant Consequently, at order OŽe the cancellation of the cosmological constant reads ½ 5 1 K X X 3 XX w w V'e W qk W q W qk W y3 W qož e s0 Ž 5 K From this equation it can be already noticed at first sight that K X cannot be much larger than K XX, otherwise the first term above cannot be cancelled by the y3< W < term Actually, from Ž 5 it is possible to get non-trivial

87 ( ) T Barreiro et alrphysics Letters B X XX bounds on the relative Ž and absolute values of K, K and thus on the relative size of the various contributions to the soft terms It is important to keep in mind that j syk X d r and that both yk X, K XX GS have positive sign Ž the former by assumption, the latter from positivity of the kinetic energy Eq Ž 5 translates into where qqq f j qj s3j, Ž 6 ž N c / 4j f Ž j '1q XX Ž 7 K d GS K < < w w w K XX y1 X X S The 1 in the right hand side corresponds to the F se W qk W contribution while the other term comes from the < F < s e Ž K W q KW contribution Note that fž j G 1, reflecting the fact that both contributions are positive definite We can treat Eq 6 as a quadratic equation in j which has fž j -depen- dent solutions given by ) 3 4f Ž j qqq j s 1" 1y Ž 8 3 N ž / 4 f Ž j c The existence of solutions requires a positive square root Hence, fž j is constrained to be within the range 3N c 1Ff Ž j F, Ž 9 4Ž qqq which in particular implies that qqq 3 F Ž 30 N c 4 < < OŽ 1 TeV to guarantee reasonable soft terms Since the perturbative and non-perturbative contributions to W Žsee Eq Ž 6 are of the same size at the minimum, we may apply this condition to the non-perturbative piece, ie Actually, q q q rnc is also bounded due to phenomenological reasons Namely m 3r ; W rmp should be ž / 1r N y1 c < y Ž qqq W < np s Ž Ncy1 e Srd GS ;OŽ 1 TeV, Ž 31 t Ž yrž N c y1 where we have used Eq 7 and everything is expressed in Planck units Using t R OŽ 1 we get Ž qqq ReSrd Ž N y1 R34 Since ReS,, we finally get GS c qqq ;9d Ž 3 N c GS Now, a non-trivial result about the relative sizes of K X and K XX can be derived from the right hand side of the Eq Ž 9 By writing fžj explicitly in terms of these derivatives of the Kahler potential and using Eq Ž 3 we get X 18 K qqq 3 3 XX F 4y F 4, Ž 33 K N c

88 88 ( ) T Barreiro et alrphysics Letters B which results in the general bound X K 1 XX F 4 34 K XX This bound means in particular that the K < < K X < assumption made in Ref wx 6 is clearly inconsistent with the cancellation of the cosmological constant As mentioned at the end of Section, this assumption was crucial for the results obtained in that paper concerning the relative size of the F and D contributions to the soft terms This suggests that those results must be revised, as we are about to do X XX Besides Eq Ž 34, Eq Ž 6 provides interesting separate constraints on K, K Namely, from Eq Ž 6 we can write the inequality qqq qj F3j, ž N c / which implies ( ( ž / ž / qqq qqq 1y 1y Fj F 1q 1y Ž 35 N N c X X This translates into an allowed range for K, since j syk d r On the other hand, Eq Ž 6 GS also implies the inequality 4j qqq XX qj F3j K d ž N / GS c and, thus / qqq XX 4 K G 3 ;OŽ 100 Ž 36 ž N c d GS Let us notice that the bounds Eq Ž 35 and Eq Ž 36 are a consequence of imposing that neither the < F < w nor the < F < contributions Ž both positive may be larger then the Ž negative sign contribution 3< W < in Eq Ž 5 S It is also interesting to discuss in which cases the < F < contribution dominates over the < F < S w one or vice-versa The < F < contribution is maximized when fž j S is as large as possible, ie when the upper bound in Eq 9 gets saturated Then j sž q q q rn and hence Žsee Eqs Ž 5, Ž 3 c < K X < ;18 Ž 37 If < K X < is smaller Ž larger than 18, < F < becomes dominant and j tends to left Ž right hand limit of Ž 35 w Condition Ž 37 does not guarantee that the < F < contribution dominates over the < F < S w one This would require X XX fž j 41, ie yk rk 4d r and, from Eq Ž 9, Ž qqq GS rn c <3r4 The latter condition clearly shows that < F < S dominance can only happen in a very restricted region of parameter space, and therefore is unlikely to appear in explicit constructions Let us turn now to the important issue of the soft terms, and how are they constrained by the previous bounds The soft mass of any matter field, f, is given by m fsm Fqm D, Ž 38 where m se K < W < and m syq ² D: Ž 39 F D f c

89 ( ) T Barreiro et alrphysics Letters B are the respective contributions from the F and D terms to m f Here qf is the f anomalous charge and we have assumed a canonical kinetic term for f in the Kahler potential Ž as for t and w From Eq K ² D : so 1 =e mˆ e qqq 40 X So, using Eq 1 we get ž / ž / / 4 md qqq MP 18 f ž X mf Nc j K sož 1 =q,ož 1 = Ž 41 This means that for < K X < -18 the D contribution to the soft masses will be the dominant one Žcontrary to what was claimed in Ref wx 6 The < K X < )18 case cannot be excluded, but we could not implement it with the explicit example we use in the next section On the other hand, notice that, since j A< K X < Žsee Eq Ž 5, the F-I scale X becomes in this case comparable to M Ž or larger P So it is not surprising that for large K the role of gravity as the messenger of the Ž F-type SUSY breaking is not overridden by the gauge mediated Ž D-type SUSY breaking associated to the F-I term Concerning gaugino masses, these are given by FS 1 XX y K X X < < l 16 m s, Ž K e W qk W Ž 4 Ž SqS Ž XX y1 < X X < < < Ž Using K W qk W F3 W to allow Vs0 in Eq 5 we get m Q Ž K e W s Ž K m Ž 43 3 XX y1 K 3 XX y1 < < l F XX Since K GOŽ 100, the gaugino masses are strongly suppressed with respect to the scalar masses, which poses a problem of naturality Namely, since gaugino masses must be compatible with their experimental limits, the scalar masses must be much higher than 1 TeV, leading to unnatural electroweak breaking This conclusion seems inescapable in this context Summarizing, the hierarchy of masses we expect is m l<m F-m D, Ž 44 although the last inequality might be reversed in special cases 4 Explicit examples In this section we want to illustrate the points we have just been discussing with a particular example For that purpose we shall take a model of gaugino condensation in which the dilaton is stabilized by non perturbative corrections to the Kahler potential w16 19 x; in particular the ansatz we shall use is KsylogŽ ReS qk np, Ž 45 where w19x D yb Ž ' ReS y' S K s log 1qe 0 np Ž 46 B ' ReS This function depends on three parameters, S 0, D, and B, the first of which just determines the value of ReS at the minimum Since ReS,, we shall fix S0 s from now on Therefore this description is effectively made in terms of only D and B, which are positive numbers X XX XXX The ansatz Eq Ž 46 can naturally implement the hierarchy K <K <K, which, as we have seen in previous sections Žcf Eqs Ž 4, Ž 34, is required to stabilize the dilaton at zero cosmological constant Actually, there is a curve of values of D versus B for which this happens This is shown in Fig 1 for a few

90 90 ( ) T Barreiro et alrphysics Letters B Fig 1 Values of the B and D parameters Ž in logarithmic scales that give a zero cosmological constant with the ansatz of Eq Ž 46 The Ž y10 hidden gauge group is SU 5, and ms10 MP for all the curves The solid line corresponds to qq qs1 and dgss45r19p, the dashed line to qq qs5 and dgs s10r19p and the dash-dotted line to qq qs35 and dgss150r19p The thin lines show the corresponding lower bound on B obtained from Eq Ž 51 different values of Ž q q q rn Ž d being determined by imposing a correct phenomenology c GS ; all of them below the upper bound of 3r4 found in the previous section, see Eq Ž 30 The sizes of K X and K XX at the minimum are < K X < ;OŽ 10-18, < K XX < ;OŽ 100 < K X < Ž 47 The second equation satisfies the bound Eq Ž 34, while the first one has implications on the size of the soft terms as we shall shortly see Concerning the size of the different terms in the potential, it is remarkable how, for small Ž big values of the DB parameter it is the Fw term the one that dominates over FS in the potential and therefore cancels the 3< W < Given that K X -18 that means that in this region of the parameter space Žsee the discussion after Eq Ž 37 ( 4 3 ž / qqq 3 4 j s 1y 1y Ž 48 N c As B decreases Ž ie D increases the size of both F terms becomes comparable and, eventually, the value of j tends to a constant value given by Ž qqq rn This can be understood from Eq Ž 46 c in the asymptotic regime of small B Ž see Fig 1 Defining 'expwybž ReS y S x we have in this regime ( ( min 0 D yd BD X XX K < ;, K < ;, K < np min np min np min ;, Ž 49 3r B( ReSmin ReSmin Ž ReS where the subscript min denotes values at the minimum We can now evaluate fž j, which is given by X K 8ReS ( min f Ž j s1y XX ;1q Ž 50 d K d B GS GS min

91 ( ) T Barreiro et alrphysics Letters B Fig Plot of qf mfrmd versus D, with B fixed to the values given by the previous figure Same cases as before The thin lines show the corresponding lower bound on q m rm derived from Eq Ž 30 f F D Imposing the limit of Eq 30, f j Fig 1 for each example, Ž w x c ( ( -3N r 4 qqq, we get a lower bound on B, which is the one shown in 8ReSmin 96 ReSmin BG ; Ž 51 3Nc 1y1dGS d y1 4Ž qqq GSž / It can also be seen in Fig 1 that B approaches asymptotically this bound, and therefore fžj will tend to a constant value in this limit Using Eq 8 we finally obtain that j goes to an asymptotic value given by X Ž qqq rn Using Eq Ž 41 this means that the ratio q m rm tends to 1 In fact, since < K < c f F D -18 in these particular models, this same equation tells us that qf mfrmd approaches 1 from below In other words, we are in a scenario where the scalar soft masses are dominated by the D-term Žsee the discussion at the end of Section 3 All this is illustrated in Fig for the same three cases presented in Fig 1 The left hand limit of each curve is determined by the corresponding value of j given by Eq Ž 48, that is corresponds to V0 s0 driven by F w, whereas as we move in the parameter space towards larger Ž smaller values of DB the FS term contributes more to the cancellation of the cosmological constant until we reach the constant value defined by j sžq q q rn c Moreover the ratio qf mfrmd tends to its maximum value Therefore this is an ansatz for K np for which we get the different possibilities for cancelling the cosmological constant, generically with the soft scalar masses dominated by their D contribution 5 Conclusions Four-dimensional superstring constructions frequently present an anomalous UŽ 1 X, which induces a Fayet-Iliopoulos Ž F-I D-term On the other hand SUSY breaking through gaugino condensation is a desirable feature of string scenarios The interplay between these two facts is not trivial and has important phenomenologw5 7 x, based on a global SUSY picture, led to contradictory results regarding the relative contribution of the F and D ical consequences, especially concerning the size and type of the soft breaking terms Previous analyses terms to the soft terms In this paper we have examined this issue, generalizing the work of Ref wx 6 to SUGRA, which is the appropriate framework in which to deal with effective theories coming from strings We have extended the

92 9 ( ) T Barreiro et alrphysics Letters B formulation of the F-I term made in Ref wx 6 to the SUGRA case, as well as considered the complete SUGRA potential in the analysis This allows to properly implement the cancellation of the cosmological constant Ž something impossible in global SUSY, which is crucial for a correct treatment of the soft breaking terms Moreover this condition yields powerful constraints on the Kahler potential, which leads to definite predictions on the relative size of the contributions to the soft terms In particular we obtain the following hierarchy of masses m l<m F-m D, Ž 5 where m F, m D are the contributions from the F and D terms, respectively, to the soft scalar masses, and m l are X the gaugino masses These results amend those obtained in Ref wx 6 The last inequality could be reversed if K, ie the derivative of the Kahler potential with respect to the dilaton at the minimum, is large Ž R0 This is not surprising, since in this way the F-I scale Žwhich is proportional to K X can be made comparable to M P, so that the role of gravity as the messenger of the Ž F-type SUSY breaking is not overridden by the gauge mediated Ž D-type SUSY breaking associated to the F-I term This situation cannot be excluded, but in our opinion it is unlikely to happen The first inequality in Ž 5 is rather worrying since it poses a problem of naturality Namely, given that gaugino masses must be compatible with their experimental limits, the scalar masses must be much higher than 1 TeV, leading to unnatural electroweak breaking This conclusion seems inescapable in this context Finally, we have illustrated all our results with explicit examples, in which the dilaton is stabilized by a gaugino condensate and non-perturbative corrections to the Kahler potential, keeping a vanishing cosmological constant There it is shown how the various contributions to the soft terms are in agreement to the hierarchy expressed in Eq Ž 5 Acknowledgements JAC and JMM thank Michael Dine and Steve Martin for useful discussions held at the University of Santa Cruz, and the NATO project CRG that made it possible The work of TB was supported by JNICT Ž Portugal, BdC was supported by PPARC and JAC and JMM were supported by CICYT of Spain Žcontract AEN Finally, the authors would like to thank the British CouncilrAcciones Integradas program for the financial support received through the grant HB References wx 1 JP Derendinger, LE Ibanez, HP Nilles, Phys Lett B 155 Ž ; M Dine, R Rohm, N Seiberg, E Witten, Phys Lett B 156 Ž wx M Green, J Schwarz, Phys Lett B 149 Ž wx 3 M Dine, N Seiberg, E Witten, Nucl Phys B 89 Ž wx 4 JA Casas, EK Katehou, C Munoz, Nucl Phys B 317 Ž wx 5 P Binetruy, E Dudas, Phys Lett B 389 Ž wx 6 N Arkani-Hamed, M Dine, S Martin, Phys Lett B 431 Ž wx 7 G Dvali, A Pomarol, Nucl Phys B 5 Ž wx 8 LE Ibanez, GG Ross, Phys Lett B 33 Ž wx 9 P Binetruy, P Ramond, Phys Lett B 350 Ž ; P Binetruy, S Lavignac, P Ramond, Nucl Phys B 477 Ž ; P Binetruy, N Irges, S Lavignac, P Ramond, Phys Lett B 403 Ž ; JK Elwood, N Irges, P Ramond, Phys Lett B 413 Ž ; N Irges, S Lavignac, P Ramond, Phys Rev D 58 Ž ; Z Lalak, Nucl Phys B 51 Ž ; SF King, A Riotto, hep-phr w10x V Jain, R Shrock, Phys Lett B 35 Ž w11x E Dudas, S Pokorski, CA Savoy, Phys Lett B 356 Ž ; E Dudas, C Grojean, S Pokorski, CA Savoy, Nucl Phys B 481 Ž

93 ( ) T Barreiro et alrphysics Letters B w1x EJ Chun, A Lukas, Phys Lett B 387 Ž ; K Choi, EJ Chun, H Kim, Phys Lett B 394 Ž w13x RN Mohapatra, A Riotto, Phys Rev D 55 Ž ; 46 w14x AE Nelson, D Wright, Phys Rev D 56 Ž w15x AE Faraggi, JC Pati, hep-phr w16x SH Shenker, in: Proceedings of the Cargese School on Random Surfaces, Quantum Gravity and Strings, Cargese, France, 1990 w17x T Banks, M Dine, Phys Rev D 50 Ž w18x JA Casas, Phys Lett B 384 Ž ; P Binetruy, MK Gaillard, Y-Y Wu, Nucl Phys B 481 Ž w19x T Barreiro, B de Carlos, EJ Copeland, Phys Rev D 57 Ž

94 31 December 1998 Physics Letters B Statistical approach for quantum gravity fluctuations in QFT JL Acebal a,b,1, AS Chaves a,c, JM Figueiredo a, AL Mota a,d, MC Nemes a a Departamento de Fısica, ICEx, UniÕersidade Federal de Minas Gerais UFMG, CP 70, Belo Horizonte, MG, Brazil b Departamento de Matematica e Estatıstica, Pontifıcia UniÕersidade Catolica de Minas Gerais PUC, Belo Horizonte, MG, Brazil c Instituto de Fısica, UniÕersidade de Brasılia UNB, Brasılia, DF, Brazil d Departamento de Ciencias ˆ Naturais Fundaçao de Ensino Superior de Sao Joao del Rei FUNREI, Sao Joao del Rei, MG, Brazil Received 9 July 1998; revised 3 November 1998 Editor: M Cvetič Abstract The consequences of gravitational zero point quantum fluctuations at Planck scale are considered in the QFTs by adopting an effective stochastic model for space-time A kinematical approach is used instead of the usual dynamical attempts The physical quantities are associated to an average over such fluctuations In particular, the fields have their spectral representation modified The class of representative field is changed for the description of a field theory and the all order correlation functions, including the two-point functions, are cosequently changed The investigation of the effect upon QFT succeeded in generating a consistent and more regular theory In such a scheme the ultraviolet divergences are controlled Although general, the derivation is performed for scalar Fields The model is shown to be independent of the specific form of a quantum gravitation theory q 1998 Elsevier Science BV All rights reserved PACS: 1110-z; 350-z; 0460-m Although a sound theory of quantum gravitation is still lacking, it seems natural, by correspondence to other fields, to associate fluctuations to the quantum gravitational vacuum state In fact, remarkable predictions have been put forth in such direction Among those is Wheeler wx 1 who suggested that at 1r "G y35 Planck s scale, L s f 10 m, the characp ž / c 3 teristic scale for such fluctuations, it would give rise to disconnected geometries, loss of causality and determinism Hawking wx also conjectured about the loss of the locality and conservation laws in the same 1 acebal@fisicaufmgbr context Both suggested the existence of a non deterministic scale limit Ž NDS It is clear that such effect should not affect usual observable symmetries of larger scales Although it is not presently possible to devise experimental procedures to directly investigate processes in scales smaller than about 10 y17 m, there have been several theoretical attempts to model nature there Most of them consist in modelling quantum gravity by including Lp in the dynamics of the theories in different ways A recent proposal wx 3 assumes that the action in the path integral is invariant under the duality transformation over the relativistic infinitesimal length ds L prds Other more sophisticated approaches based on string theories or r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

95 ( ) JL Acebal et alrphysics Letters B Ashtekan variables wx 4 also exist Padmanabham wx 5, deals with quantum conformal fluctuations that make length scales less than Planck scale operationally ill-defined and generating a cut-off in flat space field theories Important issues have been addressed by Calzetta and Hu wx 6 and Campos and Verdaguer wx 7, who studied the manifestation of quantum field fluctuations as noise and their effect on the dynamics of background space time Also light cone fluctuations and causal structure of QFT have been considered in Refs w8,9x Along these lines, Chaves et al w10x have shown that a simple effective way to mathematically implement metric fluctuations at Planck scale is to assume that zero point quantum fluctuations generate a stochastic space metric This model succeeds in explaining the contrasts between the microscopic deterministic laws and the macroscopic behavior In the present work, we introduce the quantum gravity fluctuations by a stochastic model for the space-time length This results in a kinematical approach, that is, the fluctuations are shown to fundamentally redefine the spectral representation of physical quantities The results are independent of any eventual quantum version of gravity We show that, Ž a this model can reproduce all the standing conjectures about the zero point quantum fluctuations at Planck scale The breakdown of locality, causality and determinism in that ultramicroscopic scale is obtained in such a way that it still allows for the restoring of such symmetries in larger scales Ž b The effect of the fluctuations at Planck scale introduces an average minimal intrinsic uncertainty on the spectral field representation It causes consistent change on the class of fields available for the descriptions on QFT There are preliminary results indicating the preservation of internal symmetries Ž c As a consequence of the change on fields, the all order correlation functions of QFT, including the two-points function, are consistently changed This contrasts to other attempts to investigate the effects of the fluctuations A functional generator can be defined from which the loop terms are already finite We explicitly show the effect for a scalar field theory The approach can be easily generalized for other fields Next, we introduce the basic assumptions Ž i On the basis of the determinism lies the paradigma that a mechanism of infinite precision governs the bew10 x havior of physical systems at every scale In contrast to that, the idea has been presented that the topological irregularities at Planck scale, due its unpredictability, invalidates the hypothesis of infinitely precise localization of a space-time event Irregularities of geometry are supposed to be caused by the zero point gravity quantum fluctuations over the candidate to the vacuum ie the pseudo-euclidean space-time The localization is then limited by a structural uncertainty given by Ž Dx sa L p, Ž 1 where a is a real dimensionless constant of order unity which represents the characteristic extension of such NDS s irregularities in terms of L P Ž ii The fluctuations in smaller scales turn the concept of tangent space somewhat ill defined Irregularities become increasingly important as the scale gets smaller Therefore, the usual approach for curved manifolds becomes inappropriated We obtain the necessary correspondence between the quantities in the stochastic microscopic curved manifold to the 4-dimensional space-time by effecting an average integration over the fluctuations The fluctuations at Planck scale can be naturally introduced by assuming the general Poincare group transformation to have its parameters locally stochastic Let A be a small region in space-time with dimensions comparable to the characteristic size of the fluctuations, we can write the infinitesimal general form of the Poincare group r 1 rs QAs1qj AEr q va xresyx se r The generalized arch length z s sv rs x, should not A A r depend on the choice of the origin of coordinates and as j r, must be stochastic variables with magnitude A in the order of L Hence by using e r sz r qj r, p A A A we can write Q s1qe r E Ž 3 A A r Taking the continuous limit and the exponential form we obtain the following transformation Q x se e r Ž xe r, 4 Že where the notation QŽe stands for the dependence on the parameters e r Ž x Ž iii We assume also that the stochastic paramer ters e Ž x have their statistic behavior strongly re-

96 96 ( ) JL Acebal et alrphysics Letters B lated to the form of fluctuations However, a satisfactory microscopic theory of such fluctuations is still lacking On the other hand, we do not expect to have collective effects of such irregularities being observed in deterministic scales Ž DS It means that the statistical behavior of the fluctuations are determined locally For such purpose the gaussian statistical model is suitable ² e Ž x : s0, Ž 5a m e ² s m : < < sm e Ž x,e Ž y shž xyy g Ž 5b e The precise form of the scalar function h should arise from some quantum theory of gravitation We only assumed that h is a short range correlation function having significant values at scales Ž al p comparable to the Planck scale Notice that the forms Ž 5a and Ž 5b are necessary in order that the statistical behavior to be an invariant and to preserve the covariance of the theories A similar assumption has been used in Refs w6,7x in their semiclassical approach to quantum gravity Because of the smallness of the Planck scale compared to the smallest observable scale, the physical quantities are related to an average upon the fluctuations, as discussed in Ref w10 x The linear infinitesimal form for the General Coordinate Transformation Ž GCT correspondent to Ž 4, m m x s x q e m Ž x Že, can be used to calculate the proper length from the origin to x which becomes a statistical object, S Ž x sx m x s x m x q x m e Ž x ye Ž 0 Že Že Že m m m m qe m Ž x e Ž x qe m Ž 0 e Ž 0 m ye m Ž x emž 0 Ž 6 The first term corresponds to the pseudo-euclidian metric The presence of the stochastic terms has some important consequences The condition S Že s0 can still hold, but it does not necessarily lie on the usual light cone The light cone becomes then blurred at Planck scale This effect has also been pointed out recently by Ford wx 8 A similar assumption was made by Feynman w11,1x in order to find a relativistic cutoff for the QED Notice that the model can reproduce Wheeler s and Hawking s conjectures that m causality, determinism and conservation laws are lost In particular, we could have causal relations in space-like intervals or in reverse time-like Thus, locality is lost in the hyper cube of Planck scale dimension The result of a measurement of the proper length S for x large compared to Planck scale is Že & m Že e S Ž x s ² S : Ž x ss q ² e Ž x,e Ž x : m e Ž 7 Here the invariance ² e m Ž x e Ž x: s² e m Ž 0 e Ž: m e m 0 e has been used The tilde indicates an averaged quantity The uncertainty in position is mathematically expressed by ² m : ² m Ž Dx s x, x ey x :² e x : Že Že m Že Že m e s ² e m Ž x,e Ž x : Ž 8 m However, from the assumption Ž i, Eq Ž 1, this uncertainty in the measurement of the position is of order L Then, by using Ž 5b we have p a L S p Ž 0 ² m e Ž x,e x : m es s Ž s4hž 0 Ž 9 Thus, the self-correlation is associated to a proper length of order of the Planck scale In the following we study the effects upon fields and propagators Let us next consider the action of the transformation Ž 4 over a free field Qc e Ž x scže Ž x e d 4 k m m ˆ yik x m yik emž x sh c Ž k e e Ž 10 4 Ž p The measurable object is, by hypothesis, the statistical average of this quantity So assuming the commutation of the average and the integration, we obtain cs ² c x : Že e d 4 k m m ˆ yik x m ² yik emž x sh c Ž k e e : Ž 11 4 e Ž p The average under integration can be evaluated by ² " i x : y² x :r using the identity e s e, valid for x

97 ( ) JL Acebal et alrphysics Letters B being a stochastic gaussian variable Using the relaa tion Ž 9 and taking for simplicity s 1 we obtain d 4 k m yl p k yik x c x s cˆ k e e m H Ž 1 4 Ž p This kinematical result exhibits the change on definition of the spectral representation of the field The above field is related to the conventional one by a convolution in the configuration space-time of the ordinary wave packet to a narrow gaussian distribution function H 4 Ž4 L p c Ž x s d yd Ž xy y c Ž y, Ž 13 Ž4 where D Ž xy y L p is the Fourier transformation of e yl p k, d 4 k Ž4 ylp k yikž xyy DL Ž xyy sh e e Ž 14 4 p Ž p Hence, the effect of including the fluctuations and taking the average over the fields establishes a lower limit uncertainty for the localization of the wave packet The width D x0 of a minimal uncertainty packet is intrinsically changed to 1r Ž 0 p Dxs Dx ql 15 In this sense, the only field coordinates available in a such space are quantities related to Ž 1 or Ž 13 The differential transformation associated to these equations is yl p E m E c Ž x se m c Ž x sv Ž E c Ž 16 Ž4 Not only D Ž xyy d 4 Ž xyy L when Lp 0, p but the full theory goes to the usual theory in such limit As we can see from Ž 1, the average over the quantities under the fluctuations effect causes a dispersion associated to the vacuum that suppresses the weight of the high momentum components in the spectral field representation Eq Ž 16 introduces a non locality in a very small characteristic scale, L p, which very much resembles the Quantum Mechanics non locality shown explicitly in the Wigner representation of the one particle quantum dynamics, " E h E Eh E ir Ž R, p s sin y r Ž R, p, " ž E R E p E p E R / Ž 17 4 where rsrž R, p and hshž R, p are the Wigner representation of the density and hamiltonian operators Notice that in the above the dynamics depends on derivatives of all orders with respect to R and the case " 0 gives the classical limit The same occurs m in 16 for x and Lp 0 A propagator for a scalar field theory, consistent with the above, can be obtained by a heuristic calculation The -point quantum correlation function under fluctuations has the form t Ž x, y s² 0< T f Ž x f Ž y < 0 : Ž 18 Že Such quantity has some complicated meaning considering the disorder in small scales Hence we do not know what to expect from the time ordering However, as discussed above, the causality itself is also not well defined in the smaller fluctuating scales The symmetries we observe in nature are proper of DS The process we measure are related to DS where the statistics of the fluctuations are, of course, uncorrelated Hence, this idea can be used to calculate the average quantity ² t : Ž x, y s²² 0< T f Ž x f Ž y < 0:: Že e Že Že s² 0< T ² f :² : Že Ž x e fže Ž y e < 0 : Ž 19 By using 13 it can be written in terms of the ordinary quantum correlation function 4 4 Ž4 Ž4 H H Lp Lp t Ž x, y s d w d zd Ž xyw D Ž yyz =² 0< Tf Ž x f Ž y x < 0 :, Ž 0 where double tilde in t means that the quantity is related to two subsequent operations in the form of Ž 13 The last factor under the integration can be evaluated by the usual procedure to obtain the expression in terms of Feynman s propagator Thus, after identifying Ž t x, y s² 0<ŽTwf Ž x f Ž yx< 0: s id and using Ž 14, we get F d 4 k e y L p k yikžxyy D F Ž xyy syh e Ž 1 4 Ž p k ym e e

98 98 ( ) JL Acebal et alrphysics Letters B It must be remarked that in a field theory where the observable fields are in the form of Eq Ž 1, the equation above shows the natural candidate to the propagator The unitarity of this form is preserved once the gaussian factor under integrations just modifies the Feynman s propagator by a real positive function of k Next, we study the consequences of the above assumptions upon QFT As discussed, the symmetries and observable quantities are related to averages upon the fluctuations at Planck scale In this sense we are forced to deal with fields in the form of Ž 1 This correspond to changing the class of fields used for the description of the theory The correspondence between the classes of fields is given by Ž 1, Ž 13 or Ž 16 Hence the physically relevant quantities will be of the form ² 0< f Ž x f Ž x f Ž x < 0 : op 1 op op n Consider the classical action for a free scalar field given by 1 4 S s d xf Iqm f 0 H H 1 4 y s d xf V Iqm f 3 Feynman s formula should include the modified fields in the following way i 4 y Z sn Dfexp y d xf V Iqm f F H H Ž 4 Other terms like the interaction with an external 4 source S shd xf J J can be turned on, provided the class of fields is preserved By using it, the functional generator can be written as w x H 4 Z0 J sn Dfexp yi d x H ½ 5 f V Ž Iqm fqf J Ž 5 = 1 y The above functional generator furnishes the correlations to all orders in the form consistent with the class of fields Ž 1, through functional derivatives with respect to J So, according to the present model, the usual formal theoretical tools continue to be valid In this sense we can evaluate Z 0wJx i 4 4 sexp y Hd xhd yjž x D F Ž xyy JŽ y Ž 6 Other interaction terms can be turned on in the form 4 S s Hd x L Ž f I int and the corresponding functional for the theory is given by 1 d 4 exp Hd z L int Z wj ž / 0 x i d JŽ z Z wjxs, Ž 7 wnumerator with J 0x which satisfies the differential functional equation 1 d intž 1 d y X w x / V Ž Iqm Z J ql Z wjx i d J i d J sjž x Z wj x Ž 8 4 lf For the scalar L s theory the -point funcint 4! tion becomes l 4 t Ž x, y sid xyy y D 0 d zd F F H FŽ zyx = D F Ž yyz qož l To first order in l, the correction for the mass is d ms ildfž 0, Ž 9 which is not quadratically divergent Notice that topology of the Feynman s graphics is preserved and as a rule of thumb, we can say that the usual propagator should be substituted by the dressed 1 propagator D F Although, the effect of the zero point quantum fluctuations of space-time at Planck scale does introduce a disorder at Planck scale, the usual symmetries in action are preserved after averaged In fact, in taking into account the fluctuations and its associated degree of non locality, we get, by the above approach, a more controlled, regular and consistent QFT Acknowledgements This research was partially supported by the Conselho Nacional de Desenvolvimento Cientıfico e Tecnologico, CNPq-Brazil

99 ( ) JL Acebal et alrphysics Letters B References wx 1 JA Wheeler, Geometrodynamics, Academic Press, New YorkrLondon, 196 wx SW Hawking, Nucl Phys B 44 Ž wx 3 T Padmanabham, Phys Rev Lett 78 Ž wx 4 A Ashtekar, C Rovelli, L Smolin, Phys Rev Lett 69 Ž wx 5 T Padmanabham, Ann Phys Ž NY 165 Ž wx 6 E Calzetta, BL Hu, Physical Review D 49 Ž wx 7 A Campos, E Verdaguer, Physical Review D 53 Ž wx 8 LH Ford, Physical Review D 51 Ž wx 9 LH Ford, NF Svaiter, Physical Review D 54 Ž w10x AS Chaves, JM Figueiredo, MC Nemes, Ann Phys 31 Ž w11x RP Feynman, Physical Review 74 Ž w1x RP Feynman, Physical Review 74 Ž

100 31 December 1998 Physics Letters B a-representation for QCD Richard Hong Tuan Laboratoire de Physique Theorique et Hautes Energies 1, UniÕersite de Paris XI, Batiment ˆ 10, F Orsay Cedex, France Received 0 October 1998 Editor: R Gatto Abstract An a-parameter representation is derived for gauge field theories It involves, relative to a scalar field theory, only constants and derivatives with respect to the a-parameters Simple rules are given to obtain the a-representation for a Feynman graph with an arbitrary number of loops in gauge theories in the Feynman gauge q 1998 Elsevier Science BV All rights reserved An a-parameter representation may be useful for a variety of purposes First, it is a compact representation for Feynman amplitudes which needs only one integral Ž over the a-parameter per propagator This simplicity enables one to study the renormalization w1,x by introducing a subtraction operator acting directly on the wx 3 a-integrand It also was essential in deriving 3 string motivated one-loop rules for connecting f amplitudes to QCD amplitudes Starting from a f 3 a-parameter representation one gets a QCD a-parameter representation by multiplying the f 3 expression by a reduced factor which is calculated by methods of string-theory However, going to more than one-loop level appears difficult Here, we propose an a-representation for gauge theories which can be easily realized by simple rules involving derivatives with respect to the a-parameters and for any number of loops Such a representation may also be useful to compute Regge trajectories for QCD as we wx 3 have done 4 for f It looks very promising for the calculation of the Pomeron trajectory because no approximation like a leading-log one has to be made as it is in evaluations w5,6x done so far This is because an infinite number of loops leads to a fixing mechanism for the a-parameters which therefore do not need to be integrated over Feynman amplitudes can then be calculated without any integration being performed wx 4 We must note that an a-representation applicable to derivative couplings has already been given long ago by Lam and Lebrun wx 7 However, it is given under a complicated form, where derivatives are not taken with respect to a-parameters but to currents which are complicated functions of the incidence matrix and external momenta Consequently, its use is very difficult, whereas our rules are very simple and involve only derivatives with respect to a-parameters 1 Laboratoire associe au Centre National de la Recherche Scientifique URA D r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

101 ( ) R Hong TuanrPhysics Letters B Let us start with the amplitude for a Feynman graph G of a scalar field theory with I internal lines and V vertices wx 8 We define the incidence matrix with indices running over vertices Õ and propagators l as Õ l 1 if Õ is the starting point of l s ~ Õ l y1if Õ is the endpoint of l Ž 1 0if l is not incident with l Let us denote by PÕ sý pk Õ the sum of incoming external momenta at the vertex Õ Then, in a theory without derivative couplings, ie a scalar field theory, we get for the Feynman amplitude F of G Ž G in d s 4 dimension F swcgrs Ž Gx G dk i / I 4 V I l 4 4 H Ł 4 Ł Õ Ý Õ ž l l ls1 p k l ym lqi Õs1 ls1 Ž p d P y k, where C G contains the vertices factors dependence and S G is the symmetry factor The four-momentum k l is oriented as the line l is Now, H ` ir p ym qi s da exp ia p ym qi, Ž 3 0 I Õ Ý Õ l l H Õ Õ Õ Ý Õ ž / l l ls1 l ž / Ž p d P y k s d y exp yiy P P y k Ž 4 Inserting Ž 3 and Ž 4 in Ž and interchanging integrations order Ž regularization is needed for divergent integrals we get for each d 4 k integration l ž / ½ ž Ý / ž / 5 dkr 4 p 4 exp ia k qa y1 H l l l l Ý y Õ P Õ l k l Õ 4 4 y1 y1 H l l l l Õ Õ l Ž l Ý Õ Õ l Õ Õ s dkrž p exp ia k qž a y y 4a y Ž 5 The k l dependence is now concentrated in the first term of the exponential This is crucial for what follows Derivative couplings manifest themselves through k l m factors Uneven powers of k l cancel after making the variable substitution Ý l l Ž y1 l Õ Õ l Õ k k y a y, 6 the exponential is then simplified and any k 4 4 y1 H l Ž l l l l Ý Õ Õ l Õ l factor will transform, giving a k -dependent integral dkrž p exp ia k k y Ž a y Ž 7 However, this is precisely the kind of factor QCD will exhibit If, in Feynman gauge, a a m an n am m a m a) a m 0 Ž m n n m Ž m Ž m Ž m L sy1r4 E A ye A E A ye A y1r E A q E x E x q c ig E ym c Ž 8a L sygr f abc E A a ye A a A bm A cn yg r4 f abc f cde A a A b A cm A dn ygf abc Ž E m x a) x b A c 1 m n n m m n m a m a qg c T g c Am 8b are respectively the free and the interacting part of the QCD Lagrangian, one remembers that Em is contracted bm with A, a field upon which E does not act in Ž 8b If there is another E acting on the same propagator, but at m m l

102 10 ( ) R Hong TuanrPhysics Letters B m the other end, a dot-product EE is made Ž m contracting indices along propagators joining the ends of the first propagator on which the considered E s act and we get, Fourier transforming Ž putting aside coupling constants m yik ik m sk Ž 9 m l l l Making the variable change we therefore get Ž 7 or, eliminating odd powers of k l which give vanishing contributions, ž / 4 4 y H l Ž l l l l Ý Õ Õ l Õ dkrž p exp ia k k q Ž a y Ž 10 Now, the crucial remark is that the bracket in Ž 10 can be obtained by the deriõation with respect to ia l of the integrand of () 5 This is a very simple rule indeed to obtain the kinematical factor associated with a derivative coupling: excluding the factor exp( yi a l m l ), we simply haõe to take the deriõatiõe of the integrand with respect to i a Ž m Ž Õ l Of course, there can be other dot-products arising involving k s like k k m m Õ, Žk k m Žk k n However, because, in d dimensions, 1m n 1 H H d d k k m k n f k sž g mn rd d d k k f k Ž 11 l l l l l l l all dot-products will give factors proportional to k l and the rule given above will hold up to a d-dependent y1 coefficient For instance, for k1p k kp k1 this coefficient will be drd s d in factor of the operator ErEŽ ia ErEŽ ia 1 d If we perform the Hd k l integrals Ž no derivation being done we get, in d dimensions Žin order to allow for dimensional regularization V ` I d yž dy r ydr FG s CŽ G rsž G HŁ d yõh Łda l i Ž 4pal Õs1 0 ls1 V V l l Ý Õ Õ l l Ý Õ Õ ž / ž / Õs1 Õs1 ½ 5 = exp yi a m q y rž 4a exp yi yp P Ž 1 Making the change of variable zisyiyzv is1,,vy1 Ž 13 z sy V V V because Ý s0 Ž l connects two and only two vertices, z only appears in the last exponent of Ž 1 Õs1 Õ l V and the integration over zv yields a factor d Ž ÝÕPÕ implying energy-momentum conservation There remains V y 1 independent variables zõ to be integrated over Separating the zõ dependent factors, the zõ integral reads Vy1 Vy1 Vy1 d y1 H H Ł Õ Ž l 4 Ý l Ý Õ Õ l Ý Õ Õ Õs1 l Õs1 Õs1 I s d z H j exp yi Ž 4a z q z P P Ž 14 ž / ½ 5 where HŽj 4 represents factors coming from the derivative couplings through j s defined as l Vy1 y1 l l Ý Õ Õ l Õs1 j sž a z Ž 15 Ž As previously quoted, to k l is associated k ly j l through the change of variable 6, giving k lq j l, eliminating the term linear in k l The j l term is obtained by letting ErEŽ ia l act upon Ž 14 with HŽjl 4 s1 However, we may have other sources of j dependence occurring in HŽj 4 Let us explain this l l l

103 ( ) R Hong TuanrPhysics Letters B Ž If G were a tree, then G would have only Vy1 propagators and a variable change with Jacobian equal to one Vy1 Ý z z Ž 16 Õ Õ Õ l Õs1 would be feasible while keeping j l s independent of each other as far zõ dependence is concerned Then, any X factor j X Ž X lp j l, coming for instance from a factor k ly j l P k l y j l X, with l/ l, would factorize into independent integrals, making the evaluation of Ž 14 straightforward But, as soon as G contains loops, because around a loop L Ý Vy1 l Ý Õ Õ ž l / l Õs1 z s0, s"1 Ž 17 lgl if l and l X belong to the same loop L, j and j X l l are not independent and we have a more difficult integration to do We now treat the general case where loops are present We first eliminate the linear term in the exponential of Ž 14 through a variable change zõ zõybõ Ž 18 which yields Vy1 equations y 1r Vy1 y1 Ý b X X Õ Ý Õ l a l Õ l qp Õ s0 19 X Õ s1 l The Vy1 byvy1 symmetric matrix d Ý G defined as y1 dg Ž a ÕÕ s Õ l a l Õ l Ž l is known to be non-singular and its determinant reads ÝŁ DG Ž a s a y1 l Ž 1 T lgt where the sum runs over all spanning trees T of G Ž a spanning tree is a tree incident with every vertex of G We therefore get Vy1 y1 Ý X X Õ Õ G ÕÕ X Õ s1 b s P d Ž a which means that a factor j Pj X is transformed according to l l Vy1 Vy1 y1 Ý Ý X X l l l l l l Õ Õ l Õ Õ ž /ž l / l l Õs1 Õ X s1 j Pj X j Pj Xq 4a a X b b X qterms linear in j or j X 3 We now make an orthogonal transformation in order to diagonalize d Ž a G Then, performing the integration over the V y 1 z vectors, the terms linear in j and j X vanish and for HŽ j 4 s j P j X we get Ž Õ l l l l l two derivations of Ž 14 Ž with Hs1 with respect to ip also do the job with ½ ž / ž / 5 Õ Vy1 Vy1 Vy1 y1 y1 H 0 l l Ý Õ l G ÕÕ Õ l Ý Õ Õ l Ý Õ Õ l Õ 1,Õs1 Õ1s1 Õs1 I s I 4a a X y id d a X q b P b X 4 Vy1 ŽVy1Ž dyr Ž Vy1 dr ydr y1 0 G ½ Ý Õ1 G ÕÕ 1 Õ 5 Õ 1,Õs1 I si Ž 4p D Ž a exp i P d Ž a PP Ž 5

104 104 ( ) R Hong TuanrPhysics Letters B I being I for HŽj 4 0 H l s 1 One remarks that, of course, if l and l belong to the same spanning tree, IH X reduces to its second term in Ž 4 Now, if l and l belong to the same loop L, we get according to Ž 17, Vy1 Vy1 y1 X X X l l l l Ý Õ Õ l l Ý l Ý Õ Õ l ž / ž 1 X / Õ1s1 l 1/l Õs1 l 1gL j Pj syž 4a a z z Ž 6 Cutting L at l X, we get that all l /l X 1 belonging to L are on a spanning tree of G Then, the crossed terms in Ž 6 with l / l will not contribute, leaving us only with 1 Vy1 y1 X X l l l l Ý Õ Õ ž l / Õs1 y Ž 4a a z Ž 7 because all j l s on a spanning tree are independent As 7 is proportional to j l it can be obtained by a derivation and Ž 4 will be written Žusing Ž 14 I sy X a ra X H l l Ž l l ErE Ž ia l I0 Ž 8 as can be easily checked against Ž 4 by a direct evaluation where we remark that the first term in Ž 4 comes w xyd r from deriving DG a and the second one from deriving the exponential Now, instead of dot-products one may have tensor products for j l s coming for instance from the dot-products Ž k yj PŽ k yj Ž k yj PŽ 1 1 k1 yj However, after the variable-shift Ž 18 has been made the formula Ž 11 1 can be applied to j l s as well as to k l s We can now give the procedure for writing, in the Feynman gauge, the a-parameter expression for any QCD graph amplitude: i Write the Feynman numerator which is a polynomial in dot-products and tensor-products of P Õ s, k l s and g mn s ii Replace each internal momentum k l by k ly j l in the Feynman numerator iii Choose a spanning tree T on G, for which the j l s will be independent iv Replace each j l of a propagator not belonging to T by its expression as a function of j l s belonging to T Žuse relation Ž 17 for that purpose v For each monomial of the Feynman numerator the product of j l s and k l s should be replaced by an operator product Oˆ according to the following rules Ž n is a positive integer or zero k 0 Ž 9a l I ydr mn k lmk l g rd ErE ia l acting on Ł a l 9b ls1 n n mn m n mn l l l l l l 0 Ž j mj g rd ErE Ž ia qb b yž g rd b acting on I Ž 9c n nq1 Ž nyp Ž nyp q1 p l Ý nq1 l l l 0 ps0 Ž j C Ž yb Ž j m j acting on I Ž 9d where b lsž a l y1 Ý Vy1 b In Ž 9d Ž j Õs1 Õ Õ l l nq1 is a tensor product This gives the following expression for FG QCD in d dimensions I I ydr ` QCD L yldr Ł Ý ˆ G H l l l Ł l G 0 ls1 ˆ4 ls1 O Vy1 y1 ½ Ý Õ1 G ÕÕ 1 Õ 5 Õ 1,Õ F s CŽ G rsž G i Ž 4p i da exp yia m O a D Ž a = exp i P d Ž a PP Ž 30 X ydr

105 ( ) R Hong TuanrPhysics Letters B where m ˆ l is infinitesimal for vector particles, ie gluons and O4 is the ensemble of monomials in the Feynman numerator In Ž 9c and Ž 9d n is at most equal to two because at most three loops can share the same propagator We have used wx 9 the representation Ž 30 in the calculations of the one-loop gluon self-energy and obtained the same result as in the conventional approach We found a straightforward renormalization procedure using dimensional regularization The Ward identities were found to be satisfied as easily as in the conventional approach References wx 1 MC Bergere, ` JB Zuber, Commun Math Phys 35 Ž wx MC Bergere, ` Yuk Ming, P Lam, J Math Phys 17 Ž wx 3 Z Bern, DC Dunbar, Nucl Phys B 379 Ž wx 4 R Hong Tuan, Phys Lett B 393 Ž ; preprint LPTHE 97r38, Orsay, 1997, hep-phr wx 5 Ya Ya Balitskii, LN Lipatov, Sov I Nucl Phys 8 Ž wx 6 RE Hancock, DA Ross, Nucl Phys B 394 Ž wx 7 CS Lam, JP Lebrun, Nuovo Cimento A 59 Ž wx 8 C Itzykson, JB Zuber, Quantum Field Theory, McGraw-Hill, p 94 wx 9 R Hong Tuan, to be published

106 31 December 1998 Physics Letters B Compositeness condition for non-abelian gauge fields Keiichi Akama a, Takashi Hattori b a Department of Physics, Saitama Medical College, Kawakado, Moroyama, Saitama , Japan b Department of Physics, Kanagawa Dental College, Inaoka-cho, Yokosuka, Kanagawa 38, Japan Received 11 June 1998; revised 18 September 1998 Editor: M Cvetič Abstract We derive and solve the compositeness condition for the SUŽ N gauge boson at the next-to-leading order in 1rN Ž c f Nf is the number of flavors and the leading order in ln L Ž L is the compositeness scale to obtain an expression for the gauge coupling constant in terms of the compositeness scale It turns out that the gauge-boson compositeness Ž with a large L takes place only when NfrN c) 11r, in which the asymptotic freedom fails q 1998 Elsevier Science BV All rights reserved PACS: 160Rc; 1110Jj; 1115Pg; 1115-q Recently, the possible compositeness of gauge bosons w1 7x has attracted renewed attention from both theoretical w8 1x and phenomenological w13,14x points of view The experimental results at present do not exclude w15x and may even suggest w13x the interesting possibility that quarks, leptons and gauge bosons are composite w14 x Based on general theoretw,8 x, this idea has been widely applied ical analyses in various branches of physics In quark-lepton physics, various models have been considered in terms of composite gauge bosons w3,9 x In hadron physics, the vector mesons can be interpreted as gauge bosons with hidden local symmetry w4,10 x The gauge fields applied in condensed matter systems should in principle be taken as composite, if they are not the electromagnetic fields w5,11 x The gauge fields induced in connection with the geometric phase Ž Berry phase in molecular and other systems are also expected to become dynamical through the quantum effects of matter, and could be considw6,7,1 x ered as composite gauge bosons A gauge boson interacting with matter can be interpreted as composite under the compositeness condition Z s0 w16 x 3, where Z3 is the wave-function renormalization constant of the gauge boson Under this condition, the gauge field becomes an auxiliary field without independent degrees of freedom w17 x The quantum fluctuations, however, give rise to a kinetic term of the gauge field, so that a dynamical gauge boson is induced as a composite of the matter fields The compositeness conditions which have been investigated so far hold only in the large N limit, where N is the number of the matter fields coupling with the gauge boson However, in cases of practical interest, N is rather small Hence it is important to investigate the higher order effects in 1rN In our previous papers, we derived the next-toleading order Ž NLO contributions to the compositeness conditions in the Nambu-Jona-Lasinio model w18x and the abelian gauge theory w19x at the leading Ž order in ln L rm ŽL is the compositeness scale, r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

107 ( ) K Akama, T HattorirPhysics Letters B and m is the constituent mass scale 1 In this paper, we consider the NLO contributions to the compositeness condition in non-abelian gauge theory, and then find Ž within this approximation that the gauge boson can be composite only when the number of flavors Nf is so large that the theory is not asymptot- ically free w 0 x On the other hand, many people have argued that the asymptotically non-free theory may encounter the problem of excessively large coupling constants at some ultraviolet energy scale, suggesting the necessity for some new physics such as composw1 x Thus we observe a complementarity iteness between gauge-boson compositeness Žwith a large L and asymptotic freedom in the gauge theories The basic Lagrangian for the composite SUŽ N c a Ž gauge boson G as1,,n y1 m c dade of a pair of the fermion c Ž js1,,n j f and its anti-particle is given by 1 a a Ls Ý c j ieuymq gl Gu c j, 1 j where g is the coupling constant, m is the mass of Ž a c j In 1, the gauge field Gm misses its kinetic term, being an auxiliary field without independent dynamical degree of freedom In the quantum theory, however, quantum fluctuations of a fermion-antifermion pair give rise to the missing kinetic term, and convert the auxiliary field Gm a to an ordinary Yang-Mills gauge field, which is interpreted as a composite of the fermion-antifermion pair To see the quantum effects, we introduce the gauge fixing and the Fadeev-Popov ghost term according to the ordinary procedure w x 1 1 a a m a Ý j Ž j a Ž m j Ls c ieuymq gl Gu c y E G Ž m m qe m h a E h a ygf abc h b G c where a is the gauge fixing parameter, and h as 1,,N y 1 c is the Fadeev-Popov ghost field Now we compare this with the basic Lagrangian of the ordinary Yang-Mills gauge theory Žwith a elemen- 1 In practice, we use the dimensional regularization with corre- Ž spondence e s 4y d rs1rln L rm, where d is the number of spacetime dimensions a Ž tary gauge field written in terms of the renormalized quantities w x: Lsy Z q c iz EuyZ m q Z g l Gu c y E G qz E h a r = 1 a 4 3 Grmn 1 a a Ý r j Ž m r 1 r r rj j 1 m a m a Ž rm h r Z a 1 abc b c ž m r r r rm Z / E h y g f h G Ž 3 where the quantities with the subscript r are the renormalized ones, Z 1, Z, Z 3, Z h, and Zm are the renormalization constants, and G i rmnsem Grn a yeng rm a abc b c q Z1rZ grf GrmG rn We can see that, if Z3 s0, Ž 4 the Lagrangian Ž and Ž 3 coincide with each other, as far as the fields, the masses, and the coupling constants are appropriately related It means that we can calculate everything in the composite gauge theory Ž 1 by calculating it in the ordinary gauge theory Ž 3 and imposing the condition Ž 4 For this reason, Ž 4 is called compositeness condition Note that this is the compositeness condition of w16 x, but not that of w1 x The nature of the condition, however, spoils the perturbation expansion in g r, and compel us to use 1rNf expansion, as will be explained soon At the lowest order in gr, Z3 is determined so as to cancel the divergence in the one-loop diagrams in Fig 1 A C, where the solid, wavy, and dotted lines indicate the fermion, the gauge boson, and the Faddeev- Popov ghost propagators, respectively As is well known, this is given by wx 13 a r Z3 s1y Nfy y Nc gr I, Ž ž / where I is the divergent integral, given, in the dimensional regularization, by I s 1r16p e s 1r8p Ž 4 y d with the number of spacetime dimensions d, and, in Pauli-Villars regularization, by I s Ž ln L rm r16p A simple-minded application of the compositeness condition Z3s 0 gives 13 a r gr s1r Nfyž y / Nc I Ž To avoid the absurdity of a vanishing coupling constant, we fix 1re or L at some finite value The

108 108 ( ) K Akama, T HattorirPhysics Letters B Fig 1 The gauge-boson self-energy parts at the leading order Ž A and at the next-to-leading order Ž B H in 1rN f The solid, wavy, and dotted lines indicate the fermion, gauge-boson, and Fadeev- Popov ghost propagators, respectively The disk stands for insertion of an arbitrary number of one-fermion-loops into the gauge boson propagator result Ž 6 is inappropriate because Ž i it depends on gauge and Ž ii infinitely many higher order contributions have the same order of magnitude, spoiling the perturbation expansion in g Ž r see the arguments in w16,17 x A way to include the infinitely many contributions systematically is to rely on 1rNf expansion In the leading order in 1rN f, only the diagram a in Fig 1 contributes to Z 3, and Ž 5 is replaced by Z s1y Ng I, Ž f r and the solution Ž 6 is replaced by 3 gr s Ž 8 NI f In Ž 7 the gauge independence is preserved, as also solves the above-mentioned problem Ž i within this order Because it is physical whether something is composite or not, the compositeness condition should be gauge independent The condition Z3s 0 is equivalent to that that the Lagrangian Ž has no kinetic term of the gauge boson, and the general proof of gauge independence of the Fadeev-Popov scheme is still available in Žw x Therefore, the condition Z3 s0 should be gauge independent even though Z3 itself depends on gauge It is realized at the leading order as Ž 7 shows, and at the next-toleadingž NLO order as we will show below Now we turn to the NLO contributions in 1rN f In addition to the one-boson-loop diagrams B and C in Fig 1, the multi-loop diagrams D H in Fig 1 belong to this order, where the disks stand for insertions of an arbitrary number of one-fermion-loops into the gauge boson propagator The renormalization constant Z3 should be chosen so as to cancel all the divergence in these diagrams Though the n-loop diagram diverges like I n, it is suppressed by the factor g n A1rI n Therefore, the leading divergence of each diagram is OŽ I n OŽI yn ;OŽ 1, apart from the powers of 1rN f We denote this leading contri- bution of OŽ 1 from diagram XŽ sd H in Fig 1 to Z by ZŽ X,l 3, where l is the number of the fermion loops per diagram Different diagrams with the same X and l give ZŽ X,l exactly the same contributions We call the number of such different diagrams multiplicity of X with l The overlapping divergence in G and H is separated into two parts: f where the divergence occurs at the fermion loop subdiagram in the three boson vertex part, and m where the divergence occurs at the boson-fermion Ž mixed loop subdiagram in the boson-fermion vertex part Ž see Fig We denote these contributions as ZŽ Gf,l, ZŽ Gm,l, ZŽ Hf,l, and ZŽ Hm,l The following properties are useful in the calculation Ž a Due to the gauge symmetry, the leading Fig The separated overlapping divergence Gf, Hf: the part where the divergence occurs at the fermion loop subdiagram in the three boson vertex part Gm, Hm: the part where the divergence occurs at the boson-fermion Ž mixed loop subdiagram in the boson-fermion vertex part

109 ( ) K Akama, T HattorirPhysics Letters B divergences cancel each other in two diagrams: that with a fermion-loop inserted at a three-gauge-boson vertex in some diagram and that with a fermion-loop inserted into a gauge-boson propagator adjacent to the vertex Ž b Because the one-fermion-loop Žde- mn noted by P 0 inserted into a gauge-boson propagator Ž with momentum q is divergenceless Žie mn q P s0 m 0, the ar-dependent part of the propagator vanishes The divergent contribution ZŽ X,l is separated into the gauge independent part Z Ž X,l 0 and the part Z Ž X,l a linear in a r, while the terms higher in ar are convergent We first consider the ar-independent parts Because the lowest-order fermion self-energy part and boson-fermion-fermion vertex part converge in the Landau gauge, Z Ž D,l and Z Ž E,l 0 0 have no leading divergence Because property Ž a implies Z Ž F,l syz Ž Gf,l s Z Ž Hf,l for relevant l, and their multiplicities are lq1, l, and ly1, respectively, they cancel out when lg1 For ls0, diagrams G and H are absent, while diagram F is nothing but diagram B Žincluding the ar-dependent part, and, together with diagram C, contributes the second term in the square brackets in Ž 5 Because property Ž a implies Z Ž Gm,l syz Ž Hm,l Ž lg 0 0 and their multiplicities are l and Ž ly1, respectively, diagrams Gm and Hm contribute the following terms to Z 3: ` ly1 1 l lq1 yý y NN c f Ž gr I Ž 9 lž lq1 3 ls1 ž / Then we consider the ar-dependent parts Due to property Ž b, diagrams D and E have contributions only when ls1, and F, G, and H have contributions only when no fermion-loop is inserted on one of the two gauge-boson lines The multiplicities of F, Gf, and Hf, are 1,, and 1, respectively, and those of Gm and Hm are and, respectively, for any relevant l Because property Ž a implies Z Ž F,l a s yz Ž Gf,l s Z Ž Hf,l and Z Ž Gm,l syz Ž Hm,l a a a a for relevant l, they cancel out for lg For ls1, diagram H is absent, and the contributions from diagrams D, E, F, Gf, and Gm to Z are 3 1 y 3 ar NN c f gr I 10 times y1q1rnc, y1rnc, y1r, 1, and 3r, respectively, and hence Ž 10 times 1 in total The contribution from l s 0 has already been taken into account in Ž 5 Next, we renormalize the subdiagram divergences by subtracting the divergent counter parts of Ž i each fermion loop inserted into the gauge boson lines in D H, Ž ii the fermion self-energy part in D, Ž iii the fermion-boson vertex part in E, Ž iv the three-boson vertex part in Gf and Hf, and Ž v the boson-fermion-fermion vertex part in Gm and Hm The contributions from the counter parts cancel out for even l, and amount to minus twice the original terms for odd l Therefore in total they contribute the following terms to Z 3: ` ly1 1 l lq1 Ý NN c f Ž gr I lž lq1 3 ls1 ž / 1 r c f r q a NN g I Ž 11 3 Collecting all the contributions in Ž 5 and Ž 11 together, we finally obtain the compact expression 11 a r Z3 s1y Ng f r Iq NgIy c r NgI c r = 1y Ng f r I q Nc ygr I 3 N f ž / ž / ž / =ln 1y Ng f r I Ž 1 3 up to OŽ1rN f corrections In general, Z3 depends on gauge, and involves a troublesome logarithmic term However, once we impose the compositeness condition Z s 0onŽ 1, 3 we have g s3rž NI qož1rn Žsee Ž 8 r f f, which implies that the gauge-dependent and the logarithmic terms vanish within the scope of the approximation in this paper Ž NLO in 1rN and leading order in e f The compositeness condition reduces to 11 1y 3Ng f r Iq 3 NgIs0 c r 13 within the approximation As is expected Žsee the argument after Eq Ž 8, the compositeness condition is gauge-independent, despite the Z3 itself depends on gauge Now we can see that if and only if 11N c N f), Ž 14

110 110 ( ) K Akama, T HattorirPhysics Letters B has a positive solution for g r : 3 gr s Ž 15 N y11n I f c up to OŽ1rN 3 f terms Note that the allowed region of N rn by Ž 14 f c s complementary to that for asymptotic freedom in the gauge theory w0 x If this is applied to the standard model with three generations of quarks and leptons, then the weak bosons can be composite, but the gluons cannot As was shown in the previous paper w19 x, the NLO contribution is suppressed in the abelian gauge theory, making the compositeness interpretation successful This is in accordance with the above-mentioned complementarity, since the abelian gauge theory is not asymptotically free In general, in the asymptotically non-free theories with the beta function derived via perturbations in the coupling constant, the running coupling constant diverges at some ultra-violet scale Then we should introduce a momentum cutoff, since the theory gives nothing above that scale Some new physics such as compositeness is required to fill in this blank w1 x The marginal value NfrNcs11r for asymptotic freedom is derived via the renormalization group method at the leading order in g Ž r and exactly in Nc and N f All the one-loop diagrams and the renor- malization constants Z 1, Z, and Z3 were used there On the other hand, allowed region Ž 14 of compositeness in this paper is derived via the NLO perturbation in 1rN f, dnd by using multi-loop boson-self-en- ergy diagrams and Z3 only The coincidence of the marginal values NfrNcs11r, however may not be accidental for the following reasons In the latter Ž NLO in 1rN f, we implicitly used the other one-loop diagrams Ž other than the boson self-energy as subdiagrams, and also used Z and Z Ž besides Z 1 3 to renormalize the divergences due to the subdiagrams On the other hand, in the former Žthe renormalization group method, one-loop contributions at some scale could be interpreted as a sum of multi-loops at the other scale It would be interesting to confirm this complementarity at a higher order, or in other mod- This does not eliminate the possibility that the composite gauge boson with a low L is compatible with asymptotic freedom els such as those with scalar matter, or to argue from a more general point of view covering all orders We expect that these methods and results concerning NLO compositeness and the concept of complementarity presented here will be useful in pursuing the composite dynamics of nuclei and hadrons, and the possible compositeness of quarks, leptons, gauge bosons and Higgs scalars, as well as useful in induced gauge theories concerning molecular, solidstate, and other systems Acknowledgements We would like to thank Professor H Terazawa for useful discussions References wx 1 W Heisenberg, Rev Mod Phys 9 Ž ; JD Bjorken, Ann Phys 4 Ž wx I Bialynicki-Birula, Phys Rev 130 Ž ; D Lurie, AJ Macfarlane, Phys Rev 136 Ž 1964 B816; T Eguchi, Phys Rev D 14 Ž ; D 17 Ž ; H Kleinert, in: A Zichichci Ž Ed, Understanding the fundamental constituents of matter, Proceedings, 1976, Erice Summer School, Plenum Publishing Corporation, 1978, p 89; K Shizuya, Phys Rev D 1 Ž wx 3 K Akama, H Terazawa, INS-Rep-57, 1976; H Terazawa, Y Chikashige, K Akama, Phys Rev D 15 Ž ; T Saito, K Shigemoto, Prog Theor Phys 57 Ž ; OW Greenberg, J Sucher, Phys Lett B 99 Ž ; LF Abbott, E Farhi, Phys Lett B 101 Ž wx 4 See for example, JJ Sakurai, Currents and Mesons, Univ Chicago Press, Chicago, 1969; M Bando, T Kugo, S Uehara, K Yamawaki, T Yanagida, Phys Rev Lett 54 Ž wx 5 G Toulouse, Commun Phys Ž ; IE Dzyaloshinskii, GE Volovik, J de Phys 39 Ž ; F Wilczek, Phys Rev Lett 48 Ž ; 49 Ž ; SM Girvin, The quantum Hall effect, RE Prange, SM Girvin Ž Eds, Springer Verlag, 1986, Cap 10; SM Girvin, AH MacDonald, Phys Rev Lett 58 Ž ; G Baskaran, PW Anderson, Phys Rev 37 Ž ; PW Wiegmann, Phys Rev Lett 60 Ž ; I Affleck, JB Marston, Phys Rev B 37 Ž ; A Nakamura, T Matsui, Phys Rev B 37 Ž ; I Affleck, Z Zou, T Hsu, PW Anderson, Phys Rev B 38 Ž ; SC Zhang, TH Hansson, S Kivelson, Phys Rev Lett 6 Ž ; ZF Ezawa, A Iwazaki, Phys Rev B 44 Ž wx 6 CA Mead, DG Truhlar, J Chem Phys 70 Ž ; MV Berry, Proc R Soc London Ser A 39 Ž wx 7 K Akama, in: K Kikkawa, N Nakanishi, H Nariai Ž Eds,

111 ( ) K Akama, T HattorirPhysics Letters B Gauge Theory and Gravitation, Proceedings of the International Symposium, Nara, Japan, 198, Springer-Verlag, p 67; MD Maia, J Math Phys 5 Ž ; K Akama, Prog Theor Phys 78 Ž ; 79 Ž wx 8 For recent works see for example, A Hasenfratz, P Hasenfratz, Phys Lett B 97 Ž ; MA Luty, Phys Rev D 48 Ž ; K Akama, T Hattori, Phys Lett B 39 Ž : M Bando, Y Taniguchi, S Tanimura, Prog Theor Phys 97 Ž ; M Bando, J Sato, K Yoshioka, ICRR-Report , hep-phr970331, 1997; M Bando, hep-phr970537, 1997 wx 9 For recent works see for example, K Akama, T Hattori, Int J Mod Phys A 9 Ž ; BS Balakrishna, KT Mahanthappa, Phys Rev D 49 Ž ; Phys Rev D 5 Ž ; A Galli, Phys Rev D 51 Ž ; Nucl Phys B 435 Ž ; CP Burgess, J-P Derendinger, F Quevedo, M Quiros, Phys Lett B 348 Ž ; VV Kabachenko, YF Pirogov, IHEP 96-97, hep-phr96175, 1996; M Berkooz, P Cho, P Kraus, MJ Strassler, HUTP-97rA014, hep-thr , 1997; M Yasue, ` TOKAI-HEPrTH-9701, hep-phr970731, 1997 w10x For recent works see for example, M Tanabashi, Phys Lett B 384 Ž ; MC Birse, Z Phys A 355 Ž ; M Hashimoto, Phys Rev D 54 Ž ; CM Shakin, W-D Sun, Phys Rev D 55 Ž ; L-H Chan, Phys Rev D 55 Ž ; M Harada, A Shibata, Phys Rev D 55 Ž w11x For recent works see for example, C Itoi, H Mukaida, J Phys A: Math Gen 7 Ž ; Y Hosotani, J Phys A 30 Ž 1997 L757; I Ichinose, T Matsui, preprint UT-Komaba 97-13, cond-matr w1x For recent works see for example, K Kikkawa, Phys Lett B 97 Ž ; T Isse, Mod Phys Lett A 8 Ž , 947; T Hatsuda, H Kuratsuji, UTHEP-86, 1994; K Kikkawa, H Tamura, Int J Mod Phys A 10 Ž ; S Tanimura, I Tsutsui, Mod Phys Lett A 10 Ž ; EM Monte, MD Maia, UNB-MAT-FM-M495, 1995 w13x CDF Collaboration, F Abe et al, Phys Rev Lett 77 Ž , 5336; Phys Rev D 55 Ž 1997 R563; H1 Collaboration, C Adloff et al, Z Phys C 74 Ž ; ZEUS Collaboration, J Breitweg et al, Z Phys C 74 Ž w14x M Bander, Phys Rev Lett 77 Ž ; K Akama, H Terazawa, Phys Rev D 55 Ž 1997 R51; AE Nelson, Phys Rev Lett 78 Ž ; K Akama K Katsuura, H Terazawa, Phys Rev D 56 Ž 1997 R490; SL Adler, IASSNS-HEP 97-1, hep-phr970378; MC Gonzalez- Garcia, SF Novaes, IFT-P-04-97, hep-phr ; N Di Bartolomeo, M Fabbrichesi, Phys Lett B 406 Ž w15x Particle Data Group, RM Barnett et al, Phys Rev D 54 Ž w16x B Jouvet, Nuovo Cim 5 Ž ; MT Vaughn, R Aaron, RD Amado, Phys Rev 14 Ž ; A Salam, Nuovo Cim 5 Ž 196 4; S Weinberg, Phys Rev 130 Ž ; Lurie, Macfarlane, Ref wx ; Eguchi, Ref wx ; Shizuya, Ref wx w17x DJ Gross, A Neveu, Phys Rev D 10 Ž ; T Kugo, Prog Theor Phys 55 Ž ; T Kikkawa, Prog Theor Phys 56 Ž w18x K Akama, Phys Rev Lett 76 Ž w19x Akama, Hattori, Ref w8 x w0x DJ Gross, F Wilczek, Phys Rev Lett 30 Ž ; HD Politzer, Phys Rev Lett 30 Ž w1x WA Bardeen, CT Hill, M Lindner, Phys Rev D 41 Ž ; M Harada, Y Kikukawa, T Kugo, H Nakano, Prog Theor Phys 9 Ž ; Bando, Taniguchi, Tanwx 8 ; Bando, Sato, Yoshioka, Ref wx 8 ; Bando, Ref imura, Ref wx 8 wx See for example, T Muta, Foundations of Quantum Chromodynamics, An Introduction to Perturbative Method in Gauge Theories,

112 31 December 1998 Physics Letters B Absence of cross-confinement for dynamically generated multi-chern Simons theories Emil M Prodanov 1,, Siddhartha Sen 3 School of Mathematics, Trinity College, Dublin, Ireland Received 15 October 1998 Editor: PV Landshoff Abstract We show that when the induced parity breaking part of the effective action for the low-momentum region of Ž Ž 4 U 1 = PPP =U 1 Maxwell gauge field theory with massive fermions in 3 dimensions is coupled to a f scalar field theory, it is not possible to eliminate the screening of the long-range Coulomb interactions and get external charges confined in the broken Higgs phase This result is valid for non-zero temperature as well q 1998 Elsevier Science BV All rights reserved PACS: 1110Jj; 1110Kk; 1110Wx; 1115Ex Keywords: Finite temperature field theory; Spontaneous symmetry breaking; Chern Simons; Screening; Confinement 1 Introduction In a recent paper Cornalba et al wx 1 have proposed a novel topological way of confining charged particles The method uses the special properties of UŽ 1 = UŽ 1 Chern Simons gauge theory, interacting with external sources in two spatial dimensions, with a scalar Higgs field providing condensates The idea of the approach is to note that, when chargerflux constraints of a certain type are not satisfied, the fall off of the Higgs fields at infinity will not be fast enough and will lead to configurations with infinite energy; hence, such configurations are confined The analysis is based on number-theoretic properties of the couplings and charges and shows the intriguing possibility for confinement even for integral charge particles The confinement mechanism is topological in origin A Chern Simons term of the form considered in wx 1 can be dynamically generated as the parity-breaking part of the low-momentum region of the effective action of a three-dimensional UŽ 1 = PPP =UŽ 1 Maxwell gauge 1 Supported by FORBAIRT scientific research program and Trinity College, Dublin prodanov@mathstcdie 3 sen@mathstcdie r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

113 ( ) EM ProdanoÕ, S SenrPhysics Letters B filed theory with fermions, after integrating out the fermionic degrees of freedom w,3 x Indeed, we carry out this procedure for the system at non-zero temperature The effective action, obtained by us, following the approach of wx has the correct temperature dependence for the multiple UŽ 1 Chern Simons term and yields in its zero-temperature limit a multiple Chern Simons term of the form considered in w1,4,5 x Such multiple UŽ 1 gauge theories have been considered before, for example: in the study of spontaneously broken abelian Chern Simons theories w4,5 x; in the study of two-dimensional superconductivity without parity violation wx 6 Our original motivation was to investigate if the mechanism for cross-confinement, proposed in wx 1, continues to hold for the system with a temperature slightly deviated from zero and if confinement is lost for high temperature with the system still in the Higgs phase Surprisingly, with this dynamically generated parity-breaking term, the arguments of Cornalba et al wx 1 do not hold, namely, the proposed scheme of confinement is not possible This result is valid, as we show, for zero and non-zero temperatures In this model it is not possible to eliminate the screening of the long-range Coulomb interactions We claim that, if confinement occurs, it happens when the broken UŽ 1 = PPP = UŽ 1 gauge symmetry is restored in at least one of the directions of the gauge group By the standard Higgs mechanism, the gauge group is spontaneously broken down to a product of the cyclic groups Z = PPP = Z This residual symmetry represents the non-trivial holonomy of the Goldstone boson 1 N The photon fields A Ži m, is1,,n now acquire masses by their coupling to the Goldstone bosons In the broken Higgs phase the Higgs currents are proportional in magnitude to the massive vector fields and screen the Coulomb interaction and we are left with purely quantum Aharonov Bohm interactions w4,5 x In this phase, at temperature well below the critical, all conserved charges can reside in the zero-momentum mode due to the bosonic character of the particles When the temperature increases, some of the charges get excited out of the condensate and at sufficiently high temperature the condensate becomes thermally disordered and the symmetry is restored When this happens the charges introduced by the matter currents will not be screened and the energy of the Coulomb field will logarithmically diverge with distance Ž in two spatial dimensions and this will lead to confinement The Model We will determine first the parity-breaking part of the effective action for UŽ 1 = PPP =UŽ 1 Maxwell gauge field theory coupled to massive fermions and f 4 scalar field theory in 3 dimensions at finite temperature Contact with the multiple Chern Simons term, considered by w1,4,5x is made by taking the zero-temperature limit The effective action for the low-momentum region of the theory is: N N yg Ž A Ž k, M b k f Ž k Ž k HŁ k k ½ H H ½ Ý Ž k k k a a ks1 0 ks1 e s Dc Dc Dfexp y dt d x c Du c qj A q Ž D f Ž D f 5 5 ) ) ym ff yl ff 1 where Du f seuqiq Au Ž j qm are the fermionic covariant derivatives with Q, k, js1,,n being the k kj k kj matrix of the fermionic charges with respect to the N gauge groups, D se qiq A Ž k,ks1,,n are the a a k a covariant derivatives for the scalar field with qi being the charge of the scalar field with respect to the i th gauge 1 group In this action bs T is the inverse temperature and Dirac matrices are in the representation gm ss mwe have also introduced external currents coupled to the gauge fields We shall consider first the parity-breaking part of the fermionic part of the action and at this stage the scalar field is only a spectator )

114 114 ( ) EM ProdanoÕ, S SenrPhysics Letters B For this purpose we will follow the approach of Fosco et al wx The fermionic fields obey antiperiodic boundary conditions, while the gauge fields are periodic The considered class of configurations for the gauge fields is: A Ž 3 k sa Ž 3 k Ž t, A Ž 1, k sa Ž 1, k Ž x, ks1,,n Ži There is a family of gauge transformation parameters, which allow us to gauge the time-components A Ž t 3 to Ži the constants a wx This makes the Dirac operator invariant under translations in the time coordinate Žas the Ži dependence on t comes solely from the A fields 3 and therefore we could Fourier-expand ci and ci over the Matsubara modes Following steps, similar to those in wx, one finds that the parity-odd bit of the fermion part of the effective action is given by: i N q` Ž k Žj Godd s Ý Ý fn HelmQkjE l Am d x Ž 3 p k, js1 nsy` ž / Ž k v nq Qkja Ž j p where f sarctg and v sž nq1 n n b is the Matsubara frequency for fermions M k Performing the summation we get that for UŽ 1 = PPP =UŽ 1 gauge group the parity-odd part of the action is: i N bmk 1 b yg odd Ž j Ž n e shdfexp y Ý arctg thž / tgž H QkjA3 Ž t dt/ HelmQknEl Am d x ½ p 0 k, j,ns1 b a ) ) ) Ž k Ž k H H a 0 5 q dt Ž D f Ž D f ym ff ylž ff qj A d x Ž 4 As the temperature T approaches 0 Ž that is b ` this reduces to UŽ 1 = PPP =UŽ 1 Chern Simons gauge theory We will use now the effective parity-odd temperature dependent action Žwith the induced UŽ 1 = PPP =UŽ 1 parity breaking term to re-examine the confinement argument of Cornalba et al wx 1 First of all, let us perform the integration Ž using Stokes theorem of the gauge fields over the spatial co-ordinates This gives the relation with the magnetic fluxes F l : i N bmk 1 b yg odd Ž j e shdfexp y Ý arctg th tg H Q kj A 3 Ž t dt Q kn F ½ p ž / ž / n 0 k, j,ns1 b a ) ) ) Ž k Ž k H H a 0 5 q dt Ž D f Ž D f ym ff ylž ff qj A d x Ž 5 The equations of motion, obtained by varying the action with respect to the magnetic fields, are: bm k N th QklQknF i ž / n y Ý 4p b bm 1 k 1 b k,m,ns1 Ž m Ž m cos H Qkm A3 dt qth sin H Qkm A3 dt ž / ž / ž / 0 0 H H H yq fd f ) yf ) D f d xq r Žl d xs E F Žl j3 d x Ž 6 l 3 3 j where r Žl sj3 Žl are the charge densities Here we have included explicitly the contribution of the Maxwell term Ži Ži mn F F We ignore temperature dependent terms which come from OŽ A 4 mn terms in the effective action These are of higher order Ž OŽ Q 4 in the fermionic chagres The Coulomb charges on the rhs vanish because all UŽ 1 fields are massive

115 ( ) EM ProdanoÕ, S SenrPhysics Letters B Ž ) 3 Denote by u the integral over the third component of the conserved Nother current: u s H f D f y ) Žl Žl f D f d x and by C shr d x the total external charge So we have: 3 mfscyuq Ž 7 where F 0 1 q1 C Fs, qs, Cs F q N N C and 0 0 Ž 1 Ž N bm k N th QklQ i ž / kn mlns Ý Ž 8 bm 4p k,ms1 1 b k 1 b Ž m Ž m cos H Qkm A3 dt qth sin H Qkm A3 dt ž / ž / ž / 0 0 As in wx 1 there is another condition which must be satisfied by the magnetic fluxes The Higgs field f should be is Ž x completely condensed, ie f x s Õe, where s Ž x is the Goldstone boson field Žthe mass and the coupling constants of the scalar field are temperature-dependent In order that this holds we have to require that the covariant derivative of the scalar field vanishes After integration we get: p lsq F q qq F s t 1 1 N N qf Ž 9 where p l is the non-trivial holonomy of the Goldstone boson Žreflecting a topological property of the Higgs field Combining the two conditions Ž 7 and Ž 9 for the fluxes we get: mfscyuq, p ls t qf Ž 10 Following the analysis of wx 1 we identify u as a continuous parameter, representing the ability of the condensate to screen the electric charge t The matrix m can be written as ms QFŽ b Q, where FŽ b is a diagonal matrix with entries: bm thž k i / FkjŽ b s d Ž 11 bm kj 4p 1 b k 1 b m Ž m cos Q A dt q th sin Q A H km 3 H km 3 dt As F b ž / ž / ž / 0 0 is diagonal we can always write m in the form: ms t QŽ T QŽ T Ž 1 1r where Q T sf Ž b mn mj Q jn Let us now try to eliminate the screening in Ž 10 by inverting the matrix m We get that if the determinant of m is not zero and if t q m y1 qs0 Ž 13 Ž t y1 then the screening would be eliminated the condition q m q s 0 is the condition for confinement, proposed wx y1 by Cornalba et al 1 According to their analysis, if the determinant of m vanishes, then m should be interpreted as the transposed matrix of co-factors Assuming that the determinant of m is not zero, we can re-write this as: t Q Ž T q Q Ž T qs0 Ž 14

116 116 ( ) EM ProdanoÕ, S SenrPhysics Letters B where QT is the matrix of co-factors This equation shows that the vector QT q is orthogonal to itself Ž orthogonal with respect to the matrix multiplication of column vectors and, therefore, this is the null vector: Q Ž T qs0 Ž 15 This is an equation for the values of the boson field charges, which would eliminate the screening mechanism As we see, we can have a non-trivial solution if, and only if, det Q s 0, which contradicts to our initial assumption Ž det m/ 0 Therefore, we cannot eliminate the screening Otherwise, this theory would be inconsistent with the induced parity-breaking term This argument is valid for all values of the temperature Intuitively, one can expect that if condition Ž 9 is violated, after elimination of screening, there would be currents which would not fall off faster than 1rr at infinity and the resulting long-range forces will lead to diverging energies We argue that condition Ž 9 can never be violated this condition represents the fact that we are left with a residual symmetry after the spontaneous symmetry breakdown If this condition does not hold, it would mean that the symmetry is restored This, on its turn, will lead to diverging energy straight away, but not in the broken Higgs phase We conclude that confinement is not possible in the Higgs phase in the presence of the dynamically generated parity-breaking term Ž which coincides with Chern Simons term in zero-temperature limit If there are configurations with infinite energy, they must necessarily be outside the broken Higgs phase where the gauge symmetry is restored References wx 1 L Cornalba, F Wilczek, Phys Rev Lett 78 Ž wx CD Fosco, GL Rossini, FA Schaposnik, Phys Rev D 56 Ž ; CD Fosco, GL Rossini, FA Schaposnik, Phys Rev Lett 79 Ž , 496 Ž E wx 3 S Deser, L Griguolo, D Seminara, Phys Rev Lett 79 Ž wx 4 M de Wild Propitius, Nucl Phys B 489 Ž wx 5 M de Wild Propitius, Phys Lett B 410 Ž wx 6 N Dorey, NE Mavromatos, Nucl Phys B 386 Ž

117 31 December 1998 Physics Letters B Bose-Einstein condensation on closed Robertson-Walker spacetimes M Trucks Institut fur Theoretische Physik, Technische UniÕersitat Berlin, Hardenbergstraße 36, 1063 Berlin, Germany Received 4 August 1998; revised 19 November 1998 Editor: PV Landshoff Abstract In this letter we summarize our analysis of Bose-Einstein condensation on closed Robertson-Walker spacetimes In a previous work we defined an adiabatic KMS state on the Weyl-algebra of the free massive Klein-Gordon field wm Trucks, M Keyl, Phys Lett B 399 Ž , M Trucks, Commun Math Phys 197 Ž x This state describes a free Bose gas on Robertson-Walker spacetimes We use this state to analyze the possibility of Bose-Einstein condensation on closed Robertson-Walker spacetimes We take into account the effects due to the finiteness of the spatial volume and show that they are not relevant in the early universe Furthermore we show that a critical radius can be defined The condensate disappears above the critical radius q 1998 Elsevier Science BV All rights reserved 1 Introduction We analyze the possibility of Bose-Einstein condensation of a free relativistic charged Bose gas on closed Robertson-Walker spacetimes The free Bose gas is described by an adiabatic KMS state, which we defined in w1, x In the framework of algebraic quantum field theory on curved spacetimes Žsee wx 3, we proved that an adiabatic KMS state is a Hadamard state, ie a physically relevant state Furthermore, it describes a free Bose gas and the inverse temperature change was shown to be proportional to the scale parameter in the metric of the Robertson-Walker spacetime Bose-Einstein condensation is usually considered in the thermodynamic limit Singularities appear in thermodynamic quantities, eg for a nonrelativistic Bose gas, there is a cusp-like singularity in the specific heat at the critical temperature For a finite system, these singularities are smoothed out and there is no well-defined phase transition point Furthermore, macroscopic occupation of the ground state sets in, although the temperature is higher than the critical temperature of the corresponding system in the thermodynamic limit, as we will see below We can apply known results about Bose-Einstein condensation on the Einstein universe Ža closed Robertson- Walker spacetime with constant scale parameter to closed Robertson-Walker spacetimes This is justified on the one hand by the fact that adiabatic KMS states are Hadamard states and on the other hand because on every Cauchy surface an adiabatic KMS state on a closed Robertson-Walker spacetime has the form of a KMS state on an Einstein universe Bose-Einstein condensation on the Einstein universe was analyzed by Singh and Pathria r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

118 118 ( ) M TrucksrPhysics Letters B wx 4 and Parker and Zhang wx 5 In the work of Singh and Pathria the emphasis is on the finite size effects, while they are not taken into account in the work of Parker and Zhang In contrast the work of Parker and Zhang also gives some hints on Bose-Einstein condensation on closed Robertson-Walker spacetimes and a critical radius is defined We analyze Bose-Einstein condensation on closed Robertson-Walker spacetimes, take into account the finite size effects and define a critical radius If the Robertson-Walker spacetime expands, there is a radius, below which the gas is in a condensed state The condensate disappears above this radius, but due to the finite size effects, this is not a well-defined phase transition point Using realistic values for the early universe, we will see that the finite size effects are very small In the next section we calculate the charge density of the Bose gas on closed Robertson-Walker spacetimes An approximation is made for the interesting regime, where the chemical potential reaches the value of the mass parameter This formula is analyzed in Section 3 A numerical calculation shows the effects of the finite spatial volume of closed Robertson-Walker spacetimes The critical temperature of a charged relativistic Bose gas on Minkowski spacetime serves as a useful reference point, which enables us to define a critical radius This critical radius depends only on the total charge and a constant CsRT, given by the product of the temperature and the scale parameter of the Robertson-Walker metric Due to the finite spatial volume, there is a nonvanishing charge density in the ground state at the critical temperature Ž resp radius This value Žscaled by the charge density mainly depends on C The charge density In this section we calculate the charge density of a relativistic charged Bose gas described by an adiabatic KMS state An adiabatic KMS state on closed Robertson-Walker spacetimes can be defined on a Cauchy surface at time t by 4 y1 v P sz tr exp ybh m P, w where H m is the second quantization of h m s m ydrr Žx t 1r ym, b is the inverse temperature, m the chemical potential, R the scale parameter in the metric of the Robertson-Walker spacetime, Z s w x4 3 tr exp ybh m and D the Laplace operator on the three-sphere S, the spatial part of the spacetime The conserved charge, which may be the electric or any other conserved charge, is described by the operator Q s Nqy N y, where Nq resp Ny are number operators for particles with positive resp negative charge Then the state is given by 4 y1 v P sz tr exp yb HymQ P For the expectation value of the charge density in this state we have qs² Q: rvs ² N : y² N : rv It is b,m q b,m y b,m Ý e " bm ² N : " b,m s, bv k " bm e ye ( k where v k s kž kq rr Ž t qm, kgn 0, since kkq are the eigenvalues of the Laplace operator on the three-sphere Using the fact that the multiplicity of the eigenvalues is Ž k q 1, we obtain ` ` ybvk n q s sinhž bmj k e, Ý V js1 Ý ks1

119 ( ( ) M TrucksrPhysics Letters B where vk s m qk rr, m sm y1rr The sums can be simplified with the Poisson summation formula ` ` ` ` Ý ks1 H Ý fž k s fž t dtq fž t cosž p pt dt H 0 ps1 0 The following computation follows wx 4 The first integral is given in wxž Nr 4 and we have to assume R)1rm To solve the second integral, we expand the cosine in a power series and use the formula l ` Ž zt r KlqlŽ zs KlŽ z s yt Ý s lql lr ls0 ' l! s Ž s yt Žsee wxž Nr 9, to obtain ž / ` KŽ at K Ž at fž t cosž p pt dtsj b mr y Ž p pmr, 0 Ž at Ž at H 3 ( n where ts j brr q p p, asmr and K Ž x are modified Bessel functions Therefore ž / 3 qs W b,m q jsinh j bm y p pa, p p at at 3 4 m b m ` K at K at Ý 3 j, ps1 ` where we defined W b,m sý Ž j b m y1 sinhž j bm K Ž jb m js1 We can also handle the sum over j with the Poisson summation formula in the form ` ` ` ` 1 Ý Ý Ý H js1 jsy` psy` y` fž j s fž j s fž j cosž p pj dj ž / ` ` KŽ at K Ž at 3 1 X s Ý RH jsinhž j bm y Ž p pmr dj 3 0 Ž at Ž at psy` X see wxž 4 App where m smqp i prb and R denotes the real part The result is ` X X ( ~ m 1 m m ym qs W Ž b,m y R Ý p pb X lsy` exp p R( m ym ß ž / The second term comes from the finiteness of the spatial volume of the system It can be shown that all terms with l/0 can be neglected if Rrb41, an inequality surely satisfied in the early universe The term with ls0 is the significant term, giving the main contribution especially if m m So we have ( ž / m m pr m ym qs W Ž b,m y p p br exp p R( m ym y1 The charge density in the ground state is given by 1 b Ž mym y1 bž mqm y1 0 3 ž / q s e y1 y e y1 p R If m m, we approximate q by 0 1 y1 y1 1 m m q0 f Ž b Ž mym y b Ž mqm s s, 3 3 p R bp R m ym Rb y qp

120 10 ( ) M TrucksrPhysics Letters B where we defined the thermogeometric parameter ysp R m ym Žsee wx 4 The charge density in the ground state grows rapidly if y yp If m m we can expand the function WŽ b,m Žsee w4,8 x : 3 3 m m m W Ž b,m s W Ž b,m y m ym qož m ym, p p pb ( which leads to m 3 m m 3 m qf W Ž b,m y ycoth yf W Ž b,m y ycoth y Ž 1 p p Rb p p Rb ( 3 The finite size effects We have to solve formula Ž 1 for the unknown value m This can only be done numerically It is well-known that for finite systems there is no well-defined critical temperature The critical temperature of the system in the thermodynamic limit can only serve as a useful reference point For the system in the thermodynamic limit, ie for Minkowski spacetime, the critical Ž inverse temperature is given by m 3 qsqž b c,m s W Ž b c,m p Using this reference point, we get as an equation for y and also for m, p Rq b W Ž b c,m ycoth ysy 1y, m WŽ b,m ž / which can be used to determine the charge density in the ground state q We have used this equation to analyze 0 the finite size effects This is shown in Fig 1 It can be seen clearly that there is no well-defined phase transition Fig 1 Charge density of the ground state q for a closed Robertson-Walker spacetime on a Cauchy surface, where Rs10 Ž thick line 0 compared with the system in the thermodynamic limit, ie Minkowski spacetime Ž thin line and Qs001

121 ( ) M TrucksrPhysics Letters B point Enlarging the radius R forces the curve for the Einstein universe towards the curve for Minkowski spacetime The critical temperature T of Minkowski spacetime, given by c ( T s 3< q< rm c Žsee Haber and Weldon wx 9, serves as a useful reference point Using the fact that RTs:Csconst Žproved in w1, x, we can define a critical radius R, instead of the critical temperature, given by c 3< Q< R c s, Ž 3 p mc where QsqVsqp R is the total charge This was already realized by Parker and Zhang 5 Below this radius, the gas is in a condensed state If the radius becomes larger than the critical radius, the condensate disappears The question arises, whether, due to the finite size effects, there is a considerable charge density of the ground state at the critical radius and whether the lifetime of the condensate is considerably longer than indicated by R c Ž For T s T c, we have ycoth y s 0 see Eq, ie y syp, and since y s p R( m y m, m s m y3r 4R This leads to ( Ž c 3 wx ( ( ( < q < 1 T m 3 m y3r 4R 3V 4 ( 0 c c s s s m y3r 4R c c c c ( p R c 4 4 s ( m y3rž 4R c s ( m y3rž 4R c f < < < q< < q< R p qy m< q< R 3p r4 mqr < < p 3m Q mp C p C We see that the charge density in the ground state mainly depends on C, the product of R and T, which is constant as long as the gas can be considered as relativistic Since this is a large value in the early universe, the finite size effects do not play a considerable role The critical radius defined in Eq Ž 3, below which the Bose gas is in a condensed state, makes it possible to think of the following scenario If there is an abundance of one charge Ž not necessarily the electric charge in the early universe, the Bose gas is in a condensed state The condensate disappears if the radius is larger than the critical radius and the Bose gas is in its usual state We are not aware of any consequences, a condensed Bose gas would have for the early evolution of the universe, but we think it is worth investigating such consequences We give an approximative value for the electric charge Žsee also wx 5 Results of redshift observations w10x show that the present scale factor R0 G10 7 cmf10 41 GeV y1 and results for an upper bound on the present w x y4 y3 y charge density 11 give qp -10 cm f10 GeV Using a value Cf10 and msm p, the pion mass, we find R f10 5 GeV y1 f10 y9 cm c We haven taken the maximal value of the electric charge density and the minimal value for the present scale parameter So the value may differ in both directions considerably This may be taken as a challenge to find more accurate bounds, especially for the present charge density In this context it may also be interesting to find bounds for the charge density of other charges Nevertheless even if the universe is neutral with respect to all charges in question, there may be regions with positive charge density as well as other regions with a negative one, so that condensation in special regions occurs

122 1 ( ) M TrucksrPhysics Letters B Conclusions We analyzed the possibility of Bose-Einstein condensation of a free relativistic charged Bose gas on closed Robertson-Walker spacetimes, described by an adiabatic KMS state Macroscopic occupation of the ground state sets in, if the temperature is in the region of the critical temperature of a charged relativistic Bose gas on Minkowski spacetime We have shown the effects due to the finiteness of the spatial volume of closed Robertson-Walker spacetimes but also that they are not relevant in the early universe Furthermore a critical radius can be defined, above which the condensate disappears This critical radius depends only on the total charge and the constant product of the temperature and the scale parameter in the Robertson-Walker metric We also pointed out that this critical radius may have some relevance in cosmology A next step could be to consider a self-interacting charged scalar field From a cosmological point of view it may also be interesting to consider the consequences of a condensed Bose gas for the evolution of the universe Furthermore there is a challenge to find more accurate bounds on the charge density of charged bosons References wx 1 M Trucks, M Keyl, Phys Lett B 399 Ž wx M Trucks, Commun Math Phys 197 Ž wx 3 R Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, University of Chicago Press, 1994 wx 4 S Singh, R Pathria, J Phys A: Math Gen 17 Ž wx 5 L Parker, Y Zhang, Phys Rev D 44 Ž wx 6 I Gradshteyn, I Ryzhik, Table of Integrals, Series and Products, Academic Press, 1980 wx 7 A Erdeleyi et al, Higher Transcendental Functions, vol, Mc-Graw Hill, 1953 wx 8 S Singh, P Pandita, Phys Rev A 8 Ž wx 9 H Haber, H Weldon, Phys Rev Lett 46 Ž w10x I Segal, Mon N Roy Astron Soc 4 Ž w11x A Dolgov, Y Zeldovich, Rev Mod Phys 53 Ž

123 31 December 1998 Physics Letters B Massive spinning particles and the geometry of null curves Armen Nersessian a,1, Eduardo Ramos b, a Joint Institute for Nuclear Research, BogolyuboÕ Laboratory of Theoretical Physics, Dubna, Moscow Region, , Russia b Dept de Fısica Teorica, C-XI, UniÕersidad Autonoma de Madrid, Ciudad UniÕersitaria de Cantoblanco, 8049 Madrid, Spain Received 4 September 1998 Editor: L Alvarez-Gaumé Abstract We study the simplest geometrical particle model associated with null paths in four-dimensional Minkowski space-time The action is given by the pseudo-arclength of the particle worldline We show that the reduced classical phase space of this system coincides with that of a massive spinning particle of spin ssa rm, where M is the particle mass, and a is the coupling constant in front of the action Consistency of the associated quantum theory requires the spin s to be an integer or half integer number, thus implying a quantization condition on the physical mass M of the particle Then, standard quantization techniques show that the corresponding Hilbert spaces are solution spaces of the standard relativistic massive wave equations Therefore this geometrical particle model provides us with an unified description of Dirac fermions Ž ss1r and massive higher spin fields q 1998 Elsevier Science BV All rights reserved 1 Introduction The search for a classical particle model that under quantization yields the Dirac equation and its higher spin generalizations has a long history By far the most popular approach is to supersymmetrize the standard relativistic particle model, whose action is given by the proper time of the particle s path It is not difficult to show that the system possessing N s s extended supersymmetry corresponds, after quantization, to a massive spinning particle of spin s Nevertheless, the search for geometrical particle models of purely bosonic character, which may lead to similar results, is interesting on its own The 1 nerses@thsun1jinrru ramos@deltaftuames reasons for this are quite simple, and they may be exemplified by Polyakov s results on Fermi-Bose transmutation in three dimensions In wx 1, Polyakov was able to show how, in the context of three-dimensional Chern Simons theory, the presence of a torsion term in the effective action for a Wilson loop was responsible for the appearance of Dirac fermions in an otherwise apparently bosonic theory thus opening the question of which is the natural counterpart, if any, in the four-dimensional case The purpose of this work is to show that there is a geometrical particle model, based purely on the geometry of null curves in four-dimensional Minkowski spacetime, that does the job, ie, that under quantization yields the wave equations corresponding to massive spinning particles of arbitrary integer or half-integer spin The action for such a model is given by r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

124 14 ( ) A Nersessian, E RamosrPhysics Letters B the pseudo-arclength, the simplest among all the geometrical invariants associated with null curves A few words are due now in order to motivate, from a more intuitive point of view, our approach to this problem How is that a particle model based on null curves, which should correspond a priori with massless particles, may be of relevance in the massive case? The main clue that this can be indeed the case comes from the Zitterbewegung associated with Dirac s equation It is a well-known result that a measurement of the instantaneous speed in the relativistic theory of Dirac must lead to a result of "c that this is in no contradiction with Lorentz invariance is explained, for example, in wx 6 hence making plausible that light-like paths may play a role in the corresponding particle system The second observation is a direct consequence of the one above: due to the null character of the path any sensible local action should depend on higher order derivatives It is clear that a model for spinning particles should contain extra degrees of freedom in order to accommodate the ones associated with spin Although some Lagrangian models have been introduced in the literature where the extra degrees of freedom are added by hand Žsee, eg, wx 4 and references therein, it would be highly desirable if they were provided by the geometry itself It is well known by now that higher derivative theories may provide the required phase space As we already commented, that is the case in three dimensions, and moreover it has also already proven to work in four dimensions, for massless particles, with a Lagrangian given by the first curvature of the particle s worldwx 5 Our choice has, then, been the obvious one, line this is to consider the simplest geometrical invariant associated with null, or light-like, paths Therefore, and with no further delay we will begin to introduce the necessary geometrical background to work out the model at hand Frenet equations for null curves in M 4 Let us start by constructing the Frenet equations associated with null curves in four-dimensional Minkowski space-time Our conventions for the signature of the metric are Ž q,y,y,y, ie, time-like vectors have a positive norm If we denote by x the embedding coordinate of the curve, the fact that the curve is null implies that x s0 Let us denote by eqsx the tangent vector to the curve From eq s0 it follows that there is a space-like vector e such that 1 e q ss e 1, Ž 1 with eqe1s0, and e1 sy1 Although, a priori, a term proportional to eq may appear in the right hand side of the above equation it is always possible to reabsorb it by a redefinition of e 1 It will now show convenient to chose a parametrization for which s s 1 This is equivalent to demanding that xxsy1, and this can always be achieved by a change of parametrization unless xx s 0, this last case being trivial will be excluded from our considerations This parametrization corresponds to choosing as our time parameter the pseudoarclength, which is defined as H t Ž X X 1r4 X t 0 s t s yx t x t dt We now pass to obtain Frenet equations in this parametrization From the definition of e1 it follows that ees0 and ee s1, hence q e 1 sae q qf y qcg, Ž 3 where the two new vectors g and f are chosen to y obey that g sy1, f s0, g e sg e sg f s y 1 q y ef 1 ys0, together with eqfys1 From the fact that Ž e,f,e,g span the tangent space at any point on q y 1 the curve the above equation trivially follows But notice that one can choose a basis Ž e,e,e,e with q y 1 1 eysf yqce q C e q, 4 e sg qceq Ž 5 that, while preserving the same orthogonality relationships, simplifies the above equation to yield e 1 sk 1 e q qe y, Ž 6 with k1 sayc r The remaining Frenet equations associated with the Frenet frame Ž e,e,e,e q y 1

125 ( ) A Nersessian, E RamosrPhysics Letters B follow from the orthogonality relations, and the whole set reads xse, Ž 7 q e q se 1, Ž 8 e 1 sk 1 e q qe y, Ž 9 e y sk 1 e 1 qk e, Ž 10 e sk e q Ž 11 Notice that in this case there are only two independent curvature functions k1 and k This is intuitively obvious due to the extra constraint on null curves coming from xxs0 3 The classical model We will consider the simplest geometrical action associated with null curves, this is, its pseudoarclength: SsaHds Ž 1 It is convenient, although not strictly necessary, to write the action in first order form One then can write H ž ( q q ž q( q q/ Ss dt a ye e qp xye ye e yleq/ Ž 13 Notice that the equation of motion for the Lagrange multipliers p and l implies that 1r4 ( ye e sž yxx, Ž 14 q q thus proving the equivalence of both actions It is now straightforward to develop the Hamiltonian formalism following Dirac s description for singular Lagrangians We will skip the details, which are completely standard After a little work one arrives to a symplectic form Vsdpndxqdp nde, Ž 15 q q endowed with the following set of primary constraints 1 q Ž q f sp q pe ya, 16 f seq, Ž 17 and secondary ones f3 spqe q, Ž 18 f4 speqya, Ž 19 f5 spp q Ž 0 The Hamiltonian is of the form q q Ž q HsÕ p qp e q pe ya, 1 where Õ should be regarded as an arbitrary function that is only fixed by choosing a particular parametrization If one chooses to parametrize the path by pseudo-arclength one gets that Õsy1ra There is only one first class constraint, the generator of reparametrizations The dimension of the reduced phase space is therefore 10 The equations of motion, in the pseudo-arclength parametrization, are given by xse, q 1 e q sy p q, Ž 3 a 1 p qs p eqyp, Ž 4 a ps0 Ž 5 Consistency of these equations of motion with Frenet equations imply that pqsyae 1, psa eyq p eqra, together with 1 k1 sy p Ž 6 a The solution of these equations of motion are particwx 3 ular examples of null helices The key observation, in order to arrive to a manageable expressions for the symplectic form on the constrained surface, is that it is possible to define a

126 16 ( ) A Nersessian, E RamosrPhysics Letters B standard free coordinate X out from the canonical variables of our model Notice that a Xsxy p Ž 7 p q has the property that its time derivative is given by a a Ẋsxy p q s p Ž 8 p p And from this it trivially follows that Xs0 It is then natural to introduce the new coordinate a Eqse qy p Ž 9 p so that the symplectic form V takes the simple form Vsdpnd Xqdp nde Ž 30 q on the constrained surface The constraints may be equally expressed in these variables, and they read pq qa s0, Ž 31 p E s0, Ž 3 q q a Eq q s0, 33 p q ppqs0, Ž 34 pe s0 Ž 35 q This constraint system suggests the introduction of the following complex coordinates ( zspqqi p E q 36 In terms of z the constraints simply read z s0, zzqa s0, and pzs0 Ž 37 Let us recall that the irreducible representations of the Poincare algebra are labeled by the values of the two Casimirs p and the square of the Pauli-Lubansky vector m 1 mnrs S s e pm, Ž 38 n rs with Mrs the generator of Lorentz transformations In our particular case the explicit expression for S m reads Sm semnrs p n pq r eq s, Ž 39 and one obtains that S sya 4, while there is no restriction on the possible values of p This implies that our phase space is not elementary, ie, the Poincare group does not act in a transitive way In order to obtain irreducible representations of the Poincare group under quantization physical states should always be decomposable into irreducible representations of the Poincare group we will study the elementary phase spaces defined through the extra constraint p sm, Ž 40 with M a free parameter with dimensions of mass A priori M could be positive, negative or zero, we will only consider the first case, because the other two correspond to unphysical irreducible representations of the Poincare group, ie, tachyonic and continuous spin representations, respectively Because of the null character of z it will show convenient to introduce a spinor parametrization of the reduced phase space Given an arbitrary complex four-vector y it can be rewritten in spinor coordinates as follows: AA y qy y qiy ' ž y yiy y yy / y s Ž 41 AA 1 m n mn so that detž y s g y y Because of the two-to-one local isomorphism between SLŽ,C and the identity component of the Lorentz group, one such Lorentz transformation on y is equivalently represented by the action of an SLŽ,C matrix acting on the undotted indices and its complex conjugate matrix on the dotted ones Raising and lowering of indices is mimicked in spinor language by contraction with the invariant antisym- Ž AB 01 metric tensors e s e AB, with e sq1, and analogous expressions for the dotted indices The null character of z now implies that ' AA A A z s aj h, Ž 4 and from the second of our constraints it must follow that j A ha s1, or equivalently that j and h form an spinor basis Notice, though, that this does not completely fix the spinors j and h because one still has the freedom to rescale both as follows 1 A A A A j a j and h h, Ž 43 a with a an arbitrary Ž nonzero real number This residual freedom will be fixed as follows Let us

127 ( ) A Nersessian, E RamosrPhysics Letters B consider the remaining constraint pz s 0 In spinor coordinates it reads p jhs0 Ž 44 AA A A or equivalently ( AA A A A p j sl p h slmh, Ž 45 with L an arbitrary Ž nonzero real number One then may fix completely the freedom to rescale the spinors by setting Ls"1r', and then one may finally write the only remaining constraint in the following form M AA p jaja s" Ž 46 ' A direct computation now shows that the symplectic form in spinor coordinates reads ' s A A Vsdpnd X"i ž paa dj ndj M A A qj dj nd paa / qcc, 47 where cc stands for complex conjugate of the previous term, and s is a dimensionless parameter defined as a ss Ž 48 M 4 The quantum theory One may now check Žsee wx that the reduced phase space can be identified with the coadjoint orbit of the Poincare group associated with a representation of mass M and spin s Quantization is then a standard exercise whose solution can be found, for example, in the excellent book on geometric quantiwx Nevertheless, because of zation by Woodhouse completeness, and the desire to simplify things a little to the less mathematically oriented reader, we will now sketch how the quantization procedure may be carried away in spinor coordinates In particular, we will show how Dirac equation may be obtained in the ss1r case One should start by choosing a polarization, or in simpler words one should demand the wave function to depend only on half of the canonical coordinates Our choice will be that the wave function will depend on p and j In order to quantize the system we should implement the remaining first class constraints as conditions on the wave function The first constraint p s M implies that the wave function has support on the mass hyperboloid In order to implement the second one Ž 46 notice that, roughly speaking, A paa j and j are conjugate variables in the induced symplectic form, therefore under quantization E A paa j Ž 49 A Ej up to some numerical factors A careful computation shows that 46 implies at the quantum level that Ec Ž p,j A j s sc Ž p,j, Ž 50 A Ej ie, the wave function is a homogeneous function of j of degree s Then single-valuedness of our wave function under j expž p i j requires s to be an integer or half-integer number Therefore one has that c Ž p,j sc Ž p j A 1 j A PPP j A s, Ž 51 A1A PPP A s where c Ž p A A PPP A has support on the mass hyper- 1 s boloid It is now straightforward to check that its Fourier transform 3r 1 A H 1A PPP A s ž / A 1 A PPP A s p s H M w x s c p =e yi px dt, Ž 5 with HM s the positive energy branch of the mass hyperboloid, and dt its invariant measure, yield positive frequency solutions of the massive wave equation Iw Ž x qm w Ž x s0 Ž 53 A1A PPP As A1A PPP A s Notice that, for example, for ss1r this equation has the same physical content that the Dirac equation This is so because the dotted component of the

128 18 ( ) A Nersessian, E RamosrPhysics Letters B four component Dirac spinor, which will be denoted by x A, may be defined trough the relation M A = AA w s x A Ž 54 ' This together with Ž 53 implies that M A = AA x s w A, Ž 55 ' thus being equivalent to the standard Dirac equation for Ž w,x Although this short discussion about the quantization of the system cannot make justice to this extensive topic, the interested reader may fill the gaps with the help of the current literature on the subject Acknowledgements ER would like to thank J Roca and JM Figueroa-O Farrill for many useful conversations on the subject The work of AN has been partially supported by grants INTAS-RFBR No95-089, IN- TAS and INTAS ext References wx 1 AM Polyakov, Mod Phys Lett A 3 Ž wx NMJ Woodhouse, Geometric Quantization, Oxford Science Publications, Oxford, 1991 wx 3 WB Bonnor, Tensor, NS 0 Ž wx 4 MV Atre, AP Balachandran, TR Govidarajan, Int J Mod Phys A Ž wx 5 MS Plyushchay, Mod Phys Lett A 4 Ž E Ramos, J Roca, Nucl Phys B 477 Ž wx 6 PAM Dirac, The principles of quantum mechanics, Oxford, Clarendon, 4th ed, 1958

129 31 December 1998 Physics Letters B Gaugino mass in the heterotic string with Scherk-Schwarz compactification Parthasarathi Majumdar a,1, Soumitra SenGupta b, a The Institute of Mathematical Sciences, Chennai , India b Physics Department, JadaÕpur UniÕersity, Calcutta , India Dedicated to:this paper is dedicated to the memory of Dr Reba Majumdar Received 7 September 1998; revised 1 November 1998 Editor: L Alvarez-Gaumé Abstract The generic observable sector gaugino mass in the weakly-coupled heterotic string compactified to four dimensions by the Scherk-Schwarz scheme Žtogether with hidden sector gaugino condensation inducing the super-higgs effect with a vanishing cosmological constant is shown to be nonzero at tree level, being of the order of the gravitino mass, modulo reasonable assumptions regarding the magnitude of the condensate and the Scherk-Schwarz mass parameters q 1998 Elsevier Science BV All rights reserved Despite possessing ingredients of phenomenological relevance, the weakly coupled E 6= E8 heterotic string theory suffers from the one key lacuna, viz, non-availability of a non-perturbative mechanism of breaking spacetime supersymmetry with a vanishingly small cosmological constant Conventionally, low energy approximations in four dimensions Žon toroidal Kaluza-Klein compactification wx 1 are derived assuming a condensation of the gauginos of the hidden sector E super-yang Mills theory wx 8, and compensating the resulting vacuum energy by ascrib- 1 partha@imscernetin soumitra@juphysernetin ing a finely-tuned expectation value to the internal components of the antisymmetric tensor field wx 3 The resulting supergravity theory undergoes super- Higgs effect with generation of a gravitino mass w4,5 x Serious malaise, however, plagues this procedure To begin with, the antisymmetric tensor field is quantized due to world sheet instanton effects wx 3 and hence can hardly compensate the continuously varying cosmological constant arising out of the gaugino condensate Augmenting it in accord with the Green-Schwarz anomaly cancellation alleviates this problem, but does not avoid the fine tuning requirement Also, the consequent low energy theory does not appear to have phenomenologically tenable softbreaking parameters In particular, a tree level mass for the generic observable gaugino appears to be precluded wx r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

130 130 ( ) P Majumdar, S SenGuptarPhysics Letters B To see this, recall that the gaugino mass is given, in Ds4, Ns1 supergravity, by the general formula wxž 7 in Planckian units Gr l y1 Ž m s² e G G k f : 1r l ab,k, Ž 1 ab where, for the E 6= E8 heterotic string, the supergravity functions G and fab of the moduli and matter scalar fields are given by wx ž / < W < Gs log, 3 SqS TqTy< C < Ž i f ss d, ab ab with S, T being E6 singlet moduli fields related to Ž i the dilaton and the axion, while the C C i are 7-plet Ž 7 -plet matter fields, and W is the superpotential, given by Wsl ijk CCC i j kqwž S Ž 3 Here, WŽ S is the effectiõe superpotential subsuming the effect of gaugino condensation in the hidden sector super-yang Mills theory, as discussed by w,5 x Its structure will be exhibited in the sequel Minimizing the resulting scalar potential and requiring that it vanishes at its minimum yields the vacuum condiwx tions 5 ² C : s0s² G : Ž 4 i S The gravitino mass arising as a consequence is given by m s² e Gr : 3r /0, so that, the ratio m1rrm3r vanishes as a result of the vacuum conditions Ž 4 Radiative corrections scarcely remedy the malady vis-a-vis phenomenological compulsions While the fundamental enigma continues, alternative scenarios Žwithin the weakly-coupled heterotic string theory have been explored in the recent past, with a variety of additional assumptions The low energy effective theory becomes the only arena of activity, since the belief that the mechanism underlying supersymmetry breaking be non-perturbative makes considerations based on classical Ž tree level heterotic string vacua Ž or perturbations around them quite irrelevant In one such non-perturbative low energy approach wx 8, the Kahler sector of the effective Ds4, Ns1 supergravity theory is enhanced to incorporate non-perturbative corrections parametrised in terms of certain undetermined functions of the string coupling The cosmological constant arising from the hidden sector gaugino condensation is compensated by fine tuning the boundary values of these functions at vanishing string coupling Extra assumptions regarding the form of these functions appear to yield acceptable soft operators at low energies While these assumptions are consistent with the boundary conditions, absence of a proper derivation of the structure of the functions does not fully resolve the underlying issues involved, as the authors of Ref wx 8 themselves observe In this paper, we follow a different path, first adapted for the heterotic string in Ref wx 9 This approach replaces standard toroidal compactification of the weak coupling heterotic string by the alternaw10x in tive route pioneered by Scherk and Schwarz the context of supergravity One begins with a standard dimensional reduction of the D s 10 N s 1 supergravity action of the massless fields of the heterotic string, on a five-torus Žsans the massive string modes ; the fifth dimension is now compactified a` la Scherk and Schwarz with a Z =Z truncation, to yield a D s 4, N s 1 supergravity theory coupled to E 6= E8 super-yang Mills theory We should remark at this point that the ansatze of wx 9 to obtain an effective low energy Ds4, Ns1 supergravity theory does not in general correspond to a tree level vacuum of the heterotic string in that the background does not satisfy the classical string equations of motion However, in anticipation of our passage to a non-perturbative vacuum in which the hidden sector gauginos condense, this is not a major drawback Such a condensate vacuum can scarcely be expected to arise out of quantum corrections to classical heterotic vacua The set of chiral superfields appearing in the effective D s 4, N s 1 supergravity now includes, over and above S, the matter 7-plets F i A, As1,,3, A X ia A A A 7-plets F, singlet moduli U,U and T,T, all of which are collectively designated as Y I A In terms of these fields, the effective Ds4, Ns1 supergravity is given by the functions < < ž W / YYYY Gslog, fss Ž 5

131 ( ) P Majumdar, S SenGuptarPhysics Letters B Here, Y 'Re S, 0 I 1 I < A< < A < A Y '1y Y q Ž Y, Ž 6 and i A j B k C SS iajbkc W'd F F F qd W, Ž 7 with ž / 1 I ž 1y' T q Ž Y / 1 I ž 1q' T q Ž Y 3 3 / m 1 I ž ' Ž 1 1 / i X I X ž iy' T y Y iq' /ž T3 i I 3 / m 1 SS 1 I D W' 1q' T q Ž Y 1 1 = = q l3 y 1q T q Y = y Ž Y qž l3 Ž 8 Thus, no additional quantum effects in the Kahler sector are assumed in this tree level approach, although the slightly more involved compactification scheme does manifest in a more complicated looking superpotential The scalar potential has several flat directions in this case; the simplest choice for the vacuum is given wx² I by 9 Y A: s0 Now if we require the vacuum energy to vanish for finite non-vanishing ² Y : 0, it turns out in this case that w11x one must choose the Scherk-Schwarz mass parameters m 1, m to obey m1qms0 It follows that the gravitino mass m1qm Gr m ' ² e : 3r s s0, Ž 9 1r ² : Y 0 implying that, although the Scherk-Schwarz mechanism does break supersymmetry in principle, in the present situation, the requirement of a vanishing cosmological constant is too stringent for this breakw11 x Using Eqs ŽŽŽ 1, 5 9, it is easy ing to survive to see that in this case, m1r s0, as of course is expected on grounds of consistency The vacuum of this effective theory has probably nothing to do with physical supersymmetry-breaking heterotic string vacua, because, as already mentioned, the dimensional reduction employed here does not correspond to compactification on backgrounds that satisfy the classical string equations of motion The prevalent prejudice, that for a genuine supersymmetry breaking vacuum structure, one needs in addition, a non-perturbative phenomenon, like hidden sector gaugino condensation, dilutes that requirement for reasons already explained One is therefore free to explore the fusion of the Scherk-Schwarz mechanism with hidden sector gaugino condensation w11 x Before turning to such a hybrid scenario, which we do next, we observe en passant that, were we to ignore the restrictions of a vanishing cosmological constant, we would get, for the vacuum choice made above, m 1r ;m 3r A fusion of the Scherk-Schwarz mechanism with hidden sector gaugino condensation in the weak coupling heterotic string has already been considered in w11 x Here we focus on the implications of such a marriage on the generic observable gaugino We follow ideas of Refs w,5x to stipulate that the primary outcome of gaugino condensation in the hidden sector, ² : 3 ll ;L, 10 Ž is to augment the superpotential 7 as follows W W'WqW Ž S Ž 11 Here, WS is the effective superpotential for the modulus S, possessing the generic structure wx 5 WŽ S ;AqBMc 3 expy3srb 0, Ž 1 with A, B constants, L, the scale of hidden sector supersymmetry breaking through hidden sector gaugino condensation, Mc the compactification scale and b 0, the coefficient of the one loop beta function of the hidden sector gauge group Ž for E, b s Notice that this generalizes straightforwardly to the case where the hidden sector gauge group has a product structure For reasons of simplicity, we confine our attention to a single unbroken gauge group We also emphasize that this is the only assumption in our paper that is non-perturbative in essence The Kahler sector continues to remain classical in contradistinction to Ref wx 8 The existence of flat direc-

132 13 ( ) P Majumdar, S SenGuptarPhysics Letters B tions in the moduli space would appear to ensure target space modular invariance, although, for illustrative purposes, a special point is chosen in this space in the sequel Perturbative corrections should change little in the scenario A discussion of non-perturbative corrections would entail strong extra assumptions which we choose to avoid in our approach Admittedly, the approach, like all other approaches, suffers from essential inadequacies The corresponding scalar potential once again has a minimum with vanishing cosmological constant w11 x; this is given by the vacuum conditions ² : ² I G s0s Y A: ² G : s ² G : s0 Ž 13 S ia A However, differentiating with respect to T A, TA X, we get ; ² : 1 W AX Ž m qm d ; AX W m qm q² W S : W A Ž m1qm d1 A ² A G : s s W m qm q WŽ S ² G : s s Ž 14 A X The d1 A, d1 AX are Kronecker deltas Notice that in this case, the requirement of a vanishing cosmological constant does not constrain the Scherk Schwarz mass parameters m 1, m, thereby allowing a non-vanishing gravitino mass m s² e Gr : 3r, at least for the particular point in moduli space given by Ž 13 The only assumption that is made is that ² S: is finite wx 5 Using once again Eqs Ž 1, ŽŽ 5 9, Ž 11 Ž 14, it is straightforward to obtain the ratio m1r WS Ž S s Ž m1qm m W ; 3 3r Ž m qm ² W Ž S : 1 S s Ž 15 3 ² : ² m qm q WŽ S G : 1 This establishes that the generic gaugino mass no longer vanishes at tree level, in contrast to what ensued in the case of the toroidally compactified heterotic string An order of magnitude estimate of the rhs of Eq Ž 15 can be obtained upon using the formula wx 5 g ² Y : y1 0 s, Ž 16 4p where g is the value of the gauge coupling constant of the hidden sector Ž super- Yang Mills theory around Planck scale The constant B in Ž 1 can be estimated by identifying the vacuum expectation value ² W Ž S: S with the hidden sector gaugino con- 1 densate, yielding B; y 3 b 0 We now make the reasonable choices A;yBL 3 expy3rb g, 0 m1qm ;L 3 expy3rb0g, Ž 17 recalling that all dimensional quantities are in Planckian units The parameters L and g may now be fine tuned to produce a ratio < m rm < 1r 3r ;1 A few disclaimers are in order First of all, the foregoing does not constitute anything beyond a feasibility study of a non-vanishing tree level gaugino mass of a desirable magnitude within the compound scenario incorporating both Scherk Schwarz compactification and hidden sector gaugino condensation Since our assumptions involve quantities arising in a strongly interacting gauge theory whose low energy dynamics is by no means well-understood, numerical estimates of quantities appearing above are not meant to be taken too seriously Secondly, as pointed out in Ref wx 9, the vacuum chosen is but a point in the moduli space of the theory, and it is not claimed that the estimate of the ratio above being of the order unity holds over the entire space Presently available technology, however, does not permit one to ascertain any preferred point in moduli space, thus reinforcing the vacuum ambiguities well-known in the case of toroidal compactification w,5 x Thirdly, a complete estimate of all possible soft operator coefficients Žthe scalar mass squared and the trilinear scalar coupling constant has to emerge consistently a problem to be soon resolved, hopefully A major problem in the standard analysis of supersymmetry breaking via gaugino condensation in the heterotic string is that the vacuum expectation value of the compact components of antisymmetric tensor field naturally acquires Planckian values, thereby forcing the gaugino condensate to also acquire values of OŽ M Pl However, this is not necessarily the case here w11 x The relevant vev is controlled by the Scherk-Schwarz mass parameters m 1,

133 ( ) P Majumdar, S SenGuptarPhysics Letters B which are not constrained to be Planckian We remark though that our entire analysis was performed within the context of the low energy effective supergravity theory which is what seems phenomenologically relevant The conclusions may change substantially once one ventures away from this realm into the string domain w1,13 x, in the form of extra restrictions that inhibit the mechanism advanced above This is presently under investigation Finally, the problem of a generic observable gaugw14,15x within the ino has recently received attention strongly coupled E = E heterotic string Ž 8 8 related to M-theory a` la Horava and Witten w16 x, using hidden sector gaugino condensation The general problem of supersymmetry breaking in M-theory using Scherk Schwarz compactification is also under purview w17 x It may be of some interest to consider this problem from the augmented standpoint adopted in this paper We hope to report on this elsewhere Acknowledgements SS acknowledges illuminating discussions with P Binetruy, K Dienes, E Dudas, P Fayet and N Sakai at the SUSYUNI meeting in Mumbai, India in December, 1997 PM thanks the Theory Group of Saha Institute of Nuclear Physics for hospitality during a visit during which this work was completed References wx 1 E Witten, Phys Lett B 155 Ž wx H Nilles, Phys Lett B 115 Ž ; S Ferrara, L Girardello, H Nilles, Phys Lett B 15 Ž ; M Dine, R Rohm, N Seiberg, E Witten, Phys Lett B 156 Ž wx 3 R Rohm, E Witten, Ann Phys 170 Ž wx 4 J Derendinger, L Ibanez, H Nilles, Nucl Phys B 67 Ž wx 5 J Breit, B Ovrut, G Segre, Phys Lett B 16 Ž ; Y Ahn, J Breit, Nucl Phys B 73 Ž wx 6 A Brignole, L Ibanez, C Munoz, Nucl Phys B 4 Ž wx 7 E Cremmer, S Ferrara, L Girardello, A van Proeyen, Nucl Phys B 1 Ž wx 8 P Binetruy et al, Phys Lett B 41 Ž wx 9 M Porrati, F Zwirner, Nucl Phys B 36 Ž w10x J Scherk, J Schwarz, Nucl Phys B 153 Ž w11x S SenGupta, Mod Phys Lett A 6 Ž w1x P Majumdar, S SenGupta, Class Quant Grav 7 Ž 1990 L67 w13x S SenGupta, P Majumdar, Mod Phys Lett A 7 Ž w14x H Nilles, M Olechowski, M Yamaguchi, Phys Lett B 415 Ž ; hep-thr w15x A Lukas, B Ovrut, D Waldram, hep-thr971008; hepthr w16x P Horava, E Witten, Nucl Phys B 460 Ž ; B 475 Ž w17x E Dudas, C Grojean, hep-thr ; E Dudas, hepthr ; L Antoniadis, M Quiros, Phys Lett B 39 Ž , hep-thr , and references therein; Z Lalak, S Thomas, hep-thr97073; K Choi, H Kim, C Munoz, hep-thr

134 31 December 1998 Physics Letters B Generating functionals of correlation functions of p-form currents in AdSrCFT correspondence WS l Yi 1 Department of Physics, Chungbuk National UniÕersity, Cheongju, Chungbuk , South Korea Received October 1998 Editor: PV Landshoff Abstract The generating functional of correlation functions of the currents corresponding to a general massless p-form potential is calculated in the AdSrCFT correspondence of Maldacena For this we construct the boundary-to-bulk Green s functions of p-form potentials The proportional constant of the current-current correlation function, which is related to the central charge dy p G Ž dy p of the operator product expansion, is shown to be cs The result agrees with the known cases such as ps1 or q 1998 Elsevier Science BV All rights reserved d d p G Ž y p 1 Introduction Ever since the introduction of Maldacena s proposal wx 1 that the classical supergravity action on Anti-de Sitter space Ž AdS can be used to determine the current-current correlation function of the dual conformal field theory Ž CFT on the boundary of AdS w,3 x, various successful applications are reported in the direction of confirming it w4 6 x Two point correlators for the case of scalar fields w,3,7 9 x, three- w10x and four-point correlators w1,13x are investigated For massless vectors it is rather simple when one benefits from the gauge transformations to eliminate the A component of the AdS vector potential wx 0 3 But for massive vector fields there are no such transformations In fact the determination of the A0 component of a vector potential is an essential element of holographic projection of massive vector fields w14,15 x Spinors w16,17,14 x, gravitinos w18,19 x, and gravitons w11,1x are also investigated The next natural step is the consideration of the p-form potential Since supergravity actions and brane theories contain various p-form potentials, it is rather one of the imperatives for the full understanding of AdSrCFT correspondence In this paper the generating functionals of current-current correlation functions of general p-form potentials are constructed using the AdSrCFT correspondence We follow Witten s intuitive methodology wx 3 to construct 1 address: wslyi@cbuccchungbukackr r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

135 ( ) WS l YirPhysics Letters B the boundary-to-bulk Green s function But rather than delving into the p-form potential we begin with the -form case The gauge transformation of -form potential is used to eliminate unnecessary components of the potential The boundary-to-bulk Green s function which is in fact a -form is constructed using the differential form language The classical action which is served as the generating functional of current-current correlation functions is determined from this -form Green s function This is presented in the next section The generalization to the general p-form case is rather straightforward This is done in Section 3 The appendix is devoted to prove that the Witten s method causes no Ward identity problem for the massless p-form cases which we consider General consideration of boundary-to-bulk Green s function According to the Maldacena s proposal the supergravity partition function Z wf x AdS a of general fields f dq 1 a ² d on AdS can be used as the generating functional exphd xw Ž x O Ž x: dq1 i i CFT of the operator product expansions of sources O Ž x i on the boundary of AdS The relation between fa and wi is that when one goes to the boundary of AdS, fa reduces to w i When the classical approximation is employed to the supergravity partition function, Maldacena s proposal implies that the classical action Iwf x a of the bulk field f a, when written in terms of boundary field w i, serves as the generating functional of the OPE This means that it is essential for the calculation of OPE of the sources O Ž x i to determined the boundary-to-bulk Green s function For our purpose it is sufficient to employ the Euclidean AdS which is characterized by the Lobachevsky 1qd 1qd space R s x, x gr < x )04 with the Poincare metric of the following form, q x 0 ds s Ž dx q Ž dx Ž 1 The x ` point and the x 0 region consist the boundary of AdS To simplify the notation we sometimes 0 0 confuse x 0 with x unless it is stated explicitly Roman characters such as i and j are used to denote the 0 indices 1,,d, and Greek characters such as m and n are preserved to denote whole indices 0,1,,d It is well known that to have correct correlation functions which preserve the Ward identity one should be careful to have all fields to approach the boundary of AdS uniformly For this an imaginative x0 se boundary is first used to compute the classical action, and then one take the e 0 limit In w15x this procedure in coordinate-space is analysed Using this algorithm one may prove that for massless p-form field we do not need to worry about the e-boundary procedure For more details, see appendix 3 The generating function of OPE of -form currents Consider a -form potential A in AdS The free action is given by 1 ) H Is Fn F, AdSdq1 where Fsd A is the field strength 3-form The classical equation of motion of A obtained from this action is d ) d As0 Ž 3 To calculate the current-current correlation function in boundary-to-bulk -form Green s function, AdSrCFT correspondence we need to know the 1 m n K x bk, x bd s Kmn x bk, xbd dxbkndx bk 4

136 136 ( ) WS l YirPhysics Letters B Here x bk is a point in AdS, and x bd is a point on the boundary of it K satisfies the same equation which A satisfies, and its component becomes the boundary-space delta function as x bk approaches x bd Using the gauge transformation, A AqdL, Ž 5 of a -form potential we may assume that all K 0 i, is1,,d, vanish For simplicity we take as x bd the boundary point at x0 ` The boundary of AdS and the Poincare metric are both invariant under the following translation, x xyx X, Ž 6 X where x is a constant vector This allows us to assume that x is Ž x,0,,0 wx bk 0 3 The general form of K under these restrictions is K`sK`Ž x0 dx i ndx j, Ž 7 where i-j The infinity symbol in this equation denotes the fact the the boundary point of K is chosen to be x0 ` The equation which is satisfied by K ` is d ) dk` s0 Ž 8 To solve this we compute dk `, dk` se 0 K`Ž x0 dx 0 dx i dx j, Ž 9 and use the relation, $ $ $ ) 0 i j iqjq1 yž dy5 0 1 i j d Ž dx dx dx sž y x0 dx dx PPP dx PPP dx PPP dx, $ where dx 0, etc, means that dx 0 should be omitted in the wedge product The result is Ž 10 y E 0Ž x0 E 0K`Ž x0 s0, Ž 11 whose solution is K` sa x0 dy4 dx i dx j Ž 1 Here a is an unknown constant For the case d)4, it is clear that as x goes to 0, K Ž x vanishes When x goes to `, K Ž x 0 ` 0 0 ` 0 diverges This means that K` looks like a boundary-to-bulk Green s function Before proving this fact explicitly we benefit from the following gauge transformation, K K qd b x dy4 x i dx j yg x dy4 x j dx i Ž 13 ` ` 0 0 The transformed K ` is given by K` s Ž aqbqg x0 dy4 dx i dx j q Ž dy4 b x0 dy5 x i dx 0 dx j y Ž dy4 g x0 dy5 x j dx 0 dx i Ž 14 By choosing the coefficients which satisfy 1 aqbqgs1, bsgsy, Ž 15 dy4 K reduces to ` K` sx0 dy4 dx i dx j yx0 dy5 x i dx 0 dx j qx0 dy5 x j dx 0 dx i Ž 16 It is still possible to rescale K ` by a constant c without losing the desired property

137 ( ) WS l YirPhysics Letters B To prove that K` in fact generates the correct boundary-to-bulk Green s function we use the following isometry of Ž 1 which is discussed in detail in w10 x, x m m x 17 x0 qx It is important to note that under this isometry the boundary point corresponding to x0 ` transforms to the boundary point Ž x, x 0 This isometry transforms K to 0 ` x0 dy4 x0 dy5 x i x0 dy5 x j i j 0 j 0 i K 0 s dx dx y dx dx q dx dx, Ž 18 dy dy dy x qx x qx x qx Ž 0 Ž 0 Ž 0 where the subscript 0 in K means that the boundary point of K corresponds to Ž x, x To get the Green s function corresponding to a general boundary point Ž0, x X we use the translational symmetry Ž 6 This gives the final Green s function, x0 dy4 x0 dy5 x i x0 dy5 x j i j 0 j 0 i cks dx dx y dx dx q dx dx, Ž 19 dy dy dy < X < < X < < X < x q xyx x q xyx x q xyx Ž 0 Ž 0 Ž 0 where c is a normalization constant On the other hand, it is not difficult to show that x0 dyn Ž d X dyn d,n X lim sc d Ž xyx, Ž 0 x0 0 Ž x q< xyx < 0 where, d d G ž yn / cd,n sp Ž 1 G Ž dyn This means that by choosing the constant c of Ž 19 which is equal to c, K of Ž 19 d, is in fact the desired boundary-to-bulk Green s function 1 i j Now let A s A x dx dx be a -form potential on the boundary of AdS Using Ž 19 ij we have the following -form bulk potential, X AijŽ x 1 dy4 d X i j d, Ž 0 0 H X dy x q< xyx < 0 c A x, x s x d x dx dx i X i X Ž x yx AijŽ x 1 dy5 d X 0 j 0 H X dy x q< xyx < 0 y x d x dx dx, where we used the anti-symmetric property of A ij The field strength 3-form F is given by A x X ij 1 dy5 d X 0 i j cd, FŽ x 0, x s Ž dy x0 Hd x dx dx dx X dy x q< xyx < Ž 0 A x X ij 1 X yž dy x0 Hd x dx dx dx X dy1 x q< xyx < dy3 d 0 i j Ž 0 Ž x j yx X j Ž x k yx X k A Ž x X ij 1 dy5 d X y4ž dy x0 Hd x dx 0 dx i dx k q Ž 3 dy1 X x q< xyx < Ž 0 where the abbreviated terms are those which do not contain dx 0

138 138 ( ) WS l YirPhysics Letters B By partial integration the bulk action becomes 1 1 ) ) I s H Fn Fsy lim H An F Ž 4 e 0 AdSbq1 x0se The following relation dx i dx j n ) Ž dx 0 dx l dx m sx yž 0 dy5 dlm ij dx 1 PPP dx d, dlm ij sdl i dm j ydm i dl j, Ž 5 is useful for the computation of the action The final result is given by A Ž x A Ž x X A Ž x A Ž x X Ž x yx X Ž x yx X ij ij ij ik j j k k d d X d d X Iw AxsycÝHd xd x q cý d xd x, Ž 6 X Ž dy H X Ždy1 < xyx < < xyx < i-j ijk where dy G Ž dy cs Ž 7 d d G ž y p / It agrees with the result of w0 x 4 The generating function of OPE of p-form currents A general p-form potential A on the boundary of AdS can be lifted to a bulk one A by using the boundary-to-bulk Green s function K, the determination of which is our next step Similarly to the -form case we assume that K sk x dx i 1 PPP dx i p ` ` Ž 0, Ž 8 where i - PPP -i K satisfies the free-field equation d ) dk s0 which reduces to 1 p ` ` E x E K Ž x s0 Ž 9 yž dy py ` 0 dy p The solution of this equation which vanishes at x0 0 is K` x0 sc0x 0, where c0 is a constant In fact it is true only for d) p But for df p one may use the dual p ) -form as p ) sdypy1 Using this p ) we have the relation dyg p ), or the relation d) p ) is satisfied To simplify the final calculation we apply the following gauge transformation, ž / dy p i1 ip dy p i1 i ip i1 i i3 i K c x dx PPP dx qd x c x dx PPP dx yc dx Px Pdx PPP dx p ` PPP pq1 i 1 i py1 i p p qy c dx PPP dx Px 30 The transformed K` is given by $ dy p i1 ip dy py1 i1 0 i1 i i K s c qc q qc x dx PPP dx qc dy p x x dx dx dx PPP dx p ` Ž 0 1 p $ dy py1 i 0 i1 i i yc dy p x x dx dx dx PPP dx p 0 q $ pq1 dy py1 i p 0 i 1 i py1 i qy cpž dy p x0 x dx dx PPP dx dx p, Ž 31 $ where dx i k, ks1,,p, means that dx i k should be omitted in the wedge product When one choose constants such as y1 dyp c1scs PPP scp s, c0 s, Ž 3 dy p dy p

139 K ` simplifies to p $ dy py1 m i i i i ` 0 ms0 ( ) WS l YirPhysics Letters B Ý K sx y x m dx 0 PPP dx m PPP dx p 33 i 0 0 Here we use the notation x s x Under the isomorphism 17 K transforms to ` p im $ dy p x x dy py1 m i i i 0 0 m p i1 ip d, p 0 0 Ý dyp dyp x0qx x0qx c K s x y dx PPP dx PPP dx s dx PPP dx ms0 p i x k $ dy py1 k 0 i1 ik ip 0 dyp x0 qx qx Ý Ž y dx dx PPP dx PPP dx, Ž 34 ks1 where the normalization constant cd, p is multiplied to get the correct K The final step for the construction of the boundary-to-bulk Green s function is to use the translation Ž 6 Consider a p-form potential on the boundary of AdS, 1 i1 i As A dx PPP dx p Ý i i Ž 35 1 p p! The bulk potential A which is lifted from this is A Ž x X p i PPP i 1 X 1 p k c A x, x s x d x dx 1 PPP dx p Ý H q Ý Ý Ž y x X dyp i i x q< xyx < p! i i ks1 dy p d i i dy py1 d, p p 1 p = H X X Ž x i yxi Ai PPP i Ž x 1 $ d X k k 1 p 0 i1 ik ip X dyp x q< xyx < p! 0 d x dx dx PPP dx PPP dx Ž 36 We confuse notation x with x i k i for convenience Using the anti-symmetric property of Ai PPP i it can be k 1 p further simplified to A Ž x X i PPP i 1 dy p d X 1 p i1 ip dy py1 cd, p AŽ x 0, x s Ý x0 Hd x dx PPP dx yp Ý x dyp 0 X i i x q< xyx < p! i i 0 1 p 1 p = H Ž x yx X A Ž x X i i i PPP i 1 d X p 0 i i d x dx dx PPP dx p Ž 37 dyp X x q< xyx < p! Ž 0 The field strength Ž pq1 -form Fsd A is given by A Ž x X i PPP i 1 X 1 p c Fs dyp x d x dx dx 1 PPP dx p Ý H yž dyp x X dyp i i x q< xyx < p! dy py1 d 0 i i dy pq1 d, p 0 0 = Ý i i H 1 p Ž 0 A Ž x X i PPP i 1 d X 1 p 0 i1 ip dy py1 d d x dx dx PPP dx y pž dyp x0 Ý Hd x X dypq1 x q< xyx < p! iji i 0 1 p p Ž = x yxx Ž x yx X A Ž x X i i j j ii PPP i 1 p 0 j i i X dx dx dx PPP dx p qf, Ž 38 dypq1 X x q< xyx < p! Ž 0 where F X are the terms which do not have dx 0 X

140 140 ( ) WS l YirPhysics Letters B The action, using the equation of motion, is given by 1 1 ) ) I s H Fn Fsy lim H An F Ž 39 0 e 0 AdSbq1 x se To compute this we make use of dx i 1 PPP dx i p n ) dx 0 dx j 1 PPP dx j p sx yž dy py1 d i 1 PPP i p dx 1 PPP dx d, Ž 40 0 j1 PPP j p where d i 1 PPP i p j 1 PPP j p denotes the sign of the corresponding permutation Using Ž 38 and Ž 39 this can be simplified to give A Ž x A Ž x X i PPP i i PPP i d d X 1 p 1 p Isyc Ý Hd xd x X Ž dyp < xyx < i - -i 1 p Ž x yx X Ž x yx X A Ž x A Ž x X i i j j ii PPP i ji PPP i d d X p p qc Ý Ý Hd xd x, Ž 41 X Ž dypq1 < xyx < ij i - -i p where dyp G Ž dyp cs Ž 4 d d G ž yp p / Acknowledgements The author is grateful to G Arutyunov for introducing the related works w11,0 x This work is supported in part by NON DIRECTED RESEARCH FUND, Korea Research Foundation, 1997, and in part by the Basic Science Research Institute Program of the Ministry of Education, Korea, BSRI Appendix A In this appendix we follow the notation of w15 x Suppose that fž x, x 0 is a field in AdS which has the following asymptotic behaviour f x0 yl as x0 0 Ž A1 Then one defines a holographically projected field f Ž x h by fhž x s lim e l fž e, x Ž A e 0 This is exactly the definition which Witten employed in his paper wx 3 This boundary field is related to the bulk field fž x, x by 0 H f Ž x, x s d d x X KŽ x, xyx X f Ž x X Ž A3 0 0 h Here KŽx, x y x X is a boundary-to-bulk Green s function which obeys the same equation which fž x, x 0 0 satisfies, and has following boundary condition lim KŽ x, xyx X sd Ž d Ž xyx X Ž A4 x 0 0 0

141 ( ) WS l YirPhysics Letters B This Green s function is clearly different from the Green s function G Ž x, x y x X e 0 defined to relate the original AdS field fž x, x and f Ž x, where 0 e f x sf e, x A5 e Even though Witten s procedure is quite practical it may give wrong results when one considers massive cases In fact one should use Ge rather than K in order to preserve the Ward identity But for the massless case there is no such problem It comes from the fact that for massless p-form field, ls0 In this case the AdS field at the boundary fž 0, x is exactly equal to f Ž x, and the two Green s functions G Žx, x y x X h e 0 and KŽ x qe, xyx X 0 are equal It thus causes no problem at all Now we prove that for p-form, ls0 Suppose f is a p-form with the equation of motion d ) dfs0 The asymptotic form of this equation, which is exactly the same as Ž 9 which the Green s function satisfies, is given by E x E f Ž x s 0, Ž A6 yž dy py ` 0 where f Ž x ` 0 is the asymptotic form of the independent component of f The general solution of this equation is f`ž x0 sc1qcx0 dy p Ž A7 It is already proven that we may assume that d) p without loss of any generality This shows that for p-form we do not need to introduce a rescaling factor such as x l In other words, ls0 0 References wx 1 J Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, hep-thr wx SS Gubser, IR Klebanov, AM Polyakov, Gauge theory correlators from noncritical string theory, hep-thr wx 3 E Witten, Anti de Sitter Space And Holography, hep-thr wx 4 S-J Rey, J Yee, Macroscopic Strings As Heavy Quarks Of Large N Gauge Theory And Anti-de Sitter Supergravity, hep-thr980300; S-J Rey, S Theisen, J Yee, Wilson-Polyakov Loop at Finite Temperature in Large N Gauge Theory and Anti-de Sitter Supergravity, hep-thr ; A Brandhuber, N Itzaki, J Sonnenschein, S Yankielowicz, Wilson Loops in the Large N Limit at Finite Temperature, hep-thr wx 5 GT Horowitz, R Myers, The AdSrCFT Correspondence and a New Positive Energy Conjecture for General Relativity, hepthr wx 6 E Teo, Black hle absorption cross-sections and the anti-de Sitter-conformal field theory correspondence, thep-thr ; M Banados, A Gomberoff, C Martinez, Anti-de Sitter space and black holes, hep-thr , and references there in wx 7 W Muck, KS Viswanathan, Conformal Field Theory Correlators from Classical Scalar Field Theory on AdS dq 1 wx 8 IYa Aref eva, IV Volovich, On Large N Conformal Theories, Field Theories in Anti-De Sitter Space and Singletons, hep-thr wx 9 IYa Aref eva, IV Volovich, On the Breaking of Conformal Symmetry in the AdSrCFT Correspondence, hep-thr w10x DZ Freedman, SD Mathur, A Matusis, L Rastelli Correlation functions in the CFTdr AdSdq1 correspondence, hep-thr w11x GE Arutyunov, SA Frolov, On the origin of the supergravity boundary terms in the AdSrCFT correspondence, hep-thr w1x H Liu, AA Tseytlin, On Four Point functions in the CFTrAdS Correspondence, hep-thr w13x DZ Freedman, SD Mathur, A Matusis, L Rastelli Comments on 4-point functions in the CFTdr AdSdq1 correspondence, hep-thr w14x W Muck, KS Viswanathan, Conformal Field Theory Correlators from Classical Field Theory on Anti-de Sitter Space II Vector and Spinor Fields, hep-thr w15x WS l Yi, Coordinate-space holographic projection of fields and an application to massive vector fields, hep-thr w16x R Leight, M Rozali, The large N limit of the Ž,0 superconformal field theory, hep-thr w17x M Henningson, K Sfetsos, Spinors and the AdSrCFT correspondence, hep-thr w18x S Corley, The Massless Gravitino and the AdSrCFT correspondence, hep-thr w19x A Volovich, Rarita-Schwinger Field in the AdSrCFT correspondence, hep-thr w0x GE Arutyunov, SA Frolov, Antisymmetric tensor field on AdS, hep-thr , to appear in Phys Lett B 5

142 31 December 1998 Physics Letters B Three brane action and the correspondence between N s 4 Yang Mills theory and anti-de Sitter space Sumit R Das a, Sandip P Trivedi b a Tata Institute For Fundamental Research Homi Bhabha Road, Mumbai , India b Fermi National Accelerator Laboratory, PO Box 500, BataÕia, IL 60510, USA Received 17 August 1998 Editor: M Cvetič Abstract Recently, a relation between Ns4 Super Yang Mills in 3q1 dimensions and supergravity in an AdS5 background has been proposed In this paper we explore the idea that the correspondence between operators in the Yang Mills theory and modes of the supergravity theory can be obtained by using the D3 brane action Specifically, we consider two form gauge fields for this purpose The supergravity analysis predicts that the operator which corresponds to this mode has dimension six We show that this is indeed the leading operator in the three brane Dirac-Born-Infeld and Wess-Zumino action which couples to this mode It is important in the analysis that the brane action is expanded around the anti-de Sitter background Also, the Wess-Zumino term plays a crucial role in cancelling a lower dimension operator which appears in the the Dirac-Born-Infeld action q 1998 Elsevier Science BV All rights reserved 1 Introduction and summary One of the interesting outcomes of recent progress in string theory has been the relationship between gauge theories and gravity, particularly in the context of black holes In particular, classical scattering of various fields from non-dilatonic black holes like extremal three branes are reproduced by correlators of the gauge theories living on the brane worldvolwx 1 Noting the fact that the near-horizon geome- ume try of such black holes is in fact a five dimensional anti-de Sitter Ž AdS space, Maldacena has conjectured that the large-n limit of a conformally invariant d- dimensional Yang Mills theory in fact contains supergravity Ž and IIB superstring theory in Ž dq1 dimensional AdS space wx This has led to some progress in understanding the strong coupling behavior of these gauge theories in the large-n limit The conjecture was further investigated in wx 3 and wx 4 where a concrete prescription was given for relating observables in the supergravity and the Yang Mills theories The idea in w3,4x was to consider AdS space together with a boundary The dependence of the supergravity action on the boundary values of fields then yields the required generating functional from which Greens functions can be calculated In this paper we will be concerned with 3 q 1 dimensional SUŽ N Yang Mills theories with Ns4 supersymmetry The proposal of wx relates this theory to ten- dimensional Type IIB supergravity compactified on AdS 5= S 5 Using the recipe outlined above, several two point correlation functions were r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

143 ( ) SR Das, SP TriÕedirPhysics Letters B calculated in w3,4x and recently some three point functions have been computed in wx 5 For some special operators, which are either chiral or protected from renormalization effects for other reasons agreement was found between the anomalous dimensions as calculated in the Yang mills and the supergravity theories For other operators this led to a determination of the anomalous dimensions in the large-n limit A large class of such operators are marginal or relevant in the Yang-Mills theory It is important to examine the relation between higher dimensional operators and supergravity modes as well Some higher dimensional operators have been studied in wx 6 These may have two different roles in the supergravity context For some supergravity modes, like the fixed scalar, these are responsible for the leading order absorption by the black 3-brane and related to the correlator in the AdS space itself For other modes, like the dilaton, they are responsible for corrections to the leading result and probe the 3-brane metric beyond the AdS throat wx 6 In this note we continue the study of this set of ideas by focussing on another supergravity mode: the two index NS NS antisymmetric tensor field with a polarization parallel to some directions of the brane 1 The propagation of this mode in AdS space was investigated in wx 7, where, after incorporating the mixing with the R R two form field it s mass was determined More recently in wx 8 the cros-section for scattering this field off a extremal black hole was calculated These studies show that the scattering of the two form field Žwith a polarization parallel to the brane should be suppressed for small energies Correspondingly the operator in the Yang Mills theory to which it couples should have a total dimension of six Determination of this operator will be the goal of this paper We should mention that this operator has been identified in w9,10 x, from considerations of superconformal symmetry 1 Another mode is obtained by interchanging the roles of the NS NS and R R two form fields, our results apply to this case as well We thank the authors of w9,10x for correspondence in this regard So far, in the literature, the principle behind a precise correspondence between various modes in the supergravity theory and operators in the Yang Mills theory has not been spelt out In this note we explore the idea that the operators of the Yang Mills theory can be obtained by expanding an action consisting of the Ž non-abelian Dirac-Born-Infeld Ž DBI and Wess-Zumino Ž WZ terms We will see that, for the antisymmetric tensor, the correct Yang Mills operator can be identified in this way, but only if the action is expanded about AdS spacetime and the accompanying five form field strength background In many ways this is the natural expectation The supergravity mode being considered is a perturbation about AdS space Thus one expects that to consistently couple it, the DBI action should also be expanded about the AdS background Related points have been recently made in w11 x The conformal symmetry of the three brane action has been studied in w1 x It should be emphasized that here the DBI plus WZ action will be used to identify the correct operators in the AdS-Yang Mills correspondence Whether higher order corrections to the correlator in the full 3-brane geometry require the gauge field dynamics to be governed by the DBI-WZ action remains to be seen Some evidence in favor of this has been prewx 6 sented in One noteworthy feature about our analysis is that the WZ term plays an important role in it The leading operator obtained from the DBI action has engineering dimension of four But the coupling to this operator is cancelled by a contribution coming from the Wess-Zumino term Thus the leading operator obtained from the whole action, which couples to this supergravity mode, has dimension six at tree level As was mentioned above, the supergravity analysis shows that this must in fact be its total dimension We learn in this way that the operator studied here does not acquire any anomalous dimension in the large-n limit This is also true of the dimension eight operator studied in wx 6 One more point about our discussion below needs to be mentioned In expanding the action we will need to decide where the DBI-WZ action lives in AdS space As was mentioned above, the basic idea in the discussion of w3,4x is that the boundary values for the supergravity modes act as sources for the

144 144 ( ) SR Das, SP TriÕedirPhysics Letters B Yang Mills field operators This suggests the DBI- WZ action should be expanded around the boundary of AdS space In fact, as we will see here, this yields a consistent answer We do not, however, take this to mean that there are D3-branes physically located at the boundary Clearly, this analysis needs to be extended for other modes as well For one class of Yang Mills operators the coupling to the supergravity modes has been written down from superconformal symmetry considerations in wx 9 It will be interesting to see if these and the couplings to all the other supergravity modes can be obtained by expanding the DBI-WZ action We hope to report more fully on these questions in the future Let us mention one final point Our discussion in this note suggests that in the conformally non-invariant cases, the supergravity theory should correspond to a Yang Mills theory not in flat space-time but in the supergravity background instead The supergravity analysis A system of N parallel D3-branes is described by an extremal black hole geometry with a metric: y1r i 1r a ds sh Ž dx qh Ž dx, 1 with R 4 Hs1q r 4 Here x i,is0, 3, refer to the four coordinates parallel to the brane world volume, x a to the coordi- nates transverse to the brane, r sž x a is the transverse coordinate distance away from the branes, and 4 R s 4p g Ž X sn a, where gs is the string coupling, related to the Yang-Mills coupling g YM by gss g YM In the near horizon region, r<r, this metric reduces to that of AdS 5 =S 5 : r R i a ds s Ž dx q Ž dx 3 R r we see that the S5 has a radius R and the geometry is smoothly varying if g YM N is large The dilaton is constant in this background, while the self dual five form field strength in the near horizon region is given by: r 3 F013r s 4 4 R Following w7,8x we now consider the propagation of the NS NS and R R two form fields in the AdS background The NS NS and R R gauge potentials will be denoted by Bmn and Amn and the correspond- ing fields strengths by Hmnr and Fmnr respectively The equations governing the propagation of these modes are w13 x: m kts = Hmnrs 3 Fnrkts F m kts mnr Ž 3 nrkts = F sy F H 5 We see that in the presence of a non-zero five-form field strength the Bmn and Amn fields mix with each other Here we only consider an s-wave mode with the NS NS two form being polarised along the brane directions For simplicity, the only component of B mn that is non-zero will be chosen to be B Eq 5 1 can now be used to solve for Fmnr in terms of B1 and gives 3 : F0 r 3 sy4 Fr 301 g 11 g B 1, 6 with all the other components of the three form RR field strength being zero In the subsequent discussion we consider a perturbation with energy v, with yi v t a resulting time dependence e From Eqs 5 and 6, we then get an equation for B : ' rr ErŽ y gg g g ErB1 yw g B 11 1 ' y g g g sy16f013r g 00 g rr g 33 g 11 g B 1 7 Substituting for F, from Eq 4 now gives: 013r ' rr ErŽ y gg g g ErB1 yw g B 11 1 ' y g g g 16 s B 8 R 1 Thus we see that the mode corresponds to a field 4 with mass m s In the correspondence between R 3 In our conventions F se A qe A qe A mnr m nr n rm r mn 1

145 ( ) SR Das, SP TriÕedirPhysics Letters B Supergravity and Yang Mills theory the region, r 4 R is particularly relevant In this region the second term on the left hand side of Eq 8 can be neglected, giving rise to two solutions with, B 1 ; r " 4 Of these the case, B 1 ;cr 4, 9 can be extended to a non-singular solution in the small r region, it will be the relevant one for the subsequent discussion We can also solve for the R R two form A, corresponding to Eq 9 From Eq 6, in the r4r region it is given by: A03 syb1 10 with all other components being zero Now that our analysis of the supergravity mode is complete we can use the prescription of w3,4x to calculate the anomalous dimension of the Yang Mills operator that corresponds to it The general formula relating the anomalous dimension, D, to the mass for a p form is: Ž DypŽ Dqpy4 sm, 11 where the mass is measured in units of R In this case, m s16, Eq 8, setting in addition ps, gives: Ds6 1 Thus the operator in the Yang Mills theory corresponding to the mode discussed here has a total dimension of 6 in the large-n limit In the next section we turn to determining this operator Before doing so though, let us pause to make contact with the general analysis in wx 7 Here we have focussed on an S-wave mode of the two index gauge field In wx 7 the propogation of all the higher Kaluza Klein harmonics arising from the two form field Ž and in fact all the other supergravity modes were analyzed as well The propogation equation for the two form, Eq 56 in wx 7, was elegantly factorised giving rise to two families, Eq 6, and Eq 64 of wxž 7 also shown in Fig 3 as the two a mn modes The S-wave mode discussed in this paper is the ks0 member of the second family, Eq 64 The first family, Eq 63, only involves ls1 and higher angular momentum modes In fact the ls1 mode of this family Žwhich transforms like a 6 of SUŽ 4 and is the mode shown in Fig 3 with a circle is discussed in wx 9 where it was identified with a dimension three operators in the Yang Mills theory 3 Identifying the operator in the Yang Mills theory So far in the literature on this subject, a precise procedure for identifying operators in the Yang Mills theory, that correspond to a particular perturbation mode in the supergravity theory, has not been given The supergravity analysis determines the dimension of the operator In some cases supersymmetry deterwx 9 mines the operator, eg the conserved currents in However, even in these cases the overall normalisation is not determined a priori For example, the relevant power of the string coupling in the normalisation cannot be determined in this manner This point becomes clear when one tries to extract absorption cross-sections from the Yang-Mills theory The leading power of energy in the Yang-Mills calculation is determined by the total dimension of the operator, but the power of string coupling depends on further details of the operator, as is clear from the analysis of w14 x Here, we will explore the idea that these operators can be obtained by expanding an action containing a DBI and WZ term 4 This method for identifying operators has been used earlier in w16x for five dimensional black holes and in wx 6 for the 3-brane, where the action was expanded about flat space-time and shown to give consistent results The present discussion will have two important new features First, as we will discuss, it will be crucial to expand the action about AdS space, rather than flat space, to identify the operator correctly In many ways, this is the natural thing to do The supergravity modes we are interested in are perturbations about AdS space Thus to couple them consistently one also expects to 4 Strictly speaking we are proposing to expand the action which governs the dynamics of D3-branes If additional terms besides the DBI and WZ terms are present in such an action, as has been suggested in w15 x, one would expect to keep them as well We note here that the additional terms discussed in w15x are of dimension 1 and higher; such high dimension operators do not alter our conclusions

146 146 ( ) SR Das, SP TriÕedirPhysics Letters B expand the brane action about the AdS background Secondly, our analysis will involve the WZ term in an important way In particular a cancellation, between two contributions, which arise from the DBI and WZ terms respectively, to the coupling of a dimension four operator, will be important in identifying the leading operator We will see that this procedure yields a consistent result In fact, the leading operator which couples to the mode of Section has dimension six The supergravity analysis showed that it s total dimension is six as well Thus, we find that the operator does not acquire any anomalous dimensions in the large-n limit It is worth drawing attention to one other aspect of the analysis at the outset The action we use can be identified with the world volume theory for a set of D3- brane probes In the discussion below we will take these branes to be placed at the boundary of AdS space, at r4r This is in line with the discuswx 3, wx 4 In these references, the idea was sion of to compute correlation functions by, roughly speaking, regarding boundary values of the supergravity fields as sources for the Yang Mills theory This implies that the Yang Mills operators should also be identified by working at the boundary of AdS space The action for a D3-brane has been studied in w x w x 5 17, 18, and is given by : Ssy ( 4 Hd j ydet GijqFij ž / H Ž4 Ž0 q Cˆ qfnaqc ˆ ˆ FnF Ž 31 The two terms above correspond to the DBI action and the WZ term respectively It is worth defining the various terms above carefully Gij refers to the induced world-volume metric, obtained as the pullback of the spacetime metric Similarly, F sf yb ˆ, Ž 3 ij ij ij where Fij stands for the gauge field on the D3-brane and Bˆ ij is the pullback of the NS NS two form potential In the W-Z term C ˆ, Aˆ and Cˆ Ž4 Ž0 refer to the pullback of the R R four form, two form and zero form fields respectively Strictly speaking we are interested here in the action for N D3 branes It has been suggested in w19x that this can be obtained by appropriately symmetrising the terms obtained from the single brane action To begin with we will work with the abelian single brane action The color factors will be introduced by appropriate symmetrisation towards the end Some of our conclusions do not depend on the details of this symmetrisation procedure In the following discussion it is useful to distinguish between three kinds of indices: j i,is0,,3, refer to the world-volume coordinates and are purely bosonic Z M, refers to ten dimensional superspace coordinates; M can stand for bosonic coordinates denoted by ms1,,10, or for fermionic coordinates denoted by m Finally, we denote frame indices by A; A can stand for bosonic tangent vectors, denoted by as1,,10, or for spinor tangent vectors denoted by a In this note we will be interested in expanding this action about AdS space to linear order in the perturbation, Eqs 10 and 9 Following, w18x we will choose a static gauge, where X m s j m,m s 0,,3 The remaining X m,m s 4,,9 will be denoted by f m In addition the kappa symmetry of Eq Ž 31 will be used to set half the fermionic coordinates to zero 6 w18 x The remaining 16 component Majorana-Weyl fermion will then be denoted by l With this notation in hand the induced metric is given by EZ M EZ N A B Gijs E E h Ž 33 i j M N AB Ej Ej with EM A being given by: r a a Ems h m, R ms0,,3 R s h a m, r ms4,,9 a a Em s Ž lg m Em a s0 Em a sdm a 5 We have set the dilaton e f s1 and chosen units for the string tension so that the coefficient in front of the DBI term is unity 6 Our conventions for spinors and Dirac matrices are the same as those in w18 x Namely, the G matrices are 3=3 real matrices m n 4 mn satisfying, G, G sh, with hsž y1,1, dots,1

147 ( ) SR Das, SP TriÕedirPhysics Letters B From Eqs Ž 33 and Ž 34 it then follows that the induced metric is given by 7 : r R Ef m Ef n Gijs hijq h i j mn R r Ej Ej ž ž / / r Ef m R El y l Gi qgm qilj i j R Ej r Ej 9 El El a q Ý lg lg Ž 34 i a j Ej Ej as0 Similarly, the pullback of the rank- tensor is defined as BijsBABEiZ M EjZ N EM A EN B Ž 35 and the superspace components of BAB are given in w17 x Using these we get r Ef m R El FijsFijy l Gi qgm yilj i j R Ej r Ej ž ž / / R k ybijq Bik lg Ejlyilj r ž / R k l i j kl r ž / y lg E llg E l B Ž 36 Adding Eqs Ž 34 and Ž 36 we get that the DBI action is given by: r 4 4 SDBIsyHd j ž / ( ydetž hijqmij Ž 37 R where R 4 R m n Mijs Eif Ejfhnmq F 4 ij r r R R 3 m yl Giq GmE if Ejl r r 3 R R q lg E l lg E l y B r q 9 a Ý Ž i ž a j / ij as0 r R 3 k k BiklG EjlyBjklG Eil r 3 R 4 k l kl Ž i ž j / r 4 y B lg E l lg E l Ž 38 This gives rise to the following terms in the DBI action linearly dependent on the NS NS two form: Hd 4 j LBI 1 r shd j y BijF q lge i jlb R 4 ji ji R 1 R 4 y wlg FmiEjlB xy Fim F Fjk B r r 4 m ij mj ki 1 R 1 R 4 ji l ml i j q BijF lg Elly FlmF FijB qppp 39 r 8 r 4 The ellipses above refer to operators of dimension eight and higher that occur in the expansion Also, for the sake of brevity we have regrouped terms so that the indices above take values in ten dimensions and are raised and lowered by the flat space metric Thus for example, Fim refers to the ten dimensional field strength 8 The action described above is for a single D3- brane While a definitive action is not known for N branes, we may adopt the symmetrized trace prew19 x, so that the various quantities above scription of have to be replaced by matrices and traced over The first term in Ž 39 then receives a contribution only from the UŽ 1 piece of UN and is subdominant in the large-n limit Thus the lowest dimension operator which couples to B in Eq Ž 39 ij has dimension 4 and is given by the first term in Eq Ž 39 The subsequent operators listed in Eq Ž 39 all have dimension 6 The contribution from the WZ term arises from the wedge product of F and Aˆ in Eq Ž 31, which can be expanded as: 1 ˆ ijkl LWZ s 4 FijAkle Ž 310 We remind the reader that in Eq Ž 310 Aˆ kl stands for the pullback of the R R two form This is given by: R ˆ k AijsAijy lg Ejl Aikyilj q PPP Ž 311 r where the ellipses again denote higher dimensional operators From Eq 10 it follows that in the 7 To avoid confusion let us note that all indices on Gamma matrices here are frame indices 8 The last two terms in 39 were not included in the first version of this paper, we regret this error

148 148 ( ) SR Das, SP TriÕedirPhysics Letters B general case, for a perturbation Bij polarised along the brane directions, the R R two form is given by 9 : 1 ij Akls eklij B 31 Substituting Eq Ž 311 and Eq Ž 31 in Eq Ž 310 then leads to: r R i ji m ij LWZ sy lg EjlB y lg Ejl Fim B R r 1 R ij l q FijB lg E ll Ž 313 r In obtaining Eq Ž 313 we have used Eq Ž 36 for F and expanded keeping only terms linear in B ijasin Eq Ž 39, we have regrouped terms and expressed them in a notation where all indices take values in ten dimensions Žthe indices are raised and lowered by the flat space metric Finally, for reasons mentioned above, we have omitted a term in Eq Ž 313 which goes like BijF ij On adding Eq Ž 39 and Ž 313 we now see that the dimension four operators that couple to Bij do indeed cancel between the two equations 10 Furthermore the two types of dimension six operators which involves the fermionic fields also cancel between Ž 39 and Ž 313 Thus the leading operators are of dimension six and of the form: ž / 4 R 1 mk ij 1 ml ij O6 sy FjmF Fki B q 8Flm F Fi jb r Ž 314 While we have suppressed the color indices here, the operators in Eq Ž 314 should be understood as being symmetrised in color space Eq Ž 314 is the main result of this paper A few comments are in order at this stage Our starting point was the assumption that the coupling between various supergravity modes and operators in the Yang Mills theory can be obtained 9 Here all indices are being summed using the flat space metric Our conventions for the e symbol are chosen so that, e 013 sq1 10 As was mentioned above, we have so far been suppressing color indices For the dimension four operator involved here there is a unique color singlet that can be formed ; the cancellation then follows in the full Non- Abelian theory as well by expanding the action consisting of the DBI and W-Z terms about AdS space The consistent answer obtained above provides evidence in support of this assumption Note in particular that the analysis above probed terms in the action of dimension six and involved the W-Z term in a non-trivial way It was crucial in the above analysis to expand the action about AdS space What would have happened if we had expanded about flat space? In this case, without any five -form field strength, the supergravity equations, Eq 5, would not have coupled the NS NS and R R two form gauge potentials together and would have lead to two massless modes Correspondingly, on expanding the brane action for the coupling to the NS NS two form, one would have only got a contribution of the form of Eq Ž 38 from the DBI action with no contribution from the WZ term Thus the leading operator in the Yang Mills theory would have had dimension four which is different from what we got above It will be interesting to study this set of ideas further for the other supergravity modes as well We have looked at the S-wave mode for the two form fields here As was mentioned above, the propagawx 7 One tion of higher partial waves was studied in would like to determine the corresponding Yang Mills operators as well 11 For one class of supergravity modes Žincluding an ls1 mode of the two form field the coupling to the Yang Mills theory has been written down in wx 9 from superconformal considerations It is worth examining if these couplings can be obtained by expanding the DBI plus WZ action Acknowledgements We would like to thank Joe Lykken for comments and especially Samir Mathur for sharing his insights and numerous discussions SRD would like to thank Center for Theoretical Physics, MIT and Physics Department of Stanford University for hospi- 11 Here we have been implicitly assuming that the action is expanded about a definite position in the S 5 In correctly determine the operators for the higher partial waves we will probably need to average this position over the whole of S 5

149 ( ) SR Das, SP TriÕedirPhysics Letters B tality during the course of this work The research of ST is supported by the Fermi National Accelerator Laboratory, which is operated by Universities Research Association, Inc, under contract no DE- AC0-76CHO3000 Note added While this revised version was being written w0x appeared, where a large class of gauge theory operators corresponding to supergravity modes, including the operator derived in this paper, have been shown to follow from the proposal of this work This paper has also noted the omission of the second term in O 6 in our earlier version References wx 1 IR Klebanov, Nucl Phys B 496 Ž , hepthr970076; SS Gubser, IR Klebanov, AA Tseytlin, Nucl Phys B 499 Ž , hep-thr ; SS Gubser, IR Klebanov, Absorption by branes and Schwinger terms in the world volume theory, hep-thr wx J Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, hep-thr97110 wx 3 SS Gubser, IR Klebanov, AM Polyakov, Gauge Theory Correlators from Non-Critical String Theory, hepthr wx 4 E Witten, anti-de Sitter Space and Holography, hepthr wx 5 D Freedman, S Mathur, A Matusis, L Tastelli, Correlation functions in the CFTdr AdSdq1 correspondence, hep- thr wx 6 SS Gubser, A Hashimoto, IR Klebanov, M Krasnitz, Scalar Absorption and the Breaking of the World Volume Conformal Invariance, hep-thr wx 7 HJ Kim, LJ Romans, P van Nieuwenhuizen, Phys Rev D 3 Ž wx 8 A Rajaraman, Two Form Fields and the Gauge Theory Description of Black Holes, hep-thr wx 9 S Ferrara, C Fronsdal, A Zaffaroni, On Ns8 supergravity in AdS5 and Ns4 Superconformal Yang Milles Theory, hep-thr98003 w10x L Andrianopoli, S Ferrara, hep-thr w11x S de Alwis, Supergravity, the DBI action and black hole physics, hep-thr w1x P Claus, R Kallosh, J Kumar, P Townsend, AV Proeyen, Conformal Theory of M, D3, M5 and D1 q D5 Branes, hep-thr w13x JH Schwarz, Nucl Phys B 6 Ž w14x SR Das, Phys Lett 41 Ž hep-thr w15x OD Andreev, AA Tseytlin, Nucl Phys B 311 Ž w16x SR Das, SD Mathur, Nucl Phys B 478 Ž , hep-thr ; CG Callan, SS Gubser, IR Klebanov, AA Tseytlin, Nucl Phys B 489 Ž , hep-thr961017; IR Klebanov, M Krasnitz, Phys Rev D 55 Ž w17x M Cederwall, A von Gussich, BEW Nilsson, A Westberg, Nucl Phys B 490 Ž , hep-thr ; E Bergshoeff, PK Townsend, Nucl Phys B 490 Ž w18x M Aganagic, C Popescu, JH Schwarz, Gauge-Invariant and Gauge -Fixed D-Brane Actions, hep-thr961080; M Aganagic, J Park, C Popescu, J Schwarz, Dual D-brane actions, hep-thr w19x AA Tseytlin, On non-abelian generalisation of the Born-Infeld action in string theory, hep-thr w0x S Ferrara, MA Lledo, A Zaffaroni, hep-thr980508

150 31 December 1998 Physics Letters B Anomalous D-brane and orientifold couplings from the boundary state Ben Craps 1,, Frederik Roose 3 Instituut Õoor theoretische fysica, Katholieke UniÕersiteit LeuÕen, B-3001 LeuÕen, Belgium Received 30 September 1998; revised 6 November 1998 Editor: PV Landshoff Abstract We compute scattering amplitudes involving both R-R and NS-NS fields in the presence of a D-brane or orientifold plane These provide direct evidence for the anomalous couplings in the D-brane and orientifold actions The D9-brane and O9-plane are found to couple to the first Pontrjagin class with the expected relative strength q 1998 Elsevier Science BV All rights reserved PACS: 115-w; 0465qe 1 Introduction Since their introduction in string theory D-branes have attracted an ever increasing interest Accordingly the effective action describing the low energy dynamics acquired various additional terms Among these the Wess-Zumino couplings to the RR-potentials appear to be the least well-studied Apart from the natural coupling to the Ž pq1 -form there are also interaction terms coupling a Dp-brane to eg the Ž py1 -and wx 4 py3 -forms, for p high enough In 1 the precise form of these was derived from anomaly arguments : T X X H ( p ˆX pa FqBˆ S s C ne n Aˆ R ˆ WZ Ž 11 k pq1 The p Ž Rˆ term in the expansion was already proposed in wx 1 based on an independent duality reasoning In Ref wx 3 a similar coupling was proposed for orientifold planes Although void of world brane dynamics, these planes do couple to bulk fields for anomalies to cancel along the lines of Ref wx 1 In this paper we will give additional evidence for some of these terms by evaluating the corresponding string diagrams 5 Scattering amplitudes in the presence of D-branes have hitherto been evaluated in two ways; people have either essentially performed a disk or annulus calculation w4,5x or relied on the boundary state formalism w6,7 x 1 Aspirant FWO, Belgium BenCraps@fyskuleuvenacbe 3 FrederikRoose@fyskuleuvenacbe 4 For explanation of the used symbols and conventions see Appendix A 5 The p Ž Rˆ terms in Refs w,3x actually came with the proposal to do the corresponding three point string amplitudes r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

151 ( ) B Craps, F RooserPhysics Letters B We opt for the last possibility although in principle the calculations could be done analogously to wx 5 A nice feature of our choice is that the results thus obtained carry over to the orientifold case almost without effort In Section we set up the basics of doing string tree level scattering involving the boundary state As a working example we derive the HBnCpy1 coupling In Section 3 we outline the procedure to find the desired Hp1n Cpy3 term, thereby relying on the basic results in the preceding section Making contact with the effective D-brane action is done in Section 4 In Section 5 we derive the tree level amplitude for the gravitational coupling of an orientifold plane Section 6 then concludes and raises a few questions concerning the origin of the studied interaction terms An appendix explaining our conventions and normalizations is added The BnC interaction pi1 Various string amplitude calculations w5,7x established the coupling of a D p-brane to the RR Ž p y 1 -form potential: ' H X pa T C nf, 1 p py1 where k s8p G From the fact that only the pa X FqB X combination is gauge invariant one can then infer N the presence of H ' T C nb ˆX p py1 This D-brane coupling to the NS-NS two form should reproduce the leading behaviour at low energies of the following string amplitude 6 : ² < Ž0 As C ;k V Ž z,k < B :, 3 py1 B R which is equivalent with the two point function on the disk of the Kalb-Ramond field B and the RR-potential C Here and in the following we adopt both the RR-state ² C < and the boundary state < B: 7 from wxž py1 py1 R 6 and we give the RR potential a momentum k In the following we carry out the evaluation of this simple-looking amplitude using the boundary state Thus we will find out about the basic manipulations involved when computing any boundary state amplitude First, as the boundary state and the RR potential already saturate the superghost anomaly wx 6 any additional vertex must be in the Ž 0,0 -picture Therefore we may take Žsee Eq Ž A39 k Ž0 m m n n VB s zmnhd zž E X qi kpcc Ž E X qi kpcc, 4 ' p where z and z run over the complement of the unit disk in the complex plane A substantial amount of work in computing amplitudes involving the boundary state goes into pulling the non-zero mode operators ` 1 Ý yn yn ns1 ` Ý ym ym ms1 exp y a PSPa ; 5 n exp ih c PSPc 6 from the boundary state to the left where they annihilate the out vacuum For amplitudes in the RR sector one then has to take care of the fermion zero modes From Eq A16 in the Appendix it is clear that those zero 6 With polarizations chosen along the brane worldvolume 7 For the precise form of the RR state and the boundary state see also Appendix A

152 15 ( ) B Craps, F RooserPhysics Letters B modes are effectively replaced by g-matrices In the conventions we are using any amplitude in the RR sector contains a factor 1 ž ' / n y1 y1 T tr Ž C M RC N L, 7 where L is a product of g-matrices corresponding to the left-over left-moving fermions, R a product of g s for the right-moving fermions and n is the total number of g s in LR The matrices M and N show up in the boundary state and the RR state respectively Note that Eq 7 is schematic: in fact, keeping track of all factors of g11 and minus-signs is crucial to obtain correct results Equivalently wx 4, one modifies the Green functions to include left-right contractions: ž / 1 ² m n : mn X Ž z X Ž w sys log 1 y 8 zw ² m n : R mn S ih 1qzw c Ž z c Ž w s 9 zwy1 ' zw Before applying the above rules to the amplitude Eq 3, let us remark that the ghost sector is discussed in the Appendix, whereas the superghost sector contributes a factor 1r wx 6 Contracting two fermions using the massive oscillators in the boundary state 8, leads to T k ykpspkykpk a 1 apy1 b1 b As e c z d zž < z< 4 a a b b ' ' H 1 py1 1 p Ž py1! < z < )1 = p kpspk Ž kpspk < z< y1, 10 < z< < z< y1 where c is the polarization tensor of the RR field Doing the Hd z integral gives p Ž kpspk BŽ a a 1qkP 1 py1 k, kpspk We thus conclude that, for small momenta, kt p a 1 apy1 b1 b As e c a a z b b 11 1 py1 1 Ž py1! We end this section with a remark which is related to the previous footnote Because of the decoupling of longitudinal polarizations, the amplitude vanishes if k Žor equivalently k X has a non-zero component along the brane directions 9 In computing cross-sections, one has to average over neighbouring momenta So in order to have a non-vanishing cross-section, one needs the possible momentum components along the brane to be discrete, ie all worldvolume directions should be compactified This suggests considering Euclidean branes In that case one can have on-shell momenta without a component along the brane worldvolume For such momenta the numerator of the expression in the footnote vanishes 8 kpspk The zero-mode contribution is proportional to kpk vanishes whenever the Bn Cpy 1 for pointing out this problem to us 9 This was pointed out by Igor Pesando for small on-shell momenta Further investigations seem to suggest that this interaction could contribute to cross-sections, as we argue in the next paragraph We thank Rodolfo Russo

153 3 The tr R nc interaction pi3 ( ) B Craps, F RooserPhysics Letters B We have to calculate the following amplitude: ² < Ž0 Ž0 As C ;k V Ž z,k V Ž z,k < B : Ž 31 py3 g 3 3 g 4 4 R Since we are looking for the tr R ncpy3 term in the D-brane effective action, we will consider the components of Cpy3 with indices along the brane This implies that the fermion zero modes have to provide the four gamma-matrices with Lorentz indices in the worldvolume directions complementary to the ones of the RR potential that is being considered As the computation is analogous to the one presented in the previous section but considerably longer, we only give a rough sketch Since each graviton vertex operator can be written as the sum of four different parts, the amplitude splits into various pieces Some integrations by parts in the relative angle are needed to let the various terms combine nicely Performing the trace of gamma-matrices, taking into account the superghost sector and the GSO-projection, one ends up with the following result Ž possibly up to a global sign : k As e c k k k PSPz k Pz 4' Ž py3!p X T p a a 1 PPP apy3b1ppp b4 a1 PPP a py3 3b1 4b 3 4 3b 3 4b4 yž k PSPk z Pz q k Pz k PSPz y Ž k Pk z PSPz = 3 4 3b 4b4 4 3b 3 4b b 4b4 k 3PSPk3 k4pspk 4 y k 3PSPk 3y k 3PSPk 4y H d z d z Ž < z < y1 Ž < z < y1 < z < < z 3<, < z 4< )1 y k 4PSPk 4y k 3PSPk 4y k 3Pk 4y k 3PSPk 4y = < z < < z yz < < z z y1< z z yz z This is the general result However, when we want to talk about the worldvolume action of a D-brane, we might be especially interested in momenta and polarizations which are along the brane worldvolume From now on we specialize to that case Because of the mass shell condition and momentum conservation in the worldvolume directions the integral above simplifies to < < y < < y < < y H < z 3<, < z 4< )1 Is d z d z z z z yz z z y1 z z yz z Ž 3 The integral depends only on the difference of the phases of z3 and z 4, so one of the angular integrations just gives a factor of p By changing variables to the cosine of the difference of the phases, it can be verified that the second angular integration results in a hypergeometric function With rs< z < < < 3, ts z 4, xs4rtr rqt and ys4rtrž rtq1 the integral becomes Ž up to a minus sign wx 8 : y1 3 H 1Ž r,t)1 Isp dr dt rt xy F,1,1,3; x, y 33 Performing a change of integration variables and manipulating the hypergeometric function, this becomes ` 3 p 1 y1r Ž n n 1 y1r 3 H Ý H Ž Is dy Ž 1yy y dx Ž 1yx F qn,1,3qn; x Ž 34 0 Ž 3 0 ns0 wx 10 In this expression, both integrals are well-known 8 The result is 3 ` 3 4 nqk Ý 3 3 k,ns0 nqk n p G Ž kq1 G Ž nq1 4p Is s Ž 35 3 G kq G nq 3 10 We thank Walter Troost for showing us how to sum the following series

154 154 ( ) B Craps, F RooserPhysics Letters B The scattering amplitude thus becomes X ' p k Tp a a 1 PPP apy3b1ppp b As e 4 ca PPP a k3b k4b Ž k4pz3b Ž k3pz 4b Ž 36 1 py Ž py3! 4 D-brane action Let us compare these results to what one expects from the D-brane action in supergravity Ž As to the amplitude in Section, the relevant part of the D-brane action is see Eq A3 T p H ˆX ˆ X py1 k pq1 C nb The amplitude calculated from this interaction thus becomes kt p a 1 apy1 b1 b A s e sugra ca a z b b 1 py1 1 Ž py1! This is twice the string result However, as suggested by R Russo, another diagram should be considered in the supergravity computation In fact, the bulk supergravity action given in the appendix is not the whole story: the field strength of a pq1-form RR potential gets a correction involving the py1-form RR potential and the NS -form This implies that there is a diagram in which the D-brane emits an intermediate p q 1-form potential This contribution should be added to the contact amplitude given above Checking the structure and the coefficient of this contribution shows that this could explain the apparent mismatch For the amplitude in Section 3 the relevant part of the action is X Ž 4pa H py3 pq1 19p Tp ˆX C n 1 tr Ž RnR ˆ ˆ k Expanding Rˆn Rˆ and writing everything in terms of canonically normalized fields Žsee Eqs Ž A34 and Ž A37 one obtains for the scattering amplitude in supergravity X ' p k T a p a 1 PPP apy3b1ppp b4 sugra a 1 PPP a py3 3b1 4b 3 4 3b 3 4b4 A s e c k k k Pz k Pz 6 Ž py3! This time, the supergravity computation seems to be missing a factor of relative to the string result Again, extra contributions might come from modifications of the RR field strengths wx 1 However, the details of this possible solution of the mismatch are left for future research 5 Orientifold planes The computations in the previous sections were done in oriented string theory From the Ž oriented type IIB theory, one can construct the Ž unoriented type I theory by modding out worldsheet parity This operation makes another kind of objects enter the story: the orientifold planes wx 9 In Ref wx 3 it was argued that orientifold planes exhibit a coupling to gravity similar to the one we computed for D-branes in Section 3 Moreover, it was claimed that the orientifold 9-plane in type I theory couples with half the strength of the 3 type IIB D-branes Žproviding the Chan-Paton factors w10x together wx As remarked in Ref 3, the presence of that interaction can be checked by computing an RP diagram with three insertions This is analogous to the computation for D-branes: rather than cutting a hole in the sphere to

155 ( ) B Craps, F RooserPhysics Letters B obtain a disk, one inserts a crosscap For the case of all Neumann boundary conditions, the crosscap state w x representing the insertion of a crosscap was discussed in Ref 11 : apart from a different normalization of RP relative to disk amplitudes Ž giving an extra factor of 3, the crosscap operator is obtained by inserting Ž y1 n in every term of the exponent of the boundary state The computation of the RP diagram is very similar to the one in Section 3 Because of the Ž y1 n insertions, every zz combination gets replaced by yz z In Eq Ž 3 i j i j this amounts to replacing the 1 by y1 Analogous calculations to the ones which led to Eq Ž 35 result in p 4 Icrosscaps, Ž 51 3 half the result of the D-brane computation Taking into account the factor of 3 from the crosscap normalizawx 3 tion, this reproduces precisely the coupling predicted in Ref 6 Discussion By evaluating the appropriate string diagrams we have derived two specific terms of the D-brane effective action This nicely illustrates how superstring theory provides its own consistency If Ž intersecting D-branes are to be included into the picture then from Ref wx 1 the terms in the effective D-brane action have to be there for the low energy theory on the brane not to suffer from anomalies Repeating the analysis for the orientifold, we have extended our results to the orientifold p Ž R 1 coupling, 1 first proposed in Ref wx 3 Thus new evidence has been provided for the relative factor between the D-brane and orientifold couplings Applying the above machinery one could in principle go beyond our results The fourth order curvature term can be treated analogously but may turn out to be a hard nut to crack, though From a more general perspective one may ask the following question: is there room for a similar gravitational coupling in the worldvolume theory of the M-theory 5-brane w1 x? In Ref w13x Sen found such a coupling in the worldvolume action of the KK-monopole in M-theory It would be interesting to see whether his procedure can be applied to the M5-brane Acknowledgements We would like to thank Marco Billo, Andrea Pasquinucci, Rodolfo Russo and Walter Troost for very helpful discussions This work was supported by the European Commission TMR programme ERBFMRX-CT Note added While this paper was circulating as a preprint, the issue of gravitational couplings on the M5-brane was addressed in Ref w19 x Appendix A Conventions and normalizations wx We normalize the boundary state as in 6 : T p R,NS X gh c R,NS sgh R,NS < B,h: s < B : < B : < B,h: < B,h:, Ž A1

156 156 ( ) B Craps, F RooserPhysics Letters B where ` 1 < : Ž d H ˆ Ý < : X yn yn ns1 n B sd Ž qyy exp y a PSPa 0;ks0, Ž A ` c0qc 0 < B : sexp c b yb c < qs1: < qs1 : gh Ý ž yn yn yn yn/, Ž A3 ns1 and, in the NS sector in the y1,y1 picture, ` < B,h: sexp ih c PSPc < 0 :, Ž A4 Ý c NS ym ym ms1r ` < B,h: sexp ih g b yb g < Psy1: < Psy1 :, Ž A5 NS Ý ž / sgh ym ym ym ym ms1r or, in the R sector in the y1r,y 3r picture, ` < B,h: sexp ih c PSPc < B,h:, Ž A6 Ý c R ym ym c ms1 ` Ž0 R Ž0 < B,h: Rsexp ih g b yb g < B,h: R, Ž A7 Ý ž / sgh ym ym ym ym sgh ms1 where, if we define Žh 0 l l M scg G 1 G p 1qihG 11 1qih, Ž A8 ž / the zero mode parts of the boundary state are < B,h: sm < A:< B :, Ž A9 c Ž0 Žh R AB Ž0 < B,h: s exp ihg b < P sy1r: < P sy3r : Ž A10 The matrix S sgh R 0 0 mn is given by S s h,y d Ž A11 mn ab ij The overall normalization factor Tp is claimed to be fixed from factorization of amplitudes of closed strings emitted from a disk w14,7 x It is the tension of the Dp-brane ' ' X 3yp T s p p a Ž A1 p The GSO projection acts as follows: B s Ž B,q y B,y, Ž A13 1 < : NS < : NS < : NS 1 < : R < : R < : R B s Ž B,q q B,y Ž A14 All this is explained in more detail in wx 6 Conventions on the RR zero modes and gamma matrices can be found in the Appendix of wx 7 Here we only note that m T Ž G syc G m C y1 Ž A15

157 ( ) B Craps, F RooserPhysics Letters B and 1 m m A B c A B s Ž G Ž C D C D < :< : < : < : 0 ' 1 m A m B 0 11 C D c < A:< B : s Ž G Ž G < C:< D : Ž A16 ' In our computation we have put a X s, Ž A17 so that the following OPEs hold w15 x: X m Ž z X n Ž w syh mn logž zyw, Ž A18 m n mn y1 c z c w syh zyw, A19 where XŽ z, z sxž z qxž z Ž A0 Expanding the fields in modes, this gives m n mn a m,an sh m d mqn, Ž A1 m n a 0,q mn syih, Ž A cm m,cn n 4 sh mn d mqn, Ž A3 where ie X m Ž z s z yny1 a m, Ž A4 Ý n Ý n ic m z s yry 1 z c A5 r r ( ) For the graõiton Õertex operator in the y1,y 1 picture we take Žy1 m V g szmn c c n e i kp X A6 This normalization is such that ² Žy1 < Žy1 : V V sytr z Ž A7 g g It can be checked wx 7 that, with this normalization, ² < Žy1 0 V < B: Ž A8 g NS reproduces the one point function of the canonically normalized graviton in supergravity, where the D-brane is described by the DBI action The canonically normalized graviton field is obtained by expanding gmnshmnqk h mn, Ž A9 7r X s N ( where ks8p a g s 8p G in the bulk supergravity action in Einstein frame yf X Ž 3yp Fr X SIIA,B sy Hd x' yg Rq Ž df q e Ž db q Ý e Ž dc 1 pq1 k Ž pq! Ž A30

158 158 ( ) B Craps, F RooserPhysics Letters B Ž w x Ž F r see, for instance 16 The DBI action is given in string frame G se g by ( T p pq1 yf X X ˆ ˆ BI ab ab ab H S sy d j e ydet G qb qpa F Ž A31 k Ž Here Gˆ denotes the pullback to the D brane worldvolume of the bulk field G etc In addition, the D-brane action contains a WZ part where T X X H ( p ˆX pa FqBˆ S s C ne n Aˆ R ˆ WZ, Ž A3 k pq1 X Ž 4pa AˆŽ Rˆ s1q tr Ž RnR ˆ ˆ q Ž A33 19p These actions can be written in terms of canonically normalized bulk fields by substituting Gse F r Ž hmnqk hmn Ž A34 Fs' kx Ž A35 ' ' X F r B s k e B A36 mn mn X C s k C Ž A37 For the normalization of the graõiton Õertex operator in the ( 0,0) picture, we follow Section 34 of w17 x: k Ž0 m m n n i kp X Vg s zmn Ž E X qi kpcc Ž E X qi kpcc e Ž A38 p X X Žwhere we put a s rather than a s1r asinw17 x For the Kalb-Ramond field we would correspondingly find k Ž0 m m n n i kp X V s z Ž E X qi kpcc E X qi kpcc e Ž A39 B ' p ' mn ŽThe factor of 1r is due to the different rescaling between the canonically normalized fields and the fields appearing in the string action The zmn in the vertex operators correspond to polarizations of canonically normalized fields Finally, we introduce a Ž canonically normalized out-state for a RR-potential wx 6: with C s C,q q C,y Ž A40 1 ² < ² < ² < mq 1 mq1 mq1 ² C,h < s e N ² A< ² B <, Ž A41 mq1 ihb0g 0 y1r ABy3r where we defined CG m 1 PPP m Ž mq 1 AB NABs c m PPP m Ž A4 ' 1 mq1 Ž mq1! Ž Here c denotes the polarization tensor of C The factor 1r is justified by computing 1 ² m 1 PPP m C < : mq 1 Cmq1 s cm PPP m c mq1 Ž A43 1 mq1 Ž mq1! w x pa X F As a check, one can derive 18 the e part of the WZ term of the D-brane action by projecting the states ² C <, ns0,1, on the boundary state with F turned on pq1y n '

159 ( ) B Craps, F RooserPhysics Letters B So far we have only computed one-point functions in the presence of a D-brane The ghost zero modes have been taken into account by not integrating over the position of the inserted vertex operator Because of the peculiar ghost structure of the boundary state, it is not so clear how the ghosts should be treated when one inserts more than one vertex operator Žie which part of the SL group volume still has to be divided out The procedure we will follow Ž and motivate below consists of integrating over the positions of all vertex operators except the one inserted at infinity To motivate this procedure, we compute the two graviton amplitude for both polarizations along the brane Ž up to powers of i : t ² < Žy1 Ž0 < : 0 Vg Vg B NS sk Tp q tr Ž z1pz Bž y,q q1, Ž A44 / Ž wx 1 where q s k PSPk and tsy k qk 1 This is the correct result see for instance 5 References wx 1 M Green, J Harvey, G Moore, Class Quant Grav 14 Ž hep-thr wx M Bershadsky, V Sadov, C Vafa, Nucl Phys B 463 Ž hep-thr9511 wx 3 K Dasgupta, D Jatkar, S Mukhi, Nucl Phys B 53 Ž , hep-thr97074; K Dasgupta, S Mukhi, J High Energy Phys 3 Ž , hep-thr wx 4 I Klebanov, L Thorlacius, Phys Lett B 371 Ž hep-thr wx 5 M Garousi, R Myers, Nucl Phys B 475 Ž hep-thr wx 6 M Billo, P Di Vecchia, M Frau, A Lerda, I Pesando, R Russo, S Sciuto, Microscopic string analysis of the D0-D8 brane system and dual R-R states, hep-thr wx 7 P Di Vecchia, M Frau, I Pesando, S Sciuto, A Lerda, R Russo, Nucl Phys B 507 Ž hep-thr wx 8 I Gradshteyn, I Ryzhik, Table of Integrals, Series, and Products, Academic Press, 1965 wx 9 A Sagnotti, Open strings and their symmetry groups, in: G Mack et al Ž Eds, Non-perturbative Quantum Field Theory, Cargese, 1987, Pergamon Press, 1988; P Horava, Phys Lett B 31 Ž ; J Dai, RG Leigh, J Polchinski, Mod Phys Lett A 4 Ž w10x J Polchinski, E Witten, Nucl Phys B 460 Ž hep-thr w11x C Callan, C Lovelace, C Nappi, S Yost, Nucl Phys B 93 Ž w1x P Pasti, D Sorokin, M Tonin, Phys Lett B 398 Ž , hep-thr ; M Aganagic, J Park, C Popescu, J Schwarz, Nucl Phys B 496 Ž , hep-thr w13x A Sen, J High Energy Phys 10 Ž hep-thr w14x M Frau, I Pesando, S Sciuto, A Lerda, R Russo, Phys Lett B 400 Ž hep-thr w15x D Friedan, Notes on String Theory and Two Dimensional Conformal Field Theory, in: M Green, D Gross Ž Eds, Unified String Theories, World Scientific, 1986 w16x C Bachas, Lectures on D-branes, hep-thr w17x M Green, J Schwarz, E Witten, Superstring Theory, vol 1, Cambridge University Press, 1987 w18x P Di Vecchia, unpublished notes w19x S Mukhi, Dualities and the SLŽ,Z Anomaly, hep-thr981013

160 31 December 1998 Physics Letters B Nonlinear s-model, form factors and universality Adrian Patrascioiu a,1, Erhard Seiler b, a Physics Department, UniÕersity of Arizona, Tucson, AZ 8571, USA b Max-Planck-Institut fur Physik ( Werner-Heisenberg-Institut ), Fohringer Ring 6, Munich, Germany Received 14 July 1998; revised 9 October 1998 Editor: H Georgi Abstract We report the results of a very high statistics Monte Carlo study of the continuum limit of the two dimensional OŽ 3 non-linear s model We find a significant discrepancy between the continuum extrapolation of our data and the form factor prediction of Balog and Niedermaier, inspired by the Zamolodchikovs S-matrix ansatz On the other hand our results for the OŽ 3 and the dodecahedron model are consistent with our earlier finding that the two models possess the same continuum limit q 1998 Elsevier Science BV All rights reserved In a recent paper wx 1 we reported some striking numerical results: the continuum limit of the two dimensional Ž D OŽ 3 nonlinear s model seems to agree as well with the form factor prediction wx inpired by Zamolodchikovs S-matrix as with the continuum limit of the dodecahedron spin model The latter, known rigorously to possess a phase transition at nonzero temperature, is almost certainly not asymptotically free Zamolodchikovs ansatz on the other hand is believed to embody asymptotic freedom, a belief which was reenforced by some remarkable scaling properties discovered by Balog and Niedermaier wx 4 Initially wx 5 we thought that the resolution of this apparent paradox must come from the absence of asymptotic freedom in the form factor approach and that the scaling proposed by Balog and Niedermaier must actually be false Soon after though we realized 1 patrasci@ccitarizonaedu ehs@mppmumpgde that the trouble was much more serious: indeed from the direct computation of Balog and Niedermaier of the,4 and 6 particle contribution it followed that the asymptotic value of the transverse current two point function JŽ p must obey the inequality JŽ ` )3651 Ž 1 wx 3 On the other hand in two recent papers w7,8x we used reflection positivity to prove rigorously an upper bound on JŽ ` in terms of b crt In those papers the bound is given only for the OŽ model, but it is straightforward to generalize it to any OŽ N For OŽ 3 it reads J ` F 3 bcrt Combining these two inequalities one would conclude that b crt)55 This lower bound is so large that irrespective of the Balog and Niedermaier scaling ansatz, it suggests that the critical point may very well be at infinity and the model asymptotically free Since, as stated above, asymptotic freedom of OŽ 3 is hard to reconcile with its having the same r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

161 ( ) A Patrascioiu, E SeilerrPhysics Letters B continuum limit as the dodecahedron spin model, we decided that we needed a more refined numerical study to decide these issues In the present paper we will report the results of our investigation regarding the continuum limit of the OŽ 3 and dodecahedron spin models We strove to achieve very high numerical accuracy and to our knowledge our accuracy, exceeding sometimes 005%, has never been achieved before Our results suggest that in fact the model defined by the form factor approach does not agree with the continuum limit of the OŽ 3 model On the other hand, the numerical evidence is consistent with the hypothesis that the OŽ 3 and the dodecahedron spin models have the same continuum limit We begin our report by specifying the models and defining the quantities measured In both models the action is nearest neighbour, the lattice square of length L, with periodic boundary conditions and the inverse temperature b The quantities measured are Žwith PsŽ p,0 : Ø Spin -point function 1 ipx GŽ p s ²< sž P < :; sž P s e sž x ˆ ˆ Ý L x Ž 3 Ø Current -point function J p ' F 0 y F p with 1 3 Ý ˆa 3L as1 FŽ p s ²< j Ž P < :; Ý jˆa m P s e ipx j a m x 4 x where j a Ž x sbe s Ž x s Ž xqmˆ m abc b c Ø Magnetic susceptibility x : xsgž 0 Ž 5 Ø Effective correlation length j : ( 1 js Ž GŽ 0 rgž 1 y1 Ž 6 sinž prl Ø Zero momentum renormalized coupling: u 4 r g sy Ž 7 xj where u4 is the invariant connected spin 4-point function at zero momentum Some comments regarding these quantities: Ø Our definition of j differs from that of Balog and Niedermaier, who define 1rj as the position of the pole in the spin -point function Žon our Euclidean lattice, that pole is inaccessible If we implement our definition upon the form factor prediction, using pj s 1 as the lowest nonvanishing momentum, we find j s , compared to 1 Ø The renormalized spin -point function is G Ž pj r sgž pj rgž 0 Žnormalized so that G Ž 0 s1 r Ø For the OŽ 3 model the definition of the current, including the factor of b, is fixed by the Ward identity The quantity J Ž pj OŽ3 is a renormalization group invariant, which describes continuum physics in the limit j `, pj fixed Ø For the dodecahedron spin model the definition of the current is ad hoc, but if in fact its continuum limit agrees with that of OŽ 3 it is possible that upon multiplication by a suitable factor cž j, cž j J Ž pj dod reaches the same limit as J Ž pj The results reported in Ref wx OŽ3 1 seemed to corroborate this assumption Ø Besides being a renormalization group invariant, gr vanishes in any free field theory Hence gr tests the nontrivial part of the S-matrix The numerics were performed on a SGI-000 machine Both models were simulated using the cluster algorithm and all quantities, except g r, mea- sured using Wolff s improved estimator wx 6 At every b and L value we performed runs consisting of 100,000 thermalization clusters and 1,000,000 clusters used for measurements These runs were repeated for OŽ 3 up to 149 times and for the dodecahedron up to 46 times We give the results for x and j in Table 1 ŽOŽ 3 and Table Ž dodecahedron As can be seen, our results for OŽ 3 are more precise Table 1 Susceptibility x and effective correlation length j for OŽ 3 b L a of runs j x Ž Ž Ž Ž Ž Ž Ž

162 16 ( ) A Patrascioiu, E SeilerrPhysics Letters B Table Susceptibility x and effective correlation length j for the dodecahedron b L a of runs j x Ž Ž Ž Ž Ž Ž Ž Ž than for the dodecahedron, in spite of the extra effort invested in the latter The errors were estimated using the jack-knife method on the results of different runs, since this led to a larger estimated error than the error estimated in each run The form factor prediction gives the continuum, infinite volume values of G Ž pj and JŽ pj Ž r how- ever truncated as to the number of intermediate particles included To relate Monte Carlo measurements to these predictions one must take both the continuum limit, j `, and the thermodynamic limit, Lrj ` In Fig 1 we present the deviations from the continuum, infinite volume limit for the free field, displaying JŽ pj for several correlation lengths j and ratios Lrj It can be seen that for pj-1 for LrjG13 the finite volume corrections Ž y4 become negligible less than =10 for jg11 Fig 1 Lattice and finite volume artefacts for the complex free field; dotted curves: Lrj f7, solid lines: Lrj f14, dashed lines: Lrj f4 the latter two are practically indistinguishable Fig Comparison of Monte-Carlo Ž J OŽ3 with form factor results Ž J ff ; the dashed line represents the continuum extrapola- tion Ž the smallest correlation length in our study We verified directly that the same is true for the OŽ 3 model by running at the same j values on Lrj approximately 7 8, and 3 5 Another lesson learned from Fig 1 concerns the approach to the continuum: at fixed pj the approach is described by arj qblogž j rj Ž 8 with a and b depending upon pj Since at small pj the OŽ 3 data are reasonably close to the free field values, we employed this ansatz in all our extrapolations to the continuum limit As stated above, our definition of j differs from that of Balog and Niedermaier Even though we could correct for this difference by applying our definition to their data, we decided that a better approach is to eliminate j altogether and plot J versus GŽ 0 rgž p For small pj the latter quantity is very close to its free field value, 1qŽ pj This plot is shown in Fig for various j values and Lrj aproximately 14 The solid lines represent a least square fit to the data of the form 4 Ý k ks1 k a J Ž pj Ž 9 free

163 ( ) A Patrascioiu, E SeilerrPhysics Letters B Fig 3 Comparison of Monte-Carlo Ž J dotted lines OŽ3 and their continuum extrapolation Ž dashed line with form factor results Ž J solid line : interacting contribution where ff ( ( ( / 1 1 1qzr4 z z Jfree s y log 1q q p p z ž 4 4 Ž 10 with z sž pj is the expression corresponding to J in the free two-component scalar field theory We also show the extrapolation of our results to the continuum limit using the ansatz in Eq Ž 3 and the form factor prediction As it can be seen, the latter disagrees with the former The disagreement is statistically significant, yet very small One may wonder how such a wonderful agreement could occur if the form factor prediction is actually false A possible answer is this: the correlation functions J and Gr are, especially at low momenta, very close to those of the free theory, deviating only by less than 5% The reason behind this is that the form factor squares contributing to the spectral functions of the correlation functions are at low momenta Ø dominated by the lowest possible number of intermediate particles Ž 1 for G, for J r Ø the squares of those lowest form factors are very close Ž in the case of G even equal r to those of the free theory Therefore, if we want to test the form factor prediction, this universal free field value should be subtracted To achieve such a subtraction we do the following: we plot x JŽ x versus GŽ 0 rgž x xspj for both the OŽ 3 model and the free field We then subtract the two curves at the same value of the abscissa and plot the difference This is shown in Fig 3, together with the form factor prediction The latter differs from the continuum value extrapolation by about 5% at ps0 Another quantity which measures directly the nontrivial part of the S-matrix is the renormalized coupling g For the OŽ r 3 model this quantity was determined to be around wx 9 and we determined it again at b s 15, j s 1106 to g r s 6565Ž 5 and at bs16 to g s6609ž 18 r Since gr is expected to be monotonically decreasing in b, the small difference between the two values is probably just due to statistical fluctuations The value of gr in the form factor approach has not yet been calculated, even though the ingredients are the same as for GŽ p We think that the comparison would be most interesting Next we would like to present the comparison of the dodecahedron spin model with the OŽ 3 model If the two models possess the same continuum limit, then one would expect that for sufficiently large j Fig 4 Comparison of G for OŽ r 3 and dodecahedron Data with j f11, 19, 34, 65 are overlaid The solid lines are fits

164 164 ( ) A Patrascioiu, E SeilerrPhysics Letters B and 19 and obtained 64Ž 84 and 633Ž 83 These values suggest that its continuum limit value is also around 65, in agreement with the value of gr for OŽ 3 Our conclusion is that whereas there is good numerical evidence that the model defined via the form factor approach does not agree with the continuum limit of the lattice OŽ 3 model, there is no evidence that this limit is different from that of the dodecahedron model These facts represent a major setback for the accepted saga regarding the special properties of the nonlinear s models with NG3 Acknowledgements Fig 5 Comparison of J for OŽ 3 and dodecahedron Žrenormal- ized Data with j f11, 19, 34, 65 are overlaid The solid lines are fits both the lattice artefacts and the finite volume corrections become identical Consequently if we use the same definition for j in both models, it should be legitimate to compare the same renormalization group invariants at the same pj value In Fig 4 we present the comparison of G r The data are consis- tent with the hypothesis that the two models have the same continuum limit The error bars are dominated by the those coming from the dodecahedron, where the fluctuations are very large We present a similar comparison in Fig 5 for J, where the renormalization was chosen so that the rms deviation between the two models in the range from pjs0 to pjs15 is minimized As stated above, the theoretical status of this procedure is not clear Finally we report on g for the dodecahdron We measured it at js11 r We acknowledge numerous discussions with JBalog and MNiedermaier AP is grateful to the Max-Planck-Institut for its hospitality Note added in proof: Since this paper was submitted, we carried out many additional Monte Carlo simulations, which show that actually gr does not decrease monotoni- cally in b Also the form factor computation of gr is being carried out All this will be reported in detail elsewhere References wx 1 A Patrascioiu, E Seiler, Phys Lett B 430 Ž wx J Balog, M Niedermaier, Phys Lett B 386 Ž wx 3 J Balog, M Niedermaier, Nucl Phys B 500 Ž wx 4 J Balog, M Niedermaier, Phys Rev Lett 78 Ž wx 5 A Patrascioiu, E Seiler, Phys Rev Lett 80 Ž wx 6 U Wolff, Phys Rev Lett 6 Ž wx 7 A Patrascioiu, E Seiler, Phys Lett B 417 Ž wx 8 A Patrascioiu, E Seiler, Phys Rev E 57 Ž wx 9 A Patrascioiu, E Seiler, Nuovo Cim A 107 Ž

165 31 December 1998 Physics Letters B Decoupling of heavy-quark loops in light-light and heavy-light quark currents AG Grozin 1 Institute of Theoretical Particle Physics, UniÕersity of Karlsruhe, D-7618 Karlsruhe, Germany Received 15 October 1998 Editor: PV Landshoff Abstract Matching of light-light quark currents in QCD with a heavy flavour and in the low-energy effective QCD is calculated in MS at two loops Heavy-light HQET currents are similarly considered q 1998 Elsevier Science BV All rights reserved Consider QCD with n light flavours and a single l heavy one having mass M Processes involving only light fields with momenta p <M can be described i by an effective field theory QCD with nl flavours plus 1rM n suppressed local operators in the Lagrangian, which are the remnants of heavy quark loops shrinked to a point This low-energy effective theory is constructed to reproduce S-matrix elements of the full theory expanded to some order in pirm This condition fixes MS coupling a X Ž m of the effective theory, its gauge parameter a X Ž m and running light quark masses m X Ž m i in terms of the parameters of the full theory Two-loop matching was considered in w1, x, and three-loop one in w3 x Operators of the full QCD can be expanded in 1rM in operators of the effective theory Coefficients of these expansions are fixed by equating on-shell matrix elements In this Letter, I consider 1 Permanent address: Budker Institute of Nuclear Physics, Novosibirsk , Russia AGGrozin@inpnsksu s y1 bilinear quark currents j m s Z Ž m n n j n0, where w m1 mnx the bare currents are jn0s q0g g q 0, and Z Ž m n are their MS renormalization constants In the low-energy theory, they are equal to C Ž m j X Ž m n n plus power-suppressed terms Žwhich will be omitted throughout this paper It is natural to match the currents at msm, because C Ž M n contains no large logarithms Currents at arbitrary normalization scales can be related by ž / s X X a Ž m g X n Ž a da n H X ž X a Ž M b Ž a a/ s a Ž m gnž a da exp yh s jnž m a Ž M bž a a s X X n sc M exp y j m, where dlog Z a a n s s nž s n0 n1ž / g a s sg qg q PPP dlog m 4p 4p Ž r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

166 166 ( ) AG GrozinrPhysics Letters B is the current s anomalous dimension two loops for generic n in wx 4, and Žobtained at dlog a s sy ybž a s, dlog m a a s s bž a sb qb q PPP, 4p 4p s 0 1ž / 11 4 b0 s 3 CAy 3TFn f, 3 1 Ž T s, C sn, C s N y1 rž N F A F, Ns3 is the number of colours; n fsnlq1; ds4y is the space-time dimension; g X, b X are the corresponding n quantities in nl-flavour QCD On-shell matrix element of j Ž m is MŽ m s y1 Z Z Ž mg q n n0, where Gn0 is the bare vertex function and Zq is the on-shell renormalization constant of the light-quark field It should be equal to C Ž m M X Ž m n n plus power-suppressed terms, where X X Xy1 X M m sz Z Ž mg n q n n0 Both matrix elements are ultraviolet-finite; their infrared divergences coincide, because both theories are identical in the infrared region Therefore Žcf w5,4 x, Z Z X q nž m Gn0 CnŽ m s X X Ž 4 Z Z Ž m G q n n0 The on-shell wave-function renormalization cony1 stant is Zqs 1ySVŽ 0, where the bare mass operator is SŽ p srps Žp V Only the heavy-quark loop contributes Ž Fig 1a, and we obtain at two loops g0 4 M y4 Ž dy1ž dy4 SVŽ 0 scftf I, Ž 5 d Ž 4p d where d d k P Ž k g0 4 M y4 icf g0h scftf I, d 4 d Ž p k Ž 4p Ž dy6 IsG Ž Ž dyž dy5ž dy7 sy3g 1y 3 q 9 q PPP, 6 and P Žk is the heavy-quark contribution to the gluon polarization operator This agrees with the decoupling relation wx 3 q sžz rz X 1r q X q n q Fig 1 Mass operator and vertex function The bare vertex function Ž Fig 1b can be calculated at any on-shell momenta of the quarks It is most easy to set both momenta to zero Žthus excludung power-suppressed terms in Ž 4 : 4 y4 g0 M Ž dyn n0 F F d Ž 4p ž d / G s1yc T I y1 Ž 7 The corresponding quantities in the low-energy theory are Z X s1, G X q n0 s1 The ratio Z a n 1 s X s1q 4 Ž gn0db0ydg n1 Z ž 4p / n a 1 s s1q 9 CFTFž / ny1 6 ny3 4p yž ny15 Ž 8 4 Žwhere Db0sy TF and Dgn1 are the contributions 3 of a single flavour can be obtained from wx 4 Using g M y G 0 Ž 1q asž M s qo a, Ž 9 dr Ž s Ž 4p 4p we finally obtain a s nž M s 1qCFTFž / j 4p ny1 85ny67 = X jnž M Ž X To all orders, j sj wx ; this is natural, because the integral of the vector current is the number of quarks minus antiquarks, which is the same in both theories X X Also mqqs mž qq to all orders wx 6 Therefore, C0 can be obtained from the mass decoupling relation w1,3 x The currents j3 and j4 differ from j1 and j0 by insertion of t Hooft-Veltman g5 HV ; they differ from the corresponding currents with the anticom-

167 ( ) AG GrozinrPhysics Letters B AC AC m muting g5 by finite renormalizations qg5 g q s HV m AC HV ZA qg5 g q, qg 5 q s ZP qg 5 q, known to three loops wx 7 : Z s1y4c A a s F 4p Fž / a s 198CFy 107CAq 4TFnf qc q PPP, 4p 9 Z s1y8c P a s F 4p Fž / a s CAq4TFnf qc q PPP Ž 11 4p 9 Insertion of g5 AC does not change the matching coefficient Hence, C sc Z X rz, C sc Z X 3 1 A A 4 0 PrZ P, where Z X, Z X A P contain nl instead of n fsnlq1 The result for n s is, to the best of my knowledge, new The HQET heavy-light current e Ž m sz y1 Ž m 0 e, e s qq can be considered similarly Ž results don t depend on its g-matrix structure Instead of Ž 4, we have now 1r 1r X Z Z Z Ž m G q Q 0 X ž / X X q Z Z Ž m G Q 0 C Ž m s Ž 1 Z ž / On-shell renormalization constant of the static quark y1 field is Z s 1y dsrdv Q vs0, where at two loops wx 4 4 y4 ds g 0 M sycftf Ž dy1 I Ž 13 d dv vs0 Ž 4p The bare vertex function at two loops is G s1 wx 0 4 The ratio Z a s 5 F F 6 s1qc T 1y Ž 14 Z 4p X ž / wx can be obtained from 8 Finally, we arrive at a 89 s X 36 F Fž / e M s 1 q C T e M 15 4p It was stated in wx 4 that it is impossible to match QCD heavy-light currents to HQET without heavyquark loops by equating exclusive matrix elements This statement is incorrect; it was based on Eq 1 of that paper, whose right-hand side should be multiplied by ŽZ X qrzq 1r in this case Taking into ac- count Ž 15, we can write the matching coefficients for HQET without heavy-quark loops in the form X 89 0 wx 4 with a fs a fq 36 Correspondingly, all numerical values of the coefficients of Ž a srp in Table 1 of wx 4 should be shifted by 0103 in this case In conclusion, light-light quark currents should be adjusted according to Ž 10 when crossing a heavyflavour threshold; the heavy-light HQET current Žsay, qb should be adjusted according to Ž 15 when crossing a threshold Ž say, of c quark Acknowledgements I am grateful to KG Chetyrkin for useful discussions, and to T Mannel for hospitality in Karlsruhe References wx 1 W Wetzel, Nucl Phys B 196 Ž ; W Bernreuther, W Wetzel, Nucl Phys B 197 Ž 198 8; W Bernreuther, Ann Phys 151 Ž ; Z Phys C 0 Ž wx SA Larin, T van Ritbergen, JAM Vermaseren, Nucl Phys B 438 Ž wx 3 KG Chetyrkin, BA Kniehl, M Steinhauser, Nucl Phys B 510 Ž wx 4 DJ Broadhurst, AG Grozin, Phys Rev D 5 Ž wx 5 KG Chetyrkin, Phys Lett B 307 Ž wx 6 KG Chetyrkin, BA Kniehl, M Steinhauser, Nucl Phys B 490 Ž wx 7 SA Larin, JAM Vermaseren, Phys Lett B 59 Ž ; SA Larin, in: DYu Grigoriev, VA Matveev, VA Rubakov, PG Tinyakov Ž Eds, Quarks-9, World Scientific, Singapore, 1993, p 01, hep-phr93040; Phys Lett B 303 Ž wx 8 DJ Broadhurst, AG Grozin, Phys Lett B 67 Ž

168 31 December 1998 Physics Letters B The infrared behavior of one-loop gluon amplitudes at next-to-next-to-leading order Zvi Bern a, Vittorio Del Duca b,1, Carl R Schmidt c a Department of Physics and Astronomy, UniÕersity of California at Los Angeles, Los Angeles, CA , USA b Particle Physics Theory Group, Dept of Physics and Astronomy, UniÕersity of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK c Department of Physics and Astronomy, Michigan State UniÕersity, East Lansing, MI 4884, USA Received 0 October 1998 Editor: M Cvetič Abstract For the case of n-jet production at next-to-next-to-leading order in the QCD coupling, in the infrared divergent corners of phase space where particles are collinear or soft, one must evaluate Ž nq1 -parton final-state one-loop amplitudes through OŽe, where e is the dimensional regularization parameter For the case of gluons, we present to all orders in e the required universal functions which describe the behavior of one-loop amplitudes in the soft and collinear regions of phase space An explicit example is discussed for three-parton production in multi-regge kinematics that has applications to the next-to-leading logarithmic corrections to the BFKL equation q 1998 Elsevier Science BV All rights reserved The quest to obtain ever increasing precision in perturbative QCD requires calculations to higher orders in the QCD coupling, a s Over the years, significant effort has been expended on computing the next-to-leading order Ž NLO contributions to multi-jet rates within perturbative QCD An important next step in the endeavor to obtain higher precision would be the computation of next-to-next-toleading-order Ž NNLO contributions to multi-jet rates As an example, although the NLO contributions to e q e y 3 jets have been computed for some time now w1, x, the NNLO contributions have yet to be obtained A calculation of these NNLO contributions would be needed to further reduce the theoretical uncertainty in the determination of as from event shape variables wx 3 Considerable effort has 1 On leave of absence from INFN, Sezione di Torino, Italy also been expended in the computation of leading and next-to-leading logarithmic contributions to the BFKL equations for parton evolution at small x w4,5 x These two problems are, of course, connected since the logarithms that are resummed in the BFKL equations also appear at fixed orders of perturbation theory Indeed, the one-loop Lipatov vertex may be extracted w6,7x from one-loop five-gluon helicity amwx 8 plitudes In order to compute a cross section at NNLO, three series of amplitudes are required in the squared matrix elements: a tree-level, one-loop, and two-loop amplitudes for the production of n particles; b tree-level and one-loop amplitudes for the production of n q 1 particles; c tree-level amplitudes for the production of nq particles For the case of NNLO q y e e 3 jets the five-parton final-state tree wx 9 amplitudes, as well as four-parton final-state one-loop amplitudes exist in both helicity w10x and squared r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

169 ( ) Z Bern et alrphysics Letters B matrix-element forms w11 x, but as we discuss below, in order to be used in NNLO computations additional terms enter because of infrared issues For the required two-loop three-parton final-state amplitudes no computations exist, as yet Indeed, no two-loop amplitude computations exist for cases containing more than a single kinematic variable, except in the special cases of maximal supersymmetry w1 x Besides these amplitudes it is also important to have a detailed understanding of the infrared singularities that arise from virtual loops and from unresolved real emission when the momenta of particles become either soft or collinear ŽFor hadronic initial states infrared divergences are also associated with initial-state parton distribution functions These infrared divergences show themselves as poles in the dimensional regularization parameter e sž 4 y D r By the Kinoshita-Lee-Nauenberg theorem w13x the infrared singularities must cancel for sufficiently inclusive physical quantities However, it is only when the various contributions are combined that the infrared singularities cancel At NLO the structure of the infrared singularities has been extensively studied The singularities occur in a universal way, ie independent of the particular particle production amplitude considered Accordingly, soft singularities have been accounted for by universal soft functions w14,15 x, and collinear singuw16 x A de- larities by universal splitting functions tailed discussion of the infrared singularities at NLO for e q e y jets may be found, for example, in Ref w17 x At NNLO the situation is less developed, although some work has already been performed to illuminate the structure of infrared divergences In particular, in the squared tree-level amplitudes, any two of the n q produced particles can be unresolved; accordingly the ensuing soft singularities, collinear singularities, and mixed collinearrsoft singularities, have been accounted for by double-soft functions w15 x, double-splitting functions and softw18 x, respectively Furthermore, splitting functions the universal structure of the coefficients of the 1re 4,1re 3 and 1re poles has also been deterw19x for the two-loop virtual contributions for mined n-particle productions In this letter we shall discuss the Ž nq1 -parton final-state case b when the soft or collinear particles are gluons In the interference term between a oneloop amplitude for the production of nq1 particles and its tree-level counterpart any one of the produced particles can be unresolved in the final state; the phase-space integration gives at most an additional double pole in e Therefore the expansion in e of the interference term starts with a 1re 4 pole, from mixed virtualrreal infrared singularities, and in order to evaluate it to OŽe 0, the Ž n q 1 -parton one-loop amplitude needs to be evaluated to OŽe ŽA similar need to evaluate one-loop amplitudes to higher orders in e has been previously noted in NNLO deep inelastic scattering w0x and in the NLL corrections to the BFKL equation w1 x For the case of NNLO corrections to e q e y 3 jets, this would be a rather formidable task given the already nontrivial analytic structure of the one-loop e q e y 4 partons helicity amplitudes presented in Ref w10x through OŽe 0 A much more practical approach is to evaluate the amplitudes to higher order in e only in the infrareddivergent regions of phase-space In the collinear and soft regions the amplitudes factorize into sums of products of n-parton final-state amplitudes multiplied by soft or collinear splitting functions ŽThe splitting functions in this letter are for amplitudes; the Altarelli-Parisi ones are roughly speaking the squares of these It is these soft or collinear splitting functions and the n-parton final-state one-loop amplitudes that must be evaluated to higher order in e This is a much simpler task than evaluating the full Ž n q 1 -parton final-state amplitudes beyond OŽe 0 Here we focus on the issue of supplementing one-loop Ž n q 1 -parton final-state amplitudes that are known to OŽe 0 with higher order in e pieces in the soft and collinear regions of phase space The one-loop splitting functions have been given through OŽe 0 w,3 x, and the one-loop soft functions through OŽe 0 may be extracted from the known four- w4x and five-parton w8,5,3x one-loop amplitudes Below, we provide the one-loop gluon splitting and soft functions to all orders in e, leaving the calculational details and a complete listing of the one-loop splitting and soft functions, including fermions, to a forthcoming paper As an example, we apply the framework outlined above to the one-loop amplitude for three-parton production in multi-regge kinematics w6 x, for which

170 170 ( ) Z Bern et alrphysics Letters B the produced partons are strongly ordered in rapidity In NNLO and in next-to-leading-logarithmic corrections to two-jet scattering, this amplitude appears in an interference term multiplied by its tree-level counterpart Because of the rapidity ordering, phase-space integration does not yield any collinear singularities; however, the gluon which is intermediate in rapidity can become soft Accordingly, the one-loop amplitude must be determined to OŽe 0 plus the contribution with the soft intermediate gluon evaluated to OŽ e w1 x To determine the soft gluon contribution we use our all orders in e determination of the soft functions together with previous all orders in e determinations of the four-gluon amplitudes w7,,8 x We first briefly review properties of n-gluon scattering amplitudes, since we use these below The tree-level color decomposition is Žsee eg Ref w9x for details and normalizations Mn tree Ž 1,, n Ý Ž ny a a a sg TrŽ T s 1 T s PPP T s Ž n sgsnrzn =m tree s Ž 1,s Ž,,s Ž n, Ž 1 n where SnrZn is the set of all permutations, but with cyclic rotations removed We have suppressed the dependence on the particle polarizations i and mo- menta k i, but label each leg with the index i The T a i are fundamental representation matrices for the Yang-Mills gauge group SUŽ N c, normalized so that Ž a b ab Tr T T s d The color decomposition of oneloop multi-gluon amplitudes with adjoint states cirw30x culating in the loop is? nr@ q1 1-loop n Mn 1,, n sg Ý Ý Grn; j s js1 sgsnrsn; j = m 1-loop n; j s Ž 1,,s Ž n, where? x@ denotes the greatest integer less than or Ž a 1 a equal to x, Gr 1' N Tr T PPP T n,gr Ž n;1 c n; j 1 str T a 1 PPP T a jy 1 Tr T a j PPP T a n for j)1, and Sn; j is the subset of permutations Sn that leaves the trace structure Grn; j invariant It turns out that at one-loop the mn; j)1 can be expressed in terms of 1-loop m w31 x, so we need only discuss this case in this n;1 letter The amplitudes with fundamental fermions in the loop contain only the m 1-loop n;1 color structures and are scaled by a relative factor of 1rN c The behavior of color-ordered one-loop amplitudes as the momenta of two color adjacent legs becomes collinear, is w,3x aib 1-loop tree la lb 1-loop n;1 Ý yl ny1;1 ls" m Split a, b m = K l qsplit 1-loop a l a,b l Ž b yl 4 = m tree K l, Ž 3 ny1 where l represents the helicity, m 1-loop n;1 and m tree n are color-decomposed one-loop and tree sub-amplitudes with a fixed ordering of legs and a and b are consecutive in the ordering, with k aszk and k bs Ž 1 y z K The splitting functions in Eq Ž 3 have square-root singularities in the collinear limit For the case of only gluons, the tree splitting functions splitting into a positive helicity gluon Žwith the convention that all particles are outgoing is Splitq tree Ž a q,b q s0, y1 tree y y Splitq Ž a,b s, (z Ž 1yz wabx z tree y q Splitq Ž a,b s, (z Ž 1yz ² ab: Ž 1yz tree q y Splitq Ž a, b s, Ž 4 (z Ž 1yz ² ab: where the remaining ones may be obtained by parity The spinor inner products w3,9x are ² ij: s² i y < j q : and wijxs² i q < j y :, where < i " : are massless Weyl spinors of momentum k i, labeled with the sign of the helicity They are antisymmetric, with norm < ² ij: < s < wijx< s( s ij, where s ij sk i Pk j The one-loop splitting functions are, Split 1-loop Ž a y,b y s Ž G f qg n Split tree Ž a y,b y, q Split 1-loop Ž a ",b sg n Split tree Ž a ",b, q 1 wabx 1-loop q q f Splitq Ž a,b syg (z Ž 1yz ² ab: q q Ž 5

171 ( ) Z Bern et alrphysics Letters B The function G f arises from the factorizing contributions and the function G n arises from the nonw33x and are given factorizing ones described in Ref through OŽe 0 by w,3x 1 N f f G s 1y zž 1yz qo Ž e, 48p ž N n G sc G y ž / c z 1yz ys / ab 1 m e p qlnž z lnž 1yz y qo Ž e, Ž 6 6 with N the number of quark flavors and f 1 G Ž 1qe G Ž 1ye cg s Ž 7 ye Ž 4p G Ž 1ye As at tree-level, the remaining splitting functions can be obtained by parity The explicit values were obtained by taking the limit of five-point amplitudes; the universality of these functions for an arbitrary number of legs was proven in Ref w33 x A listing of one-loop splitting functions through OŽe 0 also inw,3 x To volving fermions may be found in Refs OŽe 0, these splitting functions are independent of the regularization scheme parameter, 1 HV or CDR scheme, dr s ½ Ž 8 0 FDH or DR scheme, where CDR denotes the conventional dimensional regularization scheme, HV the t Hooft-Veltman scheme, DR the dimensional reduction scheme, and FDH the four-dimensional helicity scheme ŽFor further discussions on scheme choices see Refs w4,34 x We have extended the above results for one-loop gluon splitting, as well as similar ones for soft functions, to all orders in e in several ways The first way is by following the methods of Ref w33x and extending the discussion to include soft limits, but being careful to keep all contributions to higher order in e In this method the contributions are divided into the classes of factorizing contributions, that may be obtained directly from one-loop three-point Feynman diagram calculations and from non-factorizing contributions, that are linked to the e infrared-singular poles in e An important ingredient in this construction is that the set of all possible loop integral functions that may enter into an amplitude are known functions to all orders in e The method makes clear the universality of the splitting and soft functions since it does not rely on the computation of any particular amplitude As a second independent method for obtaining the values of the splitting and soft functions we have computed the amplitudes gggh using the effective ggh coupling w35x due to a heavy fermion loop, again being careful to keep all higher order in e contributions ŽFor a discussion of the calculation valid through OŽe 0 see Ref w36 x This is a convenient amplitude from which to extract the splitting and soft functions since it involves only four-point kinematics with one massive leg; the massive leg H ensures that the gggh amplitude has well defined limits when gluons are collinear or soft As a third independent check we have also verified that the non-factorizing contributions obtained for the special case of N s 4 supersymmetric amplitudes agree with the above determinations ŽThe N s 4 case has no factorizing contributions and does not provide a check of these The Ns4 four- five- and six-point ampliw37x making it straightforward to extract the collinear and tudes have been given to all orders in e in Ref soft limits in terms of limits of loop integral functions From these calculations, our results for the all orders in e contributions to the functions Ž 6 appearing in splitting functions are cg N f f G s 1yed R y Ž 3ye Ž ye Ž 1ye ž N / c = e m ž / ys ab ž / zž 1yz, / e e m 1 1yz n G scg y G Ž 1ye G ys ž ab e z yz k = 1qe q e Li, 1yz Ý kž / ks1,3,5, Ž 9

172 17 ( ) Z Bern et alrphysics Letters B where the polylogarithms are defined as w38 x Li 1Ž z sylnž 1yz z dt Li kž z sh Li ky1ž t Ž ks,3, ß t 0 ` Ý n z s k Ž 10 n ns1 It is not difficult to verify that Eq Ž 9 agrees with Eq Ž 6 through OŽe 0 Although not obvious, the n expression for G in Eq Ž 9 is symmetric in z lž1 yz ; indeed its expansion to OŽe may be written as m e n G scg ž ys ab / ½ zž 1yz ye 1 p = y qln zlnž 1yz y e 6 qln zlnž 1yz ye Li 3Ž z qli 3Ž 1yz 1 yz Ž 3 qe y ln zlnž 1yz ln zž 1yz 6 p y ln zln Ž 1yz q ln zlnž 1yz y p qo Ž e Ž The behavior of one-loop amplitudes in the soft limit is very similar to the above As the momentum k of an external leg becomes soft the color-ordered one-loop amplitudes become 1-loop " m n;1,a,k,b, < k 0 ssoft tree Ž a,k ",b m 1-loop ny1;1,a,b, qsoft 1-loop Ž a,k ",b m tree ny1,a,b,, Ž 1 with the tree-level soft functions ² ab: tree q Soft Ž a, k, b s, ² ak:² kb: ywabx tree y Soft Ž a,k,b s Ž 13 wakxwkbx Following analogous methods as for the collinear case, we have computed the one-loop gluon soft function to all orders of e, with the result, Soft 1-loop Ž a,k ",b 1 tree " sysoft Ž a,k,b cg e = ž / ak kb e m Ž ysab pe Ž ys Ž ys sinž pe Ž 14 The soft function Ž 14 does not depend on Nf or dr and through OŽe it is Soft 1-loop Ž a,k ",b tree " sysoft Ž a,k,b cg ž ž / e m Ž ys ab Ž ys Ž ys / ak kb = 1 p 7p 4 3 q q e qo Ž e Ž 15 e Through OŽe 0 this agrees with the results that may be extracted from four- w4x and five-parton w8,5,3x one-loop amplitudes that are known through OŽe 0 We now apply these results for one-loop splitting Ž 5 and soft Ž 14 functions to the case of three-gluon production in multi-regge kinematics To do so, we also need the exact four-gluon one-loop amplitude through OŽ e In fact, this is known exactly to all orders of e From the string-inspired decomposition of the Ž unrenormalized one-loop four-gluon sub-amw3,31 x, we plitude write ž / ž / c c Nf N 1-loop g f f s m4;1 sa4q 4y A4q 1y A 4, Ž 16 N N where A4 g, ya4 f, and A s 4 are the contributions from an Ns4 supersymmetric multiplet, an Ns1 chiral multiplet, and a complex scalar, respectively We can also write A4 x scg m tree 4 V x, xsg, f,s, Ž 17 with m tree 4 the corresponding tree-level subamplitude The functions V f and V s depend on the helicity configuration and can be extracted to all orders in e from Ref w8x by taking the massless limit For

173 1-loop configurations of type m Ž1 y, y,3 q,4 q 4;1 this yields, s f 3 Ds6y e V syiž s3 ye I 4, s 13 s s 3 Ds6y e V s ž 1ye / I s3 s 1 s 3 Ds8y e q e 1ye I 4, 18 s 1 1-loop while for configurations of type m Ž 4;1 1 y, q, y 3,4 q we have s3 s f 1 V s I s q I Ž 1 Ž s3 s s s1 s3 Ds6y e y Ž 1ye I 4, s13 s1 s3ž s1ys3 s Ds6y e V s y 3 e Ž I 3 Ž s 3 s13 s1 s Ds6y e 3 yi Ž s y s I Ž s with Ž s 13 1 Ds 6y e qs3 IŽ s1 y Žs1 I Ž s3 s 13 s1 s Ds6y e 3 qs I s q e I Ž Ds6y e 3 1 s 13 s = Ds6y e Ž s qi Ž s y = s 13 Ds 6y e Ds 6y e Ž I3 Ž s3 qi3 Ž s1 s s q I Ds6y e 4 s13 s1 s3 Ds8y e q e Ž 1ye I, Ž 19 s 13 e m / 1 e / ž ys / ĨŽ s s, ž ys e Ž 1ye m 1 Ds 6y e Ĩ s s, ž ys e Ž 1ye Ž 3ye m 1 Ds 4y e Ĩ3 s sy, e e 4 ( ) Z Bern et alrphysics Letters B s / e m 1 Ds 6y e Ĩ3 Ž s s, ž Ds 4y e I 4 ž / e ž s / 1ž ys e Ž 1ye Ž 1ye / s / e m s 3 s F1 ye,ye ;1ye ;1q s ž s e 3 1 m s 1 q F ye,ye ;1ye ;1q, Ds 6y e I sy I 4 4 1ye Ds4y e Ds 4y e Ds4y e q I3 s3 q I3 s 1, 1 s1 s Ds8y e 3 I sy I 4 4 3ye s13 s 3 Ds6y e q I3 s3 s 13 Ds6y e s 1 Ds6y e q I3 s 1 0 s 13 The functions I n are scalar loop integrals in the indicated dimension scaled by prefactors so as to make them dimensionless The function V g obtained from the N s 4 multiplet w7,,37x has the same functional form for either helicity configuration To all orders in e it is g Ds 4y e s V syi4 yedr V 1 Any partial amplitude of the type m 1-loop 4:3 may then be obtained from sums of permutations of the m 1-loop 4;1 w30 x We next need the dispersive part of this amplitude in the high-energy limit, s 4 t The leading color orderings of the sub-amplitudes of type m 1-loop 4;1 are w x Ž y Xq Xq y Ž y Xq y 6 A, A, B, B, A, B, B, A Xq, Ž y Xq y Xq Ž y y Xq A, A, B, B, and A, B, B, A Xq where we take the Mandelstam variables to be sss AB, tssbb X, and uss X In Eqs Ž 18 and Ž 1 AB orderings of type 1-loop m Ž y,y,q,q 4;1 occur with s1 s and s3 t, while in Eqs Ž 19 and Ž 1 orderings of type 1-loop m Ž y,q,y,q 4;1 occur with s1 u and s3 t or s1 t and s3 u However, for the second helicity configuration, the functions in Eqs Ž 19 and Ž 1 are symmetric under the exchange of s and 1

174 174 ( ) Z Bern et alrphysics Letters B s 3 Thus we can limit the analysis to one ordering for each type Using the usual prescription lnž t slnž yt yip Ž for t-0, we have e e m m ž t / ž yt / Re s cosž pe In the high-energy limit, s4yt, t F ye,ye ;1ye ;1q 1ž s 1 / t sg Ž 1ye G Ž 1qe qož s 1 / pe t s qo, sinž pe ž s / 1 ž s / 1 e m s 1 F1ž ye,ye ;1ye ;1q t / e m t ž / 1 yt ž s 1 / s F ye,1;1ye ;1q sy ž yt / t ž s 1 / e m s 1 e cž 1 yc Ž ye qln yt q O Ž 3 with s1 s or s1 u Since usysyt, for either choice of s1 the dispersive parts are the same to the required accuracy, and we can take s1 s in Eq Ž 3 Substituting Eqs and Ž 3 in Eq Ž 0, and using the identity cosž pe yp s c Ž 1qe yc Ž ye, Ž 4 sinž pe we obtain / m Ds 4y e Disp I4 sy c Ž 1qe ž e yt e s yc Ž ye qc Ž 1 qln yt ž / t q O Ž 5 s In addition, in the high-energy limit both Eq Ž 18 with s s and s t, and Eq Ž with s1 u and s t reduce to 3 ž / e ž / ž / e m 1 t f V sy qo, yt e Ž 1ye s ž / m 1 t s V s q O, Ž 6 yt e Ž 1ye Ž 3ye s g while the function V, Eq Ž 1, is obtained from Eqs Ž 5 and Ž 6 Using Eqs Ž 16, Ž 17, Ž 1, Ž 5 and Ž 6, and the fact that the proportionality factor between each tree-level subamplitude and its one-loop correction is the same for all color orderings, we obtain the unrenormalized four-gluon one-loop amplitude in the high energy limit to all orders in e, Disp M 1-loop Ž A y, A Xq, B Xq, B y 4 e m tree y Xq Xq y 4 G ž yt / sm A, A, B, B g c ½ 1 = Nc 1ye cž 1qe e Ž 1ye s yc Ž ye qc Ž 1 qln yt / 1ydR e 1ye q y4 q N f 5 Ž 7 3ye 3ye Following the methods of Ref wx 6, we can then extract the unrenormalized one-loop correction to the helicity-conserving vertex, to NLL accuracy, Disp C ggž1 Ž yp, p X yq a a ggž0 C Ž yp, p X yq a a Disp C ggž1 Ž yp, p X yq b b s ggž0 C Ž yp, p X yq b b ž / e m 1 scg Nc Ž 1ye yt e Ž 1ye = c Ž 1qe yc Ž ye qc Ž 1 ½ 1ydR e 1ye q y q N Ž 8 3ye 3ye ž f 5

175 ( ) Z Bern et alrphysics Letters B Eq Ž 8 is valid to all orders in e for dr s0 and 1; it agrees with the one-loop correction to the helicity-conserving vertex computed in Ref wx 6 to OŽe 0 and with the one computed in Ref w6,39x to all orders in e, for dr s1 In an inclusive high-energy two-jet cross section at NNLO and in the NLL corrections to the BFKL kernel, the five-gluon one-loop amplitude is multiplied by the corresponding tree-level amplitude with the intermediate gluon k integrated over its phase space Assuming multi-regge kinematics, the only infrared divergence that can arise is in the soft limit for the intermediate gluon, k 0 It gives a single pole in e Thus, in order to generate correctly all the finite terms in the squared amplitude, the five-gluon one-loop amplitude must be computed exactly to OŽe 0, and must be augmented by the OŽ e corrections in the soft limit for the intermediate gluon To achieve that, we need the dispersive part of Eq Ž 1 in the physical region, s ab) 0, s ak) 0, s kb) 0 ŽThe other leading color orderings yield the same result Using Eqs Ž 14 and Ž, and the identity Ž 4, we can write the dispersive part of the soft function to all orders in e as, Disp Soft 1-loop Ž a,k ",b c G tree " sysoft Ž a,k,b e = ž s s / ak kb e m s ab 1qec 1ye yec 1qe Ž 9 In addition, using the strong rapidity ordering, y 4y4y ; < k <, < k <, < k <, Ž 30 a b ah H bh the mass-shell condition for the intermediate gluon gives sak skb sabs Ž 31 < k H < Subsequently, Eq Ž 9 becomes, Disp Soft 1-loop Ž a,k ",b ž k H / cg m tree " sysoft Ž a,k,b e < < = 1qec Ž 1ye yec Ž 1qe Ž 3 e The soft limit for the tree-level five-gluon sub-amplitudes, m tree A y, A Xq,k ", B Xq 5 Ž, B y ssoft tree Ž A X,k ", B X m tree A y, A Xq, B Xq 4 Ž, B y, Ž 33 holds for arbitrary kinematics Thus, the unrenormalized five-gluon one-loop amplitude in the multi-regge kinematics, and in the soft limit for the intermediate gluon and to all orders in e, is obtained by using Eq Ž 1, with the four-gluon one-loop amplitude Ž 7, the loop soft function Ž 3, and Eq Ž 33, yielding 1-loop y Xq " Xq Disp M Ž A, A,k, B, B y < 5 k 0 tree y Xq " Xq sg c y G M 5 A, A,k, B, B < k 0 ež ž / ½ m 4 = N y q c Ž 1qe yt e e s yc Ž 1ye qc Ž 1 qln yt / 1 1ydR e q ž y4 e Ž 1ye 3ye / q 1ye 5 e Ž 1ye Ž 3ye N f yn ž / e m 1 c < k H < e c ž 1qec Ž 1ye yec Ž 1qe Ž 34 To OŽ e, Eq Ž 34 reads 1-loop y Xq " Xq Disp M y 5 Ž A, A,k, B, B < k 0 tree y Xq " Xq sg c M Ž A, A,k, B, B y < ½ G 5 k 0 e m 4 s = Nc y q ln qp ž / ž yt e e yt 64 dr y y qz Ž 3 ey ey dr e / b y qn q e e 9 7 fž / ž / k 3 H ž /5 e m 1 p ync y qo Ž e, Ž 35 < < e

176 176 ( ) Z Bern et alrphysics Letters B with b sž 11N y N 0 c f r3 We have checked that Eq Ž 35 agrees to OŽe 0 with the five-gluon oneloop amplitude, with strong rapidity ordering and in the soft limit for the intermediate gluon w8,7 x The above result may be used to verify the virtual nextto-leading log corrections to the Lipatov vertex for use in the BFKL equation as is done in Ref wx 7 The same type of analysis may be applied more generally to the problem of NNLO QCD corrections The one-loop gluon splitting and soft functions that we have presented here are valid to all orders in the dimensional regularization parameter, e This allows them to be used in NNLO calculations with infrared singular phase space where terms of up to two powers in e are necessary Previous explicit determinations of the one-loop collinear splitting functions w,3x were not performed to the required order in e A systematic discussion of the soft and collinear splitting functions and further calculational details, including the case of external fermions, will be presented elsewhere In particular, these functions can be used to aid in the computation of NNLO contributions to e q e y 3 jets once all the matrix elements are available However, much more remains to be done before this is realized Acknowledgements This work was supported by the US Department of Energy under grant DE-FG03-91ER4066, by the US National Science Foundation under grant is PHY and by the EU Fourth Framework Programme Training and Mobility of Researchers, Network Quantum Chromodynamics and the Deep Structure of Elementary Particles, contract FMRX- CT Ž DG 1 - MIHT The work of VDD and CRS was also supported by NATO Collaborative Research Grant CRG References wx 1 RK Ellis, DA Ross, AE Terrano, Nucl Phys B 178 Ž ; K Fabricius, I Schmitt, G Kramer, G Schierholz, Phys Lett B 97 Ž ; Z Phys C 11 Ž wx P Nason, Z Kunszt, in: Z Physics at LEP1, CERN Yellow report Ž 1989 ; G Kramer, B Lampe, Z Phys C 34 Ž ; C 4, 504 Ž E Ž 1989 ; Fortschr Phys 37 Ž ; S Catani, MH Seymour, Phys Lett B 378 Ž , hep-phr96077 wx 3 By PN Burrows, et al, in: New Directions for High Energy Physics: Proceedings Snowmass 96, DG Cassel, L Trindle Gennari, RH Siemann Ž Eds, hep-exr96101; M Schmelling, Proc of 8th International Conf on High Energy Physics, Warsaw, Z Ajduk, AK Wroblewski Ž Eds, World Scientific, 1997, hep-exr wx 4 EA Kuraev, LN Lipatov, VS Fadin, Zh Eksp Teor Fiz 71 Ž wsov Phys JETP 44 Ž x; EA Kuraev, LN Lipatov, VS Fadin, Zh Eksp Teor Fiz 7 Ž wsov Phys JETP 45 Ž x; Ya Ya Balitsky, LN Lipatov, Yad Fiz 8 Ž wsov J Nucl Phys 8 Ž wx 5 VS Fadin, LN Lipatov, Phys Lett B 49 Ž hep-phr98090 wx 6 V Del Duca, CR Schmidt, Phys Rev D 57 Ž hep-phr wx 7 V Del Duca, CR Schmidt, preprint EDINBURGH 98r1, MSUHEP-8098, hep-phr wx 8 Z Bern, L Dixon, DA Kosower, Phys Rev Lett 70 Ž hep-phr93080 wx 9 K Hagiwara, D Zeppenfeld, Nucl Phys B 313 Ž ; NK Falck, D Graudenz, G Kramer, Nucl Phys B 38 Ž ; FA Berends, WT Giele, H Kuijf, Nucl Phys B 31 Ž w10x Z Bern, L Dixon, DA Kosower, Nucl Phys Proc Suppl 51C Ž , hep-phr ; Z Bern, L Dixon, DA Kosower, S Weinzierl, Nucl Phys B 489 Ž , hepphr ; Z Bern, L Dixon, DA Kosower, Nucl Phys B 513 Ž , hep-phr w11x EWN Glover, DJ Miller, Phys Lett B 396 Ž , hep-phr ; JM Campbell, EWN Glover, DJ Miller, Phys Lett B 409 Ž , hep-phr w1x Z Bern, JS Rozowsky, B Yan, Phys Lett B 401 Ž , hep-phr97044; Z Bern, L Dixon, DC Dunbar, M Perelstein, JS Rozowsky, Nucl Phys B 530 Ž , hep-thr98016 w13x T Kinoshita, J Math Phys 3 Ž ; TD Lee, M Nauenberg, Phys Rev 133 Ž w14x DR Yennie, SC Frautschi, H Suura, Ann Phys 13 Ž ; A Bassetto, M Ciafaloni, G Marchesini, Phys Rep 100 Ž w15x FA Berends, WA Giele, Nucl Phys B 313 Ž w16x VN Gribov, LN Lipatov, Yad Fiz 15 Ž wsov J Nucl Phys 46 Ž x; LN Lipatov, Yad Fiz 0 Ž wsov J Nucl Phys 0 Ž x; G Altarelli, G Parisi, Nucl Phys 16 Ž ; Yu L Dokshitzer, Zh Eksp Teor Fiz 73 Ž wsov Phys JETP 46 Ž x w17x WT Giele, EW N Glover, Phys Rev D 46 Ž w18x JM Campbell, EWN Glover, Nucl Phys B 57 Ž , hep-phr971055; S Catani, M Grazzini, preprint CERN THr98-35, ETH THr98-7, hep-phr w19x S Catani, Phys Lett B 47 Ž hep-phr w0x EB Zijlstra, WL van Neerven, Nucl Phys B 383 Ž

177 ( ) Z Bern et alrphysics Letters B w1x VS Fadin, R Fiore, MI Kotsky, Phys Lett B 389 Ž hep-phr96089 wx Z Bern, L Dixon, DC Dunbar, DA Kosower, Nucl Phys B 45 Ž hep-phr94036 w3x Z Bern, L Dixon, DA Kosower, Nucl Phys B 437 Ž hep-phr w4x RK Ellis, JC Sexton, Nucl Phys B 69 Ž ; Z Bern, DA Kosower, Nucl Phys B 379 Ž ; Z Kunszt, A Signer, Z Trocsanyi, Nucl Phys B 411 Ž , hep-phr w5x Z Kunszt, A Signer, Z Trocsanyi, Phys Lett B 336 Ž hep-phr w6x VS Fadin, LN Lipatov, Nucl Phys B 406 Ž w7x MB Green, JH Schwarz, L Brink, Nucl Phys B 198 Ž w8x Z Bern, AG Morgan, Nucl Phys B 467 Ž hep-phr w9x M Mangano, SJ Parke, Phys Rep 00 Ž ; L Dixon, in: Proceedings of Theoretical Advanced Study Institute in Elementary Particle Physics Ž TASI 95, DE Soper Ž Ed, hep-phr w30x Z Bern, DA Kosower, Nucl Phys B 36 Ž w31x Z Bern, L Dixon, DA Kosower, Ann Rev Nucl Part Sci 46 Ž hep-phr96080 w3x FA Berends, R Kleiss, P De Causmaecker, R Gastmans, TT Wu, Phys Lett B 103 Ž ; P De Causmaecker, R Gastmans, W Troost, TT Wu, Nucl Phys B 06 Ž ; R Kleiss, WJ Stirling, Nucl Phys B 6 Ž ; Z Xu, D-H Zhang, L Chang, Nucl Phys B 91 Ž w33x Z Bern, G Chalmers, Nucl Phys B 447 Ž hep-phr w34x S Catani, MH Seymour, Z Trocsanyi, Phys Rev D 55 Ž hep-phr w35x S Dawson, Nucl Phys, B 359 Ž ; A Djouadi, M Spira, P Zerwas, Phys Lett B 84 Ž w36x CR Schmidt, Phys Lett B 413 Ž hepphr w37x Z Bern, L Dixon, DC Dunbar, DA Kosower, Phys Lett 394B Ž hep-thr w38x L Lewin, Polylogarithms and associated functions, North- Holland w39x VS Fadin, R Fiore, Phys Lett B 94 Ž ; LN Lipatov, Phys Rep 86 Ž , hep-phr961076

178 31 December 1998 Physics Letters B Higgs boson interactions in supersymmetric theories with large tanb Will Loinaz a, James D Wells b,1 a Institute for Particle Physics and Astrophysics, Physics Department, Virginia Tech, Blacksburg, VA , USA b Stanford Linear Accelerator Center, Stanford UniÕersity, Stanford, CA 94309, USA Received 4 August 1998; revised 1 November 1998 Editor: M Cvetič Abstract We show that radiative corrections to the Higgs potential in supersymmetric theories with large tan b generically lead to large differences in the light Higgs boson decay branching fractions compared to those of the standard model Higgs boson In contrast, the light Higgs boson production rates are largely unaffected We identify Wh associated production followed by Higgs boson decays to photons or to leptons via WW ) as potential experimental probes of these theories q 1998 Elsevier Science BV All rights reserved PACS: 1480Cp; 160-i; 160Fr; 160Jv Keywords: Higgs boson; Supersymmetry; Collider phenomenology It is well known that supersymmetry requires two Higgs doublets to give masses to the up-type quarks and to the down-type quarks Hence, we use the terminology up-higgs and down-higgs to indicate these two Higgs bosons, Hu and Hd respectively The ratio of the vacuum expectation values, ² H 0 u : tanb', Ž 1 0 ² : H d has both theoretical and experimental consequences Theoretically, a large value of tan b near mtrmb appears to be required in simple b y t y t Yukawa unification theories such as supersymmetric SOŽ 10 wx 1 Since the b-yukawa and the t-yukawa couplings are equal at the high scale they will be nearly equal at the low scales This is because the bottom and top quarks are distinguished only by their weak hypercharge, and renormalization group evolution is dominated by large Yukawa and QCD interactions Furthermore, a large value of tanbr30 is required for minimal gauge mediated models which solve the soft-cp problem w,3 x That is, CP violation effects are non-existent in the soft mass terms of a softly broken supersymmetric theory with gauge mediated supersymmetry breaking and with B Ž M s 0 at the messenger m 1 Work supported by the Department of Energy under contract DE-AC03-76SF r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S X

179 ( ) W Loinaz, JD WellsrPhysics Letters B scale M IfBm is zero at the messenger scale then arbitrary phases in the lagrangian can be rotated away, and CP violating effects do not get induced by renormalization group running However, renormalization group running does induce a non-zero Ž CP conserving Bm term at the weak scale which is phenomenologically viable only if tanb is large w3,4 x This can be seen readily from one of the conditions for proper electroweak symmetry breaking: Bm Ž M qdbm sinbcosbs m A Since B Ž M m s 0 and DBm is a small renormalization group induced mass, and ma must be large enough to escape detection, the right-hand side of the equation is much less than 1 Therefore, sinbcosb<1 tanb41 Ž 3 The tanb<1 solution is not viable in generating the B term, since the large top Yukawa coupling would disrupt the other condition for electroweak symmetry breaking, and make perturbative gauge coupling unification impossible Unification of the third family Yukawa couplings and CP conservation in gauge mediated models both imply a strong preference for large tanb It is important to keep in mind that these preferences survive even when there is non-universality among the scalar masses in the theory In supergravity, non-universality is even expected Furthermore, large tan b is of course accessible in parameter space even if third family Yukawa unification is not realized, or if the CP violating interactions are kept under control in a supersymmetric theory by some other means 0 The mass matrix of the scalar Higgs bosons in the H, H 0 4 basis can be parametrized by, ž / Ž A Z A Z d m Asin bqm Zcos b ysinbcosb m Aqm Z A11 A 1 M s q, Ž 4 ysinbcosb m qm m cos bqm sin b ž A1 A u / where ma is the physical pseudoscalar Higgs boson mass, and Aij are radiative corrections induced by particle and sparticle loops The dependences of Aij on the sparticle masses and mixing angles can be found in several Refs wx 5 For larger tan b and arbitrary mixing, a random scan over supersymmetric parameter space is enough to convince oneself that the off diagonal A 1 can be as large as " 40 GeV Models with large tan b and large ma allow suppression of the bb decay mode resulting from cancellation of tree-level and off-diagonal terms in the Higgs mass matrix There are two expansions that we would like to perform to illustrate this feature The first is an expansion of the Higgs mass matrix about pure Hu 0 and Hd 0 mass eigenvalues For these weak eigenstates to be mass eigenstates, mixing in the Higgs mass matrix is not allowed At tree level this is impossible since in our convention M1 is negative definite However, when radiative corrections are introduced a cancellation can occur between the tree-level off-diagonal piece and the quantum corrections For this to occur, ysinbcosb Ž m Zqm A qa1s0 Ž 5 is required Note that if tan b is very large then the tree-level contribution gets small and the radiative correction term, A, can compete with it to yield M s0 wx 6 Solving for the mass eigenvalues we obtain, 1 1 A 1 A Z m sym q Ž 6 sinbcosb m 0 sm cosbqa tanbqa Ž 7 Hd Z 1 11 A 1 0 Hu Z m sym cosbq qa Ž 8 tanb To demonstrate the behavior of these eigenvalues, in Fig 1 we plot their values in the limit that all soft squark masses are the same Ž m susy at the low scale Squark mass degeneracy is broken by mixing introduced through a common A-term at the low scale and the m term Taking tanbs40 and msas300 GeV we plot

180 180 ( ) W Loinaz, JD WellsrPhysics Letters B Fig 1 The mass eigenvalues m, m 0, m 0, and m q m 0 as a function of m under the condition of Eq Ž 5 m q m 0 A H H A H susy A H is d u d d constrained to be greater than ;14GeV from LEPII analyses The value of ma goes to zero near msusy s700gev because with the choice of parameters detailed in the text, A becomes equal to m sinbcosb 1 Z m,m 0, m 0 and m qm 0 as a function of m We have plotted m qm 0 A H H A H susy A H since it is constrained to be d u d d wx 0 greater than ;14 GeV from LEPII analyses 7 We see that Hd and A have mass, m,m 0,ym qa tanb Ž 9 A Hd Z 1 which ranges widely with msusy in this model For A1 not too small ma is naturally large for large tanb Hu 0 has a mass equal to, m sm 0,m qa, Ž 10 h Hu Z which is logarithmically sensitive to m susy 0 The H mass eigenvalue is always light Ž between m and 135 GeV u Z By assumption it cannot decay into bb q y ) Ž) Ž) 0 or t t, and is only allowed to decay into cc, gg, gg, WW, Z g, and ZZ Since Hu carries all the burden of electroweak symmetry breaking, it couples with full electroweak strength to the vector bosons, and the up-type quarks Therefore, Hu 0 acts like the standard model Higgs in this limit in every way except it does not couple to down quarks or leptons wx 8 The decay branching fractions wx 9 for this state are shown in Fig According to Eq Ž 10, the shaded region is theoretically not allowed in the high tanb limit Only in the region m Qm 0 Z H Q135 GeV does one expect to find the Hu 0 at a high energy collider u The above discussion may appear unsatisfactory since it seems to imply that a special finetuning of ma is needed for interesting effects to occur To address this concern, we expand in a second limit, the large ma limit, in order to justify our claim that large deviations occur generically in large tanb and large m A For a moment, let us ignore the radiative corrections and expand about large ma with Aijs 0 The lightest Higgs boson coupling to quarks and vector bosons compared to the standard model Higgs boson coupling to quarks and vector bosons is, Ž hdd susy s1qeaqeaq PPP Ž 11 Ž hdd sm Ž huu susy s1yeaeb y4eaeb q PPP Ž 1 Ž huu sm Ž hvv susy s1yeaeb q PPP Ž 13 Ž hvv sm where e A'm Zrm A and e b '1rtanb Ž 14

181 ( ) W Loinaz, JD WellsrPhysics Letters B u Fig Decay branching fractions of the extreme case where H is a mass eigenstate, and there is no bb coupling For a Higgs mass eigenstate with a suppressed, but not a zero bb coupling, the branching fraction lines will be between the ones presented in this figure and the SM branching fraction lines The shaded region is theoretically not allowed with high tanb There are several points to notice in the above expressions First, the coupling to up-type fermions decouples to the standard model value much faster than the coupling to the down-type fermions by a factor of 1rtan b Second, the vector boson couplings to the light Higgs boson decouples even more rapidly to the standard model result Therefore, even at tree-level, we expect the coupling to down-type fermions to deviate from the standard model much more than the couplings to the up-type fermions and vector bosons Now, if we add radiative corrections to the expansion we get an even more interesting result Keeping the first tree-level correction and the leading radiative correction terms, Ž hdd susy A1 ea s1qea y qppp Ž 15 Ž hdd m eb sm Ž huu susy 1 A1 A 1 A b 4 A A b Z s1ye e y e q e e q PPP Ž 16 Ž huu sm m Z m Z Ž hvv 1 A susy 1 s1yeaeb y ea q PPP Ž 17 Ž hvv m 4 sm Z In this case the up-type fermions and vector boson couplings to the light Higgs boson still decouples rapidly to the standard model result In large tanb models, A 1 ;m Z for much of the parameter space Further, unless ma is very heavy, we expect ea to be greater than or comparable to eb for large tanb In such models there is an OŽ 1 correction to the down-type fermion couplings to the light Higgs boson In particular, we see that if A 1,m ZeAreb then hshu 0 and the coupling to down-type fermions may be shut off completely This is the same result we concluded from the previous expansion above Here, however, we have shown that the coupling to down-type fermions is generically altered by OŽ 1 corrections in either direction depending on the sign of A 1, but the coupling to up-type fermions and vector bosons are standard model-like Thus, in the more general case in which the coupling of the light Higgs to bb is suppressed, h bb branching fraction curves depend sensitively on A1 over the parameter space and interpolate between Fig and the standard model curve At high tanb, the pseudoscalar state A 0 may be produced copiously at high energy colliders, and significant constraints obtained w10 x Here we focus on the CP-even scalar states, since one, which we are calling h, is guaranteed to be light Ž below Q 135 GeV The main production mechanisms for h at high energy hadron

182 18 ( ) W Loinaz, JD WellsrPhysics Letters B Fig 3 BŽ h VV rbž h VV Ž solid and BŽ h bb rbž h bb Ž dash susy sm susy sm as a function of the variation in the hbb coupling for mhs10gev A dotted line at the value 1 corresponds to the standard model reference X q y colliders are qq Wh, gg h, qq, gg tth, qq, gg bbh, and W W h At electron-positron colliders the q y q y q y production modes are e e hz, e e tth, gg h, and W W h Since the h couplings to the top-quark and vector bosons are very close to the standard model values, these cross-sections are not expected to deviate much from the Standard Model This is good news to not have to worry about complicated mixing between non-standard production and decay Only the branching fractions are altered in high tanb models For a Higgs boson mass near 10 GeV and below, the main decay modes are h bb and h tt If these two were the only decay modes then deviations in the down-type fermion couplings would just cancel in the branching fraction and only the standard model result would be seen However, there are other important decay modes of the Higgs boson, most notably the h gg decay mode In the standard model its branching fraction less than ;=10 y3 in the mass range 100 GeV-mh -150 GeV In supersymmetry, this branching fraction can be altered by three factors First, supersymmetry sparticle loops involving squarks and charged Higgs bosons can change the h gg amplitude Given current limits on sparticle masses and some fortuitously large loop integrals for the W boson compared to charged scalars, this effect is small over almost all of the parameter space w11 x Second, changes in the htt and hww couplings can change the h gg amplitude However, we argued above that this does not occur in large tan b models And third, large corrections to hbb and htt couplings can change the denominator in Bh gg This effect is large in the high tanb models Since G Ž h WW ) is unaffected in the large tanb region also, its branching ratio is determined only by the changes wrought by deviations in the down-type fermion couplings In Fig 3 we plot the deviation in the Ž Ž) Ž) h VV Vsg,W,Z branching fraction compared to the standard model branching fraction as a function of Ž hbb rž hbb We also plot the deviation of Bh Ž bb susy sm for reference The plot was made with mh s 10 GeV Since the Higgs boson decays predominantly into down-type fermions at this mass, the branching fraction into vector bosons is most sensitive to deviations in the hbb coupling The sensitivity Ž insensitivity increases even more for BŽ h VV Ž BŽ h bb for lighter Higgs masses The high tan b gauge mediated models with Bms 0 at the messenger scale provide a good specific illustration of the expected deviations It is conservative since the value of m and ma tends to be higher in these high tanb models than supergravity mediated theories If we take the messenger scale to be equal to 10 4 L, where L' F rs Žsee wx 3 for discussions of gauge mediation parameters, then we can self-consistently calculate S Note that m q m y h would be altered at a muon collider

183 ( ) W Loinaz, JD WellsrPhysics Letters B the value of tanb as a function of L which ensures that B Ž M m s0 We find that with chargino mass between 100 GeV and 00 GeV, 105 GeVQmh Q115 GeV Ž 18 BŽ h bb susy 103 Q Q106 Ž 19 BŽ h bb sm BŽ h VV susy 055 Q Q085 Ž 0 BŽ h VV sm As expected the deviation in the Higgs to vector bosons branching fraction is significant We have also scanned parameter space in supergravity theories with non-universal scalar masses and found "OŽ 1 deviations in the h VV branching fractions over much of parameter space An important question is whether or not experiment can see the effects of these deviations in the Higgs branching fractions to vector bosons It is probably difficult to try to see even fairly large effects in the gg rate from gg h fusion The QCD corrections to this rate are nearly a factor of two by themselves with uncertainties of at least 15% 0% w13 x A more fruitful approach may be to search for Wh associated w x ) production with the W decaying leptonically and the Higgs boson decaying to gg 14 or WW ln ln w1,6 x The total rates for these production mechanisms in the standard model at the LHC with ' s s14 TeV is given in Fig 4 The advantages to these modes is the smaller background and the ability to cancel some systematics by taking ratios of this with some other electroweak process, for example, s Ž WZ The maximum possible sensitivity to a deviation in the cross-section can be estimated by assuming a perfect detector with no background and no theoretical uncertainty in the calculation of the standard model rate The sensitivity to deviations in the branching fraction are then given by DBrB G 1r' s I where I is the time Ž y1 integrated luminosity, and s is the specific cross-section we are investigating eg, s lgg For 100 fb this sensitivity is at best 10% at the 1s level At 1000 fb y1, which would require several years of high-luminosity LHC running, the sensitivity is at best 3% at the 1s level The ATLAS and CMS technical design reports w15x provide estimates on signal efficiencies and background rates w15x Ž both real and fake to this process for the ATLAS and CMS detectors that will be installed at LHC An updated study at Snowmass 96 estimated that with Ž ' y1 LHC data and NLC data s s500 GeV and 00 fb one could determine Bh gg to Q16% accuracy w16x in the range 80Qm Q130 GeV h Fig 4 s Wh 3l solid, s Wh lgg dash, and s Wh ltt dot-dash in the standard model at the LHC with 14TeV center-of-mass energy

184 184 ( ) W Loinaz, JD WellsrPhysics Letters B Such a measurement of Bh Ž gg would be advantageous in piecing together large tan b supersymmetric theories This is even true in the gauge mediated model discussed above Gauge mediation models do not allow much mixing in the top squark sector because the A-terms are zero at the messenger scale Therefore, the Higgs masses in this scenario were rather light, and the off-diagonal A1 entry in the Higgs mass matrix was not as large as it generically is in supergravity models In the case of large mixing, the hbb coupling can be more volatile than we found in gauge mediation models and the Higgs mass can be larger In Fig 4 we see that the lgg rate falls off rather rapidly when mh is above 10 GeV, and the Wh 3l rate increases rapidly For Higgs masses above 10 GeV the most useful probe of Higgs coupling deviations may very well be the 3l signal w1,6 x Here, the background could be as large as 1 fb w6 x With the detailed background cuts applied also to the signal we estimate that for a 130 GeV Higgs boson the deviation in the 3l signal at best could be measured to within 0% at 1s with 100 fb y1 With 10 4 fb y1 the rate could be measured to perhaps better than 8% Therefore, it is extremely important, and even necessary to study the 3l signal in detail at the LHC as a probe of non-standard couplings to the light Higgs boson Interestingly, the 3l signal from supersymmetric chargino and neutralino production is highly suppressed in the large tan b region w17 x The Wh 3l signal therefore dominates the 3l signal from superpartner production It should be noted in conclusion that the gauge mediation model which we used to calculate these results is correlated with deviations in the b sg observable wx 4 Substantial deviations in b sg are generically expected in all large tan b models However, given the expected supersymmetry contribution to this amplitude, the expected experimental data, and the uncertainties in QCD corrections, it is likely that deviations will show up as anomalies at the few s level at B-factories, and so conclusive evidence for new physics would not be ensured A suppression of lgg events at LHC for the given mass range would then be helpful to support the claim of supersymmetric theories with high tan b Furthermore, there are large regions of parameter space in the MSSM not necessarily tied to any minimal model which predicts small deviations in Bb Ž sg but OŽ 1 corrections to the hbb coupling These are best investigated by the Higgs boson observables discussed above References wx 1 L Hall, R Rattazzi, U Sarid, Phys Rev D 50 Ž ; G Anderson, S Dimopoulos, L Hall, S Raby, Phys Rev D 47 Ž ; M Carena, M Olechowski, S Pokorski, C Wagner, Nucl Phys B 46 Ž wx M Dine, A Nelson, Y Nir, Y Shirman, Phys Rev D 53 Ž wx 3 K Babu, C Kolda, F Wilczek, Phys Rev Lett 77 Ž ; S Dimopoulos, S Thomas, J Wells, Nucl Phys B 488 Ž ; J Bagger, K Matchev, D Pierce, R-J Zhang, Phys Rev D 55 Ž wx 4 R Rattazzi, U Sarid, Nucl Phys B 501 Ž ; E Gabrielli, U Sarid, Phys Rev Lett 79 Ž , hep-phr wx 5 Y Okada, M Yamaguchi, T Yanagida, Prog Theor Phys 85 Ž ; H Haber, R Hempfling, Phys Rev Lett 66 Ž ; J Ellis, G Ridolfi, F Zwirner, Phys Lett B 57 Ž ; M Carena, M Quiros, C Wagner, Nucl Phys B 461 Ž wx 6 H Baer, J Wells, Phys Rev D 57 Ž wx 7 L3 Collaboration, CERN-EPr98-7, 1998 wx Finite Dmb corrections allow Hu to decay to bb However, small mixing between Hu and Hd can yield an eigenvalue h which does q y not couple to bb, and has an insignificant coupling to t t wx 9 A Djouadi, J Kalinowski, M Spira, Comp Phys Comm 108 Ž w10x A Djouadi, PM Zerwas, J Zunft, Phys Lett B 59 Ž ; JD Wells, GL Kane, Phys Rev Lett 76 Ž ; J Kalinowski, M Krawczyk, Acta Phys Polon B 7 Ž ; M Drees, M Guchait, P Roy, Phys Rev Lett 80 Ž ; 81 Ž Ž E ; JL Diaz-Cruz, H-J He, T Tait, C-P Yuan, Phys Rev Lett 80 Ž w11x G Kane, G Kribs, S Martin, J Wells, Phys Rev D 53 Ž w1x JD Wells, Phys Rev D 56 Ž w13x See, for example, M Spira, Fortschr Phys 46 Ž w14x J Gunion, Phys Lett B 61 Ž ; W Marciano, F Paige, Phys Rev Lett 66 Ž w15x CMS Technical Design Report, CERNrLHCCr94-38, 1994; ATLAS Technical Design Report, CERNrLHCCr94-43, 1994 w16x JF Gunion et al, in: New Directions for High-Energy Physics, Snowmass, 1996, p 541 w17x H Baer, C-h Chen, M Drees, F Paige, X Tata, Phys Rev Lett 79 Ž , hep-phr980441; JD Wells, hep-phr98044; V Barger, C Kao, T-J Li, hep-phr

185 31 December 1998 Physics Letters B Texture of a four-neutrino mass matrix Subhendra Mohanty a, DP Roy b,c, Utpal Sarkar a,b a Physical Research Laboratory, Ahmedabad , India b DESY, Notkestrasse 85, 607 Hamburg, Germany c Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay , India Received September 1998; revised 16 November 1998 Editor: M Cvetič Abstract We propose a simple texture of the neutrino mass matrix with one sterile neutrino along with the three standard ones It gives maximal mixing angles for ne ns and nm n t oscillations or vice versa Thus with only four parameters, this mass matrix can explain the solar neutrino anomaly, atmospheric neutrino anomaly, LSND result and the hot dark matter of the universe, while satisfying all other Laboratory constraints Depending on the choice of parameters, one can get the vacuum oscillation or the large angle MSW solution of the solar neutrino anomaly q 1998 Elsevier Science BV All rights reserved Recently the super-kamiokande experiment has confirmed the atmospheric neutrino oscillation result, suggesting nearly maximal mixing of nm with an- other species of neutrino wx 1 The same experiment has also confirmed the solar neutrino oscillation result, which suggests mixing of ne with another species of neutrino wx Moreover, the energy spectrum of the recoil electron seems to favour the large mixing-angle vacuum oscillation of ne over the MSW solutions wx, although this may have limited statistical significance in the global fit to the solar neutrino data w3,4 x They have led to a flurry of phenomenological models for neutrino mass and mixing which can account for these oscillations w5 8 x, most of which are focussed on the bi-maximal mixing angles for the atmospheric and the solar neutrinos However, almost all of these works are based on the three-neutrino formalism, involving the standard left handed neutrinos n,n and n wx e m t 5 On the other hand the inclusion of the LSND neutrino oscillation result wx 9 is known to require a fourth neutrino, which has to be a sterile one Ž n S for consistency with the observed Z-width w10 x Moreover it requires either nm or ne to oscillate into ns for explaining the atmospheric and solar neutrino anomalies, while requiring nm ne oscillation for the LSND result Thus the three-neutrino models for atmospheric and solar neutrino anomalies, based on a ne ynmyn t mixing, are in direct conflict with the LSND result While the LSND result has not been corroborated by the preliminary KARMEN data w11 x, the statistical significance of the latter is limited by its lower sensitivity in the relevant region of parameter space Indeed, with the standard statistical method the 90% cl limit of KARMEN excludes only half the parameter space of the LSND data in the Dm F ev region w1 x Hopefully, this issue will be resolved by the proposed mini-boone experiment at Fermilab along with more data from KARMEN It seems to us premature, however, to rule out the LSND result at present Therefore we have tried to construct a four-neutrino mass matrix, which can r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

186 186 ( ) S Mohanty et alrphysics Letters B account for the present solar and atmospheric neutrino data along with the LSND result It can also account for the hot dark matter content of the uniw13 x, while satisfying all laboratory and astro- verse physical constraints w14 16 x Table 1 summarises the experimental constraints on neutrino mass and mixing parameters, which are relevant for our model The large angle MSW and the vacuum oscillation solutions to the solar neutrino data w,17,18x are taken from a recent fit to the n e suppression rates along with the recoil electron specwx 3 For both trum by Bahcall, Krastev and Smirnov the solutions the fit favours the oscillation of ne into a doublet neutrino over ne n S The reason is that in the former case the NC scattering of this doublet neutrino with electron can partly account for the discrepancy between the observed suppression rates in super-kamiokande and the Homestake experiments On the other hand one can get acceptable solutions with ne ns oscillation if one makes al- lowance for a 0% normalisation uncertainty for the Homestake experiment This will also enlarge the acceptable range of Dm Therefore we shall consider both the oscillation scenarios ne ns and ne nt in our model It should be added here that the best value of sin u for the large angle MSW solution is slightly less than 1; and even there the quality of fit is rather poor when all the experimental data are put together wx 3 However one can get acceptable fit with the large angle MSW solution, including the sin us1 boundary, if one makes reasonable allowance for the uncertainty in the Boron Table 1 Present experimental constraints on neutrino masses and mixing Solar Neutrino wx 3 y5 Dm ; 08 =10 ev Ž Large angle MSW sin u ;1 Solar Neutrino wx 3 Dm ; Ž 05 6 =10 ev Ž Vacuum oscillation sin u ;1 Atmospheric Neutrino wx 1 y3 Dm ; 05 6 =10 ev sin u )08 wx LSND 9 Dm em ; 04 ev y3 y sin u em ;10 y10 Hot Dark Matter w13x Ý m ; Ž 4 5 i i ev Neutrinoless Double mn e -046 ev Beta Decay w14x w x y3 CHOOZ 15 DmeX-10 ev Ž or sin u -0 y10 ex neutrino flux w3,4 x Finally, the global fits w3,4x have also found acceptable small angle MSW solutions for both these oscillation scenarios But we do not consider them here, since the texture of our mass matrix naturally leads to bimaximal mixing as we shall see below The dark matter constraint on the sum of neutrino masses comes from a recent global fit to the spectrum of density perturbation in the universe using various cosmological models w13 x The best fit is obtained with a hot and cold dark matter model, where the former constitutes 0% of the critical density Besides there is an astrophysical upper bound on the number of neutrino species from nucleosynthesis, which allows 1 or at most sterile neutrinos w19 x We shall consider a four-neutrino mass matrix for the three doublet neutrinos and a singlet Ž sterile neutrino We shall present two scenarios, where the solar and atmospheric neutrino anomalies are explained by the oscillations Ž A ne ns and nm n t and Ž B ne nt and nm n S The corresponding mass matrices will be related to one another by suitable permutation of neutrino indices In each case maximal mixing between the oscillating neutrino pairs will be ensured by the texture of the mass matrix Moreover we shall obtain the vacuum oscillation and the large angle MSW solutions in each case depending on the choice of parameters ( A) ne ns and nm n t oscillations: In this case the texture of our neutrino mass matrix in the basis wn n n n x is 0 e m t S 0 0 a d 0 c b 0 mn s Ž 1 a b 0 0 d Note that it has only 4 parameters In comparison the earlier mass matrices considered had at least 5 paw0 x Moreover, the above mass matrix rameters is minimal in the sense that it has only one diagonal element The mass matrix of w0x has effectively 4 parameters in the case of maximal vacuum oscillation solution of the solar neutrino However it contains two equal diagonal elements, which could in general be different from one another It is clear from the mass matrix that the neutrinoless double beta decay vanishes because Ý U m sm s0; so i ei i ee

187 ( ) S Mohanty et alrphysics Letters B that the corresponding constraint w14x is automatically satisfied For the 3=3 submatrix of doublet neutrinos, we shall assume the hierarchy b4a,c Consequently the nm and nt will form a nearly degenerate pair with maximal mixing and small mass squared difference Ž ; bc to explain the atmospheric neutrino anomaly Moreover, the remaining eigenvalue of this 3=3 submatrix gets a tiny double see-saw contribution a crb, which will be much smaller than d over a wide range of the latter parameter Consequently, the ne and ns will form a nearly degenerate pair with maximal mixing and small mass squared difference to explain the solar neutrino anomaly The vacuum oscillation and the large angle MSW solutions will correspond to the choices d-a,c and d;b respectively I Vacuum oscillation solution ( b4a,c)d ):In this approximation the mass eigenvalues are given by, a c m1 sdq b c a c a c m sbq q q y b 8b b c a c a c m3 sybq y y y b 8b b a c m4 sydq Ž 3 b Ž T and the corresponding mass eigenstates n i ' n n n n are related to the weak eigenstates Ž T n ' n n n n 4 a e m t S through the mixing matrix Uia as, 1 1 ys1 s1 ' ' 0 n e n 1 s1 y s1 n ' ' m n s n 1 1 t X X n 3 s ys n S ' ' n4 1 1 y s s ' ' ß Ž 4 ' where, s1sar b and we can neglect the terms s and s X, which are of the order of ; Ž ' y5 Oacr b,adrb ;10 for our choice of parameters For the given 4=4 mixing matrix Ui a the probability of two flavour oscillation is given by, / Dm ijl Pn n sdaby4 Ý Uai Ua jub iub jsin, Ž 5 a b ž 4E j)i where, Dm ijsm iym j For our mixing matrix the flavour oscillation in each case is dominated by one mixing angle which can be determined by comparing the expression Ž 5 with the effective = flavour oscillation formula / Dm ijl Pn n s sin uab sin Ž 6 a b ž 4E Thus we get an expression for sin uab in terms of parameters of the mixing matrix U i a For illustration we shall now present the solution for a specific set of parameters, ie, as005 ev, bs15 ev, cs0001 ev and ds00001 ev There are two pairs of nearly degenerate eigenvalues m,ym,ds00001 ev n 1 n 4 m,ym,bs15ev Ž 7 n n 3 The LSND experiment can be explained by the oscillations between states with mass squared difference of the order ev which means that it can be explained by the oscillations between the n 1,4 and n,3 states To explain LSND as an oscillation between the ne and nm the effective mixing angle sin u is obtained by comparing Ž 5 and Ž em 6 and reading off the mixing matrix elements from Ž 4, sin uem sy4ue1ue Um1 Um que1ue3 Um1 Um3 qu U U U qu U U U 4 e4 e m4 m e4 e3 m4 m3 ž / 1 4a sy4=4ž ys1 s Ž 8 ' b

188 188 ( ) S Mohanty et alrphysics Letters B Similarly the other masses and the relevant mass squared differences and the corresponding mixing angles for the experiments listed in Table 1 are given by, a cd Dm sm ym s s=10 ev y10 sol n1 n4 b sin u es s1 Dm sm ym sbcs0003 ev atm n n 3 sin u mt s1 Dm sm ym sb yd sev LSND n n 1 4a sin uem s8s1 s s0004 b Ý m s < m < s3ev Ž 9 DM i i The ne ns oscillation gives the vacuum oscillation solution to the solar neutrino anomaly, while the nm nt oscillation explains the atmospheric neutrino anomaly The LSND result is explained by nm ne oscillation The contribution to dark matter is 3 ev The CHOOZ w15x and other laboratory constraints are satisfied Note that the above solution consists of two nearly degenerate pairs of maximally mixed neutrinos, separated by a relatively large mass squared gap This is known to be the favoured mass configuration for satisfying the various laboratory constraints w0,3 x The four model parameters are used to ensure that the three mass squared gaps correspond to the required values of Dm for the solar, atmospheric and LSND neutrino oscillations, and uem corresponds to the required mixing angle for LSND The hot dark matter prediction comes out as a bonus All these features are natural predictions of our mass matrix; and as such they will be shared by each of the alternative solutions discussed below It should also be noted that the underlying double see-saw mechanism is responsible for generating mass squared gaps differing by 10 orders of magnitude starting with mass parameters, which differ by only 3 4 orders of magnitude Žie, similar to the case of the up type quark mass matrix II Large angle MSW solution ( b)d4a,c ): In this approximation Ž b/ d the mass eigenstates are given by, a d a b c m1 sdy q Ž b yd Ž b yd c c a b a b c m sbq q q y 8b Ž b yd Ž b yd c c a b m3 sybq y y 8b Ž b yd a b c y Ž b yd a d a b c m4 sydq q Ž 10 Ž b yd Ž b yd and the mixing matrix has the same form as Ž 4, with Ž 1r X Ž s1s ar b q d, and ss ss abrd b q d 1r Hence like before ne ns mixing and the nm nt mixing are maximal, where as the ne nm mixing is given by the small parameter s 1 It may be added here that one gets a smooth numerical solution at b s d, although the approximation Ž 10 breaks down there In this case let us consider a choice, as005 ev, bs15 ev, cs00015 ev and ds15 ev Then the different masses and the relevant mass squared differences and the corresponding mixing angles are given by, m,ym,ds15 ev n 1 n 4 m,ym,bs15ev n n 3 a b cd Dmsolsmn ymn s s11=10 ev 1 4 Ž b yd sin u y5 es s1 Dm sm ym sbcs0004 ev atm n n 3 sin u mt s1 Dm sm ym sb yd s069 ev LSND n n 1 a y3 em 1 sin u s8s s8 s13=10 Ž b qd Ý m s < m < s55ev Ž 11 DM i i

189 ( ) S Mohanty et alrphysics Letters B The numerical values of the mass square differences have been calculated using the exact solutions of for the masses The analytical expressions for the mass square differences are from the polynomial approximation Ž 10 The numerical agreement for the mass square differences between the exact solutions and the polynomial approximation agrees upto 6 decimals in this range of parameters Thus the ne ns oscillation provides the large angle MSW solution to the solar neutrino problem The nm nt and nm ne oscillations explain the atmospheric neutrino anomaly and the LSND result respectively as before The contribution to dark matter is 5 ev It may be added here that with these parameters s1-s ; so the effective mass of ne is slightly lower than that of n S ( B) ne nt and nm n S oscillations: In this case we can use the same mass matrix as before if we make the following change of basis, Ž ne nm nt ns Ž nenm ns n t Ž 1 Note that with this change of basis the diagonal element for the sterile neutrino mss continues to remain zero, which is an important condition as we shall see below Moreover the change of basis does not affect the mass eigenvalues m 1,m,m3 and m 4 But now the nearly degenerate pair m1 and m4 represent the two maximally mixed states of ne and n t, while m and m3 represent similar admixtures of n and n Thus the solutions Ž 9 and Ž 11 m S will continue to hold with the exchange of the neutrino flavour indices t and S in ues and u mt Conse- quently they will represent the vacuum oscillation and large angle MSW solutions to the solar neutrino anomaly via ne nt oscillation, while the atmo- spheric neutrino anomaly is explained via nm ns oscillation The rest of the results remain the same as before Let us briefly discuss the possible mechanisms underlying the above mass matrix Consider first the 3 = 3 submatrix corresponding to the three lefthanded doublet neutrinos This sub-ev scale mass matrix could arise from the standard see-saw mechanism with three heavy right-handed singlet neutrinos Alternatively, one can get it without any right-handed neutrino but with an expanded higgs sector via a radiative mechanism w4x or Majorana coupling of the left-handed neutrino pairs to a heavy Higgs triplet w5 x In both cases one can naturally obtain a sub-ev scale mass matrix The extension of the mass matrix to include a light singlet neutrino has been tried recently in each of the above three models w6 8 x In the standard see-saw model it is assumed to be one of the right-handed singlets while one adds a singlet neutrino in the other two models In each case one has to impose a zero Majorana mass for this singlet, as otherwise it will naturally assume a high mass value This is the reason why we have set m SS to zero in our mass matrix Ž 1 In the standard see-saw model this has been done by assuming a singular Majorana mass matrix for the singlet neutrinos, so that one of the eigenvalues Ž m is zero wx SS 8 In the other two models the mss is made to vanish by imposing an additional symmetry w6,7 x Finally one asks if this singlet neutrino can naturally have Dirac masses in the F1 ev scale? It seems possible to get it in the Zee model wx 6 and the triplet higgs model wx 7 via the same suppression mechanisms which keep the 3 = 3 doublet mass matrix in the sub-ev range But in the case of the standard see-saw model it had to be put in by hand wx 8 We feel it is important to look for a more natural way to keep the Dirac masses small in this model To summarize, we have presented a texture of four-neutrino mass matrix which automatically ensures bi-maximal mixing between ne ns and ne nt or vice versa Thus with only four parameters it can account for the solar, atmospheric and LSND neutrino anomalies while remaining consistent with other experimental constraints The prediction of the desired hot dark matter density comes out as bonus Depending on the choice of parameters we can get both the vacuum oscillation and the large angle MSW solutions to the solar neutrino anomaly Thanks to the underlying double see-saw mechanism, one can generate the desired mass squared gaps differing by 10 orders of magnitude starting with the four mass parameters which differ by only 3 4 orders of magnitude Acknowledgements We are grateful to Profs KS Babu and Ernest Ma and also to Prof K Whisnant for pointing out a notational error in the original version of our mass-

190 190 ( ) S Mohanty et alrphysics Letters B matrix We are also grateful to Prof Ernest Ma for a critical reading of the manuscript We thank Profs V Barger, W Buchmuller, S Goswami, S Pakvasa, T Weiler and P Zerwas for discussions Two of us Ž US and DPR would like to acknowledge the hospitality of the Theory Group, DESY and US acknowledges financial support from the Alexander von Humboldt Foundation References wx 1 Super-Kamiokande Collaboration: Y Fukuda et al, Phys Rev Lett 81 Ž ; Phys Lett B 433 Ž ; B 436 Ž ; T Kajita, Talk presented at Neutrino 98, Takayama, Japan, 1998 wx Super-Kamiokande Collaboration: Y Fukuda et al, Phys Rev Lett 81 Ž ; Talk by Y Suzuki at Neutrino 98, Takayama, Japan, 1998 wx 3 JN Bahcall, PJ Krastev, AYu Smirnov, Phys Rev D 58 Ž wx 4 N Hata, PG Langacker, Phys Rev D 56 Ž wx 5 B Allanach, hep-phr980694; V Barger, S Pakvasa, TJ Weiler, K Whisnant, Phys Lett B 437 Ž ; V Barger, TJ Weiler, K Whisnant, hep-phr ; J Elwood, N Irges, P Ramond, hep-phr98076; E Ma, hepphr ; G Alterelli, F Feruglio, hep-phr ; Y Nomura, T Yanagida, hep-phr980735; A Joshipura, hepphr980861; K Oda et al, hep-phr980841; H Fritzsch, Z Xing, hep-phr98087; J Ellis et al, hep-phr ; A Joshipura, S Vempati, hep-phr98083; U Sarkar, hepphr980877; S Davidson, SF King, hep-phr980896; H Georgi, SL Glashow, hep-phr980893; RN Mohapatra, S Nusinov, hep-phr ; R Barbieri, LJ Hall, A Strumia hep-phr ; Y Grossman, Y Nir, Y Shadmi, hepphr wx 6 N Gaur, A Ghosal, E Ma, P Roy, Phys Rev D 58 Ž wx 7 U Sarkar, hep-phr wx 8 Y Chikira, N Haba, Y Mimura, hep-phr wx 9 LSND Collaboration: A Athanassopoulos et al, Phys Rev Lett 77 Ž , nucl-exr and nuclexr w10x PB Renton, Int J Mod Phys A 1 Ž w11x KARMEN Collaboration: B Zeitnitz, Talk at Neutrino 98, Takayama, Japan, 1998 w1x C Giunti, hep-phr w13x E Gawiser and J Silk, Science 80 Ž ; see also J Primack, Astro-phr w14x HV Klapdor-Kleingrothaus, in: Proc Lepton and Baryon number violatoin, Trento, April 1998; M Gunther et al, Phys Rev D 55 Ž ; L Baudis et al, Phys Lett B 407 Ž w15x CHOOZ Collaboration: M Appollonio et al, Phys Lett B 40 Ž w16x Particle Data Group, Eur Phys J C 3 Ž w17x GALLEX Collaboration: W Hampel et al, Phys Lett B 388 Ž ; SAGE Collaboration: V Gavrin et al, Neutrino 98, Takayama, Japan, 1998 w18x Homestake expt: BT Cleveland et al, Astrophys J 496 Ž ; R Davis, Prog Part Nucl Phys 3 Ž w19x CJ Copi, DN Schramm, MS Turner, Phys Rev Lett 75 Ž ; Phys Rev D 55 Ž w0x V Barger, TJ Weiler, K Whisnant, Phys Lett B 47 Ž ; V Barger, S Pakvasa, TJ Weiler, K Whisnant, hep-phr w1x SC Gibbons, RN Mohapatra, S Nandi, A Raychaudhuri, Phys Lett B 430 Ž wx E Ma, P Roy, Phys Rev D 5 Ž ; ZG Berezhiani, RN Mohapatra, Phys Rev D 5 Ž ; EJ Chun, AS Joshipura, AYu Smirnov, Phys Rev D 54 Ž ; K Benakli, AYu Smirnov, Phys Rev Lett 79 Ž ; G Cleaver, M Cvetic, JR Espinosa, L Everett, P Langacker, Phys Rev D 57 Ž ; AS Joshipura, AYu Smirnov, hep-phr w3x SM Bilenky, C Giunti, W Grimus, Eur Phys J C 1 Ž ; S Goswami, Phys Rev D 55 Ž w4x A Zee, Phys Lett B 93 Ž w5x E Ma, U Sarkar, Phys Rev Lett 80 Ž

191 31 December 1998 Physics Letters B Bi-maximal neutrino mixing in the MSSM with a single right-handed neutrino S Davidson a, SF King b a Theory DiÕision, CERN, CH-111 GeneÕa 3, Switzerland b Department of Physics and Astronomy, UniÕersity of Southampton, Southampton, SO17 1BJ, UK Received 18 August 1998; revised 3 November 1998 Editor: R Gatto Abstract We discuss neutrino masses in the framework of a minimal extension of the minimal supersymmetric standard model Ž MSSM consisting of an additional single right-handed neutrino superfield N with a heavy Majorana mass M, which induces a single light see-saw mass mn 3 leaving two neutrinos massless at tree-level This trivial extension to the MSSM may account for the atomospheric neutrino data via nm n t oscillations by assuming a near maximal mixing angle u 3;pr4 and taking Dm 3;m n ;5=10 y3 ev 3 In order to account for the solar neutrino data we appeal to one-loop radiative corrections involving internal loops of SUSY particles, which we show can naturally generate an additional light neutrino mass m n ;10 y5 ev again with near maximal mixing angle u 1;pr4 The resulting scheme corresponds to so-called bi-maximal neutrino mixing involving just-so solar oscillations We predict that the decay t eg should be observed in the near future q 1998 Elsevier Science BV All rights reserved Atmospheric neutrino data from Super-Kamiowx 1 and SOUDAN wx, when combined with kande the recent CHOOZ data wx 3, are consistent with nm nt oscillations with near maximal mixing and Dm 3 ;5=10 y3 ev 1 A critical appraisal of current neutrino data can be found in w5,6 x In practice a common standard approach to neutrino masses is to introduce three right-handed neutrinos with a 1 The data are equally consistent if nt is replaced by a light sterile neutrino, and indeed many authors have considered adding an extra light singlet neutrino state wx 4, although we shall not do so here heavy Majorana mass matrix When the usual Dirac mass matrices of neutrinos and charged leptons are taken into account, the see-saw mechanism then leads to the physical light effective Majorana neutrino masses and mixing angles relevant for experiment There is a huge literature concerning this and other kinds of approach to neutrino masses which is impossible to do justice to In Ref wx 7 we merely list a few recent papers, from which which the full recent literature may be reconstructed For older work on neutrino masses see for instance wx 8, where extensive references to earlier work may be found In a recent paper wx 9 one of us followed a minimalistic approach and introduced below the GUT r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

192 19 ( ) S DaÕidson, SF KingrPhysics Letters B scale only a single right-handed neutrino N into the standard model Žor supersymmetric standard 1 c model with a heavy Majorana mass MNN where M-M GUT With only a single right-handed neutrino the low energy neutrino spectrum consists of a light neutrino n 3fs3nm qc3nt with mass m n 3, plus two massless neutrinos consisting of the orthogonal combination n f c n y s n together with n which 0 3 m 3 t e we have here assumed for simplicity has a zero Yukawa coupling le to N 3 This is clearly suffi- cient to account for the atmospheric neutrino data In order to account for the solar neutrino w10x data via the MSW effect w11x we were forced to depart from minimality by introducing a small additional tau neutrino mass coming from additional right-handed neutrinos at the GUT scale, m ; ;few = n t M GUT y3 10 ev wx 9 The tau neutrino mass is clearly of the correct order of magnitude to lift the degeneracy of the two previously massless neutrinos by just the right amount in order explain the solar neutrino data w x via the MSW effect 11 since Dm 1 ; m n t ; 10 y5 ev In the present paper we wish to return to minimality by assuming there is only a single right-handed neutrino N below the GUT scale with no additional right-handed neutrinos at the GUT scale More generally we shall assume no additional operators arising from the GUT or string scale which give rise to neutrino masses Our starting point is then the treelevel spectrum described above involving one massive plus two massless neutrinos Our basic observation is that in general there is no symmetry which protects the masslessness of the two neutrinos, so they are expected to acquire masses beyond the tree-level To be definite we shall show that masses for n 0 and ne can be generated from one-loop SUSY corrections and lead to a neutrino mass m n ;10 y5 ev, corresponding to n fs1ne qc1n 0, while the orthogonal combination n f c n y s n remains 1 1 e 1 0 approximately massless The value of the 1 mixing Since CP conjugation converts a right-handed neutrino into a left handed anti-neutrino, it should strictly be called a gauge singlet 3 We shall use notation such as s 'sinu, c 'cosu, t 'tanu, extensively throughout this paper 3 3 m t angle is controlled by a ratio of soft flavour-violating parameters, t1 s D13rD3 which could be large, and Dm1 ;m n In this case we arrive at approximate bi-maximal flavour mixing in both the Ž n y n e 0 and Ž n yn sectors w1 x Ž m t Recall that n 0 ;nmyn t; we do not have four light neutrinos This may be compared to bi-maximal mixing in the Ž n y n e m and Ž n yn sectors w13 x m t A discussion of the different mechanisms for generating large mixing angles, and the naturalness thereof, may be found in w14 x The phenomenological mixing matrices consistent with the data are presented in w15 x Both forms of bi-maxiw16,17x vacuum mal mixing rely on the just-so oscillation description of the solar neutrino data For example a recent best-fit to solar neutrino data in the wx vacuum oscillation region requires 6 Dm 1, 65 = 10 y11 ev and a mixing angle sin u 1 ;075 Following w18 x, we assume that this is not in conflict with the spectrum of electron neutrinos detected from SN1987A w19 x The main point of this paper is to show that bi-maximal mixing can arise in a natural way from a bare minimum of ingredients: the addition of a single right-handed neutrino to the MSSM The MSSM plus one heavy right-handed neutrino N has a superpotential which contains, in the addition to the usual terms such as mh1h plus the quark and charged lepton Yukawa couplings, the following new terms involving the new N superfield, 1 Wnewsla La NH q MNN Ž 1 where the subscript a s e,m,t indicates that we are in the charged lepton mass eigenstate basis, eg Ž y y Les n el,el where e is the electron mass eigen- state and ne is the associated neutrino weak eigen- state, and le is a Yukawa coupling to the singlet N in this basis In this basis the soft terms involving sneutrinos are V sm ) ) L L qm ) NN ) soft L L a b NN a b ž a a a N / q l A H L NqMB NNqhc The potential involving sneutrinos, including F-terms, D-terms and soft terms is, in the usual notation

193 ( ) S DaÕidson, SF KingrPhysics Letters B where tanbsõrõ1 is the ratio of Higgs vacuum expectation values, 1 Vs m ) L L qlalbõ q mz cosbd a b ab ž / = nn ) q M qm ) ) a b NN NN q Õ Aala ) a a a i yl mõ cotb n Nq l Õ M n N qmb NNqhc Ž 3 N We neglect m ) NN with respect to M for the re- mainder of the paper The tree-level neutrino spectrum was discussed in Ref wx 9 and is summarised below The usual fermionic see-saw mechanism which is responsible for the light effective Majorana mass mn n nn a b can a b be represented by the diagram in Fig 1 The see-saw mechanism represented by Fig 1 leads to the light effective Majorana matrix in the n el,n m L,nt L basis: 0 l ll ll e e m e t Õ m s ll l ll nn a b e m m m t Ž 4 M ll ll l e t m t t with phase choices such that the Yukawa couplings are real The matrix in Eq Ž 4 has two zero eigenvalues and one non-zero eigenvalue mn 3 s Ý < < a la Õ rm corresponding to the eigenvector n 3 sl n r Ý < l < a a ( b b This can easily be understood from Eq Ž 1 where it is clear that only the combination n 3 couples to N with a Yukawa coupling l3 s( l b In the l e s0 limit n e is massless and the other two eigenvectors are simply ž 3/ ž 3 3 /ž t / n 0 c3 ys3 nm s Ž 5 n s c n where t3 slmrlt and the n 0 is massless at treelevel If we allow a small but non-zero l then we Fig 1 A diagrammatic representation of the see-saw mechanism M is the singlet Majorana mass, and we have defined m a ' l Õ D a Ž e denote the massless states by primes In this case the weak eigenstates are related to the mass eigenstates by wx 9 0 X n 0 e ne X nm su0 n 0 Ž 6 nt n 3 where L L 0 c ž 3 s 3 / 1 u u 1 1 U s u 0 1 Ž 7 y c s s ys c ( 3 3 where u 1 fler lmql t The unitary matrix U0 is the tree-level neutrino mixing matrix Žthe analogue of the CKM matrix This follows since the charged weak currents are given by: n e y m nm n t W e,m,t g q hc Ž 8 m L 0 L where n el,n m L,nt L are neutrino weak eigenstates which couple with unit strength to e, m, t, respectively Note that due to the massless degeneracy of n 0 and ne the choice of basis is not unique The MSSM generalisation of the see-saw mechanism was recently discussed by Grossman and Haber w0 x The SUSY analogue of the Majorana neutrino c mass mn n nn a b is the Majorana sneutrino mass a b m n n nn a b Such masses, which perhaps should more a b properly be referred to as lepton number violating masses, do not appear directly in the potential in Eq Ž 3, but can be generated by the scalar analogue of the see-saw mechanism One way w0x to understand the origin of a Majorana sneutrino mass is to separate each sneutrino into real and imaginary comy1r y1r ponents ns Ž n qin and Ns ŽN R I Rq in then observe that a lepton number conserving I

194 194 ( ) S DaÕidson, SF KingrPhysics Letters B Fig A scalar see-saw diagram We have defined m ' l Õ M a a Ž Dirac sneutrino mass is the same for real and imaginary parts of the field: m ) ) nn ) nn R I m nn s n qn Ž 9 whereas a lepton number violating Ž Majorana mass contributes with opposite sign to the mass of the real and imaginary components: ž / m nnqhc sm n yn Ž 10 nn nn R I Grossman and Haber studied the 4=4 matrix for the one-family case and showed that it separates into two independent = matrices corresponding to the CP even and CP odd states To calculate the sneutrino Majorana mass they find eigenvalues of the = matrices and take mass differences However one can equally well think of the Majorana sneutrino masses diagrammatically, as the scalar analogue of the diagrammatic representation of the usual fermionic see-saw mechanism discussed above, as follows As in the fermionic see-saw mechanism the light effective Majorana sneutrino masses come from inte- which has grating out a heavy state, in this case N the lepton number violating interactions Thus the scalar see-saw diagrams involve an N propagator To get a contribution to m nn that is OŽ 1rM n n a b, a b we need at least one power of M on top to cancel the 1rM from the propagator There are two possible diagrams, illustrated in Figs and 3 We estimate the contribution from Fig to be,ybl l Õ rm, and from Fig 3 to be, Ž a b Ab y mcot b l l Õ rm The total contribution to the a b Fig 4 One-loop diagram generating a Majorana neutrino mass The lepton number violation comes from the effective sneutrino Majorana lepton number violating mass m n n arising from a b Figs,3 which we have condensed into a single x here light effective Majorana Žs lepton number violating sneutrino mass is then m, A ymcotbyb m Ž 11 nanb b nanb which consists of a product of SUSY masses times the tree-level Majorana neutrino mass This result agrees with the result in Ref w0 x Having generated a sneutrino Majorana mass it is straightforward to see that such a mass will lead to one-loop radiative corrections to neutrino Majorana masses w0x via the self-energy diagram in Fig 4 The diagram involves an internal loop of neutralinos and sneutrinos, with the lepton number violating Majorana sneutrino mass at the heart of the diagram This diagram applies to an arbitrary basis Žassuming degenerate sneutrino masses In particular it applies to the basis in which the tree-level Majorana neutrino masses are diagonal, which is the most convenient basis for calculating loop corrections to neutrino masses In this basis the Majorana sneutrino masses are also diagonal Žassuming that Aa la is aligned with l, as is clear from Eq Ž 11 a, and consist of two zero Majorana mass sneutrinos 4 plus a non-zero Majorana mass given by Eq Ž 11 in this basis: m f Ž A ymcotbyb m Ž 1 n 3 n 3 3 n 3 where A3 is the trilinear soft parameter associated with the Yukawa coupling eigenvalue l Ž 3 ignoring flavour violation According to Grossman and Haber Fig 3 Another scalar see-saw diagram We have defined V ' b Õ A l y l mõ cotb b b b 4 The usual lepton number conserving sneutrino masses are of course non-zero

195 ( ) S DaÕidson, SF KingrPhysics Letters B w0x the one-loop correction to the tree-level neutrino mass is given by: g m n 3 n 3 Ý < < n 3 j jz n 3 d m f f y Z 'e m 3p cos u W m j Ž 13 0 where the y sm rm 0 j x j, xj is the j-th neutralino, the function f y j ;05y057 and m is an aver- age sneutrino mass Žsee Eqs Ž 14, Ž 15, and Ž 16 We have defined a quantity e whose value is typically e;10 y3, assuming that all soft SUSY breaking parameters and m Žincluding those associated with the right-handed neutrino N are of a similar order of magnitude Grossman and Haber actually consider the case that B 4 1 TeV but we shall assume here that B;1 TeV Thus we conclude that the correction to the tree-level neutrino mass is of order 01% and so is utterly negligible Since d m Am one might be tempted to conn 3 n 3 clude that the two neutrino states which are massless at tree-level remain massless at one-loop However this conclusion would be incorrect due to the effects of flavour violation which we have so far ignored There are two possible origins of flavour violation that can be relevant for neutrino masses: Ž i Soft lepton number conserving sneutrino masses which are flavour-violating; Ž ii Trilinear parameters Ab which are misaligned with the Yukawa couplings l b In the present paper we shall focus on case Ž i only since we shall see later that the mechanism responsible for generating small non-zero neutrino masses has two possible sources: Ž i or Ž i qž ii There is no significant source of small neutrino masses coming from Ž ii alone, and so for simplicity we shall ignore the effect of Ž ii in making our estimates We shall need to be able to move from one basis to another and be able to deal with flavour-violating effects in any given basis The tool for doing this which we shall use is the mass-insertion approximation According to the mass insertion approach in changing basis we must rotate the superfield as a whole, so that the gauge couplings do not violate flavour This means that in the fermion mass eigenstate basis the sfermions have off-diagonal masses, and flavour-violation is dealt with by sfermion propagator mass insertions To begin with we shall develop the mass-insertion approximation in the charged lepton mass eigenstate basis, then change basis to the tree-level neutrino mass eigenstate basis where we shall actually perform our estimates In the charged lepton mass eigenstate basis for simplicity we assume that the lepton number conserving sneutrino masses are approximately proportional to the unit matrix: 1 M ) sm ) ql l Õ q m cosbd L L L L a b Z ab a b a b sm dabqdab 14 where the Dab are small Note that D is dimension- less The inverse propagator in this basis is given by p ym d qd s p ym ab ab = m D ab ab ž p ym / d y Ž 15 so to linear order in D, the propagator becomes dab 1 1 q m Dab Ž 16 p ym p ym p ym where m is some average sneutrino Žlepton number conserving mass Now we need to relate the lepton number conserving sneutrino mass matrix in the charged lepton mass eigenstate basis to that in the tree-level neutrino mass eigenstate basis Since in the mass insertion approach we rotate each component of the superfields equally, the rotation on the sneutrinos is the same as that for the neutrinos and is given by the unitary matrix U in Eq Ž 0 7 Thus the relation between the sneutrinos in the two bases is the scalar version of Eq Ž X ne n 0 e X nm su n 0 Ž 17 nt n 3 Ž Of course the rotation acts on complete SU L doublets even though we only exhibit it for the

196 196 ( ) S DaÕidson, SF KingrPhysics Letters B neutrinos and sneutrinos The relation between the soft masses in the two bases is then given by: b ñ e 0 ) ) ) ž n,n,n / M ) ñ e m t L L m a ñ t e n X X ) X ) ) X X ) ñ e 0 3 LL ñ 3 0 s n,n,n M Ž 18 i j0 where the soft masses are related by M X ) su M ) LL i j 0 LaLb 0 U Ž 19 Thus the soft masses in the tree-level neutrino mass eigenstate basis can be written as M sm d yd Ž 0 X X ) LiLj ij ij where the D s are related by D X ijsu 0 DabU0 Ž 1 We now proceed to discuss the loop contributions to the neutrino masses which are zero at tree-level, i again working in the basis n sžn X,n X,n e 0 3 in which the tree-level neutrino masses are diagonal As discussed such corrections rely on flavour-violating effects, and non-zero masses develop at one loop via the neutralino exchange diagram with flavour-violating mass insertions along the sneutrino propagator In the basis in which we are working the only non-zero Majorana sneutrino mass is m n n, and so 3 3 the diagram must always have this single Majorana mass insertion at its centre The flavour-violating mass insertions therefore must connect the neutrino of interest to n 3, and so the mass insertions which are relevant are of the form D X i3 See Fig 5 The one-loop corrected neutrino mass matrix, to lowest non-zero order in the D X i3s, is given by: 0 D X D X D X D X Ž1 X X X X n inj D3 D31 D3 D3 n3 X X D31 D3 1 m s e m q m n 3 Fig 5 One-loop diagram generating a Majorana neutrino mass that is not aligned with the tree level mass The lepton number violation comes from the effective sneutrino Majorana Žlepton number violating mass m n n, and the misalignment from off 3 3 diagonal slepton masses D X k 3, treated here in the mass insertion approximation This diagram is in the tree level neutrino mass eigenstate basis The upper = block of the matrix clearly requires two mass insertions and so may be thought to be negligible compared to the other elements of the matrix However on the contrary this block actually gives the dominant contribution to the neutrinos which are massless at tree-level The reason is that the contributions to these masses from the third row and column of the matrix are suppressed by a sort of see-saw mechanism Thus if one only includes D to linear order, then the neutrinos which are massless at tree-level will get see-saw masses ;D e m n 3, which are to small to be interesting Žsuppressed by two powers of e Therefore to excellent approximation we may just consider the upper = block of the matrix involving two mass insertions In the basis i n sžn X,n X this is of the form e 0 ž / D X D X D X Ž mn n s e mn Ž 3 i j X X X 3 D D D which has one zero eigenvalue mn 1 s 0 and one non-zero eigenvalue m s D X qd X e m Ž 4 n 13 3 n 3 The corresponding eigenvectors are related to the tree-level massless eigenstates n X,n X by ž / ž 1 1 /ž 0/ e 0 n c ys n X e s X Ž 5 n s c n where the mixing angle is given by t sd X rd X Ž 6 Thus the SUSY radiative corrections induce a further two-state remixing of n X,n X e 0 The final one-loop cor-

197 ( ) S DaÕidson, SF KingrPhysics Letters B rected neutrino mixing matrix is obtained from Eqs ŽŽŽ 6, 7, 5, 0 n 0 e n 1 X nm su n Ž 7 nt n L 3 L where s1 c1 c1 y u1 s1 q u1 u1 t3 t3 X U s c1 s1 ys1 c3y u 1 c1 c3y u1 s3 s3 s3 s s yc s c Ž 8 In order to obtain just so vacuum oscillations we must be able to achieve two things: Ž y5 i We need the mass mn to be of order 10 ev For e,10 y3 and m,5=10 y ev, we need D X n 3 13 qd X 3, Ž ii We need a large mixing angle u 1 ; pr4 which corresponds to t sd X rd X ;1 X Using Eq Ž 1 we can relate Dij to the Dab quantities in the charged lepton mass eigenstate basis where the phenomenological bounds from lepton flavour violating processes appear For example there is an extremely strong model independent limit from y3 m eg on D -77=10 w1x Ž em for 100 GeV sleptons Since the limit on this quantity is so much stronger than on the other entries, we may assume it is zero If we also set u1 s0 then we find D X sd X 13 31sc3 Det D X sd X 3 3ss3c3 Dmmq c3 y1 Dmtys3c3 Dtt Ž 9 Assuming maximal mixing in the 3 sector, the condition for maximal mixing in the 1 sector is then 1 D t e, Ž DmmyDtt Ž 30 ' Assuming maximal mixing in the 1 and 3 sectors X X Ž c ss sc ss s1r' , we find D13qD3 sd t e To get m n ;mn ed ;10 y5 ev, we need 3 D ;4 It is also Ž just et small enough for the mass insertion approximation to be applicable, and since here we only interested in the order of magnitude of the effects our approximations are adequate In any case the Ds may be reduced slightly if e is somewhat larger than we have estimated The best experimental bound on Det comes from the non-observation of the decay t g e Following w1 x, and using the more recent limits on the branch- y6 ing ratio BR t eg -7=10 w x, we find that Det -66 for gaugino and slepton masses of 100 GeV The limit is Det-34 for mg rm s3, and Det-3 for mg rm s5 This is consistent with the value we need to get a large mixing angle; if our scenario is correct, the decay t eg should be observed in the near future Finally we briefly discuss the contribution of the second source of flavour violation, coming from the trilinear parameters, to the masses of n X,n X The e 0 basic effect comes from the fact that in the charged lepton mass eigenstate basis the trilinear parameters associated with the Yukawa couplings l,l,l may e m t be unequal A e /A m/a t This would imply that in the basis in which the Yukawa couplings are diagonal apart from the trilinear mass A3 associated with the coupling Õ A ln N there will be further Ž small trilinear parameters a 1, a associated with the cou- X X plings Õ a n N,Õ a n N 1 e 0 These couplings generate small off-diagonal sneutrino Majorana masses via the mechanism in Fig 3 However the Yukawa couplings l 1, l are zero in this basis so sneutrino masses are only generated in the third row and column of the sneutrino Majorana matrix If the same were true for the corresponding neutrino masses it would lead to negligible contributions to the masses of n X,n X e 0 due to the see-saw type suppression dis- cussed above However if we focus on the relevant upper = block of the neutrino Majorana mass matrix we can see that there is a one-loop diagram similar to Fig 5 but now involving both an off-diagonal sneutrino Majorana mass m n n and a mass 1 3 insertion D X 3 j This will lead to additional contributions to the = neutrino matrix which for simplicity we have not included in our estimates To summarise, we have shown that both the atmospheric and solar neutrino data may be accounted for by a remarkably simple model: the MSSM with the addition of a single right-handed neutrino N In the l s 0 limit when u s 0 the e 1 physics of atmospheric and solar neutrinos can be described very simply From the point of view of atmospheric oscillations the mass splitting between

198 198 ( ) S DaÕidson, SF KingrPhysics Letters B the two lightest neutrinos is too small to be important and the physics is described by the two state mixing of Eq Ž 5, maintaining the nm lnt tree-level prediction From the point of view of solar oscillations since le s0 the electron neutrino contains no component of n 3 and so the physics is described by two state mixing in Eq Ž 5 Ž dropping the primes induced by the radiative corrections The neutrino mixing matrix in Eq Ž 8 becomes: c1 s1 0 Us ys1 c3 c1c3 s3 Ž 31 s s yc s c When u1 s u 3 s pr4 it corresponds to bi-maximal mixing in the Ž ne yn 0 and Ž nmyn t sectors w 1 x More generally at tree-level the spectrum is controlled by the 4 parameters l,l,l, M where we e m t choose l e <lmflt to obtain u 3fpr4, and the eigenvalue m sl Õ rm ;5=10 y ev to obtain n 3 a Dm sm ;5=10 y3 ev as suggested by the 3 n 3 recent atmospheric data The effect of radiative corrections is model dependent since it depends on the SUSY masses, and crucially on the soft flavourviolating parameters To illustrate the mechanism we have made some simple estimates based on flavour violation due to the soft scalar masses only We have seen that it is quite reasonable to obtain Dm ; 1 10 y10 ev and a large mixing angle u1 from such effects Thus the origin of the tiny neutrino mass m n ;10 y5 ev required for just so neutrino oscil- lations may simply be due to the SUSY corrections arising from the atmospheric neutrino mass in the single right-handed neutrino model SFK would like to thank CERN and Fermilab for their hospitality during the initial and final stages of this work References wx 1 Y Fukuda et al, hep-exr , hep-exr980501, hepexr wx SM Kashara et al, Phys Rev D 55 Ž wx 3 M Apollonio et al, Phys Lett B 40 Ž wx 4 V Barger et al, Phys Rev Lett 45 Ž ; D Caldwell, R Mohapatra, Phys Rev D 48 Ž ; J Peltoniemi, J Valle, Nucl Phys B 406 Ž ; R Foot, R Volkas, Phys Rev D 5 Ž ; E Ma, P Roy, Phys Rev D 5 Ž 1995 R4780; Z Berezhiani, R Mohapatra, Phys Rev D 5 Ž ; E Chun et al, Phys Lett B 357 Ž ; J Espinosa, hep-phr ; P Langacker, hep-phr980581; AS Joshipura, AYu Smirnov, hepphr ; M Gonzalez-Garcia et al, hep-phr ; S Bilenky et al, hep-phr wx 5 R Barbieri et al, hep-phr980735; P Osland, G Vigdel, hep-phr ; RP Thun, S McKee, hep-phr ; DV Ahluwalia, hep-phr980767; G Fogli et al, hepphr wx 6 J Bahcall, PI Krastev, AYu Smirnov, hep-phr wx 7 B Allanach, hep-phr980694; G Altarelli, F Ferruccio Feruglio, hep-phr ; J Pati, hep-phr ; C Albright et al, hep-phr980314; J Ellis et al, hep-phr ; J Elwood et al, hep-phr980735; E Ma, hepphr ; I Sogami et al, hep-phr ; K Akama, K Katsuura, hep-phr ; M Drees et al, Phys Rev D 57 Ž wx 8 SM Bilensky, ST Petcov, Rev Mod Phys 59 Ž wx 9 SF King, hep-phr , CERN-THr98-08 w10x See, eg, Neutrino 98, in: Y Suzuki, Y Totsuka Ž Eds, Proceedings of XVIII International Conference on Neutrino Physics and Astrophysics, Takayama, Japan, 4 9 June 1998, to appear as Nucl Phys B Ž Proc Suppl w11x L Wolfenstein, Phys Rev D 17 Ž ; S Mikheyev, AYu Smirnov, Sov J Nucl Phys 4 Ž w1x V Barger, S Pakvasa, T Weiler, K Whisnant, hepphr ; A Baltz, AS Goldhaber, M Goldhaber, hepphr ; M Jezabek, Y Sumino, hep-phr ; Y Nomura, T Yanagida, hep-phr980735; H Georgi, S Glashow, hep-phr w13x H Fritzsch, Z Xing, Phys Lett B 37 Ž ; M Fukugita, M Tanimoto, T Yanagida, Phys Rev D 57 Ž ; hep-phr ; H Fritzsch, Z Xing, hepphr980734; M Tanimoto, hep-phr ; R Mohapatra, S Nussinov, hep-phr w14x M Tanimoto, hep-phr ; M Matsuda, M Tanimoto, hep-phr w15x SM Bilenky, C Giunti, hep-phr98001 w16x VN Gribov, BM Pontecorvo, Phys Lett B 8 Ž w17x More recently, see, eg, V Barger, K Whisnant, RJN Phillips, Phys Rev D 4 Ž ; SL Glashow, LM Krauss, Phys Lett B 190 Ž ; V Barger, RJN Phillips, K Whisnant, Phys Rev Lett 65 Ž ; 69 Ž ; N Hata, PG Langacker, Phys Rev D 56 Ž ; PI Krastev, ST Petcov, Nucl Phys B 449 Ž SL Glashow, PJ Kernan, LM Krauss, hepphr w18x S Hannestad, GG Raffelt, astro-phr98053 w19x AY Smirnov, DN Spergel, JN Bahcall, Phys Rev D 49 Ž ; B Jegerlehner, F Neubig, GG Raffelt, Phys Rev D 54 Ž w0x Y Grossman, H Haber, hep-phr97041 w1x F Gabbiani, E Gabrielli, A Masiero, L Silvestrini, Nucl Phys B 477 Ž hep-phr wx Review of Particle Properties, C Caso et al, Eur Phys J C 3 Ž

199 31 December 1998 Physics Letters B Electroweak baryogenesis in a left right supersymmetric model Tuomas Multamaki 1, Iiro Vilja Department of Physics, UniÕersity of Turku, FIN-0014 Turku, Finland Received 17 October 1998; revised 10 November 1998 Editor: PV Landshoff Abstract The possibility of electroweak baryogenesis is considered within the framework of a left right supersymmetric model It is shown that for a range of parameters the large sneutrino VEV required for parity breaking varies at the electroweak phase transition leading to a production of baryons The resulting baryon to entropy ratio is approximated to be nbrs;a 07= 10 y8, where a is the angle that the phase of sneutrino VEV changes at the electroweak phase transition q 1998 Elsevier Science BV All rights reserved The Standard Model Ž SM fulfills all the requirewx 1, for ments, including the Sakharov conditions electroweak baryogenesis wx that may be responsible for the observed baryon asymmetry of the universe However, in the SM the phase transition has been shown to be too weakly first order to preserve the generated asymmetry wx 3 Furthermore, the phase of the Kobayashi-Maskawa -matrix leads to an insuffiwx 4 which further indicates that cient CP-violation physics beyond the SM is needed to account for the baryon asymmetry One of the most popular extensions of the SM is the Minimal Supersymmetric Standard Model Ž MSSM that, with an appropriate choice of parameters, leads to a large enough baryon asymmetry, y10 n rs;10 w5,6 x B Some of the parameter bounds are, however, quite stringent wxž 7 although Riotto has recently argued wx 8 that an additional enhancement factor of about 10 is present 1 tuomul@newtontfyutufi vilja@newtontfyutufi In the MSSM the conservation of R-parity, R s 3Ž ByLq S y1, is assumed and put in by hand This is quite ad hoc since it is not required for the internal consistency of the model but for the conservation of baryon and lepton numbers To avoid eg proton decay, the R-parity breaking terms, allowed by gauge invariance, are therefore usually omitted in the MSSM or their couplings are assumed to be very small The supersymmetric left right model Ž SUSYLR based on the gauge group SUŽ = SUŽ L R= UŽ 1 explicitly conserves R-parity wx ByL 9 Considering the MSSM to be a low energy approximation of the SUSYLR it can also account for the baryon asymmetry of the universe Furthermore, the see-saw mechanism w10x that can be accommodated in the left right models, can be used to generate light masses for left-handed neutrinos and large masses for right-handed ones The possibility of a small Ž left-handed neutrino mass is therefore often seen as another motivation for the SUSYLR model By viewing the MSSM as a low energy approximation of the SUSYLR it becomes warranted to r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

200 00 T Multamaki, I ViljarPhysics Letters B 445 ( 1998) question whether some properties of the SUSYLR affect the prospect of electroweak baryogenesis This question is studied in the context of the present paper and it is argued that electroweak baryogenesis may be significantly larger in the SUSYLR than in the MSSM The gauge-invariant superpotential for the SUw13x at zero tem- SYLR model with two bidoublets perature is given by Žomitting all generation and SUŽ indices and the Q-terms : Wsh L T tft L c qh L T t xt L c f x qif Ž L T tdlql ct td c L c c c T 1 qm tr DDqD D qm tr tft x qm X tr tf T tf qm XX tr t x T t x, Ž where LŽ,1,y 1, L c Ž 1,,1 are the matter superc fields and D 3,1,, D Ž 1,3,y, DŽ 3,1,y, D c Ž 1,3,, F Ž,,0, x Ž,,0 are the Higgs superfields with their respective SUŽ = SUŽ L R= UŽ 1 ByL quantum numbers It is worth noting that since one of the bidoublets will be responsible for the electron mass the mechanism considered here requires at least two bidoublet fields to result in a significant production of baryons The scalar components of the superfields have the following component representations: 0 d q qq d n n ' c Ls, L c ž y/ s ž q/, Ds q, e e d 0 d y ' ž / ž / f1 0 f q x1 0 x q Fs, xs, y 0 y 0 f f x x 1 1 and similarly for the rest of the fields By standard methods w1x it can be shown that the Lagrangian contains the following terms Žomitting generation indices : c ) x R hhn Le, Ž 3 where H s x q y ž x 1 / and L is the left-handed chirality projection operator Of course similar term proportional to hf is present in the Lagrangian but since hf is proportional to the electron mass it is neglected here w16 x The Yukawa coupling h x, however, is proportional to the neutrino Dirac mass and is hence only bounded by an experimental upper limit w16 x Comparing with Ref w15x we note that this is exactly of the form that creates a CP-violating source in the diffusion equation at the electroweak Ž EW phase transition assuming that ² n : does not vanish and is not a constant It has been shown that in a wide class of SUSYLR models R-parity is necessarily spontaneously broken by a large sneutrino VEV w14 x Thus for electroweak baryogenesis we need to show that ² n : varies at the EW phase transition Following Ref w15x the CP-violating source can now be computed by using methods described in Ref w11 x In electroweak baryogenesis the created baryon to entropy ratio is nb ADGws sygž k i, Ž 4 s Õ s w where gž k i is a numerical coefficient depending on the degrees of freedom, D the effective diffusion 4 rate, G s 6a TŽ k s 1 ws w the weak sphaleron rate and Õ the speed of the bubble wall In Ref w15x w it was shown that H ` X X 1 ij kj Ž i k w i k k i 0 A;I 'h h du sin u yu AAyA A x X ylqu ej i k i k i k H ycosž u yu AAŽ u qu e I, Ž 5 ( where l sž Õ q Õ q 4G D rž D and ² n :' Ae iu Clearly nbrs vanishes unless A and u change when going over the bubble wall To show that electroweak baryogenesis is indeed possible in this model we must first consider the associated Higgs potential The Higgs potential in- q w w cluding soft breaking terms Žomitting the Q-terms again is w13 x: VsV qv qv 6 F -terms soft D -terms

201 T Multamaki, I ViljarPhysics Letters B 445 ( 1998) where V snl t h Fqh x t qifl t D N T F -terms f x ct c ct T T qnl t Ž hffqhxx tqifl tdn qtr Nh c T X f L L qm1f T qm1 x T N qtr Nh c T XX x L L qm1 x T qm1f T N T c ct qtr NifLL t qmdn qtr NifL L t c c c qmd N qnmn tr Ž DDqD D, Ž 7 Ž 1 V sm LLqL L soft c c l q M ynmn tr Ž D DqD c D c c c Ž tr Ž DDqD D X c c Ž q M ynmn q M tr DDqDD qhc Ž f 1 Ž x 1 Ž fx 1 q M y4nm X N trf F q M y4nm XX N tr x x q M y4nm N trf xqhc m m X T T q tr tft x q tr tftf ž q m XX T tr Ž t x t x qhc / T ct c c ž ž / T c 1 / q iõ L t DLqL t D L ql t Ž e Fqe x t L qhc, Ž 8 gl Ý m 8 m V s NL t L D -terms qtr Ž DtmDqDtmDqF tf m gr c c m Ý m 8 m qx t x N q NL t L c c c c T qtr Ž D tmd q D tmd qftmf g X T c c m qxt x N q NL L yll 8 c c c c qtr DDyD D yddqd D N Ž 9 In Ref w14x it was shown that the spontaneous breakdown of parity cannot occur without spontaneous R-parity breaking in a model with one bidoublet We shall now examine the Higgs potential using assumptions similar to those in Ref w14 x We assume that g and g X are much smaller than h and f and to ensure the required hierarchy of parity and R-parity breaking scales, the couplings involving F or x are assumed to be much smaller than those not involving them With these approximations the potential simplifies to the same form as in Ref w14 x: Vsm LLqL L qm TrŽ DDqD D c c l 1 c c c c c c qm Tr DDqD D qnhn L L LL ž / c c qnfn Ž LL qž L L q4nfn qm ž NL t D N qnl t DN / ct c T X c c Tr DDqDD qhc T ) ct qž L t Ž iõdqim fd LqL t Ž iõd qim fd L qhc / Ž 10 = c ) c c We will now select the VEVs of the slepton fields to be of the form c l X l ² : ž / ² : 0 ž 0/ L s, L s Ž 11 and using SUŽ = SUŽ L R-invariance these can be chosen real In Ref w14x it was shown that for the absolute minimum of the potential, the triplet fields must be of the Q preserving form em ž / ž / c ² : 0 0 ² : 0 0 D s X, D s, d 0 d 0 X ž 0 0/ 0 0 c 0 d D s, D s 0 d Ž 1 ² : ² : ž / The doublet and triplet VEVs can now be determined by substituting Ž 11 and Ž 1 into Ž 10 and finding the minimum The algebra is quite involved and the general solutions are messy so that it is useful to approximate M and M X 1 as small pertur-

202 0 T Multamaki, I ViljarPhysics Letters B 445 ( 1998) bations as was done in Ref w14 x The potential now takes the form Vsm Ž l ql X qm Ž l ql X l 4 4 q h l q f Ž l ql ) ) X X ) X) q l ÕdqfM d ql Õd qfm d qcc Ž 13 The parity breaking solution is w14x Õ fml X X X lsds0, d sy, d sy, 4 f M / Õ y4 f m l M M X X 4 l s, Vsyf 1y l 4 ž 8 f Ž M ym M Ž 14 If we assume that only M X f0, we find a parity X breaking solution Žgiven that l/0, l /0: l1 Õ l Õ X dsy, d sy, 4 f l qm 4 f l qm l1 M l M X dsy, d sy, Ž 15 M M and l, l X can be solved from equations Õ M1 ml y 1y f 4 f l ž / M Õ M1 ž 4ž / / M 8 f f l q f 1y q 1y l Ž 1 qh l X qo M 4 Õ M1 m y 1y f 4 f l s0 ž / ž 4ž X // M 8 f f l l X M Õ M1 X q f 1y q 1y l Ž 1 qh lqo M 4 s0 Ž 16 We find that in this case l generally does not vanish Numerical examinations, when further requiring that M X /0, support this result Note that after parity breaking the acquired VEVs may change as temperature falls and electroweak symmetry is broken by the VEVs of the F and x fields The F and x fields acquire Q VEVs, ž 1/ ž / em conserving k1 0 k 0 Fs X, xs, Ž 17 0 k 0 k ( where k1qk is of the order of ;100 GeV after the EW phase transition and k X i<k ik1 and k can be chosen real using UŽ 1 -invariance To study the behaviour of the sneutrino VEV at the phase transition, we set the fields to their VEVs after the EW breaking Žwe also approximate ds0, X X ds0 and set k sk s0 for simplicity 1 The potential Ž 6 simplifies to X X X X X x x f 1 1 Vs h lk q fld q h lk q h ll qm k X XX < < < < 1 x q mk q h l lqm k q4 mk q< f< < l< q< fl qmd < qm < l< q< l < 4 X X X l X X < < < < < < 1 qm d q M y M d ž / ž x 1 / X X X X f 1 1 qm dd qhc q M y4< m < < k < q M y4< m XX < < k < q ž / m X X X 1 kkqhc q Õl d qe lk l qhc gl < < < < < < 1 q l q k y k 8 gr X X X < < Ž < < < < < < < < 1 8 X g X X X q l q d y d q k y k q < l < y< l< q < d < y< d < Ž 18 8 X X Furthermore, assuming that f, M,Õ, e,d,d are real Ž for simplicity and setting hfs 0 the terms contain- ing l X are V < X < X X X < < < X < Xs4 f l d q4 flk d Re h l qk h l l x x q< h < < l X < < l< qk lm Re h l X qf < l X < 4 x 1 1 x X X X X X q fmd ReŽ l qm < l < l qõd ReŽ l q e lk ReŽ l X g R X 4 X X X < < < < Ž 1 8 X g X 4 X X X q l q l d yd qk yk q < l < q< l < Ž d yd 8 X y< l < < l < Ž 19

203 T Multamaki, I ViljarPhysics Letters B 445 ( 1998) X < X < ia i b Writing l s le, hxs he and differentiating with respect to the phase of the complex VEV, a, we obtain E V X l X X X y s 4hf lk d < l < qhm ll < < 1 k1 sinž aqb Ea X X X q4 fmd qõd < l < sinž a q e lk < l X < sinž a Ž 0 Ž< l X < can be assumed to be approximately constant to ensure parity breaking and the required heavy righthanded particle masses We note that for a/0 we must require that l/0 If bs0 we may easily solve Vl Xras0 for a: ž / hf lk d X qhm1lk1qelk a1 sarccos y X X X 4 fmd qõd < l < Ž 1 To examine when this is a minimum we substitute a into 19 writing only the a-dependent terms : 1 X Ž hf lkd qhm1lk1qelk X l Ž 1 X X V asa sy 4 fmd qõd X X X X y fmd l yõd l At a s 0 Ž the old minimum Ž 19 is Žagain writing only the a dependent part : V X X X X X X l as0 s4 flkd hl qk1lm1hl q fmd l qõd X l X q e lk l X Ž 3 To assure that a1 is a minimum, we must thus require that V X a -V X l Ž 1 l Ž 0 Ž 4 from which it follows that 4hf lk d X l X qhm lk l X q e lk l X 1 1 q4 fmd l X X q4õd l X X X Ž hf lkd qhm1lk1qelk X X q -0 Ž 5 4 fmd qõd If b/0 ie hx is complex, this question cannot be answered analytically and numerical methods must be utilized Provided that condition Ž 5 is satisfied, it is then clear that as F and x acquire their VEVs at the electroweak transition the sneutrino VEV, l X, while remaining approximately constant in magnitude rotates in the complex plane by an angle a 1 Expanding around the new minimum, we may write l X 'l X e i a sl X EW qae iu, Ž 6 from which it follows that ( sin a X iarctan iu l 1ycosa e cos ay1 sae 7 In the EW phase transition, A changes from 0 to l X ( 1ycosa while the change in u is Dus sina arccot 1y cosa We estimate the changes in A and u to be sinusoidal and choose that only one of the sneutrino VEVs is significantly large In Ref w15x it was shown that for Tfm;100 GeV I e j H is approximately y regardless of the considered generation so that Ž 5 can be written as I f48h l X 1 Ž 1ycosa sina = arccot ž 1ycosa / where H = ` 0 X ylqu du g Ž u g Ž u e, Ž 8 1 1ycosŽ uprl g u s = u Ž u yu Ž uyl qu Ž uyl Ž 9 and h;h 3i, is1,,3 For Ts100 GeV, Ls5rT the integral has a value of ;06 To estimate the value of I let us assume further that l X ;1 TeV and h;01 so that sina 5 I f Ž 1ycosa arccot 1=10 1 ž 1ycosa / Ž 30 The effect of adding a mass insertion can easily be calculated Taking X l to be constant, Ž 5 reduces to yh h H ` ducosž u yu ij kj i k 0 X yl u e i k i k H = AA u qu e q I j Ž 31

204 04 T Multamaki, I ViljarPhysics Letters B 445 ( 1998) If only one of the sneutrino fields, A e iu 3, has a 3 significant VEV ; 1 TeV, 31 becomes H ` X X ylqu 3 0 I f4h l duu e, Ž 3 where l X is the magnitude of the sneutrino VEV and I e j H fy as before The phase of the sneutrino field VEV is again assumed to change sinusoidally during the EW phase transition ie u 3Ž u sa gž u Ž 33 so that the integral in Ž 3 can be evaluated and is found to be p ylql q l L qp Ž 1qe a Ž 34 For Ls5rT, Ts100 GeV and lqf15 this has a value of about 083 a Taking h;01 and l X ;1 TeV, we arrive at an estimate for CP-violation during the electroweak phase transition with one mass insertion, I f=10 5 a Ž 35 Clearly, since sina Ž 1ycosa arccotž / Fa 1ycosa for a gw0,p x, I1- I indicating that further mass insertions may possibly increase the amount of CPviolation even further In comparison with I the estimate for the CPviolation in the MSSM w6,15x is ; 300 Žwithout the suppression factor wx 7 that leads to a baryon to y11 entropy ratio of n rs; 10 Because of Ž B 4 the baryon to entropy ratio is then n B y8 ;a 07=10 Ž 36 s Ž y3 so that even a change of O 10 in the phase of the sneutrino VEV may lead to significant baryon production at the EW phase transition This result is obtained after quite a number of assumptions and simplifications However, these are not carefully chosen to obtain a desired result but are used to simplify the calculation process In this paper we have considered the possibility of electroweak baryogenesis in the supersymmetric left right model As in Ref w15x the crucial property that allows for sufficient baryon creation is the large sneutrino VEV In SUSYLR this is set quite naturally by the left right parity breaking scale As electroweak breaking takes place the sneutrino VEV must necessarily change for baryogenesis to be possible We have shown that for a zero temperature approximation there exists a part of the parameter space where the sneutrino VEV rotates in the complex plane while conserving its large amplitude This process creates a CP-violating source that leads to a production of baryons at the electroweak phase tranwx 6 The amount sition through a known mechanism of baryons produced in this manner is obviously dependent on many unknown parameters and may change significantly due to finite temperature correc- Ž y3 tions but even a change of O 10 in the phase of the neutrino VEV may lead to a significant production of baryons Acknowledgements The authors thank R Mohapatra for discussions References wx 1 AD Sakharov, JETP Lett 5 Ž wx For recent reviews, see AG Cohen, DB Kaplan, AE Nelson, Ann Rev Nucl Part Phys 43 Ž ; M Quiros, Helv Phys Acta 67 Ž ; ME Shaposhnikov, preprint CERN-THr96-13 wx 3 M Shaposhnikov, JETP Lett 44 Ž ; Nucl Phys B 87 Ž ; B 99 Ž wx 4 MB Gavela, P Hernandez, J Orloff, O Pene, ` C Quimbay, Mod Phys Lett A 9 Ž ; Nucl Phys B 430 Ž 94 38; P Huet, E Sather, Phys Rev D 51 Ž wx 5 M Carena, M Quiros, CEM Wagner, Phys Lett B 380 Ž wx 6 M Carena, M Quiros, A Riotto, I Vilja, CEM Wagner, Nucl Phys B 503 Ž wx 7 T Multamaki, I Vilja, Phys Lett B 411 Ž wx 8 A Riotto, hep-phr wx 9 RN Mohapatra, Phys Rev D 34 Ž ; A Font, LE Ibanez, F Quevedo, Phys Lett B 8 Ž ; SP Martin, Phys Rev D 46 Ž w10x M Gell-Mann, P Ramond, R Slansky, in: P van Niewenhuizen, DZ Freedman Ž Eds, Supergravity, North Holland, 1979; T Yanagida, in: O Sawada, A Sugamoto Ž Eds,

205 T Multamaki, I ViljarPhysics Letters B 445 ( 1998) Proceedings of Workshop on Unified Theory and Baryon Number in the Universe, KEK, 1979 w11x A Riotto, Nucl Phys B 518 Ž w1x H Haber, G Kane, Phys Lett 117 Ž w13x M Cvetic, J Pati, Phys Lett B 135 Ž ; R Mohapatra, Prog in Part and Fields 6 Ž ; R Francis, M Frank, C Kalman, Phys Rev D 43 Ž ; Y Ahn, Phys Lett B 149 Ž w14x R Kuchimanhci, R Mohapatra, Phys Rev D 48 Ž w15x T Multamaki, I Vilja, Phys Lett B 433 Ž w16x K Huitu, J Maalampi, Phys Lett B 344 Ž

206 31 December 1998 Physics Letters B Heavy flavour mass corrections to the longitudinal and transverse cross sections in e q e y -collisions V Ravindran, WL van Neerven 1 DESY-Zeuthen, Platanenallee 6, D Zeuthen, Germany Received 14 October 1998; revised 4 November 1998 Editor: PV Landshoff Abstract We present the heavy flavour mass corrections to the order a corrected longitudinal Ž s and transverse Ž s s L T cross sections in e q e y -collisions Its effect on the value of the running coupling constant extracted from the longitudinal cross section is investigated Furthermore we make a comparison between the size of these mass corrections and the magnitude of the order as contribution to st and sl which has been recently calculated for massless quarks Also studied will be the changes in the above quantities when the fixed pole mass scheme is replaced by the running mass approach q 1998 Elsevier Science BV All rights reserved Experiments carried out at electron positron colliders like LEP and LSD have provided us with a wealth of information about the constants wx 1 appearing in the standard model of the electroweak and strong interactions One of them is the strong coupling constant as which can be extracted from various quantities Examples are the hadronic width of the Z-boson, the total hadronic cross section, event shapes of jet distributions and jet rates Žfor a review on this subject see wx Another quantity from which as can be extracted is the longitudinal cross section s which is measured in the semi inclusive reaction L e y qe q V Pq X Ž 1 Here V represents the intermediate vector bosons Z and g and X denotes any inclusive final state Furthermore P stands for a hadron which is produced via fragmentation by a quark or a gluon As we will see below the perturbation series in QCD for sl starts in order as provided the quark which couples to the vector boson is massless Therefore this quantity will become very sensitive to the value of the strong coupling constant Some time ago the perturbation series was only known up to order a Žsee w5,6 x s However it turned out that the result for the longitudinal cross section was much smaller than its experimental value measured at LEP w3,4x which indicated that the higher order corrections may become rather large The latter was confirmed by the second order corrections, computed very recently in wx 7, which amount to about 35% with respect to the 1 On leave of absence from Instituut-Lorentz, University of Leiden,PO Box 9506, 300 RA Leiden, The Netherlands r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

207 ( ) V RaÕindran, WL Õan NeerÕenrPhysics Letters B lowest order contribution to s L If the second order corrections are included the longitudinal cross section agrees now very well with experiment so that it can be used to extract a Žsee wx 8 Until now s was only evaluated for massless quarks This will be a correct approximation for the light quarks like u,d,s and probably also for c but it will certainly fail for the b-quark, leave alone for the top quark Therefore one has to compute the mass corrections to this quantity at least up to first order in a This contribution will be presented in this s paper and we discuss its effect on the value of as when extracted from experiment via s L The cross section corresponding to process Ž 1 where the hadron P emerges directly, or indirectly via the gluon, from the heavy quark H is given by d s H,P Ž x,q dst H,P Ž x,q dsl H,P Ž x,q dsa H,P Ž x,q s 8Ž 1qcos u q 4sin u q 4cosu dx dcosu dx dx dx From the equation above one can extract the transverse Ž s, the longitudinal Ž s and the asymmetric Ž s T L A cross sections Further u denotes the angle between the outgoing hadron P and the incoming electron The Bjørken scaling variable is defined by pq xs, q sq )0, 0-xF1, Ž 3 Q where the momenta p and q correspond with the outgoing hadron P and the virtual vector boson Ž Z,g respectively To obtain the total transverse and longitudinal cross sections one has to multiply Eq Ž with x After integration over x and summation over all hadron species P emerging from the heavy flavour H one obtains 1 d sk H,P H 1 sk Ž Q s ÝH dx x Ž x,q kst, L Ž 4 dx P 0 The result above can be written as follows 4m H Õ a sk Ž Q ssvvž Q hkž r qsaaž Q hkž r with rs, Ž 5 Q where m denotes the heavy quark mass The above definitions for the total cross sections should not be confused with the ones which are obtained by integrating d skrdx without multiplication by x The heavy flavour contributions to the latter quantities were computed for the first time in wxž 9 see also w10 1 x The difference between these two definitions will be pointed out below Finally the total cross section for heavy flavour production is given by s H Q ss H Q qs H Q Ž 6 tot T L Ž In Eq 5 the pointlike cross sections are defined by ZŽ Q 4pa Q Q ymz Q VV l q l q V, l V,q Ž V, l A, l V,q s s Q s N e e q e e C C q C q C C, Ž 7 3Q < Z Q < 4pa Q AA Ž V, l A, l A,q s Q s N C qc C, Ž 8 3Q Z Q where N denotes the number of colours corresponding to the gauge group SUŽ N Žin the case of QCD one has N s 3 Furthermore we adopt for the Z-propagator the energy independent width approximation ZŽ Q sq ymz qimzg Z Ž 9 L

208 08 ( ) V RaÕindran, WL Õan NeerÕenrPhysics Letters B In Eqs Ž 7 and Ž 8 the charges of the lepton and the quark are denoted by e l and eq respectively and the electroweak constants are given by 1 CA, ls, CV, ls yca, l Ž 1 y 4sin u W, CA,usyCA,dsyC A, l, CV,u sinu W 8 4 s CA, l 1 y sin u, C syc 1 y sin 3 W V,d A, l 3 u W 10 l The functions h Ž kst, L, lsõ,a in Eq Ž 5 k can be obtained order by order in perturbative QCD from the singlet quark and the gluon coefficient functions in the following way H 1 1 1yr l l,s l,s k k,q k, g 'r 0 H h Ž r s dx x C x,r,q rm q dx x C x,r,q rm Ž 11 Here m stands for the factorization as well as the renormalization scale The result in Eq Ž 11 has been derived H,P from the differential cross section d s rdx Žsee eg wx k 6 The latter can be written as a convolution of the P fragmentation densities D and the coefficient functions C Ž i s q, g Žsee Eq 4 in w14 x i k,i Because of the momentum sum rule satisfied by the fragmentation densities Žsee Eq 9 in wx 6 expression Ž 11 follows l l immediately Notice that the functions h differ from the functions f computed in w1x Ž k k see also the functions H Ž is,6 in w10 x i The latter were derived from the first moment of the non-singlet quark coefficient function and they differ from the former which are given by the second moment of the singlet quark and gluon coefficient functions presented in Eq Ž 11 However because of Eq Ž 5 the sum of the transverse and longitudinal components of both functions have to lead to the same perturbation series of the total cross section l l l l l so that one has the relation htq hls ftq fl ls Õ, a The functions hk can be expanded in the strong coupling constant as as follows n ` asž m l l,ž n hkž r s Ý hk Ž r Ž 1 4p ns0 ž / The lowest order contributions corresponding to the Born reaction V HqH, Ž 13 with Vsg,Z are given by r Õ,Ž0 Õ,Ž0 a,ž0 3r a,ž0 ht Ž r s' 1yr, hl Ž r,s ' 1yr, ht Ž r s Ž 1yr, hl Ž r s0, Ž 14 where r is defined in Eq Ž 5 The next-to-leading order Ž NLO contributions originate from the one-loop virtual corrections to reaction Ž 13 and the gluon bremsstrahlungs process V HqHqg Ž 15 We have computed the order as contributions to the quark Ck,q l,s and gluon Ck, l,s g coefficients functions for the case m/0 in Eq Ž 11 and confirmed the result obtained in appendix A of wx 6 Notice that the quark coefficient functions are derived from the process where the hadron P emerges from the quark H The gluon coefficient function corresponds with the process where the hadron is emitted by the gluon After performing the integral in Eq Ž 11 we obtain the following results for kst, L Õ,Ž1 1 3r 39 1 ht r scf r 1y3r F1 t qr 1qr F t q 3y rq r Li t ' q 16y10rq6r F3 t q 1yr F4 t q 8y6rq6r ln t ln 1qt 3 ' qy1q3ry r lnž t y5r 1yr, Ž 16

209 with ( ) V RaÕindran, WL Õan NeerÕenrPhysics Letters B Ž Ž 13 Ž 19 ' Õ,Ž1 1 3r hl r scf y r 1y3r F1 t yr 1qr F t q ry r Li t q10r 1yr F3 t ' qr 1yr F4 Ž t q 6ry8r lnž t lnž 1qt q yrq 4 r lnž t q 3q r 1yr, Ž 17 a,ž1 1 3r 103 ht r scf r 1y4r F1 t qr 1q r F t q 3y rq30r Li t 3r ' q 16y6rq16r F Ž t qž 1yr F Ž t q 8y14rq1 r lnž t lnž 1qt qy1q9ry r q r lnž t y 1 rq r 1yr, Ž 18 Ž a,ž1 1 3r hl r scf y r 1y4r F1 t y4f3 t y7li t y4ln t ln 1qt yr 1q r F t ' ' ' q rq r y r lnž t q 3q3rq r 1yr, Ž 19 1y 1yr ts Ž 0 1q 1yr Ž and the colour factor C is given by C s N y1 r N The functions F t appearing above are defined by F F i 3 1 F1 t sli t q4z q ln t q3ln t ln 1qtqt 1 3r 3r 1r 1r F t sli yt yli t qli yt yli t q3z qln t ln 1q' Ž t ' 3 ' 3 ylnž t lnž 1y t q lnž t ln 1qty t y lnž t ln 1qtq' Ž t F3Ž t sli Ž yt qlnž t lnž 1yt Ž 3 F4Ž t s6lnž t y8lnž 1yt y4lnž 1qt, Ž 4 where z Ž n, which occurs in the formulae of this paper for ns,3, represents the Riemann z-function and Li Ž x denotes the dilogarithm Using Eqs Ž 16 Ž 19 one can check that the order as contribution to the total cross section in Ž 6 is in agreement with the literature w13 x The next-to-next-to-leading order Ž NNLO contributions come from the following processes First one has to calculate the two-loop vertex corrections to the Born process Ž 13 and the one-loop corrections to Ž 15 Second one has to add the radiative corrections due to the following reactions V HqHqgqg Ž 5 V HqHqHqH Ž 6 V HqHqqqq Ž 7 where q denotes the light quark The results for h l,ž presented below are only computed for those contributions k containing Feynman graphs where the vector boson V is always coupled to the heavy quark H so that the light quarks can be only produced via fermion pair production emerging from a gluon splitting Because we are only interested in the ratios sk H Q Rk Ž Q s for kst, L Ž 8 H s Q tot we do not consider other contributions which drop out in the expression above provided we put the mass of H equal to zero in the second order correction The contributions which can be omitted are given by one- and two-loop vertex corrections which contain the triangular quark-loop graphs w15,16 x They only show up if the

210 10 ( ) V RaÕindran, WL Õan NeerÕenrPhysics Letters B quarks are massive and are coupled to the Z-boson via the axial-vector vertex Notice that one has to sum over all members of one quark family in order to cancel the anomaly Further we exclude all contributions from reaction Ž 7 which involve Feynman graphs where the light quarks q are coupled to the vector boson V Notice that interference terms of the latter with diagrams where heavy quark are attached to the vector boson vanish if the heavy quark is taken to be massless provided one sums over all members in one family The order a contributions to the quark and gluon coefficient functions have been calculated in w7,14 x s Because of the complexity of the calculation of these functions the heavy quark mass was taken to be zero This approximation is good for the charm and bottom quark but not for the top quark as we will see below Substituting these coefficient functions in the integrand of Eq Ž 11 Ž here r s 0 we obtain ht sht scf 6 qcacf y15 y 5 z 3 qnfct F f 3 q16z 3, 9 Õ,Ž a,ž ½ ž / 5 ½ ž / 5 Q Q Õ,Ž a,ž hl s hl s CFy 4 q CC A F y11ln q y z Ž 3 q nct f F f 4ln y, Ž m m where the colour factors are given by C sn and T s1r Žfor C see below Eq Ž 0 A f F Further n f denotes the number of light flavours which originate from process Ž 7 Finally m appearing in the strong coupling constant a and the logarithms in Eq Ž 30 s represents the renormalization scale Notice that the coefficient of the logarithm is proportional to the lowest order coefficient of the b-function The logarithm does not appear in ht l,ž because ht l,ž1 s0 for ms0 and lsõ,a Since ms0 there is no distinction between h Õ,Ž k and h a,ž k for kst, Lanymore unlike in the case of the first order corrections in Eqs Ž 16 Ž 19 where the heavy quark was taken to be massive Furthermore one can check that substitution of Eqs Ž 9 and Ž 30 into Eq Ž 6 provides us with the order a contribution to the total cross section which is in agreement with the results obtained in w17 x s When the quark is massless we get from Eqs Ž 5 and Ž 6 the following perturbation series for the ratio in Eq 8 up to order a s ž / ½ ž / 5 Q 74 ½ ž m / 5 asž m asž m Q H RL Ž Q s CF34 q CFy 4qCC A F y11ln q y44z Ž 3 4p 4p m qn CT 4ln y, Ž 31 f F F 3 R H Q s1yr H Q Ž 3 T L We will now show how the results for the above ratios will be modified when the Born and the first order contribution are computed for massive quarks Further we discuss the consequences of our findings for the extraction of a In order to make the comparison between the massless and massive approach to Eq Ž 8 s we Ž w x have chosen the following parameters see 18 The electroweak constants are: MZ s GeVrc, MS G s490 GeVrc and sin u s03116 For the strong parameters we choose: L s37 MeVrc Ž n s5 Z W 5 f which implies a Ž M s 0119 Ž two-loop corrected running coupling constant s Z Further we take for the renormalization scale msq unless mentioned otherwise Notice that we study R H k for Hsc,b at the CM energy QsM Z For the heavy flavour masses the following values are adopted: mcs15 GeVrc, mbs 450 GeVrc and mts1738 GeVrc The results for the bottom quark can be found in Table 1 A comparison between the first and second column reveals that on the Born level the difference between the massive and b b massless approach to RT is very small about two promille For R L it is more conspicuous but the correction due to mass effects, which equals for R Ž1 L, is still much smaller than the order as contribution which amounts to 0011 The corrections due to mass effects are also smaller than the changes caused by a different choice of the renormalization scale If we choose msqr orms Q one gets R Ž L s00509 and R Ž L s00461 respectively which differ by 0004 from the central value R Ž L s00485 We also studied the effect of the

211 ( ) V RaÕindran, WL Õan NeerÕenrPhysics Letters B Table 1 The ratio R ss rs Ž0 T Ž1 T Ž T k k tot Ž kst, L for bb-production b R T m s00 GeVrc m s450 GeVrc m M s80 GeVrc b b b Z R R R Ž0 L Ž1 L Ž L b R L R R R running quark mass on the ratio in Eq Ž 8 For this purpose one has to change the on-mass shell scheme used in Eqs Ž 16 Ž 19 into the MS-scheme This can be done by substituting in all expressions the fixed pole mass m by the running mass mž m Moreover one has to add to the first order contributions Ž 16 Ž 19 the finite counter term ž / dm / Dh smž m C 4y3ln, Ž 33 l, Ž 0 m m dhk r l,ž1 k F m ž msmž m where m stands for the mass renormalization scale for which we choose msq Further we adopt the two-loop corrected running mass with the initial condition mž m 0 s m 0 Using the relation between the MS-mass and the fixed pole mass, as is indicated by the first factor on the righthand side in Eq Ž 33, we have taken for bottom production m0 s410 GeVrc which corresponds with a pole mass mb s450 GeVrc This choice leads to m M s 80 GeVrc which is 5% above the experimental value 67 GeVrc measured at LEP w19 x b Z The results are presented in the third column of Table 1 Comparing the second with the third column we observe that the mass corrections decrease because the running mass is smaller than the fixed pole mass It is now interesting to see how the mass terms contributing to R b affect the extraction of the running coupling constant L at msqsm Z To that order we equate the mass corrected formula for R L b with the massless result presented in Ž Eq 31 The latter yields R s00499 for a s0119 Ž see first column, third row of Table 1 Using the same L s number for the massive expression of R Ž L we obtain ass01 which amounts to an enhancement of 5% for mb s450 GeVrc In the case of a running mass ie mb MZ s80 GeVrc we get ass010 so that the enhancement becomes 10% However as we have said before these mass effects are smaller than those caused by a variation in the renormalization scale Following the same procedure we equate R L at different values of m Choosing msq one infers from Table 1 that R Ž L s00499 for ass0119 This value of R Ž L can also be obtained at msqr but then one has to take ass017 If we choose ms Q the result is ass011 Therefore the uncertainty is about 6% which is larger than the mass correction The latter becomes even smaller when we also add the part coming from the bottom quark to the longitudinal cross section consisting of the contributions of the light quarks and the charm quark Concerning the latter quark we want to remark that its mass is so small that the massive approach will become indistinguishable from the massless results Since the mass effects up to the order as level are rather small even for the bottom quark we can assume that the second order corrections, derived for ms0, also apply for m/0 at least as long as Q4m This will be correct at large collider energies for all quarks except for the top quark as we will see below For tt production we will study the mass effects up to order as at a CM energy Qs500 GeVrc Here we have chosen m0 s1661 GeVrc which corresponds with a fixed pole mass of mts1738 GeVrc This choice leads to m Q s1535 GeVrc From Table we infer that for this process the mass corrections are t

212 1 ( ) V RaÕindran, WL Õan NeerÕenrPhysics Letters B Table The ratio R ss rs Ž0 T Ž1 T Ž T k k tot Ž kst, L for tt-production at Qs500GeVrc t R T m s0 GeVrc m s1738 GeVrc m Q s1535 GeVrc t t t R R R Ž0 L Ž1 L Ž L t R L R R R huge and they are much larger than the first and second order corrections Therefore the numbers in the last row, presented for RT t and R L t, are unreliable because they only hold if the mass corrections in the order as Ž contributions can be neglected as was done in Eqs 9 and 30 This implies that for the top quark RT and R Ž L have to be computed for m/0 which will be an enormous enterprise We also studied the effect of the running mass presented in the third column of Table Here the difference between the fixed pole mass and the running mass approach is larger than the one observed for the bottom quark in Table 1 Another feature is that the order as correction increases when the running mass scheme is used and its sign is reversed with respect to t the correction obtained for the fixed pole mass approach Notice that a study of the change in R Ž kst, L k under variation of the renormalization scale makes no sense because of the missing exact result in second order for m/0 Summarizing our findings we have computed the mass corrections up order as for the transverse and longitudinal cross sections which are due to heavy flavours In the case of charm there is no observable difference between the massless and massive approach For the bottom quark we see a difference which leads to an enhancement of the extracted value of a which is of the order of 5% Žfixed pole mass m s450 GeVrc s b Ž or 10% running mass m M s 80 GeVrc b Z These numbers become much smaller when the light flavour contributions are added to the cross sections A variation in the renormalization scale introduces larger effects on the value for a which are about 60% Žsee eg also wx s 8 In the case of top-quark production the mass terms cannot be neglected anymore since they are much larger than the second order corrections computed for massless quarks Acknowledgements The authors would like to thank J Blumlein for reading the manuscript and giving us some useful remarks This work is supported by the EC-network under contract FMRX-CT References wx 1 G Altarelli, R Barbieri, F Caravaglios, Int J Mod Phys A 13 Ž wx RK Ellis, WJ Stirling, BR Webber, QCD and Collider Physics, Cambridge University Press, 1996, chapter 6 wx 3 D Buskulic et al Ž ALEPH-collaboration, Phys Lett B 357 Ž wx 4 R Akers et al Ž OPAL-collaboration, Z Phys C 68 Ž wx 5 PG Altarrelli, RK Ellis, G Martinelli, S-Y Pi, Nucl Phys B 160 Ž ; R Baier, K Fey, Z Phys C Ž wx 6 P Nason, BR Webber, Nucl Phys B 41 Ž

213 ( ) V RaÕindran, WL Õan NeerÕenrPhysics Letters B wx 7 PJ Rijken, WL van Neerven, Phys Lett B 386 Ž wx 8 P Abreu et al Ž DELPHI-collaboration, CERN-PPEr wx 9 J Jersak, E Laermann, PM Zerwas, Phys Lett B 98 Ž ; Phys Rev D 5 Ž ; D 36 Ž Ž E w10x A Arbuzov, DY Bardin, A Leike, Mod Phys Lett A 7 Ž ; A 9 Ž Ž E w11x JB Stav, HA Olsen, Phys Rev D 5 Ž w1x V Ravindran, WL van Neerven, DESY , hep-phr w13x K Schilcher, MD Tran, NF Nasrallah, Nucl Phys B 181 Ž ; B 187 Ž Ž E ; JL Reinders, S Yazaki, HR Rubinstein, Phys Lett B 103 Ž ; TH Chang, KJF Gaemers, WL van Neerven, Phys Lett B 108 Ž 198 ; Nucl Phys B 0 Ž w14x PJ Rijken, WL van Neerven, Nucl Phys B 487 Ž w15x BA Kniehl, JH Kuhn, Phys Lett B 4 Ž : Nucl Phys B 39 Ž w16x RJ Gonsalves, CM Hung, J Pawlowski, Phys Rev D 46 Ž w17x KG Chetyrkin, AL Kataev, FV Tkachov, Phys Lett B 85 Ž ; M Dine, J Sapirstein, Phys Rev Lett 43 Ž ; W Celmaster, RJ Gonsalves, Phys Rev Lett 44 Ž w18x C Caso et al, Review of Particle Physics, Eur Phys J C 3 Ž , 4, 87 w19x P Abreu et al Ž DELPHI-collaboration, Phys Lett B 418 Ž

214 31 December 1998 Physics Letters B Second order QCD corrections to the forward-backward asymmetry in e q e y -collisions V Ravindran, WL van Neerven 1 DESY-Zeuthen, Platanenallee 6, D Zeuthen, Germany Received September 1998 Editor: PV Landshoff Abstract We will present the result of an analytical calculation of the second order contribution to the forward-backward asymmetry A H FB and the shape constant a H for heavy flavour production in e q e y -collisions The calculation has been carried out by assuming that the quark mass is equal to zero This is a reasonable approximation for the exact second order correction for charm and bottom quark production at LEP energies but not for top production at future linear colliders Our H result for A is a factor 6 Ž charm and 47 Ž bottom FB larger than obtained by a numerical calculation performed earlier in the literature We study the effect of the second order corrections on the above parameters including their dependence on the renormalization scale Further we make a comparison between the fixed pole mass and the running mass approach q 1998 Elsevier Science BV All rights reserved Experiments carried out at electron positron colliders like LEP and LSD have provided us with a wealth of information about the constants wx 1 appearing in the standard model of the electroweak and strong interactions One among them is given by the electroweak mixing angle defined by u W which can be very accurately extracted from the forward-backward asymmetry in heavy flavour production in particular when the flavour is represented by the bottom quark wx Recently this quantity is obtained for the charm quark wx 3 and in the future q y one also hopes to measure it for the top quark at the large linear e e -collider Žsee eg wx 4 The forward-backward asymmetry is extracted from the differential cross section given by H ds Ž Q 3 Õ a s 8 Ž 1qcos u svv Ž Q ft Ž r qsaaž Q ft Ž r dcosu 3 Õ a 3 a q 4sin u svv Q fl r qsaa Q fl r q 4cosu sva Q fa r, 1 where u is the angle between the outgoing quark H and the incoming electron The CM energy is denoted by Q and the scaling variable r is defined by rs4m rq where m stands for the quark mass Notice that the form of the cross section above is correct if only final state corrections are present which is the case for QCD 1 On leave of absence from Instituut-Lorentz, University of Leiden, PO Box 9506, 300 RA Leiden, The Netherlands r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

215 ( ) V RaÕindran, WL Õan NeerÕenrPhysics Letters B investigated in this paper For electroweak corrections Žsee wxwx 5, 6 which occur in the initial as well as in the final state, including interference terms, the above formula has to be modified The Born cross sections for quark final states appearing in Eq Ž 1 can be written as 4pa Q Q ymz Q VV l q l q V, l V,q Ž V, l A, l V,q s Q s N e e q e e C C q C q C C, 3Q Z Q Z Q 4pa Q AA Ž V, l A, l A,q s Q s N C qc C, Ž 3 3Q Z Q 4pa Q Q ymz Q VA l q A, l A,q A, l A,q V, l V,q s Q s N e e C C q4 C C C C, Ž 4 3Q Z Q Z Q where N denotes the number of colours in the case of the gauge group SU N in QCD one has Ns3 Furthermore in the expressions above we adopt for the Z-propagator the energy independent width approximation ZŽ Q sq ymz qimzg Z Ž 5 The charges of the lepton and the quark are given by e l and eq respectively and the angle u W defined at the beginning appears in the electroweak constants as follows 1 CA, ls, CV, ls yca, l Ž 1 y 4sin u W, CA,usyCA,dsyC A, l, sinu W 8 4 CV,us CA, l 1 y sin u, C syc 1 y sin 3 W V,d A, l 3 u W 6 l The functions f Ž kst, L, A; lsõ,a in Eq Ž 1 k can be computed order by order in perturbative QCD from the non-singlet quark coefficient function Ck,q l,ns as follows ž / 1 Q 4m l l,ns fkž r sh dx Ck,q x,r, with rs, Ž 7 m Q 'r where m stands for the factorization as well as the renormalization scale The quark coefficient function appears Ž in the fragmentation function Fk x,q with xs pqrq where p is the momentum of the outgoing hadron which originates from the quark These fragmentation functions describe the production of the quark and its subsequent decay into a hadron The forward backward asymmetry, denoted by A H FB, appears when we divide the expression in Eq Ž 1 by the total cross section The ratio can then be expressed in the following way 1 ds H Ž Q 3 4 H H s 1qa Q cos u qa Q cosu, Ž 8 H H Ž FB s Q dcosu 8 3qa Q with tot ž / 3 s VA Q fa a Ž r H AFB Ž Q s Ž 9 H 4 s Q Further the shape coefficient a H is defined by tot Õ Õ a a s VV Ž Q ft Ž r y fl Ž r qsaaž Q ftž r y flž r H a Ž Q s, Ž 10 Õ Õ a a s Ž Q f Ž r q f Ž r qs Ž Q f Ž r q f Ž r VV T L AA T L

216 16 ( ) V RaÕindran, WL Õan NeerÕenrPhysics Letters B and the total cross section for heavy flavour production is equal to H Õ Õ a a stot Q ss VV Q ft r qfl r qsaa Q ft r qfl r 11 The functions f l can be expanded in the strong coupling constant a k ž / n s as follows ` asž m l l,ž n fkž r s Ý fk Ž r Ž 1 4p ns0 The lowest order contributions corresponding to the Born reaction V HqH, Ž 13 with Vsg,Z are given by r Õ,Ž0 Õ,Ž0 a,ž0 3r a,ž0 ft Ž r s' 1yr, fl Ž r s ' 1yr, ft Ž r s Ž 1yr, fl Ž r s0, fa a,ž0 Ž r s1yr, Ž 14 where r is defined below Eq Ž 1 The next-to-leading order Ž NLO contributions originate from the one-loop virtual corrections to the Born reaction Ž 13 and the gluon bremsstrahlungs process V HqHqg Ž 15 The NLO contributions have been calculated by several groups in the literature for the case m/ 0 Žsee w7 10 x However there were some discrepancies between the results so that we decided to calculate them again using a different method Here they are derived from the order as contribution to the coefficient functions for heavy quarks which appears in the integrand of Eq Ž 7 Our computations lead to the same answers as given in appendix A of w11 x After performing the integral in Eq Ž 7 we obtain the following results for kst, L with ' Õ,Ž ft r scf r 1qr F t q r 1y3r F t q 3y ry r 1 Li t ' q 16y10ry r F3 t q 1yr F4 t q 8y6ry r ln t ln 1qt ' Ž qy1q9ry r lnž t q 1yr 1q r, Ž 16 ' Õ,Ž fl r scf y r 1qr F t y r 1y3r F t q ry r Li t q 10ry r 1 F3 t ' 3r 4 qr 1yr F t q6rln t ln 1qt q y7rq3r ln t q 1yr, 17 ' a,ž ft r scf r 1q r F t q r 1y4r F t q 3y rq9r 1 Li t 3 4 3r ' Ž q 16y6rq4r F t q 1yr F t q 8y14r ln t ln 1qt qy1q15ry r lnž t q 1yr 1q r, Ž 18 Ž a,ž1 1 fl r scf y r 1q r F1 t y4f3 t y7li t y4ln t ln 1qt y r 1y4r F t ' ' ' qy4rqr y r ln t q 1yr y rq r 8 4, 19 1y 1yr ts Ž 0 1q 1yr '

217 Further the colour factor C by ( ) V RaÕindran, WL Õan NeerÕenrPhysics Letters B Ž is given by C s N y1 r N The functions FŽ t appearing above are defined F F i 3 1 F1 t sli t q4z q ln t q3ln t ln 1qtqt, 1 3r 3r 1r 1r F t sli yt yli t qli yt yli t q3z qln t ln 1q' Ž t ' 3 ' 3 ylnž t lnž 1y t q ln t ln 1qty t y ln t ln 1qtq' Ž t, F3Ž t sli Ž yt qlnž t lnž 1yt, Ž 3 F4Ž t s6lnž t y8lnž 1yt y4lnž 1qt, Ž 4 where z Ž n, which appears for ns,3 in the formulae of this paper, represents the Riemann z-function and Li Ž x denotes the dilogarithm Using Eqs Ž 16 Ž 19 one can check that the order as contribution to the total cross section in Ž 11 is in agreement with the literature w1x Žfor the vector part see also w13 x Integration of the asymmetry quark coefficient function provides us with the result a,ž1 f r sc y4 yr ' 1yr G t q 4y5r G t q8ln 1qty' A F 1 Ž t y8ž 1yr ŽlnŽ 1qt Ž ' Ž ' Ž ' qln 1y t q 41y Ž r q 1yr Ž yq3r lnž t q4 ry r, Ž 5 which involves the following functions 3r 1r 1r 1 1 G1Ž t s Li Ž yt y 3Li Ž yt y 4Li Ž t y Li Ž yt y z y ln 8 Ž t, Ž 6 ' t GŽ t sli y z y lnž t lnž 1qt q ln Ž 1qt y 8ln Ž t 1qt Ž 7 ž / l,ž1 The functions f Ž kst, L; lsõ,a are related to the functions H, and H presented in Eq Ž 15 k 6 and wx a,ž1 appendix A of 9 and we agree with their result The same holds for fa which is proportional to H5 in the reference above There is also agreement with the calculation in w10 x The comparison is made by expanding the functions above and those in wx 9 up to seven powers in r The next-to-next-to-leading order Ž NNLO contributions come from the following processes First one has to compute the two-loop vertex corrections to the Born process Ž 13 and the one-loop corrections to Ž 15 Second one has to add the radiative corrections due to the following reactions V HqHqgqg, Ž 8 V HqHqHqH, Ž 9 V HqHqqqq, Ž 30 where q denotes the light quarks The results for fk l,ž presented below are computed for the contributions where the vector boson V is always coupled to the heavy quark H so that the light quarks in Eq Ž 30 are only produced via fermion pair production emerging from gluon splitting Besides these contributions there are other ones which have been treated in w14 x The latter consist of all one- and two-loop vertex corrections which contain the triangular quark-loop graphs Following the notation in w14x their contribution to the forward-backward symmetry will be denoted by FQCD 3-jet and FQCD -jet respectively They only show up if the quarks are massive and are coupled to the Z-boson via the axial-vector vertex Notice that one has to sum over all members of one quark family in order to cancel the anomaly Adopting the mass assignment in w14x we take the top to be massive and put the other quark masses, including that of the bottom, equal to zero Further in w14x one has included all terms originating from reaction Ž 30 where diagrams with light quarks attached to the vector boson V interfere with those describing the coupling of the heavy quarks to the vector boson Notice that this w x F contribution denoted in 14 by F vanishes if all quarks including H are taken to be massless provided one QCD

218 18 ( ) V RaÕindran, WL Õan NeerÕenrPhysics Letters B sums over all members in one family However we will omit that part of reaction Ž 30 where the heavy quarks are produced via gluon splitting and the light quarks q are coupled to the vector boson V This contribution Branco denoted by F in w14x QCD needs a cut on the invariant mass of the heavy flavour pair and it was computed for the first time in w 15 x Finally notice that these additional contributions, denoted by FQCD above, only show up in order a Moreover if we put m s 0 they only appear in the forward-backward asymmetry Ž s 9 but cancel l,ž between numerator and denominator in the shape coefficient 10 The results for fk follow from the transverse and longitudinal coefficient functions calculated in w16x and the asymmetry coefficient function computed in w17 x Because of the complexity of the calculation of these functions the heavy quark mass was taken to be zero This approximation is good for the charm and bottom quark but not for the top quark as we will see below Substituting these coefficient functions in the integrand of Eq Ž 7 we obtain Q Q Õ,Ž a,ž ft sft scf 4qCACF½y 3 ln q 18 y44z Ž 3 qnfct F f 3ln y 9 q16z Ž 3, ž 5 ½ m / ž m / 5 Ž 31 ½ ž / 5 ½ ž / 5 Q Q Õ,Ž a,ž fl sfl scfy54 qcacf y 3 ln q qnct f F f ln y, Ž m m 4 4 f a,ž scc y44z Ž 3 qn CT 16z Ž 3 Ž 33 A A F f F f Here the colour factors are given by C sn and T s1r Žfor C see below Eq Ž 0 A f F Further n f denotes the number of light flavours which originate from process Ž 30 Finally m appearing in the strong coupling constant a and the logarithms in Eqs Ž 31 and Ž 3 s represents the renormalization scale Notice that the coefficient of the logarithm is proportional to the lowest order coefficient of the b-function The same holds for z Ž 3 in Eqs a,ž a,ž1 31 and 33 The logarithm does not appear in fa because fa s0 atms0 Since the mass is equal to Õ,Ž a,ž zero there is no distinction anymore between f and f Ž ks T, L k k unlike in the case for the first order corrections in Eqs Ž 16 Ž 19 where the heavy quark was taken to be massive Furthermore one can check that substitution of Eqs 31 and 3 into 11 provides us with the order as contribution to the total cross section which is in agreement with the results obtained in w18 x For zero mass quarks the forward-backward asymmetry becomes equal to ž / ½ ž ž / / A Q sa Q 1y C 3 q C qc C 11ln y as m as m Q H H,Ž FB FB F 4 FŽ A F 4p 4p m Q f F F 5 m ž ž / / qn CT y4ln q Ž 34 Ž wx H For the discussion below and the notation often used in the literature see it is convenient to write AFB in the following way ž / asž m asž m H H,Ž0 AFB Ž Q safb Ž Q 1y c1 y c Ž 35 p p The order a contribution presented above has been also calculated in w14x s but then in a numerical way Unfortunately we get a different answer First we disagree with the statement above Eq Ž 48 in w14x that H reaction 9 does not contribute to A FB For the latter we obtain the following contribution to c, denoted by D Ž1 c, which equals Ž D c sy CFy CC A F 16 y q6z q8z 3 s014 36

219 ( ) V RaÕindran, WL Õan NeerÕenrPhysics Letters B The result for reactions Ž 8 and Ž 30, including the virtual corrections, can be obtained by subtracting Eq Ž 36 from Eq Ž 34 Using the notation in Eq Ž 45 w14x and choosing n s5 we find the following contributions 3 Ž 1 9 c1 s 4CF D c sy4cf 4CFqCCFqNNCqTT R, Eq Ž 46 w14x CCFs58, NNCsy310, TTRs14, our result CCFs y83, NNCs y438, TTRs 1375 From the results above we infer that the discrepancies mainly occur in the C and CC-terms of Eq Ž 34 F A F Substituting C Ž Fs 4r3 in the expression above we obtain for the coefficient of the as Q rp -term the value y949 instead of y6 quoted in Eq Ž 47 of w14x which amounts to a discrepancy of a factor of about 37 a,ž Notice that the bulk of the second order correction to Eq 34 is coming from f in Eq Ž 33 A which amounts l,ž to 916 The remaining part can be traced back to the functions f in Eqs Ž 31 and Ž 3 k leading to the contribution 0409 Finally we are now also able to present the second order correction for the shape coefficient Ž 10 in an analytical way a Q sa Q 1y C 8 q C 60 qc C ln y as m as m Q H H,Ž F 4 F A F 3 9 4p 4p m Q f F F m ž ž / / f ž / ½ ž ž / / Ž 37 qn CT y ln q, Ž 38 H which can be written in the same way as has been done for A in Eq Ž 35 so that we get ž / asž m asž m H H,Ž0 a Ž Q sa Ž Q 1y d1y d Ž 39 p p We will now discuss the effect of the order as corrections on the forward backward symmetry and the shape coefficient for the bottom and the charm quark Further we omit any effect coming from the electroweak sector Our results are obtained by choosing the following parameters Žsee w19 x The electroweak constants are: MZs91187 GeVrc, GZs490 GeVrc and sin u Ws03116 For the strong parameters we choose: MS L s 37 MeVrc Ž n s 5 which implies a Ž M s 0119 Ž two-loop corrected running coupling constant 5 f s Z Further we take for the renormalization scale msq unless mentioned otherwise Notice that we study A H FB and a H for Hsc,b at the CM energy QsM Z For the heavy flavour masses the following values are adopted: mcs150 GeVrc, mbs450 GeVrc and mts1738 GeVrc The results for the bottom quark can be found in Table 1 The values for c are obtained by adding to Eq Ž 34 the contributions mentioned below Eq Ž 30 -jet 3-jet F They were denoted by F, F and F in w14 x QCD QCD QCD For the value of the top mass given above we obtain for bottom production c,qcd -jet sy0645, c,qcd 3-jet sy018 and c,qcd F s013 In the entry for c we have also mentioned between the brackets the result obtained from the contribution of Ž 34 which is equal to c s 77r8 Notice that the second order contributions to the quantities in the tables are obtained by multiplying A Ž0 FB at ms0 with c Furthermore we have put in the tables the correction in percentages of the radiatively corrected quantities AFB H, a H with respect to their zeroth order result From Table 1 we infer that the order as as well as the order a corrections to both A b and a b are negative In the case of the forward-backward asymmetry the s FB QCD corrections are moderate in particular the second order ones The latter would be even smaller if we had taken the value c s19 quoted in w14x which is a factor 47 less with respect to our result In the case of the shape coefficient the QCD corrections are at least twice as large The latter are reduced if for the reference axis the thrust axis is taken instead of the quark axis wx Notice that the quark axis has been chosen in our calculation Further we want to mention that the zero mass approximation for the second order coefficient c is FB

220 0 ( ) V RaÕindran, WL Õan NeerÕenrPhysics Letters B Table 1 The forward-backward asymmetry and the shape constant of the bottom quark b A FB m s450 GeVrc m M s80 GeVrc b b Z c c 889 Ž Ž 963 AŽ0 FB Ž1 A 0100 Ž y304% Ž y333% FB Ž A Ž y48% Ž y456% FB a b d d Ž0 a Ž1 a 0916 Ž y785% 0910 Ž y88% Ž a 0883 Ž y11% 0877 Ž y11% quite reasonable This is revealed by the first order coefficient when the quark mass is chosen to be zero which leads to the values c s1 Eq Ž 34 and d s8r3 Eq Ž In the case of the bottom quark we observe a deviation of 11% for c whereas for d it amounts to % which is twice as large For the charm quark Ž 1 1 see Table these values become smaller ie about 75% for both coefficients If we expect that the same deviations occur for the coefficients c Eq Ž 34 and d Eq Ž 38 one gets a reasonable estimate of the theoretical error on the second order corrections We also studied the effect of the running quark mass on the forward-backward asymmetry and the shape coefficient For this purpose one has to change the on-mass shell scheme used in Eqs Ž 16 Ž 19 and Ž 5 into the MS-scheme This can be done by substituting in all expressions the fixed pole mass m by the running mass mž m Moreover one has to add to the first order contributions Ž 16 Ž 19, Ž 5 the finite counter term ž / dm / Df smž m C 4y3ln, Ž 40 l, Ž 0 m m dfk r l,ž1 k F m ž msmž m where m stands for the mass renormalization scale for which we choose msq Further we adopt the two-loop corrected running mass with the initial condition mž m 0 s m 0 Using the relation between the MS-mass and the fixed pole mass, as is indicated by the first factor on the right-hand side in Eq Ž 40, we have taken for bottom production m0 s410 GeVrc which corresponds with a pole mass mb s450 GeVrc This choice leads to m M s 80 GeVrc which is 5% above the experimental value 67 GeVrc measured at LEP w0 x b Z The results are presented in the second column of Table 1 Comparing the latter with the first column we observe Ži that the QCD corrections become a little bit more negative The values of A Ž is 0,1, FB decrease a little too The same features are also shown by a Ži except for a Ž0 which slightly increases In Table we also present results for the charm quark In the case of the running mass we have chosen m s130 GeVrc so that the pole mass becomes m s150 GeVrc This leads to a value of m Ž M 0 c c Z s 066 GeVrc which is rather low Furthermore for charm quark production the additional contributions become c,qcd -jet s1359, c,qcd 3-jet s033 and c,qcd F s011 Our result for c is about 6 times larger than the value 44 obtained in w14 x The features are the same as for the bottom quark The differences between the numbers in the left-hand and right-hand column in Table become even less The reason that the running of the mass in the case of the charm and the bottom quark hardly introduces any effect on the forward-backward asymmetry and the shape constant can be attributed to the fact that the mass of both quarks are small with respect to the CM energy so that the phase space is not much affected Moreover, as far as the dynamics is

221 ( ) V RaÕindran, WL Õan NeerÕenrPhysics Letters B Table The forward-backward asymmetry and the shape constant of the charm quark c A FB m s150 GeVrc m M s066 GeVrc c c Z c c 115 Ž Ž 963 AŽ0 FB Ž1 A 0074 Ž y347% 0073 Ž y360% FB Ž A 0071 Ž y507% Ž y50% FB a c d d Ž0 a Ž1 a 0907 Ž y930% 0903 Ž y970% Ž a 0874 Ž y16% 0870 Ž y130% concerned, these quantities are not proportional to the mass This is also revealed by the constants c1 and d1 which do not deviate very much from their zero mass values These arguments do not apply to the top quark except when Q4m t This is shown in Table 3 where we have studied the above quantities at a CM energy Qs500 GeVrc For the running mass we have chosen m0 s1661 GeVrc so that the pole mass becomes equal to mts1738 GeVrc This leads to a value mt Q s1535 GeVrc The constants c1 and d1 in the perturbation series completely differ from the ones given at m s 0 at which they become 1 and 8r3 respectively Therefore the running mass will have a large effects on these constants which is revealed by the switch of sign in Table 3 Hence the Born approximations to A t FB and a t will change while going from the fixed pole mass to the running mass approach However the order as corrected quantities are less sensitive to the choice between the running or the fixed pole mass because of the compensating term in Eq Ž 40 From the above it is clear that the zero mass approximation to c and d makes no sense in case of the top quark and we have omitted these contributions to A t FB and a t in Table 3 Finally we want to comment on the renormalization scale dependencies of the forward backward asymmetry and the shape constant If we vary the scale m between Qr and Q the changes in A Ž FB are small It introduces an error of 000 for the bottom quark and 0003 for the charm quark For a Ž one can draw the same conclusion and the error becomes 0005 and 0007 respectively In the case of the top quark a variation in the renormalization scale makes no sense because of the missing order as correction Its computation for massive quarks will be a enormous enterprise Table 3 The forward-backward asymmetry and the shape constant of the top quark at Qs500GeVrc t A FB mts1738 GeVrc mt Qs1535 GeVrc c1 y AŽ0 FB Ž1 A % 046 Ž y637% FB a t d1 y Ž0 a Ž1 a % 0435 Ž y154%

222 ( ) V RaÕindran, WL Õan NeerÕenrPhysics Letters B Summarizing our findings we have computed the order as contributions to the forward-backward asymmetry and the shape constant in an analytical way provided the heavy flavour mass is chosen to be zero Further we found a discrepancy with a numerical result calculated earlier in the literature for A HŽ FB The second order corrections are noticeable The transition from the fixed pole mass to the running mass approach does not introduce large changes in the values of A H FB and a H except for the first order constant in the perturbation series when Hst This indicates that the zero mass approach breaks down unless Q4m Also a variation of the renormalization scale does not lead to large effects The latter are almost equal to the differences between the results obtained by the fixed pole mass and the running mass approach Acknowledgements The authors would like to thank J Blumlein for reading the manuscript and giving us some useful remarks WLN would like to thank PW Zerwas for discussions concerning the importance of mass corrections to A b FB and a b This work is supported by the EC-network under contract FMRX-CT References wx 1 G Altarelli, R Barbieri, F Caravaglios, Int J Mod Phys A 13 Ž wx A Djouadi, JH Kuhn, PM Zerwas, Z Phys C 46 Ž ; A Djouadi, B Lampe, PM Zerwas, Z Phys C 67 Ž ; A Djouadi, PM Zerwas, Internal Report, DESY T-97-04; D Abbaneo et al, Eur Phys J C 4 Ž wx 3 R Barate et al, ALEPH-Collaboration, CERN-EPr wx q y 4 V Driesen, W Hollik, A Kraft, in: PM Zerwas Ed, e e collisions at TeV energies: the physics potential, Proceedings of the workshop - Annecy, Gran Sasso, Hamburg, February 1995 to September 1995, DESY 96-13D, pp wx 5 DY Bardin et al, Z Phys C 44 Ž ; Nucl Phys B 351 Ž wx 6 W Beenakker, W Hollik, SC van der Marck, Nucl Phys B 365 Ž ; W Beenakker, W Hollik, Phys Lett B 69 Ž ; W Beenakker, A Denner, A Kraft, Nucl Phys B 410 Ž wx 7 J Jersak, E Laermann, PM Zerwas, Phys Lett B 98 Ž ; Phys Rev D 5 Ž ; D 36 Ž Ž E wx 8 A Djouadi, Z Phys C 39 Ž wx 9 A Arbuzov, DY Bardin, A Leike, Mod Phys Lett A 7 Ž ; A 9 Ž Ž E w10x JB Stav, HA Olsen, Phys Rev D 5 Ž w11x P Nason, BR Webber, Nucl Phys B 41 Ž w1x K Schilcher, MD Tran, NF Nasrallah, Nucl Phys B 181 Ž ; B 187 Ž Ž E ; JL Reinders, S Yazaki, HR Rubinstein, Phys Lett B 103 Ž ; TH Chang, KJF Gaemers, WL van Neerven, Phys Lett B 108 Ž 198 ; Nucl Phys B 0 Ž w13x G Kallen, A Sabry, Dan Mat Fys Medd 9 Ž ; J Schwinger, in: Particles, Sources and Fields, Addison-Wesley, New York, 1973, vol II, ch 5 w14x G Altarelli, B Lampe, Nucl Phys B 391 Ž w15x GC Branco, HP Nilles, KH Streng, Phys Lett B 85 Ž w16x PJ Rijken, WL van Neerven, Phys Lett B 386 Ž ; Nucl Phys B 487 Ž w17x PJ Rijken, WL van Neerven, Phys Lett B 39 Ž w18x KG Chetyrkin, AL Kataev, FV Tkachov, Phys Lett B 85 Ž ; M Dine, J Sapirstein, Phys Rev Lett 43 Ž ; W Celmaster, RJ Gonsalves, Phys Rev Lett 44 Ž w19x C Caso et al, Review of Particle Physics, Eur Phys J C 3 Ž , 4, 87 w0x P Abreu et al Ž DELPHI-collaboration, Phys Lett B 418 Ž

223 31 December 1998 Physics Letters B Charmonium suppression in heavy ion collisions by prompt gluons Jorg Hufner a,b, Boris Z Kopeliovich b,c a Institut fur Theoretische Physik der UniÕersitat, Philosophenweg 19, 6910 Heidelberg, Germany b Max-Planck Institut fur Kernphysik, Postfach , 6909 Heidelberg, Germany c Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia Received 16 September 1998; revised November 1998 Editor: PV Landshoff Abstract In relativistic heavy ion collisions, also the bremsstrahlung of gluons in the fragmentation regions of the nuclei suppresses the produced charmonium states In the energy range of the SPS, the radiation of semi-hard gluons occurs in the Bethe-Heitler regime and the density of gluons and therefore the suppression goes like Ž AB 1r3, where A and B are the nucleon numbers of the projectile and target nuclei In contrast, the suppression via collisions with nucleons is proportional Ž 1r3 1r3 X to A qb Parameter free perturbative QCD calculations are in a good agreement with the data on JrC and C suppression in heavy ion collisions at SPS CERN At higher energies Ž RHIC, LHC the number of gluons which are able to break-up the charmonium substantially decreases and the additional suppression is expected to vanish q 1998 Elsevier Science BV All rights reserved 1 Challenges in charmonium production off nuclei Charmonium production in heavy ion collisions is considered as one of several sensitive probes for the formation of a quark-gluon plasma wx 1 This idea has led the NA38 and NA50 collaborations at SPS CERN to perform a series of measurements of nuclear suppression for charmonium production in pa and AB collisions We recall some remarkable observations Žsee the collection of data in wx : Ž iifjrc suppression in proton-nucleus collisions is treated as a final state absorption phenomenon one extracts an absorption cross section s Ž JrC N in f 6 7 mb from the data At the same time, the vector dominance model applied to JrC photoproduction data on protons results in a much smaller value s Ž JrC N tot f11 mb, which increases to a value of 3 4 mb for a coupled-channel analysis of the photopro- duction data wx 3 However, a discrepancy of about 3 mb still remains to be explained A part of this extra suppression of JrC may arise due to a stronger absorption of xc states which contribute to the production of JrC about 30% via decay 1 1 We are thankful to Mark Strikman for this comment r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

224 4 J Hufner, BZ KopelioÕichrPhysics Letters B 445 ( 1998) 3 31 Ž ii The C X Ž S state is expected to have a radius about twice as large as the JrC and, therefore, to have a few times stronger final state interaction However, in proton nucleus collisions both JrC and C X are suppressed by nearly the same amount This is an effect of the finite formation time of the JrC and C X and can be quantitatively explained in the coupled channel approach wx 4 X Ž iii However, in S U and Pb Pb collisions C is much stronger suppressed than JrC This is partially an effect of the formation time together with the inverse kinematics for the projectile nucleus wx 4 Nevertheless nearly a half of the observed suppression still needs to be explained Ž iv Final state absorption on nucleons well describes the data on JrC suppression in p A, O Cu, O U and S U collisions with light projectiles However, it fails in the case of Pb Pb collision where data show a much stronger suppression of JrC This observation presented at QM 96 wx 5 and published in wx 6 has led to intensive theoretical work Žfor a recent review we refer to wx 7 At present there are two main schools of thought: A number of authors consider the strong suppression in the Pb Pb data as a proof for the formation of a QGP, in the spirit of the original idea by Matsui and Satz wx 1 Other authors explain the strong suppression of JrC and C X in Pb Pb by hadronic comovers, ie via destruction of the charmonium by the hadrons which are produced in the final state of a nucleus-nucleus collision with the same velocity as the charmonium Both approaches, via QGP or comovers, introduce a number of free adjustable parameters, and it is only then that they can account for the data For this reason it is not possible yet to decide about the mechanism In this paper we are able to explain the same set of data by invoking an addition to the normal suppression via collisions with nucleons the charmonium suppression due to gluon bremsstrahlung produced in multiple nucleon interaction Since we deal with semi-hard processes, pqcd is expected to work reasonably well and no adjustable parameters have to be introduced Charmonium break-up via interaction with nucleons and gluons Final state interaction of a charmonium in nuclear medium which leads to the nuclear suppression is usually described in terms of collisions of the charmonium with undisturbed bound nucleons This is might not be fully correct A nucleus-nucleus collision is illustrated in Fig 1 on a two dimensional Ž time longitudinal coordinate plot which corresponds to the NN cm frame One can see that some of the nucleon trajectories cross the charmonium after they have interacted with nucleons from another nucleus A natural question arises, whether the charmonium interacts with such debris of a nucleon in the same way as with an intact one? Fig 1 A two-dimensional time longitudinal coordinate plot for charmonium production in a collision of nuclei A and B in the cm of the colliding nucleons The solid and dashed lines show the nucleon and gluon trajectories, respectively

225 J Hufner, BZ KopelioÕichrPhysics Letters B 445 ( 1998) To answer this question note, that a small-size cc pair interacts mostly with only one of the valence quarks in a large-size proton This is because of color screening which cuts off the gluons propagating far away from the charmonium Ž the Van der Waals forces are supposed to be cut off by confinement Thus, the charmonium acts like a counter of the number of constituents in the nucleon In the constituent quark model we expect s Ž CN f3 s Ž Cq Ž 1 in in Interacting with the debris of a wounded nucleon Ž N ) the charmonium can find more constituents in it than in the original nucleon, particularly gluons shown by dashed lines in Fig 1, which had a sufficiently short radiation time In this case the cc break-up cross section increases, ) s Ž CN ss Ž CN qs Ž Cg ² n :, in in in g where ² n : g is the mean number of gluons radiated in a NN interaction preceding the collision with the charmonium Interaction of the cc pair with wounded and intact nucleons leads to the following suppression of charmonium in AB collision, 1 S s d b d s dz r Ž s, z dz r Ž bys, z ` ` AB C H H H A A A H B B B AB y` y` H ` A B = exp ys Ž CN dz r Ž s, z ys Ž CN dz r Ž bys, z eff A eff B za y` A = exp ys Ž Cg ² n : s Ž NN dz r Ž s, z dz r Ž bys, z eff g in A B za y` H B = exp ys Ž Cg ² n : s Ž NN dz r Ž s, z dz r Ž bys, z H eff g in A B y` y` ` ` H ² n : g ` zb A B = exp s Ž NN ² s Ž Cg qs Ž Cg dz r Ž s, z dz r Ž bys, z Ž 3 in eff eff A B za y` X A, Here C denotes either JrC or C Effective absorption cross sections of the cc pair s B Ž CN eff and A, s B Ž Cg may be different for nuclei A and B Ž for nonzero x eff F and depend on where and when the JrC or X C is produced Ž see below In Eq Ž 3 the coordinates of the production point for the cc pair are assumed to be Ž s, z and Ž bys, z A B in the nuclei A and B, respectively, and b is the impact parameter for the AB collision; r Ž r A, B is the density of the nuclei A or B The first exponential in Ž 3 accounts for a suppression due to interaction of the charmonium with the nucleons, both in nuclei A and B The second and the third exponential factors include suppression of the charmonium due to interaction with the gluons radiated in the fragmentation regions of the nuclei A and B, respectively The last exponential eliminates a double counting in number of interacting nucleons Žassuming for this purpose that the charmonium is produced with x s 0 F Gluon bremsstrahlung will also affect nuclear suppression in proton-nucleus collision Reducing Fig 1 to the case of pa collision one can see that some of gluons radiated by the projectile proton cross the trajectory of the charmonium The expression for nuclear suppression is simpler than Ž 3, 1 S s d b dz r Ž b, z exp ys Ž CN dz r Ž b, z H H H ` ` pa A C A A A eff A A y` z A H ` A = exp ys Ž Cg ² n : dz r Ž b, z, Ž 4 eff g A z A z B H ` H H z B H

226 6 J Hufner, BZ KopelioÕichrPhysics Letters B 445 ( 1998) 3 31 Here we neglect the possible effect of dilution of the gluon density due to transverse motion of the gluons, since for the time interval less than one Fermi most of them stay within the nucleon size in transverse plane In order to evaluate expressions Ž 3, Ž 4, we have to discuss the input parameters, the mean number of produced gluons ² n :, the charmonium break-up cross sections s Ž CN and s Ž Cg g in in which are effective cross sections because of the formation time effects All the quantities can be estimated fairly reliably We concentrate on the kinematics of SPS which corresponds to about 10 GeV per nucleon in cm frame All the longitudinal distances in the nuclei are contracted by a factor of gs10 ² : 3 Mean number n of radiated gluons g A quark in a projectile nucleon radiates only if interacts with a target nucleon wx 8 The mean free path for a quark in nuclear medium is three times longer than for a nucleon, lq f6 fm One has a maximal number of produced gluons, namely, the transversal density of gluons proportional to Ž AB 1r3, if the gluons are radiated independently in each NN interaction in Fig 1 This is possible only if the radiation happens in the Bethe-Heitler regime, ie if the formation length l f g of the radiation in the cm frame does not exceed the mean free path of a quark Dzslqrgf06 fm: Eq a Ž 1ya g l f s FDz, Ž 5 a m qk q where Eq and mq are the initial energy and the mass of the radiating quark; a and k are the fraction of the quark light-cone momentum and the transverse momentum carried by the gluon When the condition Ž 5 is violated, the Landau-Pomeranchuk effect suppresses gluon radiation, besides, most of gluons are formed too late to cross the charmonium trajectory in Fig 1 Thus, the mean number of gluons radiated in a NN collision is given by the integral over the gluon radiation spectrum weighted by a step-function which requires the radiation to be in the Bethe-Heitler regime, ² n : g s dk da QŽ Dzyl Ž 6 s NN da dk 3 ` 1 ds qn gx g H H f in kmin amin ( Ž min min min q min min q Here a s v q v yk r E if k-v, otherwise a skr E The choice of the soft limit k has practically no influence on ² n : because the step function in Ž 6 min g cuts off the gluons with small transverse momenta Those which contribute have k) GeV These are semi-hard gluons which do not overlap with the soft gluonic field in a color-flux tube Ž string The cross section of gluon radiation as function of a and k integrated over the final transverse momentum of the quark is calculated in wx 9, s mq a k q 1q Ž 1ya Ž k qa mq 4 da dk k qa mq ds qn gx 3 a k C 9 1ya s aq Ž 7 p 4 a Here a Žk s is the QCD running coupling; C is the factor for the dipole approximation for the cross section of a qq qq pair with a nucleon, s r fcr, where r is the qq transverse separation w10,9 x T T T PQCD predicts Cf3 Note that Ž 7 is supposed to work well even for radiation of gluons with rather small k wx 9 We are left with only one parameter v in Ž 6 which brings the main uncertainty in the value of ² n : min g In accordance with the consideration in next section we try two values vmin s05 and 1 GeV, which result in ² n : g s 069 and 05, respectively Note that the radiation spectrum steeply grows with energy wx 9 since we select gluons with relatively large k According to the analysis w11,9x of HERA data for the proton structure function we expect energy T

227 J Hufner, BZ KopelioÕichrPhysics Letters B 445 ( 1998) dependence dn grda dk A srs 0 0 However, the restriction for the formation length imposed by the step function in Ž 6 suppresses ² n : g much more at high energy because of Lorentz contraction of the nuclei We compare ² n : g for AB collision calculated for vmins 05 GeV at SPS, RHIC and LHC energies including also the energy growth of the gluon spectrum y1 ' 69=10 Ž SPS, s s0 GeV ~ y3 69=10 Ž RHIC, ' s s00 GeV y3 1=10 Ž LHC, ' s s100 GeV ² n : s 8 g In the rest frame of either of the colliding nuclei the main fraction of gluons are radiated at high energies later than the charmonium is produced ( ) 4 The inelastic gluon charmonium cross section s C g in The choice of vmin deserves a special discussion since it is related to the problem of the energy dependence of JrCy g break-up cross section s Ž JrC in At high energies it is dominated by gluonic exchange in t-channel and slightly grows with energy wx 3 According to the previous discussion and Eq Ž 1 the inelastic cross section with a quark is known The one with a gluon differs by a color factor 9r4 Therefore, we expect 3 sinž JrC g f sinž JrC N Ž 9 4 In the energy range of interest a coupled-channel analysis of JrC photoproduction data gives s Ž JrC N in f4 mb Correspondingly, s Ž JrC g f3 mb For c X Ž S in we expect a larger cross section If the cross section is proportional to ² r : of the charmonium state, one has a factor of 7r3 in an oscillator model, or a factor of 4 for ² r : calculated from realistic wave functions w1 x X One could expect a vanishing s Ž CN in down to the threshold energy which is about 06 GeV lower for C than for JrC However, since radiation time is of the order of nuclear radius, the gluons cross the charmonium trajectory in Fig 1 at about the same time as the wounded nucleons, ie, when the charmonium is not yet formed One cannot require energy conservation to hold with accuracy better than DEf1rDt where Dt is the time interval between the production moment of cc Žthe productionrcoherence time is quite short at the SPS y1 energy and its crossing with a gluon The mean time of crossing in Fig 1 is Dtf 1r3 R Argf1 GeV Therefore, one can disregard energy conservation, binding energy and the difference in threshold energy between JrC and C X with accuracy DE;1 GeV For the same reason, the energy threshold has no influence on s Ž C g in There is also another mechanism of charmonium break up by gluons via direct absorption of the gluon w13x which has a pick about s Ž C g in f5 mb near the threshold with a width of the order of binding energy and falls steeply at higher energies In this case one should replace the binding energy by the DE To simplify calculations we fix s Ž JrC g in f3mbat v)v min We try vmin s05 and 1 GeV in accordance with the scale imposed by the estimated uncertainty in energy 5 Effective absorption cross sections The produced cc colorless pair propagates through the nucleus with a smaller size and a smaller absorption than that for either of JrC and C X, unless time exceeds the formation length l f C for the charmonium wave function w14,4 x, E C C l f s Ž 10 M XyM C JrC

228 8 J Hufner, BZ KopelioÕichrPhysics Letters B 445 ( 1998) 3 31 Here EC is the charmonium energy in the rest frame of the nucleus The transition between short and long times is controlled by the nuclear formfactor wx 4, which for the charmonium production point with impact parameter s reads, 1 ` iq L z FAŽ q L,s s H dz e raž s, z, Ž 11 TŽ s y` C ` where the longitudinal momentum transfer is related to 10, q s 1rl, and T b s H dz r Ž b, z L f y` A is the nuclear thickness function The break-up of the colorless cc pair during its evolution is described by the effective cross section wx 4, A A seff JrC N ssin JrC N 1qe RFA q L,s 1 e A X X A seff Ž C N ssinž C N 1q FA Ž q L,s Ž 13 rr The parameters here are defined in wx 4 We use the values given by the harmonic oscillator model, X X X rs² C < sˆ < C : r² JrC < sˆ < JrC : s7r3, es² C < sˆ < JrC : r² JrC < sˆ < JrC : sy' r3 The relative production rate of C X to JrC in a NN collision is known from experiment, Rf04 We expect different effective cross sections for absorption in the nuclei A and B, since the charmonium with ² : A x F s 015 NA38, NA50 propagates through these nuclei with different energies in their rest frames, ie ql and ql B are different In an AB collision with Elabs158 GeVrAand x Fs015 one has l f A s fm and B l s4 fm in the rest system of the projectile Ž A, target Ž B F, respectively The values for the effective cross Ž A B A X sections in a Pb Pb collision are Eqs 1 and 13 s JrC N s39 mb, s JrC N s33 mb, s ŽC N eff eff eff B X X s86 mb and s ŽC N s48 mb Žfor s Ž JrC N s4 mb and s ŽC N s93 mb eff in in Due to the uncertainty for the radiation time one can assume that the gluons radiated by the wounded nucleons cross the C trajectory at about the same points as the nucleons Therefore, we can use the same form of the effective cross sections Ž 1, Ž 13 for interaction with gluons 6 Results of calculations y3 To simplify the calculations we use a constant nuclear density r r s r0 Q R A y r with r0 s 016 fm, except the nuclear formfactor where a realistic density is used We calculate Ž 3 and Ž 4 using the values of the parameters fixed in previous sections We use two values of ² n : g s 05 and 05 to characterize of the uncertainty in our calculations The results are plotted in Fig a for JrC and Fig b for C X as function of A=B together with available experimental data for pa and AB collisions We reproduce the data for JrC suppression quite well, including the S U and Pb Pb points We obtained a reasonable agreement with the data for C X suppression as well, although the curves seem to be a bit too high In this respect we should note that the two coupled channel model we use with oscillator potential might be a poor approximation in the case of C X Indeed, as was mentioned, the ratio of mean square radiuses of C X to JrC which is 7r3 in the oscillator model, is predicted to be nearly 4 in a more realistic approach As an example we tried another value rs3 in Ž 1, Ž 13 The result is shown by the dashed curve in Fig b which is in a better agreement with the data Our results for JrC suppression are very stable against this modification Note that our curves are not straight lines even for pa collisions This is mostly due to the formation time effects, ie is a result of A-dependence of the effective cross sections Ž 1, Ž 13 Furthermore, we are able to account for the JrC suppression in pa collisions, although we use s Ž JrC N in s 4 mb The additional suppression Ž even in pa comes from the gluons with absorption cross section of 3 mb At larger values A = B for nuclear collisions our curves deviate even more from straight lines Ž in agreement with the data! due to the

229 J Hufner, BZ KopelioÕichrPhysics Letters B 445 ( 1998) Fig Ž a Nuclear suppression of JrC production in pa and AB collisions as function of the product A=B The two curves correspond to ² n : s05 and 05 The lines connect the point calculated for A=B corresponding to each experimental point Ž b The same as on Ž g a, but for C X production rescaled by factor 10 y The circles and squares correspond to pa and AB data, respectively The data points are from w,5,6 x nonlinear dependence on nuclear radius, as is brought into Ž 6 by the gluon radiation, whose density is proportional to Ž AB 1r3 Formulas Ž 3 and Ž 4 can be also used for calculation of nuclear suppression of charmonia as a function of impact parameter or of the mean length L of the total path of C in nuclear matter Our results for S U and P-b-Pb collisions are plotted in Fig 3 by dotted and solid curves respectively We use ² n : g s05 at 00 GeV and ² n : s057 at 158 GeV in accordance with Eq Ž 6 The overall normalization is free g Fig 3 Nuclear suppression of JrC relative to Drell-Yan lepton pairs as function of the mean length of total path of the charmonium in the colliding nuclei The dashed Ž round points and solid Ž triangles curves correspond to S U and Pb Pb collisions, respectively The data points are from wx 6

230 30 J Hufner, BZ KopelioÕichrPhysics Letters B 445 ( 1998) 3 31 Following recommendation of the referee for our paper we compare the calculations with the published data wx 6 depicted in Fig 3 by round and triangle points for S U and Pb Pb collisions respectively However, the assignment for L corresponding to the measured transverse energy is not trustable It is based on an oversimplified model, particularly neglecting the Landau-Pomeranchuk suppression Ž at low k and enhancement Ž at high k of the gluon radiation spectrum wx 9 The cascading of gluons radiated in the Bethe-Heitler regime which increases the transverse energy, is also neglected Besides, the normalization on the cross section of the Drell-Yan reaction brings an additional uncertainty Ž especially at high energies We plan to do our own calculations for the transverse energy produced in heavy ion collisions which would allow an unbiased comparison with data 7 Conclusions Charmonium suppression in pa and AB collisions at SPS energies can be calculated as arising from two sources: Ž i Collisions with nucleons, the cross section for which is the one deduced from photoproduction data modified because of the formation time effects The suppression from this source essentially scales like A 1r3 qb 1r3 Ž ii Collisions of charmonia with gluons which have been produced in multiple NN collisions In the Bethe-Heitler regime good at SPS energies about ² n : g s gluons contribute per NN collision This additional effect scales with Ž AB 1r3 and accounts for the anomalous JrC and C X suppression in nucleus-nucleus collisions We conclude that the results of the NA38r50 experiments do not signal about QGP formation Nonetheless, the suggested mechanism leaves room for other contributions to the charmonium suppression within the uncertainty of calculations Particularly, as was mentioned above, a wounded nucleon is not a colorless system of partons any more Therefore, color screening does not cut off Žonly the charmonium formfactor does soft gluon exchanges in contrast to C N interaction However this correction to Ž 1, which we Ž 1r3 1r3 do not expect to be large, scales with A qb and leads to a renormalization of s Ž CN in According to Ž 8 the suppression of charmonium by gluon radiation vanishes at high energies due to the formation time effect We expect practically no additional suppression by prompt gluons at RHIC, while the suppression via interaction with comovers Ž either QGP or hadrons increases with energy Therefore, one has less freedom in interpretation of data on nuclear suppression of charmonium at RHIC The main parameter which controls the charmonium suppression via interaction with prompt gluons in Ž 3 is the product ² n : s Ž CN g eff We estimated this factor and allowed it to vary within a factor of We think that this uncertainty covers other possible corrections, for instance, the contribution of xc decays to the production of JrC Acknowledgements We are grateful for useful discussions to Sverker Fredriksson, Yuri Kovchegov and Hans Pirner This work was completed when BZK was visiting at the University and INFN Trieste He thanks Daniele Treleani for hospitality and very helpful discussions The work is partially supported by a grant 06 HD74 from the BMBW

231 J Hufner, BZ KopelioÕichrPhysics Letters B 445 ( 1998) References wx 1 T Matsui, H Satz, Phys Lett B 178 Ž wx C Lourenco, in: Proc of the Quark Matter Conf 1996, Nucl Phys A 610 Ž c wx 3 J Hufner, BZ Kopeliovich, Phys Lett B 46 Ž wx 4 J Hufner, BZ Kopeliovich, Phys Rev Lett 76 Ž wx 5 M Gonin, in proc of the Quark Matter Conf 1996, Nucl Phys A 610 Ž c wx 6 MC Abreu et al, Phys Lett B 410 Ž wx 7 D Kharzeev, invited talk at the Quark Matter Conf 1997, nucl-thr wx 8 JF Gunion, G Bertsch, Phys Rev D 5 Ž wx 9 BZ Kopeliovich, A Schafer, AV Tarasov, hep-phr w10x AB Zamolodchikov, BZ Kopeliovich, LI Lapidus, JETP Lett 33 Ž w11x BZ Kopeliovich, B Povh, hep-phr w1x W Buchmuller, S-HH Tye, Phys Rev D 4 Ž w13x EV Shuryak, Sov J Nucl Phys 8 Ž w14x BZ Kopeliovich, BG Zakharov, Phys Rev D 44 Ž

232 31 December 1998 Physics Letters B Scheme dependence in polarized deep inelastic scattering Elliot Leader a,1, Aleksander V Sidorov b,, Dimiter B Stamenov c,3 a Department of Physics, Birkbeck College, UniÕersity of London, Malet Street, London WC1E 7HX, UK b BogoliuboÕ Theoretical Laboratory, Joint Institute for Nuclear Research, Dubna, Russia c Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, BlÕd Tsarigradsko Chaussee 7, Sofia 1784, Bulgaria Received 1 August 1998 Editor: R Gatto Abstract We study the effect of scheme dependence upon the NLO QCD analysis of the world data on polarized DIS The reliability of an analysis at NLO is demonstrated by the consistency of our polarized densities with the NLO transformation rules relating them to each other We stress the importance of the chiral JET scheme in which all the hard effects are consistently absorbed in the Wilson coefficient functions q 1998 Elsevier Science BV All rights reserved PACS: 1360Hb; 1388qe; 140Dh; 138-t There has been a major effort in the past decade to obtain reliable information about the polarized parton densities in the nucleon, and especially to try to determine the degree of polarization of the gluon Aside from its fundamental interest and its potential as a testing ground for QCD, such information is vital for the planning of experiments at the RHIC collider, due to come into operation in 1999 A great improvement in the quality of the data on inclusive deep inelastic scattering of leptons on nuw1 3x and, con- cleons has been achieved recently comitantly, several detailed NLO QCD analyses of the data have been carried out w4 13 x, of which the most comprehensive are in Refs w7,13 x It is well known that at NLO and beyond, parton densities become dependent on the renormalization 4 or factorization scheme employed It is perhaps less well known that there are significant differences in the polarized case, as a consequence of the axial anomaly and of the ambiguity in the handling of g 5 in n dimensions In this paper we study the effect of carrying out the data analysis in different schemes, in the MS, AB 1 eleader@physicsbbkacuk sidorov@thsun1jinrru 3 stamenov@inrnebasbg 4 Of course, physical quantities such as the virtual photonnucleon asymmetry A Ž x,q 1 and the polarized structure function g Ž x,q 1 are independent of choice of the factorization convention r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

233 ( ) E Leader et alrphysics Letters B and what is called the JET scheme, whose importance we wish to stress In the latter all hard effects are consistently absorbed into the Wilson coefficient functions The parton densities in each scheme are, by definition, related to each other by certain NLO transformation rules On the other hand, the Q evolution in each scheme is controlled by the splitting functions Žor the anomalous dimensions in Mellin n-moment space relevant to that scheme Thus, in principle, for any two schemes labelled 1 and and for any generic polarized density D f one has symbolically: NLO data analysis in Scheme 1 Df 1, Ž 1 NLO data analysis in Scheme Df, NLO transformation rules on D f D f Ž 3 1 and it is a major test of the stability of the analysis and of the consistency of the theory that the results for D f in Ž and Ž 3 should coincide In fact, built into the theoretical structure is the feature that the two results actually should differ by terms of NNLO order Hence the degree to which the results agree is a measure of the reliability of carrying out an analysis at NLO Moreover, in the set of schemes we study, the non- singlet and gluon densities are the same in all the schemes That this feature should emerge from the data analysis provides a further test of the stability and reliability of the analysis The importance of an analysis of this kind was first stressed in Ref wx 6 In the unpolarized case the most commonly used schemes are the MS, MS and DIS and parton densities in different schemes differ from each other by Ž terms of order as Q, which goes to zero as Q increases There are two significant differences in the polarized case: i The singlet densities DSŽx,Q, in two different schemes, will differ by terms of order asž Q DGŽ x,q, Ž 4 which appear to be of order a s But it is known w14,15x that, as a consequence of the axial anomaly, the first moment of the polarized gluon density DGŽ x,q H 1 y1 s 0 dx DG x,q A a Q, Ž 5 and thus grows in such a way with Q as to compensate for the factor a ŽQ in Ž 4 s Thus the difference between DS in different schemes is only apparently of order a ŽQ s, and could be quite large ii Because of ambiguities in handling the renormalization of operators involving g5 in n dimensions, the specification MS does not define a unique scheme Really there is a family of MS schemes which, strictly, should carry a sub-label indicating how g5 is handled What is now conventionally called MS is in fact the scheme due to Vogelsang and Mertig and van Neervenen w16 x, in which the first moment of the non-singlet densities is conserved, ie is independent of Q, corresponding to the conservation of the non-singlet axial-vector Cabibbo currents Although mathematically correct it is a peculiarity of this factorization scheme that certain soft contributions are included in the Wilson coefficient functions, rather than being absorbed completely into the parton densities As a consequence, the first moment of DS is not conserved so that it is difficult to know how to compare the DIS results on DS with the results from constituent quark models at low Q To avoid these idiosyncrasies Ball, Forte and Ridolfi wx 6 introduced what they called the AB scheme, which involves a minimal modification of the MS scheme, and for which the transformation equations are: AB sdsž x,q MS DS x,q 1 as Q dy qnf H DGŽ y,q MS, p y MS DG x,q AB sdg x,q 6 or, in Mellin n-moment space, AB sdsž n,q MS DS n,q as Q qn f DGŽ n,q MS, p n MS x DG n,q AB s DG n,q 7

234 34 ( ) E Leader et alrphysics Letters B That DS n s 1 AB is independent of Q to all orders follows from the Adler-Bardeen theorem w17 x In Ž 6 and Ž 7 Nf denotes the number of flavours The singlet part of the first moment of the structure function g 1 H Ž s 1 Ž s G Q ' dxg x,q Ž 8 then depends on DS and DG only in the combination a 0 Ž Q sdsž 1,Q MS as Q sdsž 1 AB ynf DGŽ 1,Q Ž 9 p and the unexpectedly small value for the axial charge a found by the EMC w18 x 0, which triggered the spin crisis in the parton model w19 x, can be nicely explained as due to a cancellation between a reasonably sized DSŽ 1 and the gluon contribution Of importance for such an explanation are both the positive sign and the large value Žof order OŽ 1 for the first moment of the polarized gluon density Ž DG 1,Q at small Q ; 1 10 GeV Note that what follows from QCD is that < DGŽ1,Q < grows Ž with Q see Eq 5 but its value at small Q is unknown in the theory at present and has to be determined from experiment Although the AB scheme corrects the most glaring weakness of the MS scheme, it does not consistently put all hard effects into the coefficient funcw0x one can define a family tions As pointed out in of schemes labelled by a parameter a: DS DS ž / s DG a ž DG/ MS as 0 zž a qg DS q mž / Ž 10 p 0 0 MS where ž / DG zqgž x;a snf Ž xy1ž ay1 qž 1yx, Ž 11 in all of which Ž 9 holds, but which differ in their expression for the higher moments ŽThe AB scheme corresponds to taking as Amongst these we believe there are compelling reasons to choose what we shall call the JET scheme Ž as1, ie z JET s N Ž 1yx Ž 1 qg f This is the scheme originally suggested by Carw15x and also advocated by litz, Collins and Mueller Anselmino, Efremov, Leader and Teryaev in Refs w1, x 5 In it all hard effects are absorbed into the coefficient functions In this scheme the gluon coefficient function is exactly the one that would appear in the cross section for pp jetž k qjetž yk qx, Ž 13 T T ie, the production of two jets with large transverse momentum kt and yk T, respectively More recently Muller and Teryaev w3x have advanced rigorous and compelling arguments, based upon a generalization of the axial anomaly to bilocal operators, that removal of all anomaly effects from the quark densities leads to the JET scheme Also a different argument by Cheng w4x leads to the same conclusion ŽCheng calls the JET scheme a chirally invariant Ž CI scheme The transformation from the MS scheme of Mertig, van Neerven and Vogelsang to the JET scheme is given in moment space by sdsž n,q MS DS n,q JET asž Q q N f DGŽ n,q MS, p nž nq1 MS DG n,q s DG n,q 14 JET Note that Ž 14 implies that the strange sea Ds is different in the two schemes Of course, Ž 7 and Ž 14 become the same for ns1 Ž1 The NLO Wilson coefficient functions DC Ž x i Ž1 and polarized splitting functions DP Ž x Ž ij or the Ž1 corresponding anomalous dimensions Dg Ž n ij for the MS and AB schemes can be found in Refs w16x and wx 6, respectively The NLO coefficient functions 5 1y x r x X There is misprint in Eq Ž 86 of w x The term ln x r x X 1y x r x should be wlnž X X y1 x x r x

235 ( ) E Leader et alrphysics Letters B and anomalous dimensions in the JET scheme are w x 6 related to those of the MS scheme by 3 : Ž1 Ž1 DC q Ž n JET sdc q Ž n MS, N Ž1 Ž1 f DCG Ž n JETsDCG Ž n MSy nž nq1, Ž 15 4N Ž1 Ž1 f Ž0 Dgqq Ž n JETs Dgqq Ž n MSq DgGq Ž n, nž nq1 4N Ž1 Ž1 f Dg qg Ž n JET sdg qg Ž n MS q nž nq1 = MS Ž0 Ž0 DgGG n ydgqq n qb 0, Ž1 Ž1 Dg Gq n JET sdg Gq n, 4N Ž1 Ž1 f Ž0 GG JET GG MS Gq Dg Ž n s Dg Ž n y Dg Ž n nž nq1 Note also that MS Ž1 Ž1 DC NS n JET sdc NS n, Ž1 Ž1 NS JET NS MS Ž 16 Dg Ž n s Dg Ž n Ž 17 In Ž 16 a superscript 0 is used for the corresponding anomalous dimensions in the LO approximation Note that the transformation of the coefficient functions and anomalous dimensions from the MS to the AB scheme is given by Eqs Ž 15 Ž 17, in which the factor rnž nq1 should be replaced by 1rn In each scheme the parton densities at Q0 s1 GeV, as in w13 x, were parametrized in the form: xdu x,q sh A x a u xu x,q, ÕŽ 0 u u ÕŽ 0 ÕŽ 0 d d ÕŽ 0 Ž 0 S S Ž 0 Ž 0 g g Ž 0 xdd x,q sh A x a d xd x,q, xdsea x,q sh A x a S x Sea x,q, xdg x,q sh A x a g xg x,q Ž 18 where on RHS of Ž 18 we have used the recent MRST unpolarized densities w5 x The normalization factors A f are determined in such a way as to ensure that the first moments of the polarized densities are given by h f The first moments of the valence quark densities hu and hd are fixed by the octet nucleon and hy- peron b decay constants w6x g A sfqds1573 " 0008, a s3fyds0579 " 005 Ž 19 8 and in the case of SUŽ 3 flavour symmetry of the sea Ž DusDdsDs at Q 0 h s0918, h sy0339 Ž 0 u d The rest of the parameters in Ž 18 a, a, h, a,h,a 4, Ž 1 u d S S g g have to be determined from the best fit to the A N Ž x,q 1 data To calculate the virtual photon-nucleon asymmetry A N Ž x,q 1 in NLO QCD and then fit to the data we follow the procedure described in detail in our previous papers w9,13 x, where the connection between measured quantities, Wilson coefficients and parton densities is given Here we note only that in all calculations we have used for the values of the QCD parameter L : L Ž n s3 MS MS f s353 MeV and L Ž n s 4 MS f s 300 MeV, which correspond to a Ž M s z s 01175, as obtained by the MRST analysis w5x of the world unpolarized data This is in excellent agreement with the world average a Ž M s z s 0118"0003 w7 x The numerical results of the fits in the JET, AB and MS schemes to the present experimental data on N Ž w x 7 A1 x,q 1 3,18,8 30 are listed in Table 1 The data used Ž 118 experimental points cover the following kinematic region: 0004-x-075, 1-Q -7 GeV In this paper we present the results of the fit to the N A data averaged over Q at each x Ž 1 In our previous work w13x we also analyzed data in Žx,Q bins Our conclusion in the present paper also hold for this type of fit The total Ž statistical and systematic errors are taken into account The results presented in Table 1 correspond to an SUŽ 3 symmetric sea Note that in this case the first moment of the strange Ž sea quarks h 'Ds 1,Q sh r6 As was shown in s 0 S 6 w x In Ref 3 these transformations are presented in Bjorken x space to all orders in a s 7 After the completion of this work new data on A p and g p have been reported by the HERMES Collaboration w31 x 1 1

236 36 ( ) E Leader et alrphysics Letters B Table 1 Results of the NLO QCD fits in the JET, AB and MS schemes to N Ž the world A data Q s1 GeV 1 0 The errors shown are total Ž statistical and systematic a s06 Ž fixed Scheme JET AB MS DOF x x rdof g what follows from the theory, namely, that they should be the same in the factorization schemes under consideration The results on the polarized gluon densities determined by the fit in the JET, AB and MS schemes are a 067" " "008 u a 014" " "0113 d as 1469" " "03 h y007"0004 y00"0005 y0049"0005 s h 050 " " "03 g DSŽ " " "0041 Ž a 1GeV 030 " " "004 0 our previous work w13 x, the flavour decomposition of the sea does not affect the quality of the fit and the results on the polarized parton densities DS, Ds and DG As in the case of the MS scheme w13 x, the value of ag is not well determined by the fits in the JET and AB schemes to the existing data, ie x rdof practically does not change when ag varies in the range:0fag F1 In Table 1 we present the results of the fits corresponding to a s06 g It is seen from Table 1 that the values of x rdof coincide almost exactly in the different factorization schemes, which is a good indication of the stability of the analysis The NLO QCD predictions are in a very good agreement with the presently available data on A1 N, as is illustrated in the JET scheme fit in Fig 1 We would like to draw special attention to the excellent fit to the very accurate E154 neutron data Ž x s11 for 11 experimental data points The extracted valence and gluon polarized densities at Q0 s1 GeV for the different schemes are shown in Fig Žsolid, dotted and dashed curves correspond to JET, AB and MS scheme, respectively Note that the valence densities x DuÕ and xddõ in the JET and AB schemes are almost identical so the dotted curves corresponding to x DuÕ and xddõ are not shown in Fig The difference between xduõ and xddõ in the MS and JET scheme is negligible The results of the fit for the polarized valence densities are in an excellent agreement with Fig 1 Comparison of our NLO results in the JET scheme for N Ž A1 x,q with the experimental data at the measured x and Q values Errors bars represent the total error

237 ( ) E Leader et alrphysics Letters B Fig Next-to-leading order input polarized valence and gluon distributions at Q s1 GeV in different factorization schemes Solid, dotted and dashed curves correspond to the JET, AB and MS scheme, respectively coincides within errors with the values of DSŽ 1 JET and DSŽ 1 AB DSŽ 1 JET s0416"0036, DSŽ 1 AB s0444"0040 Ž 4 determined directly by the fit to the data in the JET and AB schemes as presented in Table 1 The singlet densities DSŽ x,q Ž solid curve 0 JET and Ž DS x,q Ž dashed curve 0 MS JET are shown in Fig 3b Finally, we would like to draw attention to the excellent agreement between the values of the axial charge a Ž1 GeV 0 determined in the different schemes Ž see Table 1, which illustrates impressively how our analysis respects the scheme-independence of physical quantities In conclusion, we have performed a next-to-leading order QCD analysis of the world data on inclusive polarized deep inelastic lepton-nucleon scattering in the JET, AB and MS schemes The QCD also consistent The values of their first moments h g coincide within errors Ž see Table 1 However, as a consequence of the uncertainty in determining the gluon density from the present data, the central values of hg and therefore, the gluon densities them- selves, differ somewhat in the various schemes In Table 1 we also present our results in the different schemes for the first moments DSŽ 1 of the polarized singlet quark density as well as for the axial charge a 0 The obtained singlet densities DSŽ x,q 0 are shown in Fig 3a It is seen from the table that the first moments DSŽ 1 in the JET and AB schemes are in a very good agreement ŽWe recall that according to the definition of the JET and AB schemes DSŽ 1 should be the same in the both schemes The corresponding densities DSŽx,Q, however, are slightly different Ž see Fig 3a because their higher moments are not equal Our result for DSŽ 1 MS JET,AB using the values of DSŽ 1 and Ž h MS g MS from Table 1 and the NLO transformation rules Žsee Eqs Ž 7 and Ž 14 for n s 1 DSŽ 1 MS JET,ABs0476"0084 Ž 3 Fig 3 Ž a Next-to-leading order input polarized singlet distribu- tions DS x at Q s1 GeV in different factorization schemes Solid, dotted and dashed curves correspond to the JET, AB and MS scheme, respectively Ž b Comparison between the singlet density DSŽ x obtained from the fit Ž solid curve and DSŽ x JET MS JET determined by Eq 14 dashed curve, at Q s1 GeV

238 38 ( ) E Leader et alrphysics Letters B predictions have been confronted with the data on the virtual photon-nucleon asymmetry A N Ž x,q 1, rather than with the polarized structure function g N Ž x,q 1, in order to minimize higher twist effects Using the simple parametrization Ž 18 Žwith only 5 free parameters for the input polarized parton densities it was demonstrated that the polarized DIS data are in an excellent agreement with the pqcd predictions for A N Ž x,q 1 in all the factorization schemes considered Moreover, we have demonstrated the consistency between the results of the analysis in each scheme and the NLO transformation equations relating them Such consistency gives us confidence that the NLO analysis is reliable Acknowledgements We are grateful to OV Teryaev for useful discussions and remarks This research was partly supported by a UK Royal Society Collaborative Grant, by the Russian Fund for Fundamental Research Grant No a and by the Bulgarian Science Foundation under Contract Ph 510 References wx 1 SLACrE154 Collaboration, K Abe et al, Phys Rev Lett 79 Ž wx SMC, D Adeva et al, Phys Lett B 41 Ž wx 3 SLAC E143 Collaboration, K Abe et al, Phys Rev D 58 Ž wx 4 T Gehrmann, WJ Stirling, Phys Rev D 53 Ž wx 5 M Gluck, E Reya, M Stratmann, W Vogelsang, Phys Rev D 53 Ž wx 6 RD Ball, S Forte, G Ridolfi, Phys Lett B 378 Ž wx 7 G Altarelli, RD Ball, S Forte, G Ridolfi, Nucl Phys B 496 Ž ; Acta Phys Polon B 9 Ž wx 8 SLACrE154 Collaboration, K Abe et al, Phys Lett B 405 Ž wx 9 E Leader, AV Sidorov, DB Stamenov, hep-phr Ž to appear in Int J Mod Phys A w10x M Stratmann, in: J Blumlein et al Ž Eds, the Proceedings of nd Topical Workshop on Deep Inelastic Scattering off Polarized Targets: Theory Meets Experiment Ž SPIN 97, Zeuthen, Germany, 1977 Ž DESY, 1977, p 94 w11x A De Roeck et al, Eur Phys J C 6 Ž w1x C Bourrely, F Buccella, O Pisanti, P Santorelli, J Soffer, Prog Theor Phys 99 Ž w13x E Leader, AV Sidorov, DB Stamenov, Phys Rev D 58 Ž w14x AV Efremov, OV Teryaev, Dubna report E-88-87, 1988, in: J Fischer et al Ž Eds, the Proceedings of the Int Hadron Symposium, Bechyne, Czechoslovakia, 1988 ŽCzech Academy of Science, Prague, 1989, p 30 ; G Altarelli, GG Ross, Phys Lett B 1 Ž w15x RD Carlitz, JC Collins, AH Mueller, Phys Lett B 14 Ž w16x R Mertig, WL van Neerven, Z Phys C 70 Ž ; W Vogelsang, Phys Rev D 54 Ž w17x S Adler, W Bardeen, Phys Rev 18 Ž w18x EMC, J Ashman et al, Phys Lett B 06 Ž ; Nucl Phys B 38 Ž w19x E Leader, M Anselmino, Z Phys C 41 Ž w0x EB Zijlstra, WL van Neerven, Nucl Phys B 417 Ž w1x AV Efremov, OV Teryaev, SPIN-89, in: the Proceedings of III International Workshop, p 77 wx M Anselmino, AV Efremov, E Leader, Phys Rep 61 Ž w3x D Muller, OV Teryaev, Phys Rev D 56 Ž w4x Hai-Yang Cheng, Phys Lett B 47 Ž w5x AD Martin, RG Roberts, WJ Stirling, RS Torn, Eur Phys J C 4 Ž w6x Particle Data Group, L Montanet et al, Phys Rev D 50 Ž ; FE Close, RG Roberts, Phys Lett B 313 Ž w7x Partical Data Group, RM Barnet et al, Phys Rev D 54 Ž w8x SLAC E14 Collaboration, PL Anthony et al, Phys Rev D 54 Ž w9x SMC, D Adams et al, Phys Lett B 396 Ž w30x HERMES Collaboration, K Ackerstaff et al, Phys Lett B 404 Ž w31x HERMES Collaboration, A Airapetian et al, hepexr

239 31 December 1998 Physics Letters B Measurement of triple gauge WWg couplings at LEP using photonic events ALEPH Collaboration R Barate a, D Buskulic a, D Decamp a, P Ghez a, C Goy a, S Jezequel a, J-P Lees a, A Lucotte a, F Martin a, E Merle a, M-N Minard a, J-Y Nief a, P Perrodo a, B Pietrzyk a, R Alemany b, MP Casado b, M Chmeissani b, JM Crespo b, M Delfino b, E Fernandez b, M Fernandez-Bosman b, Ll Garrido b,1, E Grauges ` b, A Juste b, M Martinez b, G Merino b, R Miquel b, LlM Mir b, P Morawitz b, A Pacheco b, IC Park b, A Pascual b, I Riu b, F Sanchez b, A Colaleo c, D Creanza c, M de Palma c, G Gelao c, G Iaselli c, G Maggi c, M Maggi c, S Nuzzo c, A Ranieri c, G Raso c, F Ruggieri c, G Selvaggi c, L Silvestris c, P Tempesta c, A Tricomi c,, G Zito c, X Huang d, J Lin d, Q Ouyang d, T Wang d, Y Xie d,rxu d, S Xue d, J Zhang d, L Zhang d, W Zhao d, D Abbaneo e, U Becker e,3, G Boix e,4, M Cattaneo e, F Cerutti e, V Ciulli e, G Dissertori e, H Drevermann e, RW Forty e, M Frank e, F Gianotti e, AW Halley e, JB Hansen e, J Harvey e, P Janot e, B Jost e, I Lehraus e, O Leroy e, C Loomis e, P Maley e, P Mato e, A Minten e, L Moneta e,5, A Moutoussi e,nqi e, F Ranjard e, L Rolandi e, D Rousseau e, D Schlatter e, M Schmitt e,6, O Schneider e, W Tejessy e, F Teubert e, IR Tomalin e, E Tournefier e, M Vreeswijk e, H Wachsmuth e, Z Ajaltouni f, F Badaud f, G Chazelle f, O Deschamps f, S Dessagne f, A Falvard f, C Ferdi f, P Gay f, C Guicheney f, P Henrard f, J Jousset f, B Michel f, S Monteil f, J-C Montret f, D Pallin f, P Perret f, F Podlyski f, JD Hansen g, JR Hansen g, PH Hansen g, BS Nilsson g, B Rensch g, A Waananen g, G Daskalakis h, A Kyriakis h, C Markou h, E Simopoulou h, A Vayaki h, A Blondel i, J-C Brient i, F Machefert i, A Rouge i, M Rumpf i, R Tanaka i, A Valassi i,7, H Videau i, E Focardi j, G Parrini j, K Zachariadou j, R Cavanaugh k, M Corden k, C Georgiopoulos k, T Huehn k, DE Jaffe k, A Antonelli l, G Bencivenni l, G Bologna l,8, F Bossi l, P Campana l, G Capon l, V Chiarella l, P Laurelli l, G Mannocchi l,9, F Murtas l, GP Murtas l, L Passalacqua l, r98r$ - see front matter q 1998 Elsevier Science BV All rights reserved PII: S

240 40 ( ) R Barate et alrphysics Letters B M Pepe-Altarelli l,10, M Chalmers m, L Curtis m, JG Lynch m, P Negus m, V O Shea m, B Raeven m, C Raine m, D Smith m, P Teixeira-Dias m, AS Thompson m, E Thomson m, JJ Ward m, O Buchmuller n, S Dhamotharan n, C Geweniger n, P Hanke n, G Hansper n, V Hepp n, EE Kluge n, A Putzer n, J Sommer n, K Tittel n, S Werner n,3, M Wunsch n, R Beuselinck o, DM Binnie o, W Cameron o, PJ Dornan o,10, M Girone o, S Goodsir o, N Marinelli o, EB Martin o, J Nash o, JK Sedgbeer o, P Spagnolo o, MD Williams o, VM Ghete p, P Girtler p, E Kneringer p, D Kuhn p, G Rudolph p, AP Betteridge q, CK Bowdery q, PG Buck q, P Colrain q, G Crawford q, G Ellis q, AJ Finch q, F Foster q, G Hughes q, RWL Jones q, AN Robertson q, MI Williams q, P van Gemmeren r, I Giehl r, C Hoffmann r, K Jakobs r, K Kleinknecht r, M Krocker r, H-A Nurnberger r, G Quast r, B Renk r, E Rohne r, H-G Sander r, S Schmeling r, C Zeitnitz r, T Ziegler r, JJ Aubert s, C Benchouk s, A Bonissent s, J Carr s,10, P Coyle s, A Ealet s, D Fouchez s, F Motsch s, P Payre s, M Talby s, M Thulasidas s, A Tilquin s, M Aleppo t, M Antonelli t, F Ragusa t, R Berlich u, V Buscher u, H Dietl u, G Ganis u, K Huttmann u, G Lutjens u, C Mannert u, W Manner u, H-G Moser u, S Schael u, R Settles u, H Seywerd u, H Stenzel u, W Wiedenmann u, G Wolf u, P Azzurri v, J Boucrot v, O Callot v, S Chen v, M Davier v, L Duflot v, J-F Grivaz v, Ph Heusse v, A Jacholkowska v, M Kado v, J Lefrançois v, L Serin v, J-J Veillet v, I Videau v,10, J-B de Vivie de Regie v, D Zerwas v, G Bagliesi w,10, S Bettarini w, T Boccali w, C Bozzi w, G Calderini w, R Dell Orso w, I Ferrante w, A Giassi w, A Gregorio w, F Ligabue w, A Lusiani w, PS Marrocchesi w, A Messineo w, F Palla w, G Rizzo w, G Sanguinetti w, A Sciaba ` w, G Sguazzoni w, R Tenchini w, C Vannini w, A Venturi w, PG Verdini w, GA Blair x, JT Chambers x, J Coles x, G Cowan x, MG Green x, T Medcalf x, JA Strong x, JH von Wimmersperg-Toeller x, DR Botterill y, RW Clifft y, TR Edgecock y, PR Norton y, JC Thompson y, AE Wright y, B Bloch-Devaux z, P Colas z, B Fabbro z, G Faıf z, E Lançon z,10, M-C Lemaire z, E Locci z, P Perez z, H Przysiezniak z, J Rander z, J-F Renardy z, A Rosowsky z, A Trabelsi z,11, B Tuchming z, B Vallage z, SN Black aa, JH Dann aa, HY Kim aa, N Konstantinidis aa, AM Litke aa, MA McNeil aa, G Taylor aa, CN Booth ab, S Cartwright ab, F Combley ab, MS Kelly ab, M Lehto ab, LF Thompson ab, K Affholderbach ac, A Bohrer ac, S Brandt ac, J Foss ac, C Grupen ac, G Prange ac, L Smolik ac, F Stephan ac, G Giannini ad, B Gobbo ad, J Putz ae, J Rothberg ae, S Wasserbaech ae, RW Williams ae, SR Armstrong af, E Charles af, P Elmer af, DPS Ferguson af, Y Gao af, S Gonzalez af, TC Greening af, OJ Hayes af,hhu af, S Jin af,

241 i ( ) R Barate et alrphysics Letters B PA McNamara III af, JM Nachtman af,1, J Nielsen af, W Orejudos af, YB Pan af, Y Saadi af, IJ Scott af, J Walsh af, Sau Lan Wu af, X Wu af, G Zobernig af a Laboratoire de Physique des Particules ( LAPP ), INP3-CNRS, F Annecy-le-Vieux Cedex, France b ( ) 13 Institut de Fısica d Altes Energies, UniÕersitat Autonoma ` de Barcelona, Bellaterra Barcelona, Spain c Dipartimento di Fisica, INFN Sezione di Bari, I-7016 Bari, Italy d Institute of High-Energy Physics, Academia Sinica, Beijing, China 14 e European Laboratory for Particle Physics ( CERN ), CH-111 GeneÕa 3, Switzerland f Laboratoire de Physique Corpusculaire, UniÕersite Blaise Pascal, INP3-CNRS, Clermont-Ferrand, F Aubiere, ` France g Niels Bohr Institute, 100 Copenhagen, Denmark 15 h Nuclear Research Center Demokritos ( NRCD ), GR Attiki, Greece Laboratoire de Physique Nucleaire et des Hautes Energies, Ecole Polytechnique, INP3-CNRS, F-9118 Palaiseau Cedex, France j Dipartimento di Fisica, UniÕersita` di Firenze, INFN Sezione di Firenze, I-5015 Firenze, Italy k Supercomputer Computations Research Institute, Florida State UniÕersity, Tallahassee, FL , USA 16,17 l Laboratori Nazionali dell INFN ( LNF-INFN ), I Frascati, Italy m Department of Physics and Astronomy, UniÕersity of Glasgow, Glasgow G1 8QQ, UK 18 n Institut fur Hochenergiephysik, UniÕersitat Heidelberg, D-6910 Heidelberg, Germany 19 o Department of Physics, Imperial College, London SW7 BZ, UK 18 p Institut fur Experimentalphysik, UniÕersitat Innsbruck, A-600 Innsbruck, Austria 0 q Department of Physics, UniÕersity of Lancaster, Lancaster LA1 4YB, UK 18 r Institut fur Physik, UniÕersitat Mainz, D Mainz, Germany 19 s Centre de Physique des Particules, Faculte des Sciences de Luminy, INP3-CNRS, F-1388 Marseille, France t Dipartimento di Fisica, UniÕersita` di Milano e INFN Sezione di Milano, I-0133 Milano, Italy u Max-Planck-Institut fur Physik, Werner-Heisenberg-Institut, D Munchen, Germany 19 v Laboratoire de l Accelerateur Lineaire, UniÕersite de Paris-Sud, INP3-CNRS, F Orsay Cedex, France w Dipartimento di Fisica dell UniÕersita, ` INFN Sezione di Pisa, e Scuola Normale Superiore, I Pisa, Italy x Department of Physics, Royal Holloway & Bedford New College, UniÕersity of London, Surrey TW0 OEX, UK 18 y Particle Physics Dept, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 OQX, UK 18 z CEA, DAPNIArSerÕice de Physique des Particules, CE-Saclay, F Gif-sur-YÕette Cedex, France 1 aa Institute for Particle Physics, UniÕersity of California at Santa Cruz, Santa Cruz, CA 95064, USA ab Department of Physics, UniÕersity of Sheffield, Sheffield S3 7RH, UK 18 ac Fachbereich Physik, UniÕersitat Siegen, D Siegen, Germany 19 ad Dipartimento di Fisica, UniÕersita` di Trieste e INFN Sezione di Trieste, I-3417 Trieste, Italy ae Experimental Elementary Particle Physics, UniÕersity of Washington, Seattle, WA 98195, USA af Department of Physics, UniÕersity of Wisconsin, Madison, WI 53706, USA 3 Received 16 November 1998 Editor: L Montanet Abstract A study of events with photons and missing energy has been performed with the data sample obtained with the ALEPH detector at centre-of-mass energies from 161 to 184 GeV, corresponding to a total integrated luminosity of about 80 pb y1 The measured distributions are in agreement with Standard Model predictions, leading to constraints on WWg gauge coupling parameters Dkg and l g The results from the fit to the cross sections and to the energy and angular distributions q115 q155 of the photons are: Dk s005 stat "05 syst, l sy005 Ž stat "030Ž syst g y110 g y145 q 1998 Elsevier Science BV All rights reserved 1 Permanent address: Universitat de Barcelona, 0808 Barcelona, Spain Also at Dipartimento di Fisica, INFN Sezione di Catania, Catania, Italy 3 Now at SAP AG, D Walldorf, Germany

242 4 ( ) R Barate et alrphysics Letters B Introduction In e q e y collisions at LEP energies, the trilinear WWg and WWZ couplings can be probed with direct W-pair Že q e y W q W y, single W Že q e y Wen production or with photon production Že q e y nng Ž g w1, x In the WW channel a minimal set of five independent parameters is necessary to describe the Z and g couplings to the W, assuming C and CP conservation Usually a model-dependence is introduced to reduce this set to at most three parameters Žeg the model with the parameters a W, a Wf, abf wx 3 Although the photonic channel is less sensitive to the couplings than the W pair and single W channels wx 4, it can resolve sign ambiguities and is therefore complementary Constraints on the WWg vertex have also been obtained at the Tevatw5x within a slightly different theoretical ron framework 4 Supported by the Commission of the European Communities, contract ERBFMBICT Now at University of Geneva, 111 Geneva 4, Switzerland 6 Now at Harvard University, Cambridge, MA 0138, USA 7 Now at LAL, Orsay, France 8 Also Istituto di Fisica Generale, Universita ` di Torino, Torino, Italy 9 Also Istituto di Cosmo-Geofisica del CNR, Torino, Italy 10 Also at CERN, 111 Geneva 3, Switzerland 11 Now at Departement de Physique, Faculte des Sciences de Tunis, 1060 Le Belvedere, ` Tunisia 1 Now at University of California at Los Angeles Ž UCLA, Los Angeles, CA 9004, USA 13 Supported by CICYT, Spain 14 Supported by the National Science Foundation of China 15 Supported by the Danish Natural Science Research Council 16 Supported by the US Department of Energy, contract DE- FG05-9ER Supported by the US Department of Energy, contract DE- FC05-85ER Supported by the UK Particle Physics and Astronomy Research Council 19 Supported by the Bundesministerium fur Bildung, Wissenschaft, Forschung und Technologie, Germany 0 Supported by Fonds zur Forderung der wissenschaftlichen Forschung, Austria 1 Supported by the Direction des Sciences de la Matiere, ` CEA Supported by the US Department of Energy, grant DE-FG03-9ER Supported by the US Department of Energy, grant DE- FG095-ER40896 The purpose of this letter is to set constraints on the WWg trilinear couplings with a study of photonic events, using data collected by ALEPH at centre-of-mass energies ranging from 161 to 184 GeV and corresponding to a total integrated luminosity of about 80 pb y1 In the Standard Model, three processes contribute at tree level to the nng final state corresponding to the five diagrams shown in Fig 1 The WWg vertex is present only in the last diagram, which contributes about 03% to the total q y Standard Model e e nng cross section, but which also leads to characteristic energy and angular distributions of the final state photons A measurement of the total cross section, supplemented by a fit to these distributions, is therefore sensitive to the presence of an anomalous WWg coupling This vertex can be described wx 3 by three C and P conserving parameters, g 1 g, kg and l g, related to the following W boson properties: charge Qw s eg 1 g e 1 magneticdipolemoment m s g qk q l m w e electricquadrupolemoment q sy Ž k y l w g g g w g g m w In the Standard Model, these three parameters are equal to 1, 1 and 0, respectively, and their deviations from these values are parameterized as anomalous couplings D g 1 g, Dkg and l g Here, the electric charge of the W boson is assumed to be equal to that q Fig 1 Feynman diagrams for e e nng Only the process C is sensitive to the WWg couplings y

243 ( ) R Barate et alrphysics Letters B of the electron, thus fixing g 1 g s1, while no further assumptions are made on kg and l g The matrix q y element for the e e nng Ž g final state is a linear function of D kg and l g Its implementation in the KORALZ Monte Carlo program wx 6, including initial state radiation of additional photons, is used throughout this analysis This letter is organized as follows In Section, the aspects of the ALEPH detector relevant to this analysis are described The event selection is presented in Section 3, and the fit of the data to the presence of anomalous WWg couplings is discussed in Section 4 Section 5 gives the fitted values and the resulting constraints on D kg and l g ; the systematic uncertainties are discussed in Section 6 The ALEPH detector The ALEPH detector and its performance are described in detail in Refs w7,8 x Here only a brief description of the properties relevant to the present analysis is given The central part is dedicated to the detection of charged particles From the interaction point outwards, the trajectory of a charged particle is measured by a two-layer silicon strip vertex detector, a cylindrical drift chamber and a large time projection chamber Ž TPC The three tracking detectors are immersed in a 15 T axial field provided by a superconducting solenoidal coil Photons are identified in the electromagnetic calorimeter Ž ECAL, situated between the TPC and the coil It is a lead proportional wire sampling calorimeter segmented in 098 = 098 towers read out in three sections in depth It has a total thickness of radiation lengths and yields an energy resolution of deres018r' E q0009ž E in Gev Two independent readouts of the energy are implemented respectively on the cathode pads and on the anode wires of the ECAL At low polar angles, the ECAL is supplemented by two calorimeters, LCAL and SiCAL, principally used to measure the integrated luminosity collected by the experiment, but used also here for vetoeing purposes The iron return yoke is equipped with 3 layers of streamer tubes and forms the hadron calorimeter Ž HCAL, seven interaction lengths thick; it provides a relative energy resolution of charged and neutral hadrons of 085r' E Muons are identified using hits in the HCAL and the muon chambers; the latter are composed of two layers of streamer tubes outside the HCAL The information from the tracking detectors and the calorimeters are combined in an energy flow algorithm wx 8 For each event, the algorithm provides a set of charged and neutral reconstructed particles, called energy flow objects, used in the analysis 3 Event samples and selection The data were collected with the ALEPH detector at LEP at several centre-of-mass energies between 161 and 17 GeV in 1996, and between 181 and 184 GeV in 1997 The corresponding integrated luminosities are given in Table 1 31 Selections and cuts Photon candidates are defined as described in Ref wx 8 Only events with no reconstructed charged particle tracks and at least one photon with an energy E ) 01' g s are considered; the trigger efficiency for such events is almost 100% At most one hit is accepted in the muon chambers, to eliminate beamrelated and cosmic ray muons The loss of signal events with noisy muon chambers was estimated from events triggered at random beam crossings to be 3% The timing of the energy deposition in the ECAL is checked to be consistent with the beam crossing time All events with at least 05 GeV detected below 148 from the beam axis are rejected, in order to Table 1 Data samples Energy Luminosity N events Ž GeV Ž y1 pb Data Expected Total

244 44 ( ) R Barate et alrphysics Letters B remove radiative Bhabha events The efficiency correction factor associated with this cut was estimated from events triggered at random beam crossings to be 35% The consistency between the energy measured from the ECAL pads and from the ECAL wires is checked In case of leakage out of the ECAL, a localized energy deposit in the HCAL, E had, associ- ated to an ECAL cluster is added to E g, after correcting for the erp ratio; only events with EhadrEg -10% are kept To reduce the remaining background, all but 5 GeV of the total energy is required to come from photon candidates At least one photon candidate is required to fulfil the conditions u g) 08 and ptgre beam) 01 For multiphoton candidates, the additional photons are considered only if their energy exceeds 005' s The overall missing transverse momentum is required to be greater than 1 GeVrc The last cut removes the remaining Bhabha events with radiation at large angle Table 1 shows the data samples used in this analysis The numbers of selected events agree with the numbers expected from the SM cross sections determined with the KORALZ Monte Carlo The cross section measured from the data at 183 GeV, with the present analysis and within the global kinematic cuts, is 345"030 pb, to be compared to the SM prediction of 340"00 pb dard Model and those from anomalous matrix elements As the matrix element is linear in Dkg and lg the cross section and the differential distributions are bilinear forms of Dkg and l g For each event it is thus sufficient to store weights for only six configurations in the Ž Dk,l g g plane, in order to compute any cross section or kinematic variable as a function of D kg and l g The leading order amplitudes in- cluding these anomalous couplings are folded with higher order QED effects, following the procedure discussed in Ref w10 x The Standard Model predicts that the cross section for the radiative return to the Z resonance decreases when the centre-of-mass energy increases, while the opposite is true for the W exchange In case of anomalous contributions, the sensitivity of the cross section increases almost quadratically above 161 GeV The kinematic cuts have been chosen to optimize the sensitivity to the anomalous couplings D kg and l g Fig gives the statistical sensitivity, defined as the anomalous contribution divided by the statistical error on the SM expectation, as a function of the scaled energy variable x Es EgrE beam The minimum of the sensitivity occurs around the position of the Z return peak Ž x s075 An important obser- E 3 Monte Carlo simulation with KORALZ The simulation uses a modified version of the KORALZ program, which includes the SM expectation Ž with electroweak corrections as well as QED radiative corrections, and the contributions of anomalous coupling amplitudes with exact matrix element calculations wx 9 The overall higher-order QED correction factor is around 14, but depends on the centre-of-mass energy More than ten thousand simulated events are used for each energy To obtain a description of the anomalous couplings in the simulation, each event is assigned a weight, which is a function of Dkg and l g This method provides the smallest uncertainty, as the statistical error corresponds only to the differences between the distributions produced from the Stan- Fig Statistical sensitivity of this analysis as a function of the scaled energy x at 183 GeV centre-of-mass energy: Ž E a for Dk g, Ž b for l g The sensitivity is defined as the anomalous contribu- tion divided by the statistical error on the SM expectation The solid and dashed histograms correspond to parameter values of y10 and q10, respectively The single photon radiative return to the Z corresponds to x E s075

245 ( ) R Barate et alrphysics Letters B vation is that for Dk g)0 the differential cross sec- tion decreases almost linearly as a function of Dk g for events with x E - 075, and increases quadratically above 4 Likelihood fit In addition to the observed number of events, two kinematic variables of the photon are used in the fit: the photon polar angle ug and the scaled energy x E Fig 3 shows the distribution of x and < cosu < E g for data, compared to the Standard Model predictions for ' s s183 GeV Two kinematic regions have been chosen for the fit, excluding the region of the Z peak return where the sensitivity to the anomalous couplings is minimal Only photons with < cosu < g -09 are used for the fit to the shape of the distributions Z Defining E sž sym r' g Z s, the two kinematic regions are the following: Ø Region 1, low energy photons with -Eg Z y 3G Z The contribution from higher order radiative corrections is described by an almost constant term obtained from the Monte Carlo simulation The scaled variable x E is found to be as discriminant as the angular variable in the fit Both are used Fig 3 Inclusive distribution of Ž a the scaled energy x and Ž E b the absolute value of the cosine of the polar angle of the photons, after all selections, for data and Monte Carlo at 183 GeV Table Number of events Ž N events entering the fit in the two kinematic regions The number of expected events is estimated from the KORALZ cross sections, corrected for acceptance Kinematic N events, N events, region Cross section fit Ž x,u E fit Data Expected Data Expected Region Region for the D kg fit, whereas lg is determined only from the total cross section The sensitivity to the lg parameter in this kinematic region is very low Ø Region, high energy photons with E g)eg Z q 05 GeV In this region, the higher order radiative corrections decrease the number of expected events by 30% The scaled energy x E is more discriminant than the angular variable, both variables being used in the fit of Dkg and l g It can be observed Ž Fig that the sensitivity to lg with x E is similar to that of Dk g Limits for anomalous coupling parameters have been derived from the generalized likelihood expression: log Lslog Ý Nobs Ž1 Ž1 Nobs Ž Ž Ž1 ynth Ž ynth th th qlog Ž1 Ž N obs! N obs! N e N e Ý q log P Ž1 q log P Ž, i where Pi Ž1, Pi Ž are the probability density functions of observing event i with a given value of x E and ug in region 1 and respectively, and Nth Ž1 and Nth Ž are the expected number of events in each region, including background This likelihood formula contains two parts: the first one concerning the number of observed events, the second one being related to differential distributions for each kinematic region The number of events used in the fit and those expected from the SM are given in Table The acceptance convoluted with the experimental resolution leads to correction factors to the cross sections of 110 for the first kinematic region and 07 for the second; these correction factors are constant Ž within "% in each region as Dkg or lg vary The studies made with the KORALZ Monte Carlo show that the cross sections and distribution shapes i

246 46 ( ) R Barate et alrphysics Letters B vary differently in the two kinematic regions For low energy photons the anomalous effects result from the interference term between the SM amplitude and the anomalous amplitude; the resulting variation is monotonic and linear for Dk Ž l g g )0 and Dk Ž l g g -0 and only one solution is expected for the Dkg and lg fit For the high energy photons, the variations are quadratic Ždue to a quadratic contribution of the anomalous amplitude and one or two solutions are expected; the case of one solution corresponds to Dkgs0 orlg s0 This behaviour, important in the error determination, is discussed later when the error calibration procedure is presented 5 Results The likelihood functions are calculated globally for the cross section and on an event-by-event basis for the energy and angular distributions Fig 4 displays the variations of the log-likelihood Ž ydlog L corresponding to the fit of Dkg at lg s0, for the cross section and the distribution contributions At present energies, the contributions of the cross section and of the shape variation terms are equally important for the fit of Dk g The result for lg is dominated by the sensitivity to the shape in Region Fig 5 Likelihood curves for the fit of l at Dk s0 Ž g g solid curve and Dk at l s0 Ž dashed curve g g for the sum of the cross section and distribution shape terms Fig 5 shows the Ž ydlog L functions for Dk g fitted at lg s0, and for lg fitted at Dkgs0 when the two contributions are merged The results are: Dkgs005 q115 y110ž stat assuming lg s0 lg sy005 q155 y145ž stat assuming Dkg s0 where the errors correspond to an increase of ylog L by 05 The lower precision for lg is expected since the exchanged W s are at a rather low momentum scale and the lg term in the Lagrangian contains high powers of the W momentum Fig 4 Likelihood curves for the fit of Dkg at lg s0 for the contribution of the cross section term Ž solid curve and the shape term in x and u Ž dashed curve E Fig 6 68% and 95% confidence level contours in the Dk,l g g plane

247 ( ) R Barate et alrphysics Letters B The 95% CL limits derived from the one parameter fits are: y1-dkg- assuming lg s0 y30-lg-31 assuming Dkg s0 The validity of these 95% CL limits have been checked using 100 Monte Carlo samples corresponding to the data luminosity, the analysis procedure described for the data being applied to each Monte Carlo sample This study indicates that these errors are consistent with the frequentist interpretation, within 10% of their values, and do not benefit from favourable statistical fluctuations Fig 6 shows the 68% and 95% confidence level contours in the Ž D k, l g g plane from a two-parame- ter fit Although the two parameters are not independent, the confidence level contours are symmetric This comes from the fact that the results are very close to 0, so that only one minimum is found If the results were far away from the SM prediction, there could be several local minima, around which the two parameters would be correlated 6 Systematic uncertainties The contributions to the systematic uncertainty on the determination of D kg are summarized in Table 3 The total systematic uncertainty is much smaller than the statistical one Ø The acceptance corrections were tested with different cuts in x E and u This led to an uncertainty on the fit results as shown in Table 3 Ø The main contribution to the systematic error in the present study comes from the energy calibration of high energy photons, which has been Table 3 Contributions to the systematic uncertainty on fitted Dk g, as explained in the text Origin of uncertainty Region 1 Region Acceptance corrections "008 "008 Photon energy calibration "1% "010 "00 Background -1 event q 005 q 005 Model uncertainty -"5% "010 "015 Luminosity value "06% "003 "003 Total "00 "030 checked to be 1% with a large sample of e q e y gg events q y Ø The possible contributions to the e e ggq X channel, other than Xsnn, may come from q y radiative Bhabhas or e e gg Ž g events All such events in the Monte Carlo sample are eliminated by the angular and energy cuts Ø The KORALZ simulation of higher order effects gives a correction of about q 100% for the SM cross section in Region 1, and about y30% in Region A theoretical estimate of the error on these correction factors is about 5% However, only comparisons with complete calculations from the exact matrix elements Ž not present in KORALZ for the two and three hard bremsstrahlung photons would allow a satisfactory estimation of this uncertainty A discussion of the uncertainty due to the implementation of the matrix elements with anomalous couplings for the multiphoton events is presented in Ref w10 x The model uncertainty in introducing the anomalous couplings into the simulation has been checked The reliability of the simulation of the Standard Model is discussed in Ref w10 x Ø Another contribution to the uncertainty on the total cross section part of the fit is given by the luminosity error Other possible contributions to the systematic error, such as the statistical precision on the correction factors for muon rejection and energy deposition in the forward region of the detector, are negligible For l g, the basic errors are the same, one region only being used for the systematic error calculation 7 Conclusions The anomalous coupling parameters D kg and lg have been measured from single and multiphoton events in e q e y collisions between 161 and 184 GeV The results from the fit to the cross sections and to the energy and angular distributions of the photons are Dkgs005 q115 y110ž stat "05Ž syst lg sy005 q155 y145ž stat "030Ž syst

248 48 ( ) R Barate et alrphysics Letters B The corresponding 95% CL limits including systematic errors are: y-dkg-3 assuming lg s0 y31-lg-3 assuming Dkg s0 These results are in good agreement with the Standard Model predictions and the uncertainty is largely dominated by the limited statistics of the data sample Acknowledgements We thank and congratulate our colleagues in the CERN accelerator divisions for the successful operation of LEP We are indebted to the engineers and technicians in all our institutions for their contribution to the excellent performance of ALEPH Those of us from non-member countries thank CERN for its hospitality and support We would also like to thank J Kalinowski for discussions and Z Wa s for providing us with a Monte Carlo generator containq y ing the gauge sector for the reaction e e nng Ž g References wx 1 GV Borisov, VN Larin, FF Tikhonin, Z Phys C 41 Ž ; G Couture, S Godfrey, Phys Rev D 50 Ž wx J Abragam, J Kalinowski, P Sciepko, Nucl Phys B 339 Ž ; F Berends et al, Nucl Phys B 301 Ž wx 3 Triple Gauge Couplings, G Gounaris et al, in: G Altarelli, T Sjostrand, F Zwirner Ž Eds, Physics at LEP, CERN 96-01, 1996, p 55 wx 4 ALEPH Collaboration, Phys Lett B 4 Ž ; DEL- PHI Collaboration, Phys Lett B 43 Ž ; L3 Collaboration, Phys Lett B 413 Ž ; OPAL Collaboration, Eur Phys Journal C Ž wx 5 The D0 Collaboration, S Abachi et al, Phys Rev D 58 Ž wx 6 S Jadach, BFL Ward, Z Wa s, Comp Phys Comm 79 Ž wx 7 ALEPH Collaboration, Nucl Instr Meth A 94 Ž wx 8 ALEPH Collaboration, Nucl Instr Meth A 360 Ž wx 9 D Choudhbury, J Kalinowski, Unravelling the WWg and WWZ Vertices at the Linear Collider: nng and nn qq final states, MPI-PTH r96-73, IFT-96r17 w10x A Jacholkowska, J Kalinowski, Z Wa s, Higher-order QED q y corrections to e e nng at LEP, CERN-THr98-55, IFTr3r98, hep-phr , submitted to Eur Phys Journal C

249 7 January 1999 Physics Letters B Weak magnetism in two neutrino double beta decay 1 C Barbero a,, F Krmpotic a,, A Mariano a,, D Tadic b a Departamento de Fısica, Facultad de Ciencias Exactas, UniÕersidad Nacional de La Plata, CC 67, 1900 La Plata, Argentina b Physics Department, UniÕersity of Zagreb, Bijenicka ˇ c 3, POB 16, Zagreb, Croatia Received 9 October 1998 Editor: W Haxton Abstract We have extended the formalism for the two-neutrino double beta decay by including the weak-magnetism term, as well as other second-forbidden corrections The weak magnetism diminishes the calculated half-lives in ; 10%, independently of the nuclear structure Numerical computations were performed within the pn-qrpa, for 76 Ge, 8 Se, 100 Mo, 18 Te and 130 Te nuclei Not one of the second-forbidden corrections modifies significantly the spectrum shapes The total reduction in the calculated half lives varies from 6% up to 3%, and strongly depend on the nuclear interaction in the particle-particle Ss1,T s0 channel We conclude that the higher order effects in the weak Hamiltonian cannot be observed in the two-neutrino double beta experiments q 1999 Elsevier Science BV All rights reserved PACS: 1315qq; 1480; 160Jz; 340 The sensitivity of the double-beta Ž bb decay experiments is steadily and constantly increasing To become aware of this it suffices to remember that, between the pioneer laboratory measurement on 8 Se 76 wx 1 and the most recent on Ge wx, the statistics has been improved by a factor of ;5000 The quantity that is used to discern experimentally between the ordinary Standard Model Ž SM two-neutrino decays Ž bb n and the neutrinoless bb events not included in the SM, both without bb 0n 1 Work partially supported by the CONICET from Argentina and the Universidad Nacional de La Plata Fellow of the CONICET from Argentina and with Majoron emissions Ž bb M, is the electron energy spectrum d Grde of the decay rate G Itis usually given as a function of the sum Ž e of the energies Ž e and e 1 of the two emitted electrons The bbn and bbm decays exhibit continuous spectra in the interval FeFQ, while the bb0n spectrum is just a line at the released energy QsEIyE F In the evaluation of the bbn decay rate, the allowed Ž A approximation is usually assumed to be valid, ie, the Fermi Ž F and Gamow-Teller Ž GT operators are considered at the same level of approximation as in single-allowed b transitions Besides, as the F operator is strongly suppressed in the bb n decay by the isospin symmetry, only the dominant contribution of the axial-vector current is considered in practice Then, the bb matrix element, between n r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

250 50 ( ) C Barbero et alrphysics Letters B the initial state < 0 : in the Ž N, Z I nucleus to the final state < 0 : in the Ž Ny,Zq nucleus Ž F with enerp gies E and E and spins and parities J s 0 q I F reads: ² 0 q < s < 1 q : P² 1 q < s < 0 q : Ž0 F N N I Mn sg A Ý Ž 1 E qye q N 1 0 qqr N I Here the summation goes over the virtual states with spin and parity J p s1 q Except for the work of Williams and Haxton wx 3, all higher order terms Ž HOT in the weak Hamiltonian have been almost totally ignored in the past, simply because they are expected to be small Yet, the question is not so simple First, from the comparison of the recent experimental results for the bb 76 wx 1 half lives in Ge : T n (177P10 y, T M )167 P10 y and T 0n )1P10 5 y, it can be stated that presently are being observed effects of the order of 10 y4 at e;q and of the order of 10 y1 at e;qr Secondly, there are also several ongoing and planned experiments that are supposed to allow for measurements of still smaller effects Thus, the HOT could, in principle, become relevant In recent years we have examined the effects of the first-forbidden operators in the bb decay w4,5 x n, both the non-unique Ž FFNU and unique Ž FFU, which contribute via the virtual states J p s0 y,1 y and J p s y, respectively From these studies we have learned that: Ž a the FFNU transitions might increase the Tn up to ; 30%, yet they do not modify the A shape of the two-electron spectrum wx 4, Ž b the FFU transitions do alter the bbn spectrum shape but only at the level of 10 y6 and mainly at low two-electron energy, where most backgrounds tend to dominate wx 5 Therefore, the effects of the first-forbidden transitions could hardly screen the detection of exotic bb0n and bbm decays, which are the candidates for testing the physics beyond the SM In the present work we want to complete the question of the competition between the standard and exotic bb-decays wx 5, by scrutinizing the effects of the nuclear matrix elements, which have been intensively studied in connection with deviations of the simple-beta spectra from the allowed shape w6 9 x ŽIn particular, it should be reminded that the detection of the weak-magnetism Ž WM term in the spectrum of b " -decays of 1 N and 1 B to 1 C has provided a rather striking test of the CVC theory It is customary to denominate these HOT as secondforbidden corrections Ž SFC, although the most significant among them comes from the WM term, which obeys the same selection rules as the GT operator One also should keep in mind that the WM plays a very important role in both the bb0n and the bb decays, through the so called recoil term w M 10 1 x Proceeding in the same way as in our previous works w4,5,1 x, we express the bb decay rate as: n Ž where the symbol represents both the summation on lepton spins, and the integration on neutrino momenta and electron directions The transition amw4,5,1 x: 1 plitude reads Rn s Ý 1yPŽ e e 1yP Ž nn p N = ² 0 q < H Ž e n < N:² N< H Ž e n < 0 q : F W W 1 1 I, ENyEIqe1qv1 Ž 3 where the operator P exchanges the lepton quantum numbers e ' Ž e, p,s and n ' Ž v,q,s i i i e i i i n All i i other notations have the usual meaning wx 5 In the non-relativistic weak Hamiltonian we only retain the terms that are necessary for the evaluation of the SFC, and obtain Gg A HWŽ eini s sq Ž 3jqeiyqi X qqiž 3jqei YqŽ 3jqeiqqi Z PLŽ ein i, Ž 4 y1 Here Gs 996"000 =10 is the Fermi coupling constant Ž in natural units, j s a Zr R s 118ZA y1r3 is the Coulomb factor, and g V Xs Ž fw sqr=p, 3Mg A r ' Ys sq 8p smyž r ˆ 1, 7 1 Zs sqiž spp r, Ž 5 6M

251 are the nuclear operators, with gvs1 and f The leptonic part is: ( 1 i n 0 i e ei q1 LŽ e n ssg Ž s F Ž e x Ž s i i e i = ž / ( ) C Barbero et alrphysics Letters B W s47 spp 1y s Ž 1ysPqˆ x Ž ys n i eiq1 Ž 6 Keeping only the interference terms between s with X, Y and Z, and the linear contributions in the Ž Ž 3 lepton energies, from 3 and 4, we obtain: G Ž 0 n 6 0Ž 1 n R s f ee M 1Ž p 3 1 Ž i Ý i 1 n is1 1 q f Ž ee M 1yP Ž nn where =LŽ en PLŽ e1n 1, Ž 7 e1qe f0ž ee 1 s1, f1ž ee 1 s1q, 6jyQ e1qe fž ee 1 s1y, f3ž ee 1 s1, Ž 8 Q and ² 0 q < s < 1 q : P² 1 q < X < 0 q : Ž1 F N N I Mn s g A Ž 6jyQ Ý, E qye qqqr N 1N 0I ² 0 q < s < 1 q : P² 1 q < Y < 0 q : Ž F N N I Mn s6 g A j Q Ý, E qye qqqr N 1N 0I ² 0 q < s < 1 q : P² 1 q < Z< 0 q : Ž3 F N N I Mn s g AŽ 6jqQ Ý E qye qqqr N 1N 0I Ž 9 It might be interesting to note that, while the FFNU transitions interfere with the GT operator at the level 3 A factor of 3 has been omitted in the denominator of Eq Ž 8 of Ref wx 5 All other formulas are, however, correct Fig 1 Kinematical factors F Ž upper panel i and the spectrum 76 shape dg rde Ž lower panel n for Ge All quantities are normalized to the maximum value of the allowed shape of the transition rates, the SFC do it already at the level of the transition amplitude At the same order of approximation the differential transition rate reads: 4G 4 3 Ž0 Ži d Gn s dv M Ý f Ž ee M, Ž 10 5 n n i 1 n 15p where is0 1 5 n Ž 1 Ł k k 0Ž k k 64p ks1 dv s Qye ye p e F e de Ž 11 Finally we derive the expressions for the spectrum shape dg G 4 3 n Ž0 Ži s M F Ž e M, Ž 1 7 n Ý i n de 40p is0

252 5 ( ) C Barbero et alrphysics Letters B Table 1 Ž y1 Ži Numerical results for the kinematical factors G in units of y and for the nuclear matrix elements M Ž in natural units i n, evaluated within the pn-qrpa formalism with an effective axial charge g A s1 Ž0 Ž1 Ž Ž3 Nucleus G0sG3 G1 G Mn Mn Mn Mn 76 y0 y0 y0 Ge 549=10 64=10 33= y y18 y18 y18 Se 183=10 14=10 083= y y18 y18 y18 Mo 397=10 454=10 183= y y y Te 354=10 379=10 113= y Te y18 00=10 y18 3=10 y18 091= y00014 where 5 ey1 i i Ž 1 1 F e s Qye f e de p e p e F e H =F Ž e Ž 13 0 For the half-life we get q q q q y1 n Ž I F n Ž I F 3 y1 Ž 0 Ž i n Ý i n ž / is0 T 0 0 sln G 0 0 with the kinematical factors G 4 s M G M, Ž 14 H Q Gis def Ž e Ž 15 7 i 40p ln As seen from Ž 1 the spectrum shape mainly depends on the factors F Ž e i They are displayed for 76 Ge in the upper panel of Fig 1, as a function of the energy e F Ž e and F Ž e 0 3 exhibit the same energy dependence, while F1 Ž e shifts the A spectrum slightly to the right and F Ž e slightly to the left The spectrum shapes without and with the SFC are compared in the lower panel of Fig 1 It can be observed that they are quite similar Analogous behavior was found for other experimentally interesting nuclei, such as 8 Se, 100 Mo, 18 Te, and 130 Te Table Calculated half-lives within the allowed approximation ŽT A n and Ž Aq SFC with second-forbidden corrections included T n in units of y A n Nucleus T T AqSFC n Ge 73=10 68= Se 15=10 14= Mo 97=10 66= Te 81=10 76= Te 0 =10 0 0=10 From Ž 5, Ž 9 and Ž 10 one sees that the main effect of the WM consists in renormalizing the GT matrix element Ž 1 as: gvj f Ž0 Ž0 W Mn Mn 1q, Ž 16 ž g M A / ie, by a factor of ;105 for medium heavy nuclei, independently of the nuclear model employed The matrix elements Mn Ži were evaluated within the pn-qrpa model, following the procedure adopted in our previous works w4,1,13 x We display them in Table 1, together with the corresponding kinematical factors G i Besides the WM term, the velocity dependent matrix elements r=p and ižs P prare also important, and particularly in the case of 100 Mo 4 The moment Mn Ž is always relatively small As it is well known, within the QRPA the GT moment Mn Ž0 strongly depends on the particle-particle coupling constant in the Ss1,Ts0 channel, denoted by t in Ref w13 x This dependence is particularly pronounced in the physical region for t, where Mn Ž0 goes to zero and the QRPA collapses Mn Ž1 and Mn Ž3 also strongly depend on t, but in a slightly different way than Mn Ž0 Thus the decay rates rely on two or three rapidly varying functions of t, which Ž AqSFC makes the calculated half-lives T n to be even more sensitive on the value of t than within the A approximation ŽT A n The numerical results are shown in Table Contrarily to what happens in the case of the FFNU transitions, the SFC decrease the half-lives The reduction ranges from 6% in 18 Te up to ;3% in 100 Mo 4 From the theoretical point of view the nuclear matrix elements in 100 Mo are in same sense peculiar, because of the strong w p predominance of the 0 g n 0 g p ; J s1 q x configuration 7r 9r

253 ( ) C Barbero et alrphysics Letters B This work completes our previous inquiries w4,5x on the significance of the higher order corrections in the two-neutrino double beta decay It can be concluded that: 1 Both the first-forbidden transitions through the J p s0 y,1 y virtual states, and the second order corrections to the Gamow-Teller states, affect the transition rates in a significant way The theoretical uncertainties within the QRPA, in the evaluation of the half-lives, are of the same order of magnitude Ž or even larger than the experimental errors 3 The effect on the energy spectra of the higher order terms in the weak Hamiltonian is too small to shadow the possible exotic neutrinoless events in contemporary experiments References wx 1 SR Elliot, AA Hahn, MK Moe, Phys Rev Lett 59 Ž wx HV Klapdor-Kleingrothaus, hep wx 3 A Williams, WC Haxton, in: GM Bunce Ž Ed, Intersections between Particle and Nuclear Physics, AIP Conf Proc No 176, 1988, p 94 wx 4 C Barbero, F Krmpotic, A Mariano, Phys Lett B 345 Ž wx 5 C Barbero, F Krmpotic, A Mariano, Phys Lett B 436 Ž wx 6 B Eman, D Tadic, Glasnik Matematicko ˇ Fizicki ˇ i Astronomski 16 Ž ; Nucl Phys 38 Ž wx 7 HF Schoper, Weak Interactions and Nuclear Beta Decay, North-Holland, Amsterdam, 1966 wx 8 A Bohr, BR Mottelson, Nuclear Structure, Benjamin, New York, 1969, vol I wx 9 H Beherens, W Buring, Electron Radial Wave Functions and Nuclear Beta Decay, Clarendon, Oxford, England, 198 w10x M Doi, T Kotani, E Takasugi, Prog Theor Phys Suppl 83 Ž w11x C Barbero, JM Cline, F Krmpotic, D Tadic, Phys Lett B 371 Ž ; B 39 Ž w1x C Barbero, F Krmpotic, D Tadic, Nucl Phys A 68 Ž w13x F Krmpotic, S Shelly Sharma, Nucl Phys A 57 Ž

254 7 January 1999 Physics Letters B Effects of meson mass decrease on superfluidity in nuclear matter Masayuki Matsuzaki a,1, Tomonori Tanigawa b, a Department of Physics, Fukuoka UniÕersity of Education, Munakata, Fukuoka , Japan b Department of Physics, Kyushu UniÕersity, Fukuoka , Japan Received 5 August 1998; revised 5 October 1998 Editor: J-P Blaizot Abstract We calculate the 1 S0 pairing gap in nuclear matter by adopting the in-medium Bonn potential proposed by Rapp et al we-print nucl-thr x, which takes into account the in-medium meson mass decrease, as the particle-particle interaction in the gap equation The resulting gap is significantly reduced in comparison with the one obtained by adopting the original Bonn potential q 1999 Elsevier Science BV All rights reserved PACS: 160-n; 165qf; 660qc Keywords: Meson mass; Superfluidity; Nuclear matter Superfluidity in infinite hadronic matter has long been studied mainly in neutron matter from the viewpoint of neutron-star physics such as its cooling rates As a way of description, relativistic models are attracting attention in addition to traditional non-relativistic nuclear many-body theories Since Chin and Walecka succeeded in reproducing the saturation property of symmetric nuclear matter within the mean-field theory Ž MFT wx 1, quantum hadrodynamics Ž QHD which is an effective field theory of hadronic degrees of freedom has described not only infinite matter but also finite spherical, deformed and rotating nuclei successfully with various approxima- 1 Electronic address: matsuza@fukuoka-eduacjp Electronic address: tomoscp@mboxnckyushu-uacjp tions w,3 x These successes indicate that the particle-hole Ž p-h channel in QHD is realistic In contrast, relativistic nuclear structure calculations with pairing done so far have been using particleparticle Ž p-p interactions borrowed from non-relativistic models such as Gogny force Aside from practical successes of this kind of calculations, the p-p channel in QHD itself is a big subject which has just begun to be studied The first study in this direction was done by Kucharek and Ring wx 4 They adopted, as the particle-particle interaction Ž Õ pp in the gap equation, a one-boson-exchange interaction with the ordinary relativistic MFT parameters, which gave the saturation under the no-sea approximation The resulting maximum gap was about three times larger than the accepted values in the non-relativistic calculations r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

255 ( ) M Matsuzaki, T TanigawarPhysics Letters B w5 9 x Various modifications to improve this result were proposed w10 1x but this has still been an open problem One of such modifications is to include the p-h and the N-N polarizations, the former of which is effective for reducing the gap in the non-relativistic models w13,14 x From a different viewpoint, Rummel and Ring w15,x adopted the Bonn potential w16 x, which was constructed so as to reproduce the phase shifts of nucleon-nucleon scattering in free space, as Õpp and obtained pairing gaps consistent with the non-relativistic studies Since the single particle states are determined by the MFT also in this calculation, explicit consistency between the p-h and the p-p channels is abandoned However, assuming that the MFT simulates the Dirac-Brueckner-Hartree-Fock Ž DBHF calculation using the Bonn potential, we can consider that the consistency still holds implicitly A possible extension of this study is to take into account the in-medium changes of the properties of the mesons which mediate the inter-nucleon forces Among them, the change of the mass, that is, the momentum-independent part of the self-energy, has been discussed in terms of the Brown Rho Ž BR scaling w17 x, the QCD sum rules w18 x, and some other models both in the quark level and in the hadron level w19,0 x Although there still is theoretical controversy w1 x, some experiments seem to support the vector meson mass dew x In addition, the momentum-dependent crease part also attracts attention recently w3 6 x In the present work, we assume only the momentum-independent vector meson mass decrease A simple way to take into account this meson mass decrease in the meson exchange interactions is to assume the BR scaling Actually Rapp et al w7x showed, based on this, that the saturation property of symmetric nuclear matter and the mass decrease of the vector mesons were compatible So we adopt their inmedium Bonn potential as Õpp in the gap equation Since the gap equation takes the form such that the short range correlation is involved w8,9,7 x, we use this interaction in the gap equation without the iteration to construct the G-matrix As described in Ref wx 4, meson fields also have to be treated dynamically beyond the MFT to incorporate the pairing field via the anomalous Ž Gor kov Green s functions w30 x The resulting Dirac-Hartree- Fock-Bogoliubov equation reduces to the ordinary Fig 1 Pairing gap in symmetric nuclear matter at the Fermi surface as functions of the Fermi momentum Dotted and solid lines indicate the results obtained by adopting the Bonn-B and the in-medium Bonn potentials, respectively Note that accuracy is somewhat less around DŽ k,0 in our code F BCS equation in the infinite matter case We start from a model Lagrangian for the nucleon, the s boson, and the v meson, 1 1 m m Lsc Ž igm E ym cq Ž Ems Ž E s y mss 1 1 mn m y FmnF q mvvmv qgscsc 4 m ygvcgm v c, FmnsE mvnyenv m Ž 1 The actual task is to solve the coupled equations for the effective nucleon mass and the 1 S pairing gap: g s g L M ) ) c M smy H Õ Ž k k dk, m p 0 ' k qm ) s 0

256 56 ( ) M Matsuzaki, T TanigawarPhysics Letters B L c H DŽ p sy ÕppŽ p,k 8p 0 DŽ k = k dk, Ž e ye qd Ž k ( k kf ž / 1 ekyek Õ Ž k s 1y F, Ž e ye qd Ž k ( k kf ' ) ² 0: v e s k qm qg v, k where Õ Ž p, k pp is an anti-symmetrized matrix ele- ment of the adopted p-p interaction, & & X X Õ Ž p, k s² ps,ps < Õ < ks,ks: pp pp & & X X y² ps,ps < Õ < ks,ks :, Ž 3 pp with an instantaneous approximation Ženergy transfer s 0 and an integration with respect to the angle between pand k to project out the S-wave component, and ² v 0 : is determined by the baryon density Here we consider only symmetric nuclear matter Ž g s 4 and neutron matter Ž g s Conceptually the numerical integrations run to infinity in the present model where the finiteness of the hadron size is considered in Õ pp Numerically, however, we intro- duce a cut-off L c; its value is chosen to be 0 fm y1 so that the integrations converge If we adopt the Bonn potential as Õpp and include the r meson and the cubic and quartic self-interactions of s with choosing the NL1 parameter set w31x for the mean field, these equations reproduce the results of Ref w15 x Here we note that there are pros and cons as for the self-interaction terms w3,33 x First we developed a computer code to calculate the Bonn potential for this channel from scratch, and later confirmed that our code reproduced outputs of the one in Ref w34 x Then in the present work we adopt the in-medium Bonn potential of Rapp et al w7x although this is, as yet, unpublished To construct this potential, they applied the BR scaling after replacing the s boson in the original Bonn-B potential with the correlated and the uncorrelated p exchange processes Then they parameterized the obtained nucleon-nucleon interaction by three scalar bosons s 1, s and rests, in addition to p, h, r, v and d Here s1 and s with density-dependent masses and coupling constants simulate the correlated p processes and the rest-s simulates the uncorrelated ones Therefore the actual tasks are adding two extra scalar y1 Fig Matrix element Õpp k F,k as functions of the momentum k, with a Fermi momentum k F s 09 fm Dotted and solid lines indicate the results obtained by adopting the Bonn-B and the in-medium Bonn potentials, respectively

257 bosons to the original Bonn-B potential and applying the BR scaling, M ) m ) r,v L ) r,v r s s s1yc, Ž 4 M m L r r,v r,v 0 ( ) M Matsuzaki, T TanigawarPhysics Letters B where r0 is the normal nuclear matter density, Lr,v are the cut-off masses in the form factors of the meson-nucleon vertices, which is related to the hadron size The scaling parameter C is chosen to be 015 Aside from a slight tuning of the coupling constant of d, other parameters are the same as those in the original Bonn-B potential All the potential parameters are presented in Tables I and II in Ref w7 x Note that this scaling applies only to the quantities in Õ pp, since the MFT with its effective nucleon mass guarantees the saturation and we confirmed that the effects of pairing on the bulk properties are negligible except at very low densities in the present model As for the p-n coupling in the potential, we adopt the pseudovector one conforming to the sugw35x and actually the gestion of chiral symmetry pseudoscalar one in the Bonn potential was shown to lead to unrealistically attractive contributions in the DBHF calculation w36 x The result is presented in Fig 1 This shows that applying the BR scaling reduces the gap significantly in comparison with the original Bonn-B potential The potentials themselves are given in Fig The relation between the reduction of the gap and the change in the potential, ie, the shift to lower momenta, can be understood as follows: As discussed in Refs w6,15 x, DŽ k determined by the second equation of Ž exhibits a nodal structure similar to Õ Ž k, k pp F as functions of k with the opposite sign and some possible deviation of zeros Consequently both the low-momentum attractive part where DŽ k ) 0 and the high-momentum repulsive part where DŽ k -0 give positive contributions to DŽ k F, and Fig shows that both of these contributions are reduced in the case of the in-medium Bonn potential This is the main reason why DŽ k F is reduced by applying the BR scaling Actually we confirmed that this is mainly due to the mass decrease of the vector mesons, not of the nucleon, as shown in Fig 3 although also the latter produces some reduction of the gap Note here that the saturation in the DBHF Fig 3 Pairing gap in symmetric nuclear matter at the Fermi surface as functions of the Fermi momentum Solid, dotted and dashed lines indicate that the results obtained by reducing according to Eq Ž 4 only the nucleon mass, only the vector meson masses, and both, respectively calculation is guaranteed only when both M and mr,v are scaled The in-medium meson mass de- crease is brought about by the N-N polarization in hadronic models Although its explicit inclusion in the gap equation has not been reported yet, it is conjectured in Ref w1x that its inclusion simultaneously with the p-h one will reduce the gap as that of the p-h one in the non-relativistic models w13,14 x In this sense, the present result that applying the BR scaling reduces the gap is consistent with this conjecw13,14x is more ture, although the reduction in Refs drastic than that in the present work An additional element is that the cut-off masses in the form factors of the meson-nucleon vertices are also scaled Since they appear in the form L aym a, Ž 5 Laqq

258 58 ( ) M Matsuzaki, T TanigawarPhysics Letters B where qis the momentum transfer w16 x, simultaneous reductions of L and m Ž a s r, v a a, lead to a further reduction of the repulsion due to the vector mesons at large < q < Since the higher the background density becomes the more nucleons feel the highmomentum part of Õ pp, the in-medium reduction of Lr,v leads to an additional reduction of the gap in the high-density region Finally, use of the NL1 parameter set for the mean field brings negligible changes and the result for neutron matter is very similar to those presented here for symmetric nuclear matter To summarize, we adopted the in-medium Bonn potential, proposed by Rapp et al, which takes into account the in-medium meson mass decrease in a simple form and is compatible with the nuclear matter saturation in the DBHF calculation, as the particle-particle interaction in the gap equation The resulting pairing gap in nuclear matter is significantly reduced in comparison with the one obtained by adopting the original Bonn potential Here we should note that the uncertainty in the gap value deduced from various studies is still larger than the strength of the reduction found in the present work, and therefore, further investigations are surely necessary both relativistically and non-relativistically References wx 1 SA Chin, JD Walecka, Phys Lett B 5 Ž wx P Ring, Prog Part Nucl Phys 37 Ž wx 3 BD Serot, JD Walecka, Int J Mod Phys E 6 Ž wx 4 H Kucharek, P Ring, Z Phys A 339 Ž wx 5 H Kucharek et al, Phys Lett B 16 Ž wx 6 M Baldo et al, Nucl Phys A 515 Ž wx 7 T Takatsuka, R Tamagaki, Prog Theor Phys Suppl 11 Ž , and references cited therein wx 8 JMC Chen et al, Nucl Phys A 555 Ž wx 9 Ø Elgarøy et al, Nucl Phys A 604 Ž w10x FB Guimaraes, BV Carlson, T Frederico, Phys Rev C 54 Ž w11x F Matera, G Fabbri, A Dellafiore, Phys Rev C 56 Ž w1x M Matsuzaki, P Ring, in: Proc of the APCTP Workshop on Astro-Hadron Physics in Honor of Mannque Rho s 60th Birthday: Properties of Hadrons in Matter, 5 31 October, 1997, Seoul, Korea Ž World Scientific, Singapore, in press, we-print nucl-thr x w13x J Wambach, TL Ainsworth, D Pines, Nucl Phys A 555 Ž w14x H-J Schulze et al, Phys Lett B 375 Ž w15x A Rummel, P Ring, preprint, 1996 w16x R Machleidt, Adv Nucl Phys 19 Ž w17x GE Brown, M Rho, Phys Rev Lett 66 Ž w18x T Hatsuda, H Shiomi, H Kuwabara, Prog Theor Phys 95 Ž , and references cited therein w19x H-C Jean, J Piekarewicz, AG Williams, Phys Rev C 49 Ž w0x K Saito, AW Thomas, Phys Rev C 51 Ž w1x M Nakano et al, Phys Rev C 55 Ž wx As a recent review, I Tserruya, Prog Theor Phys Suppl 19 Ž w3x VL Eletsky, BL Ioffe, Phys Rev Lett 78 Ž w4x B Friman, HJ Pirner, Nucl Phys A 617 Ž w5x SH Lee, Phys Rev C 57 Ž w6x W Peters et al, Nucl Phys A 63 Ž w7x R Rapp et al, e-print nucl-thr w8x LN Cooper, RL Milles, AM Sessler, Phys Rev 114 Ž w9x T Marumori et al, Prog Theor Phys 5 Ž w30x LP Gor kov, Sov Phys JETP 7 Ž w31x P-G Reinhard et al, Z Phys A 33 Ž w3x J Boguta, AR Bodmer, Nucl Phys A 9 Ž w33x RJ Furnstahl, BD Serot, H-B Tang, Nucl Phys A 615 Ž w34x R Machleidt, in: K Langanke, JA Maruhn, SE Koonin Ž Eds, Computational Nuclear Physics Nuclear Reactions, Springer, New York, 1993, p 1 w35x S Weinberg, Phys Rev 166 Ž w36x R Machleidt, in: MB Johnson, A Picklesimer Ž Eds, Relativistic Dynamics and Quark-Nuclear Physics, Wiley, New York, 1986, p 71

259 7 January 1999 Physics Letters B Isospin symmetry breaking nucleon-nucleon potentials and nuclear structure H Muther a, A Polls b, R Machleidt c a Institut fur Theoretische Physik, UniÕersitat Tubingen, D-7076 Tubingen, Germany b Departament d Estructura i Costituents de la Materia, ` UniÕersitat de Barcelona, E-0808 Barcelona, Spain c Department of Physics, UniÕersity of Idaho, Moscow, ID 83844, USA Received 10 September 1998 Editor: J-P Blaizot Abstract Modern nucleon-nucleon Ž NN potentials, which accurately fit the nucleon-nucleon scattering phase shifts, contain terms which break isospin symmetry The effects of these symmetry violating terms on the bulk properties of nuclear matter are investigated The predictions of the charge symmetry breaking Ž CSB terms are compared with the Nolen-Schiffer Ž NS anomaly regarding the energies of neighboring mirror nuclei We find that, for a quantitative explanation of the NS anomaly, 1 it is crucial to include CSB in partial waves with L)0 Žbesides S 0 as derived from a microscopic model for CSB of the NN interaction q 1999 Elsevier Science BV All rights reserved PACS: 130qy; 165qf Isospin symmetry Ž or charge independence is invariance under any rotation in isospin space Due to the mass difference between up and down quarks and due to the electromagnetic interaction, this symmetry is slightly violated which is referred to as isospin symmetry breaking Ž ISB or charge independence breaking Charge symmetry is invariance under a rotation by 1808 about the y-axis in isospin space if the positive z-direction is associated with the positive charge The violation of this symmetry is known as charge symmetry breaking Ž CSB Obviously, CSB is a special case of ISB ISB of the strong nucleon-nucleon Ž NN interaction means that, in the isospin Ts1 state, the proton-proton Ž T sq1, neutron-proton Ž T s 0, z z or neutron-neutron Ž T sy1 interactions are Ž slightly z different, after electromagnetic effects have been removed CSB of the NN interaction refers to a difference between proton proton Ž pp and neutron neutron Ž nn interactions, only For reviews on these matters, see Refs w1, x In recent years, a new generation of realistic NN potentials has been developed which take ISB of the NN interaction into account and, therefore, yield very accurate fits of the proton-proton and protonneutron Ž pn scattering phase shifts w3 5 x Since these fits are based on the same phase shift analysis by the Nijmegen group wx 6 and yield a value for the x rdatum very close to one, these various potentials could be called phase-shift equivalent NN interactions The interactions by the Nijmegen group Ž Nijm1, Nijm, Reid93 wx 3, the Argonne group r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

260 60 H Muther et alrphysics Letters B 445 ( 1999) Ž ArgV18 wx 4 and the charge-dependent Bonn potential Ž CDBonn wx 5 describe the long-range part of the NN interaction in terms of the one-pion-exchange model, accounting for the mass-difference between p 0 and the charged pions p q and p y This distinction between the exchange of a neutral pion and charged pions is one origin of ISB in the resulting NN interactions It yields different phase shifts for pp scattering Ž only neutral pion exchange as compared to pn scattering in partial waves with isospin Ts1 Additional ISB terms have to be included in the 1 S0 state to achieve an accurate fit of pp and pn phase shifts in that partial wave The new NN interactions also account for the mass difference between proton and neutron This gives rise to a difference in the matrix elements of the meson-exchange interaction between two protons and two neutrons This breaks charge-symmetry, but only by a very small amount Additional CSB terms are included in the ArgV18 and CDBonn potentials to reproduce the difference in the empirical nn and pp scattering lengths, after subtracting the effects of electromagnetic interactions It is the aim of this letter to study the effects of ISB on the calculated bulk properties of nuclear systems Are these symmetry breaking effects similar in all these NN interactions? What is the influence of the ISB terms on the calculated symmetry energy and saturation properties of nuclear matter? And more specifically, do the CSB terms explain the so-called Nolen-Schiffer anomaly wx 7, the energy difference between neighboring mirror nuclei, which cannot be explained by the electromagnetic interaction? For that purpose we have performed Brueckner- Hartree-Fock Ž BHF calculations of nuclear matter, in which we distinguish between pp, pn and nn interactions As a reference we have also performed BHF calculations, in which the pp and nn interactions have been replaced by the corresponding pn interaction The Bethe-Goldstone equation has been solved by determining a self-consistent single-particle spectrum for the hole states, which is extended in a continuous way to states with momenta k larger than the Fermi momentum k For very high mo- F menta, for which this prescription would yield a repulsive single-particle potential, the single-particle energy has been assumed to be identical to the kinetic energy Such a continuous choice for the single-particle spectrum yields larger binding energies than the conventional choice Žsingle-particle energy equal to kinetic energy for kgk F and seems to reduce the effects of three-body correlations in the hole-line expansion wx 8 The interactions CDBonn, ArgV18, Nijm1, Nijm and Reid93 are quite different, although they are phase-shift equivalent and agree to a large extent in the OPE contribution This is demonstrated in Table 1, which presents some results calculated for nuclear matter at the empirical saturation density Ž r 0 s 017 fm y3, which corresponds to a Fermi momentum y1 of symmetric nuclear matter of k s 136 fm F The results for the energy of nuclear matter calculated in the Hartree-Fock Ž HF approximation ŽE HF, second column of Table 1 range from 606 MeV per nucleon obtained from the CDBonn potential to 356 MeV per nucleon in the case of the Nijm interaction The discrepancy between these HF results and the empirical value of 16 MeV per nucleon reflects the importance of NN correlations to be included in the nuclear structure calculation It is worth noting that these modern models of the NN interaction are much softer than older NN potentials: eg the Reid soft-core potential wx 9 yields a HF energy of 175 MeV per nucleon It should also be mentioned that the NN interactions, which contain non-local terms like the Nijm1 and the CDBonn potential are softer, ie yield less repulsive HF energies, than local NN potentials w10 x If effects of NN correlations are included by employing the BHF approach, the various models of Table 1 Energies calculated for nuclear matter with Fermi momentum k F s 136 fm y1 Results are listed for the energy per nucleon calculated in BHF Ž E and Hartree-Fock Ž E BHF HF approximation, the loss of energy per nucleon due to the breaking of isospin-symmetry wde as defined in Eq Žx ISB 1, the symmetry Ž np energy with E and without E Sym Sym inclusion of ISB All entries are in MeV EBHF EHF DEISB ESym ESym np CDBonn y CDBo99 y ArgV18 y Nijm1 y Nijm y Reid93 y

261 H Muther et alrphysics Letters B 445 ( 1999) the NN interaction yield results Žsee first column in Table 1 which are rather close to each other, the largest difference being 3 MeV, and also fairly close to the empirical energy The NN interactions with weaker tensor force Ž CDBonn and ArgV18 w11x yield more binding energy than those which contain a stronger tensor force and the softer nonlocal potentials CDBonn and Nijm1 tend to predict larger binding energies than the corresponding local interactions This can easily be understood: Stiff potentials and those with a large tensor component, receive a large part of the attraction in the T-matrix from terms of second and higher order in the potential V Due to Pauli and dispersion effects these attractive contributions are quenched in the G-matrix If two NN interactions fit the same NN phase shifts, therefore yield the same T-matrix, the G-matrix elements will be less attractive for a stiff potential as compared to a soft one The differences in the predictions for the binding energies are enhanced, if we compare the saturation points, ie the minima of the energy versus density curves Ž left part of Fig 1, instead of the energies calculated at the empirical saturation density Notice that the saturation points Ž solid symbols determined from these modern potentials fall on the so-called Coester band w1 x The main interest of the present study is to explore the effects of ISB in these modern NN interactions For that purpose we repeated the BHF calculations of nuclear matter replacing the pp and nn interactions by the corresponding matrix elements of the pn interaction, ie, we assume perfect isospin symmetry and identify the NN interaction with the pn interaction If we denote the energy from this isospin symmetric calculation by EBHF np, then we may define an energy correction which is due to the breaking of isospin-symmetry by DEISB sebhf yebhf np Ž 1 Results for this energy correction are displayed in Table 1 and in the right part of Fig 1 The energy correction is repulsive reflecting the fact that the pn interaction is more attractive in the partial waves with isospin Ts1 than the corresponding pp and nn interactions A main contribution of this ISB can be related to the pion-exchange contribution, which is repulsive in the dominant 1 S0 partial wave While the one-pion-exchange Ž OPE contribution to the pp and nn interaction can only be mediated by the neutral pion p 0, charged pions p " can be exchanged in the pn interaction Since the mass of the p 0 is smaller than the mass of p ", the Fig 1 The left part of this figure displays the saturation points for symmetric nuclear matter evaluated in the BHF approximation for various NN interactions Filled symbols represent the results for the interaction with inclusion of ISB, while open symbols are obtained if the pn interaction is used for all two-nucleon pairs The right part of the figure shows DE as defined in Eq Ž 1 ISB

262 6 H Muther et alrphysics Letters B 445 ( 1999) OPE contribution is more repulsive in pp and nn than in the pn interaction w13 x The ISB effect is largest and essentially identical for the CDBonn and Nijm1 potentials Žthe two curves referring to these potentials can hardly be separated in the right part of Fig 1 It is weaker by roughly 35 percent for the potentials Nijm and Reid93, while the prediction of the Argonne V18 potential is in-between Since the Argonne V18 potential is local, one would expect that it predicts DEISB very similar to the other local potentials, Nijm and Reid93 However, while the latter two potentials both predict DEISB s0 MeV, the ArgV18 result is 08 MeV, y1 at k s136 fm Ž cf Table 1 F The reason for this discrepancy is as follows Since the pp data are the most precise and reliable ones, all new high-precision ISB potentials are produced by first constructing the pp version of the NN potential with an accurate fit to the pp data The Ts1 np potential is then defined as the pp potential, but with the p 0 exchange replaced by the p 0 rp " exchanges appropriate for Ts1 np and with one proton mass replaced by a neutron mass The change in OPE explains about 50% of the empirically known difference between the pp scattering length Žcorrected for 1 electromagnetic effects and the np one, in the S 0 state The remaining 50% can to a large extent, but not completely be explained from ISB that emerges from p and pa exchanges where a denotes a heavy meson w1,13 x However, in the current highprecision NN potentials the latter is not included as derived from a microscopic model w13 x; instead, what is missing to get the 1 S0 np scattering length right, is just fitted by enhancing a parameter in the model that describes the intermediate range attraction Since the Nijmegen and the CDBonn potentials are constructed in a partial wave basis, this fitting affects only the 1 S0 state, making this state more attractive for np The procedure is different for ArgV18 The Argonne V18 potential is constructed in a Ž S, T decomposition Žwhere S denotes the total spin and T the total isospin of the two-nucleon system To get 1 the S np scattering length correct, the Ž 0 S s 0, T s 1 potential is made slightly more attractive However, in terms of a partial wave decomposition, this implies that all partial wave states with ŽSs0, Ts1, namely, S 0, D, G 4, etc, have their attraction enhanced This increases the binding energy obtained from the Argonne np potential Žas com- 1 pared to np potentials that adjust only S 0 and, thus, leads to a larger DEISB of 006 MeV for ArgV18 The discussion of the previous paragraph raises the question, what change in DEISB is obtained if the correct ISB in partial waves other than 1 S0 is applied Žinstead of just extrapolating what fits the np singlet scattering length Unfortunately, we do not have any reliable empirical information on the ISB of the nuclear force in partial waves with L) 0 Žwhere L denotes the total orbital angular momentum of the two-nucleon system, at this time However, there are microscopic models that predict ISB If those models predict the empirically known ISB of the 1 S0 scattering about correctly, then one may imply that the predictions for higher partial waves are also reasonable One such calculation is pubw13 x, which is based upon the Bonn lished in Ref full model for the NN interaction w14 x A refined version of the CDBonn potential w15x has been constructed Žthat has become known as the CDBonn99 potential w16x which takes the ISB effects as predicted in Ref w13x in partial waves with L)0 into account and, in addition, includes the ISB effects from irreducible pg exchange as derived in Ref w17 x We have used this CDBonn99 model w15x Ž in short: CDBo99 in our calculations as well and y1 obtain DE s 0370 MeV at k s 136 fm Ž ISB F see Table 1 This is to be compared to 039 MeV as obtained from the ordinary CDBonn wx 5 Thus, the refined treatment of ISB in partial waves higher than 1 S0 increases DEISB by 004 MeV A comparison with the Argonne V18 result discussed above implies that ArgV18 contains a reasonable estimate for the ISB effects from higher partial waves ArgV18 is overestimating the ISB effect from higher partial waves by roughly 50% The reason for this is presumably that the ISB term in the ArgV18 potential Žthat describes ISB beyond OPE in a phenomenological way is too long-ranged The effect of ISB is also reflected in the saturation points displayed in the left part of Fig 1 While the filled symbols refer to the saturation points predicted from the calculation with inclusion of the ISB terms, the open symbols are obtained for the corresponding E np The effect of the ISB terms on the calculated symmetry energy Ždifference between binding energy per nucleon of neutron matter and

263 H Muther et alrphysics Letters B 445 ( 1999) nuclear matter at the same density is negligible Žsee Table 1 The NN interactions CDBonn and ArgV18 have been adjusted to reproduce the differences in the scattering length and effective range parameters for pp and nn scattering Therefore these two potentials as well as the NN interaction CDBonn99 also yield significant differences in predicting the single-particle potentials for protons U Ž k and neutrons U Ž k p n as a function of the momentum k in symmetric nuclear matter It turns out that the momentum dependence of the difference DUCSBŽ k supž k yunž k, is weak Therefore we display in Fig this difference just calculated at the Fermi momentum ksk F for various densities This difference is positive, which implies that the pp interaction is less attractive than the corresponding nn interaction as one can already deduce from the difference in the 1 S0 scattering lengths Žappsy173 fm compared to anns y188 fm w18 x At the empirical saturation density, the ArgV18 and CDBonn99 predict a value for DUCSB of 031 MeV and 08 MeV, respectively, while the CD- Bonn potential yields 016 MeV These numbers Fig Difference between the single-particle potentials for prowas defined in Eq Žx tons and neutrons DU CSB calculated at ks k for various densities and three different NN potentials F should be compared with a value of 0 03 MeV, which is typical for the Nolen Schiffer anomaly, ie the energy difference between the binding energies of mirror nuclei, which is beyond the effects of the electromagnetic interaction wx It seems that the BHF predictions based on the three potentials yield the right order of magnitude At a first glance, it is disturbing that some predictions differ by almost a factor of two However, this can be explained The reason is similar to what we discussed above in conjunction with DE ISB In all models, the construction of the nn potential starts from the pp potential To fit the slightly more attractive 1 S0 nn scattering length, the attraction in the potential is slightly enhanced In the case of the Argonne V18 potential, this implies more attraction for all Ž S s 0, T s 1 nn partial waves The CD- Bonn99 potential, which contains the microscopically determined CSB effects, predicts different interactions in all Ts1 partial waves In the CDBonn, strictly only the 1 S state is made slightly more 0 attractive to match a nn This explains the larger value for DUCSB obtained for ArgV18 and CDBonn99 Obviously, CSB in partial waves higher than 1 S0 is of considerable influence on DU CSB Therefore our conclusion is the following: to reproduce the Nolen Schiffer anomaly, it is insufficient to take just the CSB in the 1 S0 scattering length into account; a distinguished knowledge of CSB in partial waves beyond 1 S0 is crucial In summary, we have calculated ISB effects in nuclear matter and find that they are generally small, but important in some cases For the binding energy per nucleon in symmetric nuclear matter, the breaking of isospin-symmetry is a very small effect For all the potentials analyzed in the paper, ISB produces a small loss of binding energy Žas compared to calculations that use the np potential throughout, which is mainly caused by the ISB in the 1 S0 partial wave The effects in the symmetry energy are essentially negligible On the other hand, if one wants to explain the Nolen Schiffer anomaly by calculating the difference between the single particle potential for protons and neutrons, which is a measure for charge symmetry breaking, one finds that, in order to have quantitative agreement, it is necessary to include CSB in partial waves beyond 1 S A natural 0 way to incorporate CSB in higher partial waves,

264 64 H Muther et alrphysics Letters B 445 ( 1999) based on microscopic calculations, is provided by the recent update of the CDBonn potential that has become known as the CDBonn99 Acknowledgements This work was supported in part by the SFB 38 of the Deutsche Forschungsgemeinschafti, the US National Science Foundation under Grant No PHY , the DGICYT Ž Spain Grant PB and the Program GRQ96-43 from Generalitat de Catalunya References wx 1 GA Miller, WHT van Oers, in: Symmetries and Fundamental Interactions in Nuclei, WC Haxton, EM Henley Ž Eds, World Scientific, Singapore, 1995, p 17 wx GA Miller, MK Nefkens, I Slaus, Phys Rep 194 Ž wx 3 VGJ Stoks, RAM Klomp, CPF Terheggen, JJ de Swart, Phys Rev C 49 Ž wx 4 RB Wiringa, VGJ Stoks, R Schiavilla, Phys Rev C 51 Ž wx 5 R Machleidt, F Sammarruca, Y Song, Phys Rev C 53 Ž 1996 R1483 wx 6 VGJ Stoks, RAM Klomp, MCM Rentmeester, JJ de Swart, Phys Rev C 48 Ž wx 7 JA Nolen, JP Schiffer, Annu Rev Nucl Sci 19 Ž wx 8 HQ Song, M Baldo, G Giansiracusa, U Lombardo, Phys Lett B 411 Ž wx 9 R Reid, Ann Phys Ž NY 50 Ž w10x L Engvik, M Hjorth-Jensen, R Machleidt, H Muther, A Polls, Nucl Phys A 67 Ž w11x A Polls, H Muther, R Machleidt, M Hjorth-Jensen, Phys Lett B 43 Ž w1x F Coester, S Cohen, BD Day, CM Vincent, Phys Rev C 1 Ž w13x GQ Li, R Machleidt, Charge-Dependence of the Nucleon- Nucleon Interaction, Phys Rev C, in press; nuclthr w14x R Machleidt, K Holinde, C Elster, Phys Rep 149 Ž w15x R Machleidt, to be published w16x D Huber, JL Friar, Phys Rev C 58 Ž w17x U van Kolck, MCM Rentmeester, JL Friar, T Goldman, JJ de Swart, Phys Rev Lett 80 Ž w18x GQ Li, R Machleidt, Phys Rev C 58 Ž

265 7 January 1999 Physics Letters B Low-mass dileptons and dropping rho meson mass EL Bratkovskaya a, CM Ko b a Institute fur Theoretiche Physik, UniÕersitat Giessen, 3539 Giessen, Germany b Cyclotron Institute and Physics Department, Texas A&M UniÕersity, College Station, TX 77843, USA Received 1 September 1998; revised 4 November 1998 Editor: J-P Blaizot Abstract Using the transport model, we have studied dilepton production from heavy-ion collisions at Bevalac energies It is found that the enhanced production of low-mass dileptons observed in the experiment by the DLS collaboration cannot be explained by the dropping of hadron masses, in particular the r-meson mass, in dense matter q 1999 Elsevier Science BV All rights reserved PACS: 575-q; 410Lx; 140Yx One of the most exciting physics one expects to learn from relativistic heavy-ion collisions is the possibility to create nuclear matter at densities larger than that in the center of normal nuclei and possibly comparable to that in the interior of neutron stars w1 3 x This thus offers the opportunity to study if the nuclear equation of state becomes softened at high densities as required from studies of supernova explosions The high density nuclear matter created in these collisions also makes it possible to study how spontaneously broken chiral symmetry, which is characterized by a large value of the quark condensate in the vacuum, is partially restored when the temperature and density of matter becomes high, as predicted by theoretical studies wx 4 Unfortunately, a direct relation of the quark condensate to physical observables has not been rigorously established wx 5 One conjecture has been suggested by Brown and Rho wx 6 that masses of non-strange hadrons such as nucleon, r, and v, are reduced in the nuclear medium and are proportional to the in-medium quark condensate Dropping hadron in-medium masses have also been found in some theoretical models, such as the QCD sum rules wx 7, the quark-meson coupling model wx 8, and the hadronic model including vacuum polarwx 9 Such scaled in-medium hadron ization effects masses have been shown to lead to significantly improved understanding of many nuclear phenomena, such as the enhanced axial charge transitions seen in heavy nuclei w10x and the quenching in the longitudinal response function of nuclei w11 x A more direct evidence for a dropping r-meson mass seems to be provided by the enhanced production of dileptons with invariant mass around MeV above known sources of dileptons in the CERES experiment of heavy-ion collisions at CERN SPS w1 x In theoretical studies w13 17x based on various models ranging from a simple thermal model to sophisticated transport models, dilepton production has been investigated by including contributions not only from the Dalitz decay of mesons and direct dilepton decay of vector mesons but also from r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

266 66 ( ) EL BratkoÕskaya, CM KorPhysics Letters B pion pion annihilation The latter is practically absent in proton-nucleus collisions and is found to account for only about half of the observed enhancement in heavy-ion collisions Including contributions from other processes such as the a decay w18x 1 and pion rho scattering w19x does not help much On the other hand, allowing for the reduction of the r meson in-medium mass in the relativistic transport model w13,14,17 x, it has been shown that it can indeed lead to a further enhancement of low-mass dileptons as observed in the experimental data On the other hand, it has also been claimed w0x that if the r meson which in free space has a broad decay width to two pions melts in the medium due to additional decay channels such as the resonancew1 3 x, would also significantly increase hole states the yield of low-mass dileptons However, in the dynamical studies of Ref w4x the r spectral function, which consists of effects due to both medium modifications of pions and the baryon-resonancenucleon-hole excitations, is introduced through pion pion annihilation to dileptons via the vector meson dominance Because of the large pion to nucleon ratio Ž about 5 in heavy-ion collisions at SPS energies, the effect of resonance-hole contributions to dilepton production is thus overestimated in these studies Dileptons have also been measured in heavy-ion collisions at the Bevalac by the DLS collaboration w5,6x at incident energies that are two orders-ofmagnitude lower than that at SPS Although the first published data w5x based on a limited data set are consistent with the results from transport model calw7,8x that include pn bremsstrahlung, culations p 0, h and D Dalitz decay and pion pion annihilaw6 x, including the full data tion, a recent analysis set, shows a considerable increase in the cross section, which is now more than a factor of five above these theoretical predictions even after including also contributions from the decay of r and v that are produced directly from nucleon-nucleon and pion nucleon scattering in the early reaction phase w9,30 x With an in-medium rho spectral function as that used in Ref w4x for dilepton production from heavy-ion collisions at SPS energies, dileptons from the decay of both directly produced r s and pion pion annihilation have been considered, and a factor of two enhancement has been obtained compared to the case of using a free r-spectral function Since the pion to nucleon ratio in heavy-ion collisions at Bevalac energies is much smaller than one Ž about 0, contrary to that in heavy-ion collisions at SPS energies, the contribution of pion pion annihilation instead of baryon resonance Dalitz decay is overestimated in the spectral function analysis in Ref w9 x As dropping hadron masses in dense matter have been seen to account for the observed enhancement of low-mass dileptons in heavy-ion collisions at SPS energies, it is of interest to see if this can also explain the enhanced production of dileptons at Bevalac energies We shall base our study on the Hadron String Dynamics Ž HSD model w31 x, which describes the collision dynamics by propagating nucleons in a mean-field potential given by the attractive scalar and repulsive vector potentials Free nucleon-nucleon cross sections are used in treating their collisions, which include both elastic and inelastic ones The latter leads to the production of both meson resonances such as r and v and baryon resonances such as DŽ 13, N ) Ž 1440, and N ) Ž 1535 The decay of baryon resonances then leads to the production of pions and h s Scattering of pions with nucleons is also included, leading to the production of meson and baryon resonances as well This model has been shown to give very good descriptions of the measured yield of pions and h s as well as the rapidity and transverse momentum distributions of hadrons w3 x To study dilepton production, we include contributions from pn and p N bremsstrahlung; Dalitz 0 decay of p, h, v, D, and N ) Ž 1440 ; and direct decays of r and v All contributions will be treated in the standard way as in Refs w7 9,33 x However, since vector meson production in nucleusnucleus collisions at Bevalac energies, which are below the production subthreshold in nucleonnucleon collisions, is dominated by the p N channels w34 x, we shall pay a special attention to this contribution In particular, it has been recently shown that the rho meson couples strongly to N ) Ž 150 w3,35 x, we shall thus include additionally this effect in rho meson production from p N scattering, as it has been overlooked in previous work Specifically, the isospin-averaged production cross section of a neutral r meson from p N scattering

267 ( ) EL BratkoÕskaya, CM KorPhysics Letters B ) through N Ž 150 at a center-of-mass energy ' s is dsp N r N Ž s, M dm ) ' N p N N p Ž sym ) N qsgtot Ž ' s ) ' N r N Ž s, M 4p sg s ss p N ) k dg = dm with 1 Ž J ) N q1 Sp N s 3 Ž J q1ž J q1 p Ž I ) N q1 =, Ž I q1ž I q1 p N N, Ž 1 where J ) s3r, J s0, J s1r, I ) N p N N s1r, Ip s 1, and I s1r In Eq Ž N 1, kp is the pion threemomentum in the center of mass of pion and nucleon ) The partial decay width of a N Ž 150 of mass ' s to a nucleon and a r meson of mass M is given in Ref w3 x, ie, dg ) ' N r N Ž s, M dm ž / ' ) N Nr N r r mr p s f 1 m s M k v qm Ž r =AŽ M F k, where f ) N Nr; 7 is the coupling constant, kr denotes the three-momentum of the r meson in the rest ) frame of N 150, and vr sm qkr is its en- ergy The r spectral function is denoted by AM and has a Breit-Wigner form, 1 mr Gtot r Ž M AŽ M s Ž 3 r p M ym q m G Ž M r r tot The form factor Fk Ž r is taken to have a monopole form L FŽ kr s, 4 L qk r with Ls15 GeV The r meson total width G r Ž M tot is given by the sum of the width due to decay to two pions and that due to absorption by nucleons and elastic scattering, ie, 3 mr kp Ž M r G Ž M sg Ž m M k Ž m ž /ž / tot r pp r p r qsrnž s, M Õgr N, Ž 5 where G Ž m r pp r ; 015 GeV and kp is the pion momentum in the rest frame of the r meson The second term corresponds to the collisional broadening width and is given by the product of r-nucleon total cross section s Ž s, M rn, the r meson velocity Õ in the rest frame of the nucleon current, the associated Lorentz factor g, and the nucleon density r N Based on the resonance model, the r-nucleon total cross section has been determined in Ref w36 x, and the result can be parameterized as y1 srn Ž s, M s6q15kr wmb x, Ž 6 with k wgevrcx r the rho meson three-momentum in the rest frame of the nucleon current At normal nuclear matter density, this gives an average rho meson collisional width of about 75 MeV whereas in central 40 Caq 40 Ca reactions at 1 APGeV the average collisional width amounts to, 140 MeV The partial decay width of a N ) Ž 150 to p N is w3x Ž ' k ' p Ž s k Ž m ) / p N lq1 G ) N p N s sg, 7 0ž with k Ž m p the pion three-momentum in the rest frame of N ) Ž 150, ls, and G0 s0095 GeV The total decay width of the N ) Ž 150 is given by the sum of the partial decay widths to pion and r, ie, G N ) sg ) qg ), where G ) tot N p N N r N N r N is obtained from Ž ' s, M ' s ym dg ) N N r N G ) ' N r N Ž s sh dm m p dm Ž 8 The dilepton production cross section from the direct decay of the rho meson produced from p N scattering through N ) Ž 150 is then ds q y p N e e NŽ s, M dm ds s, M G q y p N r N r e e Ž M s Ž 9 r dm G Ž M tot

268 68 ( ) EL BratkoÕskaya, CM KorPhysics Letters B e q e In the above, G y Ž M r is the dilepton decay width of a neutral rho meson of mass M From the vector dominance model, it is given by m 4 q y e e y6 r r 3 G Ž M s88=10 Ž 10 M For the case of free hadron masses, the dilepton invariant mass spectrum from 40 Caq 40 Ca collisions at 1 APGeV and after correcting for the experimental acceptance filter Ž version 41 is shown in the upper part of Fig 1 It is seen that with increasing dilepton invariant mass, the dominating contribution shifts from p 0 Dalitz decay to D Dalitz decay, to h Dalitz decay, and finally to direct r decay, as in Refs w9,30 x In particular, the contribution from the direct decay of rho mesons produced from p N scattering through the N ) Ž 150 Ž dot-dashed curve is most important in the mass region 035-M-075 GeVrc and exceeds that from other r production channels Ž dashed curve pp annihilation, pion baryon Ž without N ) Ž 150 and baryon-baryon collisions This is different from heavy-ion collisions at CERN SPS, where dilepton production from the pp r channel is more important than that from Fig 1 The dilepton invariant mass spectrum from Caq Ca collisions at 1 APGeV with free Ž upper part and in-medium Ž lower part hadron masses in comparison to the DLS data w6 x

269 ( ) EL BratkoÕskaya, CM KorPhysics Letters B the p N rn channel as a result of the large pion to nucleon ratio in these collisions Compared with the experimental data the theoretical results for the total dilepton spectrum are, however, about a factor of three lower in the invariant mass region 0-M- 05 GeVrc and practically the same as in the spectral function approach of Ref w9 x We note that contributions of other baryon resonances, eg, the N ) Ž 1700 which also couples strongly to the rho meson w37 x, to low-mass dileptons are negligibly small as a result of their large masses andror relatively weak coupling to the rho meson and nucleon To see how the results are modified by medium effects, we introduce the rhoromega meson inw13,14x but keep the medium masses as in Ref baryon masses unchanged, m ) rrv;mrrv Ž 1y018 rbrr 0 Ž 11 According to Ref w35 x, part of the decrease of the r meson mass in nuclear medium can be accounted for by the attractive interaction due to N ) Ž 150 -particle-nucleon-hole polarization The contribution to dilepton production from p N scattering is then computed from Eqs ŽŽ 1 10 using in-medium rho meson mass Specifically, the rho meson mass in Eqs Ž 3 and Ž 10 are replaced by the in-medium one Also, k in Eqs Ž and Ž r 6 are evaluated with the in-medium mass The reduction of omega mass enhances its production but does not significantly increase the yield of low mass dileptons due to the small partial decay width of omega meson to dileptons compared to that of rho meson Omega mesons thus contribute to dilepton production mainly at freeze out when they have regained their free mass Furthermore, omegas can be absorbed by nucleons From omega photoproduction data, analyses based on the Vector-Dominance model or the additive quark model give an omega-nucleon total cross section of 5 30 mb at omega momenta above 1 GeVrc, similar to that for the rho-nucleon total cross section obtained in the same model w38 x Also, for omega at finite momenta, its absorption cross section through the reaction v N p N can be obtained from the inverse reaction p N v N, which has an empirical value similar to that for p N rn Žsee eg w39 x On the other hand, for omega-nucleon scattering near threshold, Friman w40x has found from the p N v N data that the imaginary part of the omega-nucleon scattering length is about a factor of seven smaller than that for the rho meson, giving thus a much smaller omega absorption cross section than for the rho meson The latter result is, however, expected to change appreciably if one also takes into account final states that consists of more than one pion w41 x Since the omega contribution to low-mass dileptons is unimportant in heavy ion collisions at Bevalac energies, we have thus adopted the simple assumption that its interaction cross section with a nucleon is similar to that for a rho meson, ie, Eq Ž 6 We then find that the contribution of omega mesons to dileptons are not much affected by the change of their masses and thus remains unimportant The dilepton invariant mass spectrum from the same reaction for the case of dropping rho meson mass is shown in the lower part of Fig 1 We find that with dropping rho meson mass low-mass dilepton production from p N scattering through N ) Ž 150 is substantially reduced due to a significant increase of its width Although dileptons from other r production channels are enhanced, they are not sufficient to compensate for the reduction due to the broadening of NŽ 150 Including also the contributions from the Dalitz decays of p 0, D, h, and v as well as the direct decay of v, which are not much affected by the reduction of hadron masses, the total theoretical dilepton spectrum Ž solid curve remains about a factor of three below the experimental data for 0FMF05 GeV Similar results are obtained if we also allow the nucleon and NŽ 150 masses to decrease according to the constituent quark model, ie, a factor 3r reduction relative to that for the r meson In conclusion, we have shown using the HSD transport model that although dilepton production from pion nucleon scattering through the N ) Ž 150 is important in heavy-ion collisions at Bevalac energies, as the nucleon is the most abundant particles, including its contribution cannot explain the observed enhancement by the DLS collaboration Allowing the reduction of rho meson mass in dense matter enhances dilepton production from other rho production channels, but it reduces significantly that from p N N ) Ž 150 rn as a result of the broadening of N ) Ž 150 width, leading to a total

270 70 ( ) EL BratkoÕskaya, CM KorPhysics Letters B dilepton spectrum similar to that without dropping hadron masses Thus, unlike the enhanced low mass dileptons observed in heavy ion collisions at SPS energies, which is dominated by pion pion annihilation and can be explained by the decrease of hadron in-medium masses, the DLS result can not be explained by dropping hadron masses and remains a puzzle Acknowledgements The authors acknowledge valuable discussions with W Cassing, C Gale, M Effenberger, U Mosel, W Peters, M Post, and A Sibirtsev throughout this study The work of ELB was supported by BMBF, GSI Darmstadt, while that of CMK was supported by the National Science Foundation under Grant No PHY and PHY , the Welch Foundation under Grant No A-1358, the Texas Advanced Research Program, and the Alexander Humboldt Foundation References wx 1 H Stocker, W Greiner, Phys Rep 137 Ž wx W Cassing, V Metag, U Mosel, K Niita, Phys Rep 188 Ž wx 3 CM Ko, GQ Li, J Phys B Ž wx 4 P Gerber, H Leutwyler, Nucl Phys B31 Ž ; TD Cohen, RJ Furnstahl, K Griegel, Phys Rev C 45 Ž ; M Lutz, S Klimt, W Weise, Nucl Phys A45 Ž ; GQ Li, CM Ko, Phys Lett B338 Ž wx 5 CM Ko, V Koch, GQ Li, Ann Rev Nucl Part Sci 47 Ž wx 6 GE Brown, M Rho, Phys Rev Lett 66 Ž wx 7 T Hatsuda, SH Lee, Phys Rev C 46 Ž 199 R34; M Asakawa, CM Ko, Phys Rev C 48 Ž 1993 R56; Nucl Phys A560 Ž wx 8 K Saito, AW Thomas, Phys Lett B37 Ž ; Phys Rev C 51 Ž wx 9 HC Jean, J Piekarewicz, AG Williams, Phys Rev C 49 Ž ; H Shiomi, T Hatsuda, Phys Lett B334 Ž ; CS Song, PW Xia, CM Ko, Phys Rev C 5 Ž w10x EK Warburton, IS Towner, Phys Lett B94 Ž w11x EJ Stephenson, Phys Rev Lett 78 Ž w1x G Agakichiev et al, Phys Rev Lett 75 Ž ; Nucl Phys A630 Ž c; A Drees, Nucl Phys A610 Ž c; A630 Ž w13x GQ Li, CM Ko, GE Brown, Phys Rev Lett 75 Ž ; Nucl Phys A606 Ž ; GQ Li, CM Ko, GE Brown, H Sorge, ibid A610 Ž c; A611 Ž w14x W Cassing, W Ehehalt, CM Ko, Phys Lett B363 Ž ; W Cassing, W Ehehalt, I Kralik, Phys Lett B377 Ž w15x MK Srivastava, B Sinha, C Gale, Phys Rev C 53 Ž 1996 R567 w16x V Koch, C Song, Phys Rev C 54 Ž w17x EL Bratkovskaya, W Cassing, Nucl Phys A619 Ž w18x K Haglin, Phys Rev C 53 Ž 1996 R606 w19x R Baier, M Dirks, R Redlich, Phys Rev D 55 Ž ; J Murray, W Bauer, K Haglin, Phys Rev C 57 Ž w0x R Rapp, G Chanfray, J Wambach, Phys Rev Lett 76 Ž w1x R Rapp, G Chanfray, J Wambach, Nucl Phys A617 Ž wx F Klingl, N Kaiser, W Weise, Nucl Phys A64 Ž w3x W Peters, M Post, H Lenske, S Leupold, U Mosel, Nucl Phys A63 Ž w4x W Cassing, EL Bratkovskaya, R Rapp, J Wambach, Phys Rev C 57 Ž w5x G Roche et al, Phys Rev Lett 61 Ž ; C Naudet et al, ibid 6 Ž w6x RJ Porter, Phys Rev Lett 79 Ž w7x L Xiong, ZG Wu, CM Ko, JQ Wu, Nucl Phys A51 Ž w8x Gy Wolf, G Batko, TS Biro, W Cassing, U Mosel, Nucl Phys A517 Ž ; Gy Wolf, W Cassing, U Mosel, ibid A55 Ž w9x EL Bratkovskaya, W Cassing, R Rapp, J Wambach, Nucl Phys A634 Ž w30x C Ernst, SA Bass, M Belkacem, H Stocker, W Greiner, Phys Rev C 58 Ž w31x W Ehehalt, W Cassing, Nucl Phys A60 Ž w3x W Cassing, EL Bratkovskaya, Phys Rep 308 Ž w33x GQ Li, CM Ko, Nucl Phys A583 Ž w34x EL Bratkovskaya, W Cassing, U Mosel, Phys Lett B44 Ž w35x GE Brown, GQ Li, R Rapp, M Rho, J Wambach, nucl-thr w36x LA Kondratyuk, A Sibirtsev, W Cassing, YeS Golubeva, M Effenberger, Phys Rev C 58 Ž w37x B Friman, H Pirner, Nucl Phys A617 Ž w38x TH Bauer, RD Spital, DR Yennie, FM Pipkin, Rev Mod Phys 50 Ž w39x A Sibirtsev, Nucl Phys A604 Ž w40x B Friman, nucl-thr w41x W Cassing, A Sibirtsev, private communications

271 7 January 1999 Physics Letters B Sigma signal for hybrid baryon decay Leonard S Kisslinger a,b, Zhenping Li c a Department of Physics, Carnegie Mellon UniÕersity, Pittsburgh, PA 1513, USA b Los Alamos National Laboratory, Los Alamos, NM 87545, USA c Physics Department, Peking UniÕersity, Beijing , PR China Received 6 June 1998; revised 4 September 1998 Editor: J-P Blaizot Abstract We develop an ansatz of the sigma enhancement of the Is0, Ls0 p p scattering amplitude as arising from a low-energy glueball pole Using this picture we estimate the p 0 p 0 to p 0 branching ratio for the decays of the Roper resonance, which we previously found to be a hybrid in our QCD sum rule calculation q 1999 Elsevier Science BV All rights reserved PACS: 139Mk; 1155Hx; 140Gk The existence of hadrons composed of glue Ž glueballs and mesons and baryons with valence glue Ž hybrids is one of the most interesting and important problems in Quantum Chromodynamics Ž QCD The purpose of this note is to show that from recent theoretical and experimental investigations of scalar glueballs and mesons there might be found a signal for hybrid hadrons In particular, pionic decays of the Roper resonance, which we have predicted to be a hybrid, are calculated and an experimental test is suggested In an investigation of scalar glueballs one of us has has found wx 1 a scalar glueball solution with a low mass in the range MeV, the system of interest in the present work We also found that solutions for mixed scalar glueballs and mesons are at a much higher energy, consistent with the f Ž for the mainly-glueball solution and with the f0 Ž 1370 and a Ž for the mainly-meson solution This work uses the method of QCD sum rules, for which there is a long history of work on scalar mesons and glueballs In early work using sum rules it was predicted wx that the mass of the lightest scalar meson is that of the f Ž 980 0, which is now generally believed to be a meson meson resonance rather than a qq meson state If one considers the mixed scalar meson glueball current, making use of the low energy theorem for the coupling of the scalar meson and gluonic currents wx 4, one finds the solutions wx 1 at about 1400 MeV, and 1500 MeV for the mixed mesonrglueball scalars hadrons See Ref wx 3 for a recent review of the use of QCD sum rules for glueballs and mixed meson glueball hadrons, as well as estimates of pionic and other decays The main point of the present work is 1 that the low-mass scalar glueball can be understood as a coupled-channel glueballrsigma in the region below 1 GeV, with the sigma being a two-pion phenomenon; and that from the analysis of glueball decays and pi pi phase shifts one can extract the gluon-sigma coupling strength to predict two-pi decays of hybrids This enables us to estimate the r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

272 7 ( ) LS Kisslinger, Z LirPhysics Letters B two-pi branching ratio for experimental tests of these conjectures Recently there has been a great deal of experiw5,6 x One of the mental activity in glueball searches very interesting and we feel most important observations is the enhancement of the s mode in the decays of the glueball candidates, such as the dominance of the ss intermediate states in the decays of the glueball candidates f Ž p and f Ž p w6,7 x The recent analysis by the BES collabowx 7 also suggest that the tensor glueball candi- ration dates j Ž 30 also has a large 4p branching ratio that is dominated by the s f Ž 170 intermediate state By the s we mean in the present paper the enhancement in the p p Is0, Ls0 amplitude, for which a pole has been found in the analysis of Zou and Bugg wx 8 This broad pole occurs near the light scalar glueball found in our work wx 1, and the work of others using QCD sum rule methods wx 9 The sigma is predicted in the NJL model w10,11x Žsee Ref w1x for a review We propose that the physical origin of the s enhancement is the light scalar gluonic mode, and that this could be a good signal for the study of gluonic structure of hadrons This also provides a possible explanation of the absence of light, narrow glueball in lattice gauge calculations, since the strong coupling to the p p channel, which results in about a 700 MeV width must be included for a physical solution Note that the mixed glueball meson states, predicted to be at higher energies in the region of scalar mesons w1,3 x, are quite far from this coupled-channel sigma and would not be involved In the present work we use this picture of a scalar glueball mode coupled to the s channel for the decay of hybrid baryons Recently we carried out a QCD sum rule calculation w13x to find the hybrid baryon with quantum numbers J P s1r q, Is1r, the same as the nucleon and the P Ž , the Roper This consisted as an improvement over earlier work using the QCD sum rule methods w14,15 x Since that work we have considered currents of the form c1hn qch H, where hn and hh are the nucleon and hybrid currents, defined below, and have shown that the nucleon has a very small hybrid component w16 x, which indicates that using a pure hybrid current for the Roper is reasonable Because of convenience in treating renormalization we use a different current operator for the P q J s1r,is1r hybrid state than in Ref w13 x It is h Ž x se abc u a Ž x Cg m u b Ž x is ab g m g 5 G d Ž x H ab c d T d x, 1 = Ž where už x and dž x are operators of the u and d quarks, G d Ž x ms is a gluon field strength, a, b,c s 1,,3 are the color indices, C is the charge conjugation matrix, and T d sl d r is the generator of the SUŽ 3 color group For the calculation of the pion and sigma decays we use the external field two-point method w17 x, which has been used in many calculations of hadronic coupling constants, including the strong and weak pion-nucleon coupling constant w18 x Using a two-point function in an external JG field to represent the coupling of the current operator to the nucleon and the Roper, with the current of Eq Ž 1 leads to two independent invariant functions; st iqx s t P Ž q sihe 0 TŽ hhž x hnž 0 G 0 ² < < : sp G q qˆ st qp G q d st, 1 ˆ m N st where s,t are Dirac indices, q sž g st q m, and h is the nucleon current The details of the solution for the mass of the hybrid are given in Ref w13 x The coefficients of the operator product expansion Ž OPE are calculated to dimension 6 In addition to the leading perturbative contributions we include the two quark condensate terms, which are the largest nonperturbative terms For the pion decay the current Jp siqg 5q and the P correlator are used After the Borel transform we find the following sum rule lnlh p ymh r M P Ž M sgp Ž MH e N H Ž M qm ym e N Ž p ym r M 4 N 11 4a 8 3 s M E y M E 0, Ž where l N, lh are the QCD sum rule structure constants, and Gp is the pi Roper nucleon cou- pling The susceptibility term is small and has been

273 ( ) LS Kisslinger, Z LirPhysics Letters B neglected For the sigma decay we use the current J s, with the glue sigma coupling, g s, given by ² G a J G ab : sg ² G a G ab : Ž 4 ab s a s ab a The glue sigma coupling constant can be extracted from the p p T-matrix of Ref wx 8 After subtracting the higher resonances they find a fit to the p p amplitude of a Breit-Wigner form: ' ppžs s G s Ž s r T s qt Ž background sym qi' s sgs Ž s r Ž 5 The analysis of Ref wx 8 finds the sigma pole at 370 y 356i Using our ansatz we determine the sigma- gluonic coupling constant, gs from the width of this pole, giving g s,700 MeV Ž 6 We obtain for the P l l N H s H N P M sg = correlator Ž M ym e ye Ž p ym r M ym r M 4 N H ž / 7 a 8 3 sgs M E y M E 0 Ž In Eq Ž 7 the parameter Gs is the H N s coupling constant which appears in the numerator of the double-pole term in the phenomenological dispersion relationship From Gs and Gp the p to s branching ratio can be determined Using the standard values of the parameters, we find for the srp branching ratio G s,10%pr, Ž 8 G p p where R p is the ratio of phase space factors This is consistent with the present experimental limits w19 x Indeed, the data suggest that the width of the Roper resonance P Ž decaying into s N final states is generally an order of magnitude larger than those of other resonances This could be another signature of a hybrid Roper resonance in addition to its electrow0 x Thus, further ex- magnetic transition properties 0 0 perimental investigations of P 1440 pp N wto 11 select the I s 0 channel and eliminate D and r backgroundsx over an energy range to map out the s would be a very important channel to study the structure of the Roper resonance In general, we suggest that the measurement of s decays of baryons and mesons is a signal for the gluonic content of hadrons Acknowledgements We would like to thank Mikkel Johnson for helpful discussions This work was supported by National Science Foundation grants PHY and INT , and by the Department of Energy References wx 1 LS Kisslinger, J Gardner, C Vanderstraeten, Phys Lett B 410 Ž wx LJ Reinders, S Yazaki, HR Rubinstein, Nucl Phys B 196 Ž wx 3 S Narison, Nucl Phys B 509 Ž wx 4 VA Novikov, MA Shifman, AI Vainstein, VI Zakharov, Nucl Phys B 165 Ž ; B wx 5 Crystal Barrel Collaboration, Phys Lett B 355 Ž wx 6 DV Bugg, Phys Lett B 353 Ž wx 7 Zhu Yucan, BES JrC group, in: TW Donnelly Ž Ed, Proceedings of the 6th Conference On The Intersection of Particle and Nuclear Physics, Big Sky, Montana, 1997, p 476 wx 8 BS Zou, DV Bugg, Phys Rev D 50 Ž wx 9 CA Dominguez, N Paver, Zeit Phys C 31 Ž ; J Bordes, V Gimenez, JA Penarrocha, Phys Lett B 3 Ž ; J Liu, D Liu, J Phys G: Nucl Part Phys 19 Ž w10x Y Nambu, G Jona-Lasinio, Phys Rev 1 Ž w11x V Elias, MD Scadron, Phys Rev Lett 53 Ž w1x T Kunihiro, Prog Th Phys Supp 10 Ž w13x LS Kisslinger, Z Li, Phys Rev D 51 Ž 1995 R5986 w14x AP Martynenko, Sov J Nucl Phys 54 Ž w15x VM Braun, P Gornicki, L Mankiewicz, A Schafer, Phys Lett B 30 Ž w16x LS Kisslinger, Gluonic Hadrons, to be published in the Proceedings of the International Conference on Quark Lepton Nuclear Physics, Osaka, Japan, 1997 w17x BL Ioffe, AV Smilga, Nucl Phys B 3 Ž w18x EM Henley, W-YP Hwang, LS Kisslinger, Phys Lett B 367 Ž w19x Review of Particle Properties, Phys Rev D 45 Ž 199 w0x Zhenping Li, V Burkert and Zhujun Li, Phys Rev D 46 Ž ; Zhenping Li, Phys Rev D 44 Ž

274 7 January 1999 Physics Letters B Pair creation of black holes in anti-de Sitter space background Zhong Chao Wu 1 Dept of Physics, Beijing Normal UniÕersity, Beijing , China Received 19 October 1998; revised 0 November 1998 Editor: M Cvetič Abstract In the absence of a general no-boundary proposal for open creation, the complex constrained instanton is used as the seed for the open pair creations of black holes in the Kerr-Newman-anti-de Sitter family The relative creation probability of the chargeless and nonrotating black hole pair is the exponential of the negative of the entropy, and that of the charged and Ž or rotating black hole pair is the exponential of the negative of one quarter of the sum of the outer and inner black hole horizon areas q 1999 Elsevier Science BV All rights reserved PACS: 9880Hw; 9880Bp; 0460Kz; 0470Dy Keywords: Quantum cosmology; Constrained gravitational instanton; Black hole creation In the No-Boundary Universe, the wave function of a closed universe is defined as a path integral over all compact 4-metrics with matter fields wx 1 The dominant contribution to the path integral is from the stationary action solution At the WKB level, the wave function can be written as Cfe yi, Ž 1 where IsIrqiIi is the complex action of the solu- tion The Euclidean action is 1 1 Isy H Ž Ry LqLm y E K, 16p 8p M where R is the scalar curvature of the spacetime M, L is the cosmological constant, K is the trace of the 1 wu@axp3g9icrait E M second form of the boundary E M, and Lm is the Lagrangian of the matter content The imaginary part Ii and real part Ir of the action represent the Lorentzian and Euclidean evolutions in real time and imaginary time, respectively When their orbits are intertwined, they are mutually perpendicular in the configuration space with the supermetric The probability of a Lorentzian orbit remains constant during the evolution One can identify the probability, not only as the probability of the universe created, but also as the probabilities for other Lorentzian universes obtained through an anawx lytic continuation from it An instanton is defined as a stationary action orbit and satisfies the Einstein equation everywhere It is the seed for the creation of the universe However, very few regular instantons exist The framework of the No-Boundary Universe is much wider than that of the instanton theory Therefore, in order not to r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

275 ( ) ZC WurPhysics Letters B exclude many interesting phenomena from the study, one has to appeal to the concept of constrained instantons wx 3 Constrained instantons are the orbits with an action which is stationary under some restriction The restriction can be imposed on a spacelike 3-surface of the created Lorentzian universe The restriction is that the 3-metric and matter content are given at the 3-surface The relative creation probability from the instanton is the exponential of the negative of the real part of the instanton action The usual prescription for finding a constrained instanton is to obtain a complex solution to the Einstein equation and other field equations in the complex domain of spacetime coordinates If there is no singularity in a compact section of the solution, then the compact section of the solution is considered as an instanton If there exist singularities in the section, then the action of the section is not stationary The action may only be stationary with respect to the variations under some restrictions mentioned above If this is the case, then the section is a constrained gravitational instanton To find the constrained instanton, one has to closely investigate the singularities The stationary action condition is crucial to the validation of the WKB approximation We are going to work at the WKB level for the problem of quantum creation of a black hole pair A main unresolved problem in quantum cosmology is to generalize the no-boundary proposal for an open universe While a general prescription is not available, one can still use analytic continuation to obtain the WKB approximation to the wave function for open universes with some kind of symmetry The 4 S space model with OŽ 5 symmetry wx 4 and the FLRW space model with OŽ 4 symmetry wx have been discussed In this paper, we try to reduce the symmetry further, and study the problem of quantum pair creation of black holes in the Kerr-Newman-anti-de Sitter family Let us discuss the most general case, ie the quantum creation of the Kerr-Newman-anti-de Sitter black hole pair The Lorentzian metric of the wx black hole spacetime is 5 Ž r u ds sr D y1 dr qd y1 du y y qr J Du sin u adty Ž r qa df y y yr J Dr dtyasin u df, 3 where r sr qa cos u, Ž 4 Dr s Ž r qa Ž 1yLr 3 y1 ymrqq qp, Ž 5 Du s1qla 3 y1 cos u, Ž 6 Js1qLa 3 y1 Ž 7 and m, a,q and P are constants, m and ma representing mass and angular momentum Q and P are electric and magnetic charges The cosmological constant L is negative One can factorize Dr as follows L Dr sy Ž ryr0ž ryr1ž ryrž ryr 3, Ž 8 3 where at least two roots, say r, r, are complex 0 1 conjugates, and we assume r and r are real If this 3 is the case, then r and r must be positive One can 3 identify them as the outer black hole and inner black hole horizons, respectively The roots satisfy the following relations: Ý r i s0, Ž 9 i 3 Ý rrsy i j qa, Ž 10 i)j L 6m Ý rrrsy i j k, Ž 11 i)j)k L Ł i i 3Ž a qq qp r sy Ž 1 L The horizon areas are A s4p r qa J y1 Ž 13 i Ž i The surface gravities of the horizons are L Ł i j jž j/i i 6J ri qa Ž r yr k s Ž 14 We shall concentrate on the neutral case with QsPs0 first The Newman-anti-de Sitter case with nonzero electric or magnetic charge will be discussed later The probability of the Kerr-anti-de Sitter black hole pair creation, at the WKB level, is the exponen-

276 76 ( ) ZC WurPhysics Letters B tial of the negative of the action of its constrained gravitational instanton The constrained instanton is constructed from the complex version of metric Ž 3 by setting tsit One can have two cuts at ts"dtr between the two complex horizons r 0,r 1 Then one makes the f0-fold cover around the horizon r s r and the f -fold 0 1 cover around the horizon rsr In order to form a 1 fairly symmetric Euclidean manifold, one can glue these two cuts under the condition f0b0qf1b1s0, Ž 15 where we set the imaginary time periods b spk y1 0 0 and b spk y1 1 1 The Lorentzian metric for the black hole pair created is obtained through analytic continuation of the time coordinate from an imaginary to a real value at the equator The equator is two joint sections tsconst passing these horizons It divides the instanton into two halves We can impose the restriction that the 3-geometry characterized by the parameters m,a,q and P is given at the equator for the Kerr-Newman-anti-de Sitter family The parameter f or f is the only degree of freedom left for the 0 1 pasted manifold, since the field equation holds elsewhere Thus, in order to check whether we get a stationary action solution for the given horizons, one only needs to see whether the above action is stationary with respect to this parameter The equator where the quantum transition will occur has topology S = S 1 The action due to the horizons is p Ž ri qa Ž 1yfi Ii,horizonsy Ž is0,1 J Ž 16 The action due to the volume is f0b0l 3 3 IÕ sy Ž r1yr0qa Ž r1yr 0 Ž 17 6J If one naively takes the exponential of the negative of half the total action, then the exponential is not identified as the wave function at the creation moment of the black hole pair The physical reason is that what one can observe is only the angular differentiation, or the relative rotation of the two horizons This situation is similar to the case of a Kerr black hole pair in the asymptotically flat background There one can only measure the rotation of the black hole horizon from the spatial infinity To find the wave function for the given mass and angular momentum one has to make the Fourier transforwx mation 3 1 ` y id JJ H p y` C a,h s dd e C d,h, Ž 18 ij where d is the relative rotation angle for the half time period f0b0r, which is canonically conjugate to the angular momentum J s ma; and the factor J y is due to the time rescaling The angle difference d can be evaluated f 0 b 0 r H ds dtž V yv, Ž 19 where the angular velocities at the horizons are a Vis Ž 0 r qa i The Fourier transformation is equivalent to adding an extra term into the action for the constrained instanton, and then the total action becomes Isyp Ž r0 qa J y1 yp Ž r1 qa J y1 ž / 6 y1 y1 sp y qž rqa J q Ž r3qa J L Ž 1 In the derivation of the second equality in Ž 1 one notices from Eqs Ž 9 and Ž 10 that the sum of all horizon areas is equal to 4pL y1 for all members of the Kerr-Newman-Ž anti- de Sitter family This fact seems coincidental, but it has a deep physical significance It is crucial to note that the action is independent of the time identification period f0b0 and therefore, the manifold obtained is qualified as a constrained instanton Therefore, the relative probability of the Kerr black hole pair creation is Ž Pk fexpy p r qa J y1 qp r3 qa J y1 It is the exponential of the negative of one quarter of the sum of the outer and inner black hole horizon areas ij

277 ( ) ZC WurPhysics Letters B Now, let us turn to the charged black hole case The vector potential can be written as QrŽ dty asin u df q Pcosu Ž adtyž r q a df As 3 r We shall not consider the dyonic case in the following One can closely follow the neutral rotating case for calculating the action of the corresponding constrained gravitational instanton The only difference is to add one more term due to the electromagnetic field to the action of volume For the magnetic case, it is ž / f0b0p r0 r1 y Ž 4 J r qa r qa 0 1 and for the electric case, it is ž / f0b0q r0 r1 y y Ž 5 J r qa r qa 0 1 In the magnetic case the vector potential determines the magnetic charge, which is the integral over the S factor Putting all these contributions together one can find Isyp r qa J y1 yp r qa J y1 Ž 6 Ž 0 Ž 1 and the relative probability of the pair creation of magnetically charged black holes is written in the same form as Eq Ž In the electric case, one can only fix the integral vsha, Ž 7 where the integral is around the S 1 direction So, what one obtains in this way is CŽ v,a,h ij How- ever, one can get the wave function C Ž Q,a,h ij for a given electric charge through the Fourier transformation 1 ` i v Q C Q,a,h s dv e C v,a,h Ž 8 H ij p y` The Fourier transformation is equivalent to adding one more term to the action ž / f0b0q r0 r1 y Ž 9 J r qa r qa 0 1 ij Then we obtain the same probability formula for the electrically charged rotating black hole pair creation as for the magnetic one The duality between the magnetic and electric cases is recovered w3,6,7 x The case of the Kerr-Newman black hole family can be thought of as the limit of our case as we let L approach 0 from below Of course, if one lets the angular momentum be zero, then it is reduced into the Reissner-Nordstrom- anti-de Sitter black hole case If one further lets the charge be zero, then it is reduced into the Schwarzschild-anti-de Sitter black hole case For the Schwarzschild-anti-de Sitter black hole case, there are only three horizons, ( ( 1 r s sinhg, Ž 30 < L< 1 r sr s ysinhgyi' 1 0 Ž 3 coshg, Ž 31 < L< where we set g' arcsinh 3m< L < Ž 3 1 1r 3 The horizon rsr is the black hole horizon The surface gravity k of r is wx 5 i L k s Ž r yr Ž 33 Ł i i j 6ri js0,1,, Ž j/i And the total action is 6 Isyp Ž r0qr1 sp y qr L Ž 34 ž / Therefore, the relative probability of the pair creation of Schwarzschild-anti-de Sitter black holes, at the WKB level, is the exponential of the negative of one quarter of the black hole horizon area This contrasts with the case of pair creation of Schwarzschild-de Sitter black holes w3,8 x The relative creation probability for Schwarzschild-de Sitter black holes is the exponential of the total entropy of the universe One quarter of the black hole horizon area is known to be the entropy in the Schwarzschild-anti-de Sitter universes wx 9 One may wonder why we choose horizons r0 and r1 to construct the instanton One can also consider those constructions involving other horizons as the i

278 78 ( ) ZC WurPhysics Letters B instantons However, the real part of the action for our choice is always greater than that of the other choices for the given configuration, and the wave function or the probability is determined by the classical orbit with the greatest real part of the action wx 1 When we dealt with the Schwarzschild-de Sitter case, the choice of the instanton constructed from the black hole and cosmological horizons had the greatest action accidentally, but we did not appreciate this earlier wx 3 This point is important For example, if, instead we use r and r3 for constructing the charged or rotating instanton, then the creation probability of a universe without a black hole would be smaller than that with a pair of black holes This is physically absurd In Euclidean quantum gravity the partition function Z is identified with the path integral under the constraints If the system involves the imposed quantities, namely electric charge Q or Ž and angular momentum J, then one has to use the grand partition function Z in grand canonical ensembles for the thermodynamics study w10 x At the WKB level, one has ZsexpyI, Ž 35 where I is the effective action of a constrained instanton The effects of the electric charge and angular momentum have been taken into account by the two Fourier transformations The entropy S can be obtained be Ssy lnzqlnz, Ž 36 Eb where b is the time identification period Thus, the condition that I is independent from b implies that the entropy is the negative of the action One can use Eq Ž 36 to derive the entropy and it is the negative of the action For compact regular instantons, the fact that the entropy is the negative of the action is shown using different arguments in w11 x For the open creation case, if one naively interprets the horizon areas as the entropy, then the entropy is associated with these two complex horizons Equivalently, for the chargeless and nonrotating case one can say that the action is identical to one quarter of the black hole horizon area at r,or the black hole entropy up to a constant 6pL y1,as we learn in the Schwarzschild black hole case For the charged or Ž and rotating case, the action is identical to one quarter of the sum of the outer and inner black hole horizon areas up to the same constant Our treatment of quantum creation of the Kerr- Newman-anti-de Sitter space family using the constrained instanton can be thought of as a prototype of quantum gravity for an open system, without appealing to the background subtraction approach The beautiful aspect of our approach is that even in the absence of a general no-boundary proposal for open universes, we treat the creation of the closed and the open universes in the same way It can be shown that the probability of the universe creation without a black hole is greater than that with a pair of black holes in the anti-de Sitter background References wx 1 JB Hartle, SW Hawking, Phys Rev D 8 Ž wx SW Hawking, N Turok, Phys Lett B 45 Ž hep-thr wx 3 ZC Wu, Int J Mod Phys D 6 Ž , gr-qcr wx 4 ZC Wu, Phys Rev D 31 Ž wx 5 GW Gibbons, SW Hawking, Phys Rev D 15 Ž wx 6 RB Mann, SF Ross, Phys Rev D 5 Ž wx 7 SW Hawking, SF Ross, Phys Rev D 5 Ž wx 8 R Bousso, SW Hawking, hep-thr wx 9 SW Hawking, DN Page, Commun Math Phys 87 Ž w10x SW Hawking, in: SW Hawking, W Israel Ž Eds, General Relativity: An Einstein Centenary Survey, Cambridge University Press, 1979 w11x GW Gibbons, SW Hawking, Commun Math Phys 66 Ž

279 7 January 1999 Physics Letters B Emission of fermions from BTZ black holes Arundhati Dasgupta 1 The Institute of Mathematical Sciences, CIT Campus, Chennai , India Abdus Salam International Centre for Theoretical Physics, Strada Costiera, Trieste, Italy Received 31 August 1998; revised October 1998 Editor: L Alvarez-Gaumé Abstract The emission rate of fermions from q1 dimensional BTZ black holes is shown to have a form which can be reproduced from a conformal field theory at finite temperature The rate obtained for fermions is identical to the rate of non-minimally coupled fermions emitted from a five-dimensional black hole, whose near horizon geometry is BTZ = M, where M is a compact manifold q 1999 Elsevier Science BV All rights reserved PACS: 0460-m; 046qv; 0470-m; 0470dy; 115-w 1 Introduction Understanding Hawking radiation from black holes using a unitary microscopic theory has been a outstanding and difficult problem Nevertheless the progress made in the last few years suggests that resolution of the problem may be near In a series of papers, it has been shown that the Hawking emission rates of a number of four and five-dimensional black holes is reproducible from a 1q1 conformal field theory at finite temperature w1 5 x In case of the five-dimensional black hole which is a solution of Type II B supergravity obtained by wrapping D5- branes and D1-branes on T 4 =S 1, it is known that the entire configuration can be replaced by a long effective string along S 1 which is described by a conformal field theory at finite temperature How- 1 dasgupta@imscernetin ever for other black holes like the rotating ones, the origin of the effective string is not well understood Recently, new light has been shed regarding the origin of the underlying conformal field theory It has been shown that the above black holes can be mapped to the asymptotically anti-de Sitter q 1 dimensional BTZ black hole wx 6 times a compact manifold w7,8 x In particular w8,9 x, the near horizon geometries of these black holes have the form BTZ = M, Žor AdS 3 = M with global identifications in case of rotating black holes w10x where M is a compact manifold The underlying microscopic theory of the BTZ black hole is known to be a 1q1 dimensional conformal field theory w11 13 x Hence the conjecture that near horizon geometry explains thermodynamics of black holes, and the conformal field theory used to describe the BTZ = M geometry gives the microscopic theories of higher dimensional black holes The conjecture is supported by the evidences that the near horizon BTZ black hole r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

280 80 ( ) A DasguptarPhysics Letters B obtained has the same entropy as the higher dimensional black hole, and possesses the same decay rate for scalars as higher dimensional ones w14 x The q 1 dimensional BTZ black hole is an interesting system to study by itself The black hole is asymptotically AdS 3, and has two horizons with an ergosphere For a non-extreme BTZ black hole, there is a non-zero Hawking temperature, and an observer outside the outer horizon will detect thermal radiation However, since the metric is asymptotically anti-de Sitter, the local temperature measured by any time like observer decreases with distance at spatial infinity The situation is different from flat space-time where temperature is constant at infinity Moreover in flat space-time, the asymptotic observer measures a decay rate which is modified by the absorption coefficient or the greybody factor of the black hole The greybody factor is defined as the ratio of the total number of particles entering the horizon and the incoming flux at spatial infinity For asymptotically anti-de Sitter black holes, this is a little difficult to study, as spatial infinity for these black holes constitutes a time like surface through which information can enter and leave The usual way to deal with this is to impose boundary conditions on fields such that the surface acts like a reflecting wall In that case, the black hole is in equilibrium with thermal radiation and there is no net flow of flux across any time like surfaces and the concept of greybody factor remains obscure Nevertheless, an attempt can be made to define the Hawking rate in the same way as asymptotically flat space-times Using this in Refs w14,15 x, it was shown that the decay rate for scalars has the same form as the higher dimensional black holes, and can be reproduced from a conformal field theory at finite temperature In this paper we address the issue of fermion emission from BTZ black holes This is interesting to investigate by itself, as it is yet to be checked that the above stated result for scalars applies universally to all particle emissions from the BTZ black hole Not only that, the result constitutes important evidence for the conjecture that near horizon geometry of higher dimensional black holes encode information about its thermodynamics To define the Hawking emission rate for the BTZ black hole, we do not take an asymptotic observer, but an observer stationed at a radial distance r;l4r q, where l is related to the cosmological constant A motivation for this is that at this position, the observer in the BTZ geometry measures a local temperature equal to the Hawking temperature of the black hole This issue is discussed in Section 3 Also, for purposes of the greybody calculation, we assume that there is a flow of flux into the black hole This is perhaps a relevant physical situation to consider if we want to answer questions about higher dimensional black holes as flux can flow into near horizon geometry Ž BTZ Using the above assumptions, we show that the fermions in the BTZ geometry have exactly the same form of the Hawking emission rate as the non-minimally coupled fermions in a five-dimenwx 5 Indeed this strengthens the sional black hole conjecture The form obtained can be reproduced from a conformal field theory at finite temperature Since a satisfactory derivation of higher dimensional fermion decay rates does not exist from the effective string picture w4,5 x, this result is very useful A microscopic derivation of the rate found here by using the conformal field theory describing the BTZ black hole will finally clear the issue Moreover, three-dimensional fermions are much easier to handle than the higher dimensional ones, and this calculation can be used to predict decay rates for higher dimensional fermions In particular, the four-dimensional black hole of M-theory also has BTZ as its near horizon geometry and, it is interesting to predict the nature of the fermions of 11-D supergravity compactified to four dimensions, which have the same decay rate as the fermions considered in three dimensions In the next section, we solve the equation of motion of the fermion in the BTZ background We find that the equation of motion is exactly solvable, and yields a hypergeometric equation We can choose the ingoing solution at the horizon, and determine the flux which flows down the hole It should be mentioned that taking minimally coupled fermions instead of those considered here, does not give a meaningful result In the following section, we derive the greybody factor We determine the solutions in the region r; l and determine the flux which flows towards the black hole horizon Using this, we can calculate the greybody factor for the black hole, and hence the decay rate In the third section, we

281 ( ) A DasguptarPhysics Letters B give a comparison of the rate obtained here, and that obtained for higher dimensional black holes In the last section, we include a brief discussion Equation of motion of the fermion in BTZ background Einstein s gravity in q1 dimensions is essentially topological, and does not admit black hole solutions However, gravity with a negative cosmological constant, has non-trivial solutions The action for this is: 1 3 y Ss H d xž Rql, Ž 1 p where R is the scalar curvature and LsyŽ1rl is the cosmological constant; G s 1r8 according to the conventions of wx 6 The space with constant negative curvature, is called anti-de Sitter space This space is invariant under SOŽ, group, which is larger than the usual Poincare group of flat space time The appropriate covariant derivative for spin- half fields is: Dsg n E qv qge g a, n n na where vn is the spin connection, en a is the triad and g s 1r l is related to the cosmological constant The BTZ metric is derived by appropriate identifications of anti-de Sitter space time, and its asymptotic properties are the same as anti-de Sitter space wx 6 Hence the above covariant derivative is relevant for our purposes To write down the fermion equation in the BTZ background, we study the metric first The BTZ metric in coordinates r,t and f is Ž0- r- `,0-f-p : ž / D l r rqr y ds sy dt q dr q r df y dt, l r D lr Ž 3 Ž qž y D s r yr r yr Ž 4 Clearly, the metric represents a rotating black hole, with two horizons at rq and r y The angular momentum of the black hole is Js rqryrl, and Ž its mass is Ms rqqry rl The metric can be written in more convenient coordinates, with the radial coordinate r defined in terms of hyperbolic coordinate m The redefinition is: r sr cosh myr sinh m Ž 5 q y The spatial infinity corresponds to tanhm 1 In this coordinate, the metric takes a form: ž / dt ž y q l / dt q y ds sysinh m r yr df ql dm l qcosh m yr qr df 6 A convenient set of linear combinations of t and f gives us x q srqtrlyryf and x y syrytrlq r f, and the killing directions of the metric are E q q x and E y The triads are chosen in an appropriate local x Lorentz frame in the tangent plane, scoshm, e 1 sl Ž 7 e 0 q ssinhm, e y x x m The non zero spin connections for this are: v qsy coshms, v y x x s sinhms, l l Ž 8 ab w a here s s1r g,g b x Using the above, we substitute them in Ž and obtain the fermion equation on the BTZ space-time as 1 sinhm coshm Exqc 1 0 g Em q q cqg l ž coshm sinhm / sinhm Exyc 1 qg q cs0 Ž 9 coshm l We take the representation of gamma matrices to be: g 0 sıs, g 1 ss 1, g 3 ss 3 The killing isometry requires that E x "c syık " c, where k " are con- stants depending on the energy and azimuthal eigenvalues v and m respectively In fact they can be q determined, and k s Ž v y mv rž p lt H and y k s Ž r vy r mrl rž p l r T y q q H The Hawking Ž temperature is TH s rq y ry rp l rq and V s Jr rq We take the following form for the wave- function: e ž yıž kq x q qk y x y cs c 1 / 'sinhm coshm c

282 8 ( ) A DasguptarPhysics Letters B The radial equations for the two components are determined as ılk q 1 ılk y dm y csy y c1 Ž 10 sinhm coshm ž / ž / ılk q 1 ılk ž / ž y sinhm coshm / d q c sy q c Ž 11 m 1 Interestingly, to separate the wave functions, we have to go to a different basis of wave functions Let us call them c X and c X, defined as, 1 y1r4 X X 1 ' 1 c qc s 1ytanh m 1qtanhm Ž c qc Ž 1 y1r4 X X 1 ' 1 c yc s 1ytanh m 1ytanhm Ž c yc Ž 13 We obtain the equations in coordinates y s tanhm for the c X below as: ž / k q X y X y 1yy d c yıl qk y c y sy1yılž k q qk y 4c X Ž 14 k q X y X y ž / 1yy d c qıl qk y c y sy1qılž k q qk y 4c X Ž 15 The equations are now very easily separable The second order differential equation obtained from the above two equations can be cast in a simple form in the variable y which we denote as z for convenience 1 X X zž 1yz d c1q Ž 1y3z dc1 1 yılk q ql k q y q qılk 4 z ž / 1 y X yl k y c1 s0 16 1yz The solution to this equation is determined to be X c sz m Ž 1yz n FŽ a,b;g ; z 1, where F is a hypergeometric function For the ingoing function, the constants are as follows: ms1rqılk q r, nsy1r, and the hypergeometric parameters are: asılžk q q y Ž q y q k r q 1r,b s ıl k y k r and g s ılk q 3r The ingoing function is so chosen that at the horizon, the wave function has the dependence c; expw ıž vr4p T log z x H The solution for the other component of the wave function can be determined ilk q r easily now It is: c s z Ž 1 y z y1r Ž yž gy1 ra FŽ a y 1,b;g y 1; z, where a,b,g are constants as defined above The flux for this function as shown in the next section is negative, indicating a flow into the black hole Thus we see that the fermion equation of motion in the BTZ background is exactly solvable It is interesting to note that ns0 corresponds to a minimally coupled fermion, and in that case gsaqb The hypergeometric solution is not well behaved, and does not converge as z 1 There can be other kinds of couplings to the BTZ metric, and they will be interesting to investigate The BTZ black hole is locally anti-de Sitter space, with global identifications It will be interesting to see whether the solutions obtained here can be related to those obtained for AdS in w16 x, modulo the global identifications 3 3 Grey body factor The black hole grey body factor is also the absorption coefficient of the black hole The geometry of the black hole provides a kind of potential barrier for the fields propagating on it Only a fraction of the incoming flux at infinity is absorbed by the body, and rest is reflected back In order to determine the total Hawking radiation rate of the observer, sitting far away from the black hole, we need to calculate this absorption rate Indeed, as in ordinary quantum mechanics, the black hole absorption rate, which we denote by sabs is related to the ratio of the ingoing flux at horizon and incoming flux at infinity The fermion flux into the horizon will be determined by the current which enters the horizon Usually, to probe the black hole geometry, an incoming plane wave is taken at infinity However, here, for our purposes, we take the incoming flux in the region r;l4rq as the incident flux on the black hole This would correspond to an BTZ observer, sitting at finite r, detecting radiation Though, the physics of this picture is not very clear, there are a number of reasons for choosing this In curved space-time, an

283 observer measures a thermal spectrum depending upon his local temperature, which is THr( g 00 In asymptotically flat space time, ( g 00 1asr ` However, this is not the case in asymptotically anti-de Sitter space-time where ( g 00 ; r at spatial infinity For small mass BTZ black holes, ie r q<l, itis easily seen that, ( g 00 1 when r;l This motivates the choice of the observer Moreover, to compare our final answer with higher dimensional black hole rates, going infinitely away from the horizon would imply a modification of the near horizon BTZ geometry, and we are not interested in probing that region As r;l, the black hole metric is same as asymptotically anti-de Sitter space Solutions determined in this metric is also the same as that obtained in the vacuum solution of the black hole w17 x The metric is Ž r4r : q r l ds sy dt q dr qr df Ž 17 l r To determine the wave functions we then solve the radial equations: ž / ž / ıv l 1 ım f f r E r" c 1Ž sy " c Ž1 Ž 18 r r To separate this set, we go to a frame in which, c X f sc f qc f and c X f 1 1 sc 1 f yc f The equation can be exactly solved in this frame The solutions are determined, in terms of Bessel functions, ' Ž ı' x Ž X f c s x A J L x qıa N L x 19 X f c s AJ L x qıa N L x 0 E where Jn and Nn are Bessel functions of the first and second kind A1 and A are arbitrary constants of ' integration Also xs1rr, Lsl v l ym, Es lž v l q m rl Note that the same solutions will survive when r ` in anti-de Sitter space The interesting aspect about anti-de Sitter space is that r ` is a time like surface Hence, it is necessary to specify boundary conditions, which are either Dirichlet or Nueman on the surface These boundary conditions, also called reflective boundary conditions w18x can be realised in the set of functions defined in Ž 0 On choosing A s 0, this condition can be ensured for the above wavefunctions It is easy to ( ) A DasguptarPhysics Letters B ' check that in that case rcs0 for r ` However, here we are not interested in making the wall totally impervious Instead, we are forced to take A have non-zero values if we want the wavefunction to match the wavefunction determined in Ž 16 continued to z 1 This shows that, our choice of a net inflow of flux into the black hole ensures that we do not have reflecting boundary conditions at infinity For asymptotic anti-de Sitter space, this might be related to the transparent boundary conditions dew18 x Before we can determine the fined in Ref greybody factor using this far solution, we need to match this with the ingoing wavefunction at the horizon since we want the wavefunction to be continuous in space-time To do that, we continue the solution of Ž 16 to z 1 w19 x Now, with the scalings and redefinitions, the radial wavefunction is: E1( NŽ rqqr y ½ c1 yž1ycž a b 3r Ž r ' N yc bq1 b q4b log r qc ž ž / / 5 Ž 1 ( r q yr y c E1 ' r where / G Ž g G Ž gyayb E s 1, Ž 3 ž G Ž gya G Ž gyb C are the digamma function Žsubsequently, we write Ž b C a qc bq1 'C a,b Also Nsr q yr, and C Ž 1 syc Ž Euler s constant y The fac- tors (Ž r q"r y rn enter as this wavefunction given in terms of t, r,f coordinates is Lorentz rotated from the wavefunction obtained in x q, z, x y coordinates Note in the above, we have taken ' N rr<1 or in other words, ' N <l The wavefunctions obtained in Ž 0, can be cast in a similar form when Lrr<1 Ž for ms0, this indicates that v l<1 : 1 ıa L f c 1f A q log qc Ž 4 3r 1 ž r p r ž ž / // A f c f Ž 5 1r p ELr

284 84 ( ) A DasguptarPhysics Letters B The asymptotic constants are determined from the above equations, ( NŽ rqqr y A1sy E1Ž 1yCŽ a,b, ' p EL A s ( rqyr y E1 Ž 6 ' The fermionic flux is given by: ' r r 1 Fs ygj srce1gc 7 The incoming flux at rsl is determined as Žthe flux obtained from the above constants is multiplied by l rn for normalisations l 1 f F sy < E < y ReCŽ a,b Ž 8 8 N The absorption coefficient is defined as the ratio of total number of particles entering the horizon with the incoming flux at infinity w0 x The total number of particles entering the horizon is: H' r Psy ygjrqdf 9 This is equal to: A lr4n <Ž gy1 ra < H, where AH is the area of the horizon The absorption coefficient is then determined as Ž for ms0 : s abs A H s 1yReCŽ a,b = gy1 ž a / G Ž g G Ž gyayb G Ž gya G Ž gyb y Ž 30 Here, ReCŽ a, b syž vr4tl tanhž vr4tr y OŽ b for m s 0 Using the expressions for the hypergeometric parameters as given in the earlier section, the greybody factor can be written in a interesting form: v A H sabs s 4T 1qvr4T tanhž vr4p T = L L R expž vrth q1 expž vrtl y 1 expž vrtr q 1 Ž 31 where the quantities TL and TR are defined by: ž / ž / 1 1 ry 1 1 ry s 1y, s 1q Ž 3 T T r T T r L H q R H q The extremal limit, defined by taking rq r y, corresponds to TL4T R Clearly, taking the above limit in Eq Ž 3, the absorption coefficient reduces to AHr The Hawking radiation rate will be now a product of thermal distributions, instead of being a single fermionic distribution In fact, it is: v AH d k G H s 4TL expž vrtl y1 expž vrtr q1 Ž 33 This is precisely the form expected for emission rates from an underlying conformal theory at finite temperature wx 3 The fermion in the bulk couples to operators of the 1 q 1 dimensional conformal field theory The system is at finite temperature T H, which can be split into left and right temperatures such that 1rTLq1rTRsrT H The decay rate at finite temperature due to the coupling stated above is calculated to have the form of a product of left and right distributions The fermions are associated with rightmoving temperature, indicating that fermions considered here couple to chiral conformal operator Using the results of wx 3 it can be predicted that the conformal fermion couples to operator in the conformal theory of the form O q O y, where O q is rightmoving, and has conformal weight 1r and O y is leftmoving with weight 1 It will be interesting to determine the nature of the coupling as that fixes the coefficients exactly 4 Comparison with higher dimensional black holes One of the reasons behind the renewed interest in three-dimensional black holes, is the fact that near horizon geometries of certain higher dimensional stringy black holes are BTZ times a compact maniw9,1x and fold Here we briefly review this mapping discuss the implications The solution due to RR charged one branes and five branes wrapped on T 4 =S 1, and Kaluza Klein momenta along S 1 in 10

285 ( ) A DasguptarPhysics Letters B dimensions, has a near horizon geometry BTZ =T 4 =S 3 The radius of the S 3 direction is lsr1r, 5 where r1 and r5 are related to the one brane and five brane charges respectively The time, transverse ra- 1 dial distance r and S direction Ž f constitute the BTZ black hole coordinates The ordinary kaluzaklein reduction of the 10-D solution on T 4 =S 1 yields a 5-D black hole, which preserves N s 8 supergravity The entropy of the 5-D black hole is equal to the entropy of the near horizon BTZ black hole and scalar decay rate equals the decay rate for scalar emission from BTZ black holes Here we make a comparison for fermion decay rates In Ref wx 5, it has been shown that the SUGRA fermions of Ns8 supergravity, have a Hawking decay rate for the five-dimensional black hole as v 5 5 G sa H = H 4T L d 4 k expž vrtl y 1 expž vrtr q 1 Ž 34 Clearly, our decay rate is identical to this decay rate The temperatures of the left and right distributions 5 are exactly the same as given in 3 and AH is the area of the horizon of the five-dimensional black hole A interesting point to note is that the rates can be matched upto exact coefficients if we choose to factor out the phase space factors of S 3 and fsx5rl Ž 1 x is the S direction, with radius R 5 from the decay rates, as A 5 HrA 3 Hsp l 3 rr However, an observer in five-dimensional space detecting particles at infinity sees all the three dimensions of S 3 as uncompactified So, it is not clear what the above result implies However, it can be said that our result confirms the observation about scalar decay rates The range of frequency for both the calculations, v r 1 <1, is also the same The exact matching observed above provides a basis to predict rates for non-minimally coupled fermions which propagate on the background of the four-dimensional N s 4 SUGRA black hole obtained by compactifying M-theory Ž 11-D supergravity on T 6 =S 1 To identify the required fermion one requires to take the equation of motion in the 11 D Supergravity solution, and take the near horizon geometry limit as described above All fermions which will couple in the same way as in Eq Ž in the BTZ part, can then be predicted to have the rate obtained in this paper The metric in 11 dimensions is due to 3 M 5-branes wrapped on T 4 =S 1, ie directions x x and a boost in the x, ŽS direction The near horizon limit results in the metric splitting up into a BTZ =S =T 6, where S is the two sphere of the noncompact t, r,u,f dimensions of the four-dimensional black hole The radius of the two sphere is RslrsŽ rrr 1 3 1r3, where ri are related to the charges of the black hole As in the five-dimensional case here f s x11rr 11, r s Ž R r qr sinh s X Ž R is the radius of x , and time form the BTZ coordinates To find the relevant fermions which will have the rate as found in this paper, we start from the 11-D gravitino c M Clearly gravitino with vector polarisation along x or the other x 1, x, x3 directions will not satisfy our requirements We take a representative c 5, asinthe near horizon limit, all the torus directions are similar, apart from constant scalings In this limit, since g sconst, is49, we can split the 11-D equation ii of motion as: ž / Du 3 q Du V cs0 Ž 35 l where Du 3 and Du V are the dirac operators in the BTZ and the two sphere metrics respectively To get simultaneous eigenstates of both the operators, we multiply by the two-dimensional chirality matrix G s ı Ga G b Thus for G Du V c s lc, the equation of the fermion in the near horizon limit is: ž / l X Du 3m q cs0 Ž 36 l X For ls1r4, we have the required fermion Ž Du ' G Du It is not very difficult to solve the eigenvalue equation stated above The argument given here is heuristic, and we have not been careful about the supersymmetry preserved by the background metric It is to be checked whether the fermion taken above falls in the Ns4 multiplet, as the four-dimensional black hole preserves N s 4 super symmetry However, it is an interesting calculation, and is under further investigation at present 11

286 86 5 Discussion ( ) A DasguptarPhysics Letters B Acknowledgements In this paper we have calculated emission rate of fermions from BTZ geometry, using techniques of asymptotically flat space-time calculations, like the greybody factor However, since the physical situation we are interested in is when BTZ occurs as the near horizon geometry of higher dimensional black hole, this is justified We show that indeed the BTZ calculation reproduces the rate of the non-minimally coupled fermions in the background of a five-dimensional black hole whose near horizon geometry is BTZ =S 3 The fact that the rate observed by a BTZ observer at r;l looks identical to that of an asymptotic observer in a five-dimensional black hole is interesting The physical implications of this are still not clear, but the answer might lie in the location of the degrees of freedom of the underlying conformal field theory There are several ways to approach the problem It is known that q1 gravity can be cast in the form of Chern Simon theory, which induces a conformal field theory on the boundary However, on inclusion of matter fields the theory is no longer topological, and the same conclusions cannot be drawn about the entropy Hence, it is not clear how to study Hawking emission in the above frame work Recently, matter fields have been treated as a classical perturbation in the Chern Simons action, and the decay rate obtained for scalars w x The agreement with the black hole decay rate is remarkable, and calls for further investigation Apart from this, the BTZ black hole is asymptotically anti-de Sitter, and has a conformal field theory living on its boundary w1,13 x With the AdSrCFT correspondence, it is known now, that string theory on orbifolds of AdS 3 times a compact manifold M is dual to a super conformal field theory whose target space is symw1 x In this matter fields are metric product of M automatically included, and it will be interesting to study the decay rates, using this approach We are grateful to P Majumdar, D Page, A Sen, S Das, P Durganandini and T Sarkar for useful discussions We thank Theory Group CERN, where a part of the work was done, for hospitality References wx 1 C Callan, JM Maldacena, Nucl Phys B 47 Ž ; A Dhar, G Mandal, SR Wadia Phys Lett B 388 Ž ; SR Das, SD Mathur, Nucl Phys B 478 Ž ; J Maldacena, A Strominger, Phys Rev D 55 Ž ; Phys Rev D 56 Ž ; M Cvetic, F Larsen Nucl Phys B 506, 107 wx SS Gubser, IR Klebanov, Phys Rev Lett 77 Ž wx 3 SS Gubser, Phys Rev D 56 Ž wx 4 S Das, A Dasgupta, P Majumdar, T Sarkar, hepthr wx 5 K Hosomichi, Nucl Phys B 54 Ž wx 6 M Banados, C Teitelboim, J Zanelli, Phys Rev Lett 69 Ž wx 7 S Hyun, hep-thr ; K Sfetsos, K Skenderis, Nucl Phys B 517 Ž wx 8 J Maldacena, hep-thr wx 9 V Balasubramanian, F Larsen, hep-thr w10x M Cvetic, F Larsen, hep-thr w11x S Carlip hep-thr980606, and references therein w1x JD Brown, M Henneaux, Comm Math Phys 104 Ž w13x A Strominger, J High Energy Phys Ž w14x D Birmingham, I Sachs, S Sen, Phys Lett B 413 Ž w15x HW Lee, NJ Kim, YS Myung, hep-thr98037 w16x S Hyun, YS Song, JH Yee, Phys Rev D 51 Ž w17x M Banados, M Henneaux, C Teitelboim, H Zanelli, Phys Rev D 48 Ž w18x SJ Avis, CJ Isham, D Storey, Phys Rev D 18 Ž w19x A Erdely, Higher Transendental Functions ŽBateman Project, Vols IrII, McGraw Hill, 1953 w0x W Unruh, Phys Rev D 14 Ž w1x J Maldacena, A Strominger, hep-thr wx R Emparan, I Sachs, hep-thr98061

287 7 January 1999 Physics Letters B AdS3rCFT correspondence, Poincare vacuum state and greybody factors in BTZ black holes HJW Muller-Kirsten a,1, Nobuyoshi Ohta b,c,, Jian-Ge Zhou c,3 a Department of Physics, UniÕersity of Kaiserslautern, D Kaiserslautern, Germany b The Niels Bohr Institute, BlegdamsÕej 17, DK-100 Copenhagen Ø, Denmark c Department of Physics, Osaka UniÕersity, Toyonaka, Osaka , Japan Received 1 October 1998 Editor: PV Landshoff Abstract The greybody factors in BTZ black holes are evaluated from D CFT in the spirit of AdS3rCFT correspondence The initial state of black holes in the usual calculation of greybody factors by effective CFT is described as Poincare vacuum state in D CFT The normalization factor which cannot be fixed in the effective CFT without appealing to string theory is shown to be determined by the normalized bulk-to-boundary Green function The relation among the greybody factors in different dimensional black holes is exhibited Two kinds of Ž h,h sž 1,1 operators which couple with the boundary value of massless scalar field are discussed q 1999 Elsevier Science BV All rights reserved Recently it has been proposed wx 1 that stringrm-theory on AdS = M Ž where M is a proper compact space d is dual to a conformal field theory Ž CFT which lives on the boundary of the anti-de Sitter Ž AdS space Precise forms of the conjecture wx 1 of the AdSrCFT correspondence have been stated and investigated in Refs w,3 x The essence of this conjecture is that supposing the partition functions of the two theories are equal, the correlation functions in the CFT can be read off from the bulk theory and vice versa One of the interesting 3 examples is the duality between type IIB string theory on AdS =S =M and certain D CFT w4 8 x 3 4, since both sides of the theories are very well understood It has been shown that there is an agreement between the Kaluza-Klein spectrum of supergravity and the spectrum of certain D CFT w4,7 x On the other hand, much work on the absorption and Hawking radiation in 5D and 4D black holes has been done in the semiclassical analysis, D-brane picture, effective string model and effective D CFT w9 14 x Since 5D and 4D black holes are U-dual to BTZ black holes w15 x, the entropies of 5D and 4D black holes can be related to the entropy of the BTZ black holes w16 x Especially, in the large N limit w1 x, the geometries of 5D and 4D black holes are AdS =M Ž effectively near horizon region So one expects to find the explanation of 3 1 address: mueller1@physikuni-klde address: ohta@physwaniosaka-uacjp 3 address: jgzhou@physwaniosaka-uacjp r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

288 88 HJW Muller-Kirsten et alrphysics Letters B 445 ( 1999) greybody factors of higher dimensional black holes in the context of BTZ black holes In Refs w17,18 x, the greybody factors in BTZ black holes have been discussed from the semiclassical point of view, where the wave equations are solved In the effective string calculation of greybody factors in 5D and 4D black holes, the interaction between the scalar fields in the bulk and effective string is described by w1x H S s d xf Ž t, x, xs0 O Ž t, x, Ž 1 int where the integration is over the effective string worldsheet, x indicates its location in transverse space, and OŽ t, x is some local conformal operator in D CFT, which takes the form O Ž t, x so Ž x q O Ž x y, q y where x " st"x and Oq and Oy are primary fields of dimension hl and h R, respectively The OPE s of Oq and Oy with themselves are given by C Oq OqŽ z OqŽ w s qless singular terms, h Ž zyw L C Oy OyŽ z OyŽ w s qless singular terms, Ž 3 h R Ž zyw y q where zsix, zsix for real x and imaginary tsyit To compare with the macroscopic decay rate in 5D and 4D black holes, the initial state is usually thermally averaged, since the black hole corresponds to a thermal state This means that one must take finite temperature two-point correlation functions Thus the function Gt, Ž x defined by GŽ t, x s² O Ž yit, x O Ž 0,0 :, Ž 4 is usually chosen as T H / ž / h L h R CO CO p TL p T q y R h q h Rž L q y i sinhp TLx sinhp TRx GŽ t, x s Ž 5 Since x and x are linear in z and z, respectively, there seems to be some disagreement between Eqs 5 and Ž 3 In other words, it is not clear what kind of initial state is used to define Ž 4 Then natural questions arise how to describe the initial state of black holes in D CFT, and how to explain the greybody factors in 5D and 4D black holes in terms of those in BTZ black holes These are the problems we are going to examine In order to answer the above questions, we discuss the greybody factors in BTZ black holes from D CFT in the spirit of AdS rcft correspondence To get explicit description for Ž 4 and Ž 3 5, we calculate the two-point correlation functions in the BTZ coordinates by bulk-boundary correspondence w1 3,18 1 x, but here we should include all the coefficients in the calculation Using two-point correlation functions in BTZ coordinates, we evaluate the greybody factors in BTZ black holes, and find that they agree with the known results even for the coefficients This fact gives further evidence to the AdS3l CFT correspondence In fact, the calculation heavily relies on Witten s conjecture wx 3 The result obtained shows that the initial state of black holes can be described by Poincare vacuum state in D CFT The coordinate transformation between Poincare coordinates Ž w,w and BTZ coordinates Ž u,u induces a mapping of the operator OŽ w,w to OŽ u,u q y q y q y q y by Bogoliubov transformation, and the operator OŽ u,u sees the Poincare vacuum state Ž q y which is a natural vacuum state for D CFT as an excited mixed state After explaining the greybody factors in BTZ black holes, we find that the greybody factors in 5D and 4D black holes can be described by those in BTZ black holes in a unified way This is because in the large N limit wx 3 1, the geometries of 5D and 4D black holes turn out to be BTZ =S =M4 and BTZ =S =M 5, respectively, and the two-point correlation functions in 5D and 4D black holes can be related to those in BTZ black holes, q y Ž

289 HJW Muller-Kirsten et alrphysics Letters B 445 ( 1999) which are constant multiples of those in BTZ case The constant can be determined from the parent 10D supergravity theories w0,13,3 x It is known that the greybody factors in black holes only depend on the conformal dimension of the operator OŽ t, x, and are indifferent to the concrete form of the operator In order to understand the relation between the physical degrees of freedom in 3D pure gravity and those of D CFT, we next consider an Ž h, h sž 1,1 operator in NsŽ 4,4 super CFT Ž SCFT based on a resolution of the orbifold ŽT 4 Q 1Q 5 rsž QQ 1 5, and the corresponding operator in 3D gravity obtained by quantizing BTZ black holes in external massless scalar field By comparing two operators, we find that the contribution from different scalars x A is smeared, and gravity cannot distinguish between different CFT states with the same expectation value for the operator OŽ u,u q y, which shows that gravity is just like thermodynamics but gauge theory is like statistical mechanics w5,4 x Now let us consider two-point correlation functions of D CFT; in Poincare coordinates many studies of correlation functions in boundary CFT have been done by the bulk-boundary correspondence w19 1 x; in BTZ coordinates this was analysed in Ref w x To calculate the greybody factors in BTZ black holes, we need to keep all the coefficients in the two-point correlation functions, and then compare the greybody factors extracted from AdS3rCFT correspondence with those obtained from the semiclassical analysis As a first simple exploration, we consider two-point correlation functions of the operator coupling to the boundary value of the massive scalar field In terms of Poincare coordinates, the AdS metric is l y Ž q y 3 ds s dy q dw dw Ž 6 For simplicity, we choose l s 1 in the following discussions The Euclidean action of massive scalar field with mass m in AdS H q y' Ž m n 3 space is 1 mn S f s dydw dw g g E fe fqm f, 7 h which has solution behaving as f y,w,w y y f Ž w,w when y 0 The boundary value f Ž w,w q y 0 q y 0 q y has dimension h which couples to an operator OŽ w,w y q y of dimension hq with the parameters h" defined by Ž ' 1 h" s 1" 1qm 8 The normalized bulk-to-boundary Green function in Poincare coordinates is w0x h G Ž h y q X X q KP Ž y,w q,w y;w q,wy s X X, Ž 9 pg Ž h y1 y q Ž w yw Ž w yw which has singular behavior when y 0 as w3,0x y y h y K q q q y y y,w,w ;w X Ž,w X d Ž w yw X d Ž w yw X Ž 10 P q y q y q q y y Here we should mention that the behavior Ž 10 determines the normalization coefficient in Ž 9 The solution fž y,w,w can be expressed as q y H f Ž y,w,w s dw X dw X K Ž y,w,w ;w X,w X f Ž w X,w X, Ž 11 q y q y P q y q y 0 q y where f Ž w,w is the boundary value of the bulk field fž y,w,w 0 q y q y The metric of BTZ black holes is w15x r yr r yr r r r ds sy dt q dr q r df y dt, Ž 1 q y q y r r yrq r yry r ž /

290 90 HJW Muller-Kirsten et alrphysics Letters B 445 ( 1999) with periodic identification f;fqp, where we have chosen ls1 The mass and angular momentum of BTZ black holes are defined as Msrq qry, Jsrqr y Ž 13 wx It can be shown that the metric of BTZ black holes can be transformed to that of AdS locally by 4 3 ž / ž / 1r 1r r yrq rqyr p T u y " " p ŽTquqqTyu y " r yry r yry w s e, y s e, Ž 14 with r qr y T" s, u" sf"t Ž 15 p In the region r4r in the BTZ coordinates, Eqs Ž 14 can be approximated as " ž / 1r rqyr p T u y " " p ŽTquqqTyu y " w se, ys e Ž 16 r Since the boundary field f Ž w,w has conformal dimension Ž h, h 0 q y y y, it is easy to see that near the boundary y 0Ž r `, the bulk field fž r,u,u in BTZ coordinates behaves as q y yh y yh y hy y h q y q y q y y 0 q y f Ž r,u,u ; Ž p T Ž p T r yr r f Ž u,u Ž 17 By use of the conformal dimensions of f Ž w,w and the relations Ž 16, Eq Ž 11 0 q y in the region r4r " is cast into with H f Ž r,u,u s du X du X K Ž r,u,u ;u X,u X f Ž u X,u X, Ž 18 q y q y B q y q y 0 q y K Ž r,u,u ;u X,u X B q y q y ž / h q q q y yhq yhq Ž q Ž y h y1 r yr s p T p T p r h q p T q Ty r q yry X X w x X X ß p T u yu qt u yu =~ 4r e q q q y y y qsinhp Tq uqyu q sinhp Ty uyyu y Ž 19 In the derivation of Ž 19, we keep all the coefficients in the calculation, and use the normalized bulk-to-boundary Green function According to AdS3rCFT correspondence, the relation between string theory in the bulk and field theory on the boundary is wx 3 H f 0 O eff s ys e Žf ² B e : CFT 0 Since f is the solution to equations of motion, the bulk contribution to S Ž f eff is zero, and a boundary term contributes to it: H ' 1 rr Seffs lim duqduy g g fer f 1 r `

291 HJW Muller-Kirsten et alrphysics Letters B 445 ( 1999) Combining Ž 1 and Ž 17 -Ž 19, we find hqž hqy1 y1 y1 X X Seffsy Ž rqyry Ž p Tq Ž p Ty Hduqduyduqduyf 0Ž u q,uy p = From 0 and, one has ž sinhp T u yu / ž sinhp T u yu / hq hq p Tq p Ty X X X X 0 q y qž q q yž y y f Ž u,u / ž / q q y y hq hq hqž hqy1 p Tq p Ty GŽ t,f s² O Ž u,u O 0,0 : q y s, Ž 3 ž p sinhp T u sinhp T u 4 where we have used 15 to simplify the expression However, due to the periodic identification f; f q p, the above expression for Gt,f should be modified by the method of images as wx G Ž t,f s² O Ž u,u O Ž 0,0 : T q y ` hq hq hqž hqy1 p Tq p Ty s Ý ž q / ž y / p sinhp T fqtqnp sinhp T fytqnp nsy` Ž 4 The sum over n/0 in Ž 4 comes from the twisted sectors of operator OŽ u,u q y in the orbifold procedure u ;u qnp for the BTZ black holes wx 4 The greybody factors in BTZ black holes are given by w1,13x " " p p ippx s s dt df e G Ž tyie,f yg Ž tqie,f HH abs T T v 0 p ` ippx s HH dt df e GŽ tyie,f ygž tqie,f v y` v hqy1 hqy1 hqž hqy1ž p Tql Ž p Tyl sinh ž T H / v v s G hqqi G hqqi, vg Ž h ž 4p T / ž 4p T / q q y Ž 5 where the infinite sum in G Ž t,f T has changed the original integral region 0FfFp into y`fff` and the parameter l has been switched on The Hawking temperature T is defined by 1 1 s q Ž 6 T T T H q y Here we should point out that in the usual derivation of greybody factors in 5D and 4D black holes w10,1 x, the periodicity along the spatial direction f is ignored, because it is assumed that the length of effective string is large compared to the typical wavelength of the particle, ie, p is much larger than 1rT H However, in our method the same result can be obtained without the above assumption due to the good behavior of G Ž t,f H T 4 According to Ref w0 x, the Ward identities suggest that the factor h in Eq Ž 3 q may be modified to hqy1 due to singular nature of the two-point correlation functions This affects the following expressions, but the results for massless particles remain the same

292 9 HJW Muller-Kirsten et alrphysics Letters B 445 ( 1999) In the ml 0limit Ž h s1, the decay rate for massless scalar field is q sabsž hqs1 vp l Gs s, Ž 7 v r T v rt v rt e H y1 e qy1 e y Ž y1 which is consistent with semiclassical gravity calculations in w17,18 x We note that there is a minor difference between Ž 5 and that obtained in Ref w18x with h s h s h In Eq Ž 5 L R q there is an extra factor h q, however, when hqs 1, both coincide The agreement of greybody factors in BTZ black holes obtained from AdS3rCFT correspondence with those from semiclassical gravity calculations indicates that the usual effective string theory is nothing but the boundary D CFT of AdS3 space Let us recall that the two-point correlation functions in Poincare and BTZ coordinates can be written as 1 X X ² O Ž w,w O w,w : q y Ž q y ;, X hq X hq Ž w yw Ž w yw q q y y h Ž h y1 p T p T X X ² O Ž u,u O Ž u,u :; Ž 8 q y q y hq hq q q q y X X p sinhp TqŽ uqyu q sinhp TyŽ uyyu y " " In Ž 8, the Poincare coordinates w are related to BTZ coordinates u by an exponential transformation Ž 16 Ž Ž q y in the region r4r " Comparing 8 with Eqs 5, we find that z and z are not linear in x and x, but rather they should be related by an exponential transformation The result obtained also shows that the initial state of BTZ black holes can be described in D CFT by Poincare vacuum state The operators O Ž w q q and O Ž w in Poincare coordinates satisfy the OPE in Eq Ž 3 y y However, the nonlinear coordinate transformation Ž 16 introduces a mapping of the original operator OŽ w,w q y in Poincare coordinates to the new one OŽ u,u q y in BTZ coordinates, and this induces the Bogoliubov transformation on the operators The operator OŽ u,u q y sees the Poincare vacuum state as an excited mixed state; that is, they see the Poincare vacuum state as thermal bath of excitations in BTZ modes w4, x The usual procedure to thermally average the initial state of black holes Ž or scalar particles in the calculation of greybody factors is just to measure Poincare vacuum state by the operator OŽ u,u q y in BTZ coordinates, which was vague in the former treatment of greybody factors by effective string model in 5D and 4D black holes Having understood the greybody factors in BTZ black holes, let us now discuss the greybody factors in 5D and 4D black holes in the light of the fact that in the large N limit, the geometries of 5D and 4D black holes are 3 BTZ =S = M and BTZ =S = M, respectively w4,7,13 x 4 5 For example, consider near-horizon AdS3 structure of 5D black holes Ž boosted D1rD5 configuration in the large N limit wx 1 Its metric is w4,18x r r R r ds s Ž ydt q dx q Ž coshs dt q sinhs dx q dr q R dv q dx, Ž Ý i R R r yr0 R is1 where r is the extremality parameter, r, r and r sinhs are related to the charges of D1-brane Ž Q , D5-brane Q and momentum in 5D black holes, and R sr r The Ž t, x,r part of the metric Ž is the metric of BTZ black holes The coordinates Ž t, x in Ž 9 can be used to construct BTZ coordinates u ", from which the Poincare coordinates w can be introduced, and D CFT lives on the asymptotic boundary r ` wx " 4 In the background Ž 9, two-point correlation function is modified by a factor h w 0,13,3 x hqž hqy1 1 X X ² O Ž w,w O w,w : q y Ž q y s h 5D, Ž 30 h h X q X p q Ž w yw Ž w yw q q y y where the constant h5d can be completely determined from the parent 10D supergravity theory, that is, from the geometry of Ž 9 by the procedure in Refs w0,4,13,3 x Following the discussion in BTZ black holes, it is easy 5D

293 HJW Muller-Kirsten et alrphysics Letters B 445 ( 1999) to see that the greybody factors in 5D black holes is sabs 5D sh5d sabs BTZ Similarly this conclusion is also true for 4D black holes This indicates that the greybody factors in 5D and 4D black holes have their own origin in BTZ black holes, and the dynamical information of 5D and 4D black holes are encoded in BTZ black holes Thus the boundary dynamics of BTZ black holes, which is controlled by D CFT, looks like hologram and constrains the essential information of 5D and 4D black holes As we have seen, the conformal dimension of operator OŽ w,w q y dominates the greybody factors, and one need not determine the explicit form of the operator OŽ u,u q y Now let us consider the explicit form for the 4 h,h s 1,1 operator For D1rD5 system in type IIB string theory compactified on T, the NsŽ 4,4 D SCFT can be described by the resolution of the orbifold ŽT 4 Q 1Q 5 rsž QQ w5 x 1 5 By AdSrCFT correspondence, the Ž 1,1 operator can be determined from the symmetries w14x i j OijsE x AE x A, 31 where x A i are the scalar fields in the SCFT under consideration, As1,,,Q1Q5 and i is the vector index of Ž 4 SO 4, the local Lorentz group of T The other possible forms for Ž h,h sž 1,1 operator can be excluded by the symmetries in AdSrCFT correspondence w14 x The interaction between O and minimal scalars h Ž ij ij whose origin is the traceless symmetric deformations of the 4-torus in type IIB string compactified on T 4 is given by H i j Sints d zhije x AE x A 3 On the other hand, the Ž 1,1 operator can be introduced by quantizing BTZ black holes in 3D pure gravity In Ref w6 x, it has been shown that the gauge potentials can be parametrized by a 3 Ž uq e yr a q Ž uq Afs, r y 3 e a Ž u ya Ž u ž / ž a 3 Ž u e yr a q Ž u / e a Ž u ya Ž u q y f r y 3 y q y A sy, Ž 33 y and the asymptotic metric of BTZ black holes takes the form ds sl dr yl e r a y Ž u a q Ž u du du q PPP, Ž 34 q y q y Ž where the irrelevant subleading terms at large r have been omitted Then the action 7 is transformed into where H S s du du O Ž u,u sinž v t y nf, Ž 35 eff q y gravity q y O Ž u,u sa y Ž u a q Ž u, Ž 36 gravity q y q y and the classical solution with its asymptotic form f r,t,f s 1yie y r e iž v qty n q f q 1qie y r e iž v yty n y f, Ž 37 has been exploited to get Ž 35 with vsv yv and nsn yn Comparing Ž 36 with Ž 31 q y q y, one is led to the identification y q E x AE x Ala uq a u y, 38 i which shows that the contribution to O u q,uy from different x A is smeared as seen by the operator O Ž u,u gravity q y Namely 3D pure gravity cannot distinguish between different CFT states with the same expectation value for the operator OŽ u,u q y From these observations, one concludes that 3D gravity is a kind of thermodynamics but gauge theory is statistical mechanics w4 x

294 94 HJW Muller-Kirsten et alrphysics Letters B 445 ( 1999) In the above discussion, we have only considered the greybody factors induced by massive scalar fields It is highly nontrivial to check whether the above identifications for the initial state of black holes in D CFT hold valid also for the spinor field case in the AdSrCFT correspondence As we have seen, the Ž h,h sž 1,1 operator OŽ u,u q y can be easily obtained by quantizing BTZ black holes in 3D gravity, however, it is unclear whether we can get operators OŽ u,u q y of higher conformal dimensions by quantization of 3D gravity If not, it is worth discussing whether and how they can be induced in the context of six-dimensional supergravity on AdS 3 =S 3, since the Kaluza-Klein spectrum of 6D supergravity truncated by stringy exclusion principle matches the spectrum of D SCFT w4,7 x Recently it has been argued that the isometry group SLŽ, R of quantum gravity on AdS can be enlarged to the full infinite-dimensional 1 q 1 conformal group, and the mapping AdS AdS has been found w7 x 3 It would be also interesting to see whether it is possible to find the origin of greybody factors in AdS context as well We hope to return to these issues in near future Acknowledgements We would like to thank E Keski-Vakkuri, D Kutasov and J-L Petersen for useful discussions NO would also like to thank Niels Bohr Institute for their hospitality while part of this work was carried out The work was supported in part by the Japan Society for the Promotion of Science and Danish Research Academy References wx 1 J Maldacena, preprint, hep-thr wx S Gubser, I Klebanov, A Polyakov, preprint, hep-thr wx 3 E Witten, preprint, hep-thr wx 4 J Maldacena, A Strominger, preprint, hep-thr wx 5 E Martinec, preprint, hep-thr wx 6 K Behrndt, I Brunner, I Gaida, preprint, hep-thr ; hep-thr wx 7 F Larsen, preprint, hep-thr980508; J de Boer, preprint, hep-thr wx 8 A Giveon, D Kutasov, N Seiberg, preprint, hep-thr wx 9 A Dhar, G Mandal, S Wadia, Phys Lett B 388 Ž , hep-thr960534; S Das, SD Mathur, Nucl Phys B 478 Ž , hep-thr ; Nucl Phys B 48 Ž , hep-thr w10x J Maldacena, A Strominger, Phys Rev D 55 Ž , hep-thr960906; D 56 Ž , hep-thr w11x CG Callan, S Gubser, I Klebanov, AA Tseytlin, Nucl Phys B 489 Ž , hep-thr961017; I Klebanov, M Krasnitz, Phys Rev D 55 Ž , hep-thr961051; D 56 Ž , hep-thr970316; S Gubser, Phys Rev D 56 Ž , hep-thr ; M Cvetic, F Larsen, preprints, hep-thr ; hep-thr w1x S Gubser, Phys Rev D 56 Ž hep-thr w13x E Teo, preprint, hep-thr w14x J David, G Mandal, S Wadia, preprint, hep-thr w15x M Banados, C Teitelboim, J Zanelli, Phys Rev Lett 69 Ž hep-thr w16x K Sfetsos, K Skenderis, Nucl Phys B 517 Ž , hep-thr ; S Hyun, preprint, hep-thr w17x D Birmingham, I Sachs, S Sen, Phys Lett B 413 Ž hep-thr w18x HW Lee, YS Myung, preprints, hep-thr980800; hep-thr w19x M Henningson, K Sfetsos, preprint, hep-thr980351; W Muck, K Viswanathan, preprints, hep-thr ; hep-thr ; H Liu, AA Tseytlin, preprints, hep-thr ; hep-thr ; T Banks, M Green, preprint, hep-thr ; G Chalmers, H Nastase, K Schalm, R Siebelink, preprint, hep-thr ; A Chezelbach, K Kaviani, S Parvizi, A Fatollahi, preprint, hep-thr980516; S Lee, S Minwalla, M Rangamani, N Seiberg, preprint, hep-thr ; G Arutyunov, S Frolov, preprint, hep-thr980616; E D Hoker, DZ Freedman, W Skiba, preprint, hep-thr ; S Corley, preprint, hep-thr ; A Volvovich, preprint, hep-thr w0x DZ Freedman, SD Mathur, A Matusis, L Rastelli, preprints, hep-thr ; hep-thr

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296 7 January 1999 Physics Letters B String corrections to four point functions in the AdSrCFT correspondence John H Brodie 1, Michael Gutperle Department of Physics, Princeton UniÕersity, Princeton, NJ 08554, USA Received September 1998; revised 6 November 1998 Editor: M Cvetič Abstract In a string calculation to order a X3, we compute an eight-derivative four-dilaton term in the type IIB effective action Ž Following the AdS prescription, we compute the order g YM Nc y3r correction to the four-point correlation function involving the operator trf in four dimensional Ns4 super Yang-Mills using the string corrected type IIB action extending the work of Freedman et al Ž hep-thr In the limit where two of the Yang-Mills operators approach each other, we find that our correction to the four-point correlation functions develops a logarithmic singularity We discuss the possible cancellation of this logarithmic singularities by conjecturing new terms in the type IIB effective action q 1999 Elsevier Science BV All rights reserved PACS: 115Sq; 115Hf Keywords: String-theory; AdS-CFT correspondence 1 Introduction Despite the fact that our universe is presumed not to have a negative cosmological constant, there has been a great deal of interest lately in anti-de Sitter gravity This is because of the remarkable connection between d dimensional superconformal field theories decoupled from gravity and d q 1 dimensional gravitational theories in the AdS background w1 3 x Because the degrees of freedom of the bulk gravitational theory are described by a field theory in one less dimension living on its boundary, the AdSrCFT correspondence is an example of the holographic principle put forward by w4,5 x ŽSee also wx 6 In w,3x it was shown how to compute two-point correlation functions of operators in the conformal field theory by doing a corresponding supergravity calculation in the AdS background The prescription given for computing n-point correlation functions is to use 1 brodie@puhep1princetonedu gutperle@feynmanprincetonedu r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

297 ( ) JH Brodie, M GutperlerPhysics Letters B the bulk supergravity action as the generating function for correlation functions in the boundary conformal field theory d d ² : IIB OŽ x 1 OŽ xn s S eff, Ž 1 df Ž x df Ž x n where OŽ x is an operator in the conformal field theory and f0 is the bulk field value at the boundary Subsequently, three-point correlation functions were computed in w7 10x from the correspondence, while X leading order in a contributions to four-point correlation functions were examined in w11 13 x Unlike two-point and three- point correlation functions which are determined by the conformal symmetries, four-point functions are determined only up to a function of cross ratios x1 x34 x1 x34 rs, hs x x x x In w14x it was shown that non-perturbative corrections to four-point-correlation functions of four dimensional Ns4 super Yang-Mills can be calculated in the AdSrCFT correspondence by including stringy R 4 corrections to the effective type IIB supergravity action 1 IIB 10 yf X 3 yf r 4 eff ' X 4 4 H S s d x g e RqKa e f Ž t,t R Ž 3 a 5 Ž 4 since AdS =S is still a solution of 3 The f prefactor to the R term was argued in w15x 5 4 to be a generalized Eisenstein series t 3r 4 Ý 3 NmqntN m,n / 0,0 f Ž t,t s Ž 4 which for large e yf has the asymptotic expansion p 1 1r 3r 1r p imt yp imt t f4sz Ž 3Ž t q q4p ÝŽ Mt Ý Ž e qe 3 m = M M< m ž / ks1 ` G Ž ky1r yk 1q Ý Ž 4p Mt, Ž 5 G Ž yky1r k! where tsxqie and x is the RR scalar The first term in 5 comes from a string tree calculation, the second from a 1-loop calculation, and the infinite series from charge M D-instantons The prefactor Ž 5 has w x 4 recently been proven by 16 to be the exact coefficient to the R term Since in the AdSrCFT correspondence the energy-momentum tensor Tmn of the d dimensional boundary conformal field theory maps to the graviton in the d q 1 dimensional bulk theory, the four-energy-momentum-tensor correlation function in four dimensional 4 N s 4 SU N SYM is related to the R term of IIB superstring theory via Ž 1 after substituting Ž c 3 The L Ž 4 prefactor 5 expressed in terms of Yang-Mills parameters, l s g N s X YM c a and g YMs 4p gs takes the following form: z Ž 3 p l 1r f4ž l, Nc s q 3r 1r 4pl N Ž 4p yf Ž c 3r Ž 4p 1r ym Ž8p r g YM qiu ym Ž8p r g YMyiu 3r Ý M Nc M q Z M e qe Ž 6 Note that the last term in Ž 6 comes from Yang-Mills instantons and this indicates that D-instantons in AdS5 can be identified with instantons in the SYM w17,18 x

298 98 ( ) JH Brodie, M GutperlerPhysics Letters B Interestingly there are many other terms in the IIB effective action that are related to the R 4 term by supersymmetry We will show that one such term involving only the scalar dilaton field is 1 10 X 3 yf r m n a b H ' X 4 4 m n a b d x yg a e f Ž t,t D D f Ž x D D f Ž x D D f Ž x D D f Ž x, Ž 7 a Ž where f x is the dilaton In this note we calculate the g YM Nc y3r correction to the four-point correlation function of the trace of the Ns4 Yang-Mills field strength squared, tržf, from the term Ž 7 in the IIB 5 effective action expanded about the AdS = S background In Section, we derive Ž 5 7 by doing a string theory calculation in Minkowski space In Section 3, we compactify IIB string theory on AdS 5= S5 and in the Maldacena limit wx 1 we calculate the result for the four point correlation function of tržf using the AdSrCFT correspondence In Section 4, we show that in the limit where two operators in the CFT approach each other the contribution to the four-point functions calculated in Section 3 exhibit logarithmic singularities If these logarithmic singularities were really present this would have rather dramatic consequences for the conformal field theory; namely, it would imply that Ns4 SYM is a logarithmic CFT w19 x In Section 5, we discuss how including terms in the effective action containing the Riemann curvature and five form field strength and four scalars could cancel the logarithmic singularities Four-point functions in Minkowski space The tree four-point scattering amplitudes for the massless states of type IIB superstring theory can be expressed in a compact form Asc A Ž s,t,u KŽ z,k,,z,k K z,k,,z, Ž 8 tree where the tree level amplitude is given by 1 G Ž ysr4 G Ž ytr4 G Ž yur4 Atree s Ž 9 g G Ž 1qsr4 G Ž 1qtr4 G Ž 1qur4 st The kinematic factor KK is the same for the tree and one loop amplitude and is the product of two kinematic factor for the open string four point function This reflects the fact that the closed string can be viewed as the product of a left and rightmoving open string In the kinematic factor K for the open string the wavefunction z i for the i-th particle can either be a vector jm or a Majorana-Weyl spinor u a For four vectors the kinematic factor takes the form 8 m n l k r s t v 1 K Ž j 1,j,j 3,j4 stmnlkrstv j 1 k1j kj 3k3j4k4 sy4 Ž stj 1Pj3jPj4qsuj Pj3j1Pj4q PPP, Ž 10 where the detailed expression for t can be found in w0 x 8 The massless states of the IIB superstring are given by the tensor product zlmz R The bosonic states are given by jmm jns hž mn q bw mn x where b denotes the NS-NS antisymmetric tensor and h contains the transverse and traceless graviton together with the transverse dilaton In the following we will be particularly interested in the four point function for the dilaton f The scattering amplitude Ž 8 contains an infinite number of poles in the s,t and u channel, coming from massive intermediate string states In a low energy effective field approximation the effect of this stringy X behavior can be seen by an a expansion of Ž 8 which effectively integrates out all the massive intermediate string states at tree-level and leads to an infinite series of higher derivative contact terms This expansion was performed in w1x G Ž y1ysr8 G Ž y1ytr8 G Ž y1yur8 1 X 3 X 5 s qa zž 3 qož a Ž 11 G Ž qsr8 G Ž qtr8 G Ž qur8 stu

299 ( ) JH Brodie, M GutperlerPhysics Letters B Since the kinematic tensor in Ž 8 contains eight momenta, it is easy to see on dimensional grounds that the first term in Ž 11 corresponds to a two derivative contact term and does not give a new term in the effective action This is produced by the lowest order terms in the action like the Einstein term The next term in Ž 11 corresponds to a new eight-derivative string theory correction in the IIB supergravity effective action We will focus on the eight derivative four dilaton amplitude which will be related to the four point function of tržf in the AdSrCFT correspondence ' The dilaton polarization tensor is given by jm mjnsfr 8 hmnynmknynnk m The nm satisfy n s0,npk n s1 and are necessary to make the dilaton polarization vector transverse, ie zmn k s0 Plugging this into KK with K and K given by Ž 10 leads to f 4 a1aa3a4a5a6a7a8 b1 b b3b4b5b6 b7b a1 b1 ab a7 b7 a8b8 16 KKst t k k j PPP k k j s Ž s t qs u qt u f s Ž s qt qu 3 Ž 1 the normalization f of the dilaton vertex can be fixed by comparing the three point functions involving two dilatons and one graviton with the three point function coming from the kinetic term for the dilaton in the IIB effective action The eight derivative term which reproduces this S-matrix element is given by z Ž 3 a X 3 Ž1 10 m n r S sc d x' g D D fd DfD DfD D f f4 H m n r l l g s Ž 13 X 3 Apart from the tree level contribution Ž 9, the four point amplitudes at order a receives a one-loop and non-perturbative D-instanton contributions The exact form was conjectured in w15x to be 1r A4sKKt f4ž t,t, Ž 14 where f Ž 4 is a nonholomorphic Eisenstein series In the weak coupling expansion Ž 5 of Ž 14 4 the first two term are the tree level and one loop contributions, whereas the rest is an infinite series of D-instanton contributions and perturbative fluctuations around the D-instantons The form of the terms in the IIB effective action which obey a non renormalization theorem w16x can be encoded in an integral over half the IIB superspace wx of an appropriate power of a a constrained linearized a superfield F x,u where u,as1,,16 The superfield satisfies 4 4 DFs0, D FsD F Ž 15 and has an expansion up to order u 8 given by 8 4 Fstqulq PPP u D t, Ž 16 where the complex scalar t contains the dilaton and RR-scalar tsxqie yf The four point function can the be expressed as an integral over half the superspace ž / m n r l Re Hd uf s Ž t D t D tqt D t D t sdm Dnt D D t DDt r l D D t, 17 where we integrated by parts 3 Four point function in AdSrCFT The AdSrCFT correspondence allows one to calculate correlation functions of operators in the N s 4 SYM using the effective action of IIB superstring theory We will focus on terms in the effective action which involve eight derivatives on four dilatons The dilaton f of IIB on AdS 5= S5 corresponds to the marginal operator trž F

300 300 ( ) JH Brodie, M GutperlerPhysics Letters B X 3 We will concentrate on an a four dilaton term Ž 13 compactified on AdS 5= S5 which is given by 5 dz Ž1 m n r S s ' gd DfD DfD DfD D f, Ž 18 f4 H 5 m n r l l z 0 where we drop the S dependence Ž which only gives a volume factor 5 The prescription to calculate CFT correlators using the IIB effective action is given by W Ž f ss Ž f, Ž 19 CFT CFT IIB bulk where WCFT in the generating function for the conformal field theory Here SIIB is the IIB effective action compactified on AdS = S and the bulk and CFT fields are related Ž for scalars of dimension D by H 5 5 f Ž z sc d 4 xk Ž z, x f Ž x Ž 0 bulk D CFT A correlator of operators in the CFT associated with fcft can then be calculated by d d d d IŽ x, x, x, x s² O x O x O x O x : Ž 1 Ž 3 Ž 4 CFT s W Ž f df Ž x1 df Ž x df Ž x3 df Ž x4 Ž 1 In order to calculate the correction to the correlator Is² tržf tržf tržf tržf : coming from the higher derivative term Ž 18, and use the prescription Ž 1 which gives d 4 zdz s 0 m n r l I s D D K Ž z, x D D K Ž z, x D D K Ž z, x D D K Ž z, x H 5 m n D 1 D r l D 3 D 4 z 0 s t u Here I denotes the s-channel part of the amplitude There are also I and I which are obtained from by exchanging xlx4 and xlx3 respectively, and the complete correlator is given by IsI s qi t qi u The metric on the five dimensional euclidean AdS space is given by 1 g s d, m,ns0,,4 Ž 3 mn z 0 mn Hence the nonvanishing components of the metric connection are given by i 0 Gij s d ij, G0 jsy d ij, G00sy, Ž 4 z z z where i, js1 4 The bulk to boundary Greens function for a field of conformal dimension D is given by ž / z 0 D D z0 q Ž zyx D K sl Ž 5 For notational simplicity we leave out the normalization factor ld sg Ž D r p d r G Ž Dydr Ž 6 from Ž 5 in the following which can be easily reinstated at the end The second order covariant derivatives with respect to the connection Ž 4 of the propagator D D K Ž z, x are given by m n D z0 Dy z0 D 0 0 D y4dž Dq1 D z Dq1 q zyx z 0 0 q Ž zyx D D K sd z0 Dq q4dž Dq1, z Dq q Ž zyx Ž 0

301 ( ) JH Brodie, M GutperlerPhysics Letters B z0 Dy1 z0 Dq1 i 0 i D z Dq1 Dq q zyx z 0 0 q Ž zyx i D D K sydž Dq1 Ž zyx y DŽ Dq1 Ž zyx, z0 Dy z0 D i j i j D ij z D Dq q zyx z 0 0 q Ž zyx DDK sydd q4dž Dq1 Ž zyx Ž zyx Ž 7 mn rl To calculate g g D D K Ž z, x D D K Ž z, x we use the formulas Ž 7 and the metric Ž 3 m r D i n l D j The resulting expression can be simplified using Ž zyx PŽ zyx s1rž zyx qž zyx yž x yx 4 i j i j i j The result is given by the identity g mn g rl Dm DrKDŽ z, xi Dn DlKDŽ z, x j sd Ž D qd KDŽ z, xi KDŽ z, x j y4d Ž Dq1 x K Ž z, x K Ž z, x ij Dq1 i Dq1 j 4 ij Dq i Dq j q4d Ž Dq1 x K Ž z, x K Ž z, x Ž 8 Note that once Ž 8 is expressed in terms of K D s, all terms with extra factors of z0 cancel This fact will be important for the conformal invariance of the resulting CFT correlators s Using 8 I can be brought into the following form: s I sd Ž D qd I y4d Ž Dq1 Ž D qd x I qx I yx I where I DDDD 1 Dq1Dq1DD 34 DDDq1Dq1 1 Dq Dq DD 4 yx I q16d Ž Dq1 x x I yx x I yx 4 x I qx x I, Ž 9 D1DDD DDDq Dq 1 34 Dq Dq Dq Dq 1 34 Dq1Dq1Dq Dq 1 34 Dq Dq Dq1Dq Dq1Dq1Dq1Dq14 is defined by d 5 z I s K Ž z, x K Ž z, x K Ž z, x K Ž z, x Ž 30 H D1DDD D1 1 D D3 3 D4 4 z 0 t u I can be obtained from 9 by permuting xl x4 and the arguments Dl D4 in IDDDD and similarly I can be obtained by xlx3 and the arguments DlD3 in I DDDD Logarithmic singularities in the four point function We want to investigate the corrections to correlators of the four dimensional boundary conformal field theory coming from the eight derivative terms I s, I t and I u of the bulk supergravity theory In particular we want to analyze the behavior of these four point functions when two operators come close, ie, x ij 0 with all the other separations remaining finite The most useful representation of I for this purpose is given by Ž 51 D DDD in the appendix b Dy1 b D 4y1 4 ID DDDsCHdbdb 4 Ž 31 D DryD x qb x qb x b x qb x qb b x Ž Ž Here we denote DsÝ 4 is1d i It is easy to see that if x1 0 and all other x ij remain finite the only region of the integration which can give a singularity is b4 0 Extracting b from the second term in the denominator X Ž and redefining b s b x q b x rb gives I D DDD db D 4y1 b D qd 3 qd 4 yd ry1 4 1 schdb db 4 DryD D D Ž x1 qb4 Ž x14 qb x4 Ž x13qbx3qb4bx34rž x14qbx4 Ž 3

302 30 ( ) JH Brodie, M GutperlerPhysics Letters B It is easy to see that Ž 3 displays a divergence 1rŽx 1 k as x1 0 if D3 qd4 sdrqk and it displays a Ž logarithmic divergence ln x rx x if D3 qd4 sdr Note that if there is a power singularity of certain order in Ž 3 then expanding the rest of the integral as a power series in b4 also produces lower order singularities As we shall see only the logarithmic singularities will be important in the calculation of the four point function Setting D3qD4sDr, the singular part of the integral 3 then becomes / 1 1 x b D y1 1 I D DDD;C ln Hdb qož 1 Ž 33 D D Dr x x ž x13 x14 Ž 1qb Ž 13 Ž 14 In this limit the integral over b simply gives a Euler beta function and together with the value of C given in Ž 5 the coefficient of the log singularity is given by / p x1 I D DDD; ln q ož 1 Ž 34 D D ž Ž Dry1 Ž Dry Ž x x x 13 x 13 Ž s The behavior of I in Ž 9 as x1 0 is easy to analyze, all IDDDD which produce power singularities, ie have D4qD 4)Dr are multiplied by factors of x1 such that the result is nonsingular In addition the only logarithmic singularity comes from the first term in Ž 9 since all the other logarithmic divergent terms are also multiplied by factors of x1 On the other hand one finds that all the terms in I t and I u display logarithmic singularities as x1 0, and the complete logarithmic singularity is given by summing over all terms taking the prefactors appearing in Ž 9 and Ž 34 into account The final result is 1 1 I;c1 lnž x rx x, Ž 35 D D x x Ž 13 Ž 14 where the numerical constant c is given by D 5 pl ½ 3D D qd 3 D Dq1 D qd 16P1 D Dq1 c1 s y q, Ž Dy1Ž Dy Ž Dq1Ž DŽ Dy1 Ž Dq3Ž DqŽ Dq1 D where we reinstated the normalization l defined in Ž 6 of the propagators Ž 5 D It is easy to see that c1 does not vanish, in particular for the case of the dilaton and AdS we set ds4 and Ds4 and get 5 Ž c1 s Ž 37 6 p 77 5 Possible cancellations of logarithmic terms In the last section we found we found a logarithmic singularity in the four point function of Ns4 SYM in Ž the large N c limit at order 1r gym N c 3r Similar logarithmic singularities at leading order in four-point functions were found in w1x coming from the tree level two derivative part of the action In w1 x, it was speculated that by adding all diagrams the logarithmic singularities might cancel We will now address this 3 We drop nonsingular terms which are important for conformal invariance

303 ( ) JH Brodie, M GutperlerPhysics Letters B possibility of cancellation of logarithmic singularities in our case We found a nonzero logarithmic contribution from summing the s,t and u channel of diagrams coming from Ž 13 Hence additional four point functions 4 which might cancel the logarithmic singularities in 35 must come from other terms in the effective action This is possible because in contrast to the flat Minkowski background the AdS5 background has a nonvanishing curvature tensor as well as a nonvanishing five-form field strength 1 1 Rmnlrsy Ž gml gnry gmr g nl, Fmnlrss e mnlrs Ž 38 L L X Hence there is the possibility that terms of the same order in a as the eight derivative term Ž 13 which involve curvature tensors or five form field strengths together with four scalars contribute to the four point function in the conformal field theory once the constant background values Ž 38 are substituted Examples of such terms are four dilatons with six derivatives together with one Riemann tensor or two F5 as well as terms with two curvature tensors,four F5 or one Riemann tensor and two F5 and four dilatons with four derivatives Such terms should be in the supersymmetric completion of the R 4 term like the eight derivative term of four scalars After using Ž 38 the generic form of the additional four-scalar terms in the effective action is given by d 5 z d 5 z Ž m n r Ž3 m n S 4 ; H D fd fd DfD Df, S 4 ; H D fd fd fd f Ž 39 f 5 m n r f 5 z z m n 0 0 Note that there is an additional eight derivative term where the indices are contracted cyclicly In flat space this contraction of the derivatives is related to Ž 13 by integration by parts In the AdS background the issue of the noncommutative nature of the covariant derivatives becomes important and an integration by parts relates the two eight derivative terms up to terms involving the curvature and two less derivatives on the scalars, since l D m, Dn DrfsRmnrDlf 40 It is easy to repeat the analysis of Section 4 for the terms Ž 39 of the effective action and it follows that the logarithmic singularity of the four point correlation function of the CFT as x1 0 induced by these terms in exactly of the same form as Ž 35 The result for the numerical coefficients c and c3 multiplying the singularities is D pl 8y D D qd d 3 D Dq1 Dy c s q, ½ Ž Dy1Ž Dy D Ž DqŽ Dq1 DŽ Dy1 p l 4 D 3D 4 8D 4 8D 4 c3 s ½ y q Ž 41 Dy1 Dy D Dy1 Dq1 D For the four dilaton amplitude in AdS inserting Ds4,ds4 into Ž 41 5 gives cs, c s Ž p 7 p 7 The supersymmetric completion of the R 4 term in the ten dimensional IIB effective action should in principle uniquely determine the relative coefficients of these terms Unfortunately the details of this are not known and either the supersymmetrization of the R 4 term or a calculation of five point and six point S-matrix elements in string perturbation theory involving one or two gravitons and four dilatons around flat space to determine the relative normalizations of the three terms seems to be difficult 5 We are not able to confirm or defute the 4 We are grateful to A Rajaraman for raising this possibility 5 w x 4 See Ref 3 for a discussion of the supersymmetrizations of R in the ds10, Ns1 context

304 304 ( ) JH Brodie, M GutperlerPhysics Letters B cancellation of the logarithmic singularities coming from the a X 3 moment terms in the IIB effective action at this 6 Conclusions In this note, we computed the corrections to the four-point correlation function of the operator trf from the eight-derivative-four-dilaton term in the IIB effective action using the AdSrCFT correspondence We found that as two of the Yang-Mills operators approach each other, the four point function yields a logarithmic singularity, like the logarithmic singularities found at a lower order in w1 x This may indicate that the theory is a logarithmic conformal field theory Logarithmic conformal field theories in two-dimensional are non-unitary If this were the case in four-dimensions, it would certainly be a shocking result It would imply that AdS gravity is described holographically by a non-unitary conformal field theory! Clearly, the most satisfying solution would be if the logarithmic singularities in our four-point calculation Ž 35 were cancelled by other terms in the effective action However, the coefficients required for cancellation seem peculiar and unfortunately are difficult to calculate However, the fact that the logarithmic singularities do appear in our calculations and in w1x seems to suggest that they are a generic feature of the AdSrCFT correspondence At present there is the possibility that the CFT on the boundary of AdS either maybe logarithmic if a four dimensional analog of w19x exists or that a new scale is introduced and the theory ceases to be conformal Assuming that the logarithmic singularities do cancel, the Is,t,u terms will modify the CFT four-point function with finite terms which is in agreement with the intuition that the massive string modes which when integrated out do not modify the structure of the CFT significantly Acknowledgements MG gratefully acknowledges the hospitality of the Theory division at CERN at early stages of this work JHB and MG are grateful to the Aspen Center of Physics for providing a stimulating research environment while this work was completed Its a pleasure to thank G Chalmers, D Freedman, E Gimon, MB Green, V Gurarie, A Lawrence, H Liu, S Mathur and especially A Rajaraman for useful conversations and comments The work of MG is partially supported by DOE grant DE-FG0-91ER40671 and NSF grant PHY Appendix A The integrals which appear in the calculation of the four point amplitudes can be partially done using the method of Feynman parameters The basic formula is 1 1 shda1 PPP dand1y Ž Ý a i n Ž A1 X PPP X Ž a X q PPP a X 1 n 1 1 n n Specializing to n s 4 and taking derivatives with respect to X the following formula easily follows i 1 Ž Dy1! a D1y1 PPP a D 4y1 1 4 D D s Hda 1 PPP da 4 dž 1 y Ý a i D, X PPP X4 6Ž D1y1!PPP Ž D4y1! Ž a1x1q PPP qa4x4 Ž A

305 ( ) JH Brodie, M GutperlerPhysics Letters B where DsÝ is1d i In the supergravity amplitudes the factors Xi are given by Xisz0 q zyx i It is easy to see that Ý a X q PPP a X sz qz q aax, Ž A i j ij i-j where x ijs x iy x j The introduction of Feynman parameters makes it straightforward to do the integral over the AdS5 in the four point function The general form of such integrals is dz0d 4 z z0 Nq1 p N! Ž MyNy4! 1 IŽ N, M sh s Ž A4 5 M MyNy3 z0 Ž My1! z qz q aax aax ž 0 Ý i j ij/ ž Ý i j ij/ i-j The integrals which we want to evaluate in the AdS bulk are in general of the following form dz0 d 4 z ID DDDsH K Ž z, x K Ž z, x K Ž z, x K Ž z, x, Ž A5 5 D 1 D D 3 D z 0 where K is defined in Ž 5 Using Ž A gives D I D DDD Ž Dy1! dz d 4 z a D1y1 PPP a D 4y1 0 D 1 4 s Hda1 PPP da4dž 1 y Ý ai H z 5 D 6Ž D1y1!PPP Ž D4y1! z0 z qz q aax Hence the integral over the AdS bulk variable z can be done using A4 giving I D DDD i-j ž Ý 0 i j ij/ p Ž Dry3! Ž Dry1! a D1y1 PPP a D 4y1 1 4 s Hda 1 PPP da 4 dž 1 y Ý a i Dr Ž D 1 y1! Ž D y1! Ž D 3 y1! Ž D 4 y1! aax i-j ž Ý i j ij / i-j Ž A6 Ž A7 In w13x such integrals were evaluated by introducing new variables b1sa 1,bsa1a,b sa1a,b 3 4sa1a 4 The integral over b can than be done by using the d-function constraint 1 a D1y1 PPP a D 4y1 1 4 Hda 1 PPP da 4 dž 1y Ý a i Dr aax ž Ý i j ij / i-j b Dy1 b D3y1 b D 4y1 3 4 shdb db 3 db 4 Dr b x qb x qb x qb b x qb b x qb b x Ž Ž D y1! Ž DryD y1! b Dy1 b D 4y s db db Ž Dry1! x qb x qb x 1 = Ž b x qb x qb b x DryD H 4 D , Ž A8

306 306 ( ) JH Brodie, M GutperlerPhysics Letters B where the elementary integral over b3 was done to get the third line A useful integral representation for I is therefore given by b Dy1 b D 4y1 4 ID DDDsCHdbdb 4, Ž A9 D DryD x qb x qb x b x qb x qb b x Ž Ž where the numerical constant C depends on D i,is1,,4 p Ž Dry3! Ž DryD3 y1! Cs Ž A10 Ž D y1! Ž D y1! Ž D y1! 1 4 The representation Ž A9 will be most useful for the analysis of the logarithmic singularities involving the four-point functions The conformal covariance properties of these integrals can be made manifest by expressing one integral in terms of a hypergeometric functions which depends only on the conformally invariant cross ratios ŽŽ see w13,1 x References wx 1 J Maldacena, The Large N limit of superconformal field theories and supergravity, hep-thr wx SS Gubser, IR Klebanov, AM Polyakov, Phys Lett B 48 Ž hep-thr wx 3 E Witten, Anti-de Sitter space and holography, hep-thr wx 4 G t Hooft, Dimensional reduction in quantum gravity, in: Salamfest, World scientific, Singapore, 1993 wx 5 L Susskind, J Math Phys 36 Ž hep-thr wx 6 L Susskind, E Witten, The Holographic bound in anti-de Sitter space, hep-thr wx 7 S Lee, S Minwalla, M Rangamani, N Seiberg, Three point functions of chiral operators in Ds4, Ns4 SYM at large N, hep-thr wx 8 G Chalmers, H Nastase, K Schalm, R Siebelink, R current correlators in Ns4 superyang-mills theory from anti-de Sitter supergravity, hep-thr wx 9 DZ Freedman, SD Mathur, A Matusis, L Rastelli, Correlation functions in the CFT Ž d r AdS Ž dq1 correspondence, hep-thr w10x E D Hoker, DZ Freedman, W Skiba, Field theory tests for correlators in the AdS r CFT correspondence, hep-thr w11x H Liu, AA Tseytlin, On four point functions in the CFT r AdS correspondence, hep-thr w1x DZ Freedman, SD Mathur, A Matusis, L Rastelli, Comments on 4 point functions in the CFT r AdS correspondence, hep-thr w13x W Muck, KS Viswanathan, Phys Rev D 58 Ž hep-thr w14x T Banks, MB Green, JHEP 05 Ž hep-thr w15x MB Green, M Gutperle, Nucl Phys B 498 Ž hep-thr w16x MB Green, S Sethi, Supersymmetry constraints on type IIB supergravity, hep-thr w17x M Bianchi, MB Green, S Kovacs, G Rossi, Instantons in supersymmetric Yang-Mills and D instantons in IIB superstring theory, hep-thr w18x N Dorey, VV Khoze, MP Mattis, S Vandoren, Yang-Mills instantons in the large N limit and the AdSrCFT correspondence, hep-thr w19x V Gurarie, Nucl Phys B 410 Ž hep-thr w0x MB Green, JH Schwarz, E Witten, Superstring Theory, Cambridge University Press, 1986 w1x DJ Gross, E Witten, Nucl Phys B 77 Ž wx PS Howe, PC West, Nucl Phys B 38 Ž w3x M de Roo, H Suelmann, A Wiedemann, Nucl Phys B 405 Ž

307 7 January 1999 Physics Letters B Thermodynamics of D0-branes in matrix theory 1 J Ambjørn a,, YM Makeenko b,3, GW Semenoff c,4 a Niels Bohr Institute, BlegdamsÕej 17, Copenhagen 100, Denmark b Institute of Theoretical and Experimental Physics, B Cheremushkinskaya 5, Moscow 11718, Russian Federation c Department of Physics and Astronomy, UniÕersity of British Columbia, 64 Agricultural Road, VancouÕer, BC V6T 1Z1, Canada Received 6 October 1998 Editor: L Alvarez-Gaumé Abstract We examine the matrix theory representation of D0-brane dynamics at finite temperature In this case, violation of supersymmetry by temperature leads to a non-trivial static potential between D0-branes at any finite temperature We compute the static potential in the 1-loop approximation and show that it is short-ranged and attractive We compare the result with the computations in superstring theory We show that thermal states of D0-branes can be reproduced by matrix theory only when certain care is taken in integration over the moduli space of classical solutions in compactified time q 1999 Elsevier Science BV All rights reserved 1 Introduction Dirichlet p-branes are p q 1-dimensional hypersurfaces on which superstrings can begin and end Žsee w1,x for a review The low energy dynamics of an ensemble of N parallel Dp-branes can be described by the UŽ N supersymmetric gauge theory obtained by dimensional reduction of ten dimensional supersymmetric Yang-Mills theory to the p q 1-dimensional world-volume of the brane wx 3 The Yang-Mills theory gives an accurate perturbative 1 This work is supported in part by NATO Grant CRG , by NSERC and by MaPhySto Centre for Mathematical Physics and Stochastics ambjorn@nbidk 3 makeenko@itepru 4 semenoff@physicsubcca representation of the Dp-brane dynamics when the separations between the branes is large w4 6 x It represents a truncation of the full string spectrum to the lowest energy modes The full string theoretical interactions between a pair of Dp-branes is computed by considering the annulus diagram shown in Fig 1 The short distance asymptotics of this diagram are dominated by the open string sector whose lowest modes are the fields of ten dimensional supersymmetric Yang-Mills theory On the other hand, long distance asymptotics are most conveniently described by the dual description of this diagram as a closed string exchange and the relevant field theoretical modes are those of ten dimensional supergravity That these are also represented by the dimensionally reduced super Yang-Mills theory is a result of supersymmetry and the fact that, for fixed Dp-brane positions, the ground state is a BPS state At zero r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

308 308 ( ) J Ambjørn et alrphysics Letters B Fig 1 Annulus diagram for D0-brane interactions The bold lines represent world-lines of the D0-branes, separated by the distance L, which go along the periodic temporal direction They bound the string world-sheet finite temperature In Section 3 we perform one loop computation of the effective interaction between static D0-branes at finite temperature and show that it is attractive, and short-ranged In Section 4 we compare this result with the superstring computations and discuss the conditions under which the two computations agree Section 5 is devoted to the discussion of our results and, in particular, the origin of the divergence of the classical thermal partition function of D0-branes which is cured by quantum statistics temperature, because of supersymmetry, the interaction potential between a pair of static D0-branes vanishes independently of their separation Their effective action has been computed in an expansion in their velocities, divided by powers of the separation and is known to be w7,6x H ž 1 a X ' Ý s as1 < 1 < 4 X q yq / Seff Ž Ts0 s dt Ž q g a 15 q yq y Ž a q Ž 1 16 < < This result agrees with the effective potential for the interaction of D0-branes in ten dimensional supergravity Note that, for weak string coupling, the D0-brane is very heavy In this paper, we shall consider the description of D0-brane interactions in type IIA superstring theory using matrices Even at very low temperatures, non- BPS states are important to the leading temperature dependence We perform 1-loop computation of the effective interaction between static D0-branes in the matrix theory at finite temperature and compare with the known superstring computations We show that the results of the two computations are similar in the low temperature limit but an extra integration over the temporal component of the gauge field, is present in the matrix theory At finite temperature, because the Euclidean time is compact, the temporal gauge field cannot be removed by a gauge transformation This integration is needed in order to describe correctly thermodynamics of D0-branes both in the matrix and superstring theories The paper is organized as follows In Section we discuss the formulation of the matrix theory at Matrix theory at finite temperature We shall consider the matrix theory description wx 8 of the effective dynamics of D0-branes in a type-iia superstring theory which is derived by the reduction of ten dimensional supersymmetric Yang- Mills theory which has the action / 1 1 i SYM wa,u xs Hdt TR Fmnq ugm Dmu g YM ž 4 to zero spatial dimension: A sa Ž t, usuž t m m The thermal partition function of this theory is given by H Z s daž t duž t expž ys wa,u x, Ž 3 YM where SYM is the Euclidean action and the time coordinate is periodic The bosonic and fermionic coordinates have periodic and anti-periodic boundary conditions, A Ž tqb sa Ž t, Ž 4 m m už tqb syuž t, Ž 5 YM bs1rkbt, Ž 6 where T is the temperature and k B is Boltzmann s constant Gauge fixing will be necessary and will involve introducing ghost fields which will have periodic boundary conditions The representation Ž 3 of the thermal partition function can be derived in the standard way starting

309 ( ) J Ambjørn et alrphysics Letters B from the known Hamiltonian of the matrix theory wx 8 and representing the thermal partition function ZYM str e yb H Ž 7 via the path integral The trace is calculated here over all states obeying Gauss s law which is taken care by the integration over A in Ž 0 3 This representation of the matrix theory at finite temperature has been already discussed w9 11 x, but the temperature induced interaction between D0-branes described below was never identified In matrix theory, the diagonal components of the gauge fields, a a 'A aa, are interpreted as the position coordinates of the a-th D0-brane and they should be treated as collective variables Static configurations play a special role since they satisfy classical equations of motion with the periodic boundary conditions and dominate the path integral as g YM 0 Notice that there are no such static zero modes for fermionic components since they would not satisfy the antiperiodic boundary conditions 5 This is an important difference from the zero temperature case and a manifestation of the fact that supersymmetry is explicitly broken at non-zero temperature In the following, we will construct an effective action for these coordinates by integrating the off-diagonal gauge fields, the fermionic variables and the ghosts, H Ł a a ab effw x 0 m w xw x a/b S a ' yln da da du dghost =exp ys ys ys Ž 8 YM gf gh Generally, this integration can only be done in the a simultaneous loop expansion and expansion in the number of derivatives of the coordinates a a Such an a b expansion is accurate in the limit where a y a are large for each pair of D0-branes and where the velocities are small Since these variables are periodic in Euclidean time, small velocities are only possible at low temperatures The remaining dynami- 5 This is a difference between our computation at finite temperature and computations of the Witten index for the matrix theory where fermions obey periodic boundary conditions cal problem then defines the statistical mechanics of a gas of D0-branes, HŁ Ž w x a a ZYM s da t exp yseff a 9 t,a We expect that the zero temperature limit of S eff reduces to Ž 1 We shall find several subtleties with this formulation If the effective D0-brane action is to reproduce the results of a string theoretical computation, the integration over a0 a must be performed in both cases The effective action is a symmetric functional of the position variables a a Ž t Only the configuration of these coordinates needs to be periodic Therefore the individual position should be periodic up to a permutation The variables in the path integral Ž 9 should therefore be periodic up to a permutation and the integral should be summed over the permutations 3 One loop computation We will compute the effective action Seff in a simultaneous expansion in the number of loops and in powers of time derivatives of the D0-brane positions We decompose the gauge field into diagonal and off-diagonal parts, ab a ab ab Am sam d qgym A m, 10 where A aa m s0 so that the curvature is ab ab a ab ab ab ab Fmn sd fmnqgym Dm An ygym Dn Am ab yig YM A m, A n, 11 where fmn a sema n a yena m a Ž 1 and D ab se yi a a ya b Ž 13 m m m m In the Yang-Mills term in the action, we keep all orders of the diagonal parts of the gauge field and

310 310 ( ) J Ambjørn et alrphysics Letters B expand up to second order in the off-diagonal components, 1 g YM TRŽ Fmn 1 a ba ba ab Ž mn Ý m mn l l a g YM ab Ý s f q A d D D ba ab a b ab m n mn mn n y D D qi f yf A q We will fix the gauge ž / Ž 14 ab ab Dm Am s0 15 This entails adding the Fadeev-Popov ghost term to the action H Ý ½ ab ab ba m S s c yd c gh ab ba ab qig YM c Dm A m,c 16 There is a residual gauge invariance under the abelian transformation, A A e, a a qe x Ž 17 a b ab ab iž x yx a a a m m m m m We shall use this gauge freedom to set the additional condition E 0a0 a s0 Ž 18 and to fix the constant 6 yprb-a0 a Fprb Ž 19 The ghost for this gauge fixing condition decouples 6 These constants are related to the eigenvalues of the holonomy ž / b 1 N i b a 0 ib a H 0 Ž 0 0 Pexp i dt A t s V diag e,,e V known as the Polyakov loop winding along the compact Euclidean time It cannot be made trivial by the gauge transformation if T /0 5 Keeping terms up to quadratic order in A,c,c,u, the action is 1 a 1 ba ba ab S sh½ ÝŽ fmn q ÝAm Žydmn Dl D 4 g l YM a ab / qd D qi f yf A ba ab a b ab m n mn mn n i ba ab ab ba ab ab q c Ž D c q u g D u Ý m m m 5 ab Ž 0 The effective action obtained by integrating over A,c,c,u is HÝ S s f eff 1 a Ž mn a 4 g YM 1 ab Ý ½ ž mnž m a/b mn mn / ab 1 ab ž Ž m / Ž m m 5 q TRln yd D qiž f a yf b ytrln y D y TRln ig D 31 Leading order in time deriõatiões Ž 1 We will evaluate the determinants on the righthand-side of Ž 1 in an expansion in powers of the derivatives of až t The leading order term can be found by setting asconst In this case, fmn a s0 and Žhere we retain the tree-level term with time derivatives eff B ž Ž m / a-b ab F ž Ž m / 5 Ý ½ S s8 TR ln y D ab ytr ln y D, where the subscript B denotes contributions from the gauge fields and ghosts, whereas F denotes those from the adjoint fermions The determinants should be evaluated with periodic boundary conditions for bosons and anti-periodic boundary conditions for fermions ŽNote that, because of supersymmetry, if both bosons and fermions had identical boundary conditions the determinants would cancel This would

311 ( ) J Ambjørn et alrphysics Letters B give the well-known result that the lowest energy state is a BPS state whose energy does not depend on the relative separation of the D0-branes The boundary conditions are taken into account by introducing Matsubara frequencies, so that da0 a yseff ys0 N e s e b H p ž / 0 ž / 8 p n p a b ` q qa0 y a0 q < aay ab< b b = ŁŁ, 3 a-b nsy` p n a b q a0 y a0 q < aay ab< b Using the formula ž / ` p n bv qv ssin, Ž 4 b Ł ž / nsy` we obtain the result e ys eff se ys0 = da0 a N b H p 8 coshb < a ya < qcosbž a ya ž / Ž 5 a b a b 0 0 Ł < a b< a b a-b coshb a ya ycosb a0ya0 In order to find the effective action for a a, it is now necessary to integrate the temporal gauge fields a0 a over the domain Žyprb,prb x This integration implements the projection onto the gauge invariant eigenstates of the matrix theory Hamiltonian In the case where there is a single pair of D0- a branes, Ns, the integration over a in Ž 5 0 can be done explicitly to obtain the effective action ž / ½ 5 a b a 1 P z SeffsH dt Ý y ln, g YM b Ž 1yz where PŽ z s1q41 z q1649z 4 q54009z 6 Ž 6 q z 8 q z 10 q z 1 q z 14 q z 16 q z 18 q146779z 0 q43171 z q54919z 4 q1439z 6 q71 z 8 yz 30, Ž 7 Ž < 1 zsexp yb a ya < and we have included the tree level term, which gives the non-relativistic kinetic energies of the D0-branes The effective action has the low temperature expansion b 1 SeffsH dt Ž a ž Ý a 0 g YM a y b < a ya < y4b < a ya < Ž y 56e y16384e b q e q Ž y6 b < a ya < 3 We shall compare in the following section this result with the superstring computation of the effective interaction between D0-branes / 4 String theoretical interactions The effective interactions of D0-branes in superstring theory is given by computing the annulus diagram shown in Fig 1 This was done in Ref w1x Žand in w13x for Dp-branes The result of summing over physical Ž GSO projected superstring states gives the free energy 8 ` dl X yl lr4p a FwL,b,n xs X H e ' 3r pa 0 l where ž / ž / ynl ib ` 1qe = Q n X Ł, ynl pa l ns1 1ye Ž 9 ` yp zž qq1 r4qip Ž qq1n Q n iz s e, Ž 30 Ý qsy` L is the brane separation and n is a parameter which weights the winding numbers of strings around the periodic time direction An extra factor of accounting for the exchange of the two ends of the superwx 1 ending on each of the two D0-branes is string inserted in Ž 9 The product in the integrand represents the sum over string states, with requisite degeneracies, ynl 8 ` 1qe ` ynl 8 Ł ynl s Ý d N e, 31 ž 1ye / ns1 Ns0 8

312 31 ( ) J Ambjørn et alrphysics Letters B where dn is the degeneracy of the either superstring state at level N For the lowest few levels, d0 s8 and E slrpa X 0 Inserting Ž 31 in Ž 9 and integrating over l, the free energy has the form ` yb ENqipn 1ye Ý b N 1qe yb E N qipn Ns0 FŽ b, L,n s d ln Ž 3 where the string energies are given by the formula ( X 4p an L EN s X 1q Ž 33 pa L This results in the partition function yb E qipn d N N ` 1qe yb F ZstrŽ b, L,n 'e s Ł yb ENqipn 1ye Ns0 Ž 34 The physical meaning of the last formula is obvious: the partition function equals the ratio of the Fermi and Bose distributions with the power being twice the degeneracy of states and in playing the role of a chemical potential The factor of in the exponent d in Ž 34 is due to the interchange of the super- N string ends as is already mentioned It will provide the agreement with the matrix theory computation In order to compare with the Yang-Mills computation, we should first re-scale the coordinates so that the mass of the D0-brane appears in the kinetic term Ž as in 1 The mass is given by the formula 1 Ms X Ž 35 g ' a s and the Yang-Mills coupling g YM is related to the string coupling g by the equation s g s g YM s 36 X 3r 4p Ž a The physical coordinate of the a-th D0-brane is identified with q a spa X a a Ž 37 X < 1 Taking Ns in 5 and identifying Lspa a y a <, we see that the integrand in Ž 5 coincides with Ž 34 truncated to the massless modes Ž N s 0 provided nsbža 1 ya 0 0 It is clear that the integral over a0 a is responsible for the mismatch of the effective actions between the string theory computation and matrix theory computation of the free energy In the string theory done in the spirit of Ref w1 x, the parameter n appears in the same place as bža 1 ya 0 0 rp but is not integrated It is associated with the interaction of the ends of the open string with an Abelian gauge field background A Ž t, x : H m S s dx m A Ž 38 int m If the ends are separated by the distance L, eg along the first spatial axis, then b H ns dt A Ž t,0, ya Ž t, L, Ž 39 since x Ž t sž 1,0 m on the boundaries The matrix theory automatically takes into account the integration over the background field while in the string theory calculation of Ref w1x the background field is fixed This is just a reflection of the fact that matrix theory is an effective low-energy theory of D-branes, while the older string theory did not treat the boundaries as dynamical objects However, it is interesting to notice how close some of the earlier string papers came to such a description simply by the requirement of consistency w14 x Further, in the context of matrix theory it is natural to take the exponential of the free energy Ž 3 as in Eq Ž 34, and only integrate over n afterwards, a procedure not entirely obvious in a string theory where the boundaries are not dynamical objects This will be discussed further in the next section An exact coincidence between the matrix theory and superstring results is possible only when the higher stringy modes are suppressed Usually, the truncation of the string spectrum to get Yang-Mills theory is valid for small a X, that is when we are interested in temperatures which are much smaller X than 1r' a In fact, the condition in our case is a little different than this once the length L appears as a parameter in the spectrum Ž 33 Then, the spectrum can be truncated at the first level only when ( X / 1 L 1 L 'kbt< ž q X y X, Ž 40 b pa a pa which is the energy gap between the first two levels If the temperature is small, this condition is always

313 ( ) J Ambjørn et alrphysics Letters B satisfied unless the length L is not too large In other words the truncation of the spectrum to the lightest modes is valid for b4l Ž or TL<1 It is also interesting to discuss what happens in the opposite limit L 4 b where the interaction between D0-branes is mediated by the lightest closed string modes The superstring free energy can be evaluated in this limit by the standard modular transformation which relates the annulus diagram for an open string with a cylinder diagram for a closed string Introducing the new integration variable s s p rl, we rewrite Ž 9 as w1x FwL,b,n x 4 8p ` ds X ib s s yl r sa X 9r 3 X H ž / s e e Q n ' pa 0 s p a = 8 ` 1ye y nq1 s Ł ž y ns 1ye / ns1 Ž 41 In the limit where the brane separation is large the integration over s is concentrated in the region of large s;l Substituting the large-z asymptotics yp z r4 Q n iz cos pn e 4 and evaluating the saddle-point integral, we get X 3r Ž b y8p a FwL,b,n xacosž pn 4 L X X yl = ' b y8p a rpa e Ž 43 Exponentiating and integrating over n, we have X 3 1 b Ž b y8p a H dn Z str Ž b, L,n A 8 y1 L X X yl b y8p a rpa = e Ž 44 Taking into account Ž 37 the exponent at the low temperatures is the same as in Ž 8 but the pre-exponential differs The dependence of the pre-exponential on L in the superstring case emerges because the splitting between energy states in Ž 33 is of order 1rL and the truncation condition Ž 40 is no longer satisfied when L is large Higher stringy modes are then not separated by a gap and the continuum spectrum results in the L-dependence of the preexponential As usual, the limits of L ` and T 0 are not interchangeable in the superstring theory ' 5 Discussion Our main results concern D0-brane dynamics at finite temperatures We have computed the 1-loop effective action for the interaction of static D0-branes in the matrix theory at finite temperature and compared it with the analogous superstring computation We have seen that an extra integration over the eigenvalues of the holonomy along the compactified Euclidean time is present in the matrix theory The two computations agrees in the low temperature limit provided the superstring thermal partition function is integrated over the Abelian gauge fields a s living 0 on D0-branes The integration over a s is of course natural in 0 the context of the Yang-Mills theory, where it expresses that only gauge-invariant states should contribute to TR e yb H But it is also natural from the point of view of the D0-brane physics It can be seen as follows Suppose we make a T-duality transformation, which interchanges the Neumann and Dirichlet boundary conditions, along the compactified Euclidean time direction Then a s become the coordi- 0 nates of D-instantons on the dual circle The integration over a 0 s becomes now the integration over the positions of D-instantons The partition function should involve such an integration over the collective coordinates and since they are collective coordinates the integration appears in front of the exponential of the effective action, not in the action itself Viewed in terms of D0-branes and open strings, we have a gas of D0-branes with open strings between them The individual strings might have a winding number q Žmore precisely q q 1 in the case of superstrings, describing the winding around the finite-temperature space-time cylinder The energy of such states are Abqrpa X However, the q s satisfy Ýqs0 as a result of the integration over a 0 Physically this constraint is most easily understood by going to the closed string channel where we have closed string boundary state on the dual circle with X radius bs4p arb localized at the point Ž nb,q Passing to the momentum representation, we write ` y ipnq Ý 0 qsy` < B,q,n : s e < B,q, p sp qrb : Ž 45

314 314 ( ) J Ambjørn et alrphysics Letters B Here the temporal momentum is quantized as p 0 s X p qrbsq brpa, which lead to the same energy as the above mentioned open string states In this representation Ýq s 0 simply expresses momentum conservation in the thermal direction The effective static potential between two D0- branes emerges because supersymmetry is broken by finite temperature This effect of breaking supersymmetry is somewhat analogous to the velocity effects at zero temperature where the matrix theory and superstring computations agree to the leading order of the velocity expansion wx 6 We have thus shown that the leading term in a low temperature expansion is correctly reproduced by the matrix theory The discrepancy between the matrix theory and superstring computations, which we have observed in the limit of large distances LT 4 1, does not contradict this statement since temperature the limits of large distances and small temperatures are not interchangeable An interesting feature of the effective static potential between D0-branes is that it is logarithmic and attractive at short distances In the matrix theory, the singularity of the computed 1-loop potential occurs when the distance between the D0-branes vanishes and the SUŽ N symmetry which is broken by finite distances is restored The integration over the off-diagonal components can no longer be treated in the 1-loop approximation! In the superstring theory, the singularity is exactly the same as in the matrix theory since it is determined only by the massless bosonic modes Žthe NS sector in the superstring theory Its origin is not due to the presence of massless photon states in the spectrum Putting n 1 s n in the above D-instanton picture on the dual temporal circle, we see that the mass of the lowest states, associated with the winding numbers qq1 X s"1 is brpa The divergence at L s 0 emerges, in this picture, after summing over all the open string states since no single state has such a divergence It shows up only at finite temperature where the winding number q exists It is important to notice that the computed partition functions take into account only thermal fluctuations of superstring stretched between D0-branes but not the fluctuations of D0-branes themselves To calculate the thermal partition function of D0-branes, a further path integration over their periodic trajecto- ries až t is to be performed as in Ž 9 One might think that classical statistics is applicable to this problem since the D0-branes are very heavy as g YM 0 so that one could restrict himself by the static approximation This is however not the case due to the singularity of the effective static potential at small distances The integral over the D0-brane positions a s is divergent when the two positions coincide However, this singularity is only in the classical partition function The path integral over the periodic trajectories až t that we actually have to do cannot diverge since the -body quantum mechanical problem has a well-defined spectrum The path integral can then be evaluated as Ý expž ybe n n where En are in the spectrum of the operator HsP rm qv eff There certainly should not be the bound state energy eigenvalue at negative infinity for this quantum mechanical problem which implies the convergence of the path integral These issues which are related to thermodynamics of D0-branes will be considered in a separate publication Let us finally discuss when the 1-loop approximation that we have done is applicable The loop expansion in Yang-Mills theory computation is valid only in the limit where 3 ' a X 3 g r< a < YM ;gsž L / is small This is due to the fact that the distance L plays the role of a Higgs mass which cuts off the infrared divergences of the loop expansion in the 0 q 1-dimensional gauge theory Thus, the perturbative Yang-Mills theory computation is good when X g a <L Ž 46 1r3 s ' This can be satisfied if either the string coupling is small or if the D0-brane separation is large compared with the string length scale In the latter case, the truncation of the spectrum to the lightest modes is still valid when 1 1 kbt< < X Ž 47 1r3 L g ' a s Note that the first inequality which is independent of both the string scale and the string coupling is the one already discussed in the previous section In this

315 ( ) J Ambjørn et alrphysics Letters B case, the temperature must be less than the inverse distance between D0-branes In the case where the string coupling gs is small, the criterion for validity of truncation of the spectrum becomes ' X k T-1r a, Ž 48 B which is the usual one Acknowledgements We are grateful to P Di Vecchia, G Ferretti, A Gorsky, M Green, M Krogh, N Nekrasov, N Ohta, P Olesen, L Thorlacius and K Zarembo for very useful discussions JA and YM acknowledge the support from MaPhySto financed by the Danish National Research Foundation and from INTAS under the grant The work by YM is supported in part by the grants CRDF 96 RP1 53 and RFFI References wx 1 J Polchinski, hep-thr wx W Taylor, hep-thr wx 3 E Witten, Nucl Phys B 460 Ž hep-thr wx 4 U Danielsson, G Ferretti, B Sundborg, Int J Mod Phys A 11 Ž hep-thr wx 5 D Kabat, P Pouliot, Phys Rev Lett 77 Ž hep-thr wx 6 MR Douglas, D Kabat, P Pouliot, S Shenker, Nucl Phys B 485 Ž hep-thr wx 7 C Bachas, Phys Lett B 374 Ž hep-thr wx 8 T Banks, W Fischler, S Shenker, L Susskind, Phys Rev D 55 Ž hep-thr wx 9 N Ohta, J-G Zhou, Nucl Phys B 5 Ž hepthr w10x ML Meana, MAR Osorio, JP Penalba, hep-thr w11x B Sathiapalan, Mod Phys Lett A 13 Ž hepthr w1x MB Green, Nucl Phys B 381 Ž w13x MM Vazquez-Mozo, Phys Lett B 388 Ž hepthr w14x MB Green, Phys Lett B 66 Ž

316 7 January 1999 Physics Letters B Large squark and slepton masses for the first-two generations in ž / the anomalous U 1 SUSY breaking models J Hisano a, Kiichi Kurosawa b, Yasunori Nomura c a Theory Group, KEK, Oho 1-1, Tsukuba, Ibaraki , Japan b YITP, Kyoto UniÕersity, Kyoto , Japan c Department of Physics, UniÕersity of Tokyo, Tokyo , Japan Received October 1998 Editor: M Cvetič Abstract Considering that the soft SUSY breaking scalar masses come from a vacuum expectation value of the D-term for an external gauge multiplet, the renormalization of the scalar masses is related to the gauge anomaly Then, if the external gauge symmetry is anomaly-free and has no kinetic mixing with the other UŽ 1 gauge symmetries, the scalar masses are non-renormalized at all orders assuming that the gaugino masses are negligibly small compared with the scalar masses Motivated by this, we construct models where the sfermion masses for the first-two generations are much heavier than the other superparticles in the minimal SUSY standard model in a framework of the anomalous UŽ 1 mediated SUSY breaking In these models we have to introduce extra chiral multiplets with the masses as large as those for the first-two generation sfermions We find that phenomenologically desirable patterns for the soft SUSY breaking terms can be obtained in the models q 1999 Elsevier Science BV All rights reserved Supersymmetry Ž SUSY is a solution to the naturalness problem in the standard model Ž SM with the fundamental Higgs boson If the SUSY breaking scale is at most of the order of 100 GeV or 1 TeV, the quadratic divergent radiative correction to the Higgs boson mass is regularized by the scale On the other hand, introduction of the soft SUSY breaking terms to the minimal SUSY SM Ž MSSM may lead to various flavor problems First, flavor-changing 0 0 neutral current FCNC processes, such as K K mixing and m eg, are induced by the arbitrary squark and slepton masses Second, the CP violating phases in the SUSY breaking parameters may generate the electric dipole moments Ž EDMs of neutron and electron beyond the experimental bounds It is well known that these problems can be solved if the first and second generation sfermions are much heavier than the other superparticles in the MSSM wx 1 Also, these masses are irrelevant to the naturalness for the Higgs boson masses at one-loop level However, these mass parameters are coupled with each other in the renormalization group Ž RG equations at two-loop level This comes from the X e-scalar contribution in the DR scheme wx As a result, if the SUSY breaking masses are generated at high energy scale, such as the Grand Unified Theory Ž 16 GUT scale ; 10 GeV or the gravitational scale Ž 18 M ; 10 GeV ), the larger soft SUSY breaking masses for the first-two generations give a large negative contribution to those for the third genera r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

317 ( ) J Hisano et alrphysics Letters B tion, and the SUŽ 3 color may be broken wx C 3 This is a big obstacle to construct models where the first-two generation sfermions are much heavier than the other superparticles The structure of the RG equations for the soft SUSY breaking scalar masses can be understood by introduction of an external Ž spurious UŽ 1 gauge symmetry wx 4 The soft SUSY breaking scalar masses can be regarded as a non-vanishing D-term for the external UŽ 1 gauge multiplet As will be shown later, if the Lagrangian is invariant under the external UŽ 1 gauge symmetry and if the gauge symmetry is anomaly free for the SM gauge groups and has no kinetic mixing with them, the SUSY breaking scalar masses are RG invariant at any orders, assuming that the gaugino masses are negligibly small compared with the scalar masses In that case, the dangerous contribution in the RG equations for the sfermion masses leading to the color breaking Že-scalar contri- X bution in the DR scheme automatically vanishes In this letter we show that if the external UŽ 1 gauge symmetry is anomaly-free, the SUSY breaking scalar masses are non-renormalized at all orders neglecting the gaugino masses We construct models where the sfermions of the first-two generations are much heavier than the other superparticles in the MSSM in a framework of the anomalous UŽ 1 X mediated SUSY breaking w5,6 x In these models, we have to introduce additional chiral multiplets with the masses as large as those for the sfermions of the first-two generations First, we review the RG equations for the soft SUSY breaking scalar masses in SUSY gauge theow7,4 x Here, we use the Wilsonian Lagrangian for ries simplicity The bare Lagrangian for the SUŽ N c SUSY gauge theory with the soft SUSY breaking terms is written as 1 a Ls½ 4Hd u SW Ž V Wa Ž V qhc 5 qhd 4 uf r e Vq qru F r, Ž 1 introducing a chiral superfield S and a UŽ 1 vector superfield U as external fields Here, Fr is a chiral multiplet belonging to the r representation of the SUŽ N with the external UŽ 1 c charge of q r, and W a Ž V is the field strength for the SUŽ N c gauge Ž Ž r Ž r multiplet V V'Ý T V The generators T of a a a a Ž Ž r Žr the SU Nc are normalized as Trs1r TrTa Tb s T d r ab in the case of r being the fundamental representation The vacuum expectation value Ž VEV for the scalar component of S corresponds to the gauge coupling constant Ž g and that for the F-term to the gaugino mass Ž m as g 1 ² S: s 1ym u g Ž g ² : Also, the non-vanishing D-term VEV DU for U leads to the soft SUSY breaking scalar masses m r for F r, which are proportional to qr as 1 m sq ² yd : Ž 3 r r U If the bare Lagrangian is invariant under the external UŽ 1 symmetry and the UŽ 1 symmetry is anomaly free for the SUŽ N gauge symmetry, Ý r Ý c m T A qts0, Ž 4 r r r r r the scalar masses are RG invariant at all orders due to the Ward-Takahashi identity assuming that the gaugino mass is negligible In fact, the RG equations for the SUSY breaking scalar masses at two-loop X level in the DR scheme are dm r 8 g 4 m s C m T, Ž 5 rý s s dm Ž 16p s ignoring terms proportional to m g Here, Cr is the Casimir coefficient of the r representation The right-hand side of this equation, that is the e-scalar contribution, vanishes if relation Ž 4 is satisfied When another UŽ 1 X gauge symmetry exists and F has its charge q X r r, the conditions that the kinetic mixing between the internal and external UŽ 1 gauge symmetries is prevented are m q X A qq X s0, Ý r Ý r Ý r r r r r Ý m q X lnz A qq X lnz s0 Ž 6 r r r r r r r Here, Zr is the wave function renormalization for F r From the above discussion, if the soft SUSY breaking scalar masses satisfy the relations Ž 4, Ž 6, 1 wx The convention for the sign of the D-term is as in Ref 8

318 318 ( ) J Hisano et alrphysics Letters B the mass spectrum that the first-two generation sfermions are much heavier than the other superparticles in the MSSM is RG stable To satisfy the relations, however, we have to introduce additional chiral multiplets which belong to nontrivial representations of the SM gauge groups with the negative SUSY breaking scalar mass squared In order not to break SUŽ 3 or UŽ C 1 EM, they have to have the supersymmetric mass terms The supersymmetric masses should be of the same order as the SUSY breaking masses for the first-two generations so that the e-scalar contribution is not generated after decoupling of the additional chiral multiplets Now, we construct models which have the above properties We adopt the anomalous UŽ 1 X gauge symmetry, whose anomalies are canceled by the Green-Schwarz mechanism wx 9, as an origin of the SUSY breaking and identify it as the above external UŽ 1 symmetry As will be discussed later, non-trivial anomalies for the UŽ 1 X do not mean that the UŽ 1 X gauge symmetry has an anomalous matter content for the SM gauge groups below the breaking scale of the UŽ 1 X symmetry In the anomalous UŽ 1 X SUSY breaking models, the Fayet-Iliopoulos term j having mass dimension one is generated of Ž yž1 the order of O 10 M w10x and induces the spontaneous SUSY breaking, so that the D-term contribution to the SUSY breaking scalar masses may dominate over the F-term contribution suppressed by M ) The correct treatment of the dynamics requires to include the dilaton field S in the potential w11 x Even then, depending on the Kahler potential for the dilaton field, the SUSY breaking can be dominated by the D-term and the contribution from the dilaton field is also negligible w1 x We assume that it is the case and neglect the dilaton field S, hereafter We choose the anomalous UŽ 1 X charges for quarks and leptons in the first-two generations to be 1 and those for the other chiral multiplets in the MSSM to be 0 for simplicity The charge assignment Ž that the first-two generations have the same U 1 X In general, the Kahler potential for the dilaton field S can be Ž written as KSqSyd X Here, X is the UŽ 1 S GS X gauge multiplet and dgs is a constant The D-term dominated SUSY breaking occurs when K XX 4 K X S S, where prime denotes derivative with respect to S ) charges helps to avoid dangerous FCNC processes wx 6 The extension to other cases is straightforward As we mentioned above, we have to introduce additional chiral multiplets, cex and c ex, with the super- symmetric masses to satisfy the relations ŽŽ 4, 6 In order not to break the success of the gauge coupling unification, we introduce n5 pairs of complete SUŽ 5 GUT multiplets in the fundamental representations 3,, n5 cex 5 y4rn q cex 5 y4rn, where the numbers in parentheses correspond to representations of the SUŽ 5 GUT and the subscripts of Ž 4 parentheses denote the U 1 X charges If we take n 5 )4, the SM gauge couplings blow up below the GUT scale Before constructing explicit models, we review the anomalous UŽ 1 X mediated SUSY breaking and sketch how to realize the above idea We introduce a SM singlet f field with UŽ 1 X charge y4rn 5, and take the SUŽ N c gauge theory with one flavor Q and Q both of which have UŽ 1 charges 4rn wx X 5 5 The tree-level superpotential is given by ž / QQ f f Ws f qfc cexc ex Ž 9 M ) The dynamical superpotential for Q and Q induced by the SUŽ N gauge interaction w13x c leads to ² QQ :/ 0 Then, the D-flatness condition of the U1 Ž X j y < f < y < c < y < c < ex ex q PPP s0 n5 n5 n5 Ž 10 is not compatible with the F-flatness conditions and SUSY is broken Moreover, if fc is larger than f f, 3 Alternatively, we can also introduce, n10 cexž 10 y4 r3n qc Ž ex y4r3n10, Ž 7 or combinations of Eq Ž 7 and Eq Ž 8 maintaining the relations Ž 4, 6 4 In fact, this charge assignment satisfies only the one-loop and two-loop parts of Eqs X ŽEq Ž 6, that is, Ýrqqs0 r r and Ý Cqq X r r r rs0 Thus, the RG equations for the SUSY breaking scalar masses receive three-loop contributions of order OŽ a a a 3 Y These contributions, however, cause no phenomenological problem

319 ( ) J Hisano et alrphysics Letters B the VEVs of the cex and cex fields vanish and the f field gets the VEV of order j, so that the SM gauge groups remain unbroken It can be seen by looking that the scalar potential V behaves as QQ < < < < < < f c ex c ex ž / V; f f q f c q f c, Ž 11 M ) satisfying the condition Eq Ž 10 to the leading order of ² QQ: rm ) At the minimum of the potential, the D-term D X of the UŽ 1 X and the supersymmetric mass Mex for the extra chiral multiplets cex and cex are written by using the condensation of QQ as ² : n5 ff² QQ: ² QQ: X ex c 4 M) M) yd s, M sf The nonvanishing D-term generates the SUSY breaking scalar masses according to Eq Ž 3 Note that the squared masses for the scalar components of the c ex and c < < ex fields given by Mex y 4² y DX: rn5 are positive under the condition f c)f f We find that the masses for squarks and sleptons of the first-two generations are the same order as those of the scalar components of additional chiral multiplets cex and c, so that the relations Ž 4, Ž ex 6 are satisfied at almost all the energy from high energy scale down to the weak scale Thus, there are no large radiative corrections to the third generation sfermions due to the RG equations It is essential for the RG stability that supersymmetric masses for the f and c ex, cex fields are comparable, that is, E W E W ; Ž 1 ; ; Ef Ec Ec ex ex Then, the UŽ 1 X charge for the f field should be equal to those for the cex and cex fields as long as the couplings ff and fc are the same order Indeed, for the purpose of the SUSY breaking only, we could also introduce the superpotential ž / QQ ff f nq Ws qf c c, Ž 13 n c ex ex M Ž nq! M ) ) assigning an appropriate UŽ 1 X charge to the singlet field f In this case, however, the soft SUSY break- ( X ² : ing scalar masses ; y D, E WrEf are smaller than the supersymmetric mass M ex, E WrEc Ec for the c, c fields by a fac- ² : ex ex ex ex tor of Ž ² f: rm n ), jrm ) n As a result, the relations Ž 4, Ž 6 do not hold between the two scale ( y DX and M ex, giving large negative contributions to the third generation sfermion squared masses Although it is possible to make phenomenologically viable models in this case, we concentrate on the case where Eq Ž 1 holds with f f, fc and the relations Ž 4, Ž 6 are satisfied at almost all the energy In order to generate the complete Yukawa matrices, we have to introduce a field with UŽ 1 X charge y1 which has a VEV of the order of j, since otherwise the mixing between the first-two generations with the UŽ 1 X charge 1 and the third generation with the UŽ 1 X charge 0 is not allowed In the above simplified model, however, the f field has a UŽ 1 X charge y4rn 5, so that we have to choose n5 s4 resulting in the asymptotic nonfreedom of the SM gauge couplings Thus, we instead introduce two singlet fields f and f which have UŽ 1 1 X charges y1 and 1y8rn 5, respectively We, here, construct explicit models taking n5 s 5 i for simplicity We introduce Nf flavors Q and Qı Ž i,ıs1,,n of the SUŽ N gauge theory Ž f c Nf- N The tree-level superpotential is c i QQ ı ı ı fi 1 c i ex ex ž / Ws f ffqf c c, Ž 14 M ) and the dynamical superpotential is generated as w13x 3 NcyN L f QQ / 1 NcyNf W s N yn, Ž 15 dyn c f ž where L is the dynamical scale of the SUŽ N c gauge theory The dynamical degrees of freedom can be expressed by meson fields i i Mı sq Q ı 16 5 When n s, the charge assignment of the UŽ 5 1 X for the first-two generations and the additional chiral multiplets is the same as that of a UŽ 1 in E for 7 6

320 30 ( ) J Hisano et alrphysics Letters B Then, the full superpotential is expressed as Ws 1 M ) fftr 1 Ž ff M 3 NcyN L f det M / 1 NcyNf q N yn Ž 17 c f ž Here, we have ignored the cex and cex fields which have vanishing VEVs provided that fc is sufficiently larger than f f The D-term is yd sg j q4tr' MMy< f < y3< f <, X Xž 1 / Ž 18 along the SUŽ N c flat directions The minimum of the potential can be obtained by making an expansion in the parameter Ž3 N cyn f r N LrM c ) <1 To the leading order, the VEVs of the fields f 1, f, and M at the minimum are given by solving the following equations: < f < q3< f < sj, Ž 19 1 ž / < f < N yn < f < qn < f < 1 f c 1 f ž / s3< f < N yn < f < qn < f <, Ž 0 f c f 1 ž / 1 y1 M ı 3 NcyNf ı L N yn f c f f i i s ff 1 1 det M M ) From Eqs Ž 19 and Ž 0, we find that both f1 and f fields have nonvanishing VEVs of the order j, and the VEV of the M field is given by Eq Ž 1 as 3 N yn N yn c f c f N M c ) Nc f 1, f,j, ² M :,L j ² : ² : ž / Then, the complete Yukawa matrices are generated through the VEV of the f1 field suppressed by the suitable power of ² f : rm, jrm w14 x 1 ) ) Since we have chosen the Higgs bosons to be neutral under the UŽ 1, this explains why the third generation quarks X and leptons are much heavier than the first-two generation ones w15 x 6 The auxiliary fields are determined as 1 yf s f Tr f M, Ž 3 f1 f M ) 1 yf s f Tr f M, Ž 4 f 1 f M ) yd X NfyNc 1 s Ž < f < q< f < 1 TrŽ ff M Nc j M) Ž 5 From Eq Ž 5, Nf must be larger than Nc in order to give the positive SUSY breaking squared masses for the first-two generation sfermions Inserting Eq into Eqs Ž 3 Ž 5, we get 1 3 NcyN L f N c f1 f ž Ž NcyNf N j M f ) / 3 NcyN L f N c X Ž NcyNf Nf yf,yf, j, ž / yd, Ž 6 j M ) Note that there are the following useful relations among the D-term and the F-terms: f yf 1 f yf yd,, Ž 7 f f X ² : ² : 1 So far, we have considered that the UŽ 1 X have an anomaly-free matter content for the SM gauge groups to satisfy the relation Ž 4 However, the UŽ 1 X has to have anomalies for any gauge groups, so that there must be chiral multiplets x and x which carry the UŽ 1 X anomalies for the SM gauge groups Since the UŽ 1 X is broken, these chiral multiplets can get the supersymmetric masses of the order j through the coupling with the f Ž f andror f 1 fields as W;fxx, Ž 8 provided that these fields are vector-like under the Ž SM gauge groups and have appropriate U 1 X 6 If we assign the different charges among the first-two generation quarks and leptons, it may be possible to obtain more realistic mass matrices

321 ( ) J Hisano et alrphysics Letters B charges 7 Then, the gaugino masses are induced at one-loop level as w16x / a yf f m g,, Ž 9 4pž ² f: where a denotes the SM gauge coupling strength The sfermion masses of the same order are also generated at two-loop level This seems the hybrid scenario with the gauge-mediated SUSY breaking However, the existence of the messenger fields x and x is not the extra assumption but the natural consequence of the anomalous UŽ 1 X gauge symmetry w17 x We finally discuss the soft SUSY breaking parameters In the present models, there are three contributions, that is, UŽ 1 X D-term induced m D, gravity mediated m, and gauge mediated m pieces, ( m, yd, D X F ž / yff j yff j m F,,, m D, M M ² f: M ) ) ) ž / a yff a m G,, m D Ž 30 4p ² f: 4p Here, we have used Eq Ž 7 The SUSY breaking scalar masses for the first-two generations m 1,, those for the third generation or Higgs bosons m, 3, h gaugino masses m, and analytic trilinear couplings g A for the sfermions are given by m1,, m D q m F q m G, m 3,h, mf q m G, m, m, g A, m F G G Ž 31 From Eq Ž 30, we find that the relation md4m F, mg holds in a generic parameter region In that case, we obtain phenomenologically desirable spectrum at the low energy, which solves the FCNC and CP 7 It may be inevitable that the squared masses for the third generation sfermions get negative contributions through the RG equations above the decoupling scale of x and x However, we can make these negative contributions smaller than the positive contributions induced by the gravity or gauge mediation problems naturally by making the first-two generation sfermions much heavier than the other superparticles in the MSSM The point is that this mass spectrum is RG stable, since relations Ž 4, Ž 6 are satisfied due to the extra chiral multiplets cex and c ex In conclusion, we have considered that the soft SUSY breaking scalar masses come from a vacuum expectation value of the D-term for an external UŽ 1 gauge multiplet Then, if the external gauge symmetry is anomaly-free and has no kinetic mixing with the other UŽ 1 gauge symmetries, the scalar masses are non-renormalized at all orders, assuming that the gaugino masses are negligibly small compared with the scalar masses Motivated by this, we have constructed the anomalous UŽ 1 X mediated SUSY breaking models In a generic parameter region of the models, the sfermion masses for the first-two generations are much heavier than the other superparticles in the MSSM, which solves various flavor problems such as FCNC and CP ones We hope that the present models give a new possibility in constructing the models which have such a hierarchical mass pattern of superparticles Acknowledgements We would like to thank N Haba and H Nakano for useful discussions This work is partially supported in part by Grants-in-aid for Science and Culture Research for Monbusho, No and Priority Area: Supersymmetry and Unified Theory of Elementary Particles Ž a707 YN is supported by the Japan Society for the Promotion of Science References wx 1 M Dine, A Kagan, S Samuel, Phys Lett B 43 Ž ; S Dimopoulos, GF Giudice, Phys Lett B 357 Ž ; A Pomarol, D Tommasini, Nucl Phys B 466 Ž ; A Cohen, DB Kaplan, AE Nelson, Phys Lett B 388 Ž wx I Jack, DRT Jones, SP Martin, MT Vaughn, Y Yamada, Phys Rev D 50 Ž wx 3 N Arkani-Hamed, H Murayama, Phys Rev D 56 Ž ; K Agashe, M Graesser, hep-phr

322 3 ( ) J Hisano et alrphysics Letters B wx 4 N Arkani-Hamed, GF Giudice, MA Luty, R Rattazzi, hep-phr980390; N Arkani-Hamed, R Rattazzi, hepphr wx 5 P Binetruy, E Dudas, Phys Lett B 389 Ž wx 6 G Dvali, A Pomarol, Phys Rev Lett 77 Ž ; RN Mohapatra, A Riotto, Phys Rev D 55 Ž , 46 wx 7 J Hisano, M Shifman, Phys Rev D 56 Ž wx 8 J Wess, J Bagger, Supersymmetry and Supergravity, Princeton University Press, Princeton, New Jersey, 199 wx 9 MB Green, JH Schwarz, Phys Lett B 149 Ž w10x M Dine, N Seiberg, E Witten, Nucl Phys B 89 Ž ; J Atick, L Dixon, A Sen, Nucl Phys B 9 Ž ; M Dine, I Ichinose, N Seiberg, Nucl Phys B 93 Ž w11x N Arkani-Hamed, M Dine, SP Martin, Phys Lett B 431 Ž w1x T Barreiro, B de Carlos, JA Casas, JM Moreno, hepphr w13x I Affleck, M Dine, N Seiberg, Nucl Phys B 56 Ž w14x CD Froggatt, HB Nielsen, Nucl Phys B 147 Ž w15x L Ibanez, GG Ross, Phys Lett B 33 Ž w16x M Dine, AE Nelson, Phys Rev D 48 Ž ; M Dine, AE Nelson, Y Shirman, Phys Rev D 51 Ž ; M Dine, AE Nelson, Y Nir, Y Shirman, Phys Rev D 53 Ž w17x G Dvali, A Pomarol, Nucl Phys B 5 Ž ; S Raby, K Tobe, hep-phr

323 7 January 1999 Physics Letters B A new mechanism for baryogenesis in low energy supersymmetry breaking models R Brandenberger a,1, A Riotto b,,3 a Physics Department, Brown UniÕersity, ProÕidence, RI 091, USA b Theory Group, CERN, CH-111 GeneÕa 3, Switzerland Received 0 October 1998; revised 6 November 1998 Editor: H Georgi Abstract A generic prediction of models where supersymmetry is broken at scales within a few orders of magnitude of the weak scale and is fed down to the observable sector by gauge interactions is the existence of superconducting cosmic strings which carry baryon number In this paper we propose a novel mechanism for the generation of the baryon asymmetry which takes place at temperatures much lower than the weak scale Superconducting strings act like bags containing the baryon charge and protect it from sphaleron wash-out throughout the evolution of the Universe, until baryon number violating processes become harmless The superconducting strings are out of equilibrium configurations which admit baryon number violating processes such that all of the Sakharov criteria for baryogenesis are satisfied Our mechanism is efficient even if the electroweak phase transition in the MSSM is of second order and therefore does not impose any upper bound on the mass of the Higgs boson q 1999 Elsevier Science BV All rights reserved 1 Introduction Baryogenesis is an important channel which can relate particle physics beyond the standard model to data The challenge for particle physics is to determine how the observed baryon to entropy ratio of nbrs;10 y10 can be generated from symmetric initial conditions Ž n s 0 in the very early Universe B 1 rhb@hetbrownedu riotto@nxth04cernch 3 On leave of absence from Theoretical Physics Dept, Oxford Univ, Oxford, UK Žhere, nb and s denote the net baryon number and entropy density, respectively For baryogenesis to be possible, it is necessary for the particle physics model to admit baryon number violating processes, to have C and CP violation, and for the relevant processes to occur out of thermal equilibrium If ByL is a symmetry of the theory, then wx 1 sphaleron processes which are unsuppressed above the electroweak symmetry breaking scale h EW will erase any baryon asymmetry generated at temperatures higher than T EW, the temperature corresponding to h EW, and the presently observed nbrs ratio must be generated below T Ž EW for recent reviews of baryogenesis, see Refs w,3 x A lot of recent work Žsee eg the review articles r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

324 34 ( ) R Brandenberger, A RiottorPhysics Letters B wx 4 has focused on ways of generating the baryon asymmetry by electroweak sphaleron processes at or below T EW This is not possible in the standard model, firstly because CP violation couples to sphaleron processes with insufficient strength, secondly because the electroweak phase transition is not strongly first order and hence does not proceed by the nucleation of critical bubbles Such critical bubbles are the sites of non-equilibrium which are usually used for electroweak baryogenesis The constraints of demanding that the phase transition be strongly first order, and that baryon nonconserving processes be out of equilibrium immediately below TEW are hard to reconcile even in extensions of the standard model For instance, a successful electroweak baryogenesis within the Minimal Supersymmetric Standard Model Ž MSSM requires the Higgs boson and the right-handed stop to be light and CP-violating phases of the order of 01 wx 5 There is, however, another mechanism which can be effective in mediating baryogenesis Topological defects are out-of-equilibrium field configurations which can trigger baryogenesis in several ways: they can act as field condensates which release baryons during their decay wx 6, or they can be used as the catalysis sites of sphaleron processes in a way simiwx 7 lar to how bubble walls function In this paper, we work out an idea first suggested in wx 8 that defects may play a crucial role in baryogenesis in a class of supersymmetric theories where supersymmetry breaking occurs at a low energy scale It has been shown in previous work wx 8 that these models admit superconducting cosmic strings with baryon number carrying condensates As we will discuss, these strings are able to trap the baryon number from the time of string formation Ž TsT c until the temperature of the Universe has dropped down to values much lower than T EW, when sphaleron processes have fallen out of equilibrium The generation of the baryon asymmetry takes place when the baryon number carried by the condensate in the core of the strings is released into the thermal bath at very low temperatures Žthis mechanism is different from defect-mediated electroweak baryogewx 7, another mechanism which can be imple- nesis mented in certain supersymmetric models wx 9 The out-of-equilibrium condition is therefore naturally attained A significant advantage of defect-mediated baryogenesis is that the condition that sphaleron processes must be out of equilibrium immediately below T is no longer required wx EW 6 and therefore the stringent conditions on the MSSM Higgs spectrum may be relaxed Supersymmetric extensions are the best-motivated of the particle physics theories beyond the standard model Supersymmetry solves the gauge hierarchy problem, couples gauge theory to gravity, and may easily generate the gauge coupling unification However, the present state of the Universe is not supersymmetric, and thus supersymmetry must be broken in Nature at some scale higher than ;TeV Spontaneous supersymmetry breaking entails adding to the standard model and its supersymmetric copy some new fields acting as a supersymmetry breaking sector The supersymmetry breaking sector cannot have any renormalizable couplings to the visible sector, and is therefore often called hidden or secluded There are various ways in which supersymmetry breaking can be communicated from the hidden to the visible sector Žsee eg w10x for a recent review : supergravity mediation, gauge mediation, and mediation via UŽ 1 gauge factors Since in the most general and natural scenarios of low energy supersymmetry breaking there are two different sources of contributions to ordinary superpartner masses, gauge mediation from the messenger sector and anomalous UŽ 1 D-terms, a novel feature is the prediction of superconducting cosmic strings wx 8 The typical energy scale of these strings is about 10 TeV, and the bosonic charge carriers which render the strings superconducting are squark or slepton condensates Baryon andror lepton number are violated inside the strings Baryonic or leptonic charge may be stored in the core of the strings Since the strings are out of thermal equilibrium configurations, we see that all three of the Sakharov criteria for baryogenesis are satisfied After reviewing the particle physics model under consideration in Section, we study the evolution of the string network Ž Section 3 As we discover, the number density of strings at late times depends sensitively on whether the string loops become vortons or not In Section 4, we calculate the baryon asymmetry per string loop decay, and use the result to estimate the overall nbrs ratio generated by this mechanism

325 ( ) R Brandenberger, A RiottorPhysics Letters B Low energy supersymmetry breaking models Supersymmetry provides solutions to many of the puzzles of the standard model such as the stability of the weak scale under radiative corrections as well as the origin of the weak scale itself Since experimental observations require supersymmetry to be broken, it is essential to have a knowledge of the nature and the scale of supersymmetry breaking in order to have a complete understanding of the physical implications of these theories At the moment, we lack such an understanding and therefore it is important to explore the various ways in which supersymmetry breaking can arise and study their consequences The most common and popular approach is to implement supersymmetry breaking in some hidden sector where some F-term gets a vacuum expectation value Ž VEV and then transmits it to the standard model sector by gravitational interactions This is the so-called hidden N s 1 supergravity scenario w11 x If one arranges the parameters in the hidden sector in such a way that the typical ² F: -term is of 1r ² : 11 the order of F ;(Ž 1 TeV M Pl ; 10 GeV, where MPl s4=10 18 GeV, the gravitino mass m3r turns out to be of the order of the TeV scale In the N s 1 supergravity scenario, however, the flavor-blindness is likely to be spoiled in the Kahler potential and one expects large contributions to the Flavour Changing Neutral Currents Ž FCNC s at low energies The suppression of the electric dipole moment of the neutron Ž EDMN is also hard to explain in supergravity An alternative to hidden supergravity is provided by the so-called gauge mediated supersymmetry breaking Ž GMSB models w1 x The soft supersymmetry breaking mass terms of sfermions get contria butions at two-loop, m ; Ž 4p L, where L; 10 TeV is the typical scale of the messenger sector The GMSB models have the extra advantage that the FCNC effects are naturally suppressed Moreover, the trilinear soft breaking terms A vanish at L rendering the CP-violation problem milder Recently, an alternative approach has been proposed in which gauginos, higgsinos and the third generation of squarks are sufficiently light to stabilize the electroweak scale, but the two first generations of squarks and sleptons are sufficiently heavy to suppress the FCNC and the EDMN below the experimental bound w13,14 x This class of models is dubbed more minimally supersymmetric than the MSSM since they do not require some ad hoc suppow14 x In most of the sition of degeneracy or alignment attempts made so far along this line w13,15x the crucial feature is the existence of an anomaly-free UŽ 1 gauge group, which appears anomalous below some scale M The effective theory includes a Fayet-Iliopoulos Ž FI D-term proportional to Tr QM, where the Q s represent the UŽ 1 charges of the heavy fields which have been integrated out If the first two generations of squarks and sleptons, contrary to the third generation, carry UŽ 1 charges, the required mass hierarchy is obtained Supersymmetry models must be able to reproduce the quantitative features of the fermion spectrum and the CKM matrix In a unified picture it is desirable that the solution to all these puzzles reside in the same sector of the theory Recent investigations have focused on the possibility that a Frogatt-Nielsen flavor mechaw16x is implemented in the messenger sector nism w15 x This means that the messenger sector is also to be a Frogatt-Nielsen sector and that the messenger UŽ 1 symmetry by which the two first generation fermions get large masses is the same Frogatt-Nielsen UŽ 1 for quarks and leptons Therefore, the most general Ž and maybe natural scenario is an hybrid one where there are two different sources of contributions to ordinary superpartner masses, gauge mediation from the messenger sector and anomalous UŽ 1 D-terms w15,17 x As shown in wx 8, this class of low energy supersymmetry breaking models naturally predicts Žsuper- conducting cosmic strings Indeed, the gauge group of the secluded sector, where supersymmetry is broken, may be of the form Gm UŽ 1 m The group UŽ 1 is usually some global UŽ m 1 which is gauged and made anomaly-free The fields in the messenger sector are charged under UŽ 1 and Ž some of them m under the standard model gauge group When supersymmetry is broken in the secluded sector, some scalar fields in this sector may acquire a VEV, but leave the UŽ 1 m gauge symmetry unbroken Integrat- ing out the heavy fields belonging to the secluded sector amounts to generating a Ž say positive one-loop FI D-term j and negative two-loop soft supersymmetry breaking squared masses Žproportional to the square of the corresponding UŽ 1 charges q for m i

326 36 ( ) R Brandenberger, A RiottorPhysics Letters B Ž some of the scalar fields fi of the messenger sector The potential will read Ý Vsy qi m fi q i Ý ž / g Ý i i i i E W Ef i q q f qj, Ž 1 w ž / where g is the UŽ 1 m gauge coupling constant From the minimization of Ž 1, one can see that among the fields with F-flat directions, the one with the smallest negative charge Ž call it w will get a VEV 1 m ²< w < : s qw yj Ž q g When this happens, the residual global UŽ 1 m sym- metry gets broken leading to the formation of local strings whose mass per unit length is given by m;j We now suppose that the quark andror the lepton superfields are charged under the UŽ 1 m group, as is in the spirit of the more MSSM w14 x Let us focus of one of the sfermion fields, f with UŽ 1 m-charge qf such that sign qfs sign q w The potential for the fields w and f is written as w f V f,w syq m < w < yq m < f < g < < < < w f ž / q q w qq f qj ql< f < 4, Ž 3 where we have assumed, for simplicity, that f is F-flat The parameter l is generated from the standard model gauge group D-terms and vanishes if we take f to denote a family of fields parametrizing a D-flat direction At the global minimum ² f : s0 and the electric charge, the baryon andror the lepton numbers are conserved The soft breaking mass term for the sfermion reads Dm sq Ž q yq m, Ž 4 f f w f which is positive in virtue of the hierarchy qw-qf- 0 Consistency with experimental bound requires Dm to be of the order of Ž 0 TeV or so, which in f Ž Ž turn requires j; 4prg m ; 10 TeV Notice that Dm f does not depend upon j Let us analyse what happens in the core of the string In this region of space, the vacuum expectation value of the field vanishes, ²< w <: s 0, and nonzero values of ²< f <: are energetically preferred in the string core f f m q yg j q ²< f < : s Ž 5 g q ql f Since the vortex is cylindrically symmetric around the z-axis, the condensate will be of the form fs ih f Ž z,t f0 r,u e where r and u are the polar coordinate in the Ž x, y -plane One can easily check that the kinetic term for f also allows a nonzero value of f in the string and therefore one expects the existence of bosonic charge carriers inside the strings The latter are, therefore, superconducting w18 x Besides the electric charge, the baryonrand or the lepton numbers are also broken inside the string Hence, the strings provide a locus where all three of the Sakharov criteria for baryogenesis are satisfied Note, also, that baryonic charge may be stored in the core of the string Indeed, if the sfermion particles are identified with some squarks q Ževentually parametrizing a standard model D-flat direction, they carry a UŽ 1 baryonic global charge 3 String network evolution In the class of models considered in Section, a network of strings forms during supersymmetry breaking at a temperature T c ;10 TeV The initial string correlation length Žmean separation between strings is j Ž t c ;l y1 h y1, Ž 6 ' h; j being the energy scale of the string, and l;g being a typical coupling constant of the UŽ 1 order parameter field The evolution of a string network can be divided into two periods In the initial period following the string-producing phase transition, the string dynam-

327 ( ) R Brandenberger, A RiottorPhysics Letters B ics is friction-dominated w19 x During this period, the correlation length j Ž t increases super-luminally and catches up to the Hubble radius: 3r t j Ž t sj Ž t Ž 7 c ž t c / Žsee w0 x The friction-dominated phase ends once j Ž t ;t This happens at a time t given by f y1 c t ; Gm t, 8 f G being Newton s gravitational constant and msh the string mass per unit length After t f, the string network enters the scaling epoch during which j Ž t ;t Since in our class of models Gm;10 y8, all of the physics relevant for baryogenesis takes place deep in the friction-dominated period The rapid increase of j Ž t given by Ž 7 is achieved by string loop production Unless small-scale structure on the string network is important already in the friction-dominated epoch, the loops are produced with a radius proportional to j ŽŽ t see the reviews in w1 3 x In this case, the number density of loops created per unit time is dn dj y4 snj, Ž 9 dt dt where n is a constant of order unity In the class of models considered, the strings are superconducting and carry baryon charge Let Q B denote the charge per unit length By order of magnitude considerations, the value of QB will be of the order h Then, the charge on a correlation length of string at the time t is c Q sqj Ž t Ž 10 c c On longer pieces of string, the charge adds up as a random walk w4 x Hence, loops created at time t with radius j Ž t have a root mean square charge of 1r j Ž t QBŽ t s Q c Ž 11 j Ž t c Superconducting strings decay predominantly by the emission of electromagnetic radiation Using the w x formula 5 Pg sgem j m for the power of electro- magnetic radiation, where g is a constant and j is y1 the current in units of eg h Ž e ge is the electromag- netic coupling constant, and assuming that the current on a string network builds up as a random walk, it can be verified Žsee w0x that loops decay in less than a Hubble time The will simplify the following calculations The cosmic string network provides a means for trapping baryon charge until late times For now we write Dn sq e, Ž 1 B B for the net baryon number generated by the decay of a loop of charge Q B, where e is a constant related to the CP-violation parameter Ž see Section 4 The resulting baryon asymmetry depends crucially on whether the loops decay immediately or collapse to form vortons w6 x, which themselves decay after t EW In the former case, the only loops which contribute to the net nbrs are those produced after t EW : X 3r t t / t dn X X X nbž t sh dte QBŽ t Ž t ž, Ž 13 dt t EW where the final factor is due to the cosmological redshift The integral Ž 13 is dominated by the contribution from near t Ž EW or, more generally, the time when sphaleron processes fall out of equilibrium Inserting ŽŽ 7, 9 and Ž 11, the integral becomes 3r 9r4 tc t y3 c nb t, 3 en Qcj t c ž / 14 t ž t EW / The last factor in Ž 14 is a geometrical suppression factor resulting since baryons produced by loops which decay before tew equilibrate by sphaleron processes On the other hand, if superconducting loops form vortons which decay only after t EW, then all loops created since tc contribute to the net nb which is determined now by an integral like Ž 13 but with lower integration limit t c Since the integral is once again dominated by the lower limit we obtain 3r y3 t c B c c ž t / n Ž t,en Q j Ž t Ž 15 The geometric suppression factor which appeared in 14 is no longer present For T c ;10 TeV, baryo- genesis is more efficient by a factor of Ž T crtew 45 ;10 9 if the loops form vortons which are stable

328 38 ( ) R Brandenberger, A RiottorPhysics Letters B until at least t EW Obviously, the overall strength of our baryogenesis mechanism depends on the value of e, to the evaluation of which we now turn 4 The baryogenesis mechanism The strings in our class of theories carry baryonic charge in the form of a squark or slepton condensate Since baryon number is not conserved by the string space-time configuration, a net baryon number can be generated by the decay of the strings, in analogy to how GUT strings may generate such an asymmewx 6 Note that a tiny amount of try during their decay explicit CP violation in the interaction lagrangian is required to introduce a small difference in the decay rates of the baryon and antibaryon number condensates in the string loops Another analogy can be made with the Dine-Afw7 x In the Dine-Affleck mechanism, a baryonic scalar fleck mechanism for baryon number production condensate develops a large expectation value at the end of inflation At that point, the high-scale physics violates baryon number explicitly During the spiralling in of the baryon number condensate towards the origin in the Dine-Affleck scenario, a net baryon number asymmetry is generated w8 x, and the CP violation in the interaction lagrangian again leads to a net asymmetry between baryons and antibaryons There are two crucial features of the mechanism First, when superconducting strings are formed, one may expect that the number of strings with some baryonic charge QB is equal to the number of strings with opposite baryonic charge yq B Therefore, when string loops decay and release the baryon number in the thermal bath, the net baryon asymmetry vanishes unless some source of explicit CP-violation is present in the interactions Secondly, even though superconducting strings are formed at the temperature Tc 4T EW, the baryon charge stored in the core is not washed out by baryon number violating processes induced by sphaleron transitions Indeed, sphalerons are not active in the strings This is because any condensate f carrying SUŽ i L quan- tum numbers contributes to the sphaleron energy, E A( Ý < f < sph i i Therefore, if the sfermion conden- sate in the core of the string is charged under SUŽ, L sphaleron transitions are suppressed inside the strings Žsee also w9 x This is equivalent to saying that the superconducting strings act like bags containing baryon charge and protect it from sphaleron wash-out throughout the evolution of the Universe until baryon number violating processes are rendered harmless For sake of concreteness, let consider the case in which the sfermion condensate is formed by the third family left-handed stop t Ž L one can easily generalize this case The left-handed stop decay channels most important for our considerations are t L trq H 0, 0 0 H q tr where H, H 0 and t R are the Higgs, the Higgsino and the right-handed stop, respectively The constant e Žintroduced in Ž 1 which determines the asymmetry in baryon and antibaryon production is determined by the difference in the decay widths between t and t ) : L L G yg ) tl tl es Ž 16 G qg ) t L t L The value of e is determined by the strength of the explicit CP-violation which couples to the t L decay processes For us, the dominant contribution is the CP-violating phase of the coefficient At of the term Ahtt ) H qhc Ž 17 t t L R 0 in the interaction Lagrangian, where ht is top Yukawa coupling The contribution to e comes from the interference terms between the tree level and one loop decay diagrams Assuming that m, m Ž t the supersymme- try breaking mass we obtain ln e, h t sinf A t, Ž 18 8p < < if where we have written A s A e A t t t Notice that fa t is not constrained from the experimental upper bound on the EDMN and therefore can be as large as unity At this point, we can insert the value of e obtained above into the general expressions for the baryon number obtained in Section 3 Replacing time t by the corresponding temperature T by means of the Friedmann equations, and making use of the expression for the entropy density s in terms of the number of degrees of freedom g) in thermal equilibrium at T EW sž T ;g) T 3, Ž 19

329 ( ) R Brandenberger, A RiottorPhysics Letters B we obtain from 14 and 6 : T c / nb T 3 y1 EW ;en Qc l Ž g) Ž 0 ž s for the case when the string loops collapse and disappear immediately, and from 15 and 6 n B 45 3 y1 c Ž ) ;en Q l g 1 s for the case when stable vortons form during the initial loop collapse and survive until after t EW We conclude that in the case in which stable vortons form upon the collapse of a string loop, it is possible to obtain the observed value of n rs with- B out unnatural constraints on the coupling constants To get the feeling with the numbers, for e;10 y, l;10 y1, n;1 and g ;10, we need Q ;10 y3 ) c to produce n rs;10 y10 B therefore does not impose any stringent bound on the mass spectrum of the MSSM An important issue for future study is to investigate the probability of formation of vortons and their stability As has been shown recently w30 x, cosmological constraints from the density parameter at the present time and from nucleosynthesis are consistent with vorton formation at the scale considered in this paper Acknowledgements This work is supported in part by the US Department of Energy under contract DE-FG091ER40688, Task A RB wishes to thanks Profs W Unruh and A Zhitnitsky for hospitality at the University of British Columbia, where some of the work was completed 5 Conclusions In this paper we have proposed a new mechanism for the generation of the baryon asymmetry in the early Universe It is based on the general observation that, if superconducting cosmic strings form at scales within a few orders of magnitude of the weak scale and carry some baryon charge, the latter is efficiently preserved from the sphaleron erasure and may be released in the thermal bath at low temperatures A natural framework for this mechanism is represented by the class of models where supersymmetry is broken at low energy In such a case, the charge carriers inside the strings are provided by the scalar superpartner of the fermions which carry baryon Ž lepton number Since these scalar condensates are charged under SUŽ L, baryon number violating processes are frozen in the core of the strings and the baryon charge number can not be wiped out at temperatures larger than T EW The mechanism is very efficient if stable vortons form upon the collapse of the string loops and survive until after the electroweak phase transition Our proposal has a number of advantages with respect to the idea of bubble-mediated electroweak baryogenesis since it does not ask for a sufficiently strong electroweak first order phase transition and References wx 1 V Kuzmin, V Rubakov, M Shaposhnikov, Phys Lett B 155 Ž wx A Dolgov, Phys Rep Ž wx 3 A Riotto, Theories of Baryogenesis, hep-phr wx 4 N Turok, in: G Kane Ž Ed, Perspectives on Higgs Physics, World Scientific, Singapore, 199; A Cohen, D Kaplan, A Nelson, Ann Rev Nucl Part Sci 43 Ž ; V Rubakov, M Shaposhnikov, Phys Usp 39 Ž ; M Trodden, Electroweak baryogenesis, hep-phr Žto be publ in Rev Mod Phys wx 5 M Carena, M Quiros, A Riotto, I Vilja, CEM Wagner, Nucl Phys B 503 Ž wx 6 R Brandenberger, A-C Davis, M Hindmarsh, Phys Lett B 63 Ž wx 7 R Brandenberger, A-C Davis, Phys Lett B 308 Ž ; R Brandenberger, A-C Davis, M Trodden, Phys Lett B 33 Ž ; R Brandenberger, A-C Davis, T Prokopec, M Trodden, Phys Rev D 53 Ž wx 8 A Riotto, Phys Lett B 413 Ž 1997 wx 9 M Trodden, A-C Davis, R Brandenberger, Phys Lett B 349 Ž w10x K Dienes, C Kolda, 0 Open Questions in Supersymmetric Particle Physics, hep-phr9713 w11x For reviews of supersymmetry and supergravity, see HP Nilles, Phys Rept 110 Ž ; HE Haber, GL Kane, Phys Rept 117 Ž w1x The idea of gauge mediated supersymmetry breaking was originally discussed in M Dine, W Fischler, M Srednicki, Nucl Phys B 189 Ž ; S Dimopoulos, S Raby,

330 330 ( ) R Brandenberger, A RiottorPhysics Letters B Nucl Phys B 19 Ž ; M Dine, W Fischler, Phys Lett B 110 Ž 198 7; M Dine, M Srednicki, Nucl Phys B 0 Ž ; M Dine, W Fischler, Nucl Phys B 04 Ž ; L Alvarez-Gaume, M Claudson, M Wise, Nucl Phys B 07 Ž ; CR Nappi, BA Ovrut, Phys Lett B 113 Ž ; S Dimopoulos, S Raby, Nucl Phys B 19 Ž For a recent list of refs, see C Kolda, hep-phr ; GF Giudice, R Rattazzi, hepphr w13x G Dvali, A Pomarol, Phys Rev Lett 77 Ž ; P Binetruy, E Dudas, Phys Lett B 389 Ž w14x AG Cohen, DB Kaplan, AE Nelson, Phys Lett B 388 Ž w15x RN Mohapatra, A Riotto, Phys Rev D 55 Ž , 46; AE Nelson, D Wright, Phys Rev D 56 Ž w16x CD Frogatt, HB Nielsen, Nucl Phys B 147 Ž w17x DE Kaplan, F Lepeintre, A Masiero, AE Nelson, A Riotto, hep-phr w18x E Witten, Nucl Phys B 49 Ž w19x TWB Kibble, Acta Physica Polonica B 13 Ž ; A Everett, Phys Rev D 4 Ž ; M Hindmarsh, PhD thesis, Imperial College, unpublished, 1986 w0x R Brandenberger, A Riotto, hep-phr w1x M Hindmarsh, TWB Kibble, Rept Prog Phys 58 Ž wx A Vilenkin, EPS Shellard, Cosmic strings and other topological defects, Cambridge Univ Press, Cambridge, 1994 w3x R Brandenberger, Int J Mod Phys A 9 Ž w4x M Hindmarsh, A Everett, Magnetic Fields from Phase Transitions, astro-phr ; K Enqvist, P Olesen, Phys Lett B 319 Ž ; T Vachaspati, Phys Lett B 65 Ž w5x J Ostriker, C Thompson, E Witten, Phys Lett B 180 Ž w6x R Davis, EPS Shellard, Phys Rev D 38 Ž ; Nucl Phys B 33 Ž ; for a recent review see eg B Carter, Recent Developments in Vorton Theory, astrophr w7x I Affleck, M Dine, Nucl Phys B 49 Ž w8x M Dine, L Randall, S Thomas, Nucl Phys B 458 Ž w9x W Perkins, Nucl Phys B 449 Ž w30x R Brandenberger, B Carter, A-C Davis, M Trodden, Phys Rev D 54 Ž

331 7 January 1999 Physics Letters B New limits on the SUSY Higgs boson mass 1 Konstantin T Matchev a, Damien M Pierce b a Theory Group, Fermi National Laboratory, BataÕia, IL 60510, USA b Stanford Linear Accelerator Center, Stanford UniÕersity, Stanford, CA 94309, USA Received July 1998; revised 15 November 1998 Editor: H Georgi Abstract We present new upper limits on the light Higgs boson mass mh in supersymmetric models We consider two gravity-mediated models Ž with and without universal scalar masses and two gauge-mediated models Žwith a 5q5 or 10q10 messenger sector We impose standard phenomenological constraints, as well as SUŽ 5 Yukawa coupling unification Requiring that the bottom and tau Yukawa couplings meet at the unification scale to within 15%, we find the upper limit mh-114 GeV in the universal supergravity model This reverts to the usual upper bound of 130 GeV with a particular nonuniversality in the scalar spectrum In the 5q5 gauge-mediated model we find m - 97 GeV for small tan b h and m h,116 GeV for large tanb, and in the 10q10 model we find mh-94 GeV We discuss the implications for upcoming searches at LEP-II and the Tevatron q 1999 Elsevier Science BV All rights reserved PACS: 1130Pb; 110Kt; 1480Cp If weak-scale supersymmetry Ž SUSY exists it may be a challenge to discover The superpartners may all be so heavy that they do not appreciably affect any low energy observables, and are below threshold for production at LEP-II and the Tevatron In that case they will go undiscovered until the LHC turns on in 005 However, one of the most robust and enticing hallmarks of supersymmetric models is the prediction of a light Higgs boson At tree level, mhf M Z, but it receives large radiative corrections from top and stop loops wx 1 Naturalness suggests that the third generation squarks should not be too 1 KTM Ž DMP is supported by Department of Energy contract DE-AC0-76CH03000 Ž DE AC03 76SF00515 DMP contact information address: SLAC, MS 81, PO Box 4349, Stanford, CA 94309, USA pierce@slacstanfordedu heavy Hence, we only consider squark masses below 1 TeV Then, with a top quark mass of 175 GeV, the upper limit on mh is about 130 GeV Žincluding recent two-loop corrections w,3 x The largest possible value is obtained with heavy stops and large squark mixing Supersymmetry goes hand in hand with grand unification Grand unified theory Ž GUT models predict that the gauge couplings unify at the GUT scale In the MSSM, with superpartners below the TeV scale, this prediction is nearly satisfied wx 4 The couple of percent discrepancy in gauge coupling unification finds a ready explanation in GUTs, from GUT threshold effects In typical GUT models the bottom and tau Yukawa couplings Ž l and l b t are predicted to unify as well At leading order this only happens for either very small Ž Q or rather large r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

332 33 ( ) KT MatcheÕ, DM PiercerPhysics Letters B Ž ; m rm values of tan b Ž t b tan b is the ratio of expectation values of the two Higgs doublets We take the muon decay constant, the Z-boson mass, the fermion masses, tanb, and the strong and electromagnetic couplings as inputs to determine l b and l at the GUT scale Žthe scale where the UŽ 1 t and SU Ž gauge couplings meet We define the Yukawa coupling mismatch at the GUT scale to be ' Ž l yl b b t rl t, and, allowing for variations in the input parameters, GUT scale threshold corrections, and corrections from gravity-induced higherdimensional Planck-suppressed operators, we conservatively expect b to be less than 15% in magnitude At next-to-leading order bottom-tau unification is sensitive to the supersymmetric spectrum through radiative corrections The corrections to lb are en- hanced at large tanb and can be quite large w5 7 x They broaden the region at large tan b where exact unification is possible to 15 Q tan b Q 50 The branching ratio BŽ B X g s also receives large tanb enhanced corrections The requirements of Yukawa unification and compliance with the BŽ B X s g measurement tend to conflict with each other Depending on the model, imposing both constraints can single out a very particular parameter space, resulting in predictions for the superpartner and Higgs boson masses In this letter we examine the Higgs boson mass predictions in four supersymmetric models two gravity-mediated models Žwith and without universal scalar masses, and two gauge-mediated models Ž with a 5q5 or 10q10 messenger sector Yukawa coupling unification together with the b sg constraint has been previously discussed within the context of the gravity-mediated models in Refs w6,8 x, but no conclusions about the light Higgs boson mass were drawn Ref wx 9 uses fine-tuning criteria in addition to the BŽ B X g s constraint to derive some limits on the light Higgs boson mass In each model we randomly pick points in the supersymmetric parameter space Žthe parameter spaces are discussed below Rather than fix all the input parameters, we determine the Z-boson mass, the top-quark mass, and the electromagnetic and strong couplings at the Z-scale, in a global fit to precision data We construct a x function and minimize it with respect to the four standard model inputs The x function contains 30 electroweak precision observables, and BŽ B X g The list of s observables, the measurements we use, and further details are given in Ref w10 x We fix the remaining y5 y inputs G s = 10 GeV, m Ž pole m t s 1777 GeV, and m Ž pole b s49 GeV In the MSSM, the most volatile of the 30 observables is the b sg observable While we could exclude regions of parameter space by employing the full 30-observable x, for the purposes considered in this letter it suffices to consider the b sg observable alone as an additional constraint This serves to focus, simplify and clarify the results and discussion We impose a number of phenomenological constraints at each point in parameter space We require radiative electroweak symmetry breaking and determine the CP-odd Higgs boson mass ma and the absolute value of the Higgsino mass parameter m to full one-loop order wx 7 In the models we consider Ž except the non-universal model very large values of tanb eg tanbr60 are excluded since ma is found to be negative We also require that all Yukawa couplings remain perturbative up to the GUT scale Small values of tanb Ž ie tanbq1 are excluded by this constraint, due to the top-quark Yukawa coupling Landau pole Finally, we require that all the superpartner and Higgs boson masses are above the bounds set by direct particle searches We calculate the gauge and Yukawa couplings using the full one-loop threshold corrections wx 7 and two-loop renormalization group equations w11x Ž RGEs The parameter dependence of the lb thresh- old corrections can be understood from the simplified approximation dlb 1 mtan b 8,y g m ql A Ž 1 ž 3 g t t/ 3 l 16p m b The first Ž second term is the gluino-sbottom Ž chargino-stop loop contribution mq is an average Ž stop or sbottom squark mass, m g is the gluino mass, At is the stop-stop-higgs trilinear coupling and g Ž l is the strong Ž top Yukawa 3 t coupling In a leading-order analysis, where the corrections Ž 1 are neglected, lb and lt unify well below the GUT scale for intermediate values of tanb With m)0 In those cases there are solutions with m A)0, but tanb <1 and mb4 m t q

333 ( ) KT MatcheÕ, DM PiercerPhysics Letters B the corrections Ž 1 make this situation worse, so that with tanb) b falls in the range y0 to y60% This discrepancy is larger than can be accounted for in realistic GUT models w1,13 x Also, variations in the input parameters Dmts"3 GeV, Dmbs"015 GeV and Dass"0003 result in D bs"1%, "3%, and 3%, respectively With m- 0 the threshold corrections Ž 1 help Yukawa unification by increasing lb at the weak scale, thus delaying its unification with lt to higher scales Our conserva- tive requirement < b < - 15% restricts us to either tan b- with m of either sign, or tan b R 5 and m-0 The allowed values of At play a central role in our discussion Each model allows for a different range of values of A t, with corresponding implica- tions We start by discussing the results in the gravity-mediated model with universal soft parameters Ž msugra In this model three inputs are specified at the GUT scale They are a universal scalar mass M 0, a universal gaugino mass M 1r, and a universal trilinear scalar coupling A 0 The remaining two inputs are tanb and the sign of m Because of the large top Yukawa coupling, the value of At at the weak scale exhibits a quasi-fixed point behavior Hence, the sensitivity to its high scale boundary condition is reduced The quasi-fixed point behavior is illustrated in Fig 1Ž a, where we plot the dimensionless parameter a t' AtrMq as a function of the renormalization scale Q in the Ž msugra model M q '( M0q 4 M1r is approximately equal to the first or second generation squark mass We see that At tends to be negative at the weak scale In that case the stop-chargino contribution in Eq Ž 1 partially cancels the sbottom-gluino contribution At intermediate values of tan bž10 Q tan b Q 0 Yukawa unification requires that the correction Ž 1 be maximized This happens when A t) 0, so that the stop-chargino contribution adds constructively to the sbottom-gluino contribution Hence, at intermediate tan b large and positive values of a 0 ' A rm are necessary to achieve small < < in the 0 q b msugra model This is illustrated in Fig 1Ž b, where we show the results of a scan over the msugra parameter space The striking correlation between a0 and tan b is emphasized by requiring < < - 5% in this figure At large tan b the tan b b Fig 1 Ž a Renormalization group trajectories of at for M0 s500 GeV, M s00 GeV, tanb s0 and m-0; Ž 1r b The stop mixing parameter a versus tanb with < < -5%; Ž 0 b c The branching ratio BŽ B X g s versus a 0, with the additional constraints < < b -15% and m h)100 GeV The horizontal line indicates the y4 upper bound 4=10 Ž d The light Higgs boson mass as a function of a 0, scanned over the msugra parameter space with no additional constraints enhancement in Eq Ž 1 is by itself enough for successful Yukawa unification In fact, at some points cancellation between the two terms in Ž 1 is necessary, so small or negative values of a0 are preferred We also see from Fig 1Ž b that in the small tanb region large positive A0 is required, signifying that the corrections in Eq Ž 1 are relevant The points in this region have values of the Higgs boson mass below 100 GeV In 1995 the CLEO collaboration reported the measurement of the b sg rate w14 x The upper bound on the rate is relevant for our analysis CLEO y4 found the upper bound B B Xsg -4=10 at 95% CL In Ref w15x it has been pointed out that in the extraction of the total branching ratio from the data the CLEO collaboration used an inappropriate model for the photon spectrum below 1 GeV Using an appropriate model, Ref w15x finds the 90% CL upper bound on the rate varies from 46=10 y4 to 36=10 y4 as mb varies from 465 to 495 GeV This year the CLEO and ALEPH collaborations have reported new measurements of BŽ B X g w16 x s We use CLEO s original bound, BŽ B X g s - 4 = 10 y4, as a constraint The upper bound on BŽ B X g s imposes strong constraints on supersymmetric models If m- 0 the

334 334 ( ) KT MatcheÕ, DM PiercerPhysics Letters B chargino-stop and Higgs boson contributions to the b sg amplitude add constructively to the SM amplitude Due to the tanb enhancement of the chargino loop contribution, very large total amplitudes can result, leading to predictions for BŽ B X g well above the upper bound As a result, significant regions of the SUSY parameter space are excluded We can identify those by considering the following approximate formula for the leading supersymmetric corrections to the O operator coefficient With m- 7 0 and tanb large, we have ž t / 3tanb MW 1 Am t t d C7 Ž M W,y y, Ž < < M 16p m m where the first Ž second term is the contribution from q the t yx Ž tyh q loop Ž L we work to first order in stop and chargino mixing M Ž m W t is the W-boson Ž top-quark mass, M is the SU gaugino soft mass, and m t is the average stop mass If A t)0 there is destructive interference between the two terms, and the supersymmetric contribution to the b sg amplitude is reduced In Fig 1Ž c we show the full one-loop prediction for BŽ B X g s in the msugra model, subject to the b y t unification constraint < < b - 15% As expected, the rate is sup- pressed for large and positive values of a 0 We see from Fig 1Ž c that in the msugra model with tanb), reasonably small < < and BŽ b B X g s can occur only for relatively large and positive a Ž a ) Because of the focusing towards negative values, the resulting values of at at the squark mass scale are rather small Žy04-a - 07 Hence, top squark mixing is suppressed and the corrections to the Higgs boson mass are minimized The scatter plot in Fig 1Ž d shows mh versus a0 in the msugra model The Higgs boson mass is maximal at a0sy17 and decreases with increasing a 0 In Fig Ž a we show the scatter plot of mh versus b in the msugra model We have imposed the b sg constraint in Fig The vertical lines indi- cate the region < < b -15% We see that the Higgs boson mass is below 114 GeV in this region The msugra model suffers from the rather ad hoc assumption of scalar mass unification While we see no compelling justification for this boundary condition, if it did apply it would naturally hold at the Planck scale The effects of running between the s t Fig Scatter plots of mh versus b in four supersymmetric models: SUGRA with Ž a universal or Ž b nonuniversal scalar masses; and minimal gauge mediation with a Ž c 5q5 or Ž d y4 10q10 messenger sector In each case, we require 10=10 - y4 B B Xsg -4=10 The vertical lines on the plots delin- eate the region < < -15% b Planck scale and the GUT scale can be significant w17 x Regardless of the boundary condition at the Planck scale, a GUT symmetry will ensure that GUT multiplets remain degenerate above the GUT scale In SUŽ 5, the parameter space at the GUT scale in general includes the soft mass parameters M H, M H, 1 M5 and M 10, corresponding to the 5 and 5 representations of Higgs fields, and the 5 and 10 representations of sfermion fields, respectively ŽWe impose generation independence of the M5 and M10 masses Because of the larger parameter space of the nonuniversal SUŽ 5 model, the upper limit on the Higgs boson mass, including the BŽ B X g s and approximate bottom-tau unification constraints, reverts to the general upper limit of 15 GeV Žsee Fig Ž b Two-loop threshold corrections not taken into account in Fig Ž b increase this limit to 130 GeV wx 3 We can contrast the situation here with the standard msugra case Large splitting between MH and M10 can lead to much larger values of m and ma for a given tanb The larger value of m gives larger corrections to the bottom Yukawa coupling, making bottom-tau unification possible for smaller values of tanb The smaller values of tanb, with larger m and m A, lead to reductions in the

335 ( ) KT MatcheÕ, DM PiercerPhysics Letters B supersymmetric contributions to the b sg amplitude This, in turn, allows compliance with the BŽ B X g s upper bound with large and negative values of A 0 Such A0values result in large squark mixing contributions to the Higgs boson mass Žsee Fig 1Ž d The points with the largest mh have M 10,1 TeV, M 1r,400"100 GeV, tanb,14"3, A0 in the range y1 toy TeV and, typically, MH Q100 GeV Gauge mediation w18x is an attractive alternative to gravity-mediated supersymmetry breaking One of the nice features of the minimal gauge mediation models we consider is the automatic scalar mass degeneracy w19 x All sfermions with identical quantum numbers have the same mass at the messenger scale This provides a natural solution to the supersymmetric flavor problem In order to preserve gauge coupling unification we consider two models with full SUŽ 5 messenger sector representations, the 5q5 and 10q10 models We assume a minimal Higgs sector, where the mechanism which gives rise to the B and m terms does not give additional contributions to the scalar masses In the minimal models w19x the interactions between the dynamical supersymmetry breaking sector and a standard model singlet give rise to a vev in its scalar and F components The coupling of the singlet to the messenger fields results in supersymmetry breaking and conserving messenger masses To determine the effective theory below the messenger mass scale, the messenger fields are integrated out The MSSM superpartners then receive masses proportional to L s FrS, where FS is the singlet F-term Ž scalar vev At this order, there is no A-term generated Hence, we set A0 s 0 at the messenger scale The messenger scale determines the amount of running of the soft parameters The smallest allowed value of the messenger scale is 3 L Since we do not want the gravity-mediated contributions to the scalar masses to spoil the solution to the supersymmetric flavor problem, we suppress the gravity-mediated contributions by requiring the messenger scale to be below MGUTr10 3 To be precise, we require M)103L Setting Ms L would result in a phenomenologically excluded massless messenger squark Because A0 s0, we find the maximal Higgs boson mass in the gauge-mediated models is about 117 GeV This bound arises solely from perturbative Yukawa coupling and electroweak symmetry breaking requirements, and a 1 TeV naturalness bound on the squark masses The bound is not appreciably affected by the two-loop threshold corrections wx 3 As in the gravity-mediated models we just considered, we perform a full next-to-leading order analysis Hence we include effects in the one-loop RG evolution of the gauge couplings due to the splitting of the messenger multiplets, as well as messenger contributions to the two-loop gauge and Yukawa coupling RGE s w13 x In the msugra model we found that the BŽ B X g and bottom-tau unificas tion constraints required a ) 11 at intermediate to 0 large tan b Since the gauge-mediated models have a0 s0, one would expect that these models would not be compatible with the constraints at intermediate to large tan b if the spectrum did not significantly differ from the msugra model However, it is well known that the spectra in gauge- and gravity-mediated models can be quite different w0 x For example, in the 5 q 5 gauge-mediated model, the scalar masses and the m-term are significantly heavier for a given gaugino mass than in the msugra model Just as in the nonuniversal model, the larger m allows for bottom-tau unification with smaller values of tanb, and the reduced tanb and larger m and ma suppress the supersymmetric contribution to the b sg amplitude Hence, in the 5 q 5model there is a small amount of parameter space at intermediate tan b where the constraints are satisfied, even though At- 0 In this region we find the prediction m h,116 GeV In the small tanb region m -97 GeV The results are shown in Fig Ž c h For a given gaugino mass, larger messenger sector representations result in lighter scalar masses Hence, the 10 q 10 model has relatively lighter scalars than the 5 q 5 model The lighter scalars make Yukawa unification more difficult, and readily result in too large values of BŽ B X g s at interme- diate to large tanb As can be seen in Fig Ž d, the two constraints taken together exclude the 10 q 10 model outright for intermediate to large values of tanb The only allowed points with < < b -15% cor- respond to values of tanb- In this region mh is less than 94 GeV

336 336 ( ) KT MatcheÕ, DM PiercerPhysics Letters B Our results are particularly interesting in light of the upcoming Higgs boson searches at LEP and the Tevatron LEP-II should be able to either discover or rule out a light Higgs boson up to about 105 GeV If LEP finds a Higgs boson heaõier than 96 Ž 94 GeV, the minimal 5 q 5 Ž 10q 10 gauge-mediated model will require some modification in order to be compatible with bottom-tau unification If, on the other hand, LEP does not find a light Higgs boson, the Yukawa unification criterion excludes the minimal 10 q 10 gauge-mediated model Furthermore, if bottom-tau unification is taken seriously, upcoming runs at the Tevatron stand a chance to explore both the msugra and minimal 5 q 5 gauge-mediated models The Tevatron reach in mh as a function of its total integrated luminosity is currently under active investigation and no definite conclusions can be made at this point, but the upper limits of 114 and 116 GeV, correspondingly, can serve as important benchmarks in the design of an extended Run Finally, if the Tevatron can place a limit on the Higgs boson mass above 116 GeV, this would point towards a particular nonunified scenario in the gravity-mediated models, and exclude minimal gauge-mediation altogether Acknowledgements KTM wishes to thank the SLAC theory group for its hospitality during a recent visit References wx 1 H Haber, hep-phr970713, and references therein wx JR Espinosa, M Quiros, Phys Lett B 66 Ž ; J Kodaira, Y Yasui, K Sasaki, Phys Rev D 50 Ž ; R Hempfling, A Hoang, Phys Lett B 331 Ž ; JA Casas, JR Espinosa, M Quiros, A Riotto, Nucl Phys B 436 Ž ; HE Haber, R Hempfling, AH Hoang, Z Phys C 75 Ž ; M Carena, M Quiros, CEM Wagner, Nucl Phys B 461 Ž wx 3 S Heinemeyer, W Hollik, G Weiglein, hep-phr wx 4 PH Chankowski, Z Płuciennik, S Pokorski, Nucl Phys B 439 Ž ; J Bagger, K Matchev, D Pierce, Phys Lett B 348 Ž wx 5 R Hempfling, Z Phys C 63 Ž ; M Carena, M Olechowski, S Pokorski, CEM Wagner, Nucl Phys B 46 Ž ; M Carena, CEM Wagner, hep-phr wx 6 L Hall, R Rattazzi, U Sarid, Phys Rev D 50 Ž wx 7 J Bagger, K Matchev, D Pierce, R-J Zhang, Nucl Phys B 491 Ž wx 8 P Nath, R Arnowitt, Phys Lett B 336 Ž ; JL Lopez, DV Nanopoulos, X Wang, A Zichichi, Phys Rev D 51 Ž ; V Barger, MS Berger, P Ohmann, RJN Phillips, Phys Rev D 51 Ž ; FM Borzumati, M Olechowski, S Pokorski, Phys Lett B 349 Ž wx 9 P Chankowski, S Pokorski, hep-phr w10x J Erler, D Pierce, hep-phr980138, to appear in Nucl Phys B w11x Y Yamada, Phys Rev D 50 Ž ; S Martin, M Vaughn, Phys Lett B 318 Ž ; Phys Rev D 50 Ž ; I Jack, DRT Jones, Phys Lett B 333 Ž w1x BD Wright, hep-phr w13x J Bagger, K Matchev, D Pierce, R-J Zhang, Phys Rev Lett 78 Ž w14x CLEO Collaboration: MS Alam et al, Phys Rev Lett 74 Ž w15x AL Kagan, M Neubert, hep-phr w16x CLEO Collaboration: S Glenn et al, preprint CLEO CONF 98 17, 1998; ALEPH Collaboration: R Barate et al, Phys Lett B 49 Ž w17x N Polonsky, A Pomarol, Phys Rev Lett 73 Ž ; Phys Rev D 51 Ž w18x M Dine, W Fischler, M Srednicki, Nucl Phys B 189 Ž ; S Dimopoulos, S Raby, Nucl Phys B 19 Ž ; M Dine, M Srednicki, Nucl Phys B 0 Ž ; L Alvarez-Gaume, M Claudson, MB Wise, Nucl Phys B 07 Ž ; CR Nappi, BA Ovrut, Phys Lett B 113 Ž w19x M Dine, A Nelson, Phys Rev D 48 Ž ; M Dine, A Nelson, Y Shirman, Phys Rev D 51 Ž ; M Dine, A Nelson, Y Nir, Y Shirman, Phys Rev D 53 Ž w0x JA Bagger, K Matchev, DM Pierce, R-J Zhang, Phys Rev D 55 Ž

337 7 January 1999 Physics Letters B An example of Poincare symmetry with a central charge E Karat 1 Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA , USA Received 9 November 1998 Editor: M Cvetič Abstract We discuss a simple system which has a central charge in its Poincare algebra We show that this system is exactly solvable after quantization and that the algebra holds without anomalies q 1999 Elsevier Science BV All rights reserved 1 Introduction Central charges in symmetry algebras arise in two ways They can be anomalies of a quantized theory, like Ž in two space-time dimensions the Virasoro anomaly Ž triple derivative Schwinger term in the diffeomorphism algebra of a diffeomorphism invariw1,x or in the Ž infinite conformal algebra ant theory of a conformally invariant theory w3,4 x However, they can also arise classically, as for example in the conformally invariant Liouville model, where the Ž infinite conformal algebra possesses a center obtained already by canonical Ž non-quantal Poisson brackets wx 5 or as in the asymptotic symmetry group of anti-de Sitter space in q1 dimensions wx 6 Another instance arises in the non-relativistic field theoretic realization of the Galileo group With the appearance of a number of systems with central charges in their symmetry algebras at the classical level, it is useful to study a simple model 1 address: karat@mitedu with this behavior We examine a charged, scalar field in 1q1 dimensions, interacting with a constant external electric field The Poincare algebra of this system has a central charge appearing already at the classical level As a check on Poincare invariance, we verify the Dirac-Schwinger relation This leads to a modified energy-momentum tensor Next, we take the massless case and show that we can quantize the system and solve it exactly Finally, we look at the quantized algebra of the massless case and show that it holds without anomalies, with the electric charge operator functioning as the central charge The classical symmetries We begin with the Lagrangian in 1q1 dimensions for a complex scalar field of charge e interacting with a constant external electromagnetic field, described by a position-dependent vector potential ) m ) Ls Dm f D f ym ff r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S X

338 338 ( ) E KaratrPhysics Letters B with Dm fsemfqieamf 1 n Am syemn Fx 3 where we use a flat metric of signature Ž qy and define e 01 sye01 sq1 Because of the position dependence in the externally presented vector potential, the theory does not have manifest translation symmetry However, since physical motion is manifestly translation invariant, we recover invariance of the action by adding a connection term to the transformation law of the field f under translation Ž 4 m m 1 mn m mn dt fs E q iee xnf fs D qiee xnf f m m n ) ) T n d LsE Ž D f D fym ff Ž 5 The associated Noether current TC mn is the energymomentum tensor for this theory TC mn su mn q A n J m Ž 6 mn m ) n n ) m u s Ž D f Ž D f q Ž D f Ž D f mn a ) ) a yh Ž D f Ž D f ym ff Ž 7 m ) m m ) J sie f D f y D f f 8 Using the field equations of motion, D m D qm fs0 Ž 9 Ž m we see that our currents obey Em J m s0 Ž 10 Em u mn semu nm se na FJa Ž 11 Em TC mn s0 Ž 1 Note that TC mn is not symmetric: it is conserved in the first index m, while the second index n denotes the direction of the translation Thus we arrive at our first unconventional result: Even though the fields are spinless, their canonical energy-momentum tensor is not symmetric In addition, the Lagrangian has the usual Lorentz symmetry dfse ab x E f Ž 13 L a b d LsE e ab x L Ž 14 L b a with the associated conserved current M m seab x a Ž u mb qa b J m Ž 15 Em M m s0 Ž 16 We can define operators on linear functionals G of f m m P Ž Gwf x sg dt f Ž 17 M Gwf x sgwdfx Ž 18 L In this way, P m generates translations, and M generates Lorentz transformations Defining a bracket to be the commutator of these operators on linear functionals of f, m m m wm, P xž Gwf x sg dtdlfydldtf mn 1 a sg e Ž En q ieena x F f mn se Pn Ž Gwf x Ž 19 m n n m m n wp, P xž Gwf x sg dtdtfydtdtf mn sgw iee Ff x mn sye FQŽ Gwf x, Ž 0 where Q is the operator that multiplies by yie, soit simply commutes with all other operators that act on linear functionals of f QŽ Gwf x sgwyief xsyiegwf x Ž 1 Thus, we have a Poincare algebra in 1q1 dimensions with a central charge wx 7 w m mn M, P x se Pn m w n mn P, P x sye FQ Ž 3 This algebra can be realized using Poisson brackets with the charges H P m s dxt 0m Ž t, x Ž 4 C MsHdxM 0 Ž t, x Ž 5 QsHdxJ 0 Ž t, x Ž 6 Since the charges are the spatial integrals of the time components of conserved currents, the charges are time-independent, assuming that the field f dies off sufficiently rapidly ŽWe shall later show that the quantized versions of these operators are explicitly time-independent in the massless case We calculate

339 ( ) E KaratrPhysics Letters B p, the momentum conjugate to f, tobepsž D 0 f ) Similarly, the momentum conjugate to f ) is p ) s Ž 0 m 0m 0 D f Writing J, T C, and M in terms of these quantities, J 0 Ž t, x siež f ) p ) yfp Ž 7 1 ) ) ) ) C 1 1 J Ž t, x sie Ž D f fyf Ž D f Ž 8 T Ž t, x sp pq Ž D f Ž D f qm f ) fqxfj 0 Ž 9 01 ) ) 0 C Ž 1 Ž 1 T t, x syp D f y D f p qtfj 0 ) ) ) Ž 1 Ž 1 ) qt p Ž D f q Ž D f p ) 1 1 M t, x sx p pq D f D f qm ff Ž q x yt FJ 31 we can now calculate the equal-time Poisson brackets we need 0 0 J Ž x, J Ž y s0 Ž 3 0 0m mn X J x,tc y sye Jn y d xyy 33 Ž Ž X M x, J y s xj x ytj x d xyy Ž X TC x,tc y s TC x qtc y d xyy TC x,tc y s TC x qtc y Ž qfž xyy J 0 Ž y d X Ž xyy Ž m M x,tc y Ž n Ž C C Cn syx T nm Ž x qt nm Ž y qx m T n Ž y 1 l mn q xl x e FJn y ye mn FŽ xnyyn x l JlŽ x qy Ž x m e nl yx n e ml FJ Ž y d X Ž xyy Ž 37 l Žwhere the common time argument has been sup- pressed Thus, the charges satisfy wq, M xs0 Ž 38 wq, P m x s0 Ž 39 n w m mn M, P x se P Ž 40 n m w n mn P, P x sye FQ Ž 41 This is the same algebra that we obtained before in, 3 3 The Dirac-Schwinger relation The Dirac-Schwinger relation is a method of proving Lorentz invariance A system is Lorentz invariant if the energy-momentum tensor obeys the following condition Ž for Poisson brackets : Ž Ž X T x,t y s T x qt y d xyy The energy-momentum tensor we obtained in Ž 6 obeys Ž 4 with the indices reversed: Ž X TC x,tc y s TC x qtc y d xyy Unfortunately, TC 01 /TC 10 is not symmetric in its indices, so the Dirac-Schwinger condition fails However, if we modify the energy-momentum tensor to make it symmetric, the condition then holds By adding a superpotential e mb Eb V n,weob- tain a new energy-momentum tensor T mn stc mn qe mb Eb V n su mn qe an x FJ m qe mb E V n Ž 44 a Requiring T mn to be symmetric, we obtain E V m syfx J m Ž 45 m m Since J m is conserved, we can define a new variable h by J m se mn En h Ž 46 With this new expression for J m, we obtain E V m se Fx e mn h Ž 47 m m n For solutions to Ž 47 and Ž 46, we can take V m sfxn e mn h Ž 48 ` 1 0 h t, x sh dy e xyy J t, y 49 y` b

340 340 ( ) E KaratrPhysics Letters B where ež xyy denotes the sign function For these solutions, T mn simplifies to T mn su mn yh mn hf Ž 50 Checking our commutation relations, we see that Ž Ž X T x,t y s T x qt y d xyy so the Dirac-Schwinger condition indeed holds for T mn Alternately, we can derive the energy-momentum tensor Ž 50 by varying the metric in a generally covariant version of the Lagrangian Ž 1 We start by writing the generally covariant Lagrangian ' mn ) ) ž Ž m n / Ls yg g D f Ž D f ym ff Ž 5 with Am no longer an external quantity, but a func- tional of the metric satisfying the relation wx 8 ' E A ye A se F yg Ž 53 m n n m mn Varying the metric, we get ' ' 1 mn m dls yg umndg y ygj d Am 54 Owing to the covariant conservation of J m, we can write m mn 'yg J se En h 55 as a defining relation for h Substituting 55 in 54 in the variation of the action and integrating by parts, we get H ž ' / 1 dss d x yg u dg mn ye mn mn Emd Anh 56 Using our relation Ž 53 and simplifying, we end up with our result E L Tmns sumnygmnhf Ž 57 'yg E g mn This lends credence that this is the correct energymomentum tensor to use for the Dirac-Schwinger relation Unfortunately, the momentum associated with T mn differs from the momentum associated with T mn C xs` H Ž C H 1 xsy` dx T yt s dxe V sv sytq Ž 58 Furthermore, by taking the time derivative of this difference, we see that both momenta cannot be conserved simultaneously, except possibly in the uncharged sector ŽWe shall later calculate the charges associated with our original energy-momentum tensor TC mn in the zero-mass case and see explicitly that they are time-independent Therefore, the momentum associated with T mn is not conserved, except in the uncharged sector ( 4 Solution of the equations of motion for zero mass) For the remainder of this paper, we shall take our field f to be massless by setting ms0 We shall also absorb e into F, replacing ef by F Then, we shall use the equations of motion and cannonical commutation relations to quantize the system Putting f in the form dk fs e ` H y` p = 1 iž ky Ft x Ž k ) y1r4 yk fž T a qfž T b F Ž 59 e Ž F Ts Ž Ftyk Ž 60 1r < F < and setting the mass to zero, our equations of motion Ž 9 reduce to d ž ž dt / / qt fž T s0 Ž 61 which has a closed form solution in terms of Bessel functions fž T saž k f1ž T qbž k fž T Ž 6 1r 1 < < 1 y1r4 f T s T J T 63 1r 1 < < 1r4 f T se T T J T 64 Ak and Bk are arbitrary functions that parameterize the creation and annihilation operators, ak and b k, which satisfy the usual commutation relations We now impose canonical quantization relations Our form for f Ž 59 automatically satisfies

341 w f,f ) xs w p,p ) xs 0 The condition wf,p x ) ) s wf,p xsidž xyy requires X ) X ) f T f T yf T f T si 65 which with the help of the Bessel function identity Jy3r4Ž x Jy1r4Ž x qj3r4ž x J1r4Ž x s ( ) E KaratrPhysics Letters B ' p x Ž 66 implies ' ) ) AŽ k B Ž k ybž k A Ž k s p i Ž 67 4 Apart from this constraint Žwhich can be seen as a normalization condition, the choice of these functions remains arbitrary below If one insists on choosing a parameterization, two useful parameterizations are the constant parameterization and the parameterization which reduces to the standard freefield expression in the limit where F goes to zero p y1r Ž y3r4 A k s < F< < k< J k r< F < 4 yie Ž Fk J k r< F < Ž 68 1r4 p y1r Ž y1r4 B k syi < F< < k< J k r< F < 4 yie Ž Fk J k r< F < Ž 69 3r4 This second parameterization satisfies the require- ment 67 by the Bessel function identity 66 5 Quantization of the algebra In this section, we shall use the original energymn momentum tensor T Ž C 6 and verify that its associated charges are time-independent Inserting the solution for f Ž 59, Ž 6 into the classical expressions for the charges Ž 4 Ž 6, and normal ordering, we get expressions for the quantized charges in terms of creation and annihilation operators dk Qs e a a yb b Ž 70 ` H k k yk yk y` p ž / ` dk MsH yk r Fy ifž E AE B p p y` ' ye BE A ) a a yb b ) Ž k k yk yk 1 ) ) ) ) y ifž A E ByBE A qae B yb E A ' p Ž k k yk yk k k = a Ž E a y Ž E b b y Ž E a a Ž 1 qbyk E b yk y 8 F ak E a k yž E b b yž E a Ž E a yk yk k k qž E b yk Ž E b ykq Ž E a kak yb yk Ž E b yk ' q ifž E AByAE B ŽŽ E a k byk p ' ) qakž E b yk q ifž E AByAE B p ya Ž E b y Ž E a b Ž 71 = k yk k yk ' ` dk P sh y FŽ A E ByBE A qae B p p 0 ) ) ) y` Ž k ) 1 yb E A akakqbyk byk q if akž E a qž E b ykbyky Ž E a kakyb yk Ž E b yk ' q FŽ E AByAE B ak byk p ' ) q FŽ E AByAE B a b k yk Ž 7 p ` dk 1 P sh k akakybyk byk Ž 73 y` p We note that these charges are manifestly time-independent, as claimed earlier These charges satisfy the algebra wq, P m xs wq, M xs0 Ž 74 w m mn M, P x syie Pn Ž 75 m w n mn P, P x sie FQ Ž 76 with no anomalies As a note of caution, we formed the charges first and then calculated the commutators, rather than

342 34 ( ) E KaratrPhysics Letters B calculating the commutators of the local currents and then integrating over space In addition to the Virasoro anomaly, the currents have other anomalies These other anomalies depend on time as well as the particular parameterization chosen for Ak and Bk in Ž 6 Furthermore, when these anomalies are integrated over space, the results are ill-defined and depend on how the spatial integration is evaluated As an illustrative example, let us sketch the calculation of the commutator between the quantized charges Q and M by integrating the commutator between : J 0 : and :M 0 : w x H 0 0 Q, M s dxdy : J t, x :,: M t, y : 77 :M 0 Ž t, x :sx:u 00 Ž t, x :yt:u 01 Ž t, x : 1 0 q x yt F: J t, x : 78 dk dk X X 0 iž kyk x : J Ž t, x :sh e iž ET ye T X k k p p =SŽ k,k X,T,T X Ž 79 dk dk X X 00 iž kyk x :u Ž t, x :sh e F p p k Ž X k 1r = E E qt T X SŽ k,k X,T,T X Tk Tk k k k k dk dk X X 01 iž kyk x :u Ž t, x :sh e e Ž F F p p 1r Ž 80 = ie T yie T X S k,k X T X k T k Ž,T k,tk X k k Ž 81 S k,k X,T,T X sf f X) a X a qf ) f X b b X k k k k k k k k yk yk qf f X ab X qf ) f X) b k k k yk k k ykak X Ž 8 where T is the expression Ž 60 k for T, and the added subscript allows us to denote the same expression with the momentum appearing in the subscript substituted for k Similarly, f denotes fž k,t k k It must also be noted that the independent variables with X respect to the integrals are the momenta Žk, k, etc, x, and t; however, the independent variables with respect to the derivatives are the momenta, the T variables with respect to each of the momenta ŽT k, T X, etc, and x k The operator S satisfies X X S k,k,t,t X,S p, p,t,t X k k p p spd kyp X f f X) yf ) f X S p,k X,T,T X k p k p p k ypd pyk X f f X) yf ) f X S k, p X,T,T X qpdž pyk X pdž kyp X p k p k k p = f f X f ) f X) yf ) f X) f f X, Ž 83 k k p p k k p p where the delta functions are with respect to the same independent variables as the integration Ž momenta, t, and x As a consequence of the different collections of independent variables, derivatives of the S commutator Ž 83 may not vanish, even though Ž 83 vanishes identically We can now calculate the other commutators The source of the inconsistency will be most evident if we do not evaluate the integrals over the delta functions in the anomalous term yet 0 0 J Ž t, x, J Ž t, y s0 Ž J Ž t, x,u Ž t, y syij 1 Ž t, y d X Ž xyy dk dk X dp dp qh X p p p p Ž X =e iž kykx xqiž pyp X y ie E ye T k = F 1r E E qt T X Tp Tp X p p =pdž pyk X pdž kyp X T k = f f X f ) f X) yf ) f X) f f X Ž 85 k k p p k k p p X J t, x,u t, y syij t, y d xyy J Ž t, x, M Ž t, y syiž yj 1 Ž t, y ytj 0 Ž t, y d X Ž xyy dk dk X dp dp X X X qyh e p p p p iž kyk xqiž pyp y 1r X Ž T T X Ž T T X p p k k p p = ie E ye F E E qt T =pdž pyk X pdž kyp X = f f X f ) f X) yf ) f X) f f X Ž 87 k k p p k k p p

343 ( ) E KaratrPhysics Letters B dk dk X dp dp X X X wq, M xshdxdy ye p p p p iž kyk xqiž pyp y 1r X Ž T T X Ž T T X p p k k p p = ie E ye F E E qt T =pdž pyk X pdž kyp X = f f X f ) f X) yf ) f X) f f X Ž 88 k k p p k k p p dk dk X dp dp X X sh Ž yi pdž kyk p p p p =pd X Ž pyp X pdž pyk X =pdž kyp X iež E ye X T k = F 1r E E qt T X Tp Tp X p p = f f X f ) f X) yf ) f X) f f X Ž 89 T k k k p p k k p p This final form Ž 89 illustrates the problem with evaluation there are four delta functions of three independent quantities Žfour independent momenta, but three independent differences of momenta Evaluating the charges first is equivalent to evaluating the first two delta functions first In this order of evaluation, the term vanishes Evaluating the current commutator first amounts to evaluating the last two delta functions first In this case, the value of the term depends on the order in which we evaluate the remaining two delta functions This is equivalent to the value of the term depending on how the x and y integrations are evaluated Since we are interested in the commutators of the charges, it seems more reasonable to evaluate them completely, including the spatial integration, before introducing any commutators Fortunately, the term whose commutator gives an anomaly has a vanishing coefficient after the spatial integration In this way we can see that what seems to be an ill-defined anomalous term actually vanishes Acknowledgements I would like to thank Prof R Jackiw for suggesting this problem and for guidance throughout the work This work is supported in part by funds provided by the US Department of Energy Ž DOE under cooperative research agreement adf-fc0-94er40818 References wx 1 D Cangemi, R Jackiw, Phys Lett B 337 Ž wx C Teitelboim, Phys Lett B 16 Ž wx 3 S Fubini, AJ Hanson, R Jackiw, Phys Rev D 7 Ž wx 4 S Ferrara, R Gatto, AF Grillo Ž Eds, Conformal Algebra in Space-Time, Springer-Verlag, BerlinrHeidelbergrNew York, 1973 wx 5 E D Hoker, R Jackiw, Phys Rev D 6 Ž wx 6 JD Brown, M Henneaux, Commun Math Phys 104 Ž wx 7 A discussion of the extended Poincare group can be found in D Cangemi, R Jackiw, Annals Phys 5 Ž wx 8 D Cangemi, R Jackiw, Phys Lett B 99 Ž

344 7 January 1999 Physics Letters B CP violating phase difference between B Jrc K Jrc K p 0 from new physics S S and Xiao-Gang He a,b,1, Wei-Shu Hou b, a School of Physics, UniÕersity of Melbourne, ParkÕille, Vic 305, Australia b Department of Physics, National Taiwan UniÕersity, Taipei, Taiwan, ROC Received 8 July 1998; revised 9 November 1998 Editor: H Georgi Abstract The P-wave component of B Jrc K ) turns out to be small, hence one can measure sinb using B 0 Jrc K )0 Jrc KSp 0 as well as B 0 Jrc K S Since these modes are color suppressed, new physics enhanced color dipole bsg coupling, as hinted from the persistent B and n problems and the newly observed large B h X sl C qxs decay, may have significant impact We show that it may lead to a difference in sinb values between B 0 Jrc KS and Jrc KSp 0, measurable at the B Factories in the near future q 1999 Published by Elsevier Science BV All rights reserved PACS: 135Hw; 1130Er; 160-i The B Factories will soon turn on in 1999, to study CP violation in the B meson system The benchmark test is to measure the mixing dependent CP violating angle sinb in the gold-plated B Jrc K S mode, which is predicted to be large in the Standard Model Ž SM To enhance statistics, and as a cross check, one should study other modes such as B Jrc K ) 0 Jrc K Sp 0 decay Such modes suffer, however, from partial cancellation of CP violation effects since both CP even and odd amplitudes are present wx ) 0 The CLEO Collaboration has recently reported 1 the first full angular analysis of B Jrc K and )q Jrc K decays They find that the P-wave component is small, < P < s 016" 008" 004, hence B Jrc K ) 0 Jrc K Sp 0 decay is dominated by CP-even final states Compared to Jrc K S, one has a dilution factor ;30% but this is now already measured The mode can therefore be used to measure sinb without invoking an angular analysis wx An interesting question can now be raised: Can sinb /sinb 0 Jrc K Jrc K p? S S Naively it is hard to conceive how sinb and sinb 0 Jrc K Jrc K p could differ, since the decays are dominated S S 0 0 by the tree level b ccs process, while any change in CP violating phase due to new physics in B B mixing is just a common factor One needs new contributions to the decay amplitudes In this paper we elucidate some new physics mechanisms whereby the sinb measurements could differ between B Jrc K S 1 hexg@physntuedutw wshou@physntuedutw r99r$ - see front matter q 1999 Published by Elsevier Science BV All rights reserved PII: S

345 ( ) X-G He, W-S HourPhysics Letters B and B Jrc K Sp 0 decays, hence illustrating the importance of making and refining these separate measurements Let us first show how sin b and sin b 0 Jrc K Jrc K p could in principle differ The usual CP violation S S ) Ž < < 4 y if B 0 0 measure is Im jsim qrp A Ar A, where qrpse is from B B mixing, while A, A are B, B decay amplitudes into the same final state For B Jrc K S, the final state is P-wave hence CP odd Setting the weak phase to be f 0, one has Im j Ž B Jrc K S sysinž fbqf 0 'ysinb Jrc K S ) 0 For B Jrc K Jrc K p, the final state has both P-wave Ž CP odd and S- and D-wave Ž CP even S components If S- and D-wave have a common weak phase f, we have ½ 5 0 y if y if y if Im j B Jrc K Sp sim e B e 1 P ye 1 1y P ' 1y P sinb Jrc K S p, 1 1 < < < < < < 0 where f is the weak phase for the P-wave amplitude Eq Ž 1 1 holds true in SM and, in the Wolfenstein phase convention, one has f s b and f s f s f B 0 1 1s 0 One easily obtains the usual result of sin bjrc K S s sin b 0 Jrc K S p s sin b Clearly, both measurements provide true information about sin b within SM However, this is no longer true if one goes beyond SM Even for the case where the weak phases of S- and D-amplitudes are equal, if f /f or if f, f /f, then sinb /sinb Jrc K Jrc K p follows The difference between S S sinb and sinb 0 Jrc K Jrc K p is a true measure of CP violation beyond SM S S There are many ways where new physics may change the phases f and f 0,1 1 To lowest order they are mn through dimension 6 four quark operators, or the dimension 5 color dipole operator sis G Ž 1" g mn 5 b, where mn m G is the gluon field strength Operators of the form cg Ž 1"g csg Ž 1yg b may shift f and f 5 m 5 0,1 1 by a m common amount, whereas for cg Ž 1"g csg Ž 1qg b the shift in f 5 m 5 1 the same as for f0,1 but with opposite sign Our primary example, however, is the dimension 5 color dipole operator, since there are experimental hints that it may be large in Nature We parametrize the color dipole interaction normalized to SM, GF g ) s mn VtbVts F mb ssmng Ž 1qg5 b ' 16p In SM one has F,086, and the process b sg Ž where g is on-shell, or jet-like is only of order 0%, and is very hard to measure experimentally However, the persistent low semileptonic B branching ratio Ž B sl and charm counting Ž n problems suggest that b sg decay could be enhanced to the 10% level w3,4 x C, which implies < F < ; A bound on b sg from the recent B DDKqX study wx 5 does not yet rule this out while X the discovery of a surprisingly large semi-inclusive B h q X decay wx 6 may call for an interplay of wx s 7 the < SM SM bsg charge radius coupling F < ; 5 and the new physics dipole coupling < F < 1 ; Furthermore, this new physics enhanced dipole coupling brings in naturally a CKM-independent CP violating phase, and could lead to X rate asymmetries at the 10% level w7,8x in the m X recoil mass spectrum of the B h qxs mode s Still, how can f and f 0,1 1 be significantly changed by the color dipole interaction? Note that B Jrc K Ž K ) decays are color suppressed Assuming factorization, the decay rate is proportional to < c q c rn < S 1, which suffer from accidental cancellation for N s 3 A phenomenological fit suggests N, w9,10 x eff, or, alternatively, if one takes N s 3 from QCD, then there must be sizable color-octet and other nonfactorizable contributions The former amplitude is found to be ; 1rN,similar to the color-singlet term, while other nonfactorizable contributions are ; 1rN w10x in the large N expansion Thus, the weak phases f and f 0,1 1 should be sensitive to color octet operators such as the color dipole Although the pure dipole contribution to the ) B Jrc K rate is rather small, the shift in amplitude could be of order 10% if < F < ; Ž hence b sg;10%

346 346 ( ) X-G He, W-S HourPhysics Letters B We now turn to some details In the factorization approximation but including color octet contributions, the leading order decay amplitudes for B Jrc K Ž K ) can be written as S ½ G ) F ) m ) S cb cs c c c S m 5 A B Jrc K Ž K si V V f m Ž B q Br ² KŽ K < sg Ž 1yg bb < : ' as mb X ² ) < n q r K Ž K sis q Ž c Ž 1qg qc Ž 1yg bb < : 8 S mn , p q 5 where we have adopted c syf 8 in the operator language, and ž / c VibV is ) c4 i c6 i VibV is ) i i i i B1sc1q y Ý c3q qc5q, B8scy Ý Ž c4qc 6, ) ) N V V N N V V isu,c,t cb cs isu,c,t cb cs i where the Wilson coefficients c for tree and strong penguin operators evaluated in Ref w11x j will be used The index i indicates the quark in the penguin loop The electroweak penguin contributions have been neglected ² < m < : m The decay constant fc is defined by Jrc cg c 0 sifcm c c, and is determined from Jrc leptonic decay to be f,410 MeV The parameter r is defined as the ratio of color octet to singlet matrix elements c 8 r s, ² ) < a m a Jrc K K cg T c sg 1yg T b < B: S m 5 8 ² < m < :² ) < m Jrc cg c 0 K Ž K sg Ž 1yg bb < : S 5 and similarly for r X 8 with dipole current replacing the VyA current These two parameters in general should be different, but at the moment it is not possible to calculate them from first principles We will determine r8 by fitting experimental data Taking r X 8sr8 to be equal would then give an indication of the size of the effect We also include a possible color dipole Ž 1 y g 5 term with strength c 8 for completeness Following the notation of Ref w1 x, we parametrize m Bym 0 B K K ² < < : K sg Ž 1yg bb sf q p qp q F q yf q q Ž 3 Ž m 5 1 m m 0 1 m q Ž Ž The form factors are assumed to be of pole form F q sf 0 r 1yq rm n, and the values of F Ž i i i i 0 and mi are taken from Ref w1 x Heavy quark effective theory Ž HQET suggests that at maximum recoil, Ž Ž F q rf q scales as m w13 x This implies that the pole power for F Žq and F Žq 1 max 0 max b 1 0 differ by y1 We take n s for F1 and ns1 for F 0 ² 0 < n < : Ž At maximum recoil K sis q 1"g bb can be related to F q defined in Eq Ž 3 by HQET w13 x mn 5 i Taking m b,m B, p b,pb and extrapolating down to q (m Jrc, we find ² < < : m 0 n K sis q 1"g bb ( Pp m F q s q, 4 c mn 5 c B B 1 Ž Ž where sq s F0 q rf1 q y1 1ymKrmB yq rmb rfymcrmb is a suppression factor due to Ž the helicity structure of smn at the quark level Note that in the photon case, sq vanishes with q s0, as it should The full decay amplitude is now given by ½ 5 as m ) B AŽ B Jrc K S (igfvcbvcs fcmc F1Ž m c cppb wb1q Br 8 8xq c qc r sž m, Ž c p m c X where we have set r sr To fit experimental data, we take 1rNqr s1rn f1r w9,10x eff from the first term, giving r8 f 1r1

347 ( ) X-G He, W-S HourPhysics Letters B ) For B Jrc K, we continue to use the form factor parametrization of w1 x, VŽ q AŽ q ² ) < < : n a b m m K sg 1yg bb s ) q p ) qi ) Pq p qp ) m 5 mnab K K K B K m qm ) m qm ) B K B K A3 q ya0 q m yiž m qm A q qim Pqq, Ž 6 ) ) ) ) B K 1 K K K m q where m Ž Ž Ž ) A q s m qm ) A q y m ym ) A q and A Ž 0 sa Ž 0 K 3 B K 1 B K 0 3 We assume dipole behav- Ž Ž Ž Ž ior for V q, A q and monopole for A1 q, A0 q Pole masses and form factor values at q s0 are again taken from w1 x We parametrize the dipole matrix element as ) a b a b ² K < sis bb < : s g q ) Ž p qp ) qg q ) Ž p yp ) Ž a b Ž ) Ž ) ) B K B K K B mn mnab q K B K y K B K qh q p qp p yp Pp, 7 ² < < : form factors g and h are related to A and V in HQET at maximal recoil, which are given in Ref w13 x " 0,1, Extrapolating these down in q, the full decay amplitude is finally given by ) mn mnab 013 while K sismn g5bb can be arrived at via the identity is s 1r e sabg 5, where e s1 The ½ ž ) B K Ž c V m ) ) c ' m n a b AŽ B Jrc K si GV V fm wbq Br x ) p p ) F cb cs c c mnab c K B K m qm i A m y m qm ) A m P ) qi Pp ) Pp m qm ) B K as mb m n a b y c qc r g Ž m ) p p ) q c mnab c K B K p m c as mb 1 qi c yc r ž ž g Ž m Ž m ym qg Ž m m / P p m c B K 1 c c K c B K B/ ) ) q c B K y c c c K ž / / y g m yh m m Pp ) Pp Ž 8 q c c c c B K B5 Ž Ž Unlike the smc factor in Eq 4, the form factor gq mc in Eq 8 is not suppressed by mcrm B Thus, the dipole contribution to B Jrc K S is suppressed by a factor of m crm B compared to B Jrc K ) This suppression can be traced to the helicity structure of the smn interaction at the quark level Although we have worked with color singlet operators, we note that QCD is helicity conserving Soft gluon emissions would not change the helicity structure, hence the color dipole contribution to B Jrc K S should still be suppressed compared to B Jrc K ) If the color dipole coefficients c8 or c 8 contain CP violating phases, the measurements of sin b and sin b 0 Jrc K Jrc K p could differ S S The source of CP violation must come from physics beyond SM We consider one possibility here It has been shown w4,14x that in supersymmetric Ž SUSY models, it is possible to have large c8 in the desired range Ž< c < f 8 necessary for solving the nc and Bsl problems In these models there are additional CP violating phases w7,8x which cause c Ž c to be complex with a CP violating phase ff Ž 8 8 As has been discussed earlier, the unique feature of color dipole coupling is that its effect on B Jrc K S is small, whereas the impact on B Jrc K ) is enhanced by m Brm c f3 The numerical value depends on a Ž m, which we will vary from 0 to 05 For < c < s b 8 s and c 8s0, we find that the color dipole contribution to the decay amplitude can be as large as 8% If f/0, it would generate phases for the CP even and odd amplitudes f f Ž 00y003 sinf, f ff f Ž 006y008 sinf, Ž

348 348 ( ) X-G He, W-S HourPhysics Letters B becoming larger as a increases One would measure sin b 0 f sinž b q 01sinf and sinž s Jrc K s p b q 016sinf for a Ž m s b equal to 00 and 05 respectively Taking the present best fit value of 068 for sin b w15 x, the enhanced color dipole interaction could shift sin b 0 Jrc K s p by as much as 17% The shift in sin b is three times smaller Deviations between sin b and sin b 0 Jrc K Jrc K Jrc K p as large as discussed here S S S will be probed soon at asymmetric B factories, and in the future at the LHC, where sinb 0 Jrc K Žp can be S measured to 1% accuracy We note that, if b sg; 10% Ž hence < c < f 8 sounds extreme, b sg at 3% and 1% level cannot be easily ruled out by experimental methods given in Ref wx 5 Although one would then be decoupled from the n and B problems, the difference in sin b and sin b 0 c sl Jrc K Jrc K p would still be S S at ;60% and 30% of those discussed here, and still measurable If it turns out that c is small but < c < ;, f remains the same, but f 8 8 0,1 1 changes sign We now have sinb 0 fsinžbyf rž 1y< P < and the shift is larger by a factor of 1rŽ 1y< P < Jrc K p 1 compared with S the previous case The relative sign of f and f can be probed by performing an angular analysis wx 1 1 The difference between sinb and sinb 0 Jrc K Jrc K p is sensitive also to other forms of new physics Let us S S give an example of possible contributions from dimension 6 operators In R-parity violating supersymmetric models, exchange of charged sleptons and down type squarks can generate currents involving the right handed quark b with a new CP violating phase f, and therefore phase shifts of the type df s df sydf R R 0 1 as mentioned earlier The couplings involved are constrained by b sg, but contributions at 10% level to the amplitude is still allowed w19 x The difference between sin b and sin b 0 Jrc K Jrc K p can be of order S S y03sinfrcosf B, where we have used fb instead of b because new physics may also contribute to f B, which is in general different from b The effect is again measurable The effects discussed here is really some form of penguin pollution due to new physics As such, the estimates are not fully precise We have relied on the color octet mechanism to estimate the color dipole contribution This point needs further clarification However, we find support from inclusive B Jrc Xs using the formalism of Ref w16 x The tree level color octet contribution improves agreement with inclusive rate for N s 3 Through interference, the color dipole operator contributes about 0% to the decay rate, which is consistent with the level found in exclusive decays Hence, we feel that the numerical values obtained here are of the right order of magnitude One may worry whether final state interactions Ž FSI might ruin our result Let us briefly discuss how thing may chnage with FSI phases In general the P, S and D amplitudes have different CP violating weak and CP conserving FSI phases The most general decay amplitude for B Jrc K ) can be parameterized as m yif a n a b yif b yif As ae ) q p ) qibe ) Pqp ) qice c ) c mnab K K K K m K m, 10 where we have explicitly factored out the leading CP violating phase fa,b,c for each amplitude The S and D amplitudes are linear combinations of b and c In the most general case there are additional CP violating and CP conserving phases in the amplitudes a, b, c which can be studied by looking at angular distributions for each amplitude, which has been discussed in the literature w17,18 x The detailed information can not be extracted by the observable under consideration In the cases that the CP violating phases in a, b, c are zero, the FSI phases do not play a role even if the weak phases f are different We have a,b,c < < < < B a B b c Ž B b Imjsysin f qf P qsin f qf qf 1y P q sin f qf ysinž f qf qf U q sinž f qf ysinž f qf qf U, B b c b B c B b c c ž / B K c K c < b< m ym ) ym y4m ) m Ub s, 16m m ) < A< c K B K c K c ž / < c< m ym ) ym q8m ) m Uc s Ž 11 4m m ) < A< c K

349 ( ) X-G He, W-S HourPhysics Letters B In the limit of f sf, we obtain Eq Ž 1 The deviation from Eq Ž 1 is given by cosž f Ž f yf Ž U yu b c B b c b c which is small if fb and fc are approximately equal, as the specific models predict in this paper When additional CP violating phases are allowed in a, b, c, the FSI phases will play some role, but will be secondary and suppressed by an additional factor proportional to the CP rate asymmetries for different components It is interesting to note that, because A1, and V are all approximately equal at q,m Jrc, and because the new dipole contributions are dominated by the gq form factor, the weak phases obtained for S- and D-waves are approximately equal Thus, Eq Ž 1 is a good approximation for the color dipole case Furthermore, in the R-parity violation example, these weak phases are exactly the same, which in fact holds in general for m m cg 1"g csg 1qg b and cg Ž 1"g csg Ž 1yg 5 m 5 5 m 5 b interactions We note with interest that the CLEO wx ) study shows no evidence for FSI phases 1 in B Jrc K High statistics studies following wx 1 and wx can not only uncover FSI phases, but also detect in principle direct CP violating phase differences between ) B Jrc K and B Jrc K ) w18 x Before closing, we would like to point out that the color dipole induced CP violation can be further studied in B fk decay, where Im j Ž B fk S S sysinbf K S is also a measure of sinb in SM The difference between sinb and sinb is again a sensitive probe for new physics w19 x Jrc K f K At leading order, B fk S S S is a pure penguin process which is already at loop level, and the amplitude is not color suppressed, hence the enhanced color dipole interaction will have larger effect than in B Jrc K ) The effect can be comparable to the SM contribution and therefore the shift in sinbfk will be much larger than the case for B Jrc K Sp 0 S However, B fk S has yet to be measured, while in the parameter space of interest, direct CP violating partial rate asymmetry in B y fk y can become as large as 50% and can be more readily measured w 8,0 x Direct CP y y y 0 y 0 y q violation in B f K and other self-tagging modes such as B K p or B K p will be discussed elsewhere In conclusion, we have argued that physics beyond the Standard Model can change the prediction for sinbjrc K and sinbjrc K p 0 The shift due to large color dipole interaction is small for sinb Jrc K, but can S S S be as large as 17% for sinb 0, which can be tested at B factories The shift can be much larger for Jrc K S p sinb fk We would like to emphasis, however, that even though the effect in B Jrc K ) S is smaller than in the yet to be measured B fk decay, sinb 0 will provide a practical test for new physics beyond SM S Jrc K S p at B factories We encourage our experimental colleagues to carry out such an analysis Acknowledgements This work is supported in part by grant NSC M , NSC M , and NSC M of the Republic of China and by Australian Research Council References wx 1 CP Jessop et al, CLEO Collaboration, Phys Rev Lett 79 Ž wx I Dunietz et al, Phys Rev D 43 Ž ; AS Dighe et al, Phys Lett B 369 Ž wx 3 BG Grza dkowski, WS Hou, Phys Lett B 7 Ž wx 4 AL Kagan, Phys Rev D 51 Ž wx 5 T Coan et al, CLEO Collaboration, Phys Rev Lett 80 Ž wx 6 T Browder et al, CLEO Collaboration, Phys Rev Lett 81 Ž wx 7 WS Hou, B Tseng, Phys Rev Lett 80 Ž wx 8 AL Kagan, J Rathsman, e-print hep-phr ; AL Kagan, in the proceedings of the second International Conference on B Physics and CP Violation, Honolulu, Hawaii, March 1997 wx 9 HY Cheng, B Tseng, Phys Rev D 51 Ž

350 350 ( ) X-G He, W-S HourPhysics Letters B w10x M Neubert, B Stech, e-print hep-phr97059 w11x NG Deshpande, XG He, Phys Lett B 336 Ž w1x M Bauer, B Stech, M Wirbel, Z Phys C 34 Ž w13x N Isgur, M Wise, Phys Rev D 4 Ž w14x M Ciuchini, E Gabrielli, G Giudice, Phys Lett B 388 Ž w15x F Parodi, P Roudeau, A Stocchi, e-print, hep-phr98089 w16x P Ko, J Lee, HS Song, Phys Rev D 53 Ž ; XG He, A Soni, Phys Lett B 391 Ž w17x G Valencia, Phys Rev D 39 Ž w18x X-G He, W-S Hou, Phys Rev D 58 Ž w19x Y Grossman, M Worah, Phys Lett B 395 Ž w0x X-G He, W-S Hou, K-C Yang, e-print hep-phr98098 Ž to appear in Phys Rev Lett

351 7 January 1999 Physics Letters B Instability induced renormalization Jean Alexandre a,1, Vincenzo Branchina a,, Janos Polonyi a,b,3 a Laboratory of Theoretical Physics, Louis Pasteur UniÕersity, 3 rue de l UniÕersite Strasbourg, Cedex, France b Department of Atomic Physics, L EotÕos UniÕersity, Puskin u Budapest, Hungary Received 1 September 1998; revised 1 November 1998 Editor: L Alvarez-Gaumé Abstract It is pointed out that models with condensates have nontrivial renormalization group flow on the tree level The infinitesimal form of the tree level renormalization group equation is obtained and solved numerically for the f 4 model in the symmetry broken phase We find an attractive infrared fixed point that eliminates the metastable region and reproduces the Maxwell construction q 1999 Elsevier Science BV All rights reserved We have two systematic nonperturbative methods to handle multi-particle or quantum systems, the saddle point approximation and the renormalization group Our goal in this letter is to combine these two apparently independent approximation methods in order to obtain a better understanding of the instabilities and first order phase transitions We start with the path integral, H 1 y Swf x " Zs w Df xe, Ž 1 over the configurations f x We assume the presence of UV and IR cutoffs thus the dimension of the domain of integration is large but finite 1 alexandr@lpt1u-strasbgfr branchin@lpt1u-strasbgfr 3 polonyi@fresnelu-strasbgfr The saddle point approximation to Ž 1 coincides with its perturbative expansion when the saddle points are trivial, cs0 In this case the elementary excitations, the quasiparticles, are characterized by the eigenfunctions f Ž x n of the inverse propagator, y1 G Ž x, y sd SwfxrdfŽ x dfž y < fsc The pertur- bation expansion is applicable for fixed values of the cutoffs if the restoring force of the fluctuations to fž x s c Ž x s 0 is nonvanishing, ie the eigenvalues of the inverse propagator, l n, are positive When the absolute minimum of the action is reached at fž x sc Ž x /0 then we follow the strategy of the saddle point expansion and the elementary excitations are the fluctuations around c Ž x / 0 The saddle point expansion is applicable as long as l )0 n The zero modes, the elementary excitations with l s 0, correspond to continuous symmetries and are n integrated over exactly If the inverse propagator has too many small eigenvalues the saddle point expansion breaks down and strong fluctuations develop r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

352 35 ( ) J Alexandre et alrphysics Letters B around c as we remove one of the cutoffs This possibility brings us to the renormalization group method which is supposed to deal with such probwx 1 lems The basic idea of the renormalization group is the subsequent integration in Ž 1, H H H 1 y SN wf x " Z s df df PPP df e, 1 N where the field is expanded according to a suitable N chosen basis, f x sý ffž x ns1 n n, and N-` plays the role of the UV cutoff The effective theory for the first N modes is defined through the effective action S X N by the help of the blocking transformation for DNsNyN X number of modes, H 1 1 y SN Xwf x y SNwf x " X N q1 N " H e s df PPP df e Ž 3 The elimination step is usually followed by a rescaling in order to restore the cutoff to its original value This rescaling, together with its result, the anomalous dimension, is an important device to distinguish the trivial Ž tree-level and the dynamical Ž loop-level scale dependence The fluctuations of the IR modes with n-n can be described within the effective theory given by the action Ý S wf xs g Ž N s f,e f,if,, Ž 4 N j j m j where sj is a suitable chosen complete set of local functionals without keeping the UV modes n) N present explicitly Thus the IR modes decouple from the UV ones since the correlations generated by the latters are contained in the numerical value of the effective coupling constant g Ž N j This trivial obser- vation gives rise a powerful approximation scheme by the truncation of Ž 4 when any IR field configuration can be used to reconstruct the same overdetermined effective action In the infinitesimal form of the renormalization group method, wx, where DN < N the small parameter of the loop expansion in Ž 3 is " DNrN, thus we expect that the saddle point approximation is applicable in Ž 3 The Wegner-Hough- ton equation wx is obtained by eliminating the plane waves within the shell kydk-< p< -k of the momentum space, kw x w x w x ES f " 1 d S f 1 ds f s Trk, Dk y E k Dk dfdf " df P ž / y1 d S w f x ds w f x P dfdf df hqd dswf x y Em fp de m f yhyd dswf x q fp, Ž 5 df where the trace and the P operation is taken in the subspace of the eliminable modes The second line generates the rescaling assuming the anomalous dimension h Ž input in dimension d The second order approximation is used for the action as the functional of the eliminable field variables which is applicable only if the saddle point amplitude is O Ž " Our interest in this work is the case where there is a saddle point during the elimination of the modes and we shall see that there is no guarantee in general that the saddle point remains O Ž " during the evolution Thus we need a more reliable evolution equation which is applicable without imposing the condition O Ž " on the saddle point amplitude Systems with instabilities display fluctuations with large amplitudes which can be taken into account by means of the saddle point approximation In order to demonstrate the importance of the saddle point during the blocking on the renormalized trajectory we shall consider the f 4 model in the symmetry broken phase in a box with linear size L and with periodic boundary conditions The bare action of this model is d 1 S s Hd xw Ž E f q U Ž f x where U Ž f L m L L s 4 ym rf qgf r4! Ž L is the UV cutoff and we search for the vacuum in the presence of the conyd straint L Hd d x² fž x: s F The constrained vacuum displays spinodal instability or metastability when it is unstable against fluctuations with infinitesimal or finite amplitude, respectively The spinodal

353 ( ) J Alexandre et alrphysics Letters B instability can be detected by local methods, mode by mode inspection In fact, each fluctuation of the form ik m x m w yik m x c Ž x sc e qc e m k k k s r cos k x qa Ž 6 k m m k with infinitesimal amplitude, rkf 0, represents an independent unstable mode of the tree level theory when p -m ygf r The Legendre transform of the tree level theory suggests that the vacua with m rg -F -6m rg are metastable The vacuum is stable when F )6m rg It is difficult to identify the possible instabilities of the true vacuum where the local, mode by mode analysis is unreliable This is because the decay of the unstable vacuum undergoes large amplitude modifications either in the metastable or the spinodal instable region We show now that the renormalization group can be used to deal with the modes of the unstable vacuum in a simple one by one manner The blocking transformation we employ consists of the elimination of the modes with momentum kydk-< p< - k, where k is the current cutoff and Dk is a momentum parameter smaller than any characteristic momentum scale of the system The effective coupling constants can be defined in the leading order of the j gradient expansion by Uk f sý jgj k rj!f Since any infrared field configuration should yield the same effective action we take the simplest choice, a homogeneous field fž x s F The loop expansion for Ž 3 yields the general result SN XsÝ j" j SN Ž j X The perturbation expansion retains the term O Ž" j with j)0 and the tree level, O Ž" 0 contribution represents a non-perturbative piece what must be considered before the perturbative pieces are taken into account Since the tree level expressions are different from the loop corrections the tree level renormalization, if exists, consists of essentially new and different expressions than the loop contributions which have been considered so far wx 3 Our constraint generates non-trivial saddle points and we consider here the leading order, tree level renormalization only In this approximation the naive scaling is correct Ž hs0, Zs1, and we ignore the rescaling step of the renormalization group method for the sake of simplicity The saddle point of the blocking transformation X ipx can be written in the form ck x sýcpe where the prime denotes the summation for kydk-< p< - k The blocked potential is given by d LU wf x kydk H 1 y S k " w x sy"ln w Dh xe Fqh d 1 c 4 H ž Ž m k/ smin d x E c qu Fqc qo Ž " Ž 7 where the fluctuation h contains only the modes within the shell wkydk,k x It is worthwhile noting that Ž 7 reduces to the usual local potential approxiwx 4 if the mation of the Wegner-Houghton equation saddle point vanishes There might be several minima c in which case one should sum over them We retain only the single plane waõe saddle points, Ž 6 The motivation of this rather drastic approximation is the assumption that the saddle point Ž 6 seems to be optimal from the point of view of the energy-entropy balance To see this first note that when Dk 0 the saddle point is non-vanishing in the momentum space on the sphere with radius k only As far as the saddle point on this sphere is concerned, the introduction of other additional plane waves would increase the kinetic energy The phase ak sayk is a zero mode for each plane waves, corresponding to the translation invariance The more involved saddle points built by several plane waves have the same single translational zero mode Thus the kinetic energy-entropy balance is optimized for the plane waves The resulting tree-level blocking relation is UkyDkŽ F H 1 1 smin r4 k r q du Uk Fq rcos p u y1 H Ž 1 1 sk rkq du UkŽ Fq rkcos p u 8 y1 We followed the tree level evolution of the potential by performing numerically the minimization in Ž 8 at each blocking step The most interesting aspect

354 354 of the result is that, starting from ks m ydk, the saddle points c are nonvanishing Ž k and conse- quently the potential receives a nontrivial renormal- Ž ization for 0-F -F k s6 m yk vac rg Thus the mode coupling provided by our successive elimination method extends the spinodal instability over the vacua which are seen as metastable by the tree level Legendre transformation This can happen because once the tree level contributions are found at modes with p -m for certain values of F the saddle point contributions renormalize the potential in such a manner that spinodal instability occurs for other, larger values of F Other important results of the numerical analysis are in order: 1 the saddle point amplitude is a linear function of the field, r syfqf Ž k k vac ; we find the potential 1 3 UkŽ F sy k F y Ž m yk Ž 9 g whenever the saddle points are nontrivial, ie in the unstable region, see Fig 1, where the evolution of the potential with m s01 and gs0 is shown; 3 the blocking converges as Dk 0 despite the apparent absence of Dk in Ž 8 ; 4 we checked that the previous results are uniõersal with respect to the choice of the symmetry breaking bare potential Fig 1 The potential U Ž F k for different values of k showing the RG evolution towards the Maxwell effective potential ( ) J Alexandre et alrphysics Letters B ' It is possible to understand the result Ž iii if we write Ž 8 for two subsequent steps and take the difference: Uky Dk F sukydk F ydk krkqk rkek rk q q 1 1 H k k k y1 du E U Fq r cosž p u 1 H k k f k y1 du cosž p u E r E U ŽF q r cos p u qo Dk, 10 k which shows explicitly that the correction to the potential due to one blocking step is proportional to Dk It is also worth remarking that the potential Ž 9 is a solution of the Eq Ž 10, as can easily be verified Few remarks are in order at this point Ž a The disappearance of the metastable region is in agreewx 6 The saddle points ment with the finding of Ref of the static problem reflect the spontaneous droplet formation dynamics wx 5 : Suppose that the stable vacuum bubble in a sphere of radius R and created in the false vacuum has the energy ERs4pRsy 4p R 3 DEr3 where s and DE stand for the surface tension and the Ž free energy difference between the two vacua The critical droplet size, R c s srde, beyond which the droplets grow can be identified by the inverse of the highest momentum of the conden- sate, R s6rgžf Ž 0 yf c vac This relation estab- lishes a connection between the static and the dynamical properties Ž b The last term in the right hand side of Ž 9 was added by hand to achieve a continuous matching of the partition function at the instability As a result, Ž 9 reproduces the Maxwell construction for the effective potential V Ž F eff s U Ž F wx 7, ie it remains unchanged Ž ks0 at the tree level in the stable region, F )F Ž vac 0, and turns out to be constant and continuous for F FF Ž vac 0 Ž c The potential Ž 9 contains only one coupling Ž constant, mksefuk0s yk The correspond- ing dimensionless coupling constant, m Ž k s m k rk sy1, is trivially renormalization group invariant In other words we recover the usual scaling laws in the stable region, while in the unstable region the potential is a fixed point given by ŽŽ 9 d

355 ( ) J Alexandre et alrphysics Letters B Similar tree level potential has been found in the 4 N-component f model with smooth cutoff wx 8 where the would be unstable and the stable regime join and the naive metastable region is wiped out by the Goldstone modes While the possibility of having a homogeneous c Ž x k is an essential part of the argu- ment, our result shows the more general origin of the potential Furthermore note that the keeping of a single plane wave mode as a saddle point is not justified when a smooth cutoff is used The analytical structure of the loop corrections in the vicinity of the instability indicates the same potential as well, wx 9 The availability of such a diverse derivations and the microscopic dynamics-independent form suggests a more fundamental origin of this potential We believe that underlying reason of Ž 9 is the Maxwell construction and the actual form can easiest be identified by means of the renormalization group method where the dynamics of each mode can be dealt with individually In fact, one can show that the differentiability of the renormalization group flow, ie the existence of the beta functions, and the nonvanishing of the saddle points as Dk 0 requires the form Ž 9 To see this consider the finite difference U Ž F yu Ž F kydk k Dk k qef Uk Ž F s dxc Ž x d H k LDk EF 3 Uk Ž F 3 q Hdxck Ž x q PPP qo Ž ", d 6LDk Ž 11 where we expanded Ž 7 in the saddle point, c Ž x k The convergence of the left hand side requires either Ž 1r the smallness of the saddle point, c so Dk k,or n k q E U F s O Dk and E U Ž F s OŽ Dk F k F k Since the saddle point is nonvanishing the form Ž 9 follows Note the essential differences between the tree and the loop level renormalization: The former is nonanalytic in g and lacks the usual logarithms of the latter Ž e The result Ž i can be obtained by assuming Ž 9 in the unstable region and requiring continuity of E U Ž F F k in F The saddle point approximation includes the softest fluctuations, the zero modes, and one can obtain nontrivial contributions to the correlation functions at O Ž" 0 In fact, let us insert the product of field variables in the integrand of Ž and follow the successive elimination of the modes in the path integral The resulting tree level expression for the n-point function on the background field F is GF Ž0 p 1,,pn s n H H Ł k j Dwkˆx Dwax c Ž p H H js1 Dwkˆ x Dwa x n d dž p ÝŁ Ž P Ž j P Ž jy1 d p j P js1 Vd k Ž F s d p qp r d yipx k k d j Ž 1 where c p s Hd e c x, V stands for the solid angle, we integrate over the zero modes characterized by the unit vector kk ˆ corresponding for each value of the cutoff, k, and the phase až p The sum in the second equation is over the permutations P of the field variables and k F sm ygf r6 Whenever we have an odd number of fields or Ž F GFvac 0 or one of the pj is such that p j )m ygf r6, G Ž0 s0 The integration over the zero modes is reminiscent of the integration over the possible rearrangements of the domain walls in the mixed phase and restores the translation invariance of the correlations The two point function d dž p Ž0 GF Ž p 1, p s r dž p qp d p 1 1 Vd k Ž F shows that the Fourier transform of c p can be interpreted as the domain wall structure for a given choice of the zero modes The method put forward in this letter, the use of the saddle point approximation for the blocking transformation, can be used to handle some of the systems where large amplitude, inhomogeneous fluctuations are present We studied here an Euclidean system describing the equilibrium situation Whenever the time dependence far from equilibrium has nontrivial semiclassical limit this method can be extended to include the real time dependence, w10 x Another natural continuation of this work is the inclusion of the loop corrections in addition to the tree level pieces The correlation functions computed

356 356 ( ) J Alexandre et alrphysics Letters B in such a manner include the fluctuations in a systematic manner which is an improvement compared to Ref w11x where the master equation was used to describe the most probable values of the correlation functions Acknowledgements We thank Daniel Boyanowski for a useful discussion References wx 1 K Wilson, J Kogut, Phys Rep C 1 Ž ; K Wilson, Rev Mod Phys 47 Ž wx FJ Wegner, A Houghton, Phys Rev A 8 Ž wx 3 J Polchinski, Nucl Phys B 31 Ž ; A Hasenfratz, P Hasenfratz, Nucl Phys B 95 Ž ; C Wetterich, Nucl Phys B 35 Ž wx 4 JF Nicoll, Chang, Stanley, Phys Rev Lett 33 Ž ; A Hasenfratz, P Hasenfratz, Nucl Phys B 70 Ž ; C Wetterich, Nucl Phys B 35 Ž ; SB Liao, J Polonyi, Ann Phys Ž ; TR Morris, Nucl Phys B 458 Ž wx 5 JS Langer, in: solids Far From Equilibrium, C Godreche Ž Ed, Cambridge Univ Press, 199 wx 6 K Binder, D Staufer, Adv in Phys 5 Ž wx 7 Y Fujimoto, L O Raifeartaigh, G Parravicini, Nucl Phys B 1 Ž ; RW Haymaker, J Perez-Mercader, Phys Rev D 7 Ž ; CM Bender, F Cooper, Nucl Phys B 4 Ž ; F Cooper, B Freedman, Nucl Phys B 39 Ž ; M Hindmarch, D Johnson, J Math Physics A 19 Ž ; V Branchina, P Castorina, D Zappala, ` Phys Rev D 41 Ž ; K Cahill, Phys Rev D 5 Ž wx 8 A Ringwald, C Wetterich, Nucl Phys B 334 Ž wx 9 N Tetradis, C Wetterich, Nucl Phys B 383 Ž ; N Tetradis, Nucl Phys B 488 Ž w10x J Polonyi, Ann Phys 55 Ž w11x JS Langer, M Bar-on, H Miller, Phys Rev A 11 Ž

357 7 January 1999 Physics Letters B Equivalence principle: tunnelling, quantized spectra and trajectories from the quantum HJ equation Alon E Faraggi a,1, Marco Matone b, a Department of Physics, UniÕersity of Minnesota, Minneapolis, MN 55455, USA b Department of Physics G Galilei, Istituto Nazionale di Fisica Nucleare, UniÕersity of PadoÕa, Via Marzolo 8, PadoÕa, Italy Received 6 October 1998 Editor: M Cvetič Abstract A basic aspect of the recently proposed approach to quantum mechanics is that no use of any axiomatic interpretation of the wave function is made In particular, the quantum potential turns out to be an intrinsic potential energy of the particle, which, similarly to the relativistic rest energy, is never vanishing This is related to the tunnel effect, a consequence of the fact that the conjugate momentum field is real even in the classically forbidden regions The quantum stationary Hamilton Jacobi equation is defined only if the ratio c D rc of two real linearly independent solutions of the Schrodinger equation, and therefore of the trivializing map, is a local homeomorphism of the extended real line into itself, a consequence of the Mobius symmetry of the Schwarzian derivative In this respect we prove a basic theorem relating the request of continuity at spatial infinity of c D rc, a consequence of the qlq y1 duality of the Schwarzian derivative, to the existence of L Ž R solutions of the corresponding Schrodinger equation As a result, while in the conventional approach one needs the Schrodinger equation with the L Ž R condition, consequence of the axiomatic interpretation of the wave function, the equivalence principle by itself implies a dynamical equation that does not need any assumption and reproduces both the tunnel effect and energy quantization q 1999 Elsevier Science BV All rights reserved PACS: 03; 0365-w Keywords: Equivalence principle; Quantum potential; Trajectories; Mobius symmetry; Tunnel effect; Energy quantization; Bound states Let us consider a one-dimensional stationary system of energy E and potential V and set W'VŽ q ye Let us denote by S the quantum analogue of the Hamilton characteristic function, also called reduced action This faraggi@mnhepohepumnedu matone@padovainfnit r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

358 358 ( ) AE Faraggi, M MatonerPhysics Letters B function satisfies the Quantum Stationary Hamilton Jacobi Equation QSHJE ž / 1 E S0Ž q " qvž q yeq S 0,q4 s0, Ž 1 m E q 4m that in wx 1 was uniquely derived from the equivalence principle It states that a b a b For each pair W,W, there is a transformation q q sõ( q a ), such that W a Ž q a W aõ Ž q b sw b Ž q b wx This principle implies the existence of the trivializing map 1 q q 0 ss 0y1 (S q, reducing the system with a given W to the one corresponding to W 0 '0 Ž D If c,c is a pair of real linearly independent solutions of the Schrodinger equation, then we have i S " 0 i a wqi l e se, Ž 4 wyi l where wsc D rc, and Re l/0 A basic property of the conjugate momentum pseq S 0, is that it is real even in the classically forbidden regions wxwx 1 This fact is an important check in considering the trajectories described by Ž 1 To understand this, observe that in the conventional formulation of quantum mechanics the tunnel effect is a consequence of the axiomatic interpretation of the wave function c Actually, the fact that it describes the probability amplitude of finding the particle in the interval wq,qqdq x, implies that the tunnel effect simply arises in the cases in which c is not identically zero in the classically forbidden regions In the case at hand, there is no need for any axiomatic interpretation, it is just a consequence of reality of the conjugate momentum in the classically forbidden regions This result suggests considering if the theory reproduces the other fundamental aspect of quantum mechanics, namely quantization of the energy spectra This would be a basic check as also energy quantization is strictly related to the axiomatic interpretation of the wave function We will see that this is actually the case The properties of the Schwarzian derivative imply that the QSHJE is well defined if and only if wx 3 ˆ w/cnst, wgc Ž R, and E w differentiable on R, ˆ q Ž 5 and wq is locally invertible ;qgr, ˆ where R'Rj ˆ `4 denotes the extended real line In particular, this implies the following joining condition at spatial infinity wž q`, for wž y` /"`, wž y` s ½ Ž 6 ywž q`, for wž y` s"` It follows that w, and therefore the trivializing map, is a local homeomorphism of Rˆ into itself Let us start proving a result concerning the energy spectra If VŽ q )E, ;qgr, then, as we will see, there are no solutions such that the ratio of two linearly independent solutions of the Schrodinger equation corresponds to a local homeomorphism of Rˆ into itself The fact that this is an unphysical situation can be also seen from the fact that the case V)E, ;qgr, has not classical limit Therefore, if VŽ q )E both at y` and q`, a physical situation requires that there are at least two points in which VyEs0 More generally, if the potential is not continuous, we should have at least two turning points for which VŽ q ye changes sign Let us

359 ( ) AE Faraggi, M MatonerPhysics Letters B denote by q Ž q the lowest Ž highest y q value of the turning points In the following we will prove the following basic fact If ½ Py )0, q-q y, VŽ q yeg Ž 7 P q )0, q)q q, D then the ratio w s c r c is a local homeomorphism of Rˆ into itself if and only if the corresponding Schrodinger equation admits an L ( R) solution Note that by Ž 7 we have where H y` qy H q` dxk Ž x sy`, dxk Ž x sq`, Ž 8 qq (mž VyE ks Ž 9 " Let us first show that the request that the corresponding Schrodinger equation admits an L Ž R solution is a D sufficient condition for the ratio wsc rc to be continuous at "` Let us denote by c the L Ž R solution and by c D a linearly independent solution As we will see, the fact that c D and c are linearly independent D D imply that if cgl R, then c gu L R, in particular c is divergent both at qsy` and qsq` Let us consider the real ratio Ac D qbc ws, Ž 10 D Cc qdc where ADyBC/0 Since cgl R, we have Ac D qbc Ac D A lim w s lim s lim s, Ž 11 D D Cc qdc Cc C q "` q "` q "` that is w y` sw q` In the case in which Cs0 we have Ac D lim ws lim s"ep`, Ž 1 q "` Dc q "` where es"1 The fact that lim q "` Ac D rdc is divergent follows from the mentioned properties of c D and c It remains to understand the fact that if the limit is y` at qsy`, then this limit is q` at qsq`, and vice versa This can be understood by observing that D Ž q q y c sch dxc Ž x qd, Ž 13 c Ž q q 0 y1 for some real constants c/ 0 and d Now, since c g L R, it follows that c gu L Ž R In particular, q` y y` y H dxc x sq` and H dxc Ž x sy`, so that c D Ž y` rc Ž y` syep`syc D Ž ` rc Ž ` q q where 0 0 essgn c We now show that the existence in the case Ž 7 of an L Ž R solution of the Schrodinger equation is a necessary condition for the ratio wsc D rc to be continuous at "` To this end we recall that if V q yegp )0, q)q, then as q q`, we have Ž P )0 q q q

360 360 ( ) AE Faraggi, M MatonerPhysics Letters B Ø There is one solution of the Schrodinger equation, defined up to a multiplicative constant, that vanishes at least as e yp qq Ø Any other solution diverges at least as e Pqq The proof of this fact is based on Wronskian arguments and can be found in Messiah s book wx 4 The above result extends also to the case in which V q yegp y)0, q-q y In particular, as q y`, we have Ž P )0 y Ø There is one solution of the Schrodinger equation, defined up to a multiplicative constant, that vanishes at least as e Pyq Ø Any other solution diverges at least as e yp yq These properties imply that if there is a solution of the Schrodinger equation in L Ž R, then any solution is either in L Ž R or diverges both at y` and q` Let us show that the possibility that a solution vanishes only at one of the two spatial infinities is excluded Suppose that, besides the L Ž R solution, which we denote by c 1, there is a solution c which is divergent only at q` On the other hand, the above properties show that there exists also a solution which is divergent at y` Let us denote by c3 this solution Since the number of linearly independent solutions of the Schrodinger equation is two, we have c sac qbc, Ž for some constants A and B However, since c vanishes both at y` and q`, we have that Ž 14 1 cannot be satisfied unless c and c3 are divergent both at y` and q` This fact and the above properties imply the following If the Schrodinger equation has an L ( R) solution, then any solution has one of the following two possible asymptotic behaõiors Ø Vanishes both at y` and q` at least as e Pyq and e yp qq respectively Ø Diverges both at y` and q` at least as e yp yq and e Pqq respectively Similarly, we have If the Schrodinger equation does not admit an L ( R) solution, then any solution has one of the following three possible asymptotic behaõiors Ø Diverges both at y` and q` at least as e yp yq and e Pqq respectively Ø Diverges at y` at least as e yp yq and vanishes at q` at least as e yp qq Ø Vanishes at y` at least as e Pyq and diverges at q` at least as e Pqq Let us consider the ratio wsc D rc in the latter case Since any different choice of linearly independent solutions of the Schrodinger equation corresponds to a Mobius transformation of w, we can choose 3 and c D ; a e Pyq, c D ; a e Pqq, Ž 15 q y` y q q` q c ; b e yp yq, c ; b e yp qq Ž 16 q y` y Their ratio has the asymptotics c D c q q` q c D P y q Pqq q q q` ; cye 0, ; c e "`, Ž 17 c q y` Ž so that w cannot satisfy the continuity condition 6 at "` The above results imply that the quantized spectrum one obtains from the conventional approach to quantum mechanics arises as a consequence of the equivalence principle Actually, even in the conventional approach the 3 Here by ; we mean that c D and c either diverge or vanish at least as

361 ( ) AE Faraggi, M MatonerPhysics Letters B quantized spectrum and its structure arose by the condition that the values of E satisfying Ž 7 should correspond to a Schrodinger equation having an L Ž R solution Then, all the standard results on the quantized spectrum are reproduced in our formulation 0 0 Let n be the index Iq w x of the trivializing map This is the number of times q spans Rˆ while q spans R ˆ In other words, n is the index of the covering associated to the trivializing map Since q 0 and w are related by a Mobius transformation wx 3, we have that the index of q and w coincide 0 I q siww x Ž 18 Another property of the trivializing map is that its index depends on W but not on the specific Mobius state The Mobius states are the states with the same W but with different values of the constant l Žsee wx 3 for related aspects Observe that since p s E S does not vanish for finite values of w, it follows that Iw q 0 x q 0 coincides with the number of zeroes of w This aspect may be also understood by recalling a Sturm theorem about second-order differential equations The theorem states that given two linearly independent solutions c D and c XX D 4 of the equation c q sk q c q, between any two zeroes of c there is one zero of c This theorem, and the condition that the values of E satisfying Ž 7 should correspond to a Schrodinger equation having an L Ž R solution guarantees local homeomorphicity of the trivializing map It remains to understand the case in which VŽ q )E, ;qgr We already noticed that, since there is not the classical limit in this case, these solutions are not admissible ones We now show that these solutions do not satisfy the continuity condition for w at "` To see this it is sufficient to note that if c decreases as q y`, then by c XX rcs4mwr" )0, ;qgr, it follows that c is always convex, cgu L Ž R The absence of turning points does not modify the essence of the D above conclusions and if V q )E, ;qgr, then the ratio c rc is discontinuous at "` As an example, let us consider the equation " q E csa c Ž 19 m Any pair of linearly independent solutions has the form c D sae aq qbe yaq, csce aq qde yaq, Ž 0 where ADyBC/0 Ž 1 Their ratio c D Ae aq qb s, Ž aq c Ce qd has the asymptotics c D B c D A lim s, lim s, Ž 3 q y` c D q q` c C so that by Ž 1 neither the case wž y` sfiniteswž q`, nor wž y` sywž q` s"` can occur We now consider the cases of the potential well and of the simple and double harmonic oscillators We will explicitly see that in the case one considers the energy values for which the corresponding Schrodinger equation D has not L R solutions, the ratio c rc has a discontinuity at spatial infinity 4 Observe that this can be seen as a sort of duality between c D and c In this context we note that, while in the conventional approach one usually selects the wave function which is a particular solution of the Schrodinger equation, S and p contain both c D and c 0

362 36 ( ) AE Faraggi, M MatonerPhysics Letters B Let us consider the potential well 0, < q< FL, VŽ q s ½ Ž 4 V, < q < 0 )L, and set ' me mž V ye ( 0 ks, ks Ž 5 " " According to Ž 4, in order to determine S 0, and therefore to solve the dynamical problem, we have to find two real linearly independent solutions of the Schrodinger equation Since the potential is even, we can choose solutions of definite parity We have yaexp k Ž qql ybexp yk Ž qql, q-yl, y1 csk P~ sinž kq, < q< FL, Ž 6 aexp yk Ž qyl qbexp k Ž qyl, q)l, where for any EG0 1 k 1 k a s sinž kl y cosž kl, b s sinž kl q cosž kl Ž 7 k k For the dual solution, we have g exp k Ž qql qdexp yk Ž qql, q-yl, D c s~ cosž kq, < q< FL, Ž 8 g exp yk Ž qyl qdexp k Ž qyl, q)l, where 1 k 1 k g s cosž kl q sinž kl, d s cosž kl y sinž kl Ž 9 k k The ratio of the solutions is given by c D c ½ yžg expwk Ž qq L xqdexpwyk Ž qq L x ržaexpwk Ž qq L xq bexpwyk Ž qq L x, q-y L, s kp cotž kq, < q< F L, 30 Žg expwyk Ž qy L xqdexpwk Ž qy L x ržaexpwyk Ž qy L xq bexpwk Ž qy L x, q) L, whose asymptotic behavior is c D d lim s" k Ž 31 c b q "` Continuity at "` implies that either bs0, Ž 3 so that w y` sysgn dp`syw q`,or ds0, Ž 33 so that wž y` s0swž q` It is easy to see that Ž 7, Ž 9, Ž 3 and Ž 33 identify the usual quantized spectrum

363 ( ) AE Faraggi, M MatonerPhysics Letters B Let us now consider the double and simple harmonic oscillators The Hamiltonian describing the relative motion of the reduced mass of the double harmonic oscillator is p 1 Hs q mv Ž < q< yq 0 m Ž 34 Ž D The reduced action for this system is given by 4 with c and c real linearly independent solutions of the Schrodinger equation Let us set and ž / " E 1 y q mv < q< yq csec Ž 35 Ž 0 m E q 1 Es mq "v, 36 ( ( mv mv X z s Ž qqq 0, qf0, zs Ž qyq 0, qg0 Ž 37 " " X X m v Observe that z Ž yq syzq and that for q0 s0, z szs " q, so that the system reduces to the simple harmonic oscillator We stress that since we are considering Eq Ž 35 for arbitrary real E, we have that at this stage m is an arbitrary real number The Schrodinger equation Ž 35 is equivalent to and / E c 1 z X q mq y cs0, qf0, Ž 38 X ž E z 4 / E c 1 z q mq y cs0, qg0 Ž 39 ž E z 4 Ž wx For any m we have that a solution of 39 is given by the parabolic cylinder function see for example 5 G Ž 1r Dm z s mr e yz r4 1F1 ymr;1r;z r G Ž 1ym r ( z G Ž y1r q 1F1 Ž 1ym r;3r;z r, Ž 40 ' G Ž ymr where F is the confluent hypergeometric function 1 1 a z až aq1 z 1F1Ž a;c; z s1q q q Ž 41 c 1! cž cq1! D We are interested in considering the continuity of the ratio wsc rc at "` That is, if wž y` /"` we have to impose wž y` swž q`, while if wž y` s"`, then we should have wž y` sywž q` To study

364 364 ( ) AE Faraggi, M MatonerPhysics Letters B the continuity at "` we need the behavior of D for < z< 41, < z< 4< m < m In the case pr4-arg z-5pr4 we have ' p mp i z r4 ymy1 Ž mq1ž mq Dm Ž z ; y e e z 1q < z< 41 G Ž ym z D X m ya q0 X Dm Ž a q0 Ž mq1ž mqž mq3ž mq4 q P4 z 4 q mž my1 mž my1ž myž my3 yz r4 m qe z 1y q y, 4 z P4 z Ž 4 while for < arg z< -3pr4 mž my1 mž my1ž myž my3 Dm Ž z ; e yz r4 z m 1y q y 4 < z < 41 z P4 z Ž 43 A property of the cylinder parabolic function is that even D Ž yz is a solution of Ž 39 m In particular, if m is a 5 non-negative integer, then Dm z and Dm yz coincide Let us then consider the values of m different from a non-negative integer We have ½ D yz X qcd z X m m, qf0, cs D m Ž z qcd m Ž yz, qg0, Ž 44 where cs Ž 45 For any given m, a linearly independent solution is given by ½ yd yz X ydd z X m m, qf0, D c s Ž 46 Dm Ž z qddm Ž yz, qg0, where Dm Ž ya q0 dsy Ž 47 D Ž a q The ratio ½ m 0 D y D yz X qdd z X r D yz X qcd z X m m m m, qf0, q s Ž 48 c D Ž z qdd Ž yz r D Ž z qcd Ž yz, qg0, c y Ž m m Ž m m has the asymptotics behavior c D d lim s" Ž 49 c c q "` This shows that continuity at "` is satisfied in the case in which either cs0 ords0, which fix the standard energy quantized spectra Žsee also wx 3 5 y1 Note that in the case in which m is a non-negative integer we have G Ž ym s0, so that in this case the first term in Ž 4 cancels

365 ( ) AE Faraggi, M MatonerPhysics Letters B Observe that c in Ž 44 and the dual solution Ž 46 are not linearly independent in the case in which ms0,1,, In this case there is always a solution vanishing both at y` and q` In the case q0 s0, this situation corresponds to the harmonic oscillator Generally, for an arbitrary q0 and ms0,1,,, the solution D c in Ž 46 is replaced by yd yz X yd X D yiz X n yny1, qf0, D c s Ž 50 ½ X D n Ž z qd D yny1 Ž iz, qg0, where now DnŽ ya q0 X d sy Ž 51 D Ž yia q n 0 We note that above we used the parabolic cylinder functions with real argument Then, the fact that for ms0,1,,, D Ž z and D Ž yz are not linearly independent forced us to use D Ž yiz m m m In this context we observe that D Ž z and D Ž yiz are always linearly independent so that the dual solution Ž 50 m m can be extended to arbitrary values of m Acknowledgements Work supported in part by DOE Grant No DE FG 087ER4038 Ž AEF and by the European Commission TMR programme ERBFMRX CT Ž MM References wx 1 AE Faraggi, M Matone, hep-thr ; Phys Lett B 437 Ž , hep-thr971108; Phys Lett A 49 Ž , hep-thr ; hep-thr980915, to appear in Phys Lett B wx ER Floyd, Phys Rev D 5 Ž ; D 6 Ž ; D 9 Ž ; D 34 Ž ; Phys Lett A 14 Ž ; Found Phys Lett 9 Ž , quant-phr ; quant-phr ; quant-phr wx 3 AE Faraggi, M Matone, hep-thr wx 4 A Messiah, Quantum Mechanics, Vol 1, North Holland, Amsterdam, 1961 wx 5 W Magnus, F Oberhettinger, Formulas and Theorems for the Functions of Mathematical Physics, Chelsea, New York, 1954

366 7 January 1999 Physics Letters B Microscopic spectral density of the Dirac operator in quenched QCD PH Damgaard a,1, UM Heller b,, A Krasnitz c,3 a The Niels Bohr Institute, BlegdamsÕej 17, DK-100 Copenhagen, Denmark b SCRI, Florida State UniÕersity, Tallahassee, FL , USA c Unidade de Ciencias Exactas e Humanas, UniÕersidade do AlgarÕe, Campus de Gambelas, P-8000 Faro, Portugal ˆ Received 6 November 1998 Editor: H Georgi Abstract Measurements of the lowest-lying eigenvalues of the staggered fermion Dirac operator are made on ensembles of equilibrium gauge field configurations in quenched SUŽ 3 lattice gauge theory The results are compared with exact analytical predictions in the microscopic finite-volume scaling regime q 1999 Elsevier Science BV All rights reserved The study of the eigenvalue spectrum of the Dirac operator in QCD and related gauge theories has a long history, most notably due to the relation between zero modes, near-zero modes, topology, and chiral symmetry breaking More recently the subject has undergone a strong revival This was originally sparked by the observation made by Leutwyler and Smilga wx 1 that in sectors of fixed topological charge n one can derive exact analytical results for the distribution of small eigenvalues of the Dirac operator The idea is that there exists a finite-volume scaling regime for the euclidean gauge theory where the partition function can be computed exactly One crucial input is the assumption that chiral symmetry is spontaneously broken according to the conventional scenario The finite-size scaling regime is 1 phdamg@alfnbidk heller@scrifsuedu 3 krasnitz@ualgpt unphysical in the sense that it restricts the associated pseudo-goldstone bosons to be very light, so light that only their zero-momentum modes contribute significantly to the euclidean path integral The conditions for this simplification are two-fold First, the euclidean finite-volume partition function of QCD must be accurately described by the effective hadronic excitations, rather than the underlying quark and gluon degrees of freedom This imposes the large-volume condition 1rL QCD < L, where L is the linear extent of the four-volume Second, this linear extent L should always be insignificant compared with the Compton wavelength of the pseudo-goldstone bosons: L < 1rm While this latter condition p is inappropriate for actual, physical, no-so-light pions, it is of course just the right limit for studying spontaneous chiral symmetry breaking More importantly in this context: it is ideally suited for finitevolume studies of lattice gauge theory A major breakthrough in the understanding of the Dirac operator spectrum in the above finite-volume r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

367 ( ) PH Damgaard et alrphysics Letters B region has been the suggestion by Verbaarschot and co-workers that random matrix theory can give exact analytical predictions for the spectrum in a suitably defined microscopic regime w,3 x Gauge theories based on various gauge groups and with fermions in either fundamental or adjoint representations fall into essentially just three different universality classes, which beautifully fit into corresponding classificawx 4 In this tions of random matrix theory ensembles paper we shall restrict ourselves to SUŽ 3 gauge theory with quarks in the fundamental representation of the gauge group, where universality of all microscopic correlation functions, including the microwx 5 scopic spectral density itself, has been proven Moreover, it has recently been shown that all these exact results can also be derived directly from the finite-volume partition function of QCD w6,7 x, without recourse to random matrix theory Also the universal smallest-eigenvalue distribution can be computed in this way wx 8 So far the analytical predictions for the microscopic Dirac operator spectrum have been tested in 4-dimensional SUŽ lattice gauge theory with stagwx 9, and in 3-dimensional SUŽ 3 lat- gered fermions tice gauge theory with staggered fermions w10 x There have also been preliminary studies of the microscopic spectrum in -dimensional UŽ 1 lattice gauge theory with a fixed-point Dirac operator w11 x However, none of these cases test the most important universality class of all, namely that of Žin the language of random matrix theory the chiral Unitary Ensemble This is the universality class of 4-dimensional SUŽ N c gauge theories with NcG3, and quarks in the fundamental representation In this paper we shall present the first results for a lattice gauge theory determination of the microscopic Dirac operator spectrum of SUŽ 3 gauge theory with a staggered fermion Dirac operator The biggest advantage of using a staggered Dirac operator is that the microscopic spectrum at any value of the gauge coupling should coincide with the continuum predictions, since the universality class of this lattice-regularized Dirac operator in this case coincides with the universality class of the continuum Dirac operator Ž for fermions in the same color representation The main disadvantage is related to the fact that all comparisons with analytical results should be done in sectors of fixed topological charge n It is wellknown that although a remnant of chiral symmetry is exactly preserved in the lattice theory with staggered fermions, the interplay between gauge-field topology and fermions is a subtle issue in this formulation In particular, the existence of n fermion zero modes in a gauge field configuration with topological charge n is jeopardized Earlier studies of the microscopic Dirac operator spectrum in SUŽ lattice gauge thewx 9 have suggested that ory with staggered fermions to a large degree the effects of non-trivial gauge field topology are insignificant at feasible values of the gauge field coupling, and hence can be ignored at this stage While we feel that this issue still deserves further clarification, we shall here take the same stand as in Ref wx 9, and simply sum over all gauge field configurations, irrespective of topology Comparison should, in the same spirit, be made only with analytical n s 0 predictions Before considering the numerical algorithm that computes the Dirac operator eigenvalues in a given gauge field configuration, it is instructive to estimate the number of small eigenvalues it is meaningful to include in our study The crucial observation here is that in the microscopic scaling region near l;0, the Dirac operator spectrum will have an average level spacing of order unity Specifically, we shall be concerned with the rescaled, microscopic, spectral wx density / 1 z rsž z ' r ž, Ž 1 V VS where S s² cc : s prž 0, and rž l denotes the usual Ž macroscopic spectral density We choose a convention where the chiral condensate is defined, with a summation over the number of colors N c,by Ž for one flavor of mass m 1 E ² cc : s ZŽ m V E m Here ZŽ m is the finite-volume partition function The macroscopic spectral density is normalized so that rž l dl is the mean number of eigenvalues per unit volume, as in Ref wx 1 The exact, universal,

368 368 ( ) PH Damgaard et alrphysics Letters B analytical prediction from both random matrix theory w x w x 3,5 and the finite-size partition functions 6,7 reads z Nfqn rsž z sprž 0 J Ž z yj Ž z J Ž z, Ž 3 Nfqnq1 Nfqny1 with z'lvssprž 0 Vl This analytical expression has the following qualitative features: It vanishes at z s 0 Žconsistent with the fact that at any finite volume there is no spontaneous chiral symmetry breaking, rises sharply, and then undergoes exponentially damped oscillations around the constant value rž 0 4 The oscillations take place on a scale of order Dz;1, and, in fact, correspond directly to fluctuations over distinct sharp peaks of the first few individual eigenvalues While the analytical prediction is exact Žcan be achieved to any degree of accuracy in the infinite-volume limit with L < 1rm p, it is clear that at any finite volume the prediction will fail eventually In any case, since the distinct feature of the prediction, the oscillations, die out exponentially, we are in fact only interested in averaging over the first very few eigenvalues It would thus be considerable overkill to compute, for each gauge field configuration, more than a few lowest-lying eigenvalues We use the Ritz functional method w1x to compute the lowest 10 eigenvalues and eigenvectors of ydu, with Du the staggered Dirac operator 1 Du s h Ž x U Ž x d yu Ž y d Ý x, y m m xqm, y m x, yqm m sdu qdu Ž 4 e,o o,e Ý n - m x Here h x sy1 n m are the usual staggered Ý n x phase factors Introducing e x sy1 n, we have indicated that Du connects even, ež x s 1, with odd, ež x sy1, sites and vice versa For clarity, we recall some well known properties of the staggered Dirac operator here, see eg Ref w13 x The staggered Dirac operator is antihermitian and therefore ydu is hermitian and positive Ž semi- Table 1 Our estimates of rž 0 for our various gauge field ensembles The third column gives the number of configurations analyzed, and the forth column the bin size in the histograms b L meas Dl rž Ž Ž Ž Ž Ž Ž ˆ 5 definite ež x plays the role that g plays for the continuum Dirac operator, eg Du,e 4s 0, and it is easy to see that the eigenvalues of Du come in pairs "i l If c Ž x is the eigenvector with eigenvalue il then ež x c Ž x is the eigenvector with eigenvalue yil It is well known that ydu only couples even with even and odd with odd sites Furthermore, if c e is a normalized eigenvector of ydu with eigenvalue 1 l, non-zero only on even sites, then cos l Du o,ece is also a normalized eigenvector of ydu with eigenvalue l, non-zero only on odd sites 5 The eigenvectors of Du with eigenvalues "il are then c " Ž x s 1 c Ž x ic Ž x ' e o In any case, it is clear that to obtain the lowest 10 positive Ž imaginary eigenvalues of Du it is sufficient to compute the lowest 10 eigenvalues of ydu e,e, working only on the even sublattice, and then taking the square root This is the approach we have taken here, for which the Ritz functional method is very well suited We have generated configurations that are essentially statistically independent by employing a pure gauge algorithm of typically 3 to 4 microcanonical overrelaxation sweeps, followed by one heatbath update sweep, and repeating this process 10 times between eigenvalue computations To convert our eigenvalue measurements, or actually a histogram of all measured eigenvalues, into a microscopic spectral density, see Eq Ž 1, we need an estimate of S, orof rž 0, in the infinite volume limit On a finite lattice 4 The microscopic spectral density can be viewed as a blow-up of the point rž 0 due to the finite volume We should thus have lim r Ž z s rž 0 z ` s, a matching condition which indeed is satis- fied 5 Due to lattice artefacts the staggered Dirac operator does not have exact zero-modes on equilibrium gauge configurations at non-zero gauge coupling, g

369 ( ) PH Damgaard et alrphysics Letters B L<1rmp is thus trivially satisfied from the very beginning On the lattice, the other boundary, 1rL QCD < L, is effectively replaced by the condition that the volume should be large in physical units This means that the lattice volume in lattice units must increase steeply with the gauge coupling bs6rg if we wish to go towards weak coupling, and continuum scaling Another way of saying that is as follows: to obtain reasonable estimates of rž 0 ie for rž l to be approximately constant Žapart from the predicted oscillations, we need sufficiently large lattices in physical units Empirically we find that LR N Ž b is needed, with N Ž b t,c t,c denoting the approximate temporal extension of a system to be at the deconfinement transition point at given gauge coupling b For example, we found a 6 4 Ž 4 lattice at bs57 Nt,c 57 f4 and an 8 lattice at bs585žn Ž 585 f6 not to be sufficiently large t,c Fig 1 The distribution of the lowest eigenvalue for our different lattice ensembles of Table 1 rž l, obtained by normalizing a histogram of the lowest eigenvalues appropriately, will oscillated about the infinite volume rž 0 We thus obtain rž 0 just by averaging the finite volume rž l over the first few Ž leaving out the very first oscillations Our results, together with information about the number of measurements and the lattice size, is given in Table 1 Our results here are restricted to the quenched approximation, which, in the analytical context can be achieved simply by taking the Nf 0 limit of the formula Ž 3 While the quenched approximation obviously saves an enormous amount of computer time, it also has another unique advantage: The massless limit of the Dirac operator is taken automatically This means that the valence Goldstone boson in principle, were there no lattice artefacts, would be strictly massless One boundary of the scaling regime, Fig The microscopic spectral density for our different lattice ensembles of Table 1

370 370 ( ) PH Damgaard et alrphysics Letters B We begin our comparison between theory and Monte Carlo results with the distribution of the smallest eigenvalue Here the analytical prediction, which again has been proven to be uniõersal in the random matrix theory context, and derivable directly from the finite-volume partition functions too wx 8, reads z PminŽ z sprž 0 expž yz r4 Ž 5 Results for different lattice sizes of volume L 4 are shown in the histograms of Fig 1, which are compared with the analytical prediction Ž 5 For each b-value we have first determined rž 0 as described earlier, which means that these plots are all parameter-free The agreement is seen to be excellent indeed Next, we have extracted the spectral density rž l near l;0 by binning eigenvalues in small intervals Ž see Table 1, and normalizing the resulting density as described previously By rescaling the small eigenvalues according to z s prž 0 Vl, we obtain the microscopic spectral density as shown in Fig Again we find absolutely excellent agreement with the exact analytical prediction Ž 3, which is shown as the full line We have the poorest statistics for b s 51, L s 8 Žonly 1500 independent configurations, and indeed this is the plot for which we get the least impressive agreement However, taken together, over several different b-values and several different volume sizes, the precise prediction is very well reproduced We conclude by noting that the exact analytical predictions for the microscopic Dirac operator spectrum, and the smallest Dirac eigenvalue distribution, now have been successfully confirmed by direct numerical computation in SUŽ 3 lattice gauge theory with staggered fermions The relevant universality class is, in the language of random matrix theory, that of the chiral Unitary Ensemble For gauge group SUŽ N with N G3 and fermions in the fundamental c c representation, it coincides with the universality class of the continuum Dirac operator Acknowledgements PHD and UMH acknowledge the support of NATO Science Collaborative Research Grant No CRG The work of PHD has also been partially supported by EU TMR grant no ERBFM- RXCT97-01, and the work of UMH has been supported in part by DOE contracts DE-FG05-85ER50000 and DE-FG05-96ER40979 AK acknowledges the funding by the Portuguese Fundaç ao para a Ciencia ˆ e a Tecnologia, grants CERNrSr FAEr1111r96 and CERNrPrFAEr1177r97 References wx 1 H Leutwyler, A Smilga, Phys Rev D 46 Ž wx EV Shuryak, JJM Verbaarschot, Nucl Phys A 560 Ž wx 3 JJM Verbaarschot, I Zahed, Phys Rev Lett 70 Ž ; JJM Verbaarschot, Phys Lett B 39 Ž ; Nucl Phys B 46 Ž wx 4 JJM Verbaarschot, Phys Rev Lett 7 Ž wx 5 G Akemann, PH Damgaard, U Magnea, SM Nishigaki, Nucl Phys B 487 Ž ; PH Damgaard, S Nishigaki, Nucl Phys B 518 Ž wx 6 PH Damgaard, Phys Lett B 44 Ž ; G Akemann, PH Damgaard, Nucl Phys B 58 Ž ; Phys Lett B 43 Ž wx 7 J Osborn, D Toublan, JJM Verbaarschot, hep-thr wx 8 SM Nishigaki, PH Damgaard, T Wettig, hep-thr wx 9 ME Berbenni-Bitsch, S Meyer, A Schafer, JJM Verbaarschot, T Wettig, Phys Rev Lett 80 Ž ; ME Berbenni-Bitsch, S Meyer, T Wettig, hep-latr w10x PH Damgaard, UM Heller, A Krasnitz, T Madsen, heplatr980301, to appear in Phys Lett B w11x F Farchioni, I Hip, CB Lang, M Wohlgenannt, heplatr w1x B Bunk, K Jansen, M Luscher, H Simma, DESY Report, September 1994; T Kalkreuter, H Simma, Comp Phys Comm 93 Ž w13x J Smit, JC Vink, Nucl Phys B 86 Ž ; T Kalkreuter, Phys Rev D 48 Ž

371 7 January 1999 Physics Letters B The axial anomaly of Ginsparg-Wilson fermion Ting-Wai Chiu 1 Department of Physics, National Taiwan UniÕersity, Taipei 106, Taiwan, ROC Received October 1998; revised 0 November 1998 Editor: M Cvetič Abstract The axial anomaly of Ginsparg-Wilson fermion operator D is discussed in general for the operator R which enters the chiral symmetry breaking part in the Ginsparg-Wilson relation The axial anomaly and the index of D as well as the exact realization of the Atiyah-Singer index theorem on the lattice are determined solely by the topological characteristics of the chirally symmetric operator D in the chiral limit R 0 q 1999 Published by Elsevier Science BV All rights reserved c PACS: 1115Ha; 1130Rd; 1130Fs The Ginsparg-Wilson Ž GW relation wx 1 has been playing an important role in recent theoretical develw,3 x It opments of the chiral symmetry on the lattice was derived by Ginsparg and Wilson in 1981 as the remnant of chiral symmetry on the lattice after blocking a chirally symmetric theory with a chirality breaking local renormalization group transformation The GW relation for exactly massless Dirac fermion operator D is Dg 5qg5Ds Dg 5RD Ž 1 where the chiral symmetry breaking on the RHS of Eq Ž 1 constitutes an invertible hermitian operator R which is local in the position space and trivial in the Dirac space In topologically trivial background gauge field, y1 D exists and Eq Ž 1 can be written in terms of the fermion propagator D y1 D y1 g5qg5d y1 sg5r This equation displays explicitly that the chirality 1 twchiu@physntuedutw breaking part in the GW fermion propagator is dictated by the local operator R, in contrast to those non-local breakings due to adding a chiral symmetry breaking term explicitly to the action The GW relation, Eq Ž 1 can be rewritten as Dg 5Ž yrd q Ž ydr g5ds0 Ž 3 Then it becomes evident that the lattice action A s c Dc is invariant under the exact chiral transformawx 4 tion on the lattice : c exp iug yrd c 4 5 c c exp iu Ž ydr g5 Ž 5 where u is a global parameter Consequently the axial anomaly can be deduced from the change of fermion integration measure under the exact chiral transformation Ž 4, Ž 5, and its sum over all sites is equal to the index of D which is a well defined integer w5,4 x The Ginsparg-Wilson fermion circumwx 6 vents the Nielson-Ninomiya no-go theorem by r99r$ - see front matter q 1999 Published by Elsevier Science BV All rights reserved PII: S

372 37 ( ) T-W ChiurPhysics Letters B breaking the continuum chiral symmetry g, D4 5 s0 but keeping the exact chiral symmetry on the lattice, Eq Ž 3, hence D can be constructed to be local and free of species doubling The salient feature of the GW relation Eq Ž 1 in topologically nontrivial background gauge field is that its chiral symmetry breaking part does not have any effects on the zero modes This ensures that D has exact zero modes with definite chiralities Consequently the axial anomaly and the index of D are invariant with respect to R, and thus they are determined solely by the chirally symmetric operator D c in the chiral limit R 0, where the GW relation is completely turned off The role of the GW relation is to transform the nonlocal and sometimes singular Ž ie, divergent Dc into a class of local and well defined fermion operators D while keeping the topological characteristics of Dc invariant If we have a Dc which can possess nonzero indices, then any D constructed from this Dc using the general solution of GW relation has the same topological characteristics Conversely, if we have a Ginsparg-Wilson D which has proper indices in topologically nontrivial gauge fields, then there must exist a chirally symmetric Dc which has the same topological character- istics of D, and we can use this Dc to construct a class of GW fermion operators The purpose of this paper is to discuss the R-invariance of the axial anomaly, the index and the Atiyah-Singer index theorem on the lattice, and show that their realizations on the lattice are actually beyond the control of the GW relation but determined solely by the topological characteristics of the chirally symmetric D c in the chiral limit R 0 For the axial anomaly recently constructed by Luscher wx 7, we show that the term proportional to FF is independent of R but depends on the topological characteristics of D c, while the divergence term may depend on R in a very complicated way, but we show that its relationship to the divergence of axial vector current is R-invariant We begin by reviewing the general solution of the Ginsparg-Wilson relation which has been discussed in Ref wx 8 Our purpose here is to set up the notations The general solution of Eq Ž 1 can be formally represented by y1 y1 cž c Ž c c DsD qrd s qd R D 6 where Dc is the chirally symmetric fermion operator of Eq Ž 1 in the chiral limit R 0, Dcg 5qg5Dc s0 Ž 7 It is instructive to show that these two different forms of the general solution in Ž 6 are equal and they satisfy Eq Ž 1 The details are shown in the following D Ž qrd c y1 c y1 y1 Ž c Ž c cž c y1 y1 Ž c cž cž c y1 Ž c Dc s qd R qd R D qrd s qd R D qrd qrd s qd R Using the general solution Ž 6 to substitute D on the RHS of Eq Ž 1, and inserting Eq Ž 7, we obtain Dg 5 RD Ž y1 y1 c c 5 cž c y1 y1 c c 5 c c Ž y1 y1 c c 5 cž c Ž y1 y1 c Ž 5 c c 5Ž c y1 y1 c c 5 c c y1 y1 s qd R Dg RD qrd s Ž qd R DRg D Ž qrd q qd R Dg RD qrd q qd R g D qd g qrd s Ž qd R Ž qd Rg D Ž qrd q Ž qdcr Dg c 5Ž qrd cž qrdc sg5dqdg 5 For any D Ž hence D c satisfying the hermiticity condition g5 Dcg 5 sd c Ž 8 then D is antihermitian and it is evident from wx c 8 that there is one to one correspondence between Dc and a unitary operator V satisfying the hermitic- ity condition Ž g Vg s V 5 5 y1 D s Ž qv Ž yv, c y1 Ž c Ž c Vs D y D q 9 It is interesting to note that V exists for all D c,in particular, when Dc "i`, V q1; but when This observation is made by Herbert Neuberger in a private communication to the author

373 ( ) T-W ChiurPhysics Letters B Dc 0, V y1 The eigenvalues of V fall on a unit circle with center at zero On the other hand, D c becomes singular Ž with simple poles when V has real eigenvalues q1 This is exactly what we want it to happen when the background gauge field is topologically nontrivial However D is still well defined even when D is singular Substituting D into Ž c c 6, we obtain the general solution of D in terms of a unitary operator V satisfying the hermiticity condiwx 8 tion, y1 Ds Ž qv Ž yv qr Ž qv Ž 10 y1 s Ž yv q Ž qv R Ž qv Ž 11 The general solution Eqs Ž 10 and Ž 11 can be casted into many different forms, for example wx 3, 1 1 Ds Ž qw Ž 1 ' R ' R where W is a unitary operator which can be expressed in terms of the V and R However, since W depends on R, using this form could easily obscure some simple facts especially in the chiral limit R 0 Therefore we refrain from using Eq Ž 1 for analytic studies involving the general solution of the GW relation In general, the chirally symmetric operator Dc is constructed such that it agrees with the continuum Dirac operator g Ž E q ia m m m in the classical contin- uum limit It is obvious from Eq Ž 6 that if D has species doubling, D must have species doubling too In order to avoid species doubling for a chirally symmetric operator, Dc must be nonlocal We refer to Ref wx 9 for a few explicit examples of D Next we consider the topological characteristics of a Dirac operator D by comparing its index, indexž D to the topological charge Q of the smooth background gauge field The index of D is defined as NyyN q where N Ž N denotes the number of zero modes q y of D with positive Ž negative chirality If indexž D s0 for any smooth gauge configurations, then D is called topologically triõial If indexž D sq, then D is called topologically proper If D is not topologically trivial but indexž D / Q, then D is called topologically improper Examples of these three different cases are given below and detailed discussions in Ref w10 x From Eq Ž 6, it is obvious that any zeromode of D is a zeromode of Dc and vice versa c c Therefore the index of D is the same as the index of D c, independent of R indexwd, AxsindexwD c, Ax snywd c, AxyNqwD c, Ax Ž 13 where we have indicated explicitly that the number of zero modes and the index are functionals of D c Ž D and the gauge field A However, if Dc is well defined, its index must be zero This can be easily proved in the following Proof: Consider R s r, then indexž D s rý trwg DŽ x, xxw5,4 x x 5 In the limit r 0, D D c, if D is well defined, then Ý trwg DŽ x, xx c x 5 is finite, thus multiplying r gives indexž D s 0 This completes the proof Therefore in order for Dc to possess non-zero indices, D must be singular ( c or equiõalently, V has real eigenõalue pairs "1) in topologically nontriõwx 8 ial background gauge field To summarize above discussions, we list the necessary requirements for D to enter Eq Ž c 6 for constructing a local and well defined D such that D could possibly reproduce the continuum physics: Ž a D is chirally symmetric and agrees with g Ž c m Emq ia m in the classical continuum limit Ž b Dc is free of species doubling Ž c Dc is nonlocal Ž d Dc is well defined in topologically trivial back- ground gauge field Ž e D is singular Ž c or equivalently, V has real eigen- value pairs "1 in topologically non-trivial background gauge field It should be noted that each one of these constraints is required to be satisfied For example, if Ž b is not satisfied, then the index of Dc must be incorrect no matter other constraints are satisfied or not The most difficult task is to find a D satisfying Ž c e At this point, it is instructive to consider examples of topologically proper, trivial and improper Dc re- spectively First we consider the Neuberger-Dirac operator w11x in the chiral limit Ž R 0 qv y1r Dcs M,VsDwŽ DwDw Ž 14 yv where M is an arbitrary mass scale, Dw is the Wilson-Dirac fermion operator with negative mass ym0 and Wilson parameter r w ) 0 1 ) ) D sym q g = q= yr = = Ž 15 w 0 m m m w m m

374 374 ( ) T-W ChiurPhysics Letters B where = m and = m ) are the forward and backward difference operators defined in the following, =cž x s U Ž x c Ž xqmˆ yc Ž x m m = ) c Ž x s c Ž x yu Ž xymˆ cž xymˆ m m Ž Using Eq 6, we obtain the generalized Neuberger- Dirac operator Ds M Ž qv Ž yv q MR Ž qv y1 Ž 16 For m gž 0, r 0 w, both D and Dc are topologically proper and reproduce the exact index theorem on a finite lattice for smooth background gauge fields as first demonstrated in w13,14 x On the other hand, for m0 F0, that is, Dw with a positive mass, then D and D becomes topologically trivial w1 x c We note that in both trivial and proper cases D is free of species doubling and is local for properly chosen R For m0 G r w, both D and Dc are either topologically improper or trivial We emphasize that these ranges of m0 values depend on the gauge configurations, and the ranges given above are exact only in the free field limit We refer to Ref w10x for further discussions on the topological characteristics of the Neuberger-Dirac operator It is interesting to note that the GW relation does not play any role in determining the index of D The c GW relation only ensures that the chiral symmetry is broken in such a delicate way that the exact zero modes of D are also exact zero modes of D with c the same chiralities, and thus the index of D is equal to the index of D The role of R is to convert the c nonlocal and sometimes singular Dc into a local and well defined D, as well as to determine the locality of D If one could not find any Dc satisfying all constraints Ž a Ž e, then the GW relation does not have much use at all On the other hand, if one knows one D satisfying all constraints Ž a Ž c e, then the resulting D must have exact chiral symmetry on the lattice as well as all attractive features pointed out in w15,16 x The fermion integration measure in general is not invariant under the exact chiral transformation Ž 4, Ž 5, and this gives the axial anomaly on the lattice wx 4 qž x, A;D str g5ž RDŽ x, x Ž 17 where tr denotes the trace over Dirac indices We note that qž x, A;D is a function of x and the gauge field Am in the neighborhood around x, and a functional of D The sum of qž x, A; D over all lattice sites is shown w5,4,11x to equal to the index of D qž x, A;D s tr g Ž RDŽ x, x Ý Ý 5 x x sn wd, AxyN wd, Ax y q s indexwd, Ax Ž 18 where N Ž N q y denotes the number of zero modes of D with positive Ž negative chirality In this paper, we restrict our discussions to UŽ 1 gauge theory with single fermion flavor Since the index of D is indeweq Ž 13 x, the total axial anomaly must pendent of R be invariant under any smooth deformations of R d qž x, A;D s0 Ž 19 R Ý x Recently Luscher wx 7 has proved that for Rs1r, if the axial anomaly qž x, A; D satisfies Ý d qž x, A;D s0 Ž 0 x for any local deformations of the gauge field, then qž x, A;D sg cwdx e F Ž x F Ž xqmqn ˆ ˆ mnls mn ls qe G x, A;D 1 m m where g is a constant, cd w x is a functional of D which only depends on the topological characteristics of D, and Em GmŽ x, A;D Ý m m m s G Ž x, A;D yg Ž xym, ˆ A;D The explicit form of the current G Ž x, A;D m is sup- posed to be very complicated We note that cd w x is not shown explicitly in wx 7 Since the total axial anomaly is invariant under smooth deformations of R, Eq Ž 19, as well as for any local deformations of the gauge field, Eq Ž 0, then the theorem proved by Luscher wx 7 can be applied to the case of the general R 3, and the axial anomaly can be written in the same form as Eq Ž 1 on an infinite lattice But on a 3 This is confirmed by Martin Luscher in a private communica- tion to the author

375 ( ) T-W ChiurPhysics Letters B finite lattice, qž x, A; D must be modified by the boundary effects However, if D is local, the boundary effects are negligible, then Eq Ž 1 can be used for the axial anomaly on a finite lattice Since the locality of D is controlled by R, we can infer that all boundary effects must have dependence on R and hence they do not enter the FF term was we will show later, the FF term does not depend on R x Furthermore they must be in the form of the divergence term, otherwise Eq Ž 1 cannot be consistent with Eq Ž 8, the divergence of axial vector current on a finite lattice Therefore we conclude that Eq Ž 1 is applicable for a finite lattice as well as for an infinite lattice, and the boundary effects only enter the divergence term Now we proceed our discussions for a finite lattice embedded in a 4-dimensional torus After summing qž x, A; D over all sites, the divergence term is removed, then Eq Ž 18 gives Ý mnls mn ls x g cwdx e F Ž x F Ž xqmqn ˆ ˆ sn wd, AxyN wd, AxsindexwD, Ax Ž 3 y q Since indexwd, Ax is independent of R weq Ž 13 x, then Eq Ž 3 implies that cd w x must be also independent of R, cwdxscwd x Ž 4 c So we have proved that all R dependence in qž x, A; D only resides in the current G Ž x, A; D m Then Eq Ž 3 can be rewritten in the following Ý g cwd x e F Ž x F Ž xqmqn ˆ ˆ c mnls mn ls x snywd c, AxyNqwD c, AxsindexwD c, Ax Ž 5 Since indexwd, Ax c s0 if Dc is topologically trivial, then cwdx c also vanishes for trivial D c Now we consider two topologically proper Dc which are iden- tical except their electric charges of the fermion field Then for a given topologically non-trivial gauge configuration, Eq Ž 5 implies that the ratio of cd w x X c scorresponding to these two Dc must equal to the ratio of two integers corresponding to their inw x dices It follows from this that cd c must be an integer Therefore we can fix cd w x c s 1 for electric charge equal to one and then determine the value of the constant g by evaluating Ý e F Ž x F Žx x mnls mn ls q ˆ m q ˆ n on the LHS for the simplest nontrivial gauge configuration with constant field tensor, namely, F sprž LL and F sprž L L while other F X s are zero After restoring the unit of electric charge, we obtain g to be integer multiple of e rž3p for UŽ 1 gauge theory with single fermion flavor For Neuberger-Dirac fermion operator w11 x, this integer is determined to be unity by a recent perturbation calculation of qž x, A; D w17 x For the general Ginsparg-Wilson fermion with R s r, and D Ž Ž c satisfying constraints a e 4, our recent pertur- bation calculation w18x of qž x, A; D show that g is independent of r and equal to e rž3p Then Eq Ž 5 becomes e cwd x e F Ž x F Ž xqmqn c Ý mnls mn ls ˆ ˆ 3p x snywd c, AxyNqwD c, Ax Ž 6 where cwd c x 1 if Dc is topologically proper s ~ 0 if D c is topologically trivial integer/0,1 if D is topologically improper c Ž 7 For any chirally symmetric Dc satisfying constraints () a () e, Eq ( 6) is the exact realization of the Atiyah-Singer index theorem on the lattice and it holds for any Ginsparg-Wilson fermion operator DsD c qrd c y1 Although the explicit form of the current G Ž x, A;D in Eq Ž 1 m is supposed to be very com- plicated, however, it is instructive to compare Eq Ž 1 to the divergence of axial vector current on the lattice w1,5 x, E ² J 5 x, A;D : m m N q s s Ý q q s s qž x, A;D q f Ž x f Ž x N y t t Ý y y t y f Ž x f Ž x Ž 8 where fq s and fy t are normalized eigenfunctions of 4 In Ref w18 x, it is only required that the chiral fermion operator D Ž p c is free of species doubling and in the free field limit behaves like ig p as p 0 m m

376 376 ( ) T-W ChiurPhysics Letters B D Ž D c with eigenvalues lsslts0 and chiralities q1 and y1 respectively Eliminating qž x, A; D from both equations, we obtain Em GmŽ x, A;D ² : m m 1 5 s E J x, A;D e y cwdx emnls Fmn Ž x Fls Ž xqmqn ˆ ˆ 3p N q s s Ý q q s N y t t Ý y y t y f Ž x f Ž x q f Ž x f Ž x Ž 9 It is interesting to note that all dependences on R are in the currents Gm and Jm 5 The zero modes and the FF term do not depend on R as we have discussed above Then we obtain ² : 1 5 dr Em Gm x, A;D y Jm x, A;D s0 30 This equation provides a R-invariant constraint between the current G Ž x, A; D m and the axial vector current J 5 Ž x, A; D m The axial vector current J 5 Ž x, A; D is defined wx m 1 in terms of the kernel K 5 Ž x, y, z, A;D through the equation m Ý 5 5 Jm x, A;D s cxqy Km x, y, z, A;D cxqz 31 y, z where Km 5 is expressed in terms of the kernel K the vector current, 5 1 Km s Kmg5yg5Km 3 m for Ginsparg and Wilson wx 1 proved that the kernel for the vector current K Ž x, y, z, A;D is equal to DŽ m x qy, xqz, A times the sign of Ž yyz m and times the fraction of the shortest length paths from xqz to xqy which pass through the link from xymˆ to x Finally the axial anomaly in Eq Ž 1 becomes e qž x, A;D s cwdc x emnls Fmn Ž x Fls 3p = xqmqn ˆ ˆ qem Gm Ž x, A;D Ž 33 where cd w x is defined in Eq Ž 7 c and the diver- gence term E G Ž x, A;D m m is related to the divergence of axial vector current by Eqs Ž 9 and Ž 30 To summarize, we have clarified the role of the Ginsparg-Wilson relation in the realization of the axial anomaly and the exact Atiyah-Singer index theorem on the lattice The crucial point is the existence of a topologically proper chirally symmetric Dirac operator Dc in the chiral limit where the Ginsparg-Wilson relation is turned off If Dc satis- fies the constraints Ž a Ž e discussed in this paper, then any Ginsparg-Wilson fermion operator D constructed by Eq Ž 6 could reproduce the correct continuum physics, in particular, the correct axial anomaly and the exact index theorem From this point of view, the underlying reason why Neuberger-Dirac fermion really works is because the existence of the topologically proper D Ž or V w c Eq Ž 14x which stems from the Overlap formalism w19, x The Ginsparg-Wilson relation only provides a scheme to break the continuum chiral symmetry in such a delicate way that the exact chiral symmetry on the lattice is preserved and the axial anomaly and the index are invariant while smooth deformations of R away from zero bring the non-local and sometimes singular Dc into a class of local and well defined Ginsparg-Wilson fermion operators D The most crucial and highly non-trivial task is to find a good seed D which at least satisfies all constraints Ž c a Ž e So far there is only one good Dc found explic- itly However, there is no indications that Dc is unique and thus the search for computationally less expensive Dc is highly desirable Since the Overlap formalism did provide vital insights in discovering a good D w11 x c, it would be interesting to see whether it could guide us to find a better one Mathematically, it is interesting to understand what actually determines the topological characteristics of an operator and how to construct a topologically proper operator in a systematic way For the axial anomaly Ž 1 recently constructed by Luscher wx 7, we show that the FF term is R-invariant and the proportional constant is e rž3p times cwdx c which only de- pends on the topological characteristics of D c; and the divergence term is related to the divergence of axial vector current by an R-invariant constraint, Eqs Ž 9 and Ž 30 For any chirally symmetric D c satisfying constraints Ž a Ž e, the Atiyah-Singer index theorem can be realized exactly on the lattice, Eq Ž 6, and it holds for any Ginsparg-Wilson fermion operator DsD Ž qrd y1 c c

377 ( ) T-W ChiurPhysics Letters B Acknowledgements I would like to thank Martin Luscher and Herbert Neuberger for enlightening correspondences I also wish to thank Sergei V Zenkin for many discussions in the last few months This work was supported by the National Science Council, ROC under the grant number NSC88-11-M References wx 1 P Ginsparg, K Wilson, Phys Rev D 5 Ž wx For a recent review and references, see H Neuberger, Chirality on the lattice, hep-latr wx 3 F Niedermayer, plenary talk at Lattice 98, Boulder, July 13 18, 1998 wx 4 M Luscher, Phys Lett B 48 Ž wx 5 P Hasenfratz, V Laliena, F Niedermayer, Phys Lett B 47 Ž wx 6 HB Nielsen, N Ninomiya, Nucl Phys B 185 Ž ; B 193 Ž wx 7 M Luscher, Topology and the axial anomaly in abelian lattice gauge theories, hep-latr wx 8 TW Chiu, SV Zenkin, On solutions of the Ginsparg-Wilson relation, hep-latr wx 9 TW Chiu, CW Wang, SV Zenkin, Phys Lett B 438 Ž w10x TW Chiu, Topological phases of Neuberger-Dirac operator, hep-latr w11x H Neuberger, Phys Lett B 417 Ž w1x H Neuberger, Phys Lett B 47 Ž w13x TW Chiu, Phys Rev D 58 Ž w14x TW Chiu, Exactly massless fermions on the lattice, heplatr w15x P Hasenfratz, Nucl Phys B 55 Ž w16x S Chandrasekharan, Lattice QCD with Ginsparg-Wilson fermions, hep-latr w17x Y Kikukawa, A Yamada, Weak coupling expansion of massless QCD with a Ginsparg-Wilson fermion and axial UŽ 1 anomaly, hep-latr w18x TW Chiu, TH Hsieh, Perturbation calculation of axial anomaly of Ginsparg-Wilson fermion, NTUTH w19x R Narayanan, H Neuberger, Nucl Phys B 443 Ž

378 7 January 1999 Physics Letters B A momentum subtraction scheme for two-nucleon effective field theory Thomas Mehen 1, Iain W Stewart California Institute of Technology, Pasadena, CA 9115, USA Received 9 October 1998 Editor: H Georgi Abstract We introduce a momentum subtraction scheme which obeys the power counting of Kaplan, Savage, and Wise KSW, developed for systems with large scattering lengths, a Unlike the power divergence subtraction scheme, coupling constants in this scheme obey the KSW scaling for all m )1ra We comment on the low-energy theorems derived by Cohen and R Hansen We conclude that there is no obstruction to using perturbative pions for momenta p) m q 1999 Elsevier Science p BV All rights reserved Effective field theory is a useful method for describing two-nucleon systems wx 1 Recently, Kaplan, Savage, and Wise Ž KSW wx devised a power counting that accounts for the effect of large scattering lengths With this power counting the dimension six four-nucleon operators are treated non-perturbatively, while pion exchange as well as higher dimension operators are treated perturbatively A key feature of the KSW approach is the use of the power divergence subtraction Ž PDS renormalization scheme PDS gives the coefficients of contact interactions power law dependence on the renormalization scale, which makes the KSW power counting manifest For m R300 MeV, the power counting in PDS is no longer manifest wx R, and KSW conclude that application of the theory is restricted to momenta less than 300 MeV In Ref wx 3, it is emphasized that the power counting for large scattering lengths should be scheme independent Here we introduce a momentum subtraction scheme that is compatible with the KSW power counting A similar scheme is applied to the theory without pions in Ref wx 4 This scheme will be referred to as the OS scheme, since in a relativistic field theory it would be called an off-shell momentum subtraction scheme One attractive feature of the OS scheme is that the KSW power counting is manifest for all m R )1ra, where a is the scattering length Since the breakdown of the power counting is an artifact of the PDS scheme, the range of the effective field theory with perturbative pions may not be limited to 300 MeV Another attractive feature of the OS scheme is that the amplitudes are manifestly mr independent at each order in the expansion In PDS, any fixed order calculation has residual m dependence which is cancelled by R 1 mehen@theorycaltechedu iain@theorycaltechedu r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

379 ( ) T Mehen, IW StewartrPhysics Letters B higher order terms The rather strong mr dependence exhibited by next-to-leading order calculations of the phase shifts led authors w5,6x to conclude that PDS results have to be fine tuned to fit the data In fact, it is possible to fix this problem with PDS by modifying the treatment of C Ž m 0 R We begin our discussion by recalling the Lagrangian with pions and nucleons wx f f v ig A D m p m q q i i i 0 ž / L s Tr E S E S q Tr m Sqm S y N s je j yj E j NqN id q N 8 4 M Ž s C Ž s T Ž s T Ž s T Ž s T Ž s l yc0 Ž N Pi N Ž N Pi N q Ž N Pi N ž N P i = N/ qhc 8 Ž s j T Ž s T Ž s i i yd vtrž m N P N N P N q Ž 1 Here g A s15 is the nucleon axial-vector coupling, Ssj, fs130 MeV is the pion decay constant, j 1 m s Ž j m jqj m j, where m sdiagž m,m is the quark mass matrix, and m swž mqm q q q u d p u d The matrices Pi Ž s project onto states of definite spin and isospin, and the superscript s denotes the partial wave amplitude mediated by the operator This paper will be concerned only with S-wave scattering, so ss 1 S, 3 0 S 1 The ellipsis in Eq Ž 1 denotes terms that are higher order in the expansion Ž bare The four nucleon couplings C 0, C, and D in Eq 1 are bare parameters In general, a bare coupling, C, can be separated into a renormalized coupling, CŽ m, and counterterms: ` bare finite uv finite R Ý n ns0 R C sc yd C, C scž m y d CŽ m The counterterms have been divided into two classes The first, which have the superscript uv, contain all ultraviolet divergences and are mr independent Defining a renormalization scheme amounts to making a choice n n for the finite counterterms, d C m R The superscript n indicates that d C is included at tree level for a graph with n loops For example, at two loops, we have two loop diagrams with renormalized couplings at the vertices, one loop diagrams with a single d 1 C counterterm, and a tree level diagram with d C Let Q denote a typical momentum characterizing the process under consideration For nucleon-nucleon scattering we take p;q and m p;q, where p is the center-of-mass momentum The theory is an expansion in QrL where L is the range of the effective field theory Taking m ; Q, vertices with C Ž m R 0 R scale as 1rQ, while vertices with 0 C p or D m scale as Q A typical loop gives one power of Q, soc Ž m p 0 R vertices are included to all orders This sums all corrections that scale as Ž Qa n wx Note that since the pion has been included explicitly in the Lagrangian we expect that the scale of short distance physics, L, should not be set by m p, but by higher mass resonances which have not been included in the theory PDS is one scheme in which the KSW power counting is manifest In PDS, we first let ds4ye and uv define the counterterms d C to subtract 1re poles As in the MS scheme, the dimensional regularization n parameter m is set to m Next one takes ds3 and defines the finite counterterms, d CŽ m R R to subtract the 1rŽ dy3 poles in the amplitude Graphs which renormalize a given coupling are those whose vertices have the n n right number of derivatives and powers of m When calculating d C m and d C Ž m p 0 R R, we can take m s 0 since counterterms proportional to m renormalize coefficients like D Ž m p p R After making these subtractions, the amplitude is continued back to four dimensions, so dy3 1 1 In the S channel, exact expressions for the PDS beta functions can be obtained wx For C Ž m we have / 0 0 R 4p 1 1 C0Ž mr s y, Ž 3 ž M KymR LNN where L s 8p f rž Mg NN A s 300 MeV K is a constant fixed by the boundary condition, and choosing C Ž 0 s4p arm gives Ks1rŽ aq1rl We see that the scaling for C Ž m changes for m ;300 MeV 0 NN 0 R R R

380 380 ( ) T Mehen, IW StewartrPhysics Letters B wx A simple shift, C0 mr C0 mr yg Ar f, results in a coupling that scales as 1rmR for all m R)1ra Physically, this shift corresponds to summing the short distance Ž m s 0 p contributions from potential pion exchange to all orders For the 3 S1 channel, this summation is not possible because of ultraviolet divergences of the form p re and, in fact, there are unknown corrections to the 3 S1 beta functions at each order in Q This is demonstrated in Ref wx 7 by explicit computation of the counterterms When the beta function is not exactly known, the large m behavior is ambiguous For example, the PDS beta function for C Ž m is R 0 R E C0Ž mr MmR g A b0 smr s C0 Ž mr q C Ž m qo Ž Q Ž 4 0 R Em 4p f R Two solutions which satisfy this equation to order Q 0 are 4p 1y amrymgy 1q4mgy4mgŽ 1yamR C0Ž mr sy, M m Ž 1yam ym ( R R g g A 1 C0Ž mr sy, Ž 5 1y 1qr al exp ym f NN where m sm rl and we have chosen C Ž g R NN 0 0 s4p arm The first solution is obtained by computing the n 0 counterterms d C0 mr to order Q and summing them This solution falls as 1rmR for all m R) 1ra, and is numerically close to the g 0 solution The second solution is obtained by truncating and solving Eq Ž A 4 This solution approaches a constant as mr ` The two solutions both solve the beta function to order Q 0 but have very different large mr behavior In the OS scheme, there is no ambiguity since at a given order in Q the running of all the coupling constants that enter is known exactly In the OS momentum subtraction scheme, the renormalized couplings are defined by relating them to the amplitude evaluated at the unphysical momentum p s im R We start by dividing up the full amplitude as AsÝ` ms 0 A Ž my1 Here A Ž my1 contains the Feynman diagrams that scale as Q my 1 As in PDS, we can take uv mp 0 when computing the counterterms for C0 mr and C m R d C m is first defined to subtract all four dimensional 1re poles The definition for the renormalized coupling is then g Ž my1 m ia psim, m s0 syic mž mr Ž im R Ž 6 R p This condition is to be imposed order by order in the loop expansion so that the graphs at n loops determine n d C Ž m For instance, the couplings C Ž m and C Ž m 0 R 0 R R are defined by the renormalization condition in Fig 1 Summing the counterterms, we find C0 finite C finite yg A rž f m R C0Ž mr s, C Ž m s Ž 7 finite R 1ym C Mr4p finite R 0 1ym C Mr4p 4 Although it may seem that the piece of C mr that goes as 1rmR will spoil the power counting for low momentum, in fact, the 1rm R part dominates entirely for mrr1ra, since C0 finite ;a, C finite ;a In the OS scheme, D mr will be calculated as follows First mpre poles are subtracted The renormalized coupling is then defined by ia D psim syid mr m p, 8 R where AD contains terms in the amplitude that are proportional to m p, as well as analytic in m p Thus, at 0 Q graphs with a single D Ž m or potential pion and any number of C Ž m vertices contribute to AD R 0 R Note that we only keep terms that are analytic in m p because it seems unnatural to put long-distance nonanalytic contributions that come from pion exchange into the definition of the short distance coupling wx Ž Ž D m 6 For example, one potential pion exchange gives a m rp ln 1q4 p rm term Putting this into R p p R 0

381 ( ) T Mehen, IW StewartrPhysics Letters B Žy1 Fig 1 Renormalization conditions for C m and C m in the OS scheme ia is the four point function with C Ž m 0 R R 0 R and n 1 3 Ž0 d C0 mr vertices, evaluated between incoming and outgoing S0 or S1 states The amplitude A contains graphs with one C or one potential pion dressed with C bubbles 0 D Ž m R would give it both a branch cut at mrsmpr as well as explicit dependence on the scale m p, therefore this term is not included in AD Solving for the counterterms and summing we find ž / finite Ž R A R A R finite ž / ž / ž Ž m 8p 8p f m 8p 8p f m / 0 R C0 0 D m D M Mg m M Mg m s q ln y1 s ln, C ž / 64p f D finite where m sm0 exp 1y Ž 9 finite M g C A 0 0 With m p;q;m R, D mr m p ;Q The scale m must be determined by fitting to data The PDS solution for D Ž m R does not have the y1 in square brackets We now compare the amplitudes in the PDS and OS schemes At order Q 0, the amplitudes in the 1 S and 3 0 S1 channels have the same functional form The amplitudes in PDS are calculated to this order in Ref wx We find AsA Žy1 qa Ž0,a qa Ž0,b qo Q 1, 4p 1 Žy1 A sy, Ž 10 M 4prMC Ž m qm qip 0 R R ½ Ž0,a A p R y1 s ln y qmr tan Ž y1 f 4p mp MC0Ž mr p mp ž / ž / ž / ž / ž / 5 MC Ž m 4 p m C Ž m A g m M 1 m 4p 1 p A ž / 4p 1 4p D mr mp R 0 R p 0 R q qm yp ln 1q y, Ž 11 / PDS Ž0,b A CŽ mr p ga 1 1 gamp M sy y q, Ž y1 ž A C f f 4p 0Ž mr C0Ž mr Ž 1 OS Ž0,b A CŽ mr p ga p 1 MmR sy y 1q q A ž / C f ž m / C Ž m 4p 0Ž mr R 0 R Ž 13

382 38 ( ) T Mehen, IW StewartrPhysics Letters B Note that the last term in Eq Ž 1 has a factor of 1r instead of a1asinref wx since we have made a different finite subtraction The terms in braces are long distance pion contributions, and are the same in the PDS and OS schemes By substituting Eqs Ž 7 and Ž 9 into the OS amplitude one can verify that it is m R independent In contrast, in PDS the amplitude is only mr independent to order Q 0 To obtain a good fit to the scattering data at low momenta, two constraints must be approximately satisfied: Ž 0 1 A lim pcotd Ž p sy, s0, Ž 14 a Ž y1 p 0 A yi psg where g s 4prMC Ž m 0 R q m R The second constraint ensures that the next-to-leading order amplitude does not have a spurious double pole In the PDS scheme, A Ž0 is not mr independent, so the extracted parameters can not simultaneously satisfy the renormalization group equations and give a good fit This is the origin of the large dependence on m observed in Ref wx 5 In PDS the second constraint gives R ž / ž / ž / ž / MC m / 0 R yd Ž m g M 1 1 m 4p 1 4p 0, q q ln y qm q qmr C 4p mp MC0Ž mr MC0Ž mr R A R R m f mp m 0 R p 1 4p y, Ž 15 ž m p while in OS we find ž / ž / ž / yd Ž m g M 1 m 4p 0, q ln y qm R, Ž 16 C f 4p m mp MC0Ž mr R A R 0 mr p where terms of order g rm have been dropped With the convention for D Ž m in Ref wx p R the first term in square brackets in Eq Ž 15 is a 1, so they find that D Ž m is close to zero The last term in Eq Ž 15 p induces the large m dependence On the other hand Eq Ž 16 R is mr independent 1 3 Fits are performed to the Nijmegen phase shift wx 8 data between 7 and 100 MeV for both the S0 and S1 Ž0 Ž1 Ž channels The phase shifts have an expansion of the form dsd qd qo Q rl, where wx i pm pm A Ž0 Ž0 Ž y1 Ž1 d sy ln 1qi A, d s Ž 17 p 4p 1qiŽ pmrp A Žy1 The fits were weighted towards low momentum since the theoretical error is smallest there The results are shown in Fig We chose to fit using the amplitudes in the OS scheme, but an equally good fit is obtained in PDS The parameters C Ž m, C Ž m, and D Ž m 0 R R R were extracted for different values of mr in both schemes, and the conditions in Eqs Ž 15 and Ž 16 were found to be well satisfied For instance, taking mr s mp s 137 MeV in the OS scheme we find 1 S0 C0Ž mp sy354 fm, CŽ mp s300 fm 4, DŽ mp s0377 fm 4, 3 S1 C0Ž mp sy581 fm, CŽ mp s1016 fm 4, DŽ mp sy418 fm 4 Ž 18 The fit gives scattering lengths až 1 S sy33 fm, and až 3 S 0 1 s 541 fm Plugging the parameters extracted y from the fit into the right hand side of Eq 16 gives ;10 for both channels In the OS scheme, the renormalization group equations are obeyed to a few percent accuracy since the amplitude is explicitly m R independent It is important to realize that the fits do not unambiguously determine the values of C Ž m and D Ž m 0 R R The coefficient of the four nucleon operator with no derivatives is C0 bare qm p D bare Once we switch to

383 ( ) T Mehen, IW StewartrPhysics Letters B wx Fig Fit to the phase shift data weighted toward low momentum The solid line is the Nijmegen fit to the data 8, the long dashed line is the order 1rQ result, and the short dashed line is the order Q 0 result renormalized coefficients, it is not clear how to divide the couplings into a nonperturbative and perturbative piece In Ref wx, C Ž m is summed to all orders, while other authors w5,6x 0 treat both coefficients non-perturba- tively In fact, in order to do a chiral expansion, m D Ž m p R should be treated perturbatively Since 0 mp D m R ; Q, this is consistent with the power counting and the renormalization group equation On the other hand, there is some freedom in dividing C Ž m 0 R into nonperturbative and perturbative pieces: np p np p 0 C0 mr sc0 mr qc0 m R, where C0 m R ;1rQ and C0 m R ;Q This is simply a reorganization of the perturbative series For instance, consider the following expansion of the amplitude in the theory without pions: 4p 1 4p 1 As s M 1 M 1 y1raq r0 p q y ip y1raydqdq r0 p q y ip 1 y4p 1 r0 p qd s q q, Ž 19 M 1raqDqip Ž 1raqDqip where DQ1ra The series with Ds0 and with D/0 will both reproduce effective range theory, but differ in the location of the pole that appears at each order in the perturbative expansion In the 3 S1 channel, the pole of 3 the physical amplitude is at yips( MEd s457 MeV, where Ed is the binding energy of the deuteron For comparison, 1ras363 MeV in this channel For Ds0, the pole that appears at each order in the perturbative expansion will be off by 30% For some calculations, such as processes involving the deuteron wx 9, a better behaved perturbation series is obtained by choosing 1raqDs( ME d If we want to reproduce the expansion in Eq Ž 19 in the theory without pions then part of C Ž m must be treated perturbatively 0 R 3 3 In fact, in the S channel the fit value of C m from Eq 18 gives g s473 MeV 1 0 p

384 384 ( ) T Mehen, IW StewartrPhysics Letters B In the theory with pions and a non-vanishing C p Ž m, the amplitude can be obtained from Eqs Ž 10 Ž 13 0 R by np substituting C m C Ž m and 0 R 0 R C0 p Ž mr Žy1 Žy1 Žy1 A A y w A x Ž 0 np C0 Ž mr p w np In the OS scheme, the renormalization group equation makes C x 0 mr r C0 mr equal to a constant Therefore, C p Ž m can be simply absorbed into the definition of D Ž m Because of this, the value of D Ž m 0 R R R extracted from fits to NN scattering data may differ from the value of the renormalized coupling in the Lagrangian In the PDS scheme, the renormalization group equation for C p Ž m is p 0 R R np p R 0 R 0 R EmR 4p f 0 R E C Ž m Mm g m s C Ž m C Ž m q, Ž 1 with solution C0 p Ž mr g 1 q sconstant Ž np f np C Ž m C Ž m 0 R 0 R Therefore, breaking C Ž m 0 R into perturbative and nonperturbative pieces results in a manifestly mr independent 4 amplitude in PDS The constraint in Eq Ž 15 is now mr independent Integrating out the pion gives low-energy theorems for the coefficients Õi in the effective range expansion w10 x, 1 r pcotd Ž p sy q p qõ p qõ3 p qõ4 p q Ž 3 a np The Õ can be predicted in terms of one parameter, C Ž m, which is fixed in Ref w10x i 0 R by the condition w np 4pr MC Ž m x 0 R qmrs1ra Corrections to these predictions are expected to be 30 50% due to higher order QrL terms The Õ extracted from the phase shift data w10,11x i disagree with the low-energy theorems by factors Ž Ž0 Ž1 of order 5 In Fig 3, we see that the agreement of pcot d qd Ž solid lines 5 with the Nijmegen partial wave analysis is comparable to that of the effective range expansion with the Õ from the fits in Refs w10,11x i Ž dashed lines Note that our fit is more accurate at low momentum than the global fit in Ref wx However, keeping only the first five terms from the low-energy theorems Ž dotted lines gives larger disagreement at 70 MeV This is not surprising since the pion introduces a cut at psimpr, so the radius of convergence of the series expansion of pcotž d in Eq Ž 3 is,70 MeV At ps70 MeV, one expects large corrections from the next term in the series However, the fit values of Õi give good agreement with the data even at 70 MeV It is possible that uncertainty from higher order terms in the Taylor series has been absorbed into Õ, Õ 3, and Õ4 in the process of performing the fits For this reason, the uncertainty in the values of Õi that were found from fitting to the data may be considerable To get an idea of the error in Õ, we will specialize to the 3 S1 channel The Nijmegen phase shift analysis w x Ž3S 1 1 lists two data points for p-70 MeV: ps167 MeV where d s147747"00108, and ps4845 Ž3S 1 MeV, where d s118178"0018 Using as540"0001 fm and r s1753"000 fm w11x 0 in the effective range expansion and fitting to the lowest momentum data point, we find Õ sy050"05 fm 3, where the error in a, r 0, and pcotd have been added in quadrature This differs by one sigma from both the value predicted by the low-energy theorem, Õ thm sy095 fm 3, and the value from the fit, Õ fit s004 fm 3 Since 4 We would like to thank Mark Wise for pointing this out to us 5 w Žy1 x Ž0 w Ž Žy1 x Ž 3 Ž Ž0 Ž1 Note that when expanded in Q, pcotd sipq4p r MA y4p A r M A q O Q, which differs from pcot d qd by 3 terms of order Q The latter expression is used since the parameters in Eq Ž 18 were fit using Eq Ž 17

385 ( ) T Mehen, IW StewartrPhysics Letters B Fig 3 The effective field theory and Nijmegen Partial Wave analysis wx 8 values of pcotd are compared The solid lines use Ž Ž0 Ž1 pcot d qd, the dashed lines use Eq Ž 3 with the Õ from Ref w10 x i, and the dotted lines use the values of Õi from the low-energy theorems the range of the pure nucleon theory is 70 MeV, there will also be a,01 fm 3 error in this extraction from Õ 3 and higher coefficients This error was estimated by comparing the theoretical expression for pcot d with the first three terms in its series expansion If we instead use the higher momentum point we find Õ s003"004 fm 3 with,05 fm 3 theoretical uncertainty The uncertainty in these values of Õ is too large to make a definitive test of the low-energy theorems In this paper, we have addressed the issue of mr sensitivity in perturbative treatments of the pion Amplitudes are m independent in the OS scheme, and in the PDS scheme, if part of C Ž m R 0 R is treated perturbatively Fits to NN scattering data were done which agree well at low momentum Errors at high momentum are consistent with uncertainty from higher order terms if the range of the effective field theory is R300 MeV We conclude that there is no obstruction to using perturbative pions for momenta p)m p Ina future publication wx 7, we will describe the renormalization procedure and OS scheme in greater detail The range of the theory with perturbative pions will also be investigated Acknowledgements We would like to thank Mark Wise for many useful conversations We also would like to thank H Davoudiasl, S Fleming, U van Kolck, Z Ligeti, S Ouellette and K Scaldeferri, for their comments TM would like to acknowledge the hospitality of the Department of Physics at the University of Toronto, where part of this work was completed This work was supported in part by the Department of Energy under grant number DE-FG03-9-ER References wx 1 S Weinberg, Phys Lett B 51 Ž ; Nucl Phys B 363 Ž ; C Ordonez, U van Kolck, Phys Lett B 91 Ž ; C Ordonez, L Ray, U van Kolck, Phys Rev Lett 7 Ž ; Phys Rev C 53 Ž ; U van Kolck, Phys Rev C 49 Ž ; DB Kaplan, MJ Savage, MB Wise, Nucl Phys B 478 Ž ; GP Lepage, nucl-thr970609; JV Steele, RJ Furnstahl, Nucl Phys A 637 Ž

386 386 ( ) T Mehen, IW StewartrPhysics Letters B wx DB Kaplan, MJ Savage, MB Wise, Phys Lett B 44 Ž ; DB Kaplan, MJ Savage, MB Wise, Nucl Phys B 534 Ž wx 3 U van Kolck, nucl-thr wx 4 J Gegelia, nucl-thr wx 5 J Gegelia, nucl-thr wx 6 JV Steele, RJ Furnstahl, nucl-thr98080 wx 7 T Mehen, I Stewart, nucl-thr wx 8 VGJ Stoks, etal, Phys Rev C 49 Ž Ž cf wx 9 DB Kaplan, MJ Savage, MB Wise, nucl-thr w10x TD Cohen, JM Hansen, nucl-thr w11x VGJ Stoks et al, Int Symp on the Deuteron, Dubna, Russia, 1995, nucl-thr w1x VGJ Stoks et al, Phys Rev C 48 Ž

387 7 January 1999 Physics Letters B Sheets of BPS monopoles and instantons with arbitrary simple gauge group Kimyeong Lee 1 Physics Department and CTP, Seoul National UniÕersity, Seoul , South Korea Received October 1998 Editor: M Cvetič Abstract We show that the BPS configurations of uniform field strength can be interpreted as those for sheets of infinite number of BPS magnetic monopoles, and found that the number of normalizable zero modes per each magnetic monopole charge is four We identify monopole sheets as the intersecting planes of D3 branes Similar analysis on self-dual instanton configurations is worked out and the number of zero modes per each instanton number is found to match that of single isolated instanton q 1999 Published by Elsevier Science BV All rights reserved There has been some interest in BPS field configwx 1 This back- uration with uniform magnetic field ground field exerts no force on isolated BPS monopoles as the repulsive magnetic force cancels the attractive Higgs force Thus there is a possibility of BPS configuration of an isolated monopole in such background, which has been explored recently wx Besides this, there has been a long standing interest in similar uniform configuration of four dimensional self-dual equation of Yang-Mills theories wx 3 This configuration is known to be stable and the zero modes of this configuration has been explored to understand the quantum correction to the effective action In this paper we show that the BPS configuration with uniform magnetic field can be interpreted as a two-dimensional sheet made by infinite number of 1 Electronic Mail: kimyeong@phyasnuacedu BPS magnetic monopoles lying on a plane By studying zero modes, we show that the number of normalizable zero modes per unit magnetic flux is four, identical to that for an isolated monopole Along the planar direction of the monopole sheet, the zero modes are given by the wave function of the lowest Landau level, so that the monopole sheet appears a quantum phase space or noncommutative plane By considering the case of the general gauge group which is maximally broken, we show that magnetic monopole sheets exist for each pair of positive and negative root vectors Again for unit magnetic monopole flux, there are four normalizable zero modes Especially for SUŽ N gauge group, the magnetic monopole sheets can be identified as intersecting planes of D3 branes in type IIB string theory For the instanton case, the zero modes are well-known What is new here is that the center position of normalizable zero modes is pointed out to live on four dimensional quantum phase space In theories with general gauge group, we also show that the r99r$ - see front matter q 1999 Published by Elsevier Science BV All rights reserved PII: S

388 388 ( ) K LeerPhysics Letters B number of normalizable zero modes per unit Pontryagin index is identical to four times the dual Coxeter number, 4 c Ž G, which has been obtained from the index theorem for isolated instantons wx 4 and from the counting the number of constituent 3 monopoles of instantons on R = S 1 wx 5 In a Yang-Mills Higgs theory where SUŽ UŽ 1, the field configuration for single monopole is spherically symmetric and has a core region of size 1rmW s 1rÕ with coupling constant e s 1 Inside the core the gauge field for the massive W bosons is nonzero, but vanishes exponentially outside the core When we put N) 0 BPS magnetic monopoles as close as possible, the core configuration becomes complicated For two monopoles, the core is a torus wx 6, for three monopoles, the core is tetrahedral wx 7, and so on For large N, the shape of the core shape becomes presumably more involved Whatever shape it takes, one may ask what is the rough size of the core region The answer can be found readily by considering the asymptotic behavior of the Higgs field Since the Higgs field approaches Ž Õ y Nrr asymptotically, the natural scale of the core size is r ;NrÕsNrm Ž 1 core W Now imagine configurations with an increasing number of magnetic monopoles A simple way is to put magnetic monopoles along a line with an equal distance d, which, say, is much greater than 1rm W We know this configuration cannot be axially symmetric as there exits a relative orientation between any two monopoles Since the minimum core size Ž 1 is less than the size of the configuration, monopole cores seem to be separated However there exits an additional logarithmic divergence in the monopole core size To see this, consider the value of the Higgs field for a fixed point in large N limit It would be roughly of order N 1 1 < f < yõ; Ý y ;y ln N ns1 nd d Thus, the core of magnetic monopoles will eventually merge together when N;e d Õ The large N limit can be taken only if we increase Õ simultaneously Otherwise, we will end up with just symmetric phase This picture is consistent with another view of lined magnetic monopoles Far away from the line, the UŽ 1 magnetic field will fall off like 1rr with r being the distance from the line This means the Higgs field will increase like log r Now imagin putting N monopoles on a two dimensional square lattice of lattice size d With one monopole on each lattice site, they occupy a square of size ' Nd Since the core size Ž 1 grows linearly with N, the cores of these monopoles will start to overlap for N; Ž Õd and individual monopoles becomes indistinguishable As in the linear configuration, we can take the infinite N limit of the planar configuration only if we take Õ s ` simultaneously The magnetic field far from the plane is approximately uniform, which in turn implies that the Higgs field grows linearly There may be several BPS field configurations which can be interpreted as those corresponding the sheet of BPS magnetic monopoles It appears hard to find any of such BPS configurations In this paper, we argue that the well-known homogeneous BPS configuration Bi a sdif a sbda3 di3 Ž 3 is one of such configurations It is obvious in retrospective that it should be so The corresponding Higgs and gauge fields are a a3 f sbd zyz 0, 4 b a a3 Ai s d Ž yy, x,0 Ž 5 in the symmetric gauge Note that the origin of the x, y plane is not special as there is a translation symmetry on that plane However, the plane zsz 0 is very special as the gauge symmetry is restored only on that plane The naive W boson mass increases as one moves away from the plane Also the gauge invariant UŽ 1 magnetic field B aˆ f a sb signž i z y z 0 changes the sign as one crosses the plane Both of them implies that the plane could be interpreted as the sheet of BPS magnetic monopoles The individual characteristics of magnetic monopoles disappear, and so do the nonabelian characteristics of the core region Thus there is no core region at all Still, there emerges some individuality of monopole as we will see As single magnetic monopole carries the magnetic flux 4p and the magnetic fields of

389 ( ) K LeerPhysics Letters B opposite sign come out from the both sides of the sheet with strength b, we can assign the magnetic monopole number density per unit area to be b nmonopoles Ž 6 p We note that in the SUŽ case only one sheet is allowed The interesting question is then whether we can take out single BPS magnetic monopole away from the sheet While we do not know the answer for this question, we can begin by exploring what are the normalizable zero modes of the magnetic monopole sheet The linear fluctuations d A a i,df a satisfy the lin- earized BPS equations a abc b c ijkž j k j k/ e E d A qe A d A a abc c abc b c s Eidf qe Aidf ye f da i, 7 and the background gauge condition a abc b c abc b c Eid Ai qe Aid Ai qe f df s0 8 The background gauge is implied by the Gauss law satisfied by the slowly moving monopole in the A 0 gauge The fluctuations d A 3 i,df 3 along unbroken abelian direction are independent of the background and non-normalizable They correspond to the change in the position and orientation of the magnetic monopole sheet For the study of fluctuations along broken symmetry direction, we introduce 1 1 Wis Ž d Aiqid A i, Ž 9 ' 1 1 W4 s Ž df qidf, Ž 10 ' which describe to massive W bosons in the ordinary case The linearized BPS equations and the background gauge condition become e DW sdw yd W, DWqD Ws0, ijk j k i 4 4 i i i 4 4 Ž 11 where b DisEiyiV i, Vis Ž yy, x,0, D se yiv, V sbž zyz Ž Of course the fluctuations considered here are independent of x 4 and so E4 s0 These equations can be easily solved There are two independent families of solutions: W1syiW ib b b Ž x0yyy0x y xyx 0 q yyy 0 y zyz 0 4 sc e, 1 W syiw 3 4 ib b b Ž x0yyy0x y xyx 0 q yyy 0 y zyz 0 4 sc e, Ž 13 where c1 and c are two independent constants Then these two families and their complex conjugate form four independent families of zero modes Along the z direction, the zero modes are given by the ground state wave function of harmonic oscillator, which is concentrated on the monopole sheet zsz 0 This is due to the fact that the mass of W boson is zero on that sheet and increases linearly away from that Along the xyy direction, the zero modes are given by the wave function of the lowest Landau level Since we have chosen the symmetric gauge for the uniform magnetic field, the wave function can be chosen to be concentrated on the position Ž x, y 0 0 Now let us recall the well known physics of the Landau levels The conserved x and y translations of the wave function do not commute each other, making the xyy plane a quantized phase space, or noncommutative plane The conserved generators of translations are p syie yebyr and p syie x x y y q ebxr, satisfying the commutation relation, wp,p xsieb Ž 14 x y Thus, the parameters x 0, y0 appearing in the wave function overcount the number of independent ground wave function This can be seen easily by choosing the Landau gauge and compactifying the x y y plane The degeneracy is not infinite in spite that x 0, y0 are continuous Rather, there is a minimum area preb per one state Thus, in our notation es1, Bsb, the number density of independent ground states per unit area is given by b nlandaus Ž 15 p

390 390 In terms of the magnetic length lb s1r b, one can understand the quantum cell p lb as the size of the minimum quantum volume of the phase plane Also the total magnetic flux in this quantum cell is the Dirac quantum flux p, which is the minimum magnetic flux one can put consistently in the compactified xyy space Coming back to our magnetic monopole problem, the number density of the zero modes is four times that of the ground state Landau level, 4n Landau Since the number density of monopole in Eq Ž 6 is identical to the n Landau, the number of zero modes per unit magnetic monopole becomes four This is exactly identical to the number of zero modes for single BPS monopole in SUŽ gauge group For one monopole, three of them account its position and one does for its internal phase angle In our case monopoles are completely dislocalized on the sheet, but we can imagine figuratively single quantum cell as one monopole As our analysis is at linear level, we do not know exactly how these zero modes will develop nonlinearly It would be interesting to find the full nonlinear distortion of the monopole sheet and the geometry of the infinite dimensional moduli space of zero modes Due to the fact that the x 0, y0 space form noncommutative plane, the moduli space itself may be noncommutative Let us now generalize the above consideration to Yang-Mills Higgs theory of arbitrary simple gauge group G of rank rg and dimension dg We split the generators of the Lie algebra to the Cartan subalgebra and the raising and lowering operators We normalize the generators so that in the adjoint representation tr Ž HH a b scž G d ab, Ž 16 tr E E sc Ž G d, Ž 17 a b ab where a, b s 1,,rŽ G The normalization factor c Ž G is the quadratic Casimir for the adjoint representation and becomes the dual Coxeter number when the longest root vectors is normalized to have length one ŽSee for example Ref wx 4 Among the commutation relations, ones we need are wh, E xsa E, Ž 18 a a ( ) K LeerPhysics Letters B ' where H and a are rg dimensional vectors The eigenvalues of the matrix HH a b consist of aab for all roots and so Eq Ž 16 implies an identity Ý a a b sc Ž G d ab Ž 19 a The BPS equation in this general group becomes B sd f, Ž 0 i where i 1 Bis eijk EjAkyE k A jyi A j, A k, 1 D fse fyiw A,f x i i i The uniform solution of the BPS equation can be chosen to lie in the Cartan subalgebra, BisDifsbiPH, Ž 3 where b are r-dimensional vectors for each i s 1,,3 i The corresponding field configuration in the symmetric gauge is 1 j Aisyeijkx bkph, i fsx biphqhph 4 There are still remaining gauge transformations: the unbroken UŽ 1 ržg transformations and the Weyl reflections which shuffle the generators in the Cartan subalgebra The linearized BPS equations for d A i,df in this background is eijkž Ejd Akyi A j,d Ak / seidfyi A i,df ye4 d Aiqi f,d A i Ž 5 The background gauge condition is Eid Aiyi A i,d Ai qe4 dfyi f,df s0 Ž 6 Again, as there is no x 4 dependence, E4 s0 As in the SUŽ case, the background field A i,f lies on the Cartan subalgebra and the equations are linear, and so the linear fluctuations along the generators do not mix The fluctuations d A i,df along unbroken gauge groups is independent of the background fields and unnormalizable They are associated with d b i and d h

391 ( ) K LeerPhysics Letters B For the study of fluctuations lying along a raising operator E, we introduce W such that a d A sw E, dfsw E Ž 7 i i a 4 a The Wm equations become identical to those in Eq Ž 11 if we rotate the coordinate so that bipa points to zˆ direction and put bs< b Pa <, z syhpa Ž 8 i 0 Then the solution will be identical as those in Eq Ž 13 Since zero modes are confined on the plane x i bipaqhpas0, Ž 9 we interpret this plane as the a monopole sheet Note that this plane is invariant under a ya Similarly to the SUŽ case, the number density of a monopole is < b Pa < i nas, Ž 30 p and the number of zero modes per unit quantum cell or BPS monopole is again four When there is unbroken nonabelian subgroup so that the planes Ž 9 do not overlap, the number of monopole sheet is that of positive roots ŽdGy rg r Contrast to the case of finite number of magnetic monopoles where there is fundamental magnetic monopoles corresponding to simple roots wx 8, all monopole sheets in our case are in equal footing The D-brane picture of Ns4 dimensional Yang- Mills theory in four dimension helps to understand our result w10 x Monopoles appear as D-strings connecting parallel D3 branes in type B string theory Following the work by Callan and Maldacena wx 9 these D-strings can be regarded as continuous deformation of D3 branes Our work suggests that as the number of magnetic monopoles increases to infinite, something drastic happens The D3 branes get tilted and appear to intersecting each other Since D3 branes are self-dual, there is no distinction between intersecting and contacting of two D3 branes Since the field configuration is self-dual, this configuration of interacting D3 branes should be supersymmetric In the SUŽ case there is an obvious duality between the z direction and f direction in the D-brane picture, but it is not obvious in the field theory picture In SUŽ N case, we can rewrite the Higgs m field in Eq Ž 4 in the N dimensional Hermitian matrix form Its i-th diagonal component then indicates the position of the i-th D3 brane Thus, one can identify each monopole sheet associated with a given root aseiyej with the plane of intersection be- tween the i-th and j-th D3 branes Let us now change our attention to the instanton case Instantons are the solutions of self-dual equations, B sf Ž 31 i i4 in Euclidean four dimensional Yang-Mills theory The self-dual configuration with homogeneous field strength is BisFi4 sbiph Ž 3 The Pontryagin index density of this configuration is 1 bipbi n s tr F F Pont ž / s Ž 33 mn mn 3p cž G 8p Once we choose the symmetric gauge for Bi and the Landau gauge for Fi4 so that E4 Ais 0, the background field is identical to those in Eq Ž 4 with A4 s f Contrast to the magnetic monopole case, the plane Ž 9, which is now a three volume, is not special any more as hpa can be changed by the unbroken gauge transformations Rather for each bipa, Bi is the magnetic field through the two-di- sional x i ;bipa and x 4 plane The linear fluctuation equations for d A mensional plane defined by x i bipas0 and x 4 s0, and F is the magnetic field through the two-dimeni4 m will be 4 identical to Eq 5, once we allow the x dependence The background gauge condition is then identical to Eq Ž 6 The fluctuations along the unbroken abelian direction is again independent of the background and nonrenormalizable They changes the parameters bi and h On the other hand, for the fluctuations along a raising operator E, we use Eq Ž 7 a to introduce W With the x 4 dependence, e ip4 x 4 m, and a spatial rotation such that bipa lying along the z direction, the zero mode solutions are identical to monopole case Ž 13 once we use Eq Ž 8 and replace z0 by z0 qp 4 rb This shows that the zero mode is not confined on the plane Ž 9 Rather the four zero mode solutions are the product of two wave func-

392 39 ( ) K LeerPhysics Letters B tions of the lowest Landau level, one on the xyy plane and another on the zyx 4 plane First is written in the symmetric gauge and second is in the Landau gauge As far as these zero modes are concerned, the translation along the x direction and those along the y direction do not commute This makes again the x y y plane a phase space or a quantum plane with unit quantum cell of area prb Similarly the zyx 4 plane becomes a quantum plane of unit quantum cell of area prb Thus on the R 4 space the number density of zero modes for each positive root a is / bipa nas4 ž Ž 34 p The number density of zero modes from all positive roots is the sum bipbi nzero s Ý na s cž G, Ž 35 p a)0 where Eq 19 is used The number of zero modes per unit Pontryagin index is then n zero s4=c G 36 n Pont This number is exactly identical to what is obtained from the index theorem wx 4 and also from the considwx 5 eration of constituent monopoles of calorons Instantons appear as the bound states of D0 branes on overlapped D4 branes of the type A theory Our case will be the limit where the number of D0 branes is infinite There is a work recently on this limit w11 x The D-brane picture of our homogeneous solution for instanton is much less clear than that for monopoles While we assume that homogeneous self-dual configuration can be obtained by arranging hte infinite number of instantons right, we do not know how they are put together for the homogeneous configuration In this paper we explored the homogeneous BPS field configuration as the sheets of infinite number of BPS magnetic monopoles The normalizable zero modes is confined at the magnetic monopole sheet, even though their position on the plane is not localized Their position on plane are described by the quantized phase space or noncommutative two-plane We showed that the number of normalizable zero modes per unit magnetic monopole charge is four For instantons, the number of normalizable zero modes per unit Pontryagin number is identical to that of single instanton without uniform background The position space R 4 of the normalizable zero modes become noncommutative four space We also discussed the D-brane pictures of these homogeneous configurations There are several questions arising naturally First question is whether single magnetic monopole can be separated from the infinite sheet The aforementioned work on the SUŽ case by Lee and Park suggests that it cannot be done without singular behavior in other side of the monopole sheet We think this needs a further clarification On the other hand, there exists a remarkable Minkowski solution w1x in the instanton case It describes the instanton lump in the uniform background When bpas0 for a root a, one can embed the corresponding SUŽ monopole or instanton solutions In the monopole case with the SUŽ N gauge group, two D3 branes are parallel and any finite number of D string can be inserted In the instanton case, we do not know what arrangement of D4 and D0 makes such a special case Note that the number of zero modes per unit Pontryagin index is independent whether bipas0 or not Second, it would be interesting to find out the moduli space dynamics of zero modes of these monopole and instanton sheet The moduli space would be now infinite dimensional As the position space of these zero modes are noncommutative, the moduli space may be noncommutative and the low energy Lagrangian could be some sort of a field theory of noncommutative variables Third, let us ask how SUŽ monopole or instanton sheets can be obtained from ADHM and Nahm formalisms w13 x Since there are infinite number of monopoles and instantons, the SUŽ ` group appears naturally Ward s work w14x on Nahm s equation should be relevant to the magnetic monopole sheet But we do not know the corresponding ADHM formalism or the ADHMN construction of solutions In addition, for instanton case, our work seems to be related to the recent work on the noncommutative geometry w15 x It remains to be clarified

393 ( ) K LeerPhysics Letters B Acknowledgements I thank Bum-Hoon Lee, Choonkyu Lee, Nick Manton, Soo-Jong Rey, and Erick Weinberg for useful discussions This work is done partly while I was visiting the SNU-CTP during the 1998 summer This work was also supported by KOSEP 1998 Interdisciplinary Research Program and SRC program of SNU-CTP, and Ministry of Education BSRI References wx 1 D Bak, C Lee, K Lee, Phys Rev D 57 Ž wx C Lee, QH Park, Exact BPS monopole solution in a self-dual background, Seoul N University Preprint, 1998, SNUTP wx 3 LS Brown, WI Weisberger, Nucl Phys B 157 Ž ; H Leutwyler, Nucl Phys B 179 Ž ; P Schwab, Phys Lett B 109 Ž ; G t Hooft, Comm Math Phys 81 Ž wx 4 CW Bernard, NH Christ, AH Guth, EJ Weinberg, Phys Rev D 16 Ž wx 5 K Lee, Phys Lett B 46 Ž wx 6 RS Ward, Phys Lett B 10 Ž ; MF Atiyah, NJ Hitchin, The Geometry and Dynamics of Magnetic Monopoles, 1988, Princeton University Press, Princeton, NJ wx 7 HJ Hitchin, NS Manton, MK Murray, Nonlinearity 8 Ž ; CM Houghton, PM Sutcliffe, Nucl Phys B 464 Ž wx 8 EJ Weinberg, Nucl Phys B 167 Ž wx 9 CG Callan, JM Maldacena, Nucl Phys B 513 Ž w10x E Witten, Nucl Phys B 460 Ž ; AA Tseytlin, Nucl Phys B 469 Ž ; MB Green, M Gutperle, Phys Lett B 377 Ž w11x E Keski-Vakkuri, P Kraus, Nucl Phys B 510 Ž w1x P Minkowski, Nucl Phys B 177 Ž w13x MF Atiyah, NJ Hitchin, VG Drinfeld, Yu I Mannin, Phys Lett B 185 Ž ; W Nahm, in: NS Craigie et al Ž Eds, Monopoles in quantum field theory, World Scientific, Singapore, 198 w14x RS Ward, Phys Lett B 34 Ž w15x N Nekrasov, A Schwarz, Instantons and noncommutative 4 R, and Ž,0 superconformal six-dimensional theory, hepthr980068

394 7 January 1999 Physics Letters B b ssd decay in two Higgs doublet models Katri Huitu a, Cai-Dian Lu b,d,1, Paul Singer c, Da-Xin Zhang a,d a Helsinki Institute of Physics, POBox 9, FIN UniÕersity of Helsinki, Helsinki, Finland b II Institut fur Theoretische Physik, UniÕersitat Hamburg, 761 Hamburg, Germany c Department of Physics, Technion Israel Institute of Technology, Haifa 3000, Israel d The Abdus Salam International Centre for Theoretical Physics, PO Box 586, Trieste, Italy Received 9 October 1998 Editor: PV Landshoff Abstract In the Standard Model, the box-diagram mediated decay b ssd is predicted to occur with an extremely small branching ratio of below 10 y11, thus providing a safe testing ground for exposing new physics We study this process in several Two Higgs Doublet Models and explore their parameter space, indicating where this process becomes observable q 1999 Elsevier Science BV All rights reserved 1 In the study of possible virtual effects in B decays which are due to physics beyond the Standard Model Ž SM, a crucial task is to subtract reliably the SM contributions For the best studied rare b decay process, b sqg, many efforts have been made in the last decade to calculate it as accurately as possible within SM However, since the signals of new physics in b decays emerge unluckily with branchy7 ing ratios at the level of 10 or less wx 1, the commonly focused processes of which b sqg is the prototype are very problematic for testing new physics against the SM background Moreover, other yet unobserved B decays, like various B t processes, have also been shown to be rather insensitive to a large class of new physics models wx In a recent letter wx 3, we have emphasized an alternative approach to the challenge of identifying 1 Alexander von Humboldt research fellow virtual effects from new physics in b decays, by the consideration of processes which have negligible strength in the SM Such processes could thus serve as sensitive probes for new physics, relatively free of SM pollution In wx 3 we focused on the b ssd transition, which is a box-diagram induced process in the SM with a very small branching ratio of below 10 y11, and we studied possible effects from the minimal supersymmetric standard model Ž MSSM and from a supersymmetric model with R-parity violation We have shown that the existing limits on parameters of MSSM and of R-parity violating interactions allow this process to occur well in excess of the SM rate, thus providing an unusual opportunity for stricter limits or, hopefully, for discovering new physics effects In the present letter we undertake a study of the occurrence of the b ssd transition in Two Higgs Doublet Models Ž THDMs w4 6 x, which are frequently considered as likely candidates for the extension of the SM We shall study two models in which the charged Higgs exchange is contributing to r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

395 ( ) K Huitu et alrphysics Letters B box diagrams, which we denote as Model I wx 4 and Model II wx 5, respectively, as well as a third model allowing a tree level transition mediated by neutral Higgs bosons wx 6 We begin with some comments on the calculation of the W-box diagrams in the SM Due to strong cancellations between the contributions from the top, the charm and the up quarks in the loops, the leading order SM result for the b ssd decay rate is m Gs 48Ž p 5 b GF ) mwvtbv 3 ts p ) ) = VtdVts fž x qyvcdvcs gž x, y, Ž 1 where 1y11 xq4 x 3 fž x s y ln x, 3 4 xž 1yx 1yx 8 xy4 x y1 4xy1 gž x, y syln yq ln xq, 41yx 41yx Ž 3 with xsm rm, ysm rm The first term in 1 is suppressed by the small Cabibbo-Kobayashi- ) Maskawa CKM matrix elements VtdV ts, while the second term is suppressed by ysm crmw and con- tributes about a half of the CKM suppressed term In this second contribution, we have neglected a kinematics dependent term when we perform the integral over the loop momentum This amounts to neglect- Ž W t c W Ž ing a small m rm lnžfž p rm c W c contribution with fž p a function depending on the external momenta p We have checked the effect of the neglected dependence numerically and found that it never exceeds 10% of the m crmw term in the whole kine- matic region Since the involved CKM matrix elements are not well bounded and the relative phase of the two terms is not fixed, we can only determine a range for the branching ratio of b ssd, which turns out to be always below 10 y11 in the SM Note that We note that this dependence does not appear in the calculation of the KK mixing, where the external momentum squares are of the order of m s and can be safely neglected compared to m c, nor in the case of BB mixing, where even the contributions of the order of m rm are sub-dominant b W QCD corrections may not change the value by orders of magnitudes, if compared with the analogous pro cesses BB and K K mixing wx 7 All these features combine to single out this process as a very sensitive one to new physics The THDMs are the minimal extensions of the SM With one more Higgs doublet, one has to suppress the tree level flavor changing neutral current Ž FCNC interactions due to neutral Higgs bosons to be consistent with the data The Model I allows only one Higgs doublet to couple to both the up- and the down-type quarks wx 4 In the Model II one Higgs doublet is coupled only to up-type quarks while the other doublet is coupled only to down-type quarks wx 5, and thus the Higgs content in the Model II is the same as in the MSSM In both the Model I and Model II, discrete symmetries are enforced to forbid the more general couplings and thus the tree level FCNC interactions are absent In another model, the Model III wx 6, the tree level FCNC is assumed to be small enough to lie within the experimental bounds is The relevant Lagrangian in the Model I and II 1 g Ls cotb u Ž M V ij d ' M W i, R U j, L q yj ui, L MDV ijdj, R H qhc, 4 where V represents the 3=3 unitary CKM matrix, MU and MD denote the diagonalized quark mass matrices, the subscripts L and R denote left-handed and right-handed quarks, and i, j s 1,,3 are the generation indices For model I, jscotb; while for Model II, jsytanb In the charged Higgs mediated box diagrams, the couplings proportional to the up quark masses dominate the contribution to b ssd decay In both models we have m Gs 48Ž p ) = td ts 5 b GF ) mwvtbv 3 ts p V V fž x qaž x, z ) qyvcdvcs g x, y qb x, z, 5

396 396 ( ) K Huitu et alrphysics Letters B where cot b 1y4 x AŽ x, z s ž xž 1yzŽ 1yx 3 x q ln x Ž 1yx Ž zyx z y4 xz q ln z xž 1yz Ž zyx / ž / cot 4 b 1qz z q q ln z, 3 4 x Ž 1yz Ž 1yz Ž 6 4 xyz BŽ x, z scot b ln z ž Ž zyxž 1yz x/ y 3 x 1yx Ž zyx ln ž / y 4 cot b q ln z, Ž 7 1yz Ž 1yz with z s m qrm The functions AŽ x, z H t and BŽ x, z denote the new contributions from the charged Higgs Because we have dropped in Ž 5 the terms which are proportional to a factor less than mbmsrmw at the amplitude level, the limit cotb 0 corresponds to the SM case As in the SM, the result depends on the relative CKM phase between the terms with the m crmw and the CKM suppressed contributions In Fig 1 we plot R as the function of cotb, where Rs Ž 8 < ) yv V g x, y qb x, z < cd cs < ) V V fž x qaž x, z < td ts is the ratio between the amplitudes induced by the m crmw and by the CKM suppressed terms It is clear that the SM is the limit with the maximal importance of the m crmw contribution, while in the limit of large cot b this contribution is negligible In the numerical calculation, we use mb s 45 GeV, m s 15 GeV, < V < s 004, < V < c ts tds 001 and < V < cd s0 We set the relative phase between the two terms in Ž 5 as zero which corresponds to a maximal constructive interference In Fig, we Fig 1 The ratio R of the amplitude induced by the m crmw term over that by the CKM suppressed one The solid line, dashed line, dash-dotted line and dotted line correspond to mhqs100, 00, 400, 800 GeV, respectively show the numerical results for branching ratio of b ssd decay as a function of cotb for different charged Higgs masses We have taken m q H as GeV while cotb ranging from 01 to 5 Some semi-quantitative considerations suggest a small cotb Ž -, 3 in which region the decay branching ratio for b ssd decay is unobservable However, phenomenologically a cotb as large as 5 is still consiswx 9 and the direct tent with the low energy data search of the Higgs boson in top decays In this parameter space of a large cot b the charged Higgs box diagrams are sizable We note that in the MSSM cotb-1 is preferred in model building, corresponding to an unobservable contribution from the charged Higgs box diagrams in this model wx 3 As can be seen in Fig, the branching ratio in the Model I and II can be of the order 10 y8 at large cot b region The searching of the decay b ssd and its hadronic channels B " K " K " X inbexperiments will further constrain the parameters in the THDMs 3 wx We refer the reader to 8 for more detailed discussions

397 ( ) K Huitu et alrphysics Letters B Fig The branching ratio of b ssd as a function of cotb The solid line, dashed line, dash-dotted line and dotted line correspond to mhqs100, 00, 400, 800 GeV, respectively Regions above the lines are excluded 3 In the THDM III, there exists the Yukawa wx interaction 6 LsjijQi, Lf Dj, R qž the up quark sector qhc Ž 9 Here, the scalar Higgs doublet f mediates the FCNC transitions dil d j at the tree level, if the coupling jij is nonzero As discussed in the litera- ture, FCNC effects induced by the loop diagrams are always negligible compared to the tree level ones in the Model III w10 x, hence the box diagrams with charged Higgs can be safely dropped in the present decay We will neglect the unimportant QCD correction since no GIM-like cancellation happens in the process The couplings in Ž 9 are constrained strongly from the neutral meson mixing by the requirement that the FCNC contribution to the mixing from the interactions in Ž 9 does not exceed its experimental value In the KK and BsBsmixing, the dominant contribu- tions are proportional to jsd and jsb, respectively w10x ž / MS F MP F F sq mh ma DM sj q, Ž 10 with FsK, B, and qsd,b, respectively, and S ž / ž / 1 ff MF 3 F MS s 6 ffmfq m qm, s 1 11fF MF 3 F MP s 6 ffmfq Ž 11 m qm s Here mh and ma are the masses of the neutral scalar and pseudoscalar Higgs bosons Note that we have taken jijsjji while it is not difficult to generalize to the case without this assumption In numerical calculations, we use fks160 MeV, fb s s00 MeV, y15 DM s 3491 = 10 GeV w11 x K Taking mas mh 'm H, we obtain the bound j sd y8 y1-83=10 GeV Ž 1 m H from DM K The experimental lower limit from Bsy Bs mixing y1 DM )5=10 GeV w1 x, Ž 13 B s gives no constraint on jsbrm H, since the contribu- tion in the SM itself exceeds this number Furthermore, free from any assumption made for the FCNC Higgs couplings in the lepton sector, the bounds Fig 3 The branching ratio of b ssd in THDM III as a function < < of j j rm bs sd H q q

398 398 ( ) K Huitu et alrphysics Letters B from B l l and B Kl l Ž l,l se,m ort s i j i j i j do y1 not exclude even j rm as large as 10 w13 x sb H In the presence of the interactions Ž 9, b ssd can be induced by a tree diagram exchanging the neutral Higgs bosons h Ž scalar and A Žpseudo- scalar, with the amplitude i 1 1 As jsbjsd Ž sbž sd y Ž sg 5bŽ sg 5d ž m m / h A Ž 14 The decay rate is thus m 5 b < < 3 sb sd Gs j j 307Ž p = ½ ž / 5 h A h A q q Ž m m m m The numerical results are shown in Fig 3 as a < < function of jsbjsd rm H From Fig 3, it can be seen that the decay b ssd < < is observable in the Model III, if jsbjsd rm H) 10 y10 GeV y, which corresponds to a branching y9 ratio around 10 This requires at least < j rm < sb H ) 10 y3 GeV y1 Corresponding to this number, the neutral Higgs contribution to DMB is 10 6 s times larger than its present lower limit To our knowledge, such a large DM is difficult to exclude B s 4 In summary, we have presented a study of the b ssd process in several THDMs Firstly, we confirmed that the charged Higgs box contribution in MSSM is indeed negligible wx 3, while on the other hand in Models I wx 4 and II wx 5 this contribution can induce observable effects at the 10 y8 level for the branching ratio A large BB s s mixing, which is at least 10 6 times larger than its present lower limit, is required in Model 3 wx 6 for this process to be observable Combining the above results with those of Ref wx 3, we reemphasize that the search for the processes wx y y y q we recommended 3, like B K K p, will serve immediately to get better limits on the parameters of R-parity violating theories; at the later stage, when lower experimental limits are obtainable, it would constitute a direct check of the various Non-Standard Model theories investigated here and in Ref wx 3 Acknowledgements The work of KH and DXZ is partially supported by the Academy of Finland Ž no The work of PS is partially supported by the Fund for Promotion of Research at the Technion References wx 1 For reviews, see A Masiero, L Silvestrini, hep-phr970944, in proceedings of the nd International Conference on B Physics and CP Violation Ž BCONF 97, Honolulu, HI, Mar 1997, TE Browder, FA Harris, S Pakvasa Ž Eds, World Scientific Ž 1998 ; JL Hewett, hep-phr , in the proceedings of the 8th International Conference on High Energy Physics, Warsaw, Poland, July 1996, Z Ajduk, AK Wroblewski Ž Eds, World Scientific Ž 1997 wx D Guetta, E Nardi, Phys Rev D 58 Ž ; see here for an extensive list of previous references wx 3 K Huitu, C-D, Lu, P Singer, D-X Zhang, hepphr , preprint DESY , HIP rTH, TECHNION-PH-11, to appear in Phys Rev Lett wx 4 HE Haber, GL Kane, T Sterling, Nucl Phys B 161 Ž wx 5 SL Glashow, S Weinberg, Phys Rev D 15 Ž wx 6 TP Cheng, M Sher, Phys Rev D 35 Ž ; M Sher, Y Yuan, Phys Rev D 44 Ž wx 7 AJ Buras, M Jamin, PH Weisz, Nucl Phys B 347 Ž wx 8 K Kiers, A Soni, Phys Rev D 56 Ž wx 9 V Barger, JL Hewett, RJN Phillips, Phys Rev D 41 Ž w10x D Atwood, L Reina, A Soni, Phys Rev D 55 Ž w11x Particle Data Group, Phys Rev D 54 Ž w1x R Barate et al, ALEPH Collaboration, preprint CERN-PPE w13x See M Sher, hep-phr and references therein

399 7 January 1999 Physics Letters B Quark lepton mass hierarchies and the baryon asymmetry W Buchmuller a, T Yanagida a,b a Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany b Department of Physics, UniÕersity of Tokyo, Tokyo, Japan Received 14 October 1998 Editor: PV Landshoff Abstract The mass hierarchies of quarks and charged leptons as well as a large n m nt mixing angle are naturally explained by the Frogatt-Nielsen mechanism with a nonparallel family structure of chiral charges We extend this mechanism to right-handed neutrinos Their out-of-equilibrium decay generates a cosmological baryon asymmetry whose size is quantized in powers of 1r y the hierarchy parameter e For the simplest hierarchy pattern the neutrino mass mns mnm n ; 10 ev, which is m t inferred from present indications for neutrino oscillations, implies a baryon asymmetry nbrs;10 y10 The corresponding baryogenesis temperature is T B;10 10 GeV q 1999 Elsevier Science BV All rights reserved In the standard model left- and right-handed quarks and leptons can be grouped into the SUŽ 5 Ž c c ) Ž c c multiplets 10s q L,u R,e R, 5 s d R,lL and 1sn R, such that the Yukawa interactions with the Higgs fields take the form, Lsh Žu ij 10i10 jh1ž 5 qh Ž ij d 5 i ) 10 jhž 5 ) qhij Žn 5 i ) 1 jh1ž 5 qh Ž ij s 11S i j Ž 1 Ž 1 Here i, js1 3 are generation indices The expectation values of the Higgs multiplets H1 and H generate ordinary Dirac masses of quarks and leptons, whereas the expectation value of the singlet Higgs field S generates the Majorana mass matrix of the right handed neutrinos The masses of up-quarks, down-quarks and charged leptons approximately satisfy the following mass relations, m t:m c:m u,1:e :e 4, m b:m s:m d,m t:m m:m e,1:e :e 3, Ž 3 where, for masses defined at the unification scale L GUT, e ;1r300 According to the Frogatt-Nielsen mechanism wx 1 these mass hierarchies are related to a spontaneously broken symmetry Each SUŽ 5 multiplet of fermions ci carries a corresponding charge Q i, and the Yukawa couplings then scale like h Ae Q iqq j ij Ž 4 It is usually assumed that the mass hierarchy is generated by the expectation value of a singlet field F with charge QF sy1 via a nonrenormalizable interaction with a scale L)L, ie es² F : GUT rl The observed mass hierarchy of up-quarks and the large t-quark mass yield for the 10-plets uniquely the charges listed in Table 1 Here we have assumed that Yukawa couplings are not larger than OŽ 1 From the mass hierarchy of the charged leptons one infers that the second and third generation 5 ) -plet have the same charge Ž a, which differs by one unit from the charge of the first generation 5 ) -plet The value of r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

400 400 W Buchmuller, T YanagidarPhysics Letters B 445 ( 1999) Table 1 Chiral charges for the fermions of the standard model; as0 or 1, and 0F bfcf d ) ) ) c i Q 0 1 a a aq1 b c d i the b-quark mass allows as0 oras1 wx Such a nonparallel family structure has previously been identified in a detailed study of UŽ 1 generation symmetries wx 3 Note, that the UŽ 1 may be replaced by an anomaly-free discrete Z5 symmetry with the charge assignment in Table 1 The difference between the observed down-quark mass hierarchy and the charged lepton mass hierarchy can be accounted for by introducing an addiwx 4 The following tional 45-plet of Higgs fields discussion does not depend on this and could also be carried out directly for the quark and lepton multiplets of the standard model gauge group The chiral charges of the lepton doublets would then correspond to the charges of the 5 ) -plets given in Table 1 Masses for the standard model neutrinos are generated by the unique dimension-5 operator LDLs sfij 5 i ) 5 j ) H1Ž 5 H1Ž 5, Ž 5 which is induced by the exchange of heavy particles in generic seesaw models wx 5 The chiral charges of ) the 5 plets lead to the neutrino mass matrix ŽÕs ² H :, 1 0 e e e mn sfijõ A e 1 1, Ž 6 ij e 1 1 where we have only listed the order in e for the different matrix elements Diagonalization clearly yields a large n m nt mixing angle, which is needed to explain the atmospheric neutrino deficit by n m nt oscillations wx 6 As described above, this is a direct consequence of the mass hierarchy of the charged leptons which leads to the charge assignment given in Table 1 The masses of the two eigenstates nm and nt depend on factors of order one, which are omitted in Ž 6, and may easily differ by an order of magniwx 7 They can therefore be consistent with the tude y6 y5 mass differences Dm,4P10 1P10 ev wx n e n m 8 inferred from the MSW solution of the solar neutrino wx Ž y4 y3 problem 9 and Dm n n, 5P10 y6p10 ev m t associated with the atmospheric neutrino deficit wx 6 In the following we shall use for numerical estimates the average of the neutrino masses of the second and 1r y third family, mn s mn m n ;10 ev m t Consider now the simplest class of seesaw models where three right-handed neutrinos nr i are added to the standard model According to Ž 1 and the charge assignment in Table 1, the Dirac neutrino mass matrix has the form 0 Ae dq1 Be cq1 Ce bq1 Žn a md sh Õs e d De c Ee b Fe Õ, d Ge c He b Ke Ž 7 where A,,K are constants of order one Correspondingly, the Majorana mass matrix of the righthanded neutrinos reads, 0 ae d 0 0 Ž s Msh ² S: s 0 c be 0 ² S : Ž ge b Here a,,g are constants of order one and, without loss of generality, we have chosen a basis where M is diagonal and real Note, that this diagonalization does not change the hierarchy structure of the Dirac mass matrix Ž 7 The corresponding mass eigenstates are the Majorana neutrinos N i,nr i q c n In terms of the Dirac mass matrix Ž R i 7 and the Majorana mass matrix Ž 8 the Majorana mass matrix of the light neutrinos is given by wx 5 1 T mn smd m D Ž 9 M In this expression for the light neutrino masses the dependence on the chiral charges of the heavy neutrinos drops out and one obtains the hierarchy structure Ž 6 Heavy Majorana neutrinos are likely to play a crucial role for the cosmological baryon asymmetry At high temperatures, above the critical temperature of the electroweak phase transition, baryon and lepton number violating processes are in thermal equiw10 x As a consequence, a primordial lepton librium asymmetry generated by the out-of-equilibrium decay of heavy Majorana neutrinos is partially trans-

401 W Buchmuller, T YanagidarPhysics Letters B 445 ( 1999) formed into a baryon asymmetry w11 x The dominant contribution to the lepton asymmetry is produced in the decays of N 1, the lightest of the heavy Majorana neutrinos The corresponding baryon asymmetry is given by, n B 1 YB s sk C Ž 10 s g ) Here 1 is the CP-asymmetry in the decay of N 1,C is the ratio of lepton and baryon asymmetry, which equals y8r15 in the Ž supersymmetric standard model with Higgs doublets, g ;100 is the num- ) ber of effectively massless degrees of freedom and k is a dilution factor Its value reflects the effect of the various lepton number conserving and violating processes in the plasma In order to determine k reliably one has to solve the full Boltzmann equations To obtain a large asymmetry, the number density of the heavy neutrinos at high temperatures has to be large enough, they have to fall out of equilibrium at T;M, and the DLs1 and DLs washout pro- 1 cesses have to be sufficiently suppressed Typical values of the dilution factor are then k;10 y1 10 y, which corresponds to a baryon asymmetry Y B ; Ž 10 y3 10 y4 1 Ž 11 The CP asymmetry 1 can be computed in terms of the neutrino mass matrices For degenerate heavy neutrinos, ie M A 1, 1 vanishes We therefore assume hierarchical heavy neutrino masses, with d G1 In this case M <M, and one obtains w1,13 x, 1,3 3 1 M 1 1 s Ý Im Ž mdm D 1i 16p Õ Ž mdmd 11 is,3 Mi Ž 1 Note, that the CP asymmetry depends on md only through m Dm D Hence, 1 is not directly related to the n m nt mixing angle in the leptonic charged current From Eqs ŽŽ 7 9 and Ž 1 one easily obtains the dependence of the CP asymmetry and the light neutrino masses on the hierarchy parameter e, 1 ;10 y1 e Ž aqd, Ž 13 Õ Ž aqd m n;e 14 M 1 y Using m n;10 ev, the corresponding mass of N 1, the lightest of the heavy neutrinos, is given by M 1 ;e Ž aqd GeV Ž 15 Note, that the values of 1 and M1 do not depend on the charges b and c of the heavy neutrinos N3 and N, respectively From Eqs Ž 11 and Ž 13 one obtains for the baryon asymmetry, Y B ; Ž 10 y4 10 y5 e Ž aqd Ž 16 Hence, the baryon asymmetry is quantized in powers of the hierarchy parameter e Its magnitude is fixed by the neutrino charges Consider first the simplest case of hierarchical neutrino masses with Yukawa couplings of the third family OŽ 1, ie asbs0, cs1, ds One then has M,10 10 GeV, M,10 15 GeV ; 1 3 Y B ;10 y9 10 y10 Ž 17 These are precisely the parameters chosen in w14 x Solving the Boltzmann equations indeed yields an asymmetry Y B ;10 y10, the baryogenesis tempera- ture is T ;M ;10 10 GeV and ByL is broken at B 1 the GUT scale The same results are obtained for cs0 Alternatively, one may have smaller Yukawa couplings together with a smaller mass hierarchy This corresponds to as0, bscs1, ds oras1, bscs 0, ds1 In both cases baryogenesis temperature and baryon asymmetry remain the same, but M ;M ; GeV Hence, the scale of ByL breaking may be below the GUT scale In all cases considered so far the CP asymmetry ;10 y6 One may think that the charge assign- 1 ment asbscs0, ds1, which yields a CP asymmetry ;10 y4, may lead to a larger baryon asym- 1 metry However, one then has M 1 ;10 1 GeV The dependence of the baryon asymmetry on m s Ž m m D D 11rM1 and M1 has been studied in the non-supersymmetric and in the supersymmetric case by solving the full Boltzmann equations w15 x It turns out that the dependence on M1 is model dependent For the above choice of parameters, with m 1 ;m n ; 10 y ev, the washout processes in the supersymmetric case are too strong, and the generated asymmetry 1

402 40 W Buchmuller, T YanagidarPhysics Letters B 445 ( 1999) lies far below the observed value Y B ;10 y10 Hence, the observed baryon asymmetry corresponds to the maximal asymmetry The baryogenesis temperature T B ;10 10 GeV is naturally obtained in hybrid models of inflation For supersymmetric theories such a large reheating temperature imposes stringent constraints on the mass spectrum of superparticles, in particular on the gravitino mass Consistency with primordial nucleosynthesis requires either a very light gravitino, mg - 1 kev w16 x, a heavy gravitino, m ) TeV w17 x G, or a gravitino in the mass range mg s Ž GeV as LSP w18 x, with a higgsino type neutralino as next-tolightest superparticle In the last case gravitinos could be the dominant component of dark matter There are other mechanisms of baryogenesis In particular in supersymmetric theories a very large baryon asymmetry may be produced by the coherent oscillation of scalar fields carrying baryon and lepton number w19 x The generated asymmetry then depends on the initial configuration of scalar fields This is in contrast to leptogenesis, as described above Here, the baryon asymmetry is entirely determined by neutrino masses and mixings which, at least in principle, can be measured in precision experiments probing the leptonic charged current Crucial tests of leptogenesis are the Majorana nature of neutrinos and CP violation in the leptonic charged current Note, however, that the expected electron neutrino mass m n ;10 y5 e ev is rather small Its detection in neutrino-less bb-decay is an experimental challenge Leptogenesis requires that the neutrino mass m 1 ;m ns mn mn 1r is very small in order to satisfy m t the out-of-equilibrium condition for the decaying heavy neutrino According to present indications for y neutrino oscillations, which yield m n; 10 ev, this is indeed the case For the simplest hierarchy pattern one obtains a baryon asymmetry Y B ;10 y10, in agreement with observation References wx 1 CD Frogatt, HB Nielsen, Nucl Phys B 147 Ž wx J Sato, T Yanagida, Talk at Neutrino 98, hep-phr wx 3 J Bijnens, C Wetterich, Nucl Phys B 9 Ž wx 4 H Georgi, C Jarlskog, Phys Lett B 86 Ž wx 5 T Yanagida, in: Workshop on unified Theories, KEK report 79-18, 1979, p 95; M Gell-Mann et al, in: P van Nieuwenhuizen, D Freedman Ž Eds, Supergravity, Proc of the Stony Brook Workshop 1979, North-Holland, Amsterdam, 1979, p 315 wx 6 Super-Kamiokande Collaboration, Y Fukuda et al, Phys Rev Lett 81 Ž wx 7 N Irges, S Lavignac, P Ramond, Phys Rev D 58 Ž wx 8 N Hata, P Langacker, Phys Rev D 56 Ž wx 9 SP Mikheyev, AY Smirnov, Nuovo Cim C 9 Ž ; L Wolfenstein, Phys Rev D 17 Ž w10x VA Kuzmin, VA Rubakov, ME Shaposhnikov, Phys Lett B 155 Ž w11x M Fukugita, T Yanagida, Phys Lett B 174 Ž w1x L Covi, E Roulet, F Vissani, Phys Lett B 384 Ž ; M Flanz, EA Paschos, U Sarkar, Phys Lett B 345 Ž ; B 384 Ž Ž E w13x W Buchmuller, M Plumacher, Phys Lett B 431 Ž w14x W Buchmuller, M Plumacher, Phys Lett B 389 Ž w15x M Plumacher, Z Phys C 74 Ž ; Nucl Phys B 530 Ž w16x H Pagels, JR Primack, Phys Rev Lett 48 Ž w17x M Kawasaki, T Moroi, Progr Theor Phys 93 Ž w18x M Bolz, W Buchmuller, M Plumacher, hep-phr w19x I Affleck, M Dine, Nucl Phys B 49 Ž

403 7 January 1999 Physics Letters B A useful approximate isospin equality for charmless strange B decays Harry J Lipkin 1 Department of Particle Physics, Weizmann Institute of Science, RehoÕot 76100, Israel School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel High Energy Physics Division, Argonne National Laboratory, Argonne, IL , USA Received 1 October 1998 Editor: H Georgi Abstract A useful inequality is obtained if charmless strange B decays are assumed to be dominated by a D Is0 transition like that from the gluonic penguin diagram and the contributions of all other diagrams including the tree, electroweak penguin and annihilation diagrams are small but not negligible The interference contributions which are linear in these other amplitudes are included but the direct contributions which are quadratic are neglected q 1999 Elsevier Science BV All rights reserved It is now believed that charmless strange B decays are dominated by the gluonic penguin diagram If all other diagrams are negligible, the branching ratios for sets of decays to states in the same isospin multiplets are uniquely related by isospin because the gluonic penguin leads to a pure Is1r final state, and there is no simple mechanism in the standard model that can give CP violation If the tree diagram produced by the b uus transition at the quark level also contributes, the isospin analysis becomes non-trivial, much more interesting and CP violation can be observed This case 1 Supported in part by The German-Israeli Foundation for Scientific Research and Development Ž GIF and by the US Department of Energy, Division of High Energy Physics, Contract W ENG-38 HJL@axp1hepanlgov has been considered in great detail by Nir and Quinn wx 1 Our purpose here is to point out and discuss an intermediate case, where the amplitude from the tree diagram is not negligible but still sufficiently small by comparison with the gluonic penguin that its contribution to branching ratios need be considered only to first order, and higher order contributions can be neglected An interesting approximate sum rule is obtained which can check the validity of this approximation and guide the search for CP violation We consider here the B Kp decays Exactly the same considerations hold for all similar decays into a strange meson and an isovector nonstrange meson An extension incorporating also the decay into the isoscalar strange meson; eg v is presented below for all ideally mixed nonets The gluonic penguin diagram leads to a pure isospin 1r state Its contributions to all B Kp r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

404 404 ( ) HJ LipkinrPhysics Letters B amplitudes, denoted by P are simply related by isospin PŽ B q K q p o sypž B o K o p o 1 o q y s PPŽ B K p ' 1 q o q s PPŽ B K p 'P Ž 1 ' The tree diagram has two independent contributions corresponding to the color-favored and color-suppressed couplings of the final quark-antiquark pairs Their amplitudes, denoted by Tf and Ts are also simply related by isospin 1 o q y q q o PTfŽ B K p stfž B K p 'Tf ' Ž a TsŽ B q K q p o stsž B o K o p o 'Ts Ž b In the approximation where we consider the contributions of the tree amplitudes only to first order, we obtain: BRŽ B o K q p y f P q4pptf Ž 3a BRŽ B o K o p o fp y PPTs Ž 3b BRŽ B q K o p q f P Ž 3c BRŽ B q K q p o fp q PPTfq PPTs Ž 3d This leads to the approximate equality BRŽ B o K q p y ybrž B o K o p o fbrž B q K q p o ybrž B q K o p q f4pptfq4ppts s4pž TfqTs PcosŽ fpyftyfs Ž 4a where fp and ft are the weak phases respectively of the penguin and tree amplitudes and fs is the strong phase difference between the two Note that both the left-hand and right hand sides vanish independently for any transition like the pure gluonic penguin that leads to a pure I s 1r state The relation Ž 4 therefore is due entirely to interference between the penguin I s 1r amplitude and the Is3r component of the tree amplitude This immediately leads to the correspnding expression for the charge-conjugate decays, o y q o o o ybrž B K p BR B K p y y o y o y fbrž B K p ybrž B K p f4p T qt PcosŽ f yf qf Ž 4b f s P T S The direct CP violation is seen to be given by y y o q q o ybrž B K p BR B K p y o y q o q qbrž B K p ybr B K p y y o q q o = BRŽ B K p qbrž B K p y o y q o q ybrž B K p ybr B K p ftanž fpyft tanž fs Ž 5 The same equality Ž 4 is easily seen in the formalism used by Nir and Quinn wx 1 using their three amplitudes denoted by U, V and W The W amplitude is the penguin amplitude and the the two tree amplitudes U and V are linear combinations of our Tf and Ts amplitudes, defined by isospin properties rather than quark diagrams BRŽ B o K q p y ybrž B o K o p o sywpž UqV qž V yu fywpž UqV Ž 6a BRŽ B q K q p o ybrž B q K o p q sywpž UqV yž V yu fywpž UqV Ž 6b It is interesting to note that here also one sees the apparent miracle that the same linear combination of the two tree amplitudes, U q V appears in both expressions Ž 6a and Ž 6b and that the approximate equality follows from the approximation that quadratic terms in U and V are negligible in comparison with the product of the linear terms and the dominant penguin amplitude W Ž V yu <WP Ž UqV Ž 7 However, a simple isospin analysis shows that this is no miracle and that the approximate equality Ž 4 holds also when contributions of annihilation diagrams, diagrams in which a flavor-changing final state interaction like charge exchange follows the weak tree diagram and electroweak penguins We first note that both the annihilation diagram and the charge exchange diagram which proceeds via the quark annihilation and pair creation transition uu y1

405 ( ) HJ LipkinrPhysics Letters B gluons dd lead to pure Is1r final states and their contributions cancel in the sum rule Ž 4 The electroweak penguin amplitudes contain both I s 1r and I s 3r components This amplitude which we denote by PEW can be written as the sum of a contribution in which the electroweak boson creates an isoscalar qq pair and one which we denote by Pu in which the electroweak boson creates a uu pair The contribution with the isoscalar pair is proportional to the gluonic penguin Thus P sjppqp Ž 8 EW u where j is a small parameter To first order in P EW the contribution of the j P P term satisfies the approximate equality Ž 4, since the contributions of both sides vanish The Pu term arises from the quark transition b suu This then leads to the hadronic transition B bq suuq Ž 9a Ž 9b But these are exactly the same as the tree transitions Ž except that the color favored and color suppressed transitions are reversed Here the uu is a color singlet and the transitions in which this uu pair combine to make a p o are color favored We can therefore write Pu as the sum of a color-favored and color suppressed term, PusPufqP us Ž 10 and note that they satisfy the isospin relations analogous to Ž, 1 o q y q q o PP Ž B K p sp Ž B K p 'P ' us us us PufŽ B q K q p o spufž B o K o p o 'Puf Ž 11a Ž 11b We can therefore generalize the equalities Ž 4 by replacing Tf and Ts by T T X 'T qp Ž 1a f f f us T T X 'T qp s s f uf Ž 1b We thus see that the inclusion of the electroweak penguin has the effect of adding a term with the same isospin properties as the gluonic penguin and terms with the isospin properties of the color-suppressed and color-favored tree diagrams, and therefore do not affect the approximate equality Ž 4 The reason for the equality is easily seen by noting the isospin properties of all charmless strange decays The weak interaction produces either a D Is 0 transition which leads to an Is1r final state or a D Is1 transition which can lead to both Is1r and I s 3r final states wx There are therefore three independent isospin amplitudes The four decay branching ratios therefore depend in the general case on five parameters, three magnitudes and two relative phases and no simple equality between the branching ratios is obtainable However, in the approximation where only linear terms in the D Is1 amplitudes are considered, a phase convention can be chosen in which the dominant D Is0 amplitude is real and only the real parts of the D Is1 amplitudes contribute The four decay branching ratios now depend on only three real parameters, and the equality Ž 4 is therefore obtained This equality has recently been used by Gronau and Rosner wx 3 The same approach can be applied to charmless decays to a strange pseudoscalar and a nonstrange vector meson Here the ideal mixing of the r and v enables inclusion of the K v final states in the analysis if the OZI rule is assumed wx 4 We first o express the r and v states in terms of their uu and dd components, denoted respectively as Vu amd V d, o 1 r :' PŽ Vu: y V d: ; ' 1 v :' PŽ Vu: q Vd: Ž 13 ' The analogs of Eqs Ž 1, Ž have a particularly simple form when written in terms of the flavor eigenstates Vu amd V d, PŽ B q K q V spž B o K o V u spž B o K q r y q o q ' o q y q q ' f f u f d spž B K r ' PP Ž 14a T B K r st B K V ' PT Ž 14b T B q K q V st B o K o V 'T Ž 14c s u s u s

406 406 ' The negative signs and factors which appear in Eqs Ž 1, Ž as isospin Clebsch-Gordan coefficients are completely absent in this quark-flavor representation These factors appear in the quark model description only in the wave functions of the r o and p o This description enables extending the treatment beyond isospin and also including the v This implicitly asssumes the OZI rule which discards the penguin diagram in which three gluons create the omega wx 4 It also neglects the similar electroweak diagrams in which the electroweak boson creates the v Substituting the v and r wave functions from Eq Ž 13 into the relations Ž 14 brings them to the form of Eqs Ž 1, Ž with the additional predictions for the Kv decays PŽ B q K q r o spž B q K q v sypž B o K o r o spž B o K o v 1 o q y s PPŽ B K r ' 1 q o q s PPŽ B K r 'P ' Ž 15a 1 o q y q q o PTfŽ B K r stfž B K r ' stfž B q K q v 'Tf Ž 15b T Ž B q K q r o st Ž B q K q v s s ( ) HJ LipkinrPhysics Letters B stsž B o K o r o st Ž B o K o v 'T Ž 15c s In the approximation where we consider the contributions of amplitudes other than gluonic penguin only to first order, we obtain: BRŽ B o K q r y f P q4pptf Ž 16a BRŽ B o K o r o fp y PPTs Ž 16b BRŽ B q K o r q f P Ž 16c BRŽ B q K q r o sbrž B q K q v fp q PPT q PPT f s s Ž 16d BRŽ B o K o v fp q PPT Ž 16e s This leads again to an approximate equalities BRŽ B o K q r y ybrž B o K o r o fbrž B q K q r o ybrž B q K o r q f4pptfq4ppts s4p T qt PcosŽ f yf yf Ž 17a f s P T S o y q o o o ybrž B K r BR B K r y y o y o y ybrž B K r fbr B K r f4p T qt PcosŽ f yf qf Ž 17b f s P T S where fp and ft are again weak phases respectively of penguin and tree amplitudes and fs is the strong phase difference between the two The relation Ž 17 is again due entirely to interference between the penguin Is1r amplitude and the Is 3r component of other amplitudes The inclusion of the v gives the well-known equality wxž 4 16d and a new approximate equality, BRŽ B o K o r o qbrž B o K o v fbrž B q K o r q f P Ž 18 Acknowledgements It is a pleasure to thank Yosef Nir for helpful discussions and comments and in particular to thank Jonathan L Rosner for discussions and a prepublicawx 3 which pointed out that the tion copy of reference validity of the equality also for electroweak penguins had been overlooked in the previous version of this paper References wx 1 Y Nir, H Quinn, Phys Rev Lett 67 Ž wx M Gronau, O Hernandez, D London, JL Rosner, Phys Rev D 5 Ž wx 3 M Gronau, JL Rosner, hep-phr , submitted to Phys Rev Letters wx 4 HJ Lipkin, Phys Lett B 433 Ž hep-phr970853

407 7 January 1999 Physics Letters B Textures for atmospheric and solar neutrino oscillations 1 Riccardo Barbieri a, Lawrence J Hall b, A Strumia c a Scuola Normale Superiore, and INFN, sezione di Pisa, I-5616 Pisa, Italy b Department of Physics and Lawrence Berkeley National Lab, UniÕersity of California, Berkeley, CA 9470, USA c Dipartimento di Fisica, UniÕersita di Pisa, and INFN, sezione di Pisa, I-5617 Pisa, Italy ` Received 1 October 1998 Editor: H Georgi Abstract A large umt mixing angle, together with hierarchies in the neutrino Dm and in the charged lepton masses, place a significant restriction on the textures of the lepton mass matrices In theories with three light neutrinos, a simple hypothesis leads to a complete list of four zeroth-order textures for lepton mass matrices is found: three for neutrinos which naturally yield umt f1 with vanishing uet and Dm(, and one for charged leptons which naturally yields umtf1 with me smm suet s0 These textures provide suitable starting points for constructing theories which account for both atmospheric and solar neutrino fluxes Using flavor symmetries, two schemes for such lepton masses are found, and example constructions exhibited q 1999 Elsevier Science BV All rights reserved 1 This work was supported in part by the US Department of Energy under Contracts DE-AC03-76SF00098, in part by the National Science Foundation under grant PHY and in part by the TMR Network under the EEC Contract No ERBFMRX - CT The Super-Kamiokande Collaboration has provided strong evidence that atmospheric nm are de- pleted as they traverse the Earth w1, x For example, the uprdown event ratio shows a 6s statistical significance, and could only be explained by some systematic effect an order of magnitude larger than those already considered With two flavor nm nt oscillations, the mixing angle is large, sin u)08, and Dm atm is within a factor of three of 15=10 y3 ev at the 90% CL wx In this letter we study lepton masses in theories with three light neutrinos With the assumptions given below, we find a complete list of five zerothorder textures, which can account both for this atmo- spheric data and for the presence of solar neutrino oscillations at some frequency Dm( <Dm atmre- quiring the textures to follow from symmetries, we are able to find only two zeroth-order schemes for the light lepton mass matrices which give both the observed atmospheric and solar neutrino fluxes Both charged and neutral lepton mass matrices involve small dimensionless parameters: merm t< mmrm t <1 and Dm( rdm atm<1 In the flavor basis, the lepton mass matrices, l LmE l R and nl T mlln L, can be written as a perturbation series in a set of small parameters e: Ž 0 Ž 1 mesõ le qle e q PPP 1 Õ Ž 0 Ž 1 LL n n m s l ql Ž e q PPP M where Õ is the electroweak symmetry breaking scale and M is some new large mass scale, so that the l Ži are dimensionless contributions at ith order in per r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

408 408 ( ) R Barbieri et alrphysics Letters B turbation theory The matrices l Ž0 E and l Ž0 n have entries which are either zero or of order unity, and the possible forms for these matrices, which we call zeroth-order textures, are the subject of this letter Diagonalization of me and mll gives both the lepton mass eigenvalues and the leptonic mixing matrix V, defined by the charged current interaction T m n V g l W m Ignoring phases, which are irrelevant for our purposes, we define the three Euler angles by V s V V s R Ž u R Ž u R Ž u e n The rotation matrices V e,n, which diagonalize m E, LL, are the same functions of the angles ue,n ij as V is of u ij We require that the diagonalization of l Ž0 E,n leads to zeroth-order eigenvalues m Ž0 E1 sm Ž0 E s0 Ž 3 m Ž0 E3,mt Ž 4 Dm( Ž0 'm n 1 ym n s0 Ž 5 In most cases l Ž0 n will lead to a non-zero value for Dm atm; however, this is not a strict requirement We further require that l Ž0 E,n may be diagonalized to give zeroth-order leptonic mixing angles u 3 Ž0 f1 Ž 6 and u13 Ž0 s0 Ž 7 The requirement u13 Ž0 s0 follows from recent fits to the Super-Kamiokande atmospheric data wx 3, which find u13 less than approximately 08, for a hierarchy of Dm values such as we assume Furthermore, for Dm atm )P10 y3 ev, the CHOOZ experiment re- quires u We therefore take the view that a 3 non-zero value of u13 can be at most of order e Many theories of flavor with hierarchical fermion masses yield small mixing angles as in the wellknown example of the Cabibbo angle u c f Ž m rm 1r Conversely, theories with large mixing d s angles typically do not have mass hierarchies To avoid this typical situation, the textures for l Ž0 E,n become highly constrained One must search either for a l Ž0 n which gives a large un Ž0 3, while maintaining the degeneracy Dm Ž0 s0; or a l Ž0 which gives a ( E large ue3 Ž0, while maintaining the charged lepton mass hierarchy m Ž0 s0 4 There are few solutions e,m to this puzzle, and therefore few zeroth-order textures 3 In searching for textures for ln, Ž0 E, we allow two types of relations between otherwise independent, non-zero matrix elements Ø Any two entries of a matrix may be set equal This can frequently result from a symmetry, but we do not require that a symmetry origin can be found Ø The determinant of each matrix, or any of its = sub-determinants, can be set to zero We do not allow other precise relations The determinantal conditions arise naturally when heavy states are integrated out, as in the seesaw and Froggatt- Nielsen mechanisms This may be due to zeroth order textures of the light-heavy couplings for any Ž non singular mass matrix of the heavy states, or to a mass hierarchy of the heavy states themselves Suppose that a heavy charged lepton couples to combinations Ý ae and Ý be of left- and i i Li i i Ri right-handed charged lepton flavor eigenstates On integrating out the heavy state, l Ž0 Aab and has Eij i j only a single non-zero eigenvalue Integrating out each heavy state leads to a single eigenvalue in the Our classification of textures does not assume any flavor symmetry However, perturbation series of the type Ž 1 and Ž could result from a flavor symmetry which defines the flavor basis, allows the terms l Ž0 E,n, and leads to higher order corrections as a power series in the small flavor symmetry breaking parameters e 3 Suppose a zeroth-order texture gives u13 Ž0 f1 Since perturbations may give ue1 f1, it is conceivable that on computing V one finds a precise cancellation, so that u13 s0 In general such cases are fine tuned, and we do not consider them further For the textures which do give u13 Ž0 s0, it is important that the perturbations to le do not induce ue1f1, since this will in general lead to u13 f1 4 It has been noted wx 4 that the atmospheric data does not exclude a conventional interpretation, with small angles following from mass hierarchies, because the relevant mass hierarchies may not be large With u Ž0 E,n 3s0, at order e one typically finds < u < 1r < < n 3 s m( rmatm and ue3 s mmrm t 1r For Dm( s 10 y5 ev and Dm atm s10 y3 ev, and opposite signs for un 3 and u E3, one finds sin u 3s08, at the edge of the Super- Kamiokande 90% CL allowed region It is very interesting that this case is not excluded; but it is not favored, and we study the more revolutionary case that at least one of un, Ž0 E3 is non-zero With a suitable see-saw structure < u < n 3 s m( rmatm 1r4 is also possible wx 5

409 ( ) R Barbieri et alrphysics Letters B light matrix If there is a hierarchy amongst the contribution of various heavy states, then there will be a hierarchy of light eigenvalues Only the dominant contributions are included in ln, Ž0 E, which may therefore have zero determinants and sub-determinants 4 We first consider zeroth-order textures for neutrino mass matrices which give a large un Ž0 3 while maintaining the degeneracy Dm( s0 and un Ž0 13s0 There are only three such textures, which we label I, II and III: ž / 0 0 B A Ž0 I ln s B 0 0, Ž0 II ln s 0 B ra B, A B A / ž A 0 0 Ž0 III ln s 0 0 A Ž 8 0 A 0 Texture I leads to a heavy pseudo-dirac neutrino: I: Pseudo-Dirac mn1 Ž0 sm Ž0 n 4m Ž0 n 3 s0, Ž 9 while texture II gives the zeroth-order hierarchical eigenvalue pattern: II: Hierarchical m Ž0 n 3 4m Ž0 n 1 sm Ž0 n s0 Ž 10 The third texture leads to complete degeneracy at zeroth-order: III: Degeneracy mn1 Ž0 sm Ž0 n sm Ž0 n 3 /0 Ž 11 For textures I and II, BsA is an important special case giving un Ž0 3s458 Textures I and II can both be obtained by the seesaw mechanism A simple model for texture II has a single heavy Majorana right-handed neutrino, N, with interactions l,3 NHqMNN, which could be guaranteed, for example, by a Z symmetry with l,3, N odd and l1 even A simple model for texture I has two heavy right-handed neutrinos which form the components of a Dirac state and have the interactions lnhql 1 1,3NHqMN1N These interactions could result, for example, from a UŽ 1 symmetry with N 1,l,3 having charge q1, and N,l1 having charge y1 In both cases, the missing right-handed neutrinos can be heavier, andror have suitably suppressed couplings The search for charged lepton textures is very similar to that for neutrinos, except that the zerothorder eigenvalues must be hierarchical, not pseudo- Dirac or degenerate There is only one zeroth-order charged lepton mass matrix texture which satisfies Ž 3, Ž 4, Ž 6 and ŽŽ 7 with u u E, which we label IV / ž 0 0 C Ž0 IV le s 0 0 B Ž A which is the charged analogue of the hierarchical neutrino texture II We have rotated the right-handed charged leptons so that the entries of the first two columns vanish As in Ž 8, we take A and B to be non-zero and comparable, since they both occur at zeroth order It is important to realize that texture IV can give the mixing matrix V either in the form R Ž u R Ž u or R Ž u R Žu X We require that the perturbations which generate mm uniquely select the first form Texture IV has several interesting special cases: Cs0; Cs0 and AsB, giving ue3 Ž0 s458; and Ž0 y1 A s B s C giving u s u s tan Ž 1r' E3 dem, 358 In the last case, often considered in the literature wx 6, the texture, in an alternative basis for the righthanded leptons, takes the democratic form: / ž A A A Ž0dem le s A A A Ž 13 A A A This gives u13 Ž0 s0 only when ue1 Ž0 s458, corresponding to the perturbations identifying the muon and electron states as Ž 1,1,y and Ž 1,y 1,0, respectively Note that a democratic texture is not allowed for the neutrino mass matrix, since it gives Ž0 Ž n13 dem u su,358, and clearly violates 7 Texture II has been obtained by the seesaw mechanism by adding an extra singlet neutrino, or by adding R parity violating interactions in supersym- metric theories wx 7 The possibility of complete dewx 8 A generacy via texture III has also been noted search for theories giving the observed atmospheric and solar neutrino fluxes with three light neutrinos, but with more stringent assumptions than those adopted here, found both textures I and II wx 9 It is remarkable that all theories, which explain both atmospheric and solar neutrino fluxes with three light

410 410 ( ) R Barbieri et alrphysics Letters B neutrinos, must yield one of these zeroth-order textures Any exception must involve more complicated relations between matrix elements for example, with entries differing by a multiple or a sign w10 x Texture III is unique: it is the only case where the zeroth-order texture does not provide D m Ž atm for neutrinos or m Ž for charged leptons Apparently t this texture does not guarantee a large value for u : 3 at zeroth-order the three neutrinos are all degenerate, so that degenerate perturbation theory may lead to a large value for u 3 Ž1, perhaps cancelling u 3 Ž0 This is not correct a small perturbation to the Dirac mass yields a pseudo-dirac neutrino with a mixing angle Ž0 close to 458 texture III yields the result u 3 s 458, and this receives only small corrections from higher order This is to be compared with the Super- Kamiokande data: u 3 s458"138 at 90% CL 5 In listing textures I IV, we have not discussed the form of the lepton mass matrix which is not explicitly displayed A neutrino texture from Ž 8 could be paired with a diagonal charged lepton mass matrix, or with texture IV It is straightforward to find symmetries which yield textures I and II; an example of Le ylmylt was given in Ref wx 9 Any symmetry yielding these textures at zeroth order does not distinguish lm from l t, and hence requires both of these textures to be paired with the charged lepton mass matrix of texture IV We find two zeroth-order schemes for lepton masses: Scheme A l Ž0 I [l Ž0 IV Ž 14a n E Scheme B l Ž0 II [l Ž0 IV Ž 14b n E For these schemes the special case AsB, in both charged and neutral matrices, gives u 3 Ž0 s0, and is not allowed Even if A s B in just one of the matrices, neither scheme can claim to predict u 3 near 458, as suggested by the Super-Kamiokande data Theories with textures III arising from flavor symmetries will be discussed in a future paper The democratic texture Ž 13 follows from the symmetry group S = S wx 6 With l Ž l 3 L 3 R L R transforming as Ž0 1 [ 1 [ of S S, l saž 1 1 L L R R 3 L 3 R E L R, which is the democratic form of Ž 13 However, a scheme for both charged and neutral lepton masses does not result at zeroth order The neutrino mass matrix Ž0 involves two invariants: l s BŽ 11 q CŽ n L L L L, and is diagonalized by the rotation matrix Vn Ž0 sve Ž0, giving V Ž0 si Even if B and C are fine-tuned, or a suitably extended symmetry is spontaneously broken w x Ž0 in a special direction 11, so that ln AI, the lepton mixing angles are then all determined by the higher order perturbations The only zeroth-order schemes for lepton masses which we can justify from symmetries are A and B In scheme A the solar neutrino oscillations are large angle, while in B the angle may be large or small, and is entirely determined by the perturbations Neutrino hot dark matter and a bb0n signal are possible only in texture III, and hence do not occur in either scheme Models for the zeroth-order schemes A and B, in which the neutrino masses arise from the seesaw mechanism, are easy to construct, and can also be extended to SUŽ 5 unification Scheme B occurs in theories with a single heavy Majorana right-handed neutrino, N, and any flavor symmetry with N,l, e, H transforming trivially, but l, e,3 3 u,d 1 1, transforming non-trivially such that le 1 1,Hd is forbidden An example is provided by a UŽ 1 symmetry with l 1, e1, positively charged This can be extended to a theory with three right-handed neutrinos by introducing N1 and N, with equal and opposite charges, such that the only allowed renormalizable operator is the mass term M X NN 1 Scheme A occurs in a theory which has two heavy right-handed neutrinos, which form the components of a Dirac state, and the interactions lnh 1 1 u ql,3 N Hu qmn1nql,3e3h d For example, this follows from a UŽ 1 flavor symmetry with N,l 1,3 having charge q1, N,l,e, H having charge y1, 1 1, d H and e being neutral This can be extended to a u 3 theory with three right-handed neutrinos by introducing the neutral state N 3 In our example UŽ 1 theories, for both schemes A and B, in the trivial extensions to SUŽ 5 where each field is replaced by its parent SUŽ 5 multiplet Že 3 T etc 3, the only additional invariant interaction is TT 3 3H, u yielding the top quark Yukawa coupling at zeroth order 6 We have studied textures for lepton mass matrices with three light neutrinos, which at zeroth-order give a large mixing angle for the oscillation nm n t,

411 ( ) R Barbieri et alrphysics Letters B and a vanishing mixing angle for nm n e We allow textures with zero determinants and sub-determinants, and we allow independent entries to be set equal In the case of the neutrino mass matrix, requiring two degenerate neutrinos to allow Dm ( < Dm, there are just three such zeroth-order texatm tures; while in the charged case, requiring a zerothorder mass for mt but not for mm or m e, there is a single possibility For a flavor symmetry to yield both charged and neutral textures with these zerothorder properties, only two schemes are possible, both of which can be obtained in simple seesaw models Many other theories of neutrino masses exist which can account for both atmospheric and solar neutrino fluxes They involve more than three light neutrinos, textures with entries related in some special way, or theories where m andror u are t 3 determined only at higher order However, the zeroth-order textures and schemes which we have found appear particularly simple Acknowledgements We thank Graham Ross and Andrea Romanino for useful discussions This work was supported in part by the US Department of Energy under Contracts DE-AC03-76SF00098, in part by the National Science Foundation under grant PHY References wx 1 Study of the atmospheric neutrino flux in the multi-gev energy range, The Super-Kamiokande Collaboration, hepexr , May 1998 wx Evidence for oscillation of atmospheric neutrinos, The Super-Kamiokande Collaboration, hep-exr , July 1998 wx 3 V Barger, TJ Weiler, K Whisnant, hep-phr ; GL Fogli, E Lisi, A Marrone, G Scioscia, hep-phr wx 4 GK Leontaris, S Lola, G Ross, Nucl Phys B 454 Ž , hep-phr950540; KS Babu, JC Pati, F Wilczek Link between Neutrino Masses, Super-Kamiokande Result and Proton Decay, to appear wx 5 M Fukugita, M Tanimoto, T Yanagida, Prog Theor Phys 89 Ž wx 6 H Harari, H Haut, J Weyers, Phys Lett 78B Ž ; Y Koide, Phys Rev D 8 Ž ; 39 Ž ; H Fritzsch, Z Xing Phys Lett B37 Ž , hepphr ; M Fukugita, M Tanimoto, T Yanagida, Phys Rev D57 Ž , hep-phr ; K Kang, S Kang, hep-phr98038; M Tanimoto, hep-phr wx 7 Z Berezhiani, A Rossi, Phys Lett B367 Ž , hepphr ; M Drees, S Pakvasa, X Tata, T ter Veldhuis, Phys Rev D57 Ž , hep-phr97139; S King, hep-phr wx 8 GK Leontaris, S Lola, C Scheich, JD Vergados, Phys Rev D53 Ž , hep-phr ; P Binetruy et al, Nucl Phys B496 Ž , hep-phr wx 9 R Barbieri, L Hall, D Smith, A Strumia, N Weiner, to appear, hep-phr w10x F Vissani, hep-phr ; H Georgi, S Glashow, hepphr w11x R Mohapatra, S Nussinov, hep-phr ; G Ross, private communication

412 7 January 1999 Physics Letters B Just so neutrino oscillations are back Sheldon L Glashow a, Peter J Kernan b, Lawrence M Krauss b,c a Lyman Laboratory for Physics, HarÕard UniÕersity, Cambridge, MA 0138, USA b Department of Physics, Case Western ReserÕe UniÕersity, CleÕeland, OH , USA c Department of Astronomy, Case Western ReserÕe UniÕersity, CleÕeland, OH , USA Received 11 June 1998 Editor: H Georgi Abstract Recent evidence for oscillations of atmospheric neutrinos at Super-Kamiokande suggest, in the simplest see-saw interpretation, neutrino masses such that just so vacuum oscillations can explain the solar neutrino deficit Super-K solar neutrino data provide preliminary support for this interpretation We describe how the just-so signal an energy dependent seasonal variation of the event rate, might be detected within the coming years and provide general arguments constraining the sign of the variation The expected variation at radiochemical detectors may be below present sensitivity, but a significant modulation in the 7 Be signal could shed light on the physics of the solar core including a direct measure of the solar core temperature q 1999 Elsevier Science BV All rights reserved A decade ago we argued wx 1 that seasonal modulation of the solar-neutrino signal seen in various detectors could be a unique signature of just-so vacuum neutrino oscillations wx, with the typical solar-neutrino oscillation length comparable to the Earth-Sun distance The resulting oscillations could provide the large suppression needed to explain the 37 Cl data w3,1 x Since then, much attention has been paid to matter-enhanced Ž MSW oscillations, for which large suppressions of the signal are obtained for a broad range of masses, and with which virtually any combination of suppressed signals at different detectors is explicable In the intervening decade, solar neutrinos have been studied at many facilities, yet a definite resolution to the puzzle has not been identified We argue that the just-so scenario, although it requires finely tuned masses, is both more easily testable and more readily falsifiable The Super-Kamiokande Collaboration has anwx 4 Evidence for vac- nounced a remarkable result uum oscillations between muon neutrinos and another species Ž not n e was reported at a highly significant level and with a nearly maximum mixing angle The required squared-mass difference is < < Dm ' m1ym,0005 ev With the plausible hypothesis that the oscillation involves nm and n t, and that the mass difference is dominated by the tau neutrino mass, this suggests m f07 ev n t There are several reasons to reconsider just-so solar neutrino oscillations in light of this result While we have no direct empirical information on the neutrino mass spectrum, an elegant yet simple origin for the Majorana masses of light neutrinos is wx the so-called see-saw mechanism 5, which yields: m fm rm, Ž 1 n i q i,li X r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S X

413 ( ) SL Glashow et alrphysics Letters B where mq,l are the masses of the associated up-type i i quark or charged lepton in family i, and MX is a heavy mass scale associated with the symmetry breaking responsible for neutrino masses Taking the quark masses to be relevant, we expect mž n m f07 = Ž 15r175 f7=10 y6 ev, and a much smaller mž n e These neutrino masses are just right to pro- vide a just-so explanation of the solar-neutrino flux Given our lack of understanding the origin of even the charged lepton masses, one might find the above reasoning suspect It has been argued, for example, that maximal vacuum oscillations with a much smaller wavelength, resulting in a uniform halving of the observed neutrino signal, might resolve the solar neutrino problem, especially if there 37 is still some systematic error in the Cl data wx 6 On the other hand and rather remarkably, the Super-K collaboration reports tentative evidence, based on their measurements of the shape of the recoil electron spectrum, for just-so oscillations with masses provisionally in the range 6FDm F60=10 y11 ev wx 7 If the just-so solution is correct, must we wait for confirmation from observations of neutral-current events at SNO or for unambiguous results from Borexino? We argue that this may not be necessary because of the unique signature of vacuum oscillations: an energy and time dependent variation of the signal as a function of the distance to the source In our analysis, we re-examine the seasonal variation of both the overall flux Ž ie, integrated over energy and the recoil electron spectrum for the mass regions now of interest ŽNote: before the most recent Super- K result several other groups explored various just-so signatures Žie w8 10 x Of course, all explanations of solar neutrino deficit are subject to a small seasonal variation of the overall event rate due to the eccentricity of Earth s orbit Additional effects characteristic of just-so solutions are energy dependent and can average out when the signal is averaged over energies Thus, the rate measurements at Super-K and at the radiochemical detectors, which have failed to detect seasonal variations, cannot exclude just-so oscillations Depending on the neutrino mass we shall show that just-so oscillations can lead to seasonal variations in the spectrum of recoil energies wx 1 These variations can be radically energy dependent and are potentially detectable at Super-K In cases where the energy dependence of the seasonal variation is weak, the integrated event rate can vary seasonally over 5% in addition to its expected variation due to the earth s eccentricity This effect should soon be detectable The sign of the variation may be telling as well Lower masses lead to uniformly larger seasonal variations than expected in the absence of oscillations, while larger masses can result in variations which cancel out the variation due to the earth s eccentricity Žalthough, as we derive below, this possibility is disfavored by the Super-K spectral information Our neutrino mass model, motivated by Super-K data, involves three left-handed mass eigenstates Mixing between the flavor eigenstates is governed by three angles u1 u, and u 3 We assume the heaviest mass eigenstate, that playing a role in atmospheric neutrino oscillations, to have a mass of 07 ev Oscillations between a solar electron neutrino and this state, if present, will undergo many oscillations on their way to the detector sun and average out to yield an essentially energy-independent suppression Oscillations between electron and muon neutrinos, however, are assumed to be just-so Thus we take m 1;0, m ;10 y5 ev, m 3;01eV Adopting the conventional KM parameterization, we write the flavor eigenstates as: n 'c c n qc s n qs e yi d n, e n 'y csqs sce id n m q ccys sse id n qs cn from which we obtain: P Ž n n e e solar sin u s1y ycos 4 u sin u sin m Rr4E 3 For atmospheric oscillations, for which m small to play much of a role, we obtain: P Ž nm atmospheric s1y4sin u1cos u sin m 3Rr4E ½ = sin u Ž ne cos u cos u Ž n 1 t is too

414 414 ( ) SL Glashow et alrphysics Letters B Note that the CP-violating phase d plays no role Super-K Ž and CHOOZ suggest that nm ne is small for atmospheric neutrinos In the limit u s 0: P solars1ysin u 3sin mrr4e, P atmospheric s1ysin u 1 sin m3rr4e We use the general form here, but take sin u -01 to assure that atmospheric muon neutrinos oscillate mostly into n t We can readily determine the resulting energy-dependent seasonal suppression of the neutrino signal at Super-K as a function of neutrino mass Our numerical routine uses the B neutrino spectrum from Bahcall and Pinnsoneault w11x and standard neutrino-electron scattering cross sections We assume a sharp detector threshold of 65 MeV and integrate over the Super-K electron energy resolution function Predicted rates are multiplied by a normalization factor of ; 09 to account for detector efficiency, deadtime, etc, thereby ensuring that the observed rate of 135 eventsrday corresponds, as rew11 x Preliminary analyses of the Super-K spectrum, ported, to ;47% of its SSM value and the rates at other solar neutrino experiments, lead to a variety of estimates for parameters that best explain the data w1,13 x: Spectral measurements by Super-K suggest assuming u s 0 Dm, 4 = y10 10 ev w1 x A global analysis of event rates at all neutrino detectors, combined with the Super-K spectrum, suggests again with u s0 Dm,7= y11 10 ev w13 x Our analysis spanned both these estimates as well as the smaller value suggested by the see-saw mechanism: FDm F60=10 y11 ev, with mixing angles chosen to fit the observed rate of 135 eventsrday within uncertainties The predicted seasonal dependence of the integrated Super-K signal as a function of Dm can be understood as follows: For small mass differences the oscillation length far exceeds the Earth-Sun distance As Dm increases, oscillations kick in and the survival probability of an electron neutrino decreases monotonically, so that as the Earth-Sun distance Fig 1 Upper: Seasonal variation predicted for the solar neutrino signal in Super-K for maximal just-so oscillations Žno oscillations dashed curve versus neutrino Dm The number of events within 90 days of perihelion is divided by the number of events more than 90 days removed from perihelion Lower: Overall event rate vs Dm

415 ( ) SL Glashow et alrphysics Letters B increases from perihelion to aphelion the ne flux at the detector decreases via the inverse-square law and through the effect of just-so oscillations For F Dm F6=10 y11 ev, the seasonal variation of the signal significantly exceeds its no-oscillation value For Dm,6=10 y11 ev, the mean oscillation probability at Earth is largest and the electron neutrino flux reaches a minimum For slightly larger values, oscillations tend to increase the ne flux as the Earth-Sun distance increases For this case the just-so and inverse-square effects oppose one another and the overall event rate can even increase As Dm is further increased, the seasonal effect oscillates in sign At the same time, the seasonal effect becomes energy dependent so that the signals in different energy bins at Super-K would differ dramatically from one another For even larger Dm, the just-so signal washes out due to averaging over the different neutrino energies: the seasonal variation becomes conventional and energy independent We have explored these effects quantitatively using several different numerical analyses To maximize statistics and sensitivity to seasonal variation, the relevant quantity to consider is not the day of the year, but rather the Earth-Sun distance, which is probed twice over the course of the year Seasonal variation should be plotted as a function of days since perihelion Fig 1 shows the ratio R of the predicted number of Super-K events within 90 days of perihelion to those in the rest of the year, versus Dm and with maximal oscillations The lower part of the figure displays the predicted overall event rate at Super-K versus Dm The effects outlined earlier are evident For small Dm the seasonal variation is enhanced compared to the no-oscillation case For larger values, the seasonal variation oscillates in and out of phase with the 1rr variation Moreover, the shift in the sign of the seasonal variation Žwhere the curve crosses the no-oscillation point occurs precisely at Fig Predicted seasonal variation for the solar neutrino signal in Super-K as a result of just-so oscillations, for Dm s5=10 y11 ev, and mixing angle chosen to reproduce the Super-K observed event rate Ž within quoted uncertainties On the left is shown the variation of the number of eventsrdayrmev predicted at various electron energies On the right is the variation of the total event rate Ž eventsrdayždashed curve s no-oscillations

416 416 ( ) SL Glashow et alrphysics Letters B the maxima and minima of the event rate curve, validating the picture discussed above Figs 4 show the time variation of the differential and integrated event rates at Super-K for three different values of Dm At the left, we display the temporal variation of the predicted neutrino signal at various energies between 6 and 14 MeV, with oscillation parameters as specified At the right, for the same parameters, we display the temporal variation of the overall event rates We also indicate the expected variation in the signal in the absence of oscillations The results validate our a priori reasoning and provide some new insights For smaller values of Dm, as in Fig, the integrated and differential seasonal variations are similar as expected because none of the detected neutrinos have completed one full oscillation In this case, the overall seasonal variation can exceed its no-oscillation value by an additional 5 6% For intermediate values of Dm, Fig 1 shows the just-so effect canceling the seasonal variation and the event rate increasing with Dm, and hence decreasing with neutrino energy This behavior is evident in Fig 3, where there is virtually no seasonal variation and the event rate drops precipitously at high energies However, observations at Super-K suggest an upturn in the neutrino signal at high energies Thus, these values of Dm are disfavored The just-so oscillations indicated by Super-K data suggest that the seasonal Õariation of the oõerall eõent rate is in phase ( or at least, not out of phase) with the inõerse square Õariation Žsee also w10 x At the higher end of the mass range, as in Fig 4, the difference in seasonal modulation at different energies is noticeable This is true even when the integrated event rate variation does not differ significantly from that expected for no oscillations This unique and unambiguous signature of just-so oscillations means that measurements of the seasonal variation of the differential spectrum can be of great utility when combined those of the overall event rate Note that if the Super-K energy resolution Žs f 058' E MeV could be sharpened, the above effect would be dramatically enhanced! Note Žie see wx 1 that the seasonal variation in both the Cl and Ga signals can be less than 10% for just-so oscillations because of the integration over Fig 3 Same as Fig, for Dm s1=10 y10 ev, unfavored as described in the text

417 ( ) SL Glashow et alrphysics Letters B Fig 4 Same as Fig, for Dm s5=10 y10 ev energies inherent in chemical detectors Žthis effect is most striking for the Cl detector While the Ga signal is expected to have a larger seasonal variation than Cl, it may be small enough to have been missed Finally, a larger annual variation is expected in a detector such as Borexino, which is strongly sensitive to the 086 MeV 7 Be neutrino line Seasonal variations as large as 40% can be possible at this energy w1,9 x Perhaps even more exciting is the fortuw14x that the phase of time-dependent itous result oscillations in the Be neutrino signal, combined with the amplitude of the oscillations, can give a direct measurement of the central solar core temperature While the possibility of utilizing oscillations for this purpose exists only over a limited range of masses and mixing angles in the just-so range, it would be remarkable if nature allowed such a measurement The just-so solution of the solar neutrino problem regained credibility as a result of recent Super-K data Furthermore, the technique that Super-K used to provide its strongest evidence for atmospheric neutrino oscillations a time and energy dependent modulation of the neutrino signal may be exploited to resolve one of the longest standing puzzles in particle astrophysics This work was supported in part by the DOE and the National Science Foundation under grant number NSF-PHYr LMK thanks the Aspen Center for Physics, and A Smirnov for useful comments References wx 1 SL Glashow, LM Krauss, Phys Lett B 190 Ž wx SM Bilenky, B Pontecorvo, Phys Rep 41 Ž wx 3 V Barger, K Whisnant, RJN Phillips, Phys Rev D 4 Ž wx 4 T Kajita, to appear in the Proceedings of the XVIII International Conference on Neutrino Physics and Astrophysics, Takayama, Japan, June 1998; Super-Kamiokande Collaboration, hep-exr wx 5 M Gell-Mann, P Ramond, D Slansky, in: P van Nieuwenhuizen, D Freedman Ž Eds, Supergravity, North-Holland, Amsterdam, 1979 wx 6 See JN Bahcall PI Krastev, AYu Smirnov, hepphr980716, for a criticism of this possibility

418 418 ( ) SL Glashow et alrphysics Letters B wx 7 Y Suzuki, to appear in the Proceedings of the XVIII International Conference on Neutrino Physics and Astrophysics, Takayama, Japan, June 1998 wx 8 PI Krastev, S Petcov, Nucl Phys B 449 Ž wx 9 B Faid et al, Phys Rev D 55 Ž ; also hepphr w10x A correlation between spectral shape and seasonal variation was analyzed for earlier preliminary Super-K data, by SP Mikheyev, AYu Smirnov, Phys Lett B 49 Ž w11x JN Bahcall, S Basu, MH Pinnsoneault, astro-phr , Phys Lett B, in press w1x Y Suzuki, in: WEIN 98, Proceedings, to appear; also Y Suzuki in: Neutrino 98, Proceedings, to appear in Nucl Phys B Ž Proc Suppl ; see also Y Fukuda et al, Phys Rev Lett, in press w13x JN Bahcall, PI Krastev, AYu Smirnov, hep-phr980716, 1998 w14x LM Krauss, F Wilczek, Phys Rev Lett 55 Ž

419 7 January 1999 Physics Letters B A measurement of the proton-antiproton ( total cross section at s s18 TeV E-811 Collaboration C Avila a,b, WF Baker c, R DeSalvo d,1, DP Eartly c, C Guss a,, H Jostlein c, MR Mondardini a,3, J Orear a, SM Pruss c, R Rubinstein c, S Shukla c,4, F Turkot c a Cornell UniÕersity, Ithaca, NY 14853, USA b UniÕersidad de Los Andes, Bogota, Colombia c Fermi National Accelerator Laboratory, BataÕia, IL 60510, USA d CERN, GeneÕa, Switzerland Received 1 November 1998 Editor: L Montanet Abstract We report a measurement of the pp total cross section at s s18 TeV at the Fermilab Tevatron Collider, using the luminosity independent method Our result is sts7171" 0 mb We also obtained values of the total elastic and total inelastic cross sections q 1999 Published by Elsevier Science BV All rights reserved ' ' At energies above s ; 0 GeV, the pp total cross section st increases with increasing energy w1 8x Žsimilar to all other measured hadron-hadron total cross sections Accurate measurements of s T Žtogether with those of r, the ratio of the real to imaginary part of the forward scattering amplitude at the highest available energy should give information on how fast this increase is with energy, and 1 Present address: California Institute of Technology, Pasadena, California 9115 Present address: 11 Pine Ave, Apt 800, Montreal PQ, H3G 1A9, Canada 3 Present address: INFN, Univ di Roma I, Roma, Italy 4 Present address: Sonat Inc, Birmingham, Alabama 350 whether or not the rate of increase is levelling off so that st may approach a constant value at even higher energies So far, there have been two previous measurements of st at the Fermilab Tevatron Collider at the currently highest available energy of ' s s18 TeV, differing by some standard deviawx 9 78"31 mb and CDF tions; E-710 has reported has reported w10x 8003"4 mb The aim of the present experiment, Fermilab E-811, is to measure this quantity to higher accuracy than previously; it has many features in common with our earlier E-710 experiment The experiment was also designed to measure r; that analysis is still in progress We use the luminosity independent method to obtain s, as in Refs w7,9 x Using the optical theo- T r99r$ - see front matter q 1999 Published by Elsevier Science BV All rights reserved PII: S X

420 40 ( ) C AÕila et alrphysics Letters B rem, the total cross section can be expressed in terms of the forward differential number of nuclear elastic events el 1 16p "c dn st s Ž 1 L 1qr dt ts0 where L is the integrated luminosity Another expression for the total cross section is 1 st s Ž NelqN inel, L where Nel and Ninel are the total elastic and total inelastic number of events respectively By combining Eqs Ž 1 and Ž, we obtain el 16p "c dn 1 st s Ž 3 dt N qn 1qr el inel ts0 In Eq Ž 3, st is determined independently of luminosity, which often has a large uncertainty We measure Nel and Ninel simultaneously; the previously measured nuclear elastic distribution dnel dnel s expž yb< t< dt dt ts0 with Bs1698"0, the mean from Refs w9,11 x, is used to extrapolate our elastic data to ts0 ŽOur data are consistent with this distribution, but with a larger uncertainty in the value of B because of the limited t range of this experiment Nel is obtained by integrating the fit to our elastic data from ts0 to tsy`, using the above value of B The elastic detectors in this experiment were placed at the same Tevatron locations as those used for the lowest < t< measurements of E-710 w9,1,13 x, and the accelerator magnetic lattice in the region of the experiment was unchanged between the two experiments However, to detect the elastic events, this experiment used scintillating fiber high resolution detectors inside the Tevatron beam vacuum pipe Each detector was a bundle of 15,000 scintillating fibers each of 100m diameter, 45 cm long parallel to the Tevatron circulating proton and antiproton beams, and the detectors were moved until the closest fibers were only ; mm from the center of the beams The light from the fibers was transmitted by a glass fiber rod to two planar focused intensifiers in cascade, followed by a specially designed CCD, which was read out following a trigger from scintillation counters in line with the scintillating fiber bundles The CCD signals were digitized and compressed using VDAS Ž Videao Data Acquisition System modules w14 x The particle resolution obtained in the detectors was measured to be "38 mm Measured detector efficiencies were always greater than 097, as were the measured trigger scintillator efficiencies Descriptions of the detectors are given in Refs w15,16 x There were two detectors at each location, one above the circulating beam and one below Elastic events required a particle in the upper detector on the scattered proton side in coincidence with one in the lower detector on the scattered antiproton side, or vice versa; the timing of the particles was required to be consistent with them coming from a collision at the interaction point In the analysis stage, collinearity of the two scattered particles in both the horizontal and vertical planes was required For the results reported here, data were analyzed over the range Q< t< Q 0036 Ž GeVrc ; backgrounds varied from 0% in the bin closest to the beam to 1% at the farthest bin The background, mainly due to a beam halo particle in the antiproton detector in the accidental coincidence with a beam halo particle in the proton detector, was experimentally determined by 3 different methods, which gave consistent results Two of the methods obtain the background from the shape of the distributions of non-collinear events in a pair of detectors, while the third used events from upperupper or lower-lower detector pair combinations In order to obtain the nuclear elastic scattering, we subtract from the observed elastic distribution the distributions from Coulomb and Coulomb-nuclear interference At our lowest < t< bin, this was a 6% correction and rapidly became smaller for larger < t< bins We used a value for r of 0145 w17x in this correction The inelastic rate, N inel, was obtained using an annular scintillation counter system and method essentially identical to that of E-710 Žsee Ref w13x and Fig 1 Coincidences Ž LR of any one of 3 sets of annular counters on the left Ž L and any one of 3 sets of counters on the right Ž R of the interaction point, covering 5-< h< - 65, were made The time of flight registered by the counters was required to be

421 ( ) C AÕila et alrphysics Letters B Fig 1 Ž a Results for s from this experiment and from previous w1 5,7,8x experiments; Ž T b Results for sel from this experiment and from previous experiments w4,5,7,9,0,1 x consistent with particles from the interaction point, and pulse heights were required to be at least as large as those from minimum ionizing particles Background Ž ; 1% due to accidental coincidences was subtracted We use the same method as that in E-710 w13 x, which is similar in principle to that of UA4 wx 7, to obtain those inelastic events not observed by the LR coincidences; the method has few assumptions, and almost all quantities are measured in this experiment To the LR events wž 13503"64 =10 x we added the single arm L and R events Ž03"004 of the LR rate, and a small extrapolation Ž0011"0006 of the LR rate to account for those single arm events not observed because of the limited h coverage of the counters; this gives Ninel s " 57 = 10 events From our measured 8,000 elastic events, we dn obtain N sž 5081" 35 = 10 events and el < el dt ts0 s 868" 603 = 10 Ž GeVrc y Our final result, obtained by averaging the values from our 10 runs, and assuming a r value of 0145 w17 x, is s s7171"0 mb T where the error is predominantly statistical We also obtain sel s 1579" 087 mb and sinels 559" 119 mb, and selrst s00"00078 As in E-710, the single arm data used above for obtaining N has a significant Ž ; 93% inel back- ground caused by, for example, beam-gas interactions It was determined by using data taken with some of the antiproton and proton bunches missing, which allowed a statistically accurate determination of the background rate without colliding antiprotonproton events As a check of our result, we also obtain consistent results by two alternate methods, given below, which do not use this technique Ž i We can estimate the LR rate that we would have if we had 4p coverage by studying in the analysis stage the addition to our LR rate of other annular counters Ž 39-< h< - 5 and an extrapolation to h regions not covered The Ž 39-< h< -5 region adds 01005"00016 of the LR rate, and the extrapolation adds 001" 0014 of the LR rate Note that both of these additions are small because the high multiplicity of inelastic events leads to about 90% of all LR events with at least one particle

422 4 ( ) C AÕila et alrphysics Letters B in the range 5-< h< - 65 on both sides If we make the assumption that the inelastic cross section is composed of only double-arm events and single diffraction Žas in Ref w10 x, and use ' s s 18 TeV world average w13,18,19x single diffraction cross section of 945"04 mb, we obtain st s7141"198 mb Ž ii We can use the CDF Monte Carlo calculation for their LR acceptance w10x for inelastic events in the range 3-< h< -67 We have counters covering a somewhat similar range Ž 39-< h< -65; however, the different beam pipe configurations of the two experiments, leading to different particle distributions due to interactions in the beam pipes, preclude a precise comparison Nevertheless, we use the CDF Monte Carlo calculation to obtain our number of LR events for a 4p coverage, and add the world average single diffraction cross section Žthus with the same assumption as in Ž i above The result is s s73 mb We quote as our final result st s7171"0 mb; this is shown in Fig 1 together with earlier data w1 5,7,8 x Also shown in the figure is s el, together with earlier data w4,5,7,9,0,1 x Our result is in good agreement with that of E-710 wx 9, and differs by about 8 standard deviaw10 x; the confidence level that tions from that of CDF all 3 results are compatible is only 16% Our result, together with the lower energy data, is consistent with a logž s variation of st with energy, rather than a log Ž s variation Also, using the approximate asymptotic dispersion relation wx p ds T rf Ž 4 s dž log s T our value of s T, together with lower energy data, gives a r value consistent with those measured w9,3x in this energy range In summary, we have made a new measurement of the proton-antiproton total cross section at ' s s 18 TeV, with higher precision than previously available Our result, st s 7171" 0 mb, together with lower energy data, is consistent with a logž s variation of s T T Acknowledgements This work was supported by the US Department of Energy and the US National Science Foundation We are grateful for support and initial detector development by the Small Angle Subgroup of the CERN LAA Lab, especially C Davia, M Lundin and A Zichichi We received invaluable aid from the Fermilab Accelerator, Computing, and Research Divisions, and the Physics Section We thank A Baumbaugh and K Knickerbocker for the design, construction, and testing of the fiber detector readout system, and also N Amos, C McClure, and N Gelfand for their interest and assistance References wx 1 W Galbraith, Phys Rev 138 Ž 1965 B913 wx SP Denisov et al, Phys Lett B 36 Ž ; Nucl Phys B 65 Ž wx 3 AS Carroll et al, Phys Lett B 61 Ž ; B 80 Ž wx 4 NA Amos, Nucl Phys B 6 Ž wx 5 M Ambrosio, Phys Lett B 115 Ž wx 6 G Arnison, Phys Lett B 18 Ž wx 7 M Bozzo, Phys Lett B 147 Ž wx 8 GJ Alner, Z Phys C 3 Ž wx 9 NA Amos, Phys Rev Lett 68 Ž w10x F Abe, Phys Rev D 50 Ž w11x F Abe, Phys Rev D 50 Ž w1x NA Amos, Phys Rev Lett 61 Ž w13x NA Amos, Phys Lett B 43 Ž w14x A Baumbaugh et al, Fermilab Report FN 439, 1986, unpublished w15x C Avila, Nucl Instr Meth A 360 Ž w16x C Avila, PhD thesis, Cornell University, May 1997, unpublished w17x MM Block, RN Cahn, Rev Mod Phys 57 Ž ; Phys Lett B 188 Ž ; see also MM Block et al, Phys Rev D 41 Ž ; D 45 Ž w18x NA Amos, Phys Lett B 301 Ž w19x F Abe, Phys Rev D 50 Ž w0x DS Ayres, Phys Rev D 15 Ž w1x F Abe, Phys Rev D 50 Ž wx MM Block, K Kang, AR White, Int Journ Mod Phys A 7 Ž w3x C Augier, Phys Lett B 316 Ž

423 7 January 1999 Physics Letters B Study of the ABC enhancement in the d d a X 0 reaction R Wurzinger a,b, O Bing c, M Boivin b, P Courtat a, G Faldt d, R Gacougnolle a, A Gardestig d, F Hibou c, Y Le Bornec a, JM Martin a, F Plouin b,e, B Tatischeff a, C Wilkin f,1, N Willis a, J Yonnet a,b, A Zghiche c a Institut de Physique Nucleaire, INP3rUniÕersite Paris-Sud, F Orsay Cedex, France b Laboratoire National Saturne, F Gif-sur-YÕette Cedex, France c Institut de Recherches Subatomiques, INP3-CNRS r UniÕersite Louis Pasteur, BP8, F Strasbourg Cedex, France d Nuclear Physics DiÕision, Uppsala UniÕersity, Box 535, Uppsala, Sweden e LPNHE, Ecole Polytechnique, F-9118 Palaiseau, France f UniÕersity College London, London WC1E 6BT, UK Received 19 October 1998 Editor: L Montanet Abstract The d d a X 0 reaction at beam energies close to the h threshold shows very strong structure in the missing mass corresponding to the ABC enhancement The deuteron tensor analysing power A yy, and the slope of the vector analysing power A y with respect to angle, have been measured for this reaction around the forward direction Both signals are small, and their variations with the a-particle momentum are in broad agreement with a theoretical model in which each pair of nucleons in the projectile and target deuterons undergoes pion production through the NN dp reaction q 1999 Elsevier Science BV All rights reserved PACS: 545-z; 470qs; 540Qa Keywords: ABC effect; Analysing powers; Two-pion production The discovery of a sharp enhancement in the missing mass spectrum of the pd 3 He X 0 reacwx 1 excited great interest tion almost forty years ago since it seemed to indicate either an enormous pionpion scattering length Žtypically 3 fm wx or a resonance in that system at only 30 MeV above the p threshold The absence of any effect in the parallel 1 cw@hepuclacuk pd 3 H X q case showed that the anomalous behaviour was associated with isospin-zero pion pairs and hence presumably s-wave However, the weight of evidence is that the isoscalar scattering length is small wx 3, so that the ABC anomaly cannot be an intrinsic property of the two-pion system but must be associated with the presence of other particles The characteristics of the ABC phenomenon were independently confirmed in the pd 3 He X 0 reacwx 4 and exactly the same type of effect was tion 0 wx 0 observed in both pn dx 5 and pp pp X at r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

424 44 ( ) R Wurzinger et alrphysics Letters B low pp excitation energies wx 6 The most splendid ABC manifestation is in the dd 4 He X 0 reaction in the 1 GeV region, where it dominates a large fraction of the differential cross section, with sharp peaks associated with its production in the forward and backward cm hemispheres w7,8 x Since the masses and widths of these peaks change somewhat with beam energy and production angle, this reinforces the idea that the ABC might be some kind of kinematic enhancement The simplest model for ABC production in pn dx 0 involves the excitation of both nucleons into D-isobars through one pion exchange where, after the decay of the two D s, the final neutron-proton pair sticks together to produce the observed deuteron w9,10 x A theoretical description actually looks simpler in the case of dd 4 He X 0, where double-pion production can be achieved by two pairs of beam and target nucleons undergoing a pn dp 0 reaction Ž or similarly for charged pion production, as illustrated in Fig 1 w11 x Each of the single-pion-production amplitudes can be driven by virtual D excitation, and this allows the large momentum transfer to be shared evenly amongst all the nucleons If the Fermi momenta in the initial deuterons are neglected, then the dd cm system is also that in the np channel, which means that the produced pions have the same energy The dominantly p-wave nature of the pn dp 0 amplitudes, combined with the form factor coming from the requirement that both deuterons stick to form an a-particle, then leads naturally to enhancements when the two pion momenta are parallel These correspond to the narrow ABC peaks with masses around 310 MeV and widths about 40 MeV The model also predicts an enhancement when the pion momenta are antiparallel, though this is somewhat suppressed by Fig 1 Double NN dp model for the dd 4 He X 0 reaction the much smaller sticking factor when the two recoil deuterons are produced back to back w11 x The resultant broad central structure at the maximum missing mass of the reaction is a notable feature in all cases where the ABC is seen clearly w4,5,7,8 x By taking the dominant pn dp 0 amplitudes into account in the calculation, a good description of the dd 4 He X 0 spectrum and its angular dependence could be obtained w11 x In order to provide extra tests on this and other theoretical models, we report here upon an experiment to measure the deuteron vector and tensor analysing powers in d d 4 He X 0 in a small angular region around the forward direction The experiment was carried out at the SPESIII magnetic spectrometer at the Laboratoire National SATURNE Ž LNS as a by-product of one to measure h production in the d d 4 He h reaction near threshold Data were therefore collected at several energies above the h-threshold at Tds 111 MeV, but also with one background run just below this energy The experimental conditions were identical to those described in ref wx 8 and so only the essential details are reported here The SATURNE accelerator delivered four consecutive pulses of deuterons with different linear combinations of tensor and vector polarisations quantised in the vertical Ž y direction These had maximum values of r0 s 0649" 0011 and r sy0405" 0011 respectively w1 x 10 The wire chambers placed close to the focal plane of the spectrometer measured the production angle of a-particles in the horizontal Ž x plane up to u s "50 mrad about the forward direction, but the only information on the y-coordinate of the track was provided by collimators In most of the runs these subtended angles uys"0 mrad and so the data were integrated over this vertical angular domain In Fig is shown a scatter plot of the momentum per unit charge prz and horizontal production angle ux of identified a-particles arising from the interac- tions of unpolarised deuterons just below the h threshold The vertical bands close to the kinematic limits, and the less intense one in the middle of the plot, are clear indications of the ABC peaks and central bump respectively Equally clear are the signs of the granularity and inefficiencies in the detector system It is important to note that these are geometric in nature and rest unchanged for different polarisation states of the incident beam Summing this x

425 ( ) R Wurzinger et alrphysics Letters B angular integration domain is small, such that A y varies linearly with ux and the differential cross section is essentially constant at its value in the forward direction To first order in Ž u,u x y the form is not disturbed by the non-measurement of y-component of the a-particle production angle in SPESIII This is a reasonable assumption in the region of the Fig Two-dimensional scatter plot of the momentum per unit charge prz and horizontal production angle ux of identified a-particles arising from the unpolarised dd 4 He X 0 reaction just below the h threshold at Td s11167 MeV spectrum over u x, the experimental counting rate for the dd 4 He X 0 reaction is shown in Fig 3a, from which it is easier to identify the ABC peaks and central bump Although the vector analysing power A y has to vanish at ux s0, the experiment shows a clear left- right asymmetry over the horizontal opening angle of "50 mrad which is associated with the state of vector polarisation of the beam In the regions of the ABC peaks, where the statistics are very good, it is possible to fit this directly with the form A ysc u x, allowing us to deduce the slope c of the analysing power near the forward direction However, since the typical A signal is 005 or less, we attempt to y compensate for this and the inefficiencies of the system by defining an average slope through da X y A y s du ž / ž / d s d s shdvux Ay HdV Ž ux P dv dp dv dp Ž 1 The identification of this quantity with the slope of the analysing power at us0 is valid provided the Ž 4 0 Fig 3 a Raw counting rate for the unpolarised dd He X reaction at Td s11167 MeV, integrated over "0 mrad in the vertical and "50 mrad in the horizontal directions, as a function of the a-particle momentum The curve is the unnormalised prediction of the double-d model w11,14 x, integrated over the experimental acceptance The top scale shows the values of the Ž X missing mass M X b Average slope A y of the deuteron vector analysing power of the d d 4 He X 0 reaction defined as in Eq Ž 1 The experimental data are an average over six beam energies with 11167FTdF117 MeV, whereas the solid curve is the prediction of the double-d model w11,14x at Tds11 MeV An arbitrary displacement of the predictions by rad y1 gives the dashed curve, which is a much better representation of the data Ž 4 0 c Deuteron tensor analysing power A yy of the d d He X reaction at Tds11167 MeV, averaged over the aperture of SPESIII The theoretical curve is the prediction of the double-d model w11,14 x

426 46 ( ) R Wurzinger et alrphysics Letters B ABC peaks, where the maximum horizontal and vertical cm angles are about 1758 and 68 respectively However in the central bump, where the cm momenta are quite small, the integration in Eq Ž 1 is over most of the available phase space and this has to be borne in mind in any subsequent interpretation By replacing the integrals in Eq Ž 1 by sums over the experimental events, using the polarised and non-polarised data respectively, the experimental X y value of A is deduced and shown in Fig 3b The data set taken just below the h threshold contained less than 10% of the total available counts and, in order to increase the statistical significance, results at six different energies above and below the h thresh- old Ž 11167FT F117 MeV d were summed in Fig 3b As a consequence the experimental points in the middle of the central bump, 000 F p F 100 MeVrc, include some h-production Only statistical errors in A are shown in Fig 3b The results depend weakly upon a possible off-set in the angular reading of 4 mrad and, apart from a reduction in statistics, are not changed significantly by cutting the range in ux from "50 mrad to "30 mrad The influence of potential systematic effects, arising from detector inefficiencies or angular offset, could be tested by applying the same angular aver- age as in Eq Ž 1 to evaluate the slope of the tensor analysing power da yyrdu, which should vanish in the forward direction Within statistical errors this was found to be consistent with zero for all values of X p with a mean value of ² A : sy005" Ž 010 yy y1 X rad The overall systematic error on A y is hard to quantify but could be rather larger than this due to the smallness of the signal In Fig 3c are shown the values of the tensor analysing power A yy integrated over the whole hori- zontal and vertical aperture of SPESIII Since a slope determination is not required, the statistics from the single run at Tds MeV, ie below the h threshold, are here sufficient In addition to the statistical error bars, there is a systematic uncertainty of only "% arising from the polarisation of the beam w1 x; it should be noted that the value deduced for A yy for h production in this experiment is in agreement with the threshold theorem to this accuwx 8 Detector inefficiencies in the ABC peaks racy could however lead to slightly larger uncertainties In this context it should be noted that at strictly 08 the X y values of A yy for the production of a 0 q state, such as the ABC, should be identical in the forward and backward peaks The difference in the peaks of Fig 3c of 005"00 could in part arise from integration over the finite SPESIII angular acceptance The analysing power signals are small and fluctuate with a similar frequency to the counting rate shown in Fig 3a The shape of the latter is well reproduced by the double-d model of ref w11x where the theoretical calculation has been integrated over < u < F 50 mrad, < u < x y F 0 mrad The predictions in the central bump are the most uncertain since, in this region, the dd a sticking factor is required for high relative momenta Furthermore, as noted in the top scale, the missing mass is well above the threepion threshold and extra pions could be produced In the original double-d calculation of ref w11 x, only the dominant NN dp partial wave amplitude was considered and this gives zero for both the vector and tensor analysing powers By including all the significant amplitudes of the C500 solution of ref w13 x, the very encouraging estimates shown in Figs 3b, 3c at 11 MeV could be achieved, where the results have been averaged over the experimental acceptance w14 x The refined model reproduces all X the main features of both A yy and A y, the frequency of their fluctuations and their strengths The only X gross discrepancy is an overall displacement of A y by about rad y1, but this quantity is very sensitive to any small extra terms in the theoretical model Our results on the vector and tensor analysing powers in d d 4 He X 0 give strong quantitative support to the idea that p 0 production in this reaction is the result of single pion production occurring twice through the pn dp 0 reaction being repeated, or similarly for charged pions Whether this remains true closer to threshold, where the ABC peaks are not seen w15 x, would require the measurement of analysing powers over a greater range in energy This is no longer possible at Saturne following its final closure on December nd, 1997 Acknowledgements We are grateful for all the help offered by colleagues at Saturne over many years

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428 7 January 1999 Physics Letters B Searches for scalar top and scalar bottom quarks in e q e y ( interactions at 161 GeV F s F 183 GeV L3 Collaboration M Acciarri ab, O Adriani q, M Aguilar-Benitez aa, S Ahlen l, J Alcaraz aa, G Alemanni w, J Allaby r, A Aloisio ad, MG Alviggi ad, G Ambrosi t, H Anderhub aw, VP Andreev g,al, T Angelescu n, F Anselmo j, A Arefiev ac, T Azemoon c, T Aziz k, P Bagnaia ak, L Baksay ar, S Banerjee k, Sw Banerjee k, K Banicz at, A Barczyk aw,au, R Barillere ` r, L Barone ak, P Bartalini w, A Baschirotto ab, M Basile j, R Battiston ah, A Bay w, F Becattini q, U Becker p, F Behner aw, J Berdugo aa, P Berges p, B Bertucci ah, BL Betev aw, S Bhattacharya k, M Biasini ah, A Biland aw, GM Bilei ah, JJ Blaising d, SC Blyth ai, GJ Bobbink b, R Bock a, A Bohm a, L Boldizsar o, B Borgia r,ak, D Bourilkov aw, M Bourquin t, S Braccini t, JG Branson an, V Brigljevic aw, IC Brock ai, A Buffini q, A Buijs as, JD Burger p, WJ Burger ah, J Busenitz ar, A Button c, XD Cai p, M Campanelli aw, M Capell p, G Cara Romeo j, G Carlino ad, AM Cartacci q, J Casaus aa, G Castellini q, F Cavallari ak, N Cavallo ad, C Cecchi t, M Cerrada aa, F Cesaroni x, M Chamizo aa, YH Chang ay, UK Chaturvedi s, M Chemarin z, A Chen ay, G Chen h, GM Chen h, HF Chen u, HS Chen h, X Chereau d, G Chiefari ad, CY Chien e, L Cifarelli am, F Cindolo j, C Civinini q, I Clare p, R Clare p, G Coignet d, AP Colijn b, N Colino aa, S Costantini i, F Cotorobai n, B de la Cruz aa, A Csilling o, TS Dai p, R D Alessandro q, R de Asmundis ad, A Degre d, K Deiters au, D della Volpe ad, P Denes aj, F DeNotaristefani ak, M Diemoz ak, D van Dierendonck b, F Di Lodovico aw, C Dionisi r,ak, M Dittmar aw, A Dominguez an, A Doria ad, MT Dova s,1, D Duchesneau d, P Duinker b, I Duran ao, S Easo ah, H El Mamouni z, A Engler ai, FJ Eppling p, FC Erne b, P Extermann t,bm Fabre au, R Faccini ak, MA Falagan aa, S Falciano ak, A Favara q, J Fay z, O Fedin al, M Felcini aw, T Ferguson ai, 1 Also supported by CONICET and Universidad Nacional de La Plata, CC 67, 1900 La Plata, Argentina r99r$ - see front matter q 1999 Published by Elsevier Science BV All rights reserved PII: S

429 ( ) M Acciarri et alrphysics Letters B F Ferroni ak, H Fesefeldt a, E Fiandrini ah, JH Field t, F Filthaut r, PH Fisher p, I Fisk an, G Forconi p, L Fredj t, K Freudenreich aw, C Furetta ab, Yu Galaktionov ac,p, SN Ganguli k, P Garcia-Abia f, M Gataullin ag, SS Gau m, S Gentile ak, N Gheordanescu n, S Giagu ak, S Goldfarb w, J Goldstein l, ZF Gong u, A Gougas e, G Gratta ag, MW Gruenewald i, R van Gulik b, VK Gupta aj, A Gurtu k, LJ Gutay at, D Haas f, B Hartmann a, A Hasan ae, D Hatzifotiadou j, T Hebbeker i, A Herve r, P Hidas o, J Hirschfelder ai, WC van Hoek af, H Hofer aw, H Hoorani ai, SR Hou ay,ghu e, I Iashvili av, BN Jin h, LW Jones c, P de Jong r, I Josa-Mutuberria aa, RA Khan s, D Kamrad av, JS Kapustinsky y, M Kaur s,, MN Kienzle-Focacci t, D Kim ak, DH Kim aq, JK Kim aq, SC Kim aq, WW Kinnison y, A Kirkby ag, D Kirkby ag, J Kirkby r, D Kiss o, W Kittel af, A Klimentov p,ac, AC Konig af, A Kopp av, I Korolko ac, V Koutsenko p,ac, RW Kraemer ai, W Krenz a, A Kunin p,ac, P Lacentre av,1,3, P Ladron de Guevara aa, I Laktineh z, G Landi q, C Lapoint p, K Lassila-Perini aw, P Laurikainen v, A Lavorato am, M Lebeau r, A Lebedev p, P Lebrun z, P Lecomte aw, P Lecoq r, P Le Coultre aw, HJ Lee i, JM Le Goff r, R Leiste av, E Leonardi ak, P Levtchenko al, C Li u, CH Lin ay, WT Lin ay, FL Linde b,r, L Lista ad, ZA Liu h, W Lohmann av, E Longo ak,wlu ag, YS Lu h, K Lubelsmeyer a, C Luci r,ak, D Luckey p, L Luminari ak, W Lustermann aw, WG Ma u, M Maity k, G Majumder k, L Malgeri r, A Malinin ac, C Mana aa, D Mangeol af, P Marchesini aw, G Marian ar,4, A Marin l, JP Martin z, F Marzano ak, GGG Massaro b, K Mazumdar k, RR McNeil g, S Mele r, L Merola ad, M Meschini q, WJ Metzger af, M von der Mey a, D Migani j, A Mihul n, AJW van Mil af, H Milcent r, G Mirabelli ak, J Mnich r, P Molnar i, B Monteleoni q, R Moore c, T Moulik k, R Mount ag, GS Muanza z, F Muheim t, AJM Muijs b, S Nahn p, M Napolitano ad, F Nessi-Tedaldi aw, H Newman ag, T Niessen a, A Nippe w, A Nisati ak, H Nowak av, YD Oh aq, G Organtini ak, R Ostonen v, C Palomares aa, D Pandoulas a, S Paoletti ak,r, P Paolucci ad, HK Park ai, IH Park aq, G Pascale ak, G Passaleva r, S Patricelli ad, T Paul m, M Pauluzzi ah, C Paus r, F Pauss aw, D Peach r, M Pedace ak, YJ Pei a, S Pensotti ab, D Perret-Gallix d, B Petersen af, S Petrak i, A Pevsner e, D Piccolo ad, M Pieri q, PA Piroue aj, E Pistolesi ab, V Plyaskin ac, M Pohl aw, V Pojidaev ac,q, H Postema p, J Pothier r, N Produit t, D Prokofiev al, Also supported by Panjab University, Chandigarh , India 3 Supported by Deutscher Akademischer Austauschdienst 4 Also supported by the Hungarian OTKA fund under contract numbers T38 and T06178

430 430 ( ) M Acciarri et alrphysics Letters B J Quartieri am, G Rahal-Callot aw, N Raja k, PG Rancoita ab, M Rattaggi ab, G Raven an, P Razis ae, D Ren aw, M Rescigno ak, S Reucroft m, T van Rhee as, S Riemann av, K Riles c, A Robohm aw, J Rodin ar, BP Roe c, L Romero aa, S Rosier-Lees d, S Roth a, JA Rubio r, D Ruschmeier i, H Rykaczewski aw, S Sakar ak, J Salicio r, E Sanchez aa, MP Sanders af, ME Sarakinos v, C Schafer a, V Schegelsky al, S Schmidt-Kaerst a, D Schmitz a, N Scholz aw, H Schopper ax, DJ Schotanus af, J Schwenke a,g Schwering a, C Sciacca ad, D Sciarrino t, L Servoli ah, S Shevchenko ag, N Shivarov ap, V Shoutko ac, J Shukla y, E Shumilov ac, A Shvorob ag, T Siedenburg a, D Son aq, B Smith p, P Spillantini q, M Steuer p, DP Stickland aj, A Stone g, H Stone aj, B Stoyanov ap, A Straessner a, K Sudhakar k, G Sultanov s, LZ Sun u, GF Susinno t, H Suter aw, JD Swain s, XW Tang h, L Tauscher f, L Taylor m, C Timmermans af, Samuel CC Ting p, SM Ting p, SC Tonwar k, J Toth o, C Tully aj, KL Tung h, Y Uchida p, J Ulbricht aw, E Valente ak, G Vesztergombi o, I Vetlitsky ac, G Viertel aw, S Villa m, M Vivargent d, S Vlachos f, H Vogel ai, H Vogt av, I Vorobiev r,ac, AA Vorobyov al, A Vorvolakos ae, M Wadhwa f, W Wallraff a, JC Wang p, XL Wang u, ZM Wang u, A Weber a, SX Wu p, S Wynhoff a,jxu l, ZZ Xu u, BZ Yang u, CG Yang h, HJ Yang h, M Yang h, JB Ye u, SC Yeh az, JM You ai, An Zalite al, Yu Zalite al, P Zemp aw, Y Zeng a, ZP Zhang u, B Zhou l, GY Zhu h, RY Zhu ag, A Zichichi j,r,s, F Ziegler av, G Zilizi ar,4 a I Physikalisches Institut, RWTH, D-5056 Aachen, Germany, and III Physikalisches Institut, RWTH, D-5056 Aachen, Germany 5 b National Institute for High Energy Physics, NIKHEF, and UniÕersity of Amsterdam, NL-1009 DB Amsterdam, The Netherlands c UniÕersity of Michigan, Ann Arbor, MI 48109, USA d Laboratoire d Annecy-le-Vieux de Physique des Particules, LAPP, INP3-CNRS, BP 110, F Annecy-le-Vieux CEDEX, France e Johns Hopkins UniÕersity, Baltimore, MD 118, USA f Institute of Physics, UniÕersity of Basel, CH-4056 Basel, Switzerland g Louisiana State UniÕersity, Baton Rouge, LA 70803, USA h Institute of High Energy Physics, IHEP, Beijing, China 6 i Humboldt UniÕersity, D Berlin, Germany 5 j UniÕersity of Bologna and INFN-Sezione di Bologna, I-4016 Bologna, Italy k Tata Institute of Fundamental Research, Bombay , India l Boston UniÕersity, Boston, MA 015, USA m Northeastern UniÕersity, Boston, MA 0115, USA n Institute of Atomic Physics and UniÕersity of Bucharest, R Bucharest, Romania o Central Research Institute for Physics of the Hungarian Academy of Sciences, H-155 Budapest 114, Hungary 7 p Massachusetts Institute of Technology, Cambridge, MA 0139, USA q INFN Sezione di Firenze and UniÕersity of Florence, I-5015 Florence, Italy r European Laboratory for Particle Physics, CERN, CH-111 GeneÕa 3, Switzerland s World Laboratory, FBLJA Project, CH-111 GeneÕa 3, Switzerland 5 Supported by the German Bundesministerium fur Bildung, Wissenschaft, Forschung und Technologie 6 Supported by the National Natural Science Foundation of China 7 Supported by the Hungarian OTKA fund under contract numbers T019181, F0359 and T04011

431 ( ) M Acciarri et alrphysics Letters B t UniÕersity of GeneÕa, CH-111 GeneÕa 4, Switzerland u Chinese UniÕersity of Science and Technology, USTC, Hefei, Anhui 30 09, China 6 v SEFT, Research Institute for High Energy Physics, PO Box 9, SF Helsinki, Finland w UniÕersity of Lausanne, CH-1015 Lausanne, Switzerland x INFN-Sezione di Lecce and UniÕersita Degli Studi di Lecce, I Lecce, Italy y Los Alamos National Laboratory, Los Alamos, NM 87544, USA z Institut de Physique Nucleaire de Lyon, INP3-CNRS, UniÕersite Claude Bernard, F-696 Villeurbanne, France aa Centro de InÕestigaciones Energeticas, Medioambientales y Tecnologicas, CIEMAT, E-8040 Madrid, Spain 8 ab INFN-Sezione di Milano, I-0133 Milan, Italy ac Institute of Theoretical and Experimental Physics, ITEP, Moscow, Russia ad INFN-Sezione di Napoli and UniÕersity of Naples, I-8015 Naples, Italy ae Department of Natural Sciences, UniÕersity of Cyprus, Nicosia, Cyprus af UniÕersity of Nijmegen and NIKHEF, NL-655 ED Nijmegen, The Netherlands ag California Institute of Technology, Pasadena, CA 9115, USA ah INFN-Sezione di Perugia and UniÕersita Degli Studi di Perugia, I Perugia, Italy ai Carnegie Mellon UniÕersity, Pittsburgh, PA 1513, USA aj Princeton UniÕersity, Princeton, NJ 08544, USA ak INFN-Sezione di Roma and UniÕersity of Rome, La Sapienza, I Rome, Italy al Nuclear Physics Institute, St Petersburg, Russia am UniÕersity and INFN, Salerno, I Salerno, Italy an UniÕersity of California, San Diego, CA 9093, USA ao Dept de Fisica de Particulas Elementales, UniÕ de Santiago, E Santiago de Compostela, Spain ap Bulgarian Academy of Sciences, Central Lab of Mechatronics and Instrumentation, BU-1113 Sofia, Bulgaria aq Center for High Energy Physics, AdÕ Inst of Sciences and Technology, Taejon, South Korea ar UniÕersity of Alabama, Tuscaloosa, AL 35486, USA as Utrecht UniÕersity and NIKHEF, NL-3584 CB Utrecht, The Netherlands at Purdue UniÕersity, West Lafayette, IN 47907, USA au Paul Scherrer Institut, PSI, CH-53 Villigen, Switzerland av DESY-Institut fur Hochenergiephysik, D Zeuthen, Germany aw Eidgenossische Technische Hochschule, ETH Zurich, CH-8093 Zurich, Switzerland ax UniÕersity of Hamburg, D-761 Hamburg, Germany ay National Central UniÕersity, Chung-Li, Taiwan, China az Department of Physics, National Tsing Hua UniÕersity, Taiwan, China Received 1 August 1998 Editor: K Winter Abstract Searches for scalar top and scalar bottom quarks have been performed at center-of-mass energies between 161 GeV and 183 GeV using the L3 detector at LEP No signal is observed Model-independent limits on production cross sections are 0 0 determined for the two decay channels t1 cx 1 and b1 bx 1 Within the framework of the Minimal Supersymmetric 0 extension of the Standard Model mass limits are derived For mass differences between t1 and x 1 greater than 10 GeV a 95% CL limit of 815 GeV is set on the mass of the Supersymmetric partner of the left-handed top A supersymmetric partner of the left-handed bottom with a mass below 80 GeV is excluded at 95% CL if the mass difference between b 1 and x 0 is greater than 0 GeV q 1999 Published by Elsevier Science BV All rights reserved 1 1 Introduction 8 Supported also by the Comision Interministerial de Ciencia y Technologia In the Minimal Supersymmetric extension of the Standard Model Ž MSSM wx 1 for each helicity state Standard Model Ž SM quark q there is a correspond-

432 43 ( ) M Acciarri et alrphysics Letters B L,R ing scalar SUSY partner q Generally, the mixing between left q L and right q R eigenstates is propor- tional to the corresponding quark mass The heavy top quark enhances t yt L R mixing leading to a large splitting between the two mass eigenstates This is usually expressed in terms of the mixing angle, u LR The lighter scalar top quark t st cosu qt sinu Ž 1 1 L LR R LR can well be in the discovery range of LEP Large ratio of the vacuum expectation values of the two Higgs fields, tanbr 10, results in a large b yb L R mixing that may also lead to a light sbottom b 1 In the present analysis R-parity conservation is assumed which implies that SUSY particles are produced in pairs; the heavier sparticles decay into lighter ones and the Lightest Supersymmetric Particle Ž LSP is stable In the MSSM the most plausible LSP candidate is the weakly interacting lightest neutralino, x 0 1 The squark production at LEP proceeds via the exchange of virtual bosons in s-channel The production cross section is governed by two free parameters: the squark mass, m q, and the mixing angle, u LR wx At cosu ; 057 Ž 039 the stop Ž sbottom LR decouples from the Z and the cross section is mini mal It reaches the maximum at cosu s1 when LR t1 is the weak eigenstate t L The decay mode of the squark depends mainly on its mass and the masses of the decay products At LEP energies the most important channels for the 0 q stop are: t cx,t bx, and t bn lrb l l n l, where x 1 0 and x q 1 are the lightest neutralino and chargino, respectively, and l, n l are the supersym- metric partners of the charged lepton and neutrino The scalar top analysis is performed assuming 100% 0 branching ratio for the decay channel t1 cx 1 For 0 sbottom the most important decay mode b1 bx 1 is investigated under the same assumption The t 1 bx " 1 decay channel is the dominant one when kine- matically allowed However, the current limits on chargino mass wx 3 preclude this decay to occur, except for tan b R 10, 60 GeV - m0-90 GeV Ž common scalar mass, and very large mixing A t values, which are very unlikely according to the theoretical predictions wx The stop decay t cx is a second order weak decay and the lifetime of the t 1 is larger than the typical hadronisation time of 10 y3 s Thus the scalar top first hadronises into a colourless meson or baryon and then decays For the sbottom the situation depends on the gaugino-higgsino content of the neutralino: the hadronisation is preferred for a gaugino like neutralino In the present analysis we follow this scenario Though hadronisation does not change the final event topology, it does affect event multiplicity, jet properties and event shape Previous searches for stop and sbottom have been performed at LEP wx 5 and at the TEVATRON wx 6 Data samples and simulation The data used in the present analysis were collected in 1996 and 1997 at ' s s161 GeV, 17 GeV and 183 GeV with integrated luminosities of 107 pb y1, 101 pb y1 and 55 pb y1, respectively The description of the L3 detector and its performance can be found in Ref wx 7 Monte Carlo Ž MC samples of signal events are generated using a PYTHIA wx 8 based event generator and varying the stop Ž sbottom mass from 45 GeV up to the kinematical limit and the x 0 1 mass from 1 GeV to M GeV Ž M 5 GeV t b About events are generated at each mass point The following MC programs are used to generate Standard Model background processes: PYTHIA for e q e y q y q y q y qq, e e gzrgz and e e Ze e, KORALZ wx q y q y w x q y 9 for e e t t, KORALW 10 for e e q y w x q y " W W, EXCALIBUR 11 for e e W e n, w x q y q y PHOJET 1 for e e e e qq and DIAG36 w13x for e q e y e q e y t q t y The number of simulated events for each background process exceeds 100 times the statistics of the collected data samples q y q y except for the process e e e e qq, for which three times more MC events are generated The detector response is simulated using the GEANT 315 package w14 x It takes into account effects of energy loss, multiple scattering and showering in the detector materials and in the beam pipe Hadronic interactions are simulated with the GEISHA program w15 x

433 ( ) M Acciarri et alrphysics Letters B Event preselection 0 0 The signal events, t1 cx 1 and b1 bx 1, are characterised by two high multiplicity acoplanar jets containing c- or b-quarks The two neutralinos in the final state escape the detector leading to missing energy in the event A common preselection was applied to obtain a sample of unbalanced hadronic events and to reduce the background from two-photon interactions and from dilepton production The events have to fulfil the following requirements: more than 4 tracks; at least 10 but not more than 40 calorimetric clusters; event visible energy, E vis, larger than 3 GeV; an energy deposition in the forward calorimeters less than 10 GeV and a total energy in the 308 cone around the beam pipe less than 05= E vis; a missing momentum greater than 1 GeV After the preselection 900, 95 and 4378 data events are retained in the 161, 17 and 183 GeV data samples, respectively This is to be compared with 1175, 1088 and 4517 events expected from the SM processes The dominant contribution comes from two-photon interactions in which we observe a 10 0% normalisation uncertainty Fig 1 shows the distributions of some kinematical variables for the data sample, the SM background expectations and the signal events at ' s s 183 GeV after preselection The distributions of the event b-tagging variable D btag and a b-tagging Neural Network output for a jet NNbjet are shown in Fig D btag is defined as the Fig 1 Distributions of Ž a multiplicity of tracks passing some quality criteria, Ž b visible mass M, Ž c jet acollinearity and Ž vis d normalised miss ' q y missing parallel energy EI revis for data and Monte Carlo events at s s183 GeV after preselection Contributions from e e qq, qq and other backgrounds, dominated by W q W y production, are given separately For illustration the expected stop signal for M t1 s80 GeV, Mx 0 s40 GeV and cosu s1 is shown together with the cut values obtained by the optimisation procedure 1 LR

434 434 ( ) M Acciarri et alrphysics Letters B Fig Distribution of Ž a the b-tagging event discriminant, D btag, defined as the negative log-likelihood of the probability for the event being consistent with light quark production, and Ž b b-tagging Neural Network output, NN bjet, for data and Monte Carlo events at ' s s183 GeV after preselection Contributions from q y q y e e qq, qq and other backgrounds, dominated by W W production, are given separately For illustration the expected sbottom signal for M t s80 GeV, Mx 0 1 s40 GeV and cosu LR s1 is 1 shown together with the cut value on Dbtag obtained by the optimisation procedure negative log-likelihood of the probability for the event being consistent with light quark production w16 x After preselection the data and MC are in a fair agreement 4 Selection optimisation The kinematics of the signal events depend strongly on the mass difference between the squark and the neutralino, D MsMq ym x 0 In the low D M 1 region, between 5 and 10 GeV, visible energy and track multiplicity are low Therefore the signal events are difficult to separate from two-photon interactions For high D M values, between 50 and 70 GeV, large visible energy and high track multiplicity cause the signal events to be similar to WW, Wen or ZZ final states The most favourable region for the signal and background separation appears at D M s 0 40 GeV Searches are performed independently in different D M regions At ' s s161 GeV and 17 GeV three different selections have been designed for three D M regions, whereas for ' s s 183 GeV four selections have been optimised to account for the wider kinematical range The most discriminating sets of cuts are obtained using an optimisation procedure which minimises the following sensitivity function w17 x: ` ks Ý k n P B Ž n re is0 Here k n is the 95% CL upper limit for n observed events It is calculated without subtraction of expected background B when optimising the cuts for D M region of 5 0 GeV, and with background subtraction for higher D M values Žsee Results section P Ž n B is the Poisson probability function of n observation with the mean value of B and e is the signal selection efficiency w18 x The following kinematical variables are used in the selections: the visible energy E vis, the visible mass, M vis, and the sum of the two jets transverse momenta These variables allow to discriminate between signal and two-photon background A further reduction of this background is achieved by rejecting events with a pair of collinear tracks To discriminate hadronic W and Z decays an upper cut on Evis is applied W q W y events where one W decays leptonically and W " e n events are suppressed by vetoing energetic isolated leptons A cut on the normalised parallel missing momentum EI miss revis removes most of the qq events The remaining qq contribution can be suppressed by applying cuts on jet acollinearity and acoplanarity A veto on the energy deposition in the 58 azimuthal sector around the missing momentum direction suppresses the t q t y events For sbottom selection we make use of b-quarks appearance in the final state and apply additionally a cut on the event b-tagging variable D btag The exact cut values for each D M region are chosen by the optimisation

435 ( ) M Acciarri et alrphysics Letters B procedure described above As an example the cut values obtained for a stop signal of M s80 GeV t1 and M 0 x 1 s40 GeV are indicated by arrows in Fig 1 The cut applied on D btag in the selection of sbottom signal of M s80 GeV and M 0 b x s40 GeV is shown 1 1 in Fig The achieved signal selection efficiencies for stop Ž sbottom range from 5% Ž 0% to 45% Ž 50% depending on D M The efficiencies are lowest at low 0 D M values The b quark from b1 bx 1 forms a long-lived hadron which decays at distances up to 3 mm from the interaction point Use of this information in the discriminant variable D btag results in 0 higher efficiencies for the b1 bx 1 channel com- 0 pared to t cx Statistical and systematic errors The errors arising from signal MC event statistics vary from 3% to 8% for stop and from 3% to 7% for sbottom depending on selection efficiencies The main systematic errors on the signal selection efficiency arise from the uncertainties in the t1t1 Ž b b production, stop-ž sbottom- 1 1 hadron formation and the decay scheme We have studied in detail the following sources of systematic errors: Ø The mixing angle cosu LR between the left and right eigenstates The stop Ž sbottom signals have been generated assuming cosu LR s 1 However, as the coupling between t Ž b 1 1 and Z depends on cosu LR, the initial state radiation spectrum is also mixing angle dependent The maximal influence of this source has been evaluated by generating signal samples with the values of cosu LR when stop Ž sbottom decouples from Z The largest uncertainty in the selection efficiency, 4% for stop and 6% for sbottom, is observed at low D M;5 10 GeV With increasing D M the selection efficiencies are less affected by this source of systematics At D M;70 GeV the error is estimated to be negligible Ø The Fermi motion parameter of the spectator quarkž s in the t -Ž b hadron The invariant mass available for spectator quarkž s has been assumed to be M s05 GeV w19 x eff The hadronic energy and track multiplicity of the event depend on the value of this variable so that a variation of M eff from 05 GeV to 075 GeV w19x results in 4 1% relative change in efficiency for stop and 6 8% for sbottom Ø The Peterson fragmentation function parameter e For the t -Ž b - b 1 1 hadron the Peterson fragmenta- tion scheme w0x was used with e Ž e t b propagated from e so that e se m rm, e s00035 w1x b q b b q b and mbs5 GeV The eb was varied in the range from 000 to 0006 w1 x This induces a 5 1% and 6% systematic effect in selection efficiencies for stop and sbottom, respectively 0 Ø For the t1 cx 1 decays the uncertainty on the c-quark fragmentation parameter ec results in a 1 4% change in efficiency when ec is varied from 00 to 006 w1 x The central value is chosen to be e s 003 w1 x c 0 For the t1 cx 1 channel all sources of systemat- ics have larger impact on lower D M selections This is because the energy available for c-quarks is low and the variation of the cosu LR, M eff, eb and ec has a relatively high influence on the event kinematics 0 In contrast, for the b1 bx 1 decays, the systematic errors, except the one related to cosu LR, increase with increasing D M This is because the sbottom selection relies strongly on the b-tagging at high D M, whereas at low values of D M the b-tagging is not applied The b 1 hadronisation and decay scheme, especially the uncertainty on e b, have a noticeable Table 1 Relative statistical error on stop and sbottom selection efficiencies and contribution from various systematic uncertainties for 183 GeV The lower part of the table shows the overall systematic error in different D M regions Source Dere Dere Ž 0 t cx,% Ž 0 b bx,% Statistical error Spectator Fermi motion Uncertainty on e b Uncertainty on e 1 4 c Mixing angle u LR Overall systematic error D Ms5 10 GeV D Ms10 0 GeV D Ms0 60 GeV D M G60 GeV

436 436 ( ) M Acciarri et alrphysics Letters B Table all Number of observed events, N, and Standard Model background expectations, N, for the stop and sbottom selections at ' data MC s s161 Ž q y Ž q y " q y GeV, 17 GeV and 183 GeV The contribution of two-fermion qq, t t, four-fermion W W, W e n, ZZ, Ze e and two-photon Ž q y q y q y e e qq, e e t t processes are given separately The errors are due to MC statistics only two-fermion four-fermion two-photon all Channel Ndata NMC NMC NMC NMC t cx, s s GeV 0035" " " 06 13" 03 t cx, ' s s183 GeV " " " 045 9" 05 b bx, ' s s GeV 1 045" " " 07 8" 07 b bx, ' s s183 GeV 0010" " " 08 31" ' impact on the b-jet track multiplicity and hardness, and consequently on the signal efficiency for the D btag cut The overall systematic error ranges from 7% to 18% for stop The D M;5 10 GeV is the region of highest systematic uncertainty of about 15 18% Above D M;10 0 GeV the error decreases to 7% For sbottom the highest overall uncertainty of about 10 1% is observed at very low, ;5 10 GeV, and high, R 60 GeV D M regions In the intermediate D M region the systematic error amounts to 7 10% The summary on statistical and systematic errors for 0 0 t1 cx 1 and b1 bx 1 channels is given in Table 1 for 183 GeV Similar numbers are found also at 161 and 17 GeV In the final results the systematic error is incorporated using the method described in Ref w x 6 Results Table summarises the number of selected data 0 and expected background events for t1 cx 1 and 0 b1 bx 1 channels The contribution of two-fermion Ž q y Ž q y " qq, t t, four-fermion W W, W e n, ZZ, q y Ž q y q y q y Ze e and two-photon e e qq, e e t t processes are given separately No evidence for stop or sbottom was found and the upper limits on their production cross sections are derived Due to the uncertainties in the simulation of two-photon interactions these contributions, conservatively, are not subtracted from data when deriving limits The modelindependent cross section limits for both scalar quarks in terms of Ž M, M & q are given in Fig 3 No 0 x 1 scaling of the production cross section has been applied when combining the 161 GeV, 17 GeV and q y q y Fig 3 Upper limits on Ž a e e t t and Ž b e e bb production cross sections In both cases the branching ratios are assumed to be 100%

437 ( ) M Acciarri et alrphysics Letters B with the obtained 95% CL limit on production cross section we determine the excluded mass regions for t and b Fig 4Ž 1 1 a shows the excluded mass regions as a function of M and M 0 q x for stop at cosu LRs1 1 and 057 The region excluded by the D0 experiment is also shown wx 6 The corresponding exclusion plot for sbottom is given in Fig 4Ž b for cosu LR s1 and cosu LR s039 For a mass difference of D Ms15 Ž 35 GeV the excluded stop Ž sbottom masses as a function of cosu LR are shown in Fig 5 Independent of cosu LR the stop pair production is excluded at 95% CL for M less than 75 GeV if t1 the mass difference between stop and neutralino is Ž Ž Fig 4 95% CL exclusion limits for a stop and b sbottom as a function of the neutralino mass for maximal and minimal cross section assumptions For comparison we show also result on stop searches from the D0 experiment 183 GeV analyses Thus the evaluated limits correspond to luminosity weighted average cross section 7 MSSM Interpretation In MSSM the stop and sbottom production cross sections depend on the squark mass Mq and the mixing angle cosu LR The cross section is highest for the SUSY partner of left-handed stop Ž sbottom, ie cosu LR s1, and has its minimum at cosu LR, 057 Ž 039 By comparing the theoretical prediction Ž Ž Fig 5 95% CL exclusion limits for a stop and b sbottom masses as a function of cosu LR