b-delayed neutron decay of 104 Y, 112 Tc, 113 Tc and 114 Tc:

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1 3 May 999 Physics Letters B b-delayed neutron decay of 04 Y, Tc, 3 Tc and 4 Tc: test of half-life predictions for neutron-rich isotopes of refractory elements JC Wang, P Dendooven, M Hannawald, A Honkanen, M Huhta, A Jokinen, K-L Kratz, G Lhersonneau, M Oinonen, H Penttila,,) K Perajarvi, B Pfeiffer, and J Aysto Department of Physics, UniÕersity of JyÕaskyla, FIN-4035 JyÕaskyla, Finland Institut fur Kernchemie, UniÕersitat Mainz, D-558 Mainz, Germany Received December 998; received in revised form 0 March 999 Editor: RH Siemssen Abstract Beta-decay gross properties of neutron-rich isotopes 04 Y and,3,4 Tc produced in 5 MeV proton-induced fission of 38 U have been measured Decays of 04 Y with a half-life of 80"60 ms and of 4 Tc with a half-life of 50"30 ms are,3,4 reported for the first time Beta-delayed neutron emission probabilities of Tc are determined as Ž 5" 0 %, " 03 % and Ž 3" 04 %, respectively Comparison of the observed Pn values with a recent theoretical calculation based on deformed quasiparticle random phase approximation Ž QRPA shows good overall agreement for Tc isotopes However, comparison of beta-decay half-lives of neutron-rich isotopes of Y to Rh with the QRPA model and the refined gross theory shows only modest agreement, pointing to inadequate treatment of structure of these very neutron-rich nuclei lying in a region of rapidly changing deformation q 999 Elsevier Science BV All rights reserved Introduction Determination of gross properties of neutron-rich nuclei far from the valley of b-stability represents an important first step towards better understanding of the evolution of nuclear structure and of possible new nuclear phenomena predicted to appear near the neutron drip-line wx Studies on extremely neutronrich nuclei also provide input required for explaining ) Corresponding author Fax: q ; aysto@physjyufi important questions in astrophysics wx Since production cross sections of neutron-rich nuclei far from the valley of b-stability are generally low, efficient and selective experimental techniques are needed The combination of the ion-guide isotope separator on-line Ž IGISOL technique and the proton-induced fission of 38 U has shown to be a powerful and universal tool to search for and study neutron-rich isotopes of all medium-heavy elements, including refractory ones w3,4 x However, without any physical separation for Z it is difficult to detect the decay of most neutron-rich members of an isobaric chain by r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

2 ( ) JC Wang et alrphysics Letters B b- and g-spectroscopic methods due to background caused by more copiously produced isobaric contaminants Presently, the high sensitivity requested in order to identify exotic neutron-rich isotopes is achieved by b-delayed neutron spectroscopy The combination of a usually high b-delayed neutron emission probability and high-efficiency neutron counting offers an excellent means to probe the gross decay characteristics of exotic neutron-rich nuclei produced in fission The b-decay half-life Ž T r and the delayed neutron-emission probability Ž P n are the easiest mea- surable gross b-decay properties for nuclei very far from stability produced in low production yields They contain important nuclear-structure information; Tr is sensitive to low-lying b-strength and the Pn-value carries information on strength just above the neutron separation energy Indications of nuclear-structure changes associated with neutron excess may be obtained by comparing experimental values with theoretical predictions using the available nuclear models In addition, properties of nuclei on the expected r-process path can be predicted by extrapolation on the basis of the systematics of experimental Tr and Pn-values In our previous work b-decay half-lives and delayed-neutron emission probabilities of neutron-rich wx Y to Tc isotopes were measured 4 Beta decays of four new isotopes were reported and b-delayed neutron-emission was discovered for new isotopes of yttrium, niobium and technetium These results showed delayed-neutron emission probabilities as high as 30 40% for 09 Nb and 0 Nb indicating importance of this decay mode already for nuclei with NrZ s 6 As a continuation of this systematic investigation, b-decay and b-delayed neutron emission of even more neutron-rich nuclei were searched for and studied further in this work in substantially improved yield conditions as compared to the previwx ous work in Ref 4 Experimental method Radioactive isotopes of interest were produced in 5 MeV proton-induced fission of 38 U and mass separated using the upgraded ion-guide isotope separator IGISOL at the Ks30 MeV cyclotron at the Accelerator Laboratory of the University of Jyvaskyla wx q 5 Protons were provided as a 50 MeV H ion beam with a typical intensity of 0 ma Recent fission ion guide development and increasing accelerator-beam intensity have substantially increased the yields of mass-separated ions wx 6 At present, isobaric yields of the order of 0 5 ionsrs are routinely available for medium-mass fission-product beams The 40 kev ion beam of short-lived mass-separated nuclei was implanted into a collection tape, which was moved at preset time intervals to reduce the b- and g-background caused by longer-lived isobaric activities As in our previous work wx 4, the Mainz 4p neutron long counter and a thin plastic scintillator were used to detect neutrons and b-particles, respectively The neutron counter included 4 3 He ionization chamber tubes arranged in three concentric rings in a polyethylene block surrounding the implantation point The efficiency of the neutron counter is nearly energy independent due to the fact that neutrons have to be thermalized before their actual detection It was determined off-line with calibrated AmrLi and 5 Cf neutron sources and on-line with the well-known beta-delayed neutron 95 emitter Rb wx 7 to be about 5% For the determination of the Pn values, the required ratio between the efficiency of the b-detector and the efficiency of the neutron detector was determined using the well known neutron branching ratio, Pn s 873" 00%, of 95 Rb In addition, a 3% HPGe detector, placed inside the neutron counter, was used to detect g-rays simultaneously with b-particles and neutrons The use of the Ge detector allowed continuous monitoring of the mass purity in order to obtain an unambiguous mass-number assignment for the recorded b- and n-spectra It also provided the means to measure simultaneously the isobaric yield via the intensities of the g-transitions Signals for gamma-ray energy, as well as logic signals for the observation of neutrons and b-particles were associated with the time of occurrence within the measurement cycle to form an event for the list mode recording The time spectra gated by b-, g-, or neutron signals could then be used to obtain the half-life and the production rate from the growth and the decay curves of the activities collected using the pulsed beam of radioactive ions

3 ( ) JC Wang et alrphysics Letters B Results 3 Beta-decay half-liões in some detail The decays were fitted with a linear component plus a single exponential component corresponding to the short half-life of the b-delayed Half-lives of b-delayed neutron precursors were determined from the single neutron time spectra by fitting to the growth-in and the decay periods The growth and decay curves for the neutron singles measured for A s 04,, 3 and 4, which were attributed to 04 Y,,3,4 Tc, respectively, are shown in Fig In all cases, only one component and a constant background were required for a satisfactory fit In the present work, the time spectrum recorded for neutrons at As04 could be fitted with a single half-life component of 80"60 ms However, in the previous study reported in Ref wx 4 the A s 04 time spectrum for neutrons showed weak componentž s with a substantially longer decay half-life The spectrum was therefore fitted as a superposition of two components with the known half-lives of 04m Nb 04g Ž T s0 s and Nb Ž T s50 s r r We did not find any evidence of these activities in the present experiment, see Fig As a matter of fact, in the previous experiment neutrons giving rise to a component with a half-life of a few seconds could have been due to a mass contaminant, such as 05 Nb Ž T s 8 s r This points out the importance of continuous monitoring of the beam composition by simultaneous g-spectroscopy In this way, a possible 03 mass contamination due to Y Ž T s 30 ms wx r 4 could be definitely excluded in this experiment, because no transitions from nuclei with A s 03 were observed in the g spectra The corresponding time spectra for b-singles are more complex Since their detailed analysis is crucially important for extracting information on neutron branching ratios this procedure is described here Fig Time distribution of neutron singles events within the measurement cycle The growth-in and decay curves are shown for the four mass numbers studied in this work The solid curves represent the fits to the experimental points The horizontal line represents the time independent background In cases when the half-life is not very much shorter than the duration of the beam-off period, a cumulation of the activity is observed that is due to the non-removal of the source after each separator pulsing cycle This effect has been taken into account in the fit function

4 4 ( ) JC Wang et alrphysics Letters B neutron precursor The linear component simulates the contributions of all long-lived activities plus a constant background and is based on the approximation e ylt fylt, iflt- This approximation is valid when the cycle time Ž here the decay period is a fraction of the half-life of the shortest-lived contaminant This assumption is valid for all cases studied in this work The validity of this approximation was proven by the measurements on the 95 Rb, Tc and 3 Tc decays In this way, the half-life of 95 Rb was determined as 380"0 ms in excellent agreement with the previously measured value of 377"6 ms wx 7 Moreover, the half-lives for Tc and 3 Tc were also found consistent with the values previously measured at IGISOL using the g-spectrow3,8 x As an example, the analysis of scopic methods the decay part of the time spectrum for b-singles at As is shown in Fig In this case the fit with a linear-plus-exponential curve resulted in 80" 0 ms for the short-lived component in agreement with Table b-decay half-lives measured in this work T Ž experiment T Ž theory r Isotope This work Others QRPA gross theory w5,6x w7,8x r 04 Y 80" Tc 90"0 80"30 wx "0 wx 4 3Tc 70"0 30"50 wx Tc 50" the neutron half-life and the previously reported value for Tc wx 8 All half-lives deduced from the singles neutron time spectra are given in Table 3 Neutron-branching ratios Beta-delayed neutron branching ratios of Tc, 3 Tc and 4 Tc were determined from the ratio of the corresponding neutron and b-intensities after unfolding the decay curves of Fig for neutrons and the decay curves similar to the one given in Fig for b-particles The low counting rate for A s 04 did not allow to extract a meaningful value for the P -value of 04 n Y The resulting Pn-values are sum- marized in Table The values for 3 Tc and 4 Tc are reported for the first time in this work The present P -value measured with good statistics for n Tc, as shown in Fig, is somewhat smaller than wx our previous less-precise value reported in ref 4 33 Yields Independent production yields of neutron emitters were deduced directly from the observed b-delayed neutron activities Simultaneously measured b-gated Fig An example of the analysis of a b-decay curve measured for As Ž a The linear component is shown as a solid line Ž b The residual part is fitted by one half-life, which is consistent with the half-life of Tc obtained from the fit of the neutron timespectrum shown in Fig The vertical scale on the left is for the curve Ž a and the one on the right for the curve Ž b Table b-delayed neutron branching ratios Pn measured in this work Comparison is made with the values calculated using the QRPA theory w5,6 x P Ž experiment in % P Ž theory in % n wx Isotope This work Ref 4 QRPA Tc 5"0 6" Tc " Tc 3"04 49 n

5 g spectra provided a means to obtain the yield information of the other isobars closer to the valley of b-stability For the extraction of independent production rates, b-filiation was included according to the following formalism: Ig FiŽ l 0 0 P Ig s Fi b Ž l,l FiŽ l 0 s P F Ž l,l,l F Ž l,l F Ž l P Ig 3 i bb 3 ib 3 i 3 3 ( ) JC Wang et alrphysics Letters B The symbol Ig i is the peak area divided by the absolute g-decay branching, the detection efficiency and the number of cycles during the measurement Thus Ig i represents the number of nuclei of type i observed to decay during a single acquisition cycle as determined by the selected gamma transition P, P and P3 are independent production rates in ionsrs for nuclei with Ž Z, A, Ž Zq, A and Ž Zq, A and with decay constants l, l and l3 respectively The functions F i, Fib and Fi bb take into account differ- ent production modes of these nuclei They correspond to independent production, productions by the decays of the parent and grandparent during the cycle duration These functions have the dimension of time They depend on the decay constants and on the beam-on and beam-off durations in the beam pulsing cycle The relevant formulae have been prewx 9 In the conven- sented recently by Kudo et al tional case of measurement with continuous implantation during the whole cycle, the F-functions can be approximated by the cycle duration, giving simple solutions for P s by successive subtractions The independent yields for A s, 3, 4 isobars are shown in Fig 3 The yields versus Z were fitted by single Gaussian distributions whose widths sz were taken as a constant of 07 according to a recent experimental study, see Ref wx 9 The resulting Z-values corresponding to the maximum yields were 455, 458 and 460 for As, 3 and 4, respectively These values are systematically higher by 0 to 03 units than those measured in the previous experiments using 0 MeV protons, where excitation energy was lower w0, x It is worthwhile to note that the yield points of Tc, 3 Tc and 4 Tc fall nicely on the Gaussian curves, which can be taken as a proof of consistency of the analysis In general, the experimental sensitivity of IGISOL in decay studies of neutron-rich fission Fig 3 Independent yields of neutron-rich nuclides with As,3,4 Data for As and As3 have been rescaled in order to improve the reading The open and filled symbols refer to the values obtained from g-ray and delayed neutron data, respectively The arrow at As3 shows the lower production limit for 3 Rh if assuming no ground state feeding Solid lines represent fits by Gaussian functions with fixed width taken from Ref wx 9 products has reached the level of 0 to ionrs, which corresponds to a distance of 5 to 6 neutrons away from the bottom of the valley of stability This compares well with the recent experimental results obtained in fission of relativistic 38 U projectiles w x For As3, the b-decay ground state branching 3 of Rh is unknown w3x and only a lower limit for the yield of 3 Rh can be obtained This experimental point can be brought on the Gaussian curve if we assume for the 349 kev transition a branching of bg f5% per decay, which converts to a b ground- state branching of 60% The logž ft -value deduced for this transition would then be 50 in agreement with the allowed character of the decay between the 7r q ground state of 3 Rh and the 5r q ground 3 w x state of Pd 4 4 Discussion Experimental Tr and Pn-values determined in this work are listed in Tables and The half-life values are compared with the recently published predictions of the deformed quasiparticle random phase approximation Ž QRPA w5,6x and the semi-

6 6 ( ) JC Wang et alrphysics Letters B gross theory Ž SGT of beta decay w7,8 x The QRPA predictions are based on the calculation of the Gamow Teller strength distribution with single particle levels and wave functions at the calculated nuclear shapes as input quantities The derivation of the most important adjustable constants in the folded Yukawa single particle model are the diffuseness and spin orbit constants Their derivation is discussed in detail in Ref w5,6 x The semi-gross theory is an improved version of the original gross theory refined to take into account shell effects of parent nuclei derived from experimental masses or from the mass systematics w9 x In both cases the calculations employ the decay energy values Ž Q b derived from the finite range droplet model Ž FRDM as given in Ref w5,6 x In case of the half-lives of Tc isotopes the QRPA-approach leads to a good agreement while the refined gross theory overestimates the half-lives by a factor of two to three Instead, the situation for 04 Y is different; the half-life is underestimated by the QRPA but well reproduced by the semi-gross theory For the Pn-values of Tc isotopes agreement with QRPA is reasonably good and is in accordance with our previous observation for neutron-rich Nb isotopes with an exception for the neutron number Ns65, where a specific difficulty is encountered in half-life calculation as discussed in our previous paper wx 4 The Pn-values are also compared with the values derived using the empirical formula of Kratz and Herrman and similar agreement is observed No calculation for the Pn-values is yet available using the semi-gross theory Due to new experimental data, it is now of interest to compare the half-life predictions over a wider range in Z and N This is done in Fig 4 for neutron-rich Y Ž Z s 39 through Rh Ž Z s 45 isotopes ranging in neutron number from 6 to 7, ie in the middle of the closed Ns50 and 8 neutron shells Overall agreement between theory and experiment is fairly good, especially for Nb, Tc and Rh isotopes, but is worse for Mo, Zr and Y isotopes In addition to the systematic kink at N s 65, fluctuations in QRPA predictions are observed in Z and N They may well be related with the difficulty in exactly reproducing theoretically the experimentally known rapid changes in the energies and ordering of single particle orbitals in this region, where changes in deformations as well as coexistence and degener- Fig 4 Comparison of theoretical to experimental beta-decay half-lives of neutron-rich nuclei from Y to Rh The solid squares refer to the QRPA calculation w5,6x and the open squares to the semi-gross theory w7,8 x Circles are used to represent half-lives of beta-decaying isomers The vertical scale has to be read alternatively on the left and right-hand sides acy of different structures are known to exist The systematic underestimation of the half-lives for N s 65 nuclei can derive from the competition of the w x y w x q 53 5r and the 43 5r neutron valence orbitals in the parent nuclei at deformation of about 03 The positive parity neutron state when coupled to a 5r q proton state can lead to allowed ground state transition resulting in a short half-life, as observed for Ns65 However, for example, it is known experimentally that a nearby Ns63 nucleus 03 Zr y has a 5r ground state w0 x Consequently, the change to this neutron configuration in the ground states leads to calculated half-lives that are about 3, 7, 0 and 5 times longer for 04 Y, 05 Zr, 06 Nb and

7 ( ) JC Wang et alrphysics Letters B Mo, respectively These values are in substantially better agreement with the experiment In general, a somewhat better predictive power is observed for the semi-gross theory in the studied region The smooth behaviour of semi-gross theory with respect to deformation is also observed in other regions of changing deformations w7,8 x This general property of the semi-gross theory could be related to the fact that the single-particle energies used in the calculation are derived from experimental or systematic mass data tabulated in Ref w9 x Both models used show a diverging behaviour from experiment for Mo isotopes towards increasing neutron number It is at least partly related with underestimation of their Qb-values in the FRDM calculation Unlike in all other cases in this comparison, the predicted decay energies of 09 Mo and 0 Mo are much lower, ie by 3 and MeV than those given in the 995 update to the atomic mass evaluation w9 x In order to perform reliable calculations for the most exotic decays, better knowledge of the level structure of mother and daughter nuclei is required While only gross properties can be obtained for the activities far out of the valley of stability, detailed spectroscopic investigations are indeed feasible for their isobars with or 3 neutrons less They would provide firm grounds for modelling the nuclear structure of the most exotic decays within current experimental reach 5 Outlook The present measurements confirm the experimental possibilities for spectroscopy of very neutron-rich isotopes at IGISOL However, more detailed information on the energies of beta-delayed neutrons is required in order to extract needed nuclear structure information This is especially true for the multineutron quasiparticle structures known to be populated strongly in b-decay as shown for nuclei closer to the valley of b-stability These quasineutron structures could provide information on the evolution of pairing energies of very neutron-rich nuclei Among the known neutron-rich Sr, Zr, Ru and Pd nuclei the lowest two quasineutron states occur at excitation energies near 8 MeV Ž Sr, Zr and 5 MeV Ž Ru, Pd w 4 x It can be anticipated that when the neutron binding energy of the daughter nucleus approaches these energies, a distinct increase in the P -value will occur due to favourable b-feedn ing of these states Such an increase may already 09 0 wx have been seen in the case of Nb and Nb 4 Acknowledgements This work was supported by the Deutsche Akademische Austauschdienst and Academy of Finland and by the Access to the Large Scale Facilities programme under the TMR programme of the European Union Finally, the authors wish to thank prof Takahiro Tachibana for providing the theoretical half-lives used in this work References wx J Dobaczewski et al, Phys Rev Lett 7 Ž wx B Chen et al, Phys Lett B 355 Ž wx 3 J Aysto et al, Phys Rev Lett 69 Ž wx 4 T Mehren et al, Phys Rev Lett 77 Ž wx 5 H Penttila et al, Nucl Instr Methods in Phys Res B 6 Ž wx 6 M Huhta et al, Nucl Instrum Methods in Phys Res B 6 Ž wx 7 G Rudstam, K Aleklett, L Sihver, At Data and Nucl Data Tables 53 Ž 993 wx 8 J Aysto et al, Nuclear Physics A 55 Ž wx 9 H Kudo et al, Phys Rev C 57 Ž w0x M Leino et al, Phys Rev C 44 Ž wx P Jauho et al, Phys Rev C 49 Ž wx C Donzaud et al, Eur Phys J A Ž w3x H Penttila et al, Phys Rev C 38 Ž w4x RB Firestone et al, Table of Isotopes, Eight Edition, Wiley, New York Ž 996 w5x P Moller, J Randrup, Nucl Phys A 54 Ž 990 w6x P Moller et al, At Data and Nucl Data Tables 66 Ž w7x H Nakata, T Tachibana, M Yamada, Nucl Phys A 65 Ž w8x Private communication by T Tachibana and M Yamada w9x G Audi, AH Wapstra, Nucl Phys A 595 Ž w0x G Lhersonneau et al, Phys Rev C 54 Ž wx B Pfeiffer et al, Z Phys A 353 Ž 995 wx JL Durell et al, Phys Rev C 5 Ž w3x R Capote et al, J Phys G Nucl Part Phys 4 Ž w4x J Aysto et al, Nuclear Physics A 480 Ž

8 3 May 999 Physics Letters B Triaxial superdeformed bands in 64 Lu and enhanced E decay-out strength S Tormanen a, SW Ødegard a,b, GB Hagemann a, A Harsmann a, M Bergstrom a, RA Bark a, B Herskind a, G Sletten a, PO Tjøm b, A Gorgen c, H Hubel c, B Aengenvoort c, UJ van Severen c, C Fahlander d,g, D Napoli d, S Lenzi e, C Petrache e,cur e, HJ Jensen a,f, H Ryde g, R Bengtsson g, A Bracco h, S Frattini h, R Chapman i, DM Cullen j, SL King j a NBI, UniÕ of Copenhagen, BlegdamsÕej 7, DK-00 Copenhagen Ø, Denmark b Dept of Physics, UniÕ of Oslo, Oslo, Norway c Inst fur Strahlen- und Kernphysik, UniÕ of Bonn, Nussallee 4-6, D-535 Bonn, Germany d INFN, Laboratori Nazionale di Legnaro, Legnaro, Italy e Dipartemento di Fisica and INFN, Sezione di PadoÕa, PadoÕa, Italy j i f FZ Julich, Germany g Department of Physics, UniÕ of Lund, Lund, Sweden h Dipartemento di Fisica and INFN, Sezione di Milano, Milano, Italy Department of Elec Ing and Physics, UniÕ of Paisley, Paisley PA 8E, UK OliÕer Lodge Lab, Department of Physics, UniÕ of LiÕerpool, LiÕerpool L69 7ZE, UK Received 0 December 998; received in revised form 7 February 999 Editor: RH Siemssen Abstract In a search for exotic structures in odd-odd 64 Lu, performed as one of the first Euroball experiments eight new, presumably triaxial, superdeformed bands were found For the first time, evidence is presented for superdeformation in an odd-odd Lu isotope for which theory predicts large triaxiality The data are compared to expectations from calculations with the code Ultimate Cranker Two of the bands are connected to normal-deformed structures and the E strength of the decay-out appears to be enhanced over single particle expectations, possibly due to octupole correlations q 999 Elsevier Science BV All rights reserved PACS: 0yk; 30Lv; 570yz; 770qq Keywords: Triaxial superdeformation; Odd-odd nucleus; Enhanced E decay out; PES calculations; Configuration assignments Nuclei with N;94 and Z;7 constitute a new region of exotic shapes, coexisting with normal prow 4 x They provide a unique possi- late deformation bility of studying superdeformed Ž SD shapes with a pronounced triaxiality Two such cases have recently 63,65 been found in Lu w3,4 x Large Qt values, corre- sponding to ; 04 Ž SD with g; q88, were 63 derived for the triaxial SD band in Lu wx 4 from r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

9 S Tormanen et alrphysics Letters B 454 ( 999) both Recoil Distance and Doppler Shift Attenuation Method measurements This band was interpreted as w x q most likely corresponding to the p i3r 660 r configuration In a later measurement, a band with transition energies, identical within -3 kev to those of the SD band in Lu, was found in Lu wx 3 and, based on the similarity, was also interpreted as a w x q p i 660 r band Calculations wx 3r 3 with the Ultimate Cranker Ž UC code have revealed that large deformation minima are actually expected for all 63,65 elements of the symmetry group Ž p, a in Lu That is, the large deformation is due not only to the deformation-driving effect of the p i3r intruder orbital, but also is the result of a re-arrangement of the core The neutron number N;94 is crucial, as a gap in single particle energy appears at large values of g Ž ;"08 Therefore, large triaxial deformations are expected as a general phenomenon In an odd-odd nucleus, the possibility exists to exploit the various combinations of proton and neutron orbitals which are expected to sample the triaxial minima differently Calculations of the potential energy surfaces for the lowest expected configurations in 64 Lu with positive and negative parity are shown in Fig In addition to the minima at normal deformation the calculations show local minima with large deformation and g;"08 A search for triaxial SD structures in 64 Lu was performed as one of the first Euroball experiments 39 using the reaction LaŽ 9 Si,4n with thin self-supporting targets at a beam energy of 45 MeV The 9 Si beam was provided by the Legnaro XTU tandem accelerator and the g-rays were detected with the Euroball array consisting, at the time of the experiment, of 3 clusters, 5 clovers and 8 single element tapered detectors Altogether, ;38P0 9 events requiring six or more coincident Ge signals before Compton suppression were collected After presorting, ;3P0 9 clean, three- or higher-fold events were sorted into gated matrices for DCO analysis, cubes and a 4D-hypercube The present Euroball data-set contains rotational bands in 63,64 Lu as the main exit channels, in Extensive use of the program NUSMA has been applied in wx the analysis of the UC calculations 5 addition to several Yb and Tm isotopes The analysis has provided a major extension of the known w6,7x normal-deformed Ž ND level scheme in addition to eight new, presumably triaxial SD, bands in 64 Lu See Fig The SD bands have dynamic moments of 63,65 inertia similar to those found in Lu w3,4 x, and they may belong to both of the calculated minima with g;"08 Two of the strongest populated bands, SD and SD3, have been connected to known 64 w6,7 x, ND bands in Lu The band SD, decays to several states of both positive and negative parity, which suggests the spin and parity assignments shown in Fig These assignments imply that the strongest transition of 8 kev is of stretched dipole character, which agrees with the measured DCO ratio of 0Ž The theoretical values for the Euroball geomwx 8 etry using summed gating on all detector angles are about 08 and 4 for stretched dipole and quadrupole transitions, respectively A double gated g-ray spectrum corresponding to this band is shown in Fig 3a For band SD3, the single decay of 53 kev to the yrast ND 4 y state with a DCO ratio of 09Ž is assigned as a stretched dipole transition, which determines the spin of band SD3 Since a pure M transition of such high energy is very unlikely, we assume it is of E character, and SD3 is therefore assigned positive parity As can be seen from Fig 3b, the new bands in 64 Lu have quite similar dynamic moments of inertia, J Ž, at low frequency, and a small variation in the moderate increase in J Ž with increasing frequency From the g-ray energies, bands SD and SD, and bands SD3 and SD4 could be signature partners, at least in a limited frequency range The excitation energies of the remaining SD bands, Ž SD, and SD4-SD8, could not be established as the transitions linking them to the rest of the level scheme could not be found In general, the relative population of states after heavy-ion induced fusion-evaporation reactions is strongly related to the excitation energy above the yrast line In order to estimate the relative excitation energies of these bands, which all have dynamic moments of inertia J Ž similar to SD and SD3, we have assumed that they also have similar alignments, i Thereby the spin values I;J Ž vqi can be estimated A comparison of the g-ray intensities at I;30", and the assumption of the same dependence of population

10 0 S Tormanen et alrphysics Letters B 454 ( 999) 8 4 Fig Calculated UC potential energy surfaces at I s 35 and 36 for the lowest configuration with positive left and negative right parity in 64 Lu intensity on excitation energy as found for the ND band structures w0 x, for which the spin and excitation energy are known, can then be used to provide an estimate of the excitation energies of these bands This assumption agrees with the measured relative intensities for bands SD and SD3 The results of this estimate are far from accurate, but give a rough idea of the relative placement of the bands The population of band SD suggests that it is placed around 00 kev above SD A change in spin value of " in the assumed spins of band SD would change the slope by 300 kev in the spin range Is0 40 " As can be seen from Fig 4a, such a change would cause SD to have an unrealistic slope relative to SD and SD3 keeping the relative excitation at I;30" For the bands SD4-SD8, the excitation energies are most likely within 00 kev above SD3 From this information it appears that all the new bands are found within an energy range of ;400 kev at I ;30" The new bands may belong to both of the calculated minima with g;"08, which have close to identical shapes but rotation around the smallest or intermediate axis The calculated kinematic and dynamic moments of inertia, J Ž and J Ž, do not show a large difference between bands of positive and negative g values, whereas an appreciable difference is found in the transition quadrupole moments for the two minima At present, without measured transition quadrupole moments, only indirect evidence for the sign of g can be given Experimental and calculated excitation energies are compared in Fig 4 According to the calculation, the lowest excitation energies of the triaxial SD bands are expected for a deformation with positive g values The bands with negative g deformation lie about 05 MeV higher than those for positive g The SD configuration with Ž p, a sy,0 corresponds to the lowest excitation with g;q08 which is in agreement with the parity and spin determined for the lowest-energy SD band The lowest positiveparity SD band is expected about ;00 kev higher in energy Bands SD and SD3 are therefore in qualitative agreement with the lowest calculated bands for g; q08 although the experimental alignments apparently are larger than calculated for the triaxial SD bands This could be caused by a strong deformation dependence of the aligning n i3r configurations in the calculations In 63,65 Lu the identical triaxial SD bands were assigned to the lowest p i3r orbital with a gradual alignment of the first 64 pair Ž AB of i3r quasineutrons In Lu the lowest calculated configuration is obtained by coupling the i quasiproton to the lowest negative parity Ž h 3r 9r quasineutron This is supported by an alignment increase of ;" relative to the bands in 63,65 Lu,

11 S Tormanen et alrphysics Letters B 454 ( 999) 8 4 Fig Partial level scheme showing the new triaxial SD bands in 64 Lu Only the lowest-energy positive and negative parity ND bands to which the new triaxial SD bands decay are included For the hanging bands the excitation energy is estimated from their intensities See text which agrees with the expected alignment for the h9r quasineutron in the observed frequency range For SD3 with Ž p,a sq,, the alignment increase relative to the SD bands in 63,65 Lu is ;5" This band could possibly be assigned to the configuration p i m n i with a gradual second Ž BC 3r 3r i3r quasineutron alignment It should be noted that the configurations, p i3rmn h9r and p i3rmn i 3r, assigned to bands SD and SD3, both, in addition to the triaxial SD minima, have a normal deformed

12 S Tormanen et alrphysics Letters B 454 ( 999) 8 4 Ž 64 Fig 3 a Sum of double gates on transitions in the triaxial SD band connected to ND states in Lu The linking transition of 8 kev is marked by an arrow Ž b Dynamic moments of inertia for the yrast ND band Ž full symbols and all the new SD bands Ž open symbols in 64 Lu minimum close to the minima at ;0 shown in Fig, but at considerably higher excitation energy With a detailed and complex decay-out of SD and a single decay of SD3, it is interesting to compare the E strength to statistical expectations For SD the strongest E decay-out branch is the I I y transition of 8 kev The 80 kev I I and 5 kev I Iq E transitions are G5 and G0 times weaker, respectively, which may be partly 3 explained by the E dependence Ž See Fig 3a g The strength of the E decay is estimated Žassuming Q Ž SD s b t from the out-of-band to in-band y4 branching to be B E ;08P0 e fm Žs 04P y4 0 WU for both bands, which is around 400 times faster than the E-decay found for the Žaxially sym- 94 metric SD to ND states in Hg w x, and only ;6 times slower than octupole-enhanced E transitions between some of the ND bands in the same nucleus,

13 S Tormanen et alrphysics Letters B 454 ( 999) Ž 64 Fig 4 a Excitation energy versus spin for three SD and selected ND bands in Lu The strongest E decay branches from SD to ND are indicated The spin and excitation energy of SD are rough estimates based on alignments and population intensities See text Ž b 64 Calculated Ž UC excitation energy for bands in Lu corresponding to the ND and the triaxial SD local minima with gs";08 shown in Fig 64 Lu w0 x The E decay from SD is associated mainly with an h9r SD to i3r ND quasineutron transition, whereas the E-decay from SD3 is associated with an i3r SD to hr ND quasiproton transition Octupole enhancement w x is found between ND bands of similar structure in odd-n and odd-z rare earth nuclei and may therefore be present in both of these different E transitions The measured g-ray branching ratio for the 6 y 5 y, 45 kev and 6 y 5 q, 8 kev transitions is ;03 From a similar estimate, Žassuming negligible M contribution the values of BŽ E,SD ND are times reduced compared to the SD in-band BŽ E values This is quite different from the decay

14 4 S Tormanen et alrphysics Letters B 454 ( 999) 8 4 of the triaxial SD band in 63 Lu In that nucleus the decay to the normal deformed structures takes place through an isolated mixing at I s r with an interaction strength of around 0 kev of the SD with w x q the 4 r configuration wx 9 The measured difference in energy between the bands SD3 and SD in 64 Lu is ;00 kev, in agreement with the calculated energy difference for g; q08 at I ; 30" In contrast, the measured excitation of the bands SD and SD3 relative to the ND yrast band shows that the triaxial SD minima in 64 Lu appear 05 MeV lower in excitation energy than calculated for g;q08, as shown in Fig 4 For the p i SD band in 63 3r Lu the calculated excitation energy relative to the yrast ND p h r band is also considerably higher than measured wx 9 It should be noted that the UC calculations are based on generally accepted best Nilsson parameters w3 x However, the most important deformation driving single-particle levels Ž such as p i 3r were not included in the fitting of the Nilsson parameters for nuclei in the deformed rare earth region The position of those levels may therefore be incorrect, resulting in systematic deviations in the calculated excitation energies for rotational bands in the SD minimum Alternatively, a more speculative explanation could possibly be found in a different rotational scheme Inspecting Fig, it is clear that the two SD minima are very similar in shape The real minimum with the same shape might be found at a lower excitation energy rotating around an axis tilted away from the principal axes In summary, the present letter reports on eight, new presumably triaxial, SD bands in 64 Lu The structure of the two lowest bands for which spin and parity have been determined correspond most likely to shapes with g; q08 Configurations have been assigned based on UC calculations and alignments From the measured E and E decay out of the SD bands, relative to the in-band decay, the E decay strength from SD to ND states is found to be strongly enhanced over expectations from single particle estimates Excitation energies of the new bands are lower than obtained from UC calculations which might imply problems with single particle intruder levels in the cranking calculations, or could be an indication for tilted rotation Acknowledgements This project has been supported by the Danish Natural Science Foundation, the EU TMR project no ERBFMBICT9607, the Swedish Natural Science Research Council, the Research Council of Norway, BMBF Germany and the UK EPSRC The dedicated help from staff and Euroball support groups at the INFN laboratory in Legnaro is highly appreciated References wx S Aberg, Nucl Phys A 50 Ž c, and refs therein wx I Ragnarsson, Phys Rev, Lett 6 Ž wx 3 H Schnack-Petersen, Nucl Phys A 594 Ž wx 4 W Schmitz et al, Nucl Phys A 539 Ž 99 ; Phys Lett B 303 Ž wx 5 See wx 6 X-H Wang, Nucl Phys A 608 Ž wx 7 MA Cardona, Phys Rev C 56 Ž wx 8 M Palacz, Nucl Phys A 65 Ž wx 9 J Domsheit et al, to be published w0x A Harsmann et al, to be published wx GB Hagemann, I Hamamoto, Phys Rev C 47 Ž wx G Hackman, Phys Rev Lett 79 Ž w3x T Bengtsson, Nucl Phys A 5 Ž 990 4

15 3 May 999 Physics Letters B Nature of excited 0 q states in 54 Sm R Krucken a, CJ Barton b, CW Beausang a, RF Casten a, G Cata-Danil a, JR Cooper a, J Novak a, L Yang a, M Wilhelm a, NV Zamfir a,b,c, A Zilges a,d a AW Wright Nuclear Structure Laboratory, Yale UniÕersity, New HaÕen, CT 0650, USA b Clark UniÕersity, Worcester, MA 060, USA c National Institute for Physics and Nuclear Engineering, Bucharest-Magurele, Romania d Institut fur Kernphysik, Technische UniÕersitat Darmstadt, Germany Received 5 January 999; received in revised form 4 March 999 Editor: JP Schiffer Abstract Lifetimes of the 0 q and q states in 54 Sm were measured using the Doppler-shift Attenuation Method following g Coulomb excitation The experiment employed the gamma-ray detector array YRAST Ball at Yale University in conjunction with an array of photo-electric cells 54 Sm was identified as one of the few deformed nuclei where the first excited 0 q state is the b-vibration of the ground state The 0 q level has no collective decay to the ground state band and is interpreted as a 3 spherical shape coexisting state based on its decay properties and calculations in the framework of the interacting boson model IBM and geometric collective model GCM q 999 Elsevier Science BV All rights reserved PACS: 0Re; 0Tg; 60Ev; 30Js; 770qq The nature of the lowest excited 0 q levels in deformed even-even nuclei remains poorly understood despite intensive investigation In the traditional view the first excited 0 q state is interpreted as a b-vibration built on the ground state wx but recently this interpretation has been subject of intense discussions w 6 x In particular the available experimental data Žsee, eg, Refs w7 9x and references therein and new theoretical studies Žsee, eg, Refs w0,x have established that in many nuclei the first excited 0 q state is connected only by a weak transition to the ground state band while one often finds a strong electric quadrupole Ž E transition to the band head of the first excited K p s q band, the gamma or quasi-gamma band While this situation, which contradicts the traditional interpretation, seems to be present in several deformed nuclei, only in a few cases, such as the 0 level in Er 8, has it has been possible to identify strong candidates for a b-vibrational 0 q state q 66 wx 4 Recently the nature of the first excited 0 q state in 5 Sm has been investigated experimentally wx and theoretically w3 x While at first glance the strong transition of this level to the ground state band seems consistent with an interpretation as a b-vibrational 5 state w x, it has been argued that Sm presents a r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

16 6 R Krucken et alrphysics Letters B 454 ( 999) 5 very special case w3 x, since this nucleus happens to lie in the middle of a very sharp ground state shape transition region between the spherical 50 Sm and the deformed nucleus 54 Sm The strong reduced transi- q q tion probability B E between the 0 and the states in 5 Sm can be explained by a shape mixing of a deformed state Ž0 q, similar to the ground state 54 in Sm, and a spherical state Ž0 q, similar to the ground state in 50 Sm In 5 Sm these shape coexisting states are sufficiently close to each other that a shape mixing occurs, leading to the strong transition between the resulting states With the strong interest generated by the new results in 5 Sm, it becomes of high interest to study the neighbouring isotopes in order to understand the evolution of structure in this transition region Therefore, the purpose of the present Letter is to report on an investigation of the nature of the excited 0 q states in 54 Sm In this nucleus the first excited 0 q level is well established at 099 kev and the q,4 q, and 6 q members of this K p s0 q band have been identified w x q q 4 While the 59 kev 4 is the only known in-band transition, the branching ratios of the various members of the K p s0 q band to members of the ground state band are well established Previously, a Ž q q B E; 0 value of e b has been reported by Veje et al w5x from inelastic scattering of a-particles However, as already pointed out by the authors of Ref w5 x, the incident energy of the a-particles was close to the value of the Coulomb barrier, which led to a significant deviation of the excitation cross section from that of pure Coulomb excitation and therefore to uncertainties in the BŽ E value An earlier work using NaI detectors by Yoshizawa et al w6x found a large excitation probability for the K p s0 q band in Coulomb excitation of 54 Sm with an 6 O beam However, they could not resolve the excitation cross-sections for individual members of this band nor did they take multiple Coulomb excitation into account The previous studies of the K p s0 q band and its interpretation as a b-vibrational excitation have to be viewed in the light of the limitations of these experiments as well as their analysis The new experimental results presented in this Ž q letter led to the discovery of a large B E; 0 q value and the clear identification of the 0 q level as the b-vibrational excitation of the ground state They Fig Schematic drawing of the arrangement of the eight element array of photoelectric cells in the Coulomb excitation experiment of 54 Sm The arrangement shown was mounted inside the target chamber of the YRAST Ball array at WNSL also show that the nearby 0 q 3 level has completely different properties: it is interpreted as a shape coexisting spherical 0 q level similar to the 0 q state in 5 Sm Therefore, 54 Sm represents one of the rare cases where the lowest 0 q state of a deformed nucleus appears to represent a collective b-vibrational mode built on the ground state States in 54 Sm were Coulomb excited by a 65 MeV 6 O beam incident on a 3 mgrcm 98% isotopically enriched 54 Sm foil, which was backed by a 5 mgrcm Au layer to stop nuclei recoiling in the forward direction The beam was delivered by the ESTU Tandem accelerator of the AW Wright Nuclear Structure Laboratory Ž WNSL at Yale University The g-rays emitted by the Coulomb excited nuclei were detected by the YRAST Ball array w7 x, which at the time of the experiment comprised 3 unshielded segmented Clover detectors, a 75% efficient Compton shielded coaxial HPGe detector and 8 Compton shielded coaxial HPGe detectors with efficiencies ranging from 7% to 5% Additionally, backward scattered beam particles were detected by an array of eight cm = cm large photo-electric cells arranged as indicated in Fig and positioned 4 cm upstream from the target position The photoelectric cells were operated without bias voltage and were directly connected through a charge sensitive preamplifier to a spectroscopic amplifier Details on the performance of this array of photo-electric cells will be published in a forthcoming article w8 x The cells covered angles from 608 to 738 with respect Type LC805 by Silicon Sensors Inc

17 R Krucken et alrphysics Letters B 454 ( 999) 5 7 to the beam direction Detection of g-rays in coincidence with 6 O particles scattered at these angles selected 54 Sm nuclei recoiling forward into the Au layer in a cone of 8 to 08 This allowed the observation of Doppler-shift lineshapes for transitions from levels with a lifetime on the order of the average stopping time Ž f ps of the recoils The detectors of YRAST Ball can be grouped into 4 different rings, where all the detectors of a given ring are positioned at the same angle with respect to the beam axis In this experiment there were 3 coaxial detectors at 638, 8 coaxial detectors at 58, 3 clover detectors at 908, and 8 coaxial detectors at 508 Fig shows spectra for the 440 kev q g 0 q and 0 kev 0 q q transitions in 54 Sm as ob- served in the detectors at 508 and 58 The lineshapes obtained were analysed using the standard Doppler Shift Attenuation Method Ž DSAM analysis package by Wells et al w9 x The complete stopping was modelled using the prescription discussed in detail by Gascon et al w0 x The code was slightly modified to accommodate the different momentum transfer to the recoiling nuclei in the Coulomb excitation process as compared to the fusion evaporation reactions it was written for The electronic stopping powers by Northcliffe and Schilling wxincluding shell corrections were used in the simulation of the stopping process and the nuclear component to the stopping was treated according to the theory of Lindhard et al w x In the lineshape analysis a direct population of the two levels was assumed The fact that the beam energy was about 95% of the value of the Coulomb barrier could introduce population of the levels through higher lying states after direct reactions However, this effect has to be small, since the gamma-gamma coincidences did not reveal any feeding transitions and an upper limit of about 0% can be set on the total feeding intensity Therefore it is reasonable to assume prompt population via the Coulomb excitation process The fitted lineshapes are included in Fig and excellent agreement between experimental and calculated lineshapes can be observed Lifetimes of 3Ž 3 ps and 06Ž 4 ps were determined for the 0 q and q levels in 54 g Sm, respectively While the given uncertainties represent statistical uncertainties one cannot exclude the possibility of systematic uncertainties of up to about 5% due to the insufficient knowledge of the stopping process The lifetime of the q g level measured in this work is slightly higher than the value of 040Ž 7 ps extracted Ž q q 4 from the B E;0 g of e fm in Ref w x q 5 The lifetime of the 0 level is established for Ž q the first time and results in a B E; 0 q value of e fm The 0% limit on the unobserved feeding could only result in a f0% reduction of the level lifetime and thus a small increase in the BŽ E values Therefore, the discussion below is independent of the assumption of prompt feeding The transitions from the q and 4 q members of the lowest excited K p s0 q band to the ground state band did not exhibit any Doppler-shift lineshapes This is of particular interest since we would have expected to observe a broadened lineshape for the transitions from the q level if its reported lifetime of Ž 5 ps w5x were correct Again assuming prompt feeding for this level we can set a lower limit of 35 ps for its lifetime The known feeding of this level from the 4 q at 338 kev was determined to be only of the order of 3% and therefore does not play a significant role We have performed Coulomb excitation calculations, including multiple step excitations, and found that the relative populations of the q and 0 q members of the lowest K p s0 q band can be reproduced with a lifetime of the q level of the order of 35 ps However, due to the possible effects of direct reactions in the population of the q level as well as the ambiguous relative signs of various matrix elements in the Coulomb excitation calculations, we did not feel confident enough to extract a Ž q B E; 0 q values from the intensity We rather conclude that the Coulomb excitation calculations as well as the fact that not broadened lineshapes were observed consistently indicate that the lifetime of the q level at 78 kev is at least 35 ps and not 05 ps as reported earlier w5 x The resulting BŽ E values for the various transitions from the q g and 0 q to lower lying states are shown in Fig 3 It is assumed that there is no significant E0 0 q 0 q transition and that the q g q transition is of pure E character Unfortunately there is no experimental data available to put these assumptions on a firmer ground However, if the 0 q is indeed the b-vibrational state one expects an E0 Ž matrix element of r E0 s spu using Eq 9

18 8 R Krucken et alrphysics Letters B 454 ( 999) 5 q q q q 54 Fig Spectra at forward and backward angles for the 440 kev 0 top and 08 kev 0 Ž bottom g transitions in Sm The lineshape fits obtained by the DSAM analysis are also shown w x Ž q of Ref 4, which would reduce the B E;0 q value by only 0% The situation in 54 Sm regarding the nature of the excited 0 q states is particular interesting since it is the only nucleus in the rare earth region where two excited 0 q states are established well below the energy of the q g level In this Coulomb excitation experiment the 0 q level w 5 x was only weakly popu- 3

19 R Krucken et alrphysics Letters B 454 ( 999) Fig 3 Partial experimental level scheme for Sm Ž middle in comparison with IBM- calculations by Scholten et al w3x Ž left and GCM calculations Ž right See Ref w3x and text for parameters used in the calculations For each transition the BŽ E value in Weisskopf units is q q given For the IBM calculations the B E values were normalised to the experimental value of 74 Wu for the 0 ground state transition lated although it is only 03 kev above the 0 q level Both, the 0 q and 0 q levels are mostly populated in 3 two-step Coulomb excitation: one-step excitations are impossible and Coulomb excitation calculations show that the probability for higher order multi-step excitations is significantly reduced In order to reproduce the apparent population intensities of these two levels, measured by the intensities of the 0 kev 0 q 3 q and 08 kev 0 q q transitions, the Ž q B E; 0 q must be 40 times stronger than the Ž q q q B E; 03 This assumes that the 03 level is only populated via the q level while the 0 q level is populated also via the q and q levels However, 3 even if we were to add large matrix elements from other known q levels to the 0 q 3 level, which might not be physical, the BŽ E ratio above can at most be reduced by a factor of Using a BŽ E ratio of 40 the lifetime result for the 0 q level then allows the deduction of a reasonable upper limit of 03 Wu for Ž q the B E;0 q 3 value Due to the proximity of the 0 q and 0 q 3 levels it is natural to assume that there might be a significant mixing between them This was already suggested in the discussion of the Ž t,p cross-sections by Bjerrew6 x With the reasonable assumption of gaard et al an upper limit of 0 kev for the interaction strength between the two excited 0 q states the original separation of the two levels must have been at least 95 kev In the present situation one of the mixed transitions carries virtually all of the BŽ E strength, which can only be achieved if the ratio of the BŽ E values of the unmixed states is essentially equal to the ratio of the squared mixing amplitudes of the mixed wavefunctions Using this relation and the above values for interaction strength and separation one obtains an upper limit of 004 for the smaller of the two squared mixing amplitudes Therefore, the BŽ E values of the unmixed states to the ground state band must differ by at least a factor of 4 to lead to a concentration of virtually all the BŽ E strength in one transition from the mixed states This shows that the two excited 0 q states in 54 Sm can only weakly mix and that the 0 q level has no collective transition 3 to the ground state band At the same time the strength of the 0 q q transition is clearly consis- tent with the 0 q level being the b-vibration built on the ground state There is no indication in the data for the existence of a 38 kev q g 0 q 3 transition although the upper limit for the intensity of this transition is insufficient to exclude a collective character of such a transition Žfor example a transition between gyg and g-vibra- tional modes, which would be greatly inhibited by the small transition energy However, the relatively

20 0 R Krucken et alrphysics Letters B 454 ( 999) 5 q large t,p cross section to the 0 state w6x 3 is inconsistent with an interpretation of this level as a member of the double-gamma phonon multiplet The experimental data were compared with theoretical calculations in the framework of the interacting boson model Ž IBM w7x and the geometric collective model Ž GCM w8 30 x Fig 3 shows a partial 54 level scheme of Sm including BŽ E values in comparison with results of IBM calculations by Scholten et al w3x and GCM calculations carried out in the course of this work The GCM calculations used the simplified Hamiltonian proposed by Zhang et al w3x with parameters of C sy75 MeV, C 3s 50 MeV, C 4s900 MeV, Bs68=0 y4 MeV s Žall other parameters were set to zero following the approach of Ref w3 x The agreement for the energies and BŽ E values in the ground band, the first excited K p s0 q band and the gamma-band is reasonable while both models fail to predict a low lying 0 q 3 level It is interesting to take a closer look at the structure of the wavefunctions of the 0 q states in these calculations Fig 3 of Ref w3x clearly shows that the IBM calculations predict that the ground state as well as the 0 q state in 54 Sm have a very broad distribution in terms of the squared wave function amplitudes as a function of n d At the same time the 0 q level shows a dominant contribution for the 3 number of quadrupole bosons nd s 0, consistent with a spherical shape To understand the possible structure of the 0 q 3 level in 54 Sm it is very useful to recall the situation in the lighter even-even Sm isotopes As pointed out by Iachello et al w3x 5 Sm is positioned exactly at the point of the shape transition between 50 Sm, which is spherical in its ground state, and 54 Sm, which is strongly prolate deformed in its ground state In 5 Sm the structure of the ground state and the first excited 0 q state are a result of the mixing between the shape coexisting spherical and deformed q 54 0 states w3 x In Sm the deformed shape is energetically lowest and the coexisting spherical shape is significantly higher in energy so that little mixing occurs The decay properties in conjunction with the theoretical predictions suggest that the 0 q 3 level has a large amplitude of the spherical shape coexisting state In this case the large Ž t,p cross-secw6x to this level could be explained by tion a significant probability Ž f 30% of a spherical component in the ground state of 5 Sm The vastly different structure of the 0 q state and the b-vibra- tional 0 q state in 54 Sm accounts for the fact that these two states show no significant mixture despite their proximity However, it remains unclear why the 0 q 3 level is much lower in energy than predicted in either the GCM or IBM calculations In summary, lifetimes for the 0 q and q levels g in 54 Sm were measured using the DSAM technique after Coulomb excitation The experiment employed the WNSL YRAST Ball array and an array of eight photo-electric cells for the detection of g-rays and backscattered beam particles, respectively The lifetime of the 0 q state was measured for the first time Ž q and the extracted B E;0 q of Wu is 3 consistent with the interpretation of this level as a b-vibration built on the ground state of 54 Sm These results have established one of the few cases where the lowest excited 0 q state indeed seems to be a b-vibrational excitation In fact only in 66 Er, 7 Hf 74 Ž q and Hf have the B E;0 q values been measured and none of the 0 q levels is as collective as the one in 54 Sm In several other nuclei in the rare earth region, such as 58 Dy, 6 Dy, 68 Yb, and 74 Hf, the q member of the lowest excited K p s0 q band has collective transitions to the ground state band However, these values are more difficult to interpret since band mixing has to be taken into account From the Coulomb excitation cross-sections of the 0 q and 0 q 3 levels it was possible to determine an Ž q upper limit for the B E;0 q of 03 Wu 3 Therefore a significant mixing of the two close lying excited 0 q states can be excluded The low-spin 54 structure and BŽ E values of Sm were well reproduced in the IBM and GCM calculations However, both models fail to reproduce the low excitation energy of the 0 q 3 level The IBM model predicts a large nd s0 wavefunction amplitude in the 0 q 3 level suggesting that this state is the spherical shape coexisting state that is related to the ground state in 50 Sm and to the 0 q state in 5 Sm, which mixes 5 Ž q into the ground state in Sm The small B E;0 q value as well as the relatively large Ž t,p crosssection to the 0 q 3 level support this interpretation Collective models, such as the IBM or GCM for most of their parameter space predict that the lowest excited 0 q states in deformed nuclei do not have a 3

21 R Krucken et alrphysics Letters B 454 ( 999) 5 large BŽ E value to the ground state band This seems to suggest that either the b-vibrational mode does not in fact exist at low energies or is fragmented To date 54 Sm presents one of the few counter examples to this, where the experimental results show a collective transition from the first excited 0 q to the ground state band which is also predicted by the IBM and GCM calculations However, it has to be stressed that these theoretical predictions can only be obtained in a relatively small parameter space of both models Putting the present results in the perspective of the rare earth region suggests that the nature of the excited 0 q states is complex, with several competing degrees of freedom, any one of which can lie lowest in energy for a given Ž N,Z value Understanding the interplay of these degrees of freedom is one of the key challenges in the microscopic interpretation of the elementary collective modes of deformed nuclei Acknowledgements The authors would like to acknowledge useful discussions with J Saladin This work is supported by the US-DOE under contract numbers DE-FG0-9ER and DE-FG0-88ER-4047 References wx A Bohr, B Mottelson, Nuclear Structure, Benjamin, New York 969, 975, and World Scientific, New York, 998, vol II wx RF Casten, P von Brentano, Phys Rev C 50 Ž 994 R80 wx 3 K Kumar, Phys Rev C 5 Ž wx 4 DG Burke, PC Sood, Phys Rev C 5 Ž wx 5 RF Casten, P von Brentano, Phys Rev C 5 Ž wx 6 C Gunther, S Boehmsdorff, K Freitag, J Manns, U Muller, Phys Rev C 54 Ž wx 7 PE Garrett, M Kadi, CA McGrath, V Sorokin, Min Li, Minfang Yeh, SW Yates, Phys Rev Lett 78 Ž wx 8 PE Garrett, M Kadi, CA McGrath, V Sorokin, Min Li, Minfang Yeh, SW Yates, Phys Lett B 400 Ž wx 9 H Lehmann, J Jolie, F Corminboeuf, HG Borner, C Doll, M Jentschel, RF Casten, NV Zamfir, Phys Rev C 57 Ž w0x RF Casten, P von Brentano, Phys Rev C 50 Ž 994 R80 wx Takaharu Otsuka, Ka-Hae Kim, NV Zamfir, RF Casten, Phys Lett B 35 Ž wx RF Casten, M Wilhelm, E Radermacher, NV Zamfir, P von Brentano, Phys Rev C 57 Ž 998 R553 w3x F Iachello, NV Zamfir, RF Casten, Phys Rev Lett 8 Ž w4x J Morikawa, Z Phys A 343 Ž w5x E Veje, B Elbek, B Herskind, MC Olesen, Nucl Phys A 09 Ž w6x Y Yoshizawa, B Elbek, B Herskind, MC Olesen, Nucl Phys 73 Ž w7x CW Beausang et al, to be published w8x R Krucken et al, to be published w9x JC Wells, N Johnson Ž private communication Modified from the original code by JC Bacelar w0x J Gascon, Nucl Phys A 53 Ž wx LC Northcliffe, RF Schilling, Nucl Data Tables 7 Ž wx J Lindhard, K Dan Vidensk Selsk Mat Fys Medd 33 Ž w3x O Scholten, F Iachello, A Arima, Annals of Physics 5 Ž w4x K Heyde, RA Meyer, Phys Rev C 37 Ž w5x SA Elbakr, IJ van Heerden, BC Robertson, WJ McDonald, GC Neilson, WK Dawson, Nucl Phys A Ž w6x JH Bjerregaard, O Hansen, O Nathan, S Hinds, Nucl Phys 86 Ž w7x F Iachello, A Arima, The Interacting Boson Model, Cambridge University Press, 987 w8x G Gneuss, U Mosel, W Greiner, Phys Lett 30 B Ž w9x G Gneuss, W Greiner, Nucl Phys A 7 Ž w30x G Gneuss, U Mosel, W Greiner, Phys Lett B 3 Ž w3x Jing-ye Zhang, RF Casten, NV Zamfir, Phys Lett B 407 Ž 997 0

22 3 May 999 Physics Letters B Pre-bangian origin of our entropy and time arrow G Veneziano Theory DiÕision, CERN, CH- GeneÕa 3, Switzerland Received 3 February 999 Editor: R Gatto Abstract I argue that, in the chaotic version of string cosmology proposed recently, classical and quantum effects generate, at the time of exit to radiation, the correct amount of entropy to saturate a Hubble Ž or holography entropy bound Ž HEB and to identify, within our own Universe, the arrow of time Demanding that the HEB be fulfilled at all times forces a crucial branch change to occur, and the so-called string phase to end at a critical value of the effective Planck mass, in agreement with previous conjectures q 999 Elsevier Science BV All rights reserved The origin of the present entropy of our Universe, S 0, is one of the deepest cosmological mysteries The 7 K cosmic microwave background Ž CMB, if it indeed fills our observable Universe uniformly, contributes a gigantic 0 90 to S 0 However, as repeatedly emphasized by many people, most notably by Roger Penrose wx, such an amount falls very short of what entropy could have been expected to be, even if we go back to the Planckian era, ie to tst P ;0 y43 s after the big bang Since entropy can only grow, the entropy of our Universe at tst P, S P, must be smaller than S ; yet, on the basis of the energy 0 content and of the size of the Universe R at P t;t P'lPrc, we might have expected S ;E R rc";r R ; Ž R rl ;0 Ž P P P P P P P The fact that the entropy of our Universe must have been at least 30 orders of magnitude smaller Throughout this paper we will stress functional dependences while ignoring numerical factors than the value in Ž would nicely explain our arrow of time, by identifying the beginning of the Universe with this state of incredibly small entropy near the Planck time wx In order to solve the problem, Penrose wx invokes, without much justification, a new Weyl-curvature hypothesis The expected value given in Eq Ž coincides with the so-called Bekenstein entropy bound Ž BEB wx, which states that, for any physical system of energy E and physical size R, entropy cannot exceed SBB s ERrc" This bound is saturated by a black hole of mass E and size equal to its Schwarzschild radius RsGE If the newly born Universe were a single black hole its Schwarzschild radius would have been much larger than R P, and an even higher entropy, Ž 80 O 0, would have resulted What could have made the initial entropy much smaller than SBB is instead the possibility that the Universe, right after the big bang, was already in a very ordered, homogeneous state But this is just restating the puzzle in terms of the usual homogeneity problem of standard wx non-inflationary cosmology r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

23 ( ) G VenezianorPhysics Letters B The way the two problems are related can be made explicit by introducing a stronger bound on entropy, which, unlike Bekenstein s general bound, should apply to the special case of Ž fairly homogeneous cosmological situations We shall call it the Hubble entropy bound and formulate it as follows: Consider a sufficiently homogeneous Universe in which a Ž local Hubble expansion Ž or contraction rate can be defined, in the sinchronous gauge, as: H;r6 E Ž log g, g'det Ž g, t with H varying little Ž percentage-wise over dis- Ž y y tances O H In this case H, the so-called Hubble radius, is known to correspond to the scale of causal connection, ie to the scale within which microphysics can act In such a context it is hard to imagine that a black hole larger than H y can form, since, otherwise, different parts of its horizon would be unable to hold together Thus, the largest entropy we may conceive is the one corresponding to having just one black hole per Hubble volume H y3 Using the Bekenstein Hawking formula for the entropy of a black hole leads to our proposal of a Hubble entropy bound Ž HEB : Ý S-S 'l y H y ;n S, Ž 3 HB P i H H i ij where the sum runs over each Hubble-size region The last estimate in Ž 3 assumes a fairly constant H throughout space, and defines nh as the number of Hubble-size regions, each one carrying maximal entropy SHsl y P H y This bound appears to be related, at least in some cases, to the one recently proposed by Fischler and Susskind wx 4 on the basis of the so-called holography principle We may thus take SHB to stand for Hubble or Holography entropy bound according to taste If one applies the HEB to the initial Universe, one finds wx 4 that, unlike the BEB, it is practically saturated It is widely fulfilled thereafter wx 4 This letter aims at explaining why saturation of the HEB naturally takes place at the big bang in the context of pre-big bang Ž PBB cosmology wx 5 Before proceeding, we note that, in ordinary inflation, the entropy problem is solved wx 3 by invoking a nonadiabatic reheating process occurring after inflation Since inflation has already made the Universe homogeneous, after thermal equilibrium is reached entropy is given by its standard thermodynamic relation to temperature Ž here the reheating temperature as S RH ;TRH 3 RRH 3 ;S 0 Ž 4 One thus naturally obtains the correct value However, unlike what will be shown to be the case in the PBB scenario, SRH fails to saturate the HEB since: SHB Ž tstrh sl y P HRH RRH 3 sh y RHTRH 4 RRH 3 s Ž TRHrHRH SRH4S RH Ž 5 In order to discuss various forms of entropy in the PBB scenario, let us recall some basic ideas, which have emerged from recent studies of the latter Žsee wx 6 for a review It now looks quite certain that generic though sufficiently weak initial conditions lead to a form of stochastic PBB, which, in the Einstein-frame metric, can be seen as a sort of chaotic gravitational collapse w7,8 x Black holes of different sizes form but, for an observer inside each horizon measuring distances with a stringy meter, this is experienced as a pre-big bang inflationary cosmology in which the t s 0 Ž hopefully fake big bang singularity is identified wx 8 with the Žhopefully equally fake black hole singularity at r s 0 We are thus led to identifying our observable Universe as what became of a portion of space that was originally inside a large enough black hole In general, if we want to achieve a very flat and homogeneous Universe, we should better identify our present Universe with just a tiny piece of the collapsingrinflating region For the purpose of this note, however, this would only complicate the equations without adding new physical information This is why, hereafter, we shall identify our present Universe with the whole interior of a single initial black hole It is helpful to follow the evolution of various entropies with the help of Fig At time t s t i, corresponding to the first appearance of a horizon, Note that, while we shall work in the string frame throughout, the same results would also follow in the Einstein frame

24 4 ( ) G VenezianorPhysics Letters B Fig At the beginning of the DDI era Ž tst i the entropy of the just-formed black hole, S coll, coincides with both the BEB, S BB, and the HEB, S HB, while the entropy in quantum fluctuations, S qf, is completely negligible At the beginning of the string phase, tst s, both Scoll and SHB still have their common initial value SBB and Sqf have grown considerably, but the latter is still negligible if the string coupling is still small at tst s During the string phase, Sqf catches up with Scoll first, and with SHB later, ie when the energy in the quantum fluctuations becomes critical and exit to radiation is expected Ž tst r Finally, during the radiation and matter-dominated phases, SHB grows towards S BB, while our own entropy Stot lags far behind and increases only slowly as the result of dissipative phenomena and growth of inhomogeneities we can use the Bekenstein Hawking formula to argue that y S coll; Ž Rinrl P,in ; Ž HinlP,in ssbbss HB, Ž 6 where we have used the fact wx 8 that the initial size of the black-hole horizon determines also the initial value of the Hubble parameter Thus, at the onset of collapserinflation, there is no hierarchy between the two bounds and entropy is as large as allowed by them Furthermore, since the collapsing region is large in string Ž and a fortiori in Planck units, Eq Ž 6 corresponds to a large number Incidentally, this number is also close to the number of quanta needed for the collapse to occur wx 8 We have also assumed the initial quantum state to be the ground state Because of the small initial coupling and curvature, quantum fluctuations around it are very small wx 9, initially, and contribute a negligible amount Sqf to the total entropy After a short transient phase, dilaton-driven inflation Ž DDI should follow and last until t s, the time at which a string-scale curvature OŽ M s is reached We expect the process not to generate further entropy Žunless more energy flows into the black hole, but this would only increase its total comoving volume, but what happens to the two bounds? This is the crucial observation: while the HEB also stays constant, the BEB grows, causing a large discrepancy between the two at the end of the DDI phase In order to show this, let us recall one of the equations of string cosmology wx 5, the conservation law: yf 3 yf y Ž ' tž Ž ' E e g H se gh e H t setž nhsh s0 Ž 7 Ž Ž Comparing with 3, we recognize that 7 simply expresses the time independence of the HEB during the DDI phase While at the beginning of the DDI phase n s, and the whole entropy is in a single H Hubble volume, as DDI proceeds the same total amount of entropy becomes equally shared among very many Hubble volumes until, eventually, each one of them contributes a relatively small number By contrast, it is easy to see that the BEB is increas- Ž ing fast during the DDI phase since, using 7, S BB ;MR;rR ;H e g Rsconst= Ž HR, Ž 8 4 yf '

25 ( ) G VenezianorPhysics Letters B and both R and H grow during DDI Also, having assumed that the string coupling is still small at the end of DDI, we can easily argue that the entropy in quantum fluctuations remains at a negligible level during that phase Something interesting happens if we now consider the string phase 3, characterized by a constant H and f It is easy to find that, if f)3h, the HEB starts to decrease while for f-3h it increases Clearly, the first alternative leads to a contradiction with the HEB, since Scoll cannot decrease We are thus led to the amusing result that the HEB demands f'fy 3HF0 during the string phase as opposed to the ḟ) 0 condition that characterizes the DDI phase Thus, the HEB implies a branch change occurring between the DDI and the string phase, a well known necessary condition for achieving a graceful exit w x The condition f-0 for the string phase also follows from Ž apparently independent arguments based on the study of late-time attractors w,3 x When will the final exit to the FRW phase occur? It has been assumed w4x that it does when the energy in the quantum fluctuations becomes critical, ie when r ;N H 4 se yf exit qf eff max M s Hmax, Ž 9 where Neff is the effective number of particle species produced Taking H max ;M s, fixes the value of the dilaton at exit, e f exit ;rn eff Using known results on entropy production due to the cosmological squeezing of vacuum fluctuations w5 x, we find: S ex ;N H 3 V;e yf exit qf eff max M s 3 V ; l rl Vl y3 ;S Ž ex, Ž 0 s P exit s HB ie saturation of the HEB by S qf Unless exit occurs at this point, the HEB will be violated at later times We thus arrive, generically, at the situation shown in Fig At tst ex't r, the entropy in the quantum fluctuations has catched up with Žand possibly overcome that of the classical collapse and has become equal to the HEB, S ; Ž HR 3 ; 0 90 By then, HB 3 We concentrate here on the standard PBB scenario wx 5 and not on its variant w0x in which the DDI phase flows directly into a low-energy M-theory phase SBB is a factor HR larger, which is precisely the factor 0 30 that we are running after From there on, the story is simple: our entropy remains, to date, roughly constant and around 0 90, while SHB keeps increasing with somewhat different rates during the radiation and the matter-dominated epochs S BB always remains a factor HR above S HB, but this factor, originally huge, shrinks to unity today, by definition In conclusion, the entropy and arrow-of-time problems are neatly solved, in PBB cosmology, by the identification of our observable Universe with Ž part of the interior of an original black hole As such, its initial entropy saturates both the HEB and the BEB and is large because of the assumed large size Ž in string or Planck units of the initial black hole From there on, there is a natural mechanism to provide saturation of the HEB at the beginning of the radiation-dominated phase, ie when the BEB lies some thirty orders of magnitude higher This is precisely what is needed to account for the initial entropy of our Universe, and to unambiguously identify its time arrow We do not wish to conceal the fact that our choice of the initial size of the collapsingrinflating region can be objected to, along the lines of Refs w6 x, as representing a huge amount of fine-tuning Our answer to this objection has been expressed elsewhere wx 7 : the classical collapserinflation process is a scale-free problem in General Relativity; as such, it should lead to a flattish distribution of horizon sizes, extending from the string length to very large scales, including those appropriate for giving birth to our Universe No other dimensionless ratio is tuned to a particularly large or small value as evidentiated in Fig by the three upper curves all originating from the same point at tst i Finally, we wish to stress again that the entropy considerations discussed in this note appear to provide new general arguments supporting previous conjectures on the way pre-big bang inflation should make a graceful exit into standard, post-big bang FRW cosmology Note added After completion of this work: i I became aware of a very recent paper w7x which reaches similar conclusions on the role of H y in cosmological entropy bounds

26 6 ( ) G VenezianorPhysics Letters B ii I was informed of some old work by J Bekenw8x extending his 98 bound wx to cosmol- stein ogy His proposal coincides with that of Ref wx 4 and Ž y with mine if the particle horizon is O H However, the three proposals all differ from each other when dp 4H y, as it is the case after a long inflationary phase I am grateful to A Feinstein for pointing out Ref w8x to me and to J Bekenstein for useful discussions about it Acknowledgements Useful discussions with M Bowick, R Brustein, M Gasperini, A Ghosh, F Larsen, R Madden and E Martinec are gratefully acknowledged References wx See, for instance, R Penrose, The Emperor s new mind, Oxford University Press, New York, 989, Chapter 7 wx JD Bekenstein, Phys Rev D 3 Ž 98 87; D 49 Ž 994 9, and references therein wx 3 EW Kolb, MS Turner, The early Universe, Addison-Wesley, Redwood City, CA, 990; AD Linde, Particle physics and inflationary cosmology, Harwood, New York, 990 wx 4 W Fischler, L Susskind, Holography and cosmology, hepthr ; see also D Bak, S-J Rey, Holographic principle and string cosmology, hep-thr98008; AK Biswas, J Maharana, RK Pradhan, The holography principle and prebig bang cosmology, hep-thr9805; SK Rama, T Sarkar, Holographic principle during inflation and a lower bound on density fluctuations, hep-thr98043 wx 5 G Veneziano, Phys Lett B 65 Ž 99 87; KA Meissner, G Veneziano, Phys Lett B 67 Ž 99 33; Mod Phys Lett A 6 Ž ; M Gasperini, G Veneziano, Astropart Phys Ž ; Mod Phys Lett A 8 Ž ; Phys Rev D 50 Ž An updated collection of papers on the pre-big bang scenario is available at wwwtoinfnit/gasperin/ wx 6 G Veneziano, Inflating, warming up, and probing the prebangian Universe, CERN-THr99-, hep-thr wx 7 G Veneziano, Phys Lett B 406 Ž ; A Buonanno, KA Meissner, C Ungarelli, G Veneziano, Phys Rev D 57 Ž , and references therein wx 8 A Buonanno, T Damour, G Veneziano, Pre-big bang bubbles from the gravitational instability of generic string vacua, hep-thr wx 9 A Ghosh, G Pollifrone, G Veneziano, Phys Lett B 440 Ž w0x M Maggiore, A Riotto, D-branes and cosmology, hepthr98089; see also T Banks, W Fishler, L Motl, Duality versus singularities, hep-thr9894 wx R Brustein, G Veneziano, Phys Lett B 39 Ž ; N Kaloper, R Madden, KA Olive, Nucl Phys B 45 Ž , Phys Lett B 37 Ž ; R Easther, K Maeda, D Wands, Phys Rev D 53 Ž wx M Gasperini, M Maggiore, G Veneziano, Nucl Phys B 494 Ž w3x R Brustein, R Madden, Phys Lett B 40 Ž 997 0; Phys Rev D 57 Ž w4x G Veneziano, in: A Zichichi Ž Ed, Effective theories and fundamental interactions, Erice, 996, World Scientific, Singapore, 997, p 300; A Buonanno, KA Meissner, C Ungarelli, G Veneziano JHEP0 Ž w5x M Gasperini, M Giovannini, Phys Lett B 30 Ž ; Class Quant Grav 0 Ž 993 L33; R Brandenberger, V Mukhanov, T Prokopec, Phys Rev Lett 69 Ž ; Phys Rev D 48 Ž w6x M Turner, E Weinberg, Phys Rev D 56 Ž ; N Kaloper, A Linde, R Bousso, Phys Rev D 59 Ž w7x R Easther, DA Lowe, Holography, cosmology and the second law of thermodynamics, hep-thr w8x JD Bekenstein, Int J of Th Phys 8 Ž ; see also M Schiffer, Int J of Th Phys 30 Ž 99 49

27 3 May 999 Physics Letters B Ns, Ds0 tensionless superbranes II P Bozhilov BogoliuboÕ Laboratory of Theoretical Physics, JINR, 4980 Dubna, Russia Received February 998 Editor: PV Landshoff Abstract We consider a model for tensionless Ž null p-branes with Ns global supersymmetry in 0-dimensional Minkowski space-time We give an action for the model and show that it is reparametrization and kappa-invariant We also find some solutions of the classical equations of motion In the case of null superstring Ž ps, we obtain the general solution in arbitrary gauge q 999 Published by Elsevier Science BV All rights reserved Introduction The null p-branes are the zero tension limit of the tensionful ones The correspondence between this two types of branes may be regarded as a generalization of the massless-massive particles relationship Null branes with manifest space-time or worldwx and wx volume supersymmetry are considered in respectively In a previous paper wx 3, we began the investigation of a tensionless p-brane model with N s supersymmetry in ten dimensional flat spacetime Starting with a Hamiltonian which is a linear combination of first and mixed Ž first and second class constraints, we succeed to obtain a new one, which is a linear combination of first class, BFVirreducible and Lorentz-covariant constraints only Work supported in part by the National Science Foundation of Bulgaria under contract f y60r996 bojilov@thsunjinrru; permanent address: Deptof Theoretical Physics, Konstantin Preslavsky Univ of Shoumen, 9700 Shoumen, Bulgaria This was done with the help of the introduced auxilw4,5 x Then we gave manifest iary harmonic variables expressions for the classical BRST charge, the corresponding total constraints and BRST-invariant Hamiltonian In this letter, we continue the investigation of the model Here, we consider the corresponding action, establish its symmetries, and present some solutions of the classical equations of motion Our initial Hamiltonian is wx 3 H p 0 j a H0s d s m T0qm Tj qm D a, where the constraints T 0, Tj and Da are defined by the equalities: T0 spm pnh mn, diagž hmn sž y,q,,q, Ž m,ns0,,,9, Tj spn Ej x n qpuaeju a, EjsErEs j, Ž js,,,p, D syip y Ž puu pu sp s n, a ua a ab n ab Ž as,,, r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

28 8 ( ) P BozhiloÕrPhysics Letters B Ž n Here x,u a are the superspace coordinates, Ž p, p n ua are their canonically conjugated momenta, u a is a left Majorana-Weyl space-time spinor, and m s are the ten dimensional Pauli matrices Žour spinor conventions are as in wx 3 The Hamiltonian Ž is a generalization of the Hamiltonians for the bosonic null p-brane and for the N s Brink- Schwarz superparticle Solutions of the equations of motion The equations of motion which follow from the Hamiltonian H are Ž E s ErEt : Ž t j 0 t E ym j E x n sm 0 p n y Ž ms n u, Ž Et ym j Ej pn s Ž Ej m j p n, Ž t j E ym j E u a sim a, Ž t j ua Ž j ua a E ym j E p s E m j p q Ž m pu Ž 0 j a In, one can consider m, m and m as depending Ž only on s s s,,s p but not on t Žthis is a consequence from their equations of motion In the gauge when m 0, m j and m a are constants, the general solution of Ž is Ž n n 0 n n x t,s sx z qt m p z y ms u t,s Ž n 0 n n sx z qt m p z y ms u z, p t,s sp z, n n u a Ž t,s su a Ž z qitm a, p t,s sp z qtž ms n p z, Ž 3 ua ua a n where x n Ž z, p Ž z, u a Ž z and p Ž n ua z are arbitrary functions of their arguments z j sm j tqs j In the case of tensionless strings Ž ps, one can write explicitly the general solution of the equations 0 0 of motion in arbitrary gauge: m sm s, m 'ms a ms, m sm a Ž s This solution is given by 0 s m Ž s n n n x Ž t,s sg Ž w yh dsf Ž w m Ž s q H s H a m Ž s n ds szž w mž s s n a Ž ms a Ž s s m Ž s H mž s mž s yi ds ds, p Ž t,s sm y Ž s f Ž w, n a s m Ž s a a u Ž t,s sz Ž w yih ds, mž s p t,s ua n n s Ž ms a Ž s y sm Ž s ha Ž w yh dsfn Ž w mž s Ž 4 Here g n Ž w, f Ž w, z a Ž w and h Ž w n a are arbitrary functions of the variable s ds wstq H mž s When p s, the solution Ž 3 differs from Ž 4 by the choice of the particular solutions of the inhomogeneous equations As for z and w, one can write for Ž 0 a example m, m, m are now constants p Ž t,s sm y f Ž tqsrm n n y y sm fn m mtqs spn z a and analogously for the other arbitrary functions in the general solution of the equations of motion 3 Lagrangian formulation Taking into account the equations of motion for x n and u a, one obtains the corresponding Lagrangian density j j t j t j Ls E ym E xqius E ym E u 4m 0

29 ( ) P BozhiloÕrPhysics Letters B Indeed, one verifies that the equations of motion for the Lagrange multipliers m 0 and m j give the constraints T0 and T j The remaining constraints follow from the definition of the momenta p ua To establish the invariances of the action, it is useful to rewrite L in the form LsV J V K Y n Y, Ž J, Ks0,,,p, where J Kn ž / ( ( m j J 0 j V sž V,V s y, 0 0 m m and YJ n sejx n qiž us n EJu Then, the action H J Kn Ss d pq j V J V K Y n Y, j J s j 0,j j s t,s, has global super-poincare` symmetry, local worldvolume reparametrization and k-invariances Let us show that this is indeed the case Before doing this, we note that actions of this type are first given in wx 6 for the case of tensionless superstring Ž p s and in wx 7 for the bosonic case Ž Ns0 The global Poincare` invariance is obvious Under global infinitesimal supersymmetry transformations, the fields x m Ž j, u a Ž j and V J Ž j transform as follows du h a sh a, dh x m siž us m du h, dhv J s0 As a consequence, dhyj n s0 and hence dhlsdhs s0 also To establish the invariance of the action under infinitesimal diffeomorphisms, we first write down the corresponding transformation law for the Ž r,s - type tensor density of weight a J J r J w x J r L J w x J d T a sl T as E T r wax K K s K Ks L K Ks J J r K qt waxe q KK Ks K J J r K qt waxe K KsyK Ks JJ J r J yt waxe y K Ks J J yt ry J waxe K Ks J J J J r L qat waxe, Ž 5 K Ks L J r where L is the Lie derivative along the vector field Using Ž 5, one verifies that if x m Ž j, u a Ž j are world-volume scalars Ž a s 0 and V J Ž j is a world-volume Ž,0 -type tensor density of weight n n asr, then YJ is a 0, -type tensor, YJ YKn is a Ž 0, -type tensor and L is a scalar density of weight as So, H d Ss d pq je Ž JL J and the variation d S of the action vanishes under suitable boundary conditions Let us now check the kappa-invariance We dea fine the k- variations of u Ž j, x n Ž j and V J Ž j as follows: a a J a du k siž Gk siv Ž Yu J k, dk x n syiž us n du k, dkv K sv K V L Ž ELuk Ž 6 Therefore, k a Ž j is a left Majorana-Weyl space-time spinor and world-volume scalar density of weight asyr Ž From 6 we obtain: n dk YJ YKn syi EJu Yu KqEKu Yu J du k and J n K K L dk LsV YJ YKn dkv yv V Ž ELuk s0 The algebra of kappa-transformations closes only on the equations of motion, which can be written in the form: EJŽ V J V K YKn s0, V J V K Ž EJu Yu K a s0, V J YJ n YKn s0 Ž 7 As usual, an additional local bosonic world-volume symmetry is needed for its closure In our case, the Lagrangian, and therefore the action, are invariant under the following transformations of the fields: du l Ž j slv J EJu, dlx n Ž j syiž us n du l, dlv J Ž j s0 Now, checking the commutator of two kappa-transformations, we find: a a d k,dk u Ž j sdku Ž j qterms A eqs of motion, n n d k,dk x Ž j s Ž dk qd qdl x Ž j qterms A eqs of motion, J J d k,dk V Ž j sd V Ž j qterms A eqs of motion

30 30 ( ) P BozhiloÕrPhysics Letters B Here k j, lj and j are given by the expressions: a K a a k syv EK uk k y EK uk k, ls4iv K Ž kyu K k, J syv J l G We stress that s V J Yu ab J ab Ž in 6 has the following property on the equations of motion G s0 This means that the kappa-invariance of the action indeed halves the fermionic degrees of freedom as is needed Finally, we give the expression for world-volume stress-energy tensor T J s V J Y n yd J V L Y n V M Y, K K K L Mn TrŽ T s Ž yp L Ž 8 From Ž 7 and Ž 8 it is clear, that TK J s0 Ž 9 on the equations of motion It is natural, because the equality Ž 9 is a consequence of pq of the constraints 4 Conclusions In this letter we consider a model for tensionless Ž null p-branes with N s global supersymmetry in 0-dimensional Minkowski space-time We give an action for the model and show that it is reparametrization and kappa-invariant As usual, the algebra of kappa-transformations closes only on-shell and it halves the fermionic degrees of freedom In proving the kappa-invariance, we do not use any specific ten dimensional properties of the spinors Hence, the model is extendable classically to other space-time dimensions There exist also the possibilwx 8 ity of its generalization to N supersymmetries The properties of the model in nontrivial backwx 9 grounds are also under investigation In this letter we also find some solutions of the classical equations of motion In the case of null superstring Ž p s, we obtain the general solution in arbitrary gauge Acknowledgements The author would like to thank B Dimitrov for careful reading of the manuscript References wx A Zheltuchin, Yader Fiz 48 Ž ; 5 Ž ; Teor Mat Fiz 77 Ž Ž in russian I Bandos, A Zheltuchin, Fortsch Phys 4 Ž wx P Saltsidis, Phys Lett B 40 Ž 997, hep-thr97008 wx 3 P Bozhilov, Phys Lett B 440 Ž , hep-thr wx 4 E Sokatchev, Phys Lett B 69 Ž ; Class Quant Grav 4 Ž wx 5 E Nissimov, S Pacheva, Phys Lett B 0 Ž wx 6 U Lindstrom, B Sundborg, G Theodoridis, Phys Lett B 53 Ž wx 7 S Hassani, U Lindstrom, R von Unge, Class Quant Grav Ž 994 L79 wx 8 P Bozhilov, Ds0 Chiral Tensionless Super p-branes, in preparation wx 9 P Bozhilov, D Mladenov, Ds0 Tensionless Super p-branes in Nontrivial Backgrounds, under investigation

31 3 May 999 Physics Letters B A note on modified Veselov-Novikov hierarchy Kengo Yamagishi Akatsuka , Itabashi-ku, Tokyo , Japan Received 9 September 998; received in revised form 0 March 999 Editor: M Cvetič Abstract Because of its relevance to lower-dimensional conformal geometry, known as a generalized Weierstrass inducing, the modified Veselov-Novikov Ž mvn hierarchy attracts renewed interest recently It has been shown explicitly in the literature that an extrinsic string action a` la Polyakov Ž Willmore functional is invariant under deformations associated to the first member of the mvn hierarchy In this note we go one step further and show the explicit invariance of the functional under deformations associated to all higher members of the hierarchy q 999 Elsevier Science BV All rights reserved PACS: 7; 0L; 0G Introduction In a last decade we have witnessed a tremendous flow of applications of Ž q -dimensional exactly soluble models in -d CFT, -dimensional gravity both continuum and matrix model approach Ž super string theories, etc Specifically, KdV, KP hierarchy and their modified cousins played important and remarkable roles in various occasions We owe this success mainly to the existence of fascinating mathematical structures underlying such exactly soluble models, ie an infinite dimensional symmetry, known as W-algebras, including Virasoro algebra In the case of higher dimensional exactly soluble models Žsee eg w, x, however, even though their importance was pointed out some time ago wx 3, due to their mathematical complexity, not much application has been explored until recently Veselov-Novikov Ž VN hierarchy wx 4 and its modified cousin Ž mvn wx 5 are demonstrated as another type of Ž q -dimensional extension of KdV and mkdv, as compared to the well-known KP-hierarchy One interesting feature of this higher dimensional generalization is that in the mvn case one deals with a deformation problem of Dirac operators in dimensions wx 5, rather than that of quadratic differential operators as in ordinary cases In addition, thanks to the contributions of the authors of the recent literature w6 8x we now know the important relevance of mvn to conformally Euclidean immersion of -surfaces into 3 Ž or higher dimensional Euclidean Ž or Minkowski manifold w9 x There the potential term in the mvn equation is interpreted as mean curvature of the immersed surface Ž times g, and the first integral of the mvn equation Ž ie -st member of the hierarchy is shown w,7x to be in agreement with Polyakov s extrinsic string action w3x ŽWillmore functional w4x in Euclidean signature at classical level r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S '

32 3 ( ) K YamagishirPhysics Letters B Purpose of this note is to show explicitly that the first integral is also invariant under the deformations associated to the rest of the members of the hierarchy This confirms the statement suggested in the literature wx obtained from general argument of the soluble system The derived transformation laws for the potential will also be useful, following similar methods to ordinary Ž intrinsic string cases, to pin down the algebraic structure of the infinite symmetries in the extrinsic strings ŽFor the current status of the extrinsic strings in connection with QCD, see eg w5x and references therein The mvn and generalized Weierstrass inducing We first review generalized Weierstrass inducing and see how the mvn hierarchy is involved there The Weierstrass representation is the construction of the conformally Euclidean minimal surface Žie that with vanishing mean curvature in Euclidean 3-space ŽSee also w6 x The generalized Weierstrass inducing considered here is the extension of this construction to non-minimal surfaces To explain this, here we follow the notations of Kenmotsu wx 9 Let x :S R 3 Ž js,,3 be a conformally Euclidean immersion of an oriented -surface S Ž j coordi- 3 natized locally by z, zgc into R Then following wx 9 we have: l EE x js Ž hqh e 3 j, Ž 4 l E x jselp eqie jq hyhqih e 3 j, 4 l E x jselp eyie jq hyhyih e 3 j 3 4 Here l )0 is related to the conformal factor of the induced metric and h 3 E x j ds ' Ý Ž dx sl dz d z, or l s s < g < j Ý, Ž 4 E z ij js are related to the mean H and Gaussian K Ž ; Ricci scalar curvatures j Hs h qh, 5 KsH y< f <, f' Ž h yh yih, Ž 6 respectively The quantities e Ž a s,,3 a are normalized tangent and normal vectors of the immersed surface S, whose components are defined as follows: E x j E xj i E x j E x j e js ž q /, e js ž y /, e3is eijk e j e k Ž 7 l E z E z l E z E z Next we introduce c wx, 8 r c ' E Ž x qix, Ž 8 r c ' ye Ž x qix Ž 9 ( As a decade-old subject, there are a huge number of worksrcontributions to the subject The references cited in this Letter may not reflect all of them

33 ( ) K YamagishirPhysics Letters B Then by direct calculation, using above formulas, we find that c i s have to satisfy lh Ecs c, Ž 0 lh Ecsy c Ž and f E Ž crl s c, Ž f E Ž c rl sy c Ž 3 It is also easy to show ls< c < q< c < Ž 4 It is remarkable that Eqs Ž 0 and Ž guarantee the integrability of forms V ", V 3 defined by ž / Vqsc dzyc dz, Vys V q, V 3sy cc dzqcc dz, 5 namely, we have dv v s0 Now we can consider a converse problem, given a solution to Eqs Ž 0 and Ž as a system of differential equations with respect to c s Since forms V are all integrable, comparing with ŽŽ i v 8, 9, we observe that Eqs 3 0 and induces an immersion Xj z, z js,,3 of conformally Euclidean d surface in R via relations H H X ix s V, X s V Ž 6 " 3 3 G G Here G is an appropriate integration contour ending at Ž z, z The last relation comes from imposed conformally Euclidean property g zzs E X3 q E X q E X s 0, and g zzs0 The original Weierstrass inducing corresponds to the special case H s 0, ie induction for minimal surfaces An important observation made in the recent literature w6 8,7x is the relevance of Eqs Ž 0 and Ž to the Ž q dimensional exactly soluble mvn system The mvn hierarchy is defined as a deformation problem Ž associated to -dimensional Dirac operator times some g-matrix L with a potential p s pž z, z : ž / ž c / E yp c Ls, L s0 Ž 7 p E We note that Eqs Ž 0 and Ž correspond to taking special potential p s lhr The n-th deformation in the hierarchy is defined via d c c sa n, Ž 8 d t c c n ž / ž / where the deformation operator A takes the form n ny ny nq Ži i nq Ži i A s E q X E q E q X E, Ž 9 n Ý is0 Ý is0 We would rather use following form Ž 7 of the operator for computational simplicity We could have used ordinary form for the Dirac operator Only B in Eq Ž 0 get changed under such redefinition In any event essential point is unaffected n

34 34 ( ) K YamagishirPhysics Letters B Ži with X, X Ž j Ž i, js0,,ny being = matrices These matrices are completely determined, together with the other matrix-valued differential operator B in Eq Ž 0, from the compatibility condition: n d ya n, L sb n L Ž 0 d t n The operator Bn has a similar expression as in the case A n: ny ny Ži i Ži i B s S E q S E, Ž n Ý is0 Ý is0 Ži Ž j with = matrices S, S i, js0,,ny The compatibility condition 0 also gives a deformation equation for the potential p in the form d p nq nq se pqe pq PPP d t n The first case ns is known to yield modified Veselov-Novikov equation Here we just write down the result We will provide more technical details for higher mvn case in later sections d p se pq3v E pq p EvqE pq3v E pq p E v, Ev'E p d t It is remarkable that we have a simple first integral of this deformation, which is obtained from the relation d p sež E p y3ž E p q3 p v qež E p y3ž E p q3 p v / Ž 3 d t Namely, the integral S H Ss p dzdz 4 does not change its value under the first deformation drd t Ž if p is localized This conserved quantity has special meaning in the generalized Weierstrass inducing discussed before Substituting p s lhr, we find that S is nothing but Polyakov s extrinsic string action H (< < Ss g H d x Ž That is known as a Willmore functional in the mathematics literature To sum, Polyakov s extrinsic string action is invariant under the deformation associated to the Ž st mvn equation 3 Second mvn deformation We write deformation operators for the nd mvn as d c c Žq Žy sa, AsA qa, Ž 5 d t c c ž / ž / where = matrix-valued operators A Ž" are defined as Žq 5 3 Žy 5 3 A se qve qwe qxeqz, A se qve qwe qxeqz 6

35 ( ) K YamagishirPhysics Letters B Žq Žy Then all we have to do is to work out the compatibility condition 0 for B sb qb, here represented as Žq 3 Žy 3 B sqe qre qseqt, B sqe qre qs EqT We will perform this independently on q and y parts of the compatibility conditions d d d d Ž " Ž" ya, L sb L, s q Ž 7 " q y d t d t d t d t Some care must be taken with regards to matrix components V,W, X, and V,W, X For instance, X and X give rise to a term in the deformation ž / d c XŽ Ec ypc ;, Ž 8 d t X Ec qpc ž / c which vanishes on-shell, so we set X sx s0 by hand Similarly we set V sw sv sw s0 In the ( ( course of calculation we also get Z sz sconst, and ZsZq PPP This type of deformation ž / ž / d c ( c ; Z d t c c ( is merely an overall constant scalar transformation, so we set Z s0 as well The same is true for Z After all this is done, the deformation matrices are uniquely determined For the operator A Žq we obtain 0 ž / Ž pevyve pye p 0 y5e p 0 y5e pq5p v Vs ž /, Ws, Xs, 0 5v ž 0 5Evr / 0 5 v q3 E Ž vqz 0 5 p Ž v qzqe v qve pq E p Ev Ez'E p vy Ž E p, Zs Ž 9 5 For the operator B Žq the result is 0 E v qzqe v ž 5 E p 0 / ž / 0 0 Ž Ž Z p Ž v qzq3e v q5ve pq5e p Evq5E p 0 0 y5e p 0 Ž W Qs, Rs, 0E pq5p v 0 0 Ž X Ss 5, 3 Ž 3 pevq6ve pq4e p 0 Ts Ž 30

36 36 ( ) K YamagishirPhysics Letters B Žy Žy The -components should be taken from those in Eq 9 The results for operators A and B are given by the general rule ž / ž / ž / ž / Žy 0 y Žq 0 Žy 0 y Žq 0 A s A, B s B, Ž 3 0 y 0 0 y 0 Žq Žq where A and B imply taking complex conjugate of respective components of the matrix operators Finally the deformation equation for the potential p is obtained as d p q d t se pq5v E pq Ev E pq E p v q3 E vqz q p E v qzqe v, 3 y q and d prd t sd prd t, respectively At this stage we can check explicit invariance of our first integral Ž 4 From our deformation Eq Ž 3 we immediately have d p 4 5 q se E p y5e Ep qq5 E p q5ve p y5v E p q Ev E p q5p v qzqe v, d t Ž 33 y q and similarly for d p rd t sd p rd t Thus we find that for localized p the integral Ž 4 remains invariant under deformations associated to our nd member of mvn hierarchy " Actually, if the purpose is only to show d p rd t s Ž total divergence, the following is more straightforward We just inspect - and -components of every terms from the compatibility conditions Ž 7, which provide us 5E pqv s0, 0E pqe VypVqWs0, 0E 3 pqe WypWqXs0, 5E 4 pqe X ypxqzs0, E 5 pqezqve 3 pqwe pqxe psd prd t q Ž 34 Eventually we end up with a simpler expression d p 4 3 se E p qv E pqw E pqx E pqz p Ž 35 q Ž d t This agrees with the previous result Ž 33 after substitutions of the matrix-components from Eqs Ž 9 The argument for the other deformation d p rd t y goes completely in parallel From these calculations we can naturally infer that the similar structure persists in higher mvn deformations, and we can state quite safely that we have generally ž / ž / d p ny ny n Ž i i n Ž i i se E p q Ý X E p qe E p q Ý X E p Ž 36 d t n is0 is0 in our original notation Ž 9 On the physics side this indicates the invariance of the extrinsic action a` la Polyakov under all deformations associated to mvn hierarchy 4 Discussions We have derived nd member of the mvn hierarchy, and checked its consistency to the first integral Žone of the so-called Kruskal integrals derived from the st member of the hierarchy Even though the result is not unexpected from the general argument of exactly soluble models of this sort, writing down a correct form of explicit deformation equation is very important in various respects First of all, we would like to know the

37 ( ) K YamagishirPhysics Letters B complete symmetry structure of the Polyakov s extrinsic string In order to pin down such an infinite symmetry we have to be aware of the algebraic structure of the Poisson algebra associated to this system In the case of KdV, KP we have successfully identified their Poisson structures w8x to be W -algebra family w3 x ` The latter has played an important role in other string analyses Higher Kruskal integrals are also important there The derived deformation equations here are crucial to deduce such Poisson structures Secondly, though the immediate relevance found is to Polyakov s extrinsic string, we have pointed out several years ago wx 3 the importance of - Ž or higher dimensional exactly soluble system Žthat reduces to KdV in one dimension in its application to 4-dimensional self-dual gravity The Ž m VN is one of the simplest extension of Ž m KdV, other than KP Clarifying its symmetry structure through associated Poisson algebra is an important step towards that goal as well We would like to know the possible relevance of mvn to that problem, before proceeding to more complex Davey-Stewartson hierarchy Though calculations become more involved in higher dimensional exactly soluble system, we wish to report on our analyses of these issues in future publications References wx MJ Ablowitz, PA Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 99 wx BG Konopelchenko, Solitons in Multidimensions, World Scientific, Singapore, 993 wx 3 K Yamagishi, Phys Lett B 59 Ž ; K Yamagishi, G Chapline, Class Quantum Grav 8 Ž wx 4 AP Veselov, SP Novikov, Sov Math Dokl 30 Ž wx 5 LV Bogdanov, Theor Math Phys 70 Ž wx 6 BG Konopelchenko, Stud Appl Math 96 Ž wx 7 BG Konopelchenko, IA Taimanov, Generalized Weierstrass formulae, soliton equations and Willmore surfaces I, preprint Univ Bochum Nr 87 Ž 995 dg-gar95060 wx 8 IA Taimanov, Amer Math Soc Transl Ž 79 Ž wx 9 K Kenmotsu, Math Ann 45 Ž w0x DA Hoffman, R Osserman, Proc London Math Soc 50 Ž wx LP Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Dover Publisher, New York, 909 wx R Carroll, BG Konopelchenko, Int J Mod Phys A Ž w3x AM Polyakov, Nucl Phys B 68 Ž w4x TJ Willmore, Riemannian Geometry, Chap 7, Clarendon Press, Oxford, 993 w5x P Horava, ˇ On QCD string theory and AdS dynamics, preprint CALT hep-thr9808 w6x R Parthasarathy, KS Viswanathan, Int J Mod Phys 7 Ž w7x BG Konopelchenko, G Landolfi, Generalized Weierstrass representation for surfaces in multidimensional Riemannian spaces, math DGr w8x VG Drinfel d, VV Sokolov, Sov Math Dokl 3 Ž ; J Sov Math 30 Ž

38 3 May 999 Physics Letters B Superspace action for the first massive states of the superstring Nathan Berkovits, Marcelo M Leite Instituto de Fısica Teorica, UniÕersidade Estadual Paulista, Rua Pamplona 45, Sao Paulo, SP, Brazil Received 5 January 999 Editor: M Cvetič Abstract Using the manifestly spacetime-supersymmetric version of open superstring field theory, we construct the free action for the first massive states of the open superstring compactified to four dimensions This action is in Ns Ds4 superspace and describes a massive spin- multiplet coupled to two massive scalar multiplets q 999 Published by Elsevier Science BV All rights reserved Introduction The conventional action for open superstring field theory contains contact-term problems caused by the presence of picture-changing operators wx Recently, a new action was proposed for open superstring field theory where these picture-changing operators are absent wx In addition to eliminating the contact-term problems, this new action can be written in a form which is manifestly N s D s 4 spacetime-supersymmetric After compactifying the open superstring on a Calabi-Yau manifold to four dimensions, it is easy to show that the massless compactification-independent contribution to the free part of this action reproduces the usual Ns Ds4 superspace action for super- Maxwell In this paper, the first massive contribution to the free part of this action will be computed in Ns Ds4 superspace nberkovi@iftunespbr mmleite@iftunespbr In an earlier paper wx 3, we used open superstring vertex-operator arguments to show that the massive Ds4 spin-two multiplet can be described in Ns D s 4 superspace by a vector superfield satisfy- a m ing the constraint D s V sž E n E q aa m n Vms 0 It will be shown here that these constraints come from gauge-fixing the equations of motion of a superspace action involving not just V, but also a spinor and m two scalar superfields, V, B and C The propagating a degrees of freedom of this action describe a massive spin- multiplet as well as two massive scalar multiplets which are always present in Calabi-Yau compactifications of the open superstring to four dimensions Before constructing the open superstring field theory action for these massive states, we shall review the construction of the action for the massless states Construction of the free action for the massless states The topological description of the superstring, developed by one of the authors with Vafa in Ref r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

39 ( ) N BerkoÕits, MM LeiterPhysics Letters B wx 4, is particularly suitable for formulating open suwx In this approach, the BRST perstring field theory current is replaced by a spin-one generator G q, the h ghost Žwhich comes from bosonizing the RNS super-reparameterization ghosts as bsej e yf and f g s he is replaced by a spin-one generator G q, and the ghost number current is replaced by a spinwx 4, these three one generator J As discussed in generators form part of a small twisted N s 4 algebra The usual condition for physical vertex operators is QV s0 where V is independent of the j zero mode, ie hž V s0 Note that GV always means the contour integral of G around V Since the h cohomology is trivial, one can always find an operator F such that VshŽ F So the condition for F to be physical is that G q ŽG q Ž F s 0 where F is defined up to the gauge invariance dfsg q Ž L q q G Ž L These linearized equations of motion and gauge invariances are easily obtained from the string field theory action q Ss² F G G q Ž F : where ²: is the two-point correlation function on a sphere As usual in open superstrings, the states carry Chan-Paton factors which will be suppressed throughout the paper The manifestly spacetime-supersymmetric description of the superstring is related to the usual RNS description by a field redefinition For compactifications of the superstring on a Calabi-Yau threefold, this field redefinition allows Ns Ds4 super-poincare invariance to be made manifest The field redefinition takes the left-moving worldsheet matter and ghost fields of the RNS description into Ž m five free bosons x,rž where ms0 to 3 which satisfy the OPE s m n < < mn x Ž y x Ž z log yyz h, r Ž y r Ž z logž yyz, Ž a a a a eight free fermions u, u, p, p Žwhere a and a s or which satisfy the OPE s b b y pa y u z da yyz, b b a y p Ž y u Ž z d Ž yyz, a and a field theory for the six-dimensional compactification manifold which is described by the cs3 ˆ w q y N s superconformal generators T,G,G, J x C C C C This Ns superconformal field theory is twisted so that GC q has conformal weight q and GC y has conformal weight q As usual, right-moving fields are related to the left-moving fields through boundary conditions The small N s 4 superconformal generators are defined in terms of these free fields by Ts E x E x qp Eu qp Eu q ErEr m a ȧ m a a i y E rqt C, q i r a q y yi r a y a C a C G se d d qg, G se d d qg, q y i rqihc a yi rqihc y G se d d aqe G C, y i ryih C a i ryih c q G se d daqe G C, JsiErqiJ C, J qq se yi rqih C, J yy se qi ryih C, 3 where i ȧ a a aa 4 a 8 a d sp q u E x y u Eu q u E u, i a a a aa 4 a 8 a d sp q u E x y u Eu q u E u, 4 JCsE H C, and we use the bispinor convention Õaȧs m m aa saa Õ m Our Pauli matrices, saa and s m, are those of Wess and Bagger wx 5, but we choose to define Ž a Ž ȧ u s uu and u s uu a ȧ, and to use the mostly-negative Minkowski metric h mn s Ž m n mn diag q,y,y,y so that Tr s s sh Although our hermiticity conditions on the superm ) m Ž a ) ȧ space variables are x s x and u s u as usual, our hermiticity conditions on the chiral bosons are r ) s ryh, H ) s3ry H, C C C

40 40 ( ) N BerkoÕits, MM LeiterPhysics Letters B defined to commute with the spacetime supersymmetry generators and to satisfy the OPE s q ) q so that G sg Note that da and dȧ have been P aȧ da Ž y dȧ Ž z i, yyz yieabeub da Ž y Pbb Ž z 5 yyz where Paa se xaa qiu aeu aqiu aeu a To make Ns Ds4 supersymmetry manifest in the string field theory action, one integrates over the non-zero modes of the worldsheet fields in the correlation function of, but leaves the integration m a ȧ over the zero modes of x, u, and u explicit in the action The result is the Ns Ds4 superspace action H ² : 4 q q Ss d xd u d u FG G F 6 where ²: now does not include the contribution of m a ȧ the zero modes of x, u, and u As our first example, we shall consider the contribution to the above action from the massless compactification-independent states Since there is no tachyon, the massless states at zero momentum are described by string fields of conformal weight zero The string field is independent of the compactification manifold and is of neutral UŽ -charge Žie zero ghost number, so it is described by a scalar superfield VŽ x,u,u Plugging FsV into the action of 6, one reproduces the super-maxwell action H 4 ȧ Ss d xd u d u VDȧ D D V 7 E i a m where D s q us E and D s E a a Eu aa m a q Eu ȧ i a m u saa E m Furthermore, the gauge invariances df s q q G Ž L q G Ž L reproduce the expected gauge invariances d VsD lqd l yi r i ryih where Lse l and Lse C l 3 Action for first massive states The field theory action of 6 will now be used to compute the contribution of the first massive states of the open superstring We shall consider a generic Calabi-Yau three-fold and will not consider states which depend on the explicit structure of the compactification manifold However, we will allow states which can be constructed from the Calabi-Yau UŽ current E H C As will be shown later, the presence of such states in the action is necessary for gauge invariance Since we want to describe states of Ž mass s, our string field should contain conformal weight q at zero momentum Furthermore, it should have no UŽ charge and should only depend on the compactification manifold through the UŽ current The most general such string field is a ȧ Fsd Wa x,u,u qd Wȧ x,u,u m a qp Vm x,u,u qeu Va x,u,u ȧ qeu V a x,u,u qi EryE HC B x,u,u qž E HC y3er CŽ x,u,u Ž 3 where B and C are real superfields q q This field transforms as dfsg L qg Ž L under the gauge transformations parameterized by yi r a a m ž a a m Lse d C qd E qp B qerfq a ȧ qeu Ba qeu H ȧ, i ryihc a a m ž a a m Lse d E qd C qp B / / a ȧ q EryE HC FqEu HaqEu B ȧ q q Note that dfs0 when LsG V or LsG V Defining y i r a a ȧ ž a aȧ Vse E u M q Eu NqEu d P a ȧ qpaȧ Eu Q, / one can see that the transformations parameterized by C a, B m, F, and Ha can be ignored Furthermore, the transformations parameterized by Ba and Bȧ can be used to algebraically gauge-fix Wa swȧ s0 in the string field F

41 ( ) N BerkoÕits, MM LeiterPhysics Letters B In this gauge, the remaining string fields transform under the remaining gauge transformations as: d V syiž s D E yiž s D E, a a a a m m aa m aa ȧ d Va s4eay DDEqiE a aa DE, a d CqiB sid D E a 3 Note that B and C are not gauge-invariant, so they can not be dropped from the action without breaking gauge invariance In our earlier analysis of massive vertex operators wx 3, it was not necessary to include B and C since we were working in a fixed gauge Finally, the string field action S s Hd 4 xd u ² qž q =d u FG G Ž F : in the gauge Wa swȧ s0 is given by 4 m SsHd xd u d u yv D D Vmy4Vm n aa aa Ž m ž a a a a/ m aa Ž m a a / qiž s En Da DaVm y4i s DVqD V y8e B q s D, D C ž a a a ž a a a aa a Ž 8iCq B / qv y4d D V y DVq4E D C yd D a a ž a aa qv y DVy4E D C qd D ȧ Ž 8iCy B/ m Ž m y3c D Dy6E E y8 C a a qbž yd Dy8 Bq6BEaȧ D, D C Ž 33 To evaluate the propagating degrees of freedom, it is convenient to use the gauge transformations parameterized by E and E to gauge-fix V sv s a a a a 0 Note that this gauge-fixing involves derivatives, so unlike the gauge-fixing of W a, it cannot be per- formed directly in the action Furthermore, d Va s0 when the gauge parameter E satisfies E s i a Ž n E DE and EE y aa n Ea s0 This implies that there is a residue gauge transformation even after gauge-fixing Va s 0 a a In the gauge Va svȧ s0, the equations of motion from the action of Ž 33 are D, D 4V y4v ye E V y4e B n m m n m m aa m a a q6ž s D, D Cs0, Ž 34 4iŽ s aa D V m q4e aa D C m a a qda D y8icy B s0, 35 y8e V y6by D, D 4B m m a a q6eaȧ D, D Cs0, 36 4 aa m ysm D a, Da V q48cy33 D, D C q96e E Cy6E D, D Bs0 Ž 37 m m a a aȧ Plugging Ž 35 into Ž 36 and Ž 37 one learns that i Bs wd, D xc Ž m 6 E Em y Cs6 D, D C 39 The equation of motion for C has two solutions: a D Cs0, E m E y Cs0; Ž 30 m b D D D Cs0, Ž E E q Cs0 3 a m a m Using the residue gauge transformations of E a, any solution of Eq Ž 30 can be gauged away Therefore, the only physical degrees of freedom of C come from solutions of Ž 3, which are of the form: CsFqF Ž 3 where F is a chiral superfield satisfying m m D a FsDa Fs E Emq Fs E Emq Fs0 Although F is a single chiral superfield, it describes two massive scalar multiplets since it does not satisfy the on-shell equation D Fs0 Expanding in components, a FsXqu jaq u Y where X and Y are complex bosons satisfying ŽE m E q X s ŽE m E q m m Y s 0 Defining cȧ s i a y Eaȧ j, the equations of motion for the fermion ' are ' a ȧ Eaa j si c a, Eaa c si ja '

42 4 ( ) N BerkoÕits, MM LeiterPhysics Letters B A Ž a which describe a massive Dirac spinor L s j, c ȧ Plugging Ž 3 and Ž 38 into Ž 34 and Ž 35, one obtains D a V yie D a Fs0, Ž 33 aa aa D, D 4V y4v ye E V y6iž FyF s0 Ž 34 n m m n m Finally, shifting Vˆ sv yie Ž FyF m m m, one can rewrite Ž 33 and Ž 34 as a ˆ m D V s0, Ž E E q Vˆ s0, Ž 35 aa m m which were shown in wx 3 to be the gauge-fixed equations of motion of a massive spin-two multiplet Therefore, the propagating degrees of freedom described by the action of Ž 33 are a massive spin-two multiplet and two massive scalar multiplets On-shell, these multiplets contain bosonic and fermionic degrees of freedom The bosonic degrees of freedom can be identified with the following Neveu-Schwarz light-cone gauge vertex operators: < : < : < : m m n J K L by 3 0, by ay 0, vjklby by by 0, J K L m J J vjklby by by 0, by gjjby by 0, < : < : < : < : J J J J gjjby ay 0, gjjby ay 0, 36 where m,n, ps,3 are the Ds4 light-cone indices, g JJ and vjkl are the metric and holomorphic 3-form used to define the Calabi-Yau manifold, and b m and N a m are the oscillator modes of the light-cone c m and N E X m Similarly, the fermionic degrees of freedom can be identified with the following Ramond light-cone gauge vertex operators: b m < 0 : qqqq, b m < 0 : yyyy y y, a m < 0 : yqqq, a m < 0 : qyyy, y y J J qyyy J J JJ y 0 JJ y 0 J J yqqq J J JJ 0 y JJ 0 y yyyy g b b < 0 :, g a b < 0 :, qqqq g b b < 0 :, g b a < 0: Ž 37 where < 0: """" is the light-cone spinor in SUŽ 4 notation where the last 3 " signs refer to the six Calabi-Yau directions Although there might be other states of Ž mass s in the open superstring spectrum which come from Calabi-Yau excitations, the q states described above are always present in any Calabi-Yau compactification Acknowledgements NB would like to acknowledge partial support from CNPq grant number 30056r94-9 and MML would like to acknowledge support from FAPESP grant number 96r References wx E Witten, Nucl Phys B 76 Ž 986 9; C Wendt, Nucl Phys B 34 Ž wx N Berkovits, Nucl Phys B 450 Ž , hep-thr wx 3 N Berkovits, MM Leite, Phys Lett B 45 Ž , hep-thr wx 4 N Berkovits, C Vafa, Nucl Phys B 433 Ž 995 3; see also N Berkovits, A New Description of the Superstring, proceedings of the VIII JA Swieca Summer School on Particles and Fields, World Scientific Publishing, 996, hep-thr96043; Nucl Phys B 43 Ž , hep-thr94046 wx 5 J Wess, J Bagger, Supersymmetry and Supergravity: Notes from Lectures given at Princeton University, Princeton Univ Press, 98

43 3 May 999 Physics Letters B Algebraic structure in 0- c F open-closed string field theories Daiji Ennyu a,, Hiroshi Kawabe b,, Naohito Nakazawa c,3 a Hiroshima National College of Maritime Technology, Toyota-Gun, Hiroshima , Japan b Yonago National College of Technology, Yonago , Japan c Department of Physics, Faculty of Science and Engineering, Shimane UniÕersity, Matsue , Japan Received 9 February 999 Editor: M Cvetič Abstract We apply stochastic quantization method to Kostov s matrix-vector models for the second quantization of orientable strings with Chan-Paton like factors, including both open and closed strings The Fokker-Planck hamiltonian deduces an orientable open-closed string field theory at the double scaling limit There appears an algebraic structure in the continuum F-P hamiltonian including a Virasoro algebra and a SUŽ r current algebra q 999 Published by Elsevier Science BV All rights reserved The explicit construction of non-critical string field theories via the double scaling limit of the matrix models wx provides not only the basis for the non-perturbative analysis of string theories but also the clear understanding of the origin of the constraints realized in the algebraic structure of the string field theoretic hamiltonian w 0 x The algebraic structure in non-critical string field theories has been investigated in this context for orientable closed strings w 4 x, non-orientable closed strings w6 x, oriw7,8x and non-orientable entable open-closed strings open-closed strings w0 x For orientable D surfaces with boundaries, an interesting algebraic structure, slž r,c = slž r,c chiral current algebra including SUŽ r current algebra, has been found in a field ennyu@hiroshima-cmtacjp kawabe@yonago-kacjp 3 nakazawa@ifserikoshimane-uacjp theoretic hamiltonian of orientable open-closed str ings wx 9 The observation indicates the relation between the algebraic structure and a Chan-Paton like realization of gauge groups in open string theories While the SOŽ r current algebra has been found in the string field theoretic hamiltonian for non-orientaw0 x In this note, we construct non-critical orientable ble open-closed strings open-closed string field theories for 0- c F by applying stochastic quantization method to Kostov s matrix-vector model w7, x To introduce Chan-Paton like factors, we slightly modify Kostov s model The continuum limit of the string field theory is taken at the double scaling limit of this matrix-vector model We introduce the scale parameters on the boundaries which characterize the scaling behaviour of the time evolution along the boundaries We show that the string field theoretic hamiltonian, which is the continuum limit of the loop space Fokker-Planck hamilw3,6 x, appears in tonian in stochastic quantization the r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S X

44 44 ( ) D Ennyu et alrphysics Letters B form of a linear combination of three deformation generators of orientable strings which satisfy algebraic relations including a Virasoro algebra and a SUŽ r current algebra For orientable D surfaces with boundaries, the sum of triangulated surfaces is reconstructed by the hermitian matrix-vector models w, x To introduce the orientable open-closed string interactions for 0- c F, we consider Kostov s matrix-vector model with Chan-Paton like factors The action is given by wx 7 ž S str A X A Ý X xx x x X x, x g y A X A X XX A XX, 3 ' N X XX x, x, x Ý xx x x x/ x a g ž B ' N / r a) a S s V d Xy A X V X ÝÝ x xx xx x, X x, x as where the interactions of matrices, A xx X, take place only for < x y x X < F in the target space Z, ie x, x X gz Namely, the relevant matrices are A xxij, which are N=N hermitian matrices, and A xxq ij sa xq xij Vxi a and Vxi a) are N-dimensional complex vectors with a Chan-Paton like factor a which runs from to r The partition function, Z, and the free energy, F, of this model are given by Z' dadvdv ) e ys ys H, r Ł a a L B S x B F'lnZs g g N, as where S and L a B are the total area and the total length of the boundary with the colour index a of the D surface, respectively x is the Eular number of the triangulated D surface with boundaries, x' yaž handles yaž boundaries By integrating out all the non-hermitian matrices, A xxq ij and A xxq ji, we obtain an effective action as a function of the 'N hermitian matrix M xij' A xxijy d ij, and the vec- g tors, Vxi a and Vxi a) We denote the effective action as S Ž M,V,V ) wx eff 7 We define the time evolution, M Ž t q Dt xij ' M Ž t q DM Ž t and V a Ž t q Dt ' V a Ž t xij xij xi xi q DV a Ž t, in terms of Ito s calculus w3x xi with respect to the stochastic time t which is discretized with the unit time step Dt The time evolution is given by the following Langevin equations, E ) DMxijŽ t sy Seff Ž M,V,V Ž t Dt E M xji qdjxijž t, E a a ) DVxiŽ t sylx Seff Ž M,V,V Ž t Dt a) E V xi qdhxi a Ž t, E a) a ) DVxi Ž t sylx Seff Ž M,V,V Ž t Dt a E V xi qdhxi a) Ž t Ž 3 We have introduced the scale parameter l a x in the Langevin equation of vector variables, Vxi a and Vxi a) At the double scaling limit, the parameter l a x defines the time scale for the stochastic time evolution of the open string end-point along the boundaries with the colour index a The correlations of the white noise Djxij and Dhxi a are defined by ² Dj Ž t Dj X Ž t : j s Dtd d d X, xij x kl il jk xx ² a) b : a ab Dh Ž t Dh X Ž t sl Dtd d d X Ž 4 xi x j h x ij xx We will see later that the l a x-independence of the equilibrium is realized as the algebraic relation between the constraints in the string field theoretic hamiltonian which is equivalent to the integrability condition of the multi-time evoluti on of the Langevin equations Now we define the closed strings, f Ž L x s y Ž LNy r M x ab N tr e, and the open strings, c Ž L x s y Ž a) LN y r M N V e x V b Following to Ito s calculus, we calculate the time development of these string variables wx 6, L X X X x ½H x x 0 ` qýh dl X C Ž c X X X f LqL f X xx x x L x X 0 ` a b X X Ž o q Ý g gh dllc X B B xx Ng X a,b, x 0 =c ab Ž LqL X c ba X Ž L X x x r a aa q Ý g B c x Ž L y f x Ž L N as 4 g Df L sdt L dlf L f LyL

45 E qg fxž L qdzxž L, E L 5 gb a E ab a a ab x x B x ½ ž / Dc Ž L sl Dt yq qg c Ž L g E L ` a c X Ž o Ý H X B B xx g X c, x 0 X Ž L X x x 5 q g g dl C =c cb LqL X c ac ½ ž / gb b E b b ab x B x ql Dt yq qg c Ž L g E L ` b c X Ž o Ý H X B B xx g X c, x 0 X Ž L X x x 5 q g g dl C =c ac LqL X c cb q Dtl a xd ab fxž L L E ab ab qdt ½y cx Ž L qlg cx Ž L 4 g E L ` ÝH X Ž c ab X X X X xx x x x X 0 ql dlc c Ž LqL f Ž L Ý H c L X ac X cb X B x x 0 q g dlc Ž L c Ž LyL c ( ) D Ennyu et alrphysics Letters B c d gbgb L ` ` X XX XXX Ž o X Ý H H H xx c,d, x X g q dl dl dl C =c ad Ž L X ql XX c cb Ž LyL X ql XXX x x =c dc X Ž L XX ql XXX x L X X ab X X H x x 5 0 q dl Lc L f LyL qdzx ab Ž L Ž 5 Here we have introduced the adjacency matrices, Žc Žo C XsC Xsd X qd X In Ž xx xx xxq xxy 5, the new noise variables are defined by Ž x Dz L 'LN y3r tr e LNy r M xdj, x ab y3r L X a) X yr LN M x x x Dz Ž L 'N H dlv e 0 =Dj e Ž LyLX N y r M xv b x x qn y x V a) e LNy r M xdh b qdh a) e LNy r M xv b Ž 6 x x 4 x ² : ² ab We notice that Dz L s Dz Ž L: x jh x jhs 0 hold in the sense of Ito s calculus Especially, from Eqs Ž 5, the l-independence of the equilibrium limit deduces a S-D equation which ensures the SUŽ r current algebra We can describe a class of matrix-vector models for the central charge 0-cF with the following adjacency matrices wx 7 Žc C Ẋscos Ž p Ž d X qd X, C xx 0 xxq xxy Ž o Ẋ scos Ž p r Ž d X qd X Ž 7 xx 0 xxq xxy The case, p0 s rm, corresponds to the central 6 charge, csy mž mq The stochastic process is interpreted as the time evolution in a string field theory The corresponding non-critical orientable open-closed string field theory is defined by the F-P hamiltonian operator In terms of the expectation value of an observable OŽ f,c,a ab function of f L s and c Ž L x x s, the F-P hamilto- nian operator Hˆ is defined by wx 6 FP yt Hˆ FP Ž ˆ ˆ ² Ž jh jh : jh ² f 0,c 0 < e O f,c < 0: ' O f Ž t,c Ž t Ž 8 In RHS, f Ž t and c Ž t jh jh denote the solutions of the Langevin Eqs Ž 5 with the appropriate initial ab configuration f L;t s 0 and c Ž L;t s 0 x x In LHS, Hˆ FP is given by the differential operator in the well-known Fokker-Planck equation for the expectation value of the observable OŽ f,c by replac- Ž ab ing the closed open string variable f L c Ž L x x to the creation operator fˆ Ž L Žcˆ ab Ž L x x and the E E differential Ef Ž L Ec abž L to the annihilation opera- x x Ž ab tor ˆ p L p Ž L x ˆx, respectively The continuum limit of Hˆ FP is taken by introducing a length scale e which defines the physical length of the strings as l'le At the double scaling limit e 0, we keep the string coupling constant, G'N y e y D, to be finite The continuum stochastic time is given by dt'e yqd Dt While we define the scaling of the scale parameter l a by l a ' x e yryd r l a We also redefine field variables as x follows F Ž l 'e f Ž L, P Ž l 'e p Ž L, yd ˆ yqd x x x ˆ x ab yryd r ˆ x x ab C l 'e c L, P ab Ž l 'e yrqd r ˆ p ab Ž L Ž 9 x x

46 46 ( ) D Ennyu et alrphysics Letters B X X x x X ab cd X ac bd X X xx x x The commutation relations are P l,f l s d d l y l and P l,f l s d d - d Ž X X d l y l xx Then we obtain the continuum F-P hamiltonian, H, from H ˆ, FP FP ` ` ÝH Ý X X X XH X FP ½ xx x x 0 X x x 0 H s dll C dl F Ž lql F Ž l H ` ' Ý X X ab X ba X X X xx x x a,b, x X 0 l X X X H x x 0 q G C dl lc Ž lql C Ž l q dl F l F lyl ` H X X X X x x x 0 qg dl lf Ž lql P Ž l P Ž l Ý H ` Ý q dl l C ½ X xx 0 X a,b, x x = ` X ab X X H dlc Ž lql F X x x Ž l 0 l ` ` ad q Ý C X xxhdlh dlh dl3c x lql c,d, x X =C cb Ž lyl ql C dc X Ž l ql x 3 x 3 l X X ab X X H x x 0 q dl lc l F lyl H ` a Ý X cb X ac X X X xx x x c, x X 0 5 ql C dlc Ž lql C Ž l H ` b Ý X ac X cb X X X xx x x c, x X 0 ql C dlc Ž lql C Ž l qž l a ql b d ab F Ž l P ab Ž l x Ý ' H H H H 5 x ` ` X X l l X q G dl dl dl dl a,b,c,d, x =C ad Ž l ql X C cb Ž lql X yl yl X x x =P ab Ž l P cd Ž l X x Ý ' H H x ` ` X q G dl dl a,b,c, x 0 0 x x x = l a C cb Ž lql X P ab Ž l P ca Ž l X ql b C ac Ž lql X P ab Ž l P bc Ž l X 4 x x x ` ` X X ab X Ý H H x qg dl dl llc Ž lql a,b, x 0 0 =P Ž l P ab Ž l X Ž 0 x x The continuum F-P hamiltonian 0 takes the form of a linear combination of three continuum generators of string deformation, ÝH ` H s dll L Ž l P Ž l FP x x x 0 ` a ab b ba) q Ý H dl l Jx l ql Jx l a,b, x 0 =Px ab Ž l ` ab q dl l Kx l a,b, x 0 Ý H 4 ½ l X X cb X ac X ÝH Ž x x c 0 y dl l J Ž l C Ž lyl qj ca) Ž l X C bc) Ž lyl X P ab Ž l Ž x x x The explicit forms of these generators are given by H ` Ý X X X X X x xx x x x X 0 L Ž l s C dlf Ž lql F Ž l H ` ' Ý X X ab X ba X X X xx x x a,b, x X 0 q G C dl lc Ž lql C Ž l l X X X H x x 0 ` H X X Ž X Ž X x x 0 ` ÝH X X ab X ab X x x a,b 0 q dl F l F lyl qg dl lf lql P l qg dl lc Ž lql P Ž l, H ` ab Ý X ab X X X X x xx x x x X 0 K Ž l s C dlc Ž lql F Ž l ` H X X abž X Ž X x x 0 ` lql Ý XH H ac xx x c,d, x X 0 0 qg dl lc lql P l q C dl dl C Ž l =C db lql yl C cd X x Ž x Ž l ' ÝH H ` lql ac x 0 0 q G dl dl C Ž l c,d =Cx db Ž lqlyl Px dc Ž l l X ab X X H x x 0 q dlc Ž l F Ž lyl, 5

47 ( ) D Ennyu et alrphysics Letters B J ab Ž l sd ab F Ž l x x H ` Ý X cb X ac X X X xx x x c, x X 0 ` ' ÝH X cb X ca X x x c 0 q C dlc Ž lql C Ž l q G dlc Ž lql P Ž l Ž These generators satisfy the following algebra including the Virasoro algebra and the SUŽ r current algebra, X X X L l, L X l sy lyl Gd X x x xx LxŽ lql, ab X X ab X L Ž l, K X Ž l syž lyl Gd X K Ž lql x x xx x qgd X xx ÝH l duž lyu c 0 x x = J cb Ž lql X yu C ac Ž u qj ca) lql X x Ž yu Cx bc) Ž u 4, ab X X ab X L Ž l, J X Ž l sl Gd X J Ž lql, x x xx x ab cd X ad cb X X ' X x x xx x J l, K l s G d d K lql ' ÝH l ad y G d d X xx du e 0 = J ec) Ž lql X yu C be) Ž u ' x y G d X xx H 0 l du = J ad Ž lql X yu C cb Ž u, x x ab cd X ad cb X X ' X x x xx x cb ad X y' G d d X xx J Ž lql, X X X l l yu ab cd ' X xx ÝH H e 0 0 J Ž l, J Ž l s G d d J Ž lql K l, K l s G d du dõ = J eb lql X x Ž yuyõ =Cx ad Ž u Cx ce Ž Õ qj ea) lql X x Ž yuyõ =C de) Ž u C cb Ž Õ4 ' x x l lyu X xx ÝH H e 0 0 y G d du dõ = J ed lql X x Ž yuyõ =Cx cb Ž u Cx ae Ž Õ qj ec) lql X x Ž yuyõ =C ad Ž u C be) Ž Õ Ž 3 x x 4 x In the precise sense, these commutation relations are not an algebra because the open string creation operab ators C Ž l appear in the RHS of Ž 3 The algebraic structure Ž 3 is understood as the consistency condition for the constraints on the equilibrium expectation value or integrable condition of the stochastic time evolution The hermitian part of the ab Ž ) ba Ž current, J l q J l, generates the S-D equation which implies the l-independence of the equilibrium distribution While the anti-hermitian part of ab Ž ) ba the current, J l y J Ž l, satisfies SUŽ r current algebra The similar algebraic structure is also found in Ref wx 9 in a discretized form without continuum limit In conclusion, we have derived the continuum F-P hamiltonian which defines the non-critical orientable open-closed string field theory for 0-cF byapplying SQM to Kostov s matrix-vector model The origin of the algebraic structure Ž 3 can be traced to the noise correlations which generate the time evolution in the Langevin equation Namely, the noise correlations realize the deformation of open and closed strings which are equivalent to those gener- Ž ab ated by the three constraints, L l, J Ž l and ab K Ž l We have shown that the structure to be universal representing the scale invariance in the non-critical orientable open-closed string theories We hope that our approach is also useful to the non-perturbative analysis of superstring theories such as type IIA, IIB and type I theories for which matrix models are proposed w4,5x by the large N reduction of super Yang-Mills theory Acknowledgements NN would like to thank I Kostov for valuable comments This work was partially supported by the Minisitry of Education, Science and Culture, Grantin-Aid for Exploratory Research, , 998 References wx E Brezin, V Kazakov, Phys Lett B 36 Ž ; M Douglas, S Shenker, Nucl Phys B 335 Ž ; D Gross, A Migdal, Phys Rev Lett 64 Ž 990 7; Nucl Phys B 340 Ž

48 48 ( ) D Ennyu et alrphysics Letters B wx N Ishibashi, H Kawai, Phys Lett B 34 Ž ; B 3 Ž wx 3 A Jevicki, J Rodrigues, Nucl Phys B 4 Ž wx 4 M Ikehara, N Ishibashi, H Kawai, T Mogami, R Nakayama, N Sasakura, Phys Rev D 50 Ž wx 5 Y Watabiki, Phys Lett B 346 Ž ; Nucl Phys B 44 Ž wx 6 N Nakazawa, Mod Phys Lett A 0 Ž wx 7 I Kostov, Phys Lett B 344 Ž ; B 349 Ž wx 8 T Mogami, Phys Lett B 35 Ž wx 9 J Avan, A Jevicki, Nucl Phys B 469 Ž w0x N Nakazawa, D Ennyu, Phys Lett B 47 Ž wx I Kostov, Nucl Phys B 376 Ž ; VA Kazakov, I Kostov, Nucl Phys B 386 Ž wx VA Kazakov, Phys Lett B 37 Ž 990 w3x K Ito, Proc Imp Acad 0 Ž ; K Ito, S Watanabe, in: K Ito Ž Ed, Stochastic Differential Equations, Wiley, 978 w4x T Banks, W Fischler, S Shenker, L Susskind, Phys Rev D 55 Ž w5x N Ishibashi, H Kawai, Y Kitazawa, A Tsuchiya, Nucl Phys B 498 Ž ; M Fukuma, H Kawai, Y Kitazawa, A Tsuchiya, hep-thr97058

49 3 May 999 Physics Letters B Conformal Ns0 ds4 gauge theories from AdSrCFT superstring duality? Paul H Frampton, William F Shively Department of Physics and Astronomy, UniÕersity of North Carolina, Chapel Hill, NC , USA Received March 999; received in revised form March 999 Editor: M Cvetič Abstract Non-supersymmetric ds4 gauge theories which arise from superstring duality on a manifold AdS =S rz are 5 5 p cataloged for a range FpF4 A number have vanishing two-loop gauge byfunction, a necessary but not sufficient condition to be a conformal field theory q 999 Published by Elsevier Science BV All rights reserved The relationship of the Type IIB superstring to conformal gauge theory in d s 4 gives rise to an interesting class of gauge theories w 4 x Choosing the simplest compactification wx on AdS 5= S5gives rise to an Ns 4 SUŽ N gauge theory which has been known for some time wx 5 to be conformal due to the extended global supersymmetry and non-renormalization theorems All of the RGE b y functions for this Ns 4 case are vanishing in perturbation theory One of us Ž PHF has recently wx 6 pursued the idea that an Ns 0 theory, without spacetime supersymw7 6x on the metry, arising from compactification orbifold AdS = S rg Žwith G;u SUŽ could be conformal and, further, could accommodate the standard model In the present note we systematically catalog the available Ns0 theories for G an abelian discrete group GsZ We also find the subset which p has bg Ž s0, a vanishing two-loop byfunction for the gauge coupling, according to the criteria of wx 6 In a future publication, we hope to find how many if any of the surviving theories satisfy by Ž s0 and bh Ž s0 for the Yukawa and Higgs self-coupling two-loop RGE b y functions respectively Note that the one-loop byfunctions satisfy by Ž s0 and bh Ž s 0 because they are leading order in the planar expansion w7 0 x All one-loop Ns 0 calculations coincide with those of the conformal Ns 4 theory, beyond one-loop this coincidence ceases, in general The ideas in Frampton wx 6 concerning the cosmological constant and model building beyond the standard model provide the motivation as follows At a scale sufficiently above the weak scale the masses and VEVs of the standard model obviously become negligible Consider now that the standard model is promoted by additional states to a conformal theory of the ds4 Ns0 type which will be highly constrained or even unique, as well as scale invariant Low energy masses and VEVs are introduced softly into this conformal theory such as to preserve the desirable properties of vanishing vacuum energy and hence vanishing cosmological constant Since no supersymmetry breaking is needed and provided the introduction of scales is sufficiently mild it is expected that a zero cosmological constant can be retained in this approach r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

50 50 ( ) PH Frampton, WF ShiÕelyrPhysics Letters B The embedding of GsZp in the complex three- dimensional space C3 can be conveniently specified by three integers ais Ž a,a,a 3 The action of Zp on the three complex coordinates Ž X, X, X 3 is then: Z p a a a3 3 3 Ž X, X, X Ž a X,a X,a X Ž where asexpž p irp and the elements of Zp are a r Ž0FrFŽ py The general rule for breaking supersymmetries is that for G;SUŽ, there remains Ns supersymmetry; G;SUŽ 3 leaves Ns supersymmetry; and for G;u SUŽ 3, no supersymmetry Ž Ns0 survives To ensure that G;u SUŽ 3 the requirement is that a q a q a 3 / 0 Ž mod p Each ai can, without loss of generality, be in the range 0Fa FŽ py i Further we may set afa F a3 since permutations of the ai are equivalent Let us define n Ž p k to be the number of possible Ns0 theories with k non-zero a Ž FkF3 i Since a sž 0,0,a i 3 is clearly equivalent to ais Ž 0,0, pya the value of n Ž p is 3 n p s pr where? x@ is the largest integer not greater than x For n Ž p we observe that a sž 0, a, a i 3 is equivalent to a sž 0, pya, pya i 3 Then we may derive, taking into account Eq Ž that, for peõen n p s Ý rs 4 p py 4 rs while, for p odd p Ý 4 rs n Ž p s rq s Ž py Ž 5 For n Ž p 3, the counting is only slightly more intricate There is the equivalence of a sž a,a,a i 3 with Ž pya, pya, pya as well as Eq 3 to contend with In particular the theory a sž a, pr, pya i is a self-equivalent Ž SE one; let the number of such theories be n Ž p Then it can be seen that n Ž p SE SE spr for p even, and n Ž p s0 for p odd With SE regard to Eq Ž, let n Ž p p be the number of theories with Ýa sp and n Ž p i p be the number with Ýais p Then because of the equivalence of Ž a,a,a 3 with Ž pya, pya, pya, it follows that n Ž p 3 p sn Ž p p The value will be calculated below; in terms of it n Ž p is given by 3 n 3 p s n p ynp p qn SE p 6 where n Ž p is the number of unrestricted Ž a,a,a 3 satisfying Fa FŽ py i and afafa 3 Its value is given by py py n p s Ý Ý a s 6 pž p y 7 a3s a3s It remains only to calculate n Ž p given by p 3 py a p Ý as n p s ya q 8 The value of n Ž p p depends on the remainder when p is divided by 6 To show one case in detail consider ps6k where k is an integer Then ž / k 3a Ý ž / ky 3a npž p s Ý 3kq y a sodd q 3kq q a seven s3k s p 9 Hence from Eq Ž 6 p n 3Ž p s 6pŽ p y y 6p q p s Ž p ypq Ž 0 Taking n Ž p from Eq Ž 3 and n Ž p from Eq Ž 5 we find for ps6k n TOTALŽ p sn Ž p qn Ž p qn 3Ž p p s Ž p q pq Ž For ps6kq orps6kq5 one finds similarly 3 n p s py pq, Ž ps6kq or 6kq5 Ž

51 ( ) PH Frampton, WF ShiÕelyrPhysics Letters B n TOTAL s py pq pq, Ž ps6kq or 6kq5 Ž 3 For ps6kq orps6kq4 n 3 p s pq p y pq4, Ž ps6kq or 6kq4 Ž 4 3 n TOTAL s p q p q pq4, Ž ps6kq or 6kq4 Ž 5 and finally for ps6kq3 3 n 3 p s p yp ypy3, ps6kq3 3 n TOTAL s p q p ypy6, ps6kq3 Ž 6 Ž 7 The values of n Ž p, n Ž p, n Ž p, n Ž p 3 TOTAL and p Ý Ž X X n p p s TOTAL for FpF4 are listed in Table The next question is: of all these candidates for conformal Ns 0 theories, how many if any are conformal? As a first sifting we can apply the critewx 6 from vanishing of the two-loop rion found in Ref RGE byfunction bg Ž s0, for the gauge coupling The criterion is that aqasa 3 Let us denote the number of theories fulfilling this by n Ž p alive If p is odd there is no contamination by selfequivalent possibilities and the result is py alive Ý 4 rs n s pyr s py, psodd Ž 8 For p even some self equivalent cases must be subtracted The sum in Eq Ž 8 is pž py 4 and the number of self-equivalent cases to remove is? pr4@ with the results nalive s 4 p py3, ps4k 9 nalive s 4 py py, ps4kq 0 In the last two columns of Table are the values of p n p and Ý Ž X X n p alive p s alive Table p Values of n p, n p, n p, n p, Ý X n Ž p X 3 TOTAL p s TOTAL, p n p and Ý X n Ž p X for F pf4 alive p s alive p n n n n Ýn n Ýn 3 TOTAL TOTAL alive alive () () () () () () p p p p p p Asymptotically for large p the ratio n Ž p alive r n Ž p TOTAL ; 3rp and hence vanishes although n Ž p alive diverges; the value of the ratio is eg 08 at p s 5 and at p s 4 is 0066 It is being studied how the two-loop requirements by Ž s0 and bh Ž s0 select from such theories That result will further indicate whether any n Ž p alive can survive to all orders

52 5 ( ) PH Frampton, WF ShiÕelyrPhysics Letters B Acknowledgements One of us PHF thanks David Morrison for a discussion This work was supported in part by the US Department of Energy under Grant No DE- FG0-97ER-4036 References wx J Maldacena, Adv Theor Math Phys Ž 998 3, hepthr9700 wx SS Gubser, IR Klebanov, AM Polyakov, Phys Lett B 48 Ž , hep-thr98009 wx 3 E Witten, Adv Theor Math Phys Ž , hepthr98033 wx 4 L Susskind, E Witten, hep-thr98054 wx 5 S Mandelstam, Nucl Phys B 3 Ž wx 6 PH Frampton, hep-thr987 wx 7 S Kachru, E Silverstein, Phys Rev Lett 80 Ž , hep-thr98083 wx 8 S Ferrara, A Kehagias, H Partouche, A Zaffaroni, Phys Lett B 43 Ž 998 4, hep-thr wx 9 JR Russo, Phys Lett B 435 Ž , hep-thr98087 w0x M Schmaltz, hep-thr98058 wx M Berkooz, S-J Rey, hep-thr98074 wx J Distler, F Zamaro, hep-thr98006 w3x JA Harvey, Phys Rev D 59 Ž , hepthr98073 w4x IR Klebanov, AA Tsyetlin, hep-thr9900 w5x J Kakushadze, Phys Rev D 59 Ž , hepthr w6x AA Tsyetlin, K Zarembo, hep-thr w7x M Bershadsky, Z Kakushadze, C Vafa, Nucl Phys B 53 Ž , hep-thr w8x M Bershadsky, A Johansen, Nucl Phys B 536 Ž 998 4, hep-thr w9x A Lawrence, N Nekrasov, C Vafa, Nucl Phys B 533 Ž , hep-thr w0x N Nekrasaov, SL Shatashvili, hep-thr9900

53 3 May 999 Physics Letters B String unification at intermediate energies: phenomenological viability and implications GK Leontaris a, ND Tracas b a Physics Department, UniÕersity of Ioannina, GR-450 Ioannina, Greece b Physics Department, National Technical UniÕersity, Zografou, Athens, Greece Received 3 February 999 Editor: R Gatto Abstract Motivated by the fact that the string scale can be many orders of magnitude lower than the Planck mass, we investigate Ž 0 3 the required modifications in the MSSM b-functions in order to achieve intermediate 0 GeV scale unification, keeping the traditional logarithmic running of the gauge couplings We present examples of string unified models with the required extra matter for such a unification while we also check whether other MSSM properties Žsuch as radiative symmetry breaking are still applicable q 999 Published by Elsevier Science BV All rights reserved Introduction Recent developments in string theory have revealed the interesting possibility that the string scale Mstring may be much lower than the Planck mass M According to a suggestion wx P the string scale could be identified with the minimal unification scenario scale M string ;0 6 GeV It was further noted that, if extra dimensions remain at low energies w 8 x, unification of gauge couplings may occur at scales as low as a few TeV w5,6 x However, it is not trivial to reconcile this scenario with all the low energy constraints w9 x Recently w 4x it was further proposed that in the weakly-coupled Type I string vacua the string scale can naturally lie in some intermediate energy, GeV, which happens to be the geometrical mean of the MP and weak, M W, Ž scales ie M ; M M ; 0 GeV string ( W P It is a rather interesting fact that the possibility of intermediate scale unification was also shown to appear in the context of Type IIB theories w5 x This scenario has the advantage that this intermediate scale does not need the power-like running of the gauge couplings in order to achieve unification Appearance of extra matter, with masses far of being accessible by any experiment, could equally well change the conventional logarithmic running and force unification of the gauge couplings at the required scale Of course, intermediate scale unification could in principle trigger a number of phenomenological problems, such as fast proton decay Also, some nice features of the Minimal Supersymmetric Standard Model 6 MSSM unification at 0 GeV, among them the radiative electroweak breaking, could be problematic in principle In this short note we would like to investigate the changes that the MSSM beta functions should suffer in order to achieve gauge coupling unification at r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

54 54 ( ) GK Leontaris, ND TracasrPhysics Letters B GeV We further determine the extra matter fields which make gauge couplings merge at an intermediate energy and show that such spectra may appear in the context of specific string unified models which can in principle avoid fast proton decay We also examine the conditions in order to achieve radiative breaking of the electroweak symmetry, keeping of course the top mass within its experimental value In the context of heterotic superstring theory, the value of the string scale is determined by the relation MstringrMPs( asr8 where as is the string coupling This relation gives a value some two orders of magnitude above unification scale predicted by the MSSM gauge coupling running On the contrary, in the case of type I models, for example, this ratio depends on the values of the dilaton field and the compactification scale Choosing appropriate values for both parameters it may be possible to lower the string scale The gravitational and gauge kinetic terms of the Type-I superstring action are d 0 x y f SsH e I R 7 ž X 4 Ž p a yf I qe F Ž 9 q PPP 4 a / X 3 where a X sm y I, with the string scale now being denoted by M, while e f I I 'l I is the dilaton coupling and fi the dilaton field Consider now 6 of the 0 dimensions compactified on a 3 two-torii T =T =T with radii R, R, R 3 Then, the compactification volume is Vs Ł Ž p R Assuming the simplest case with i i isotropic compactification R s R s R ' R, with 3 the compactification scale M s rr, the 4-d effec- C tive theory obtained from the above action is ž 4 6 d x Ž p R y f SsH e I R 7 X 4 Ž p a 6 Ž p R yf I qe FŽ 9 q PPP Ž 4 a / X 3 In the above, FŽ9 is the 9-brane field strength of the gauge fields while dots include similar terms for 7,5, etc branes The gauge fields and the various massless states arising from non-winding open strings live on the branes while graviton lives in the bulk w6 x From the action one obtains the following expressions The gravitational constant GN is related to the first term and is given by 6 8 8Ž p R 8 M yf I ' M s e I P s Ž 6 6 X 4 G N Ž p a li MC The gauge coupling is extracted from the field strength term FŽ9 of the gauge fields in the 9-brane in Ž and is given by 6 6 4p 4p Ž p R M yf I ' s e I s Ž X 3 g9 a9 Ž p a li MC where g9 is the 9-brane coupling constant Combining the above two equations, one also obtains the relation li a9 li X GN s s a a Ž M I It can be checked that for a p-brane in general, the Ž w x formula 3 generalizes as follows 3,4 / py3 li MC a p s Ž 5 ž M I Then, the formula Ž 4 for the gravitational coupling constant becomes / I 9yp li M X C GNs s a a Ž 6 p ž M 4 M P The string unification scale may be also given in terms of the compactification scale and the p-brane coupling as follows ž / 6yp a p MC MIs MP Ž 7 ' M I From the last three expressions, it is clear that the compactification scale M C ; rr is rather crucial for the determination of the string scale We may explore the various possibilities by solving for M I and li in terms of the compactification scale and obtain the following relations ž / a p 6yp I C P r 7yp M s M M Ž 8 ' / C py3 a p M P 7yp lisa Ž 9 pž ' M

55 ( ) GK Leontaris, ND TracasrPhysics Letters B In terms of l, the string scale for any p-brane is I also written as follows ž / l I 3 4 I P C M s M M Ž 0 ' where all the p-dependence is absorbed in l I In order to remain in the perturbative regime, we should impose the condition l FOŽ From the last ex- I pression it would seem natural to assume M I;MC and demand that l I< to obtain a small string scale However, this is not a realistic case since from relation Ž 5 we would also have a p <, ie, an extremely low initial value for the gauge coupling From Ž 9 it can be seen that viable cases arise either for pf3 orp)7 In what follows, we wish to elaborate further the case where the string scale lies in the intermediate ( P W energies define by the geometric mean M M ; 0 GeV The weak coupling constraint on li above suggests that the effective field theory gauge symmetry is more naturally embedded in a 3- or 9-brane Taking into consideration these remarks, the corresponding compactification scale can be extracted from the above formulae In Table we give some characteristic values of the M, M and l for the 3- I C I and 9-brane case We assume that a p ;r0 which, as we will see in the next section, is indeed the correct value of the unified coupling for a unification 0 3 scale around M U ;0 GeV From the above table, it is clear that the requirement to remain in the perturbative regime is satisfied in all cases considered above However, for the case of 9-branes, the dilaton coupling is extremely small On the contrary, in the case of 3-branes this coupling takes reasonable values, in fact its value is fixed through the rela- tion Ž 5, l sa, being independent of the ratio Table I 3 p l log M log M I 0 I 0 C 3 r r0 3 3 r0 4 3 r < 8 9 < 9 < 67 0 MCrM I Therefore, the embedding of the gauge group in the 3-brane looks more natural w3,4 x Renormalization group analysis In this section we will explore the possibility of modifying the MSSM b-functions in order to implement the intermediate scale unification scenario Next, we will give examples of matter multiplets which fulfill the necessary conditions For simplicity, we will assume in the following that the compactification scale is the same as the string scale We begin by writing down the Ž one-loop running of the gauge couplings bi MU bi NS MSB s q log q log, a Ž M a p M p M i U SB is,,3 Ž where MU is the unification scale and MSB is the SUSY breaking scale and we have of course M U ) M SB)M In the equation above, we have assumed that Ø the three gauge couplings unify at M Ža Ž M U i U sa U Ø extra matter, possibly remnants of a GUT, appears in the region between MU and M SB: bis S S b qdb, where b sž 33r5,,y3 is the MSSM i i i b functions, and Ø in the region between MSB and MZ we have the Ž non-susy SM Žalthough with two higgs instead NS of one and the corresponding b sž i 4,y 3, y7 functions By choosing M'M in Ž Z, we can solve the system of these three equations with respect to Ž M, M,a U SB U as functions of the db i s, taking the values of a Ž M i Z from experiment In this sense, the db i s are treated as free continuous parameters However, when a specific GUT is chosen, these free parameters take discrete values depending on the matter content of the GUT surviving under the scale M Solving therefore Ž we get t SB U již jk jk jkž ji ji db pdž a y ji qdb NS t y db pdž a y jkqdb NS t s NS NS ydb db y db q db db y db jk ji Z ji jk Z

56 56 t s U ji pd a y qdb NS t y db NS ydb t ( ) GK Leontaris, ND TracasrPhysics Letters B ji Z ji ji SB db ji bi NS bi s y Ž tsbytz y Ž tuytsb a a p p U i where tsb,u,z is the logarithm of the corresponding scales, d pijspjypi and i, j,k should be different Although we have not written explicitly the unknowns wrt the db s Ž only t is given explicitly i SB it is obvious that t U, tsb and au depend only on the differences of b i s Therefore, if a certain solution Ž t,t, a U SB U is obtained by using specific values for Ž db,db,db 3, the same solution is obtained for Ž db q c,db q c,db q c 3 where c is an arbitrary constant By putting the following constraints 0 0 -MUrGeV-0 3, 0 3 -MSBrGeV-3P0 3 Ž we plot in Fig the acceptable values of Ž db,db for db3s0 The four lines correspond to the four combinations Ž a3ž M Z,sW u Ž MZ s Ž 0,033, Ž 0,036, Ž 0,033, Ž 0,036 Translating the lines by an amount c in both directions, the corresponding figure for db3s c appears Fig The allowed region of the Ž db,db space, for db3s0, in order to achieve unification in the region GeV, while the supersymmetry breaking is in the region 3 TeV Four Ž different a, sin u pairs are shown 3 W Fig The inverse of the unified gauge coupling as a function of Ž db,db, for db3s0 and the same constraints from MU and MSB as in Fig In Fig we plot the inverse of the unification y coupling, a, versus Ž db,db U, for db3s 0 Again, since au is one of the three unknowns of Ž, we can easily have the required au for any value of db 3 We see therefore a slight increase of the unification coupling with respect to the MSSM one Ž ;r4 As far as the unification scale M U and the SUSY breaking scale MSB are concerned, there is a tendency to decrease as db gets bigger, while the opposite happens for db Let us try to find the acceptable values for a specific GUT model, namely the SUŽ 4 =SUŽ L = SUŽ R In this case, we assume that the breaking to the standard model occurs directly at the string scale MIs M U, so that the gauge couplings g L, g R, g4 attain a common value g U The massless spectrum of the string model in addition to the three families and the standard higgs fields decomposes to the following SUŽ 3 = SUŽ = UŽ L Y representations w7x n Ž,,"r, n X Ž,,"r n Ž 3,,"r3, n X Ž 3,,"r6 3 3 n Ž 3,," r3, n Ž,,0 3 L In the above, n, n 3,, represent the number of each multiplet which appears in the corresponding parenthesis with the quantum numbers under SUŽ 3

57 ( ) GK Leontaris, ND TracasrPhysics Letters B =SUŽ =UŽ L Y In this case, the db i s are given explicitly n n X n n X dbs q q q q n 3, n qn n qn X L 3 3qn3 dbs, db3s Ž 3 In the specific GUT model the above n s are even integers Therefore, we see that db,3 are integers while db can change by steps of r6 In that case only the following 3 points are acceptable in all the region allowed by the constraints on sin u Ž M W Z and a Ž M having put earlier Ž keeping db s 0 3 Z 3 Ž db,db,db s Ž 4,,0, Ž 65,3,0, Ž 85,4,0 3 Several possible sets of n s can generate the above changes in the b-functions Again, as was mentioned above, acceptable values for higher db3 can be obtained in a straightforward manner Ž 4qc,qc,c, Ž 65qc,3qc,c, Ž 85qc,4qc,c where c is an integer but not any integer, since Eq X Ž should be satisfied for even ns It is easy to see from these equations that we need to change db3 by 3 units to find an acceptable solutions for the n X s Ž 4,,0 Ž 7,5,3 Ž 0,8,6 Ž 65,3,0 Ž 95,6,3 Ž 5,9,6 Ž 4 Ž 85,4,0 Ž 5,7,3 Ž 45,0,6 Of course, to these values correspond different sets of n s and obviously as the db i s increase more and more possible sets appear We give the possible n X s for the three acceptable cases with db s0 in Table 3, while of course n sn sn X 3 3 3s0 We have also checked whether the radiative breaking of the electroweak symmetry is still applicable In other words, we have used the coupled differential equation governing the running of the mass squared parameters of the scalars and checked H that only m becomes negative at a certain scale This scale depends of course on the chosen db i s, but stays in the region between GeV In conclusion, we have checked the possibility of Ž 0 3 intermediate scale 0 GeV gauge coupling unification, using the traditional logarithmic running, ie without incorporating the power-law dependence Table X n nl n Ž 4,, Ž 65,3, Ž 85,4, on the scale coming from the Kaluza-Klein tower of states We have showed that this kind of unification can be achieved with small changes of the b-functions of the MSSM gauge couplings, which can be attributed to matter remnants of superstring models We have applied the above to the successful SUŽ 4 = SUŽ = SUŽ model Ž L R which is safe against proton decay even in this intermediate scale, and found the necessary extra massless matter and higgs fields needed Finally we have checked that the radiative electroweak breaking of the MSSM still persists, driving the mass squared of the higgs to negative values at the scale ;0 5 7 GeV, while all others scalar mass squared parameters stay positive References wx E Witten, Nucl Phys B 47 Ž wx I Antoniadis, Phys Lett B 46 Ž ; I Antoniadis, K Benakli, M Quiros, Phys Lett B 33 Ž wx 3 N Arkani-Hamed, S Dimopoulos, G Dvali, Phys Lett B 49 Ž , hep-phr wx 4 JD Lykken, Phys Rev D 54 Ž wx 5 K Dienes, E Dudas, T Gherghetta, Phys Lett B 436 Ž ; hep-phr98069; D Ghilencea, GG Ross, Phys Lett B 44 Ž wx 6 I Antoniadis, N Arkani-Hamed, S Dimopoulos, G Dvali, hep-phr ; I Antoniadis, S Dimopoulos, A Pomarol, M Quiros, hep-phr98040 wx 7 C Bachas, hep-phr980745; wx 8 G Shiu, S-HH Tye, Phys Rev D 58 Ž ; Z Kakushadze, SH Tye, hep-phr980947; Z Kakushadze, hep-thr990080

58 58 ( ) GK Leontaris, ND TracasrPhysics Letters B w9x N Arkani-Hamed, S Dimopoulos, G Dvali, hepphr w0x S Abel, SF King, hep-phr ; K Benakli, S Davidson, hep-phr98080 wx P Nath, M Yamaguchi, hep-thr99033 wx K Benakli, hep-phr w3x CP Burges, LE Ibanez, F Quevedo, hep-phr w4x LE Ibanez, C Munoz, S Rigolin, hep-phr98397 w5x I Antoniadis, B Pioline, hep-thr w6x J Polchinski, S Chaudhuri, CV Johnson, hep-thr96005; J Polchinski, hep-thr96050 w7x I Antoniadis, GK Leontaris, ND Tracas, Phys Lett B 79 Ž 99 58

59 3 May 999 Physics Letters B ADE spectra in conformal field theory AG Bytsko a,, A Fring b, a StekloÕ Mathematics Institute, Fontanka 7, St Petersburg 90, Russia b Institut fur Theoretische Physik, Freie UniÕersitat Berlin, Arnimallee 4, D-495 Berlin, Germany Received 30 December 998; received in revised form 0 February 999 Editor: PV Landshoff Abstract We demonstrate that certain Virasoro characters Ž and their linear combinations in minimal and non-minimal conformal models which admit factorized forms are manifestly related to the ADE series This permits to extract quasi-particle spectra of a Lie algebraic nature which resembles the features of Toda field theory These spectra possibly admit a construction in terms of the Wn-generators In the course of our analysis we establish interrelations between the factorized characters related to the parafermionic models, the compactified boson and the minimal models q 999 Elsevier Science BV All rights reserved Introduction It is well known, that a large class of off-critical integrable models is related to affine Toda field theories wx or RSOS-statistical models wx, which possess a rich underlying Lie algebraic structure Since these models can be regarded as perturbed conformal field theories, it is suggestive to recover the underlying Lie algebraic structure also in the conformal limit Of primary interest is to identify the conformal counterparts of the off-critical particle spectrum One way to achieve this is to analyze the quasi-particle spectrum, which results from certain expressions of the Virasoro characters x Ž q or their linear combinations Hitherto this analysis was mainly performed wx 3 for formulae of the form q l t A lqbpl const x q sq Ý l Ž q l Ž q lr Here r is the rank of the related Lie algebra g, the matrix A coincides with the inverse of the Cartan matrix, B l characterizes the super-selection sector, q :s Ł Ž y q k l ks, and there may be certain restrictions on the summation over l Following the prescription of wx 3 one can always obtain a quasi-particle spectrum once a bytsko@pdmirasru fring@physikfu-berlinde r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

60 60 ( ) AG Bytsko, A FringrPhysics Letters B character admits a representation in the form of Eq Ž It should be noted that such spectra can not be obtained form the standard form of the Virasoro characters Ž 0 In the following we will demonstrate that one also recovers Lie algebraic structures in certain Virasoro characters or their linear combinations which admit the factorized form q const y X X q y N 4 y M 4 y x ;;x x ;;x, 4 where we adopt the notations of wx 4 ` n " xqky " " y Ł n y Ł a y ks0 as x 4 :s "q, x ;;x 4 :s x 4 In many cases Žsee w4,5x for details expressions of the type can be rewritten in the form Ž, but now A is entirely absent or, at most, is a diagonal matrix There are no restrictions on the summation over l, and we allow terms of the form Žq y in the denominator Žwhich may be regarded as an anionic feature wx l 5 Unlike the conventional form for the Virasoro characters Ž 0, formulae Ž and Ž allow to extract the y leading order behaviour in the limit q by means of a saddle point analysis, see eg w3,5 x For a slightly generalized version of Ž, in the sense that all Ž q are replaced by Žq y, this analysis leads to l r 6 r y Ž A ijqaji zi s Ł Ž yz j, ceffs Ý LŽ z i Ž 3 yp js is This means solving the former set of equations for the unknown quantities z i, we may compute the effective central charge thereafter by means of the latter equation in terms of Rogers dilogarithm LŽ x Recall that the effective central charge is defined as c scy4h X, where h X eff is the lowest conformal weight occurring in the model There exist inequivalent solutions to Eqs Ž 3 leading to the same effective central charge corresponding either to the form Ž or Ž When treating these equations as formal series, such computations give a first hint on possible candidates for characters Alternatively, with regard to factorization, we can exploit the essential fact that the blocks x4 y " are closely related to the so-called quantum dilogarithm and we can easily compute their contributions to the effective central charge As explained in Ref wx 4, each block Žx4 y " and Žx4 q " in expressions of type Ž y y contributes Dceff s, and Dceffs", Ž 4 y y respectively In the course of our argument, ie when we consider the difference of the Virasoro characters, we will also need the notion of the secondary effective central charge csy4h XX, Ž 5 where h XX is the next to lowest conformal weight occurring in the model l ADE structure Let g be a Lie algebra of rank r and h be its Coxeter number We define the following function related to g q const g J x,q s y, 6 x ;;xr4 h q

61 ( ) AG Bytsko, A FringrPhysics Letters B Table Effective central charges for minimal affine Toda field theories Ž Ž Ž Ž Ž Ž g An Dn E6 E7 E8 An n 6 7 n ceff nq3 7 0 nq3 with x obeying the condition x qx shrq, as,,r, Ž 7 a rqya h and for odd r we put x rqs 4 q Our aim is to find conformal models such that their characters or possibly Ž const linear combinations coincide with 6 for appropriately chosen q and x Such conformal models have quasi-particle spectra, generated by Ž 6 for the related sectors, with the number of different particle species equal to the rank r The question arises for which conformal models can we expect Ž 6 to be a character? Exploiting Ž 4, we readily find the corresponding effective central charge r r ceff Ž g s s Ž 8 hq dim gqr On the other hand, the analysis of the ultra-violet limit of the thermodynamic Bethe ansatz wx 6 for the ADE related minimal scattering matrices of affine Toda field theory leads to the following effective central charges Ž see Table 3 Thus, we see that, upon substitution of the related Lie algebraic quantities of the simply laced algebras Žsee eg wx 7, Eq Ž 8 recovers all the effective central charges in Table Furthermore it turns out that for g from this table corresponding to minimal models or cs models we are always able to identify several J g Ž x,q with single Virasoro characters or specific linear combinations of them In addition, there exist characters which exhibit even stronger Lie algebraic features They are given by Ž 6 with the values of x chosen as follows Žwhich is a particular case of Ž 7 a x yse, as,,r, Ž 9 a a where e 4 a stands for the set of the exponents of the Lie algebra g We denote this particular character as J g Ž q Minimal models The minimal models wx 8 are parameterized by a pair Ž s,t of co-prime positive integers and the corresponding s,t central charge is cs,t s y 6 syt Labeling the highest weights as h s ntyms y syt, with the st restrictions FnFsy and FmFty, the usual form of the characters of irreducible highest weight wx representations reads 9 ` s,t s,t stk n,m n,m Ý kž ntyms kž ntqms qnm ksy` x Ž q sh q q yq Ž 0 s,t h y y n,m s,t cž s,t n,m 4 4 Here we abbreviated the ubiquitous factor h :s q r by an analogy with the eta-function The Ž secondary effective central charge is easy to find Žsee eg wx ceff Ž s,t sy, c Ž s,t sy Ž st st n,m 4 st 3 Ž k Žk Ž Ž For a twisted affine Lie algebra of type XN one introduces h the Coxeter number and h s kh We should use h in 8 for A n

62 6 ( ) AG Bytsko, A FringrPhysics Letters B Thus, the values of c eff which are less than can be matched as follows c A Ž sc E Ž sc Ž 3,4, Ž eff eff 8 eff c A Ž sc Ž 5,6 sc Ž 3,0 sc Ž,5, Ž 3 eff eff eff eff c E Ž sc Ž 6,7 sc Ž 3,4 sc Ž,, Ž 4 eff 6 eff eff eff c E Ž sc Ž 4,5, Ž 5 eff 7 eff c A Ž sc Ž,nq3 Ž 6 eff n eff We see, that the matching is, in general, not unique For Ž 6 it depends on n the first non-unique representations occur for n s 6 and n s 9 and coincide with Ž 3 and Ž 4, respectively Therefore we might have for instance relations between A Ž ;A Ž and E6 Ž ;A Ž 8 Some of these apparent ambiguities are easily explained as the consequence of a symmetry property of the characters For instance we observe that Eq Ž 0 a s,t s,a t 6,5 3,0 possesses the symmetry: x q sx q, for instance x q sx Ž q a n,m n,a m,m,m Of course Eqs Ž Ž 6 are only to be understood as a first hint on a possibility for characters in the corresponding models to be of the form J g Ž x,q In order to make the identifications more precise, we have to resort to more stringent properties of the characters We shall be using previously obtained results w4,0x on representation of characters of minimal models in the form Ž In Ref w0x it was proven, that for Ms0 and x /x for i/j in Ž the only possible factorizable single characters are i j n,t n,t x Ž q sh nm;nt;ntynm 4 y, Ž 7 n,m n,m nt y y nt 4 nt 3n,t 3n,t x Ž q sh nt;nm;ntynm4 ntynm;ntqnm 4 Ž 8 n,m n,m Combining now characters in a linear way, the property to be factorizable remains still exceptional It was wx argued in Ref 4 that x s,t Ž q "x s,t Ž q Ž 9 n,m n,tym are the only combinations of characters in the same model which have a chance to acquire the form Ž with y reasonably small N and M The limit q of the upper and lower signs in Ž 9 is governed by ceff and c, respectively, which can be seen form their properties with respect to the S-modular transformation wx 4 The following factorizable combinations of type Ž 9 where found in Ref wx 4 5 ½ 5 y " y nt ntynm ntqnm 3n,t 3n,t 3n,t n,m n,m n,m 4 nt ½ nt ; nt 4 4 x Ž q "x Ž q sh nm;ntynm, Ž 0 5 y nt nt 4 n,t 4 n,t 4 n,t xn,m Ž q yx3n,mž q sh n,m ½ ; ynm;nm nt, Ž 5 nt nt nt 4 n,t 4 n,t 4 n,t x q qx q sh nm;nt;ntynm 4 y n,m 3n,m n,m nt ½ ynm; qnm;, nt y nt 6 n,t 6 n,t 6 n,t x Ž q yx Ž q sh nt;nm;ntynm4 ntynm;ntqnm 4 Ž 3 n,m 5n,m n,m y nt q

63 ( ) AG Bytsko, A FringrPhysics Letters B Table Representation of characters in the form of Ž 6 and for differences of the type Ž 3 in the form Ž 4 The replacement of the blocks x4 y typed in bold by x4 q yields the corresponding differences of characters g sectors x ;;xr4y E x ;3;4;5;;;3;44 3,4 4y x, ;3;5;7;9;;3;5 6 3,4 4y x,3 ;4;6;7;9;0;;5 6 3,4 4y A x, 4,5 4y E7 x, ;3;4;5;6;7;9 0 4,5 4y x, ;;3;5;7;8;9 0 5,6 5,6 3 4y A x, q x,4 ; 5r 5,6 5,6 4y x, q x,4 ; 5r 5,6 5,6 4y x, y x,5 ;8 0 5,6 5,6 4y x, y x,5 4;6 0 6,7 6, y E6 x,q x4, ; ;3;4; ;6 7 6,7 6,7 3 4y x, q x4, ; ;;5; ;6 7 6,7 6,7 3 4y x,3 q x 4,3 ;;3;4;5; 7 6,7 6,7 4y x, y x,6 ;3;4;0;; 4 6,7 6,7 4y x, y x,5 ;4;6;8;0;3 4 6,7 6,7 4y x,3 y x,4 ;5;6;8;9; 4 3,4 4y G x, ;3 4,7 4y F4 x, ;3;4;5 7,7 4y x, ;3;4;6 7,7 4y x ;;5;6 3,4 y 8, 6,3 7 y Remarkably, as we demonstrate in Table, all identifications presented in Ž Ž 6 can be realized in terms of characters with the help of Ž 7 Ž 3 In particular, we identify J g Ž q with the following characters J A Ž J A Ž J E Ž 6 J E Ž 7 Ž q sx 3,4 Ž q, J AŽ nž q sx,nq3 Ž q,,,nq Ž q sx 5,6 Ž q qx 5,6 Ž q sx 3,0 Ž q qx 3,0 Ž q,,,4,,8 Ž q sx 6,7 Ž q qx 6,7 Ž q sx 3,4 Ž q qx 3,4 Ž q,,,6,, Ž q sx 4,5 Ž q, J E 8 Ž Ž q sx 3,4 Ž q,,3 Since these identifications hint on the connection with massive models, ie affine Toda field theories, it is somewhat surprising that also the non-simply laced algebras G and F4 occur in Table 4 No connection is known between G - and F -affine Toda models and Ž 3,4 and Ž,7 4 minimal models, respectively At present it seems to be just an intriguing coincidence It is interesting to notice that, as seen from Table, the differences of type Ž 3 can also be of the form q const y Ž 4 x ;;xr4 b 5,6 5,6 6,7 6,7 For instance, x q yx q corresponds to bshq4, x s e q, and x q yx Ž q,,5 a a,,5 corresponds to bshq, x se q, where h and e 4 are the Coxeter number and exponents of A and E, respectively a a a 6 4 wx G 3,4 F 4,7 We thank W Eholzer for pointing out to us that our formulae in Ref 5 may include J q s x q and J q s x Ž q,, as well

64 64 ( ) AG Bytsko, A FringrPhysics Letters B Eq Ž 6 does not exhaust all manifest Lie algebraic functions in which combinations of characters of the type Ž 9 can be represented For instance, the following representation q hq ½ 5 hq 8 g const 4 J Ž w,q sq y Ž 5 w ;;w 4 ry h q with w obeying the condition w qw shrq also occurs Ž see Table 3 Compactified free Boson a rya In general the Fock space of a free boson may simply be constructed from a Heisenberg algebra and the corresponding Virasoro central charge equals cs The character of the Heisenberg module ŽUˆ Ž -Kac Moody yr4 is simply the inverse of the h-function, q r 4 y When compactifying the boson on a circle of rational square radius R s' srt one can associate highest weight representations of a UˆŽ k-kac Moody algebra to this theory The UˆŽ -algebra has an integer level, which is ksst The corresponding characters read wx 7 k ` k k kl qml k y q m m Ý m 4 k 4 k lsy` ˆ x Ž q shˆ q shˆ k kym;kqm, Ž 6 k y k hmy k m 4 where we denoted ˆm h :sq r 4 The highest weight may take on the values h m s 4k with ms0,,,k y As Table indicates, the Dn-affine Toda models are related to compactified bosons In order to recover the D Dn-structure at the conformal level, we shall find realizations of J nž q in terms of Ž 6 Similar to the case of minimal models, it will be helpful to study factorization of Ž combinations of the Kac Moody characters n First, choosing the constant in Ž 6 as h y we identify for even n n r 4 y q n4 n q n y y n Dn n n n n n ˆn r y ˆ 4 n ½ 5 ˆ 4 n n n r n r ½ 5 ˆn r n J Ž q sh sh n sh n sx Ž q Ž 7 ½ 5 n Formally this expression also holds for odd n, albeit in this case the right hand side may not be interpreted as D the character related to a compactified boson Therefore we need another way to construct J nž q in case n is n n odd For this purpose we consider the combinations ˆ x q " x Ž q which are analogues of Ž 9 m ˆnym In y particular, the q limit of these sums and differences is governed by c scs and c According to Ž eff 5 and the possible values for the highest weights we have csy6rn Table 3 Representation of characters in the form of Ž 5 For the arguments in bold the same convention applies as in Table, including the numerator g sectors w ;;w 4 y ry y 3,4 3,4,,3 4,5 4, y 7,,4 0 4,5 4, y x, q x,3 ;4; ; ;6; 0 A x q x E x q x ;; ; ;8;

65 ( ) AG Bytsko, A FringrPhysics Letters B wx Exploiting the identity 3 obtained in Ref 4, we find for 0-m-nr y n n m n m " n n n x q "x q sh n y ; q n m nym m Ž ˆ ˆ ˆ ½ 5 ½ 5 The counting, based on Ž 4, gives the expected values of c and c Notice that for the upper sign the rhs can be identified as the product side of Ž 6 : ˆ x n Ž q qxˆ n Ž q sxˆ n r4 Ž q Ž 9 m nym mr D Comparison with 7 then yields a formula for J nž q valid for both odd and even n J D nž q sxˆ 4 n Ž q qxˆ 4 n Ž q Ž 30 n 3n Finally, it is interesting to observe that, employing 7 0, we may express 6 and 8 entirely in terms of the minimal Virasoro characters: x Ž q sx Ž q, Ž 3 x,n,3n q n 3,n ˆm,m x,n,mž q x Ž q " x Ž q s x Ž q " x Ž q Ž 3 x,n,3n q n n 3,n 3,n m nym,m,m,n x,mž q ˆ ˆ Here ns6l", lgn if we really regard all components on the rhs as characters of irreducible Virasoro representations This restriction can be omitted if we regard Ž 7 Ž 0 just as formal series With regard to the central charge Eqs Ž 3 -Ž 3 imply c D Ž sc Ž 3,n qc Ž,3n yc Ž,n Ž 33 eff n eff eff eff Thus, the connection with minimal models is more subtle than one would expect at first sight from a simple Ž Ž matching of the central charges, eg c D sc Ž 3,4 3 Parafermions eff n eff Ž The A -series of affine Toda theories is known to be related in the ultra-violet limit Žsee eg wx n 6 to the Z -parafermions w x The corresponding central charge, cks Ž ky rž kq nq and characters may be ˆŽ ˆŽ k obtained from the SU ru -coset, where ksnq Introducing the quantity D sjž jq rž kq ky j,m y m rk the characters of the highest weight representation, which appear as branching functions in the coset, acquire the form wx k ` h rž rq sž sq k j,m rqs rsž kqq q xj,mž q s y Ý Ž y q 4 r, ss0 r jqm qsž jym r kqyjym qsž kqyjqm qkqy j = Ž q yq, Ž 34 k cž k k D j, my 4 4y where h j, m :sq r The labels are restricted as yjfmfkyj, 0FjFkr and Ž jym gz In k k particular the x 0, m q are the characters of the parafermionic currents c m The characters possess the symmetries x k Ž q sx k Ž q sx k Ž q Ž 35 j,m j,ym kryj,k rym From our observations made above for characters of the ADE related conformal models one may expect that expressions Ž 34 exhibit A -type structures Ž n eg possess n quasi particles and moreover acquire the form of the A type J nž q in some of the cases when they admit a factorized form We shall now discuss this issue in detail for several of the lowest ranks

66 66 ( ) AG Bytsko, A FringrPhysics Letters B As we have seen above, factorization of linear combinations of characters occurs usually only for the specific k k y type of combinations Now the analogue of 9 is xj, m q "x j, kym q One expects that the q limit of XX k these sums and differences is governed by c s ck and c, respectively Since h s D, Eq Ž eff, 5 yields csc Ž kž ky6 rk This is confirmed by all the examples given below For the parafermionic formulae Ž 34 we do not have such powerful analytical tools Žanalogues to the factorization formulae Ž 7 -Ž 3 at hand as in the case of the minimal models Therefore, as a first step, we resort to an analysis with Mathematica Typically we expand the characters up to q 00 A : In this case there are only three distinct Žup to the symmetries Ž 35 characters and they can be matched with those of the Ž 3,4 minimal model: x0,0 Ž q sx, 3,4 Ž q, x0, Ž q sx,3 3,4 Ž q, Ž 36 3,4 A x Ž q sx Ž q sj Ž q Ž 37,, A Thus, all the characters in this case are factorizable and moreover J Ž q is present among them A : There are four distinct characters in this case and they can be matched with those of the Ž 5,6 minimal model: x0,0 3 Ž q sx, 5,6 Ž q qx,5 5,6 Ž q, x0, 3 Ž q sx,3 5,6 Ž q, Ž ,6 3 5,6 5,6 x Ž q sx Ž q, x 3Ž q sx Ž q qx Ž q Ž 39,3,,5,, 3 3 Only x q and x Ž q are factorizable Ž see subsection IIA 0,, A 3: Since cs, it is suggestive to try to relate the characters to those of the compactified bosons This turns out to be possible for all the characters Ž thus factorizability is guaranteed : x 4 Ž q sxˆ Ž q, x 4 Ž q qx 4 Ž q sxˆ 3 Ž q, Ž 40 0, 6 0,0 0, 0 x,0 4 Ž q sxˆ 3 Ž q, x, 4 Ž q sxˆ 3 Ž q, Ž x Ž q s ˆ x Ž q, x 3Ž q s ˆ x Ž q Ž 4 3,, Furthermore, some of linear combinations can be expressed in terms of the characters of the Ž 3,4 minimal model Žnotice that cs, csyr for A and csr, csy for the Ž 3,4 minimal model : ,4 y x0,0 q y x0, q s Ž x, q, ,4 3,4 y x q " x 3 q s Ž x, q x,3 q,, 44 A 4: No characters or linear combinations factorize A : Several characters and combinations are factorizable and can be expressed via those of the Ž 3,4 5 minimal model and D n, for instance 6 3,4 4 4 x 3 Ž q sx Ž q ˆ x Ž q yxˆ Ž q,,, 6 8 ˆ ˆ 6 3,4 4 4 x 3, q sx, q x8 q yx6 q, x0, 6 Ž q "x 0, 6 Ž q s x, 3,4 Ž q "x,3 3,4 Ž q ˆ x9 4 Ž q xˆ5 4 Ž q, x, 6 Ž q "x, 6 Ž q s x, 3,4 Ž q "x,3 3,4 Ž q ˆ x3 4 Ž q xˆ 4 Ž q 6 6 5r96 Also we notice that x q yx 5,, q sq Such an identity can occur only in this parafermionic model since it requires cs0 A 6: no combinations factorize and the only factorizable single characters are y y 343 m;7ym; x,mž q sh y y 4 3m;y3m4, ms,,3 Ž 45

67 ( ) AG Bytsko, A FringrPhysics Letters B Summarizing these data, we see that, apart from the A case, none of the factorizable Ž combinations of A characters provided by Eq 34 for An can be identified as J nž x, q However, it is plausible to speculate that A in general the J nž x, q might be identifiable as characters of other conformal models having the central charge nrž nq3 This conjecture is supported by the A case Ž see Table and A case, in which we can identify 3 J A 3Ž q sxˆ Ž q qxˆ Ž q Ž To conclude the discussion on factorizable parafermionic characters, we notice an intriguing fact some characters in the A case exhibit an E structure Ž cf Tables and 3 : ,5 x0,ž q qx0,3ž q s x, Ž q, 8 8 4,5 x,ž q qx,3ž q s x, Ž q, ,5 4,5 x0,0ž q " x0,ž q qx0,4ž q s x, Ž q "x,4 Ž q, ,5 4,5 x,0ž q " x,ž q qx,4ž q s x, Ž q "x,3 Ž q This is the first case in which we have to combine three characters in order to obtain a factorized form A more detailed account on the factorization of A -related characters will be presented elsewhere n 3 One particle states q 4 y Ž y x4y The functions x and can be written as double series in q with coefficients being P n, m or QŽ n,m the number of partitions of an integer ng0 into m distinct Ž or smaller than mq non-negative integers Žsee eg w3,4 x Applying this fact to a character of the type Ž with x i/ x j, we obtain it in the form of a series ` k i x q sý m q, where the level k admits the partitioning, ksý Ý p a, into parts of a specific form Ž ks0 k a i a eg a i 47 and 48 below The interpretation of the p a a as momenta of massless particles gives rise to a quasi-particle picture Ždeveloped originally for characters of the form Ž in Ref wx 3, where a character is yb E k y pbõrl regarded as a partition function, x q sýkmke Here qse, with Õ being the speed of sound, and L the size of the system A quasi-particle spectrum constructed in this way is in one-to-one correspondence to the corresponding irreducible representations of the Virasoro algebra or some modules related to linear combinations It is crucial to stress that this procedure is not applicable to the standard representation of the characters Žie of the type Ž 0 and is a very specific feature of the representations Ž and Ž Note that the modules which are of the form Ž do in general Žif they do, they give rise to Rogers Ramanujan type identities wx 4 not factorize, such that the spectra related to do not only differ in nature from the ones obtainable from Ž, but are also related to different sectors As just explained, a quasi-particle representation can be constructed for any factorizable character of the type Ž provided that x / x For instance, the characters Ž 7 i j related to a compactified boson admit a representa- tion with Ž k q particles However, since we are particularly interested in spectra with Lie algebraic features, it is most natural to perform the quasi-particle analysis for the characters which admit the form J g Ž x, q In this way we obtain the following fermionic spectrum Žif we employ the series involving PŽ n, m in the units of prl / ž / h h i i paž m sx aq ž q Ž yma q q Na, Ž 47 4

68 68 ( ) AG Bytsko, A FringrPhysics Letters B Table 4 5,6 5,6 Bosonic spectrum for x q q x Ž q k denotes the level and m its degeneracy,,4 k i 5i i 3 5i k k m p sq, p s q 0 - < 0 p : 3 < 0 p : < 0 0 p, p : 5 < 0 0 p, p : < 0 0 0: < p, p, p, p, p : 7 < 0 0 0: < p, p, p, p : 4 3 < : < 0 0 0: < p, p, p, p, p, p, p, p : or bosonic spectrum Žif we use the series with QŽ n,m h i p sx q q N i a a ž / a Ž 48 i Here x, m and N parameterize the possible states In Eq 47 the numbers Na are distinct positive integers such m a i that Ý N s N, whereas in Eq Ž 48 is a a they are arbitrary non-negative integers Notice that for the combination of characters the levels may be half integer graded, such that also the momenta take on half integer values in this case A sample spectrum is presented in Table 4 which illustrates how the available momenta of the form Ž 48 are to be assembled in order to represent a state at a particular level Naturally the questions arise if we can interpret these spectra more deeply and if we can possibly find alternative representations for the related modules First of all we should give a meaning to the particular combinations which occur in our analysis In Ref wx 4 we provided several possibilities In particular the 5,6 5,6 combination x q q x Ž q,,4 is of interest in the context of boundary conformal field theories, since this combination of characters coincides with the partition function ZA, F for the critical 3-state Potts model with boundaries w4x Ž F denotes the free boundary condition It is intriguing that this combination possesses a 5,6 5,6 manifestly Lie algebraic quasi-particle spectrum The combination x q q x Ž q,,4, which coincides with ZBC, F in the same model possesses a slightly weaker relation to A To answer the question concerning possible representations, we recall the fact that the fields corresponding to the highest weight states satisfy the quantum equation of motion of Toda field theory w5 x It is therefore very suggestive to try to identify the presented spectra in terms of the W-algebras w6 x For J g Ž q we can make this more manifest Changing the units of the momenta to prl, we obtain from Ž 48 pa i seaqq Ž hq Na i Ž 49 Here ea belongs to the exponents of the Lie algebra Since the generators of the W-algebras Wsq are graded by the exponents plus one w7 x, we may associate the following generators to this quasi-particle spectrum i Na i a a a rya p ;W W W 50 In particular, the critical 3-state Potts model with boundaries would be related to the W -algebra We leave it for 3 the future to investigate this conjecture in more detail Acknowledgements We would like to thank W Eholzer for useful discussions AB is grateful to the members of the Institut fur Theoretische Physik, FU-Berlin for hospitality AF is grateful to the Deutsche Forschungsgemeinschaft Ž Sfb88 for partial support

69 ( ) AG Bytsko, A FringrPhysics Letters B References wx AV Mikhailov, MA Olshanetsky, AM Perelomov, Commun Math Phys 79 Ž ; G Wilson, Ergod Th Dyn Syst Ž 98 36; DI Olive, N Turok, Nucl Phys B 57 Ž wx G Andrews, R Baxter, P Forrester, J Stat Phys 35 Ž wx 3 R Kedem, TR Klassen, BM McCoy, E Melzer, Phys Lett B 304 Ž ; B 307 Ž wx 4 AG Bytsko, A Fring, Factorized Combinations of Virasoro Characters, hep-thr Ž 998 wx 5 AG Bytsko, A Fring, Nucl Phys B 5 Ž wx 6 TR Klassen, E Melzer, Nucl Phys B 338 Ž wx 7 VG Kac, Infinite dimensional Lie algebras, CUP, Cambridge, 990 wx 8 AA Belavin, AM Polyakov, AB Zamolodchikov, Nucl Phys B 4 Ž wx 9 BL Feigin, DB Fuchs, Funct Anal Appl 7 Ž 983 4; A Rocha-Caridi, in: J Lepowsky et al Ž Eds, Vertex Operators in Mathematics and Physics, Springer, Berlin, 985 w0x P Christe, Int J Mod Phys A 6 Ž wx VA Fateev, AB Zamolodchikov, Sov Phys JETP 6 Ž 985 5; Nucl Phys B 80 Ž wx VG Kac, D Petersen, Adv Math 53 Ž 984 5; J Distler, Z Qiu, Nucl Phys B 336 Ž w3x GE Andrews, The Theory of Partitions, CUP, Cambridge, 984 w4x JL Cardy, Nucl Phys B 34 Ž w5x A Bilal, JL Gervais, Nucl Phys B 38 Ž ; Z Bajnok, L Palla, G Takacs, Nucl Phys B 385 Ž 99 39; T Fujiwara, H Igarashi, Y Takimoto, Phys Lett B 430 Ž 998 0; Y Takimoto, H Igarashi, H Kurokawa, T Fujiwara, Quantum Exchange Algebra and Exact Operator Solution of A-Toda Field Theory, hep-thr98089 w6x VA Fateev, SL Lukyanov, Int J Mod Phys A 3 Ž ; Sov Phys JETP 67 Ž w7x VA Fateev, AB Zamolodchikov, Int J Mod Phys A 5 Ž

70 3 May 999 Physics Letters B Gauge unification in nonminimal models with extra dimensions Christopher D Carone Nuclear and Particle Theory Group, Department of Physics, College of William and Mary, Williamsburg, VA , USA Received 6 February 999 Editor: M Cvetič Abstract We consider gauge unification in nonminimal models with extra spacetime dimensions above the TeV scale We study the possibility that only a a subset of the supersymmetric standard model gauge and Higgs fields live in the higher dimensional bulk In two of the models we present, a choice for bulk MSSM matter fields can be found that preserves approximate gauge unification This is true without the addition of any exotic matter multiplets, beyond the chiral conjugate mirror fields required to make the Kaluza Klein excitations of the matter fields vector-like In a third model, only a small sector of additional matter fields is introduced In each of these examples we show that gauge unification can be obtained without the necessity of large string scale threshold corrections We comment briefly on the phenomenology of these models in the case where the compactification scale is as low as possible q 999 Elsevier Science BV All rights reserved Introduction Recently, Dienes, Dudas and Gherghetta Ž DDG w,x have suggested the intriguing possibility that grand unification may occur at intermediate or even low energy scales in models with extra spacetime dimensions compactified on orbifolds of radii R Above the compactification scale m0 s rr, the vacuum polarization tensors for the gauge fields of the minimal supersymmetric standard model Ž MSSM receive finite corrections from a tower of Kaluza Klein Ž KK excitations which contribute at one loop As a consequence, the gauge couplings develop a power-law rather than logarithmic dependence on the ultraviolet cut off of the theory, L w,3 x While this is not running in the conventional sense, the gauge couplings nonetheless evolve rapidly as a function of carone@physicswmedu L, so that it is possible to achieve an accelerated unification In the minimal scenario proposed by DDG, all the non-chiral MSSM fields, the two Higgs doublets and the gauge multiplets, live in a 4 q d dimensional spacetime, and have an associated tower of KK excitations The chiral MSSM fields are assumed to lie at fixed points of the orbifolds, and thus have no KK towers This is the simplest way of avoiding the difficulties associated with giving mass to chiral KK states DDG demonstrated that an approximate gauge unification could be achieved in this scenario, at a grand unification scale MGUT that is much smaller than its usual value in supersymmetric theories, = 6 0 GeV However, as pointed out in Ref w,4 x, strict comparison to the low energy data reveals that for TeV-scale compactifications, the DDG model predicts a value for a Ž m 3 z that is higher than the prediction in conventional unified theories, which is r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

71 ( ) CD CaronerPhysics Letters B already ; 5 standard deviations higher than the experimental value Assuming that the unification point coincides with the string scale, then a specific model of string scale threshold corrections is required before one can claim that unification in the DDG model is actually achieved It is the purpose of this paper to point out that there are a number of simple variations on the DDG proposal that achieve gauge unification much more precisely than the minimal scenario described above To begin, however, let us consider the variations that are as successful as the minimal case As pointed out by DDG, one possibility is to allow h generations of matter fields to experience extra dimensions, and to add to the theory their chiral conjugate mirror fields, so that suitable KK mass terms may be formed Assuming that the orbifold is S rz, then one may take the mirror fields to be Z odd, so that unwanted zero modes are not present in the low-energy theory Given that the KK excitations of the matter fields form complete SUŽ 5 multiplets, it is perhaps not surprising that if unification is achieved for hs0, it will also be preserved for h/0, at least for some range of m 0 What is less obvious is that the KK excitations of the matter fields may be chosen to form incomplete SUŽ 5 multiplets, and an approximate unification may still be preserved if only some of the MSSM gauge and Higgs fields experience extra dimensions In Section, we will present three models that demonstrate Ž i that it is not necessary for all gauge groups to live in the higher dimensional bulk in order to achieve unification, and Ž ii a precise unification may sometimes be obtained without introducing any exotic matter, beyond the mirror fields described above 3 In Section 3 we study these cases quantitatively, taking into account weak-scale threshold corrections, and two-loop running up to the compactification scale For TeV scale compactifica- tions w0 x, we will see that all of the nonminimal scenarios we present in this paper unify more pre- Using the same two loop code and input values described later in this paper, we find that a Ž m 3 z f076 in conventional supersymmetric unified theories, compared to the world average, 09"0008 wx 5 3 For other possibilities, see Ref wx 6 cisely than the minimal DDG scenario, and do not require large threshold corrections at the unification scale In Section 4 we summarize our conclusions, and make some brief comments on the phenomenological implications of these models when the compactification scale is low Three scenarios We assume 4 q d spacetime dimensions, with d dimensions each compactified on a Z orbifold of radius rm 0 The fields that experience extra dimensions are periodic in the d new spacetime coordinates y y d, and are either even or odd under y yy For example, in the case where ds, these bulk fields have expansions of the form ` ny Ž n m Fqs Ý cosž / F Ž x, ns0 R ` ny Ž n m y Ý ž R / ns F s sin F Ž x where n indicates the KK mode The only other fields in the theory are those which live at the orbifold fixed points ys0 or ysp R, and have no KK excitations The effect of a tower of KK states on the running of the MSSM gauge couplings was computed by DDG, and is given in a useful approximate form by wx / ž / b L b i i L y y ai Ž L sai Ž m z y ln q ln p ž m z p m0 ž / d bi Xd L y y 3 pd m 0 Here the b i are the beta function contributions of a single KK level, and X is given by p dr d Xd s 4 dg Ž dr

72 7 ( ) CD CaronerPhysics Letters B Table Contributions to b i gauge Ž 0,y4,y6 H q H Ž 3r5,,0 u d QqQ UqU Dq D Lq L Eq E Ž r5,3, h Ž 8r5,0, h Ž r5,0, h Ž 3r5,,0 h Ž 6r5,0,0 h total: 3r5,y3,y6 4,4,4 h For the scenarios considered by DDG, these beta functions are 33 3 bis Ž 5,,y3, bis Ž 5,y3,y6 qhž 4,4,4, 5 where h is the number of generations of matter fields that experience extra dimensions DDG observe that a sufficient condition for gauge unification to be preserved is that the ratios b yb i j Bijs 6 b yb i j be independent of i and j Thus, they point out that in the scenario above B B 7 3 s 77 f094 and s f09 7 B B 3 3 We will now show that there are a variety of other models, each with a different set of MSSM fields living at the orbifold fixed points, that lead to B rb f B rb f First, notice that the b i of the minimal scenario can be decomposed into the contributions from the KK excitations of each MSSM field, as shown in Table An overline denotes a mirror field required so that the given KK tower is vector-like This table is useful in that it allows us to mix and match We will do so taking into account the string constraint that bulk matter may only transform under bulk gauge groups For example, consider a model with all leptons and gauge fields living in the bulk, but with Higgs fields and quarks at the fixed points The b are given by i b is Ž 0,y4,y6 q3p Ž 9r5,,0 s Ž 7r5,y,y6 8 In this case we find B B s 33 f096 and s 0 f095 9 B B 3 3 As we will confirm explicitly in the next section, this scenario achieves unification more precisely than the minimal one In Table, we present three scenarios with B ij ratios that are significantly better than in the minimal scenario We indicate the gauge group when the corresponding gauge multiplet is a bulk field, H for both MSSM Higgs fields, and nf for n generations of an MSSM matter field F' Ž Q,U, D, L, or E Note that it is possible in scenario to exchange an L for an H; the vector-like tower of KK excitations associated with a zero-mode left-handed lepton field have the same effect on the b as the tower associated with the MSSM Higgs fields Scenarios and 3 demonstrate that it is not necessary to assume that both SUŽ and SUŽ 3 gauge multiplets live in the higher dimensional bulk in order to obtain a successful unification As far as we are aware, this point has not been made in the literature Note that only the third scenario involves extra matter, two SUŽ 5 5q5 pairs in which only the leptons live in the bulk We assume that the exotic matter zero modes have a mass of ; mtop for the purpose of our subsequent analysis More strikingly, Scenarios and demon- Table Three scenarios Ž for explanation of the notation, see the text Scenario Bulk MSSM fields Exotic fields BrB3 B3rB3 minimal SUŽ 3, SUŽ,UŽ, H none SUŽ 3, SUŽ, UŽ, 3E, 3L none SUŽ 3, UŽ, U, D, 3E none SUŽ, UŽ, H, 3L, E Two 5q5 wr blk leptons 00 00

73 ( ) CD CaronerPhysics Letters B strate that is possible to achieve an improved unification in nonminimal models without the addition of any exotic matter multiplets, beyond the mirror fields required to render the KK towers of the matter fields vector-like We will now consider all three scenarios quantitatively, and show that none require large threshold corrections at the unification scale 3 Numerical results Our numerical analysis of gauge unification in the scenarios listed in Table is quite conventional We adopt the MS values for the Ž GUT normalized gauge couplings a Ž m s 5899 " 004 and a Ž m z z s 957"003 that follow from data in the 998 Rewx 5 We run these up to the view of Particle Physics top quark mass, where we then assume the beta functions of the supersymmetric standard model, and where we convert the gauge couplings to the DR scheme We take into account threshold effects, due to varying superparticle masses, at the one-loop level, and running between mtop and the compactification scale m at the two loop level We then use Eq 3 0 above the scale m0 to determine the unification point Thus, our procedure is similar to Ref wx 4, except that we allow for greater freedom in our choice of weak scale threshold corrections This procedure is iterated with trial values of a Ž m 3 z until a suitable three coupling unification is achieved For each of the given scenarios we obtain a prediction for a Ž m 3 z assuming no threshold corrections at the unification scale While such high scale threshold corrections should be present generically, our approach allows us to test the assumption that these need not be large In Fig, we show the qualitative behavior of unification in scenarios, and 3 by plotting the Fig Unification in Scenarios, and 3, with L in GeV The minimal scenario is provided for comparison

74 74 ( ) CD CaronerPhysics Letters B running couplings above a compactification scale of TeV, assuming the experimental value of a Ž m 3 z s 09" 0008 wx 5 Table 3 presents predictions for a Ž m 3 z assuming either an intermediate or low unification scale We display results for ds and, and for m0 s TeV and 0 8 GeV These choices are sufficient to understand the qualitative behavior of the results: as d increases, the predictions for a Ž m 3 z increase monotonically, while for increasing values of m 0, the predictions approach that of the MSSM without extra dimensions Table 4 provides the predictions for a Ž m 3 z including representative weak scale threshold effects, in which we have either placed all the non-colored MSSM superpartners at TeV, with the rest at m top, or vice versa Let us consider the results for each of the scenarios in turn: Ø Minimal scenario: This is the hs0 scenario of DDG, which we include as a point of reference The beta functions for this scenario are given in Eq 5 Note that for ds and m0 s TeV the low energy value of a Ž m 3 z is ; 30 standard deviations above the experimental central value, 09"0008 wx 5, and improves to ;6 standard deviations 4 if one assumes that colored MSSM superpartners at the weak threshold are all at mtop while noncolored sparticles are at ; TeV These results agree qualitatively with those in Ref wx 4, where a different approximation for weak scale threshold effects was used Ø Scenario : In this scenario, the gauge fields and leptons live in the bulk, while the Higgs and quarks live at orbifold fixed points The beta functions for this scenario were given in Eq 8 Notice that our previous observation that this scenario satisfies the relation BrB3 s B3rB3 s more accurately than the minimal case does translate into a better predictions for a Ž m 3 z For ds and m0 s TeV, and assuming the same choice for weak scale threshold corrections applied to the minimal scenario above, we find agreement with the experimental value of a Ž m at the 4 standard deviation level 3 z 4 Had we used value of a Ž m 3 z in the 996 Review of Particle Physics, we would obtain results too high by about 98 standard deviations The experimental determination of a3 has since improved Table 3 Predictions for a Ž m 3 z, assuming no weak scale threshold corrections Scenario d s d s d s d s m0s m0s m0s0 m0s0 TeV TeV GeV GeV 8 8 minimal Ø Scenario : In this scenario, the SUŽ gauge multiplet and Higgs fields are confined to the fixed point, while precisely one generation of right-handed up and down quarks, and three generations of right-handed leptons live in the bulk The KK beta function contributions are given by b is Ž 8r5,0,y4 Ž 3 This scenario achieves unification much more precisely than the minimal one In the case where m0 s TeV and d s, the predicted value of a3 Ž m z is only 8 standard deviations below the experimental central value, ignoring all threshold corrections Allowing the superparticle spectrum to vary as in Table 4, we find this prediction varies between 005-a3Ž m z -06 Ž 3 Thus, unification can be achieved in this model without any string scale threshold corrections Ø Scenario 3: In this example, the SUŽ 3 gauge multiplet is confined to the orbifold fixed point, and hence we are only allowed to place leptons and Higgs fields in the bulk If we let he and hl represent the number of right-handed and lefthanded KK lepton excitations Žincluding their chiral conjugate partners, then the constraint B s B s B implies that h s Ž 3h y E L r, which has no solution for only three generations of left-handed lepton fields However, there is a simple way to circumvent this problem Notice that if we add SUŽ 5 5q5 pairs in which only the lepton compontents live in the bulk, then the condition on he and hl given above will hold, since differences in zero mode beta function pairs will remain unchanged A solution may then be obtained by choosing hls5 and hes, which implies the existence of two such 5 q 5 pairs,

75 ( ) CD CaronerPhysics Letters B Table 4 Predictions for a Ž m, assuming Ž A all noncolored MSSM superpartners have TeV masses while the rest have masses of m, and Ž B 3 z top all colored MSSM superpartners have TeV masses while the rest have masses of m top Scenario ds ds ds ds m s TeV m s TeV 8 m s0 GeV 8 m s0 GeV case A minimal case B minimal when one generation of right-handed and three generations of left-handed MSSM lepton fields are assigned to the bulk The beta functions for this scenario are then given by b s Ž 43r5,3,y, b s Ž 4r5,,0 Ž 33 i i Notice in this case that the SUŽ 3 gauge coupling only evolves logarithmically, since b 3 s 0 Unification may be achieved at a low scale by virtue of the power-law evolution of a y and a y, as can be seen from Fig This scenario is about as successful as scenario, predicting a Ž m 3 z only 0 standard deviations below the experimental central value, ignoring all threshold corrections, and assuming that the exotic matter fields have masses of m top Allowing the sparticle mass spec- trum to vary between mtop and TeV, as in Table 4, we find that the scenario 3 prediction for a Ž m varies between 3 z 004-a Ž m -04, Ž 34 3 z for ds and m s TeV Again, unification is 0 achieved without the need for threshold corrections at the high scale Although this scenario does indeed involve some new matter fields, the choice is relatively minimal, and may be completely natural from the point of view of string theory 4 Discussion What is interesting about the scenarios we have presented is that gauge unification can be achieved in so many different ways Each of our scenarios unifies more precisely than the minimal DDG model, and none requires large Ž or in two cases any threshold corrections at the unification scale These models illustrate two other interesting points as well: Ž One can achieve unification when some of the standard model gauge groups are confined to a brane Ž There are some models that unify more precisely than DDG that do not require any additional matter fields with exotic quantum numbers, beyond the vector-like KK towers of certain MSSM fields that are chosen to live in the bulk Before concluding, we comment briefly on some of the other phenomenological implications of these scenarios when the compactification scale is low For a more complete discussion of the phenomenology of standard model KK excitations in models with TeV scale compactifiwx 9, we refer the reader to Refs w8,0 x In scenarios and the gluon has a tower of KK cation excitations, which leads to a significant bound on the compactification scale The KK gluon excitations are massive color octet vector mesons, with couplings both to zero mode gluons and to all the quarks Thus, the KK gluons are in every way identical to flavor universal colorons, and are subject to the same bounds Recall that in coloron models, one obtains a massive color octet from the spontaneous breaking of SUŽ 3 =SUŽ 3 down to the diagonal color SUŽ 3 In the case where the two SUŽ 3 gauge couplings are equal, the coloron couples to quarks exactly like a gluon, or a KK gluon The couplings of colorons or KK gluons to zero mode gluons are completely determined by SUŽ 3 gauge invariance, and hence are also the same Thus the relevant bound on the

76 76 ( ) CD CaronerPhysics Letters B lowest KK gluon excitation is given by M c )759 GeV at the 95% confidence level wx 7, which follows from consideration of the dijet spectrum at the Tevatron This constraint places a lower bound on the scale for all the KK excitations in scenarios and We can obtain a similar bound on the compactification scale in scenario 3 from the production and hadronic decay of W boson KK excitations, which have standard model couplings to the quarks: M W X) 600 GeV wx 5 Other direct collider bounds on the compactification scale require a more detailed analysis, given the nonstandard W X and Z X couplings in our models This issue will be considered elsewhere w x Scenario is particularly interesting when one takes into account that interaction vertices involving fields that all live in the higher dimensional bulk respect a conservation of KK number ŽOne can think of this as arising from the conservation of KK momentum following from translational invariance in the extra dimensions Hence, in scenario, the KK excitations of the electroweak gauge fields cannot couple to the lepton zero modes, and we obtain both Z X and W X bosons with otherwise standard couplings, that are naturally leptophobic! These states would likely be within the reach of the LHC for TeV scale compactifications The other two scenarios present a more complicated phenomenology, since some generations of a given MSSM matter field live in the bulk while others live on the brane It follows that the KK excitations of a standard model gauge field would have generation-dependent couplings to the zero mode matter fields, and may contribute to a variety of quark and lepton flavor-changing processes Finally it is worth pointing out that in scenario 3, the fact that the KK W boson excitations can t couple to zero mode left-handed lepton fields, also leads to a leptophobic W X While the purpose of the present work was to focus on gauge unification in these nonminimal scenarios, a more quantitative discussion of the TeV scale phenomenology of the scenarios described here will be presented in a sepaw x rate publication Acknowledgements We are grateful to Keith Dienes for his comments on the manuscript, and thank Marc Sher and Carl Carlson for useful discussions We thank the National Science Foundation for support under grant PHY References wx KR Dienes, E Dudas, T Gherghetta, Phys Lett B 436 Ž ; Nucl Phys B 537 Ž wx KR Dienes, E Dudas, T Gherghetta, e-print archive: hepphr98075 wx 3 G Veneziano, T Taylor, Phys Lett B Ž wx 4 D Ghilencea, GG Ross, Phys Lett B 44 Ž wx 5 Review of Particle Physics, Particle Data Group, Eur Phys J C 3 Ž 998 wx 6 Z Kakushadze, e-print archive: hep-thr9893 wx 7 B Abbott et al, D0 Collaboration, FERMILAB-Conf- 98r79-E, e-print archive: hep-exr wx 8 VA Kostelecky, S Samuel, Phys Lett B 70 Ž 99 wx 9 I Antoniadis, Phys Lett B 46 Ž w0x I Antoniadis, K Benakli, Phys Lett B36 Ž ; I Antoniadis, K Benakli, M Quiros, Phys Lett B 33 Ž wx CD Carone, in progress

77 3 May 999 Physics Letters B RG fixed points in supergravity duals of 4-d field theory and asymptotically AdS spaces M Porrati a,b,, A Starinets a,b, a Theory DiÕision CERN, CH GeneÕa 3, Switzerland b Department of Physics, NYU, 4 Washington Pl, New York, NY 0003, USA Received 8 March 999 Editor: L Alvarez-Gaumé Abstract Recently, it has been conjectured that supergravity solutions with two asymptotically AdS5 regions describe the RG flow of a 4-d field theory from a UV fixed point to an interacting IR fixed point In this paper we lend support to this conjecture by showing that, in the UV Ž IR limit, the two-point function of a minimally coupled scalar field depends only on the UV Ž IR region of the metric, asymptotic to AdS 5 This result is consistent with the interpretation of the radial coordinate of Anti de Sitter space as an energy scale, and it may provide an analog of the Callan Symanzik equation for supergravity duals of strongly coupled field theories q 999 Elsevier Science BV All rights reserved The duality between gauge theories and Ž super string geometries, first proposed for conformal field theories w 3 x, also holds in a more general setting Particularly interesting is the case when the superstring geometry is only asymptotically AdS 5 This setting describes 4-d gauge theories that are Ž super conformal only in the ultraviolet In the inw4 6 x, or reduce to another conformal field theory w7 9 x The 5-d metric that describes the latter case has two frared, they may confine andror screen charges regions, respectively far and close to the brane, where the metric is asymptotically AdS 5 The inter- massimoporrati@nyuedu andreistarinets@physicsnyuedu polating metric is still invariant under the 4-d Poincare group and it reads wx 8 : ds se fž z Ž dz qhmn dx m dx n, hmn sdiagž y,,, Ž For small z Ž far from the brane the prefactor in the metric has the following expansion: ž / z f Ž z sylog q OŽ z R UV Here, RUV is the radius of the far AdS region For large z the expansion is, instead: ž / z f Ž z sylog q OŽ rz Ž 3 R IR r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

78 78 ( ) M Porrati, A StarinetsrPhysics Letters B In Ref wx 8 it was shown that an interpolating metric as in Eq Ž exists, and that it describes mass deformations of N s 4 SUŽ N super-yang Mills theory The metric was shown to exist using some general properties of type IIB, 5-d, gauged superw0, x The proof given in Ref w8x does not gravity rely heavily on specific properties of gauged supergravity, and it is valid also in a more general context; in type 0 strings, for instance The interpolating metric was interpreted as describing the renormalization group flow from an UV N s 4 superconformal theory to an IR conformal theory This interpretation is suggested by the UVrIR connection w5,,3 x, ie by the identification of the AdS5 coordinate, z, with an appropriate length scale in the 4-d field theory A problem with a direct interpretation of the equations of motion of gauged supergravity as RG equations is that they are second order, instead of first order This seems to suggest that z cannot be identified with the renormalization scale of the 4-d field theory Purpose of this paper is to prove that this identification is nevertheless correct, namely, that the IR physics of the boundary field theory is essentially independent of z This will be proven by finding an analog of the Callan Symanzik equation of field theory, that describes the change of the two-point function of a composite operators under change of the UV cutoff This equation will show, that the low-momentum limit of the two-point function is only sensitive to the IR region of the 5-d geometry ie the region close to the brane We will also show that the high-momentum limit of the two-point function is sensitive only to the UV region of the 5-d geometry Let us start by recalling that in the geometry dual of field theory the two-point function of a composite operator, OŽ x m, with source c Žx m is found as follows 3 The field c is promoted to a 5-d field c Ž x m, z It obeys some boundary conditions, given in Refs w,3 x Here we find it convenient to follow the 3 Here, for simplicity, we will restrict ourselves to minimallycoupled scalar fields prescription of Ref wx, and to choose as boundary conditions at small z a plane 4-d wave: c Ž x m sexpžik x m m m ik m x m < k k zse c x, z se c z, c z s, lim c Ž z s0, k k m 'k )0 Ž 4 z ` k m A few comments are in order here Ž a Here e is an UV regulator, and must be chosen much smaller than any other length scale in the prob- Ž lem, in particular, ke< b When k -0 the boundary condition at large z is c Ž z k A expž ikzqik < < x m Ž c m If the 5-d geometry is not AdS5 at large z, but rather it develops a singularity at finite zsa, then the boundary condition at a is that c Ž z k is regular near z s a In this paper, we set aside the latter possibility, and assume that the large-z geometry of the 5-d space obeys Eq Ž 3 The 5-d field c Ž z k obeys free scalar equation of motion: f Ž z yezey3f z z z Ezqk qe M z ck z s0 Ž 5 Here, f Ž z ' E fž z z z The square-mass term M Ž z becomes constant both in the IR and in the UV lim M Ž z sm, lim M Ž z sm Ž 6 z 0 UV z ` Generically, MUV /MIR 3 Finally, the two-point function Ak Ž s 4 H d xexpžik x m ² OŽ x OŽ 0: is given by wx m 3f ` Ž z AŽ k s e ck ) Ž z EzckŽ z e ` EzckŽ z 3f Ž z ' e Ž 7 c Ž z k The latter form of Ak Ž is independent of the normalization condition at zse, and it is valid whenever c Ž z k obeys the correct boundary con- dition at zs` We want to prove, first of all, that the lowmomentum behavior of Ak Ž is insensitive to the small-z region of the 5-d geometry e IR

79 ( ) M Porrati, A StarinetsrPhysics Letters B The key to the proof is a first-order equation for the two-point function, somewhat reminiscent of the Callan Symanzik equation Using Eq Ž 5, and defining 3f Ž z A k, z sye Ezlog ck z, 8 one finds the equation: E AŽ k, z se y3 f Ž z A Ž k, z yk e z 3f Ž z ym Ž z e 5f Ž z Ž 9 Notice that the boundary conditions at large z ensure that lim Ak, Ž z z ` s 0 The two-point function plays the role of the initial condition for Eq Ž 9 : AŽ k,e saž k Ž 0 Notice that, analogously to the Callan Symanzik equation, Eq Ž 9 does not fix the value of the two-point function: that comes from solving the second-order equation Eq Ž 5, subject to the boundary conditions given in Eq Ž 4 Eq Ž 9 describes instead the evolution of a quantity, Ak, Ž z that coincides with the true two-point function at small z It is tempting to interpret Ak, Ž z as the two-point function computed with a cutoff z This is indeed true if for low momenta, k 0, AŽ k, z differs from Ak Ž by at most a multiplicative factor, and an additive factor either polynomial in k or of higher order in the k expansion: AŽ k, z sz Ž z AŽ k qpž k, z qo k z AŽ k, kz< Ž The multiplicative factor Z Ž z is interpreted as the wave-function renormalization of the operator OŽ x m The polynomial PŽk, z changes only the contact terms in the two-point function, without affecting its behavior at non-coincident points The last term is negligible in the infrared limit Notice that, whenever Eq Ž holds, the dependence of the two-point function on the UV geometry, ie the small-z region, is completely factored into contact terms and a wave-function renormalization constant In field theory, the same can be said Õerbatim for the dependence of the two-point function on the UV cutoff Therefore, if Eq Ž holds, the coordinate z plays exactly the role of a length cutoff This is another manifestation of the UVrIR connection for non-conformal theories More interestingly, Eq Ž says that in geometries with two AdS regions, as in the examples in Refs w8,9 x, the infrared behavior of the 4-d theory is completely described by the IR AdS geometry, given in Eqs Ž 5 Eq Ž,Eq Ž 3 To study the IR, one can ignore the behavior of the metric in the UV region kz) As a concrete application of this result, the quantity MIR is related to the IR scaling dimension of O, D IR,by the standard AdS formula w,3,4x ( DIR sq 4qMIR R IR Eq Ž is easily proven It is sufficient to notice that Ak, Ž z is a smooth function of the initial conditions Ak Ž This is a standard property of ordinary differential equations with smooth coefficients, as Eq Ž A proof of this theorem can be found in Ref w5 x We are interested in the k-dependence of AŽk, z Smoothness in the initial conditions Ž and z implies Ak, Ž z sfžažk, z,k, with FŽA, z,k a smooth function of A, z and k The field-theory interpreta- Ž tion of Ak tells us that the small-k expansion of Ž 4 Ak reads AŽ k saž 0 qqž k qck DIRy4 qož k D IRy Ž 3 Here c is a nonzero constant, positive by unitarity of Ž the 4-d IR theory; Qk is a polynomial in k, vanishing at k s0 By expanding Ak, Ž z near the initial condition AŽ 0 we find AŽ k, z sfž AŽ 0,z,k sf E F q Ž A, z,k < AsAŽ0 AyAŽ 0 E A 4 qo AyAŽ 0 Ž AŽ 0,z,0 qrž k, z E F q Ž A, z,0 < AsAŽ0 AyAŽ 0 E A 4 qo AyA 0, AyA 0 k, Ž 4 4 Here we write the expression valid for generic, non-integer D For integer D, the non-analytic term reads k Ž D IRy IR IR log k

80 80 ( ) M Porrati, A StarinetsrPhysics Letters B Ž where Rk, z is a polynomial in k, vanishing at k s0 Eq, with Z se FrE A < AsAŽ0, follows immediately from the expansion in Eq Ž 3 and the smoothness of FŽ A, z,k Positivity of the wave-function renormalization Z is proven as follows By analytic continuation in k, Ak, Ž z becomes an analytic function with a cut along the real negative axis By its definition, given in Eq Ž 8, it obeys Ak Ž ), z sa ) Žk, z By splitting Eq Ž 9 into real and imaginary part, we find the equation EzIm AŽ k, z se y3 f Ž z Re AŽ k, z Im AŽ k, z Ž 5 Expanding its solution near k s0 we find 5, thanks to Eq Ž 3 : Im AŽ k, z se H z ey3 f Žw ReAŽ 0,w dw csin D IR Ž 4D p k y4 e IR qož k D IRy Ž 6 From this equation, it follows that the wave-function renormalization factor is positive: z H ey3 f Žw ReAŽ 0,w dw e ZŽ z se Ž 7 After having studied the IR limit of the two-point function, we want to study the opposite limit, namely k ` We want to show that, in this limit, Ak Ž can be computed by approximating the metric Ž with its UV AdS form, given in Eq Ž We shall do so by writing Eq Ž 5 in the form of an integral equation and studying its Jost solution in the limit k ` Let us write fž z in the form z f Ž z sylog q hž z, Ž 8 ž / R UV where hz has properties hž z sož z for z 0, hž z R ž IR R UV / slog qož rz for z ` Ž 9 5 Again, we write an equation valid for non-integer D IRItis trivial to see that the equation for DIR integer gives the same result for the Z factor y By writing c s e f Ž z f z, Eq 5 is converted k k into the Schrodinger equation 5 MUV RUV XX fk y f yk f y f svž z f, k k k k 4 z z Ž 0 where 9 3 XX 9 X X VŽ z s h q 4Ž h y h z MUV R UV M Ž z hž z y y e z Ž Here VŽ z sož rz ž M UV / for z 0, d 3 VŽ z s qo rz for z `, z where dsmir R IRyMUV R UV The pure-ads Eq Ž 0 with VŽ z s0 has two independent solutions: ' p kz K Ž kz and ' p kz I Ž kz n n We are look- ing for the solution of the full equation defined by kz the boundary condition lim e f Ž z z ` k s Con- structing an appropriate Green s function, we can write Eq Ž 0 Žor Eq Ž 5 in the form of an integral y 3 hž z equation Let c Ž z se c Ž z, then c Ž z k k k is a solution of k z c kž z s Kn Ž kz ` dj yz H GŽ j, z ;k VŽ j c kž j, Ž 3 j z where GŽ j, z ;k sin Ž kz KnŽ kj yinž kj KnŽ kz Ž 4 We normalize the solution c Ž z k to at zse: ckž z CkŽ z,e s, Ž 5 ckž e then the two-point function is given by AŽ k s e 3f Ž z EzC kž z,e C Ž z,e k ` e E c z kž z 3f Ž z 3f Ž z 3 X s e ye h Ž z Ž 6 c Ž z k ` e

81 ( ) M Porrati, A StarinetsrPhysics Letters B All the k-dependence in Eq Ž 6 is contained in E logwc x z k It is therefore sufficient to compute E c Ž z rc Ž z The solution of Eq Ž 3 z k k is given by the series c Ž0 kž z sck Ž z qck Ž Ž z q, Ž 7 where k z Ž0 c k Ž z s Kn Ž kz, Ž 8 c k Ž nq Ž z k z ` dj Ž n sy H GŽ j, z ;k VŽ j ck Ž j z j Ž 9 It can also be written as k z c kž z saž z,k Kn Ž kz k z qbž z,k In Ž kz, Ž 30 where ` dj až z,k sq H V Ž j I kj c n kž j, k z j Ž 3 ` dj H n k k z j bž z,k sy V Ž j K Ž kj c Ž j Ž 3 Using Eq Ž 7 we can write az,k s q a q, bž z,k sb q, where ` t t a s H Vž / InŽ t KnŽ t dt, Ž 33 k m k k ` t t b sy H V Kn Ž t dt, Ž 34 k k k m ž / etc, and m s kz Here, we are interested in the m < case, since we want to study the region kz;e< The integrals are dominated by the contribution of the t;m< region where we have zvž z sož and, therefore, AŽ m a s qož rk, Ž 35 k BŽ m b sy qož rk Ž 36 k In general, we have a ;OŽrk n, b ;OŽrk n n n We arrive, therefore, at the standard Born-type series for a and b Let us see now how the two-point function depends on a and b We have E c z k nq kž KnqŽ kz yrž k, z InqŽ kz s y z K Ž kz qrž k, z I Ž kz ck n n b X X a KnŽ kz q X InŽ kz a q, Ž 37 a K Ž kz qrž k, z I Ž kz n X X where rž k, z sbž k, z raž k, z and a sez a, b s Ezb Let us consider massive and massless cases separately Massive case, n) We have K Ž kz qrž k, z I Ž kz n ny G Ž n s n Ž kz where ž n q PPP n kz G Ž yn / G Ž qn / yc Ž k, z q PPP, Ž 38 n ž rž k, z cn Ž k, z sy, Ž 39 G Ž n G Ž yn and K Ž kz yrž k, z I Ž kz nq s n G Ž nq Ž kz nq ž nq q PPP n nq kz G Ž yn / G Ž nq / yc Ž k, z q PPP, nq ž Ž 40

82 8 ( ) M Porrati, A StarinetsrPhysics Letters B so E c G Ž yn kz zg Ž n z k sy ž / c k n X / 4r k, z z r k, z kz q q zg Ž n n rž k, z ž q PPP, Ž 4 where r X Ž k, z s E rž k, z and rž k, z z s bra s OŽ rk, and dots represent higher powers of k The k-dependent part of Ak Ž is n G Ž yn ke 3 3hŽe AŽ k sruv e 4 egž n ž / X r e r = q q ž G Ž n G Ž yn n r / qo rk n nqn qo ke, 4 where n)0 The term proportional to rž k, z is the correction to the pure AdS result obtained in Ref wx Massless case, ns Here E c z k kž K3Ž kz yri3ž kz sy K Ž kz qri Ž kz ck b X X a KŽ kz q X IŽ kz a q Ž 43 a K Ž kz qri Ž kz X For a s0, as, rs0 this gives wx 3 4 z k E c kk kz z k sy sy log kqož k z c KŽ kz 4 k For nonzero r, ž / 3 4 X z k Ž 44 E c z k a z sy log k q qož k z,rk c k 4 4a Ž 45 X Here a raso rk Therefore, k 4 3 3hŽe a X e AŽ k sr e log k q qppp, Ž a UV ž / to leading order in ke and rk We can see, therefore, that in the k ` limit the two-point function can be written as Ž A k sz e AAdS k qo rk, 47 where Z Ž e s e 3hŽe depends on interpolating metric, but not on k In summary, in this paper we have shown how Green s functions of composite operators in geometry duals of strongly coupled gauge theories can be computed in the IR and UV limit We have found the pleasant result that the IR behavior of the Green s functions depends only on the near-brane geometry We have also checked that, in all theories that are UV asymptotic to Ns4, 4-d super Yang Mills, the UV Green s functions are universal It would be interesting to see if this result can be strengthened to include the case when string corrections to classical gravity become important It would be also important to see whether the formal similarity between Eq Ž 9 and the Callan Symanzik equation is an accident, or it suggests instead a way of writing the C-S equation in geometry duals of gauge theories Acknowledgements MP is supported in part by NSF grant no PHY References wx J Maldacena, Adv Theor Math Phys Ž 998 3, hepthr9700 wx SS Gubser, IR Klebanov, AM Polyakov, Phys Lett B 48 Ž , hep-thr98009 wx 3 E Witten, Adv Theor Math Phys Ž , hepthr98050 wx 4 A Kehagias, K Sfetsos, hep-thr9905 wx 5 JA Minahan, hep-thr wx 6 SS Gubser, hep-thr99055

83 ( ) M Porrati, A StarinetsrPhysics Letters B wx 7 IR Klebanov, E Witten, Nucl Phys B 536 Ž , hep-thr wx 8 L Girardello, M Petrini, M Porrati, A Zaffaroni, hepthr9806 wx 9 J Distler, F Zamora, hep-thr98006 w0x M Gunaydin, LJ Romans, NP Warner, Phys Lett B 54 Ž ; M Pernici, K Pilch, P van Nieuwenhuizen, Nucl Phys B 59 Ž wx M Gunaydin, LJ Romans, NP Warner, Nucl Phys B 7 Ž wx L Susskind, E Witten, hep-thr98054 w3x A Peet, J Polchinski, hep-thr98090 w4x C Fronsdal, S Ferrara, A Zaffaroni, Nucl Phys B 53 Ž , hep-thr98003 w5x VI Arnol d, Ordinary Differential Equations, Springer- Verlag, Berlin, Heidelberg, New York, 99

84 3 May 999 Physics Letters B Infra-red stable fixed points of R-parity violating Yukawa couplings in supersymmetric models B Ananthanarayan a, PN Pandita b,c a Centre for Theoretical Studies, Indian Institute of Science, Bangalore 560 0, India b Theory Group, Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, D 603 Hamburg, Germany c Department of Physics, North-Eastern Hill UniÕersity, Shillong 793 0, India Received 8 February 999; received in revised form 0 March 999 Editor: PV Landshoff Abstract We investigate the infra-red stable fixed points of the Yukawa couplings in the minimal version of the supersymmetric standard model with R-parity violation Retaining only the R-parity violating couplings of higher generations, we analytically study the solutions of the renormalization group equations of these couplings together with the top- and b-quark Yukawa couplings We show that only the B-violating coupling l XX 33 approaches a non-trivial infra-red stable fixed point, whereas all other non-trivial fixed point solutions are either unphysical or unstable in the infra-red region However, this fixed point solution predicts a top-quark Yukawa coupling which is incompatible with the top quark mass for any value of tan b q 999 Elsevier Science BV All rights reserved PACS: 0Hi; 30Fs; 60Jv Keywords: Supersymmetry; R-parity violation; Infra-red fixed points There is considerable interest in the study of infra-red Ž IR stable fixed points of the standard model Ž SM and its extensions, especially those of the minimal supersymmetric standard model Ž MSSM This interest follows from the fact that in the SM Ž and in the MSSM there are large number of unknown dimensionless Yukawa couplings, as a consequence of which the fermion masses cannot be predicted One may attempt to relate the Yukawa couplings to the gauge couplings via the Pendleton- Permanent Address Ross infra-red stable fixed point Ž IRSFP for the top-quark Yukawa coupling wx, or via the quasi-fixed point behaviour wx The predictive power of the SM and its supersymmetric extensions may, thus, be enhanced if the renormalization group Ž RG running of the parameters is dominated by IRSFPs Typically, these fixed points are for ratios like Yukawa coupling to the gauge coupling, or, in the context of supersymmetric models, the supersymmetry breaking tri-linear A-parameter to the gaugino mass, etc These ratios do not always attain their fixed points values at the weak scale, the range between the GUT Žor Planck scale and the weak scale being too small for the ratios to closely approach the fixed point Never r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

85 ( ) B Ananthanarayan, PN PanditarPhysics Letters B theless, the couplings may be determined by quasiwx, where the value of the fixed point behaviour Yukawa coupling at the weak sale is independent of its value at the GUT scale, provided the Yukawa couplings at the unification scale are large For the fixed point or quasi-fixed point scenarios to be successful, it is necessary that these fixed points be stable w3 5 x Since supersymmetry wx 6 necessitates the introduction of superpartners for all known particles in the SM Žin addition to the introduction of two Higgs doublets, which transform in an identical manner under the gauge group, we have additional Yukawa couplings in supersymmetric models which violate wx 7 baryon number Ž B or lepton number Ž L In the MSSM a discrete symmetry called R-parity Ž R p is invoked to eliminate these B and L violating Yukawa couplings wx 8 However, the assumption of R p con- servation at the level of MSSM appears to be ad hoc, since it is not required for the internal consistency of the model Therefore, the study of MSSM, including R-parity violation, deserves a serious consideration Recently attention has been focussed on the study of renormalization group evolution of R p violating Yukawa couplings of the MSSM wx 9, and their quasi-fixed points This has led to certain insights and constraints on the quasi-fixed point behavior of some of the R p violating Yukawa couplings, involv- ing higher generation indices We recall that the usefulness of the fixed point and quasi-fixed point scenarios is the existence of stable infra-red fixed points The purpose of this paper is to address the important question of the infra-red fixed points of supersymmetric models with R p violation, and their stability Our interest is in the structure of the infrared stable fixed points, rather than the actual values of the fixed points To this end we shall consider the supersymmetric standard model with the minimal particle content and with R P violation, and refer to it as MSSM with R-parity violation We begin by recalling some of the basic features of the model The superpotential of the MSSM is given by a b a b WsmHHq hu abqlurh q hd abqldrh a b q he abllerh, to which we add the L and B violating terms a b c X a b c L i i abc L L R abc L L R W sm LHq l L L E ql LQD, XX a b c B abc R R R W s l D D U, Ž 3 respectively, as allowed by gauge invariance and supersymmetry In Eq ŽŽ, h, Ž h and Ž h U ab D ab E ab are the Yukawa coupling matrices, with a, b, c as the generation indices The Yukawa couplings l abc and l XX abc are antisymmetric in their first two indices due to SUŽ and SUŽ L 3 C group structure Phenomenological studies of supersymmetric models of this type have placed constraints w0x on the various couplings l, l X and l XX abc abc abc, but there is still considerable room left We note that the simultaneous presence of the terms in Eq Ž and Eq Ž 3 is essentially ruled out by the stringent constraints wx implied by the lack of observation of nucleon decay In addition to the dominant third generation Yukawa couplings h ' Ž h, h ' Ž h t U 33 b D 33 and ht ' Ž h in the superpotential Ž E 33, there are 36 independent R violating couplings l and l X p abc abc XX in Eq Ž, and 9 independent l in Eq Ž abc 3 Thus, one would have to solve 39 coupled non-linear evolution equations in the L-violating case, and in the B-violating case, in order to study the evolution of the Yukawa couplings in the minimal model with R p violation In order to render the Yukawa coupling evolution tractable, we need to make certain plausible simplifications Motivated by the generational hierarchy of the conventional Higgs couplings, we shall assume that an analogous hierarchy amongst the different generations of R p violating couplings exists Thus we shall retain only the couplings l 33, l X and l XX , and neglect the rest We note that the R p violating couplings to higher generations evolve more strongly because of larger Higgs couplings in their evolution equations, and hence could take larger values than the corresponding couplings to the lighter generations Furthermore, the experimental upper limits are stronger for the R p violating Yukawa couplings corresponding to the lighter generations We shall first consider the evolution of Yukawa couplings arising from superpotentials Ž and Ž 3, which involve baryon number violation The one-loop

86 86 ( ) B Ananthanarayan, PN PanditarPhysics Letters B renormalization group equations for h, h, h XX l Ž all others set to zero are: t b t 33 dh t XX 6 t t b Ž 6p sh 6h qh ql y g dž ln m 3 y3 gy 5 g, dh b XX 6 b t b t Ž 6p sh h q6h qh ql y g dž ln m 7 y3 gy 5 g, dh t 9 6p shtž 3hb q4ht y3gy 5 g, dž ln m dl XX 33 XX XX 33 t b 33 6p sl h qh q6l dž ln m Ž and 4 y8 g3y 5 g 4 For completeness we list the well-known evolution equations for the gauge couplings, which at one-loop order are identical to those in the MSSM, since the additional Yukawa couplingž s do not play a role at this order: dg i 3 6p sbig i, is,,3, Ž 5 dž lnm with bs33r5, bs, b3sy3 With the definitions h h h l XX t b t XX 33 R ts, R bs, Rt s, R s, g3 g3 g3 g3 Ž 6 and retaining only the SUŽ 3 C gauge coupling constant, we can rewrite the renormalization group equa- Ž tions as a sg rž6p : 3 3 dr t 6 XX sa 3 R t Ž 3 qb3 y6r tyr by R, Ž 7 dt dr b 6 XX sa 3 R b Ž 3 qb3 yr ty6r byrt y R, dt Ž 8 dr t sa 3 R t wb 3 y3r b y4r t x, Ž 9 dt dr XX XX XX 3 Ž 3 t b dt sa R 8qb y R y R y6r, Ž 0 where b3sy3 is the beta function for g3 in the MSSM, and tsyln m Ordering the ratios as R i s XX Ž R, R, R, R, we rewrite the RG equations Ž 7 t b t Ž 0 in the form wx 3 dr i sa 3 R i Ž r i qb 3 y Ý S ij R j, Ž dt where risýr C R, CR is the QCD Casimir for the various fields Ž C s C s C s 4r3 Q U D, the sum is over the representation of the three fields associated with the trilinear coupling that enters R i, and S is a matrix whose value is fully specified by the wavefunction anomalous dimensions A fixed point is then reached when the right hand side of Eq Ž is 0 for all i If we were to write the fixed-point solutions as R ) i, then there are two fixed point values for each coupling: R ) i s0, or Ý ) Ž riqb3 y SijRj s0 Ž j It follows that the non-trivial fixed point solution is Ý R ) s Ž S y r qb Ž 3 i ij j 3 j Since we shall consider the fixed points of the couplings h, h and l XX t b 33 only, we shall ignore the evolution Eq Ž 9 However, the coupling ht does enter the evolution Eq Ž 8 of h b, but it can be related to h at the weak scale Ž b which we take to be the top-quark mass, since ' m t Ž m t ht Ž mt s, Ž 4 h Õcosb and t mt Ž mt hb ht Ž mt s hbž mt s06hbž m t, Ž 5 m Ž m h b b t where h gives the QCD or QED running wx b of the b-quark mass m Ž m b between m s mb and m s mt Ž similarly for h t, and tan b s ÕrÕ is the usual ratio of the Higgs vacuum expectation values in the j

87 ( ) B Ananthanarayan, PN PanditarPhysics Letters B ' yr MSSM, with Õ s GF s 46 GeV The anomalous dimension matrix S can, then, be written as 0 6 Ss 6qh, Ž 6 6 where hsh Ž m rh Ž m t t b t,036 is the factor com- ing from Eq Ž 5 We, therefore, get the following fixed point solution for the ratios: 385q76h ) XX ) R 'R s,076, 3Ž 70q3h 0 ) ) R 'R b s,0, 70q3h 0q4h ) ) R 3 'R t s,0 Ž 7 70q3h Since each of the R i s is positive, this is a theoreti- cally acceptable fixed point solution We next try to find a fixed point solution with R XX ) s0, with R b and R t being given by their non-zero solutions We need to consider only the lower right hand = sub-matrix of the matrix S in Eq Ž 6 to obtain the fixed point solutions for R b and R t in this case We then have R ) 'R XX ) s0, 35 ) ) R 'R b s,036, 3Ž 35q6h 75qh ) ) R 3 'R t s,034 Ž 8 335q6h This is also a theoretically acceptable solution, as all the fixed point values are non-negative We must also consider the fixed point with R ) s0, which is b relevant for the low values of the parameter tanb In this case, we have to reorder the couplings as R i s XX Ž R, R, R, so that we have the anomalous dimenb t sion matrix Ž in this case denoted as S 0 6qh Ss 6 Ž 9 6 Since R ) b s0, we have to determine the non-zero fixed point values for R XX and R only For this we t consider the lower right hand = submatrix of the matrix in 9 to obtain ) ) ) XX ) 9 b 4 ) ) R 'R s0, R 'R s,079, R 3 'R t s 8,0 Ž 0 which is an acceptable fixed point solution as well Since there are more than one theoretically acceptable IRSFPs in this case, it is important to determine which, if any, is more likely to be realized in nature To this end, we must examine the stability of each of the fixed point solutions The infra-red stability of a fixed point solution is determined by the sign of the eigenvalues of the matrix A whose entries are Ž i not summed over wx 3 ) A s R S, Ž ij i ij b 3 where R ) i is the set of the fixed point solutions of the Yukawa couplings under consideration, and S ij is the matrix appearing in the corresponding RG equations Ž for the ratios R i For stability, we require all the eigenvalues of the matrix Eq Ž to have negative real parts Žnote that the QCD b-function b is negative 3 Considering the fixed point solution Ž 7, the matrix A can be written as 6R ) R ) R ) ) ) ) Asy R Ž 6qh R R, 3 ) ) ) R3 R3 6R3 ) where R are given in Eq Ž 7 i The eigenvalues of the matrix Eq are calculated to be lsy6, lsy0, l3sy0, Ž 3 which shows that the fixed point Ž 7 is an infra-red stable fixed point We note that the eigenvalue l is larger in magnitude as compared to the other eigen- XX values in Ž 3, indicating that the fixed point for l 33 is more attractive, and hence more relevant Next, we consider the stability of the fixed point solution Ž 8 Since in this case the fixed point of the coupling R XX ) s0, we have to obtain the behaviour of this coupling around the origin This behaviour is determined by the eigenvalue wx ) ls Ý SjRj y Ž rqb 3, Ž 4 b 3 js where r sž C qc U D s8, the Cs are the quadratic Casimirs of the fields occurring in the B-violating

88 88 ( ) B Ananthanarayan, PN PanditarPhysics Letters B terms in the superpotential Ž 3, and the Sij is the ) matrix 6, with the fixed points R i, i s,,3 given by Eq Ž 8 Inserting these values in Eq Ž 4, we find 385q76h l s )0, Ž 5 9Ž 35q6h thereby indicating that the fixed point is unstable in the infra-red The behaviour of the couplings R b and R t around their respective fixed points is governed by the eigenvalues of the the = lower submatrix of the matrix A in Eq ) ) Ž 6qh R R y ) ), Ž 6 R 6R ž / which we find to be lsy078, l3sy056 Ž 7 Although l and l3 are negative, because of the result Ž 5, the fixed-point solution Ž 8 is unstable in the infra-red In other words, the R p conserving fixed point solution Ž 8 will never be achieved at low energies and must be rejected Finally we come to the question of the stability of the fixed point solution Ž 0 The behaviour of the coupling R ) b around the origin is determined by the eigenvalue 3 ) ls Ý SjRj y Ž rqb 3, Ž 8 b 3 js where r sž C qc s6r3, and S Q D is the matrix Ž 9 Inserting these numbers, we find 5 l s 4,0)0, 9 with the other two eigenvalues for determining the stability given by the eigenvalues of the matrix which is obtained from the lower = submatrix of the matrix S in Ž 9 This submatrix can be written as ž / ) ) 6R R ) ) 3 R3 6R3 Asy, Ž 30 ) ) where R and R are given by Eq Ž 0 3 The eigenvalues are lsy6, l3sy0 Ž 3 It follows, once again, that the fixed point solution given in Ž 0 is not stable in the infra-red and is, therefore, never reached at low-energies One may also consider the case where the couplings l XX and h attain trivial fixed point values, 33 b whereas h attains a non-trivial fixed point value In t this case we have R 3 ) 'R t ) s7r8, the well-known Pendleton-Ross wx top-quark fixed point of the MSSM To study the stability of this solution in the present context, we must consider the eigenvalues ) lis Ž Si3R3 y Ž riqb 3, is,, b 3 where S are read off from the matrix Ž 6 i3, which yields ls, ls 7 54 Since the sign of each of l and l is positive, this solution is also unstable in the infra-red region However, from our discussion of infra-red fixed point solution Ž 7, it is clear that the Pendelton-Ross fixed point would be stable in case h and l XX b 33 are small, though negligible at the GUT scale In this case, these would, of course, evolve away from zero at the weak scale, though realistically they would still be small Ž but not zero at the weak scale Thus, the only true infra-red stable fixed point solution is the baryon number, and R, violating solution Ž 7 p This is one of the main conclusions of this paper We ) note that the value of R in Ž 7 t is lower than the corresponding value of 7r8 in MSSM with R p conservation It is appropriate to examine the implications of the value of h Ž m t t predicted by our fixed point analysis for the top-quark mass From Ž 7, and a Ž m 3 t, 0, the fixed point value for the top- Yukawa coupling is predicted to be h Ž m t t, 04 This translates into a top-quark Ž pole mass of about m t,70sinb GeV, which is incompatible with the measured value w3x of top mass, m t,74 GeV, for any value of tanb It follows that the true fixed point obtained here provides only a qualitative understanding of the top quark mass in MSSM with R p viola- tion We now turn to the study of the renormalization group evolution for the lepton number violating, and

89 ( ) B Ananthanarayan, PN PanditarPhysics Letters B R, violating couplings in the superpotential Ž p Here we shall consider the dimensionless couplings l and l X only The relevant one-loop renormalization group equations are: dh t X 6 t t b p sh 6h qh ql y g, dž ln m dh b 6p sh Ž h q6h qh dž ln m b t b t X q6l y g, dh t 6p sh Ž3h q4h dž ln m t b t q4l q3l X , dl 33 X 33 t p sl 4h q4l q3l, dž ln m dl X 333 X 333 t b t 33 6p sl h q6h qh ql dž ln m Defining the new ratios l 33 X l X 333 g3 g3 Ž X q6l y g Ž 3 Rs, R s, Ž 33 we may now rewrite Eqs 3 as dr X sa 3Rwb3y4Ry3R y4r t x, Ž 34 dt X dr X 6 X sa 3R Ž 3 qb3 yry6r dt yrty6r byr t, Ž 35 dr t X sa 3 Rtwb3 y4ry3r y3r b y4r t x, Ž 36 dt dr b 6 X sa 3 R b Ž 3 qb3 y6r yrt y6r byr t, dt Ž 37 dr t 6 X sa 3 R t Ž 3 qb3 yr yr by6r t Ž 38 dt X Ordering the ratios as R sž R, R, R, R, R i t b t, we can write these RG equations as: dr i sa 3 R i Ž r i qb 3 y Ý S ij R j, Ž 39 dt where ris ÝRC R, with CR denoting the quadartic Casimir of the each of the fields, the sum being over the representation of fields that enter R i, and S fully specified by the respective wavefunction anomalous dimensions It follows that there are two fixed point values for each coupling: R ) i s0, or the non-trivial fixed point solution Ý R ) s Ž S y r qb Ž 40 i ij j 3 j We shall be interested in the fixed-point solutions of the couplings l, l X , h b, ht only, and shall not consider the ht coupling Therefore, we replace it, as we did earlier, by h Ž m s06h Ž m t t b t at the weak scale in the determination of the fixed point solutions Ž 40 The anomalous dimensions matrix can then be written as: h 0 6 6qh Ss Ž qh 0 6 This leads to the fixed point values for the ratios: 35q94h ) ) ) X ) R 'R s0, R 'R s, 366hy35 40 ) ) R 3 'R b sy, hy05 0hy05 ) ) R 4 'R t s Ž 4 366hy35 We note that R ) 'R X ) -0, and therefore, this fixed point solution is not an acceptable fixed point We, thus, see that a simultaneous fixed point for the lepton number violating couplings l, l X, and h, h does not exist b t We now consider the two L-violating couplings separately, ie, we shall take either l <l X,or j

90 90 ( ) B Ananthanarayan, PN PanditarPhysics Letters B l X <l, respectively In the case when l X is the dominant of the couplings, we order the cou- X plings as R sž R, R, R i b t, so that the matrix S that enters Eq Ž 40 for this case can be written as 0 6 6qh Ss 6 6qh Ž 43 6 Since the determinant of this matrix vanishes, there are no fixed points in this case We thus conclude that a simultaneous non-zero fixed point for the coupling l X 33, h b, ht does not exist We note that the vanishing of the determinant corresponds to a solution with a fixed line or surface If h is small Ž eg, for the case of small tanb b X we may reorder the couplings R sž R, R, R i b t, and the matrix S, to find the fixed point solution ) ) ) X ) ) ) b 3 3 t 3 R 'R s0, R 'R s, R 'R s Ž 44 In order to study the stability of this solution, we must obtain the behaviour of the coupling R ) b around the origin from the eigenvalue 3 ) l s S R y Ž r qb, Ž 45 Ý j j 3 b 3 js where r s6r3 Inserting the relevant R ) s into i 45, we get l s0, Ž 46 from which we conclude that the fixed point Eq 44 will never be reached in the infra-red region This fixed point is either a saddle point or an ultra-violet fixed point We conclude that there are no non-trivial stable fixed points in the infra-red region for the lepton number violating coupling l X 333 Finally, we consider the case when l X <l We find the fixed point solution y35y94h ) ) R 'R s, Ž 35q6h 35 ) ) R 'R b s, 335q6h 75qh ) ) R 3 'R t s, Ž q6h which is unphysical We, therefore, try a fixed point with R ) s0 We find b R ) 'R ) s0, b R ) 'R ) sy3r4, R 3 ) 'R ) t s7r8, Ž 48 which, again, is unphysical We have also checked that: Ž trivial fixed points for l33 and hb and the Pendleton-Ross type fixed point for the top-quark X Yukawa coupling, or Ž trivial fixed points for l 333 and hb and the Pendleton-Ross fixed point for the top-quark Yukawa coupling, are, both, unstable in the infra-red region We, therefore, conclude that there are no fixed point solutions for the lepton number violating coupling l 33 To summarize, we have analyzed the one-loop renormalization group equations for the evolution of Yukawa couplings in MSSM with R p violating cou- plings to the heaviest generation, taking into account B and L violating couplings one at a time The analysis of the system with R p, and the baryon number, violating coupling l XX 33 yields the surprising and important result that only the simultaneous nontrivial fixed point for this coupling and the top-quark and b-quark Yukawa couplings ht and hb is stable in the infra-red region However, the fixed point value for the top-quark coupling here is lower than its corresponding value in the MSSM, and is incompatible with the measured value of the top-quark mass The R conserving solution with l XX p 33 attaining its trivial fixed point, with ht and hb attaining non-triv- ial fixed points, is infra-red unstable, as is the case for trivial fixed points for l XX 33 and h b, with a non-trivial fixed point for h t Our analysis shows that the usual Pendelton-Ross type infra-red fixed point of MSSM is unstable in the presence of R p violation, though for small, but negligible, values of h and l XX it could be stable The system with L, b 33 and R p, violating couplings does not possess a set of non-trivial fixed points that are infra-red stable Our results are the first in placing strong theoretical constraints on the nature of R p violating couplings from fixed-point and stability considerations: the fixed points that are unstable, or the fixed point that is a saddle point, cannot be realized in the infra-red region The fixed points obtained in this work are the true fixed points, in contrast to the quasi-fixed points

91 ( ) B Ananthanarayan, PN PanditarPhysics Letters B of wx 9, and serve as a lower bound on the relevant R p violating Yukawa couplings In particular, from our analysis of the simultaneous Ž stable fixed point for the baryon number violating coupling l XX 33 and the top and bottom Yukawa couplings, we infer a lower bound on l XX R Note added After this paper was submitted for publication, another paper w4x which considers the question of infra-red fixed points in the supersymmetric standard model with R p violation has ap- peared The fixed points for l X, l XX and h t, neglecting all other Yukawa couplings, is considered Their results, where there is an overlapp, agree with ours However, unlike in the present work, the stability of the fixed points has not been considered in w4 x Acknowledgements One of the authors PNP thanks the Theory Group at DESY for hospitality while this work was done The work of PNP is supported by the University Grants Commission, India under the project No 0-6r98 SR-I References wx B Pendleton, GG Ross, Phys Lett B 98 Ž 98 9; M Lanzagorta, GG Ross, Phys Lett B 349 Ž wx CT Hill, Phys Rev D 4 Ž wx 3 BC Allanach, SF King, Phys Lett B 407 Ž wx 4 SA Abel, BC Allanach, Phys Lett B 45 Ž wx 5 I Jack, DRT Jones, Phys Lett B 443 Ž wx 6 HP Nilles, Phys Rep 0 Ž 984 ; HE Haber, GL Kane, Phys Rep 7 Ž wx 7 S Weinberg, Phys Rev D 6 Ž 98 87; N Sakai, T Yanagida, Nucl Phys B 97 Ž wx 8 G Farrar, P Fayet, Phys Lett B 76 Ž wx 9 V Barger, MS Berger, RJN Phillips, T Wohrmann, Phys Rev D 53 Ž ; H Dreiner, H Pois, hepphr95444, and references therein w0x H Dreiner, in: GL Kane Ž Ed, Perspectives on supersymmetry, World Scientific, Singapore wx AY Smirnov, F Vissani, Phys Lett B 380 Ž wx V Barger, MS Berger, P Ohmann, Phys Rev D 47 Ž w3x Particle Data Group, C Caso, Eur Phys J C3 Ž 998 w4x BC Allanach, A Dedes, HK Dreiner, hep-phr9905

92 3 May 999 Physics Letters B Relaxing axions SM Barr Bartol Research Institute, UniÕersity of Delaware, Newark, DE 976, USA Received March 999 Editor: M Cvetič Abstract A mechanism for lifting the cosmological upper bound on the axion decay constant, f a, is proposed It entails the near masslessness of the radial mode whose vacuum expectation value is f a Energy in the coherent oscillations of the axion field in the early universe gets fed into the motion of the radial mode, from which it is then redshifted away It is found that the initial value of fa can be at scales between =0 4 GeV and 0 6 GeV This evolves with time to values close to the Planck scale It is suggested that the nearly massless radial mode might play the role of quintessence q 999 Published by Elsevier Science BV All rights reserved PACS: 9880Cq; 30Er; 480Mz Introduction The axion wx idea is an extremely elegant approach to solving the Strong CP Problem However, it is not without difficulties The main difficulty is that axions have neither been observed directly in the laboratory, nor indirectly through astrophysical effects This implies that the axion decay constant, f a, must be greater than about 0 0 GeV Moreover, there is the cosmological bound, based on the sowx, that fa is less called axion energy problem than about 0 GeV This means that the axion decay constant Žor, equivalently, the scale at which the Peccei Quinn symmetry is broken cannot be at any of the mass scales at which other physics is known or suspected to be based In particular, it smbarr@bartoludeledu cannot be at the weak scale, the Planck scale, or the grand unification scale A number of attempts wx 3 have been made to weaken or remove the cosmological bound on f ain this paper we suggest an idea for doing this The idea is that if the radial mode associated with the axion Žthat is, the scalar field whose expectation value is f a has a nearly flat potential then it would have increased with time as soon as the coherent axion oscillations commenced As a result of this, energy would have drained out of the axion oscillations into the nearly massless radial mode, and from thence been redshifted away by the cosmic expansion We find that the initial value of fa can be as large as 0 6 GeV In the course of cosmic evolution this would have relaxed to a value we presume to be near the Planck scale Axions in this scenario would therefore be extremely hard if not impossible to detect However, the radial mode may play the role of quintessence wx r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

93 ( ) SM BarrrPhysics Letters B The plan of the paper is as follows In Section, a rough sketch of the axion energy problem in conventional axion models is given In Section 3, a simplified discussion of the relaxing axion scenario is given to show how energy is drained out of the axion field In Section 4, a more detailed quantitative analysis is given, still assuming, however, that the radial direction is exactly flat In Section 5, the radial potential is discussed, in particular how flat it must be and how such flatness might arise The usual axion energy problem In ordinary axion models, the Peccei Quinn symmetry is spontaneously broken by a complex scalar field, which we shall denote F The expectation value of F is called f a, while the phase of F is the axion mode Thus one may write Fsre iu, where ² r: sf a, and the axion field a is given by asfau Since the Peccei Quinn symmetry is anomalous, QCD instanton effects give the axion a mass The instanton-generated potential has the form V I s 4 ym cosž Nu 0, where m0 is of order the QCD scale, and N is an integer, which for present purposes can be taken to be When the temperature in the early universe was well above the QCD scale the instanton-generated potential was not fully turned on, whereas after the temperature fell below the QCD scale it had essentially its full zero-temperature strength Let us call the time when the instanton potential reached nearly full strength, t Since the temperature was then of 0 order m 0, t 0;MPrm 0 Assuming a prior epoch of inflation, the axion field at t was approximately spatially constant, but barring an extremely unlikely coincidence, it had no reason to be sitting at the minimum of its potential Žwhich we take to be at us0 Rather, it had some arbitrary value už t 0 s u 0 Consequently, the axion field underwent coherent oscillations about its minimum The energy density in these oscillations at time t0 would have been 4 y3 approximately mu 0 0This energy scaled as R, where R is the scale factor of the universe, and therefore the energy density in the coherent axion oscillations remained Ž after t 0 in a constant ratio to the baryon energy density Since at t0 the baryon energy density was of order 0 y0 m p m 3 0, the present 0 Ž ratio of axion to baryon energy is roughly u m r Ž y m p m 0 ;0 u 0 If the axions are not to over- close the universe, u 0 Q0 y7 This bound on u can be satisfied if f is suffi- 0 a ciently small The point is that the instanton-generated potential of the axion field did not turn on instantaneously If the potential turned on slowly then the number of quanta in the coherent axion oscillations remained constant since it is an adiabatic invariant This implies that as the mass of the axion increased the amplitude of the oscillations decreased, and therefore by t 0, when the potential reached full strength, the amplitude u could have been very small A simple estimate gives that u 0 ;farm P Thus, to satisfy the bound on r one requires that f Q0 GeV a 0 axion 3 The relaxing axion mechanism Let us suppose that the field F whose phase is the axion has a flat potential in the radial direction Then the Lagrangian density can be written 4 m 0 Ls E F qm cosu, 4 m m 0 ( E r q r E u y mu Ž As before, Fsre iu, and the axion field is asru Ignoring spatial derivatives of the fields, the equations of motion of r and u are 0srq3Hryru, ṙ 4 y 0suq 3Hq uqm0 r u r ž / The last term in the equation for the radial mode r is just the centrifugal force, and it is this that drives r to larger values as u oscillates The energy densities in the radial and axion modes are given simply by r s r and r s r u r u 4 q mu 0 The energies in a comoving volume in these modes are given by ErsR 3 rr and EusR 3 r u Using the equation of motion of u to eliminate u, it is easy to show that ṙ 3 E usr y r u q3h Ž y r u q mu 4 0 Ž 3 r

94 94 ( ) SM BarrrPhysics Letters B If the oscillator parameters evolve adiabatically, then averaged over many oscillations the kinetic and potential energy in the oscillator should be equal That 4 is, ² r u : s² mu : s ² r : 0 u Therefore, averaged over many oscillations, ṙ Ėu sy Eu Ž 4 r In a similar way, using the equation of motion of r to eliminate r, and averaging over many oscillations, one finds that ṙ Ėr s Eu y3he r Ž 5 r The interpretation is clear The increase in r caused by the centrifugal force drains energy out of the axion oscillations and into the radial mode, while at the same time energy is redshifted away from the radial mode To put it another way, as r increases the effective mass of the axions, m 0rr, decreases, while the number of axions remains constant Thus E ;rr, as implied also by Eq Ž u 4 Suppose that r;t q, and E u;t yq The exponent q can be determined by writing the equation of motion of r in terms of Eu as follows: 0srq3Hr y3 ² : y Ž qy yr Eu r The first two terms go as t, while the last goes as t yž qq3r, assuming that R ; t r Therefore q s r6 Writing E u s E Ž trt y r 6, Eq Ž 5 u 0 0 is solved by Er yr6 y3r s 8 Eu 0 trt0 q ct Thus, for large t, the radial energy is one-eighth of the energy in the coherent axion oscillations, and the energy in a comoving volume in either mode falls off as t yr6 4 A more detailed analysis So far we have not taken into account the temperature dependence of the instanton-generated potential of the axion field According to the dilute-instantonwx 5, the gas calculation of Gross, Pisarski, and Yaffe instanton potential for u at high temperature goes as m ; T expžy8p rg Ž T ; T expž Ž 3 N y Žy7q N f r3 Nf lnt ; T We shall assume hence- 4 forth that m sm 4 k 4 0 T0rT sm0 trt 0 k, for T4m0 4 4 ie t<t 0, and that m (m0 for t4t 0 The dilute-instanton-gas calculation suggests that k( 5r, but we shall keep k as a parameter If we change variables to t' trt 0, and denote ErEt by a dot, then we can write the equations of motion 3 0srq ryru, t ž / 3 ṙ 4 k 0suq q uq Ž m0t0rr tu Ž 6 t r The effective mass of the field u, then, is m Ž t u s Ž m 0t0rr t t k r Therefore it is reasonable to make the ansatz that ut has the form už t saž t e ibžt, Ž 7 where A and B are real and k r u 0 0 BŽ t sm Ž t s m t rr Ž t t Ž 8 Substituting into the equation of motion for u and taking the real and imaginary parts of the equation, one gets Ž using the fact that rrrskrtybrb 3 0sAq qk rty BrB A, 3 0syBq qk rtq ArA B Ž 9 The first of these equations is exactly solved by Ž ArB t Ž3rqk s constant, while the second is Ž yž3rqk solved exactly by BrA t s constant Eliminating B from these equations and solving gives Ž5rqk yr3 Asu t qc, 0 X Ž5rqk yr3 Ž3rqk 0 Ḃscu Ž t qc t, Ž 0 where c X, C, and u 0 are integration constants After many oscillations the integration constant C can be neglected, and one has, using Eqs Ž 7 and Ž 0, a solution for ut of the following form: u t su t yž5 r6qk r3 exp ib t Ž5r6qk r3 qib X Ž c X u 0 Ž 0 5r6 q kr3 b s Eqs 8 and 0 directly give the solution for the radial mode: rž t s m t 0 0 Ž 5r6q k r3 b 0 Žr6qk r6 =t Žagain, neglecting the integration con- stant C Note that for the case of k s 0, which corresponds to a fixed value of m 4, the radial variable increases as t r6, in agreement with the result obtained in Section 3

95 ( ) SM BarrrPhysics Letters B It must be checked that this solution for rž t and the solution for ut given in Eq Ž satisfy the equation of motion of r Assuming that u is dominated by the rapid oscillations of u rather than by the slow variation of its amplitude, one has that y Ž5r6qk r3 X u ( ib Ž 5r6 q kr3 ut expž ib t q ib Averaged over many oscillations, therefore, ² u : y s b0 5r6 q kr3 ut 0 Substituting this and the expression for rž t into the equation of motion for r Žsee Eq Ž 6, one sees that that equation is satisfied if b s ' u Thus we have that Ž q kž 4q k y 0 5q k 0 6m 0t0 Žr6qk r6 r Ž t s ut 0 Ž ( Ž qkž 4qk The solutions given in Eqs Ž and Ž apply to the period t-, in which the instanton potential was still turning on The same expressions with k set to zero apply to the period t), after the instanton potential turned on The true solution will smoothly interpolate between these in the period t; ŽNote that the k/ 0 and k s 0 solutions for u actually agree at ts, while the k/0 and ks0 solutions for r differ at t s by a factor of ( Ž qkž 4qk rf4 One is now able to estimate the energy in the axion oscillations The crucial parameter is the value of r at the time when the axion oscillations started We will call that time t i The oscillations started when the effective mass of the u field, m s B, became equal to the expansion rate of the universe, y y Ž Hs t That is, when t f m t rrž t i 0 0 i = Ž t rt k r Clearly, the smaller rt i 0 i was, the ear- lier the axion oscillations began Turning this around, Žqk r Žqk r rti fm0t0trt i 0 ;MP trt i 0 At ti it is to be expected that u was of order unity, since it had not had time to be affected by the instanton potential But according to Eq Ž, už t i yž5r6qk r3 ; u 0 tirt 0 Thus, u 0 is of order Ž t rt Ž5r6qk r3, or, in terms of rt, i 0 i 5q k 6q3k 0 i P u ; rž t rm Ž 3 As was seen in Section, the factor u 0 tells how much the energy of the coherent axion oscillations was suppressed by the time the axion potential fully u turned on at t We will call this suppression factor 0 S before 5q k qk Ž 3 S f u ; r t rm 4 before 0 i P If this were the only suppression of the axion energy, solving the axion energy problem would require that y7r u Q0 With ks5r this gives rt 0 i Q= 0 4 GeV That is, the initial value of f a can have been quite near the grand unification scale By t 0 this would have increased, according to Eq Ž, to 5 a value rt 0 ;m0t0u 0;u 0M P ;3=0 GeV One possibility, which we will call Case I, is that the radial mode stopped evolving at that point, because its potential has a minimum there Another possibility, which we will call Case II, is that r continued to increase to some final value near the Planck scale That would mean that the axion energy would have been further suppressed by the evolution of r in the period t)t Since rt 0 0 ;u 0M P, and it is assumed that rt f ;M P, there is an increase of r by a factor of u y 0 in this period It is easily shown that the energy of the axion field in a comoving volume varies inversely with r in this period Žas was seen already in Section 3, so that the further suppression of the axion energy, which we shall call S after, is given by Safterfu 0, Ž 5 or StotalsSbeforeSafterfu 0 3 Ž 6 In Case II, therefore, it is only necessary that u 0 Q y7r3 6 0, meaning that rti Q0 GeV Case I and Case II are, in a sense, the extreme cases One can consider intermediate cases as well But one sees that in general the relaxing axion scenario would have f a starting out in the range 0 4 to 0 6 GeV, near the grand unification scale, and evolving to higher values 5 The flatness of the radial potential The mechanism described in the preceding sections depends crucially on the assumption that the potential in the radial direction is nearly flat For the mechanism to work, the centrifugal term in the equation of motion for r had to have dominated over

96 96 ( ) SM BarrrPhysics Letters B the force coming from the potential energy of r X That is, ru ) V Ž r One can write this as r axion ) X rv Ž r, where raxion is the energy in the coherent axion oscillations At some point this condition was no longer satisfied and the radial field s evolution was controlled by VŽ r Unless the radial field was at that point overdamped Ž not so in the cases of interest it would have started to oscillate about the minimum of VŽ r For reasonable potentials Žwhere one assumes that the cosmological constant problem has somehow X been solved one would expect that VŽ r ; rv Ž r, and therefore the energy in the coherent oscillations of the radial mode were also of that order Consequently, when the coherent radial oscillations began, the energy in them was typically of the same order as the energy in the coherent axion oscillations Thus the coherent radial oscillations do not in themselves pose a cosmological problem However, for the mechanism to work at all it is necessary that the centrifugal term in the equation of motion of r dominated over the potential term for a sufficiently long time to solve the axion energy problem This puts a constraint on the flatness of VŽ r In Case I, it is assumed that the centrifugal term drove r until t 0, when the temperature was of order m 0 At that time raxion had to have been less than about 0 r B ;0 y8 m 3 0 m p ;0 y0 GeV 4 Thus, at t0 X y0 4 it must also have been that rv Ž r Q0 GeV In Case II, the centrifugal term is assumed to have dominated until r got to be of order M P Since r6 rt 0 ;u 0M P, and r grew as t for t)t 0, this happened at a time t f;u y6 0 t 0 Assuming a radiation 3 dominated universe, T t ; u TŽ t f 0 0 ; Stotal m 0 ; 0 y7 m 0 Therefore, at t f it must have been that X y8 3 y3 4 rv Ž r Q0 T tf m p;0 GeV How can the potential be that flat? Certainly it is trivial to arrange that the radial direction be flat in the supersymmetric limit The real problem is to insulate the radial mode from supersymmetry breaking This is easiest to do if supersymmetry is broken at low energies, as it is in models with gauge-mediawx 6 In such a model, the ted supersymmetry breaking ordinary quarks must be split from their supersymmetry partners by an amount that is of order 0 to 0 3 GeV, which scale we will call m 0 Since the axion sector must couple directly or indirectly to the quark sector for the Peccei Quinn symmetry to have a QCD anomaly, supersymmetry breaking will be fed into the radial mode of the axion sector through quark loops Typically, then, the radial mode of the axion sector will acquire a potential of the form 4 e m lnž rrm 0 0 e is model-dependent and is smaller the more indirectly and the more weakly the axion sector couples to the quark sector In Case I, one has 4 y0 4 that e m Q0 GeV, implying Ž if m ; TeV 0 0 that eq0 y In Case II, one has the more severe constraint that eq0 y43 To see how such small values of e might be achieved, consider first a conventional axion model where the radial mode has a tree-level potential The relevant terms in the superpotential would have the following general structure: W sgsqqqw, Ž 7 axion S where Q and Q are lefthanded quark and anti-quark superfields, and S is a superfield containing the axion WS is some set of terms, generally involving other fields, which has the effect of fixing ² S: to have some value M The phase of S is, however, assumed not to be fixed except by QCD instanton Ž iu effects One can write Ss MqS e, where here we mean by S the bosonic component, and the axion field is Mu The radial mode S would typically have some mass of order M Consider, now, a somewhat different model, with the corresponding terms in the superpotential being the following X X W sgsqqqw qg SAyM B Y, 8 axion S where S, A, B, and Y are all gauge singlets, and as before WS has the effect of making ² S: s M but leaving the phase of S undetermined Suppose that both M and M X are very large compared to the scale of supersymmetry breaking It is apparent that, since S has a Peccei Quinn charge, so must either A or B Therefore, if ² A: and ² B: are larger than M, they rather than ² S: control the value of f a Assume that Y has no other couplings in the superpotential Then one of the terms in the scalar X X potential is g Ž SAyM B This term fixes only the ratio of A and B and leaves them otherwise undetermined There is, therefore, a direction that is

97 ( ) SM BarrrPhysics Letters B flat in the supersymmetric limit along which A and B can become arbitrarily large This flat direction would play the role of r in this model iu i a M Writing SsŽ MqS e, Asre, and BsŽ X M r iž aquqd qb e, the aforementioned term can easily be found to give a mass to the fields S, B, and X d ' rd In particular, one finds that < g ŽSAy X X MB< sg M d qg X Ž² r: SyM X B q terms of cubic and higher order In addition, there is the mass term for S coming from W S The phase a is an exact goldstone mode, while u, since it corresponds to an anomalous UŽ by virtue of its coupling to the quarks, is the axion mode The scalar field r has Ž before supersymmetry breaking a flat potential It is easily seen that the Lagrangian terms for r and u have Ž after suitable rescaling of fields essentially the same form as those shown in Eq Ž Thus this model is an implementation of the relaxing axion mechanism The question is how large a potential r gets in this model If supersymmetry breaking is mediated from some hidden sector by Standard Model gauge interactions, then only Q and Q will directly feel it The splittings in the supermultiplet S will only arise through a quark loop, and the splittings in the multiplets A and B will only arise through diagrams involving both an S loop and a quark loop Ž X Ž One would expect that e ; g r6p g r 6p This can easily be as small as required For example, if M and M X are near the unification scale, the coupling g could even be as small as 0 y4 It seems to be possible to shield the radial mode from supersymmetry breaking even more thoroughly As an extreme case one could imagine a whole series of gauge-singlet sectors separating the quark sector from the axion sector: Squark ls llsn l S, where the S s denote various sectors, and the axion arrows represent a coupling in the superpotential between two sectors In this case, supersymmetry breaking in the radial mode would be an n-loop effect On the other hand, because of the couplings between sectors, there are fields in each sector that have non-trivial Peccei Quinn charges Thus the axion field, though removed by several steps from the quarks, will nonetheless get a potential from 4 QCD instanton effects that goes as ym cosž Nu 0, where N is an integer From these examples, it seems that there is no reason in principle why the radial mode could not be sufficiently flat to allow the relaxing axion mechanism to work It is an interesting question whether this flat radial direction can be identified with other flat directions that have been discussed in recent years Could it be, to mention two obvious examples, the dilaton of superstring theory or the quintessence field? In any event, given the increasing role that very flat potentials have played in particle physics and cosmology Žinflatons, quintessence, dilatons, moduli, and the axion itself, it is suggestive that a flat radial direction can allow the breaking of the Peccei Quinn symmetry to take place at the unification scale or even higher Acknowledgements This work was supported in part by the Department of Energy under contract No DE-FG0-9ER References wx S Weinberg, Phys Rev Lett 40 Ž 978 3; F Wilczek, Phys Rev Lett 40 Ž 978 7; for a review see, The Strong CP Problem, by RD Peccei, in: C Jarlskog Ž Ed, CP Violation, World Scientific, 989, pp wx J Preskill, M Wise, F Wilczek, Phys Lett B 0 Ž 983 7; L Abbott, P Sikivie, Phys Lett B 0 Ž ; M Dine, W Fischler, Phys Lett B Ž wx 3 P Steinhardt, M Turner, Phys Lett B 9 Ž 983 5; G Lazarides, RK Schaefer, D Seckel, Q Shafi, Nucl Phys B 346 Ž ; S-Y Pi, Phys Rev D 4 Ž ; A Linde, Phys Lett B 0 Ž ; G Dvali, IFUP-TH r95, unpublished; SM Barr, KS Babu, D Seckel, Phys Lett B 336 Ž wx 4 M Bronstein, Physikalische Zeitschrift Sowjet Union 3 Ž ; M Ozer, MO Taha, Nucl Phys B 87 Ž ; B Ratra, PJE Peebles, Phys Rev D 37 Ž ; JA Freiman, CT Hill, R Watkins, Phys Rev D 46 Ž 99 6; R Caldwell, R Dave, PJ Steinhardt, Phys Rev Lett 80 Ž wx 5 DJ Gross, RD Pisarski, LG Yaffe, Rev Mod Phys 53 Ž wx 6 For a review see A Nelson, Nucl Phys B Ž Proc Suppl 6A-C Ž 998 6

98 3 May 999 Physics Letters B Fermion mixing renormalization and gauge invariance P Gambino a, PA Grassi b, F Madricardo b a Technische UniÕersitat Munchen, Physik Dept, D Garching, Germany b Max Planck Institut fur Physik ( Werner-Heisenberg-Institut ), Fohringer Ring 6, D Munchen, Germany Received 9 November 998; received in revised form 0 March 999 Editor: PV Landshoff Abstract We study the renormalization of the fermion mixing matrix in the Standard Model and derive the constraints that must be satisfied to respect gauge invariance to all orders We demonstrate that the prescription based on the on-shell renormalization conditions is not consistent with the Ward-Takahashi Identities and leads to gauge dependent physical amplitudes A simple scheme is proposed that satisfies all theoretical requirements and is very convenient for practical calculations q 999 Elsevier Science BV All rights reserved Despite a few interesting papers have been dew 4 x, the subject has so far escaped much attention This is voted to the fermion mixing renormalization mostly due to the fact that, as a result of GIM cancelations, the radiative corrections related to the renormalization of the CKM matrix can be made very small, OG Ž m F q, where mq is the mass of a light quark They are therefore of little practical importance in the context of the Standard Model Ž SM Still, the subject has some conceptual interest in its own, not to mention the relevance of mixing in many extension of the SM In this letter, we reconsider it from a different point of view, which allows us to point out some inconsistency in previous analyses and to propose an alternative solution For definiteness, we concentrate on the case of the CKM matrix renormalization in the SM A convenient framework to study this issue is provided by the Ward Takahashi Identities Ž WTI of the theory with background fields w5 7 x Indeed, as the diagonalization of the fermion mass matrix is achieved by field redefinitions that do not commute with the gauge transformations, the CKM elements appear explicitly in the WTI, unlike masses and gauge couplings This will give us a strong constraint At the functional level, the WTI which represent the s q generator of SUŽ L are implemented by the local functional operator Wq acting on the effective ac- tion G Žsee wx 7 For our purposes, the relevant part of W is the one which contains the quark fields: q d d quark L 0 0 L Wq s Ý cu Vud y Vud c d Ž L dc dc L u,d d Here V 0 is the tree level CKM matrix Upon renormalization, the fermionic fields are rescaled by nondiagonal complex wave function renormalization 0 WFR matrices Zu and Z d; consequently, V is replaced by a renormalized CKM matrix V s ŽZ L yr V 0 ŽZ L r, where Z L is the left-handed u d u r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

99 ( ) P Gambino et alrphysics Letters B component of the WFR Expanding V at first order, we obtain for the CKM counterterm L L d V s dz XV X y V XdZ X ud Ý uu u d Ý ud d d X X u An additional constraint on the WFR comes from the requirement of unitarity for V, Ý Ý V X dz L X qdz L) X s dz L XqdZ X ud d d dd uu u u u d X X d u d L) V X Ž 3 Notice that this constraint follows also from the request that the commutation relations among the operators W q, W y, and W3 be preserved Combin- ing Eqs Ž and Ž 3 we find L) L d V sy dz ydz V Ý X X X ud u u uu u d 4 u X L L) qý V X dz X ydz X ud d d dd, 4 X d where, as expected for a unitary matrix, the renormalization of V is expressed in terms of anti-hermitian matrices Any renormalization prescription that preserves the above WTI leads to a gauge-independent definition of the CKM matrix To prove this theorem we start noticing that, as a consequence of W Ž G q s0, one has Ej WqŽ G swqž Ej G L d qý c u E j V ud L u,d dc d d L y E V c q G j ud d dc u L s0, Ž 5 where j is a gauge parameter and the ellipses stand for additional contributions not relevant for us On the other hand, the gauge variation of G is controlled by a Slavnov-Taylor Identity Ž STI, Ej G s S Ž E G w5,8 x x, where x is the anti-commuting source of the composite operator generated by the variation of j, and S is the Slavnov-Taylor operator wx 9 As Wq does not depend on x and commutes with S, the first term of Eq Ž 5 vanishes The only possibility compatible with the invariance of the theory Žie with Eq Ž is then that all the parameters in the square brackets of Eq Ž 5 are identically zero From this Ej Vs0 follows An interesting point about Eq Ž 4 is that the counterterms for physical parameters, the CKM matrix elements, are given in terms of conventional objects like the WFR constants From a rigorous point of view, the distinction between physical and conventional objects can be formulated in terms of the cohomology classes of the Slavnov-Taylor operawx 9 The physical parameters on tor of the theory which the S-matrix depends are the coefficients of the cohomology classes For example, in the absence of mixing, one cohomology class is provided by each Yukawa interaction term of the Lagrangian, which guarantees that the masses are physical objects In the case of mixing, the Yukawa couplings are complex non-diagonal matrices and can be diagonalized by a redefinition of the fields, that is a finite WFR In higher orders, however, the redefinition of fields originated by the diagonalization of the Yukawa matrix inevitably mixes with the one generated by the anomalous rescaling of the kinetic terms A formal cohomological analysis wx 7 shows that indeed the off-diagonal field redefinition contains some physical parameters, namely the CKM matrix elements On the other hand, Eqs Ž Ž 4 allow us to disentangle the physical information related to the diagonalization of the Yukawa matrix Žcontained in the CKM matrix elements and in their counterterms from the unphysical information carried by the Z factors These constraints rely on the gauge invariance of the theory and, in a consistent framework, should be all satisfied As we have seen, the definition of the CKM matrix at higher orders is conveniently expressed in terms of the anti-hermitian component of the WFR It therefore depends on the scheme chosen for the Strictly speaking, the CKM element Vud is not a physical quantity; physical observables are identified at tree level by the Jarlskog reparametrization invariants w0 x, which are constructed from physical amplitudes This scheme can in principle generalized at higher orders

100 00 ( ) P Gambino et alrphysics Letters B WFR It should be clear, on the other hand, that once the counterterm d V is calculated through Eq Ž 4, it can be used independently of the choice of the dz factors, because physical amplitudes are independent of the scheme adopted for the WFR For example, in practical applications at the level of S-matrix, it is often convenient to avoid the rescaling of the fields Ž ie the WFR altogether wx and introduce only the LSZ factors for the external fields: if mixing is present, however, one still has to renormalize the mixing parameters A number of renormalization prescriptions for the CKM matrix are indeed possible; a first convenient option is the MS subtraction: by definition, assuming gauge invariant mass renormalization and after adjusting for the possible breaking of chiral invariance, it satisfies the WTI and the STI Hence, as a consequence of the above theorem, it can be guaranteed to MS yield a gauge-independent ultraviolet pole d V wx to all orders Ža proof can also be found following w3 x On the other hand, it is well-known that the decoupling of heavy particles is not manifest in the MS scheme This means that if we work in the framework of an effective Lagrangian where the heavy fields Ž W boson and top quark are integrated out, the dimension three and four operators that mix the quarks yield contributions to the amplitude which are not suppressed by the high mass scale Moreover, as noted in w4 x, these terms are not even defined in the limit of vanishing light quark masses All this makes the MS definition unnatural in the context of effective Lagrangians As the CKM elements are mostly determined from low-energy hadronic processes, this is not very convenient Physical amplitudes calculated with an MS counterterm for the CKM would depend on the renormalization scale Žsee w5x for studies of the scale evolution of the CKM matrix and would contain OŽ a corrections Ž Ž proportional to m qm r m ym i j i j, where mi, j are the poorly known light quark masses A second possibility consists in fixing four CKM elements in terms of four physical amplitudes, eg of the four most precise experimental processes This procedure bypasses the definition of the WFR, but destroys the symmetry between the quark generations and is not practical in higher order calculations A third option is provided by the use of the on-shell renormalization conditions of Ref w6x to define the WFR constants and, through Eq Ž 4, the CKM counterterm, as has been suggested by Denner and Sack wx 3 and generalized to extended models in wx 4 This approach also implies decoupling in the sense explained before, ie dimension three and four operators are removed The renormalization conditions, which are written in the u sector in terms of the fermionic two-point functions G XŽ pu, u m G X u uu pu s0; G X pu u m X uu u s0; už mu GuuŽ pu s; pu ym u GuuŽ pu už mu s, Ž 6 pu ym u fix the masses of the u quarks and all the dz X uu by setting them equal to the LSZ factors Their use in Eq Ž 4 defines a counterterm d Vud which makes physical amplitudes finite wx 3 We now show explicitly that the latter procedure leads to gauge dependent amplitudes in one-loop calculations To this end, we consider the decay of a W boson into two arbitrary quarks u and d that was studied in Ref wx 3 We conform to the notation of that paper and write the one-loop renormalized amplitude as ž / L) ž Ý X X X u d e d sw M sv M qd q y q dz e s ud ud 0 vert W W qm dz V 0 uu ud L q V X dz X qd V, Ž 7 X d where M sygr' um eu Ž M a ÕŽ m 0 u W y d, u and Õ are the spinors of the final-state quarks, a sž " " g 5 r are the right and left-handed projectors, and e m is the polarization vector of the W boson Let us now consider the gauge dependence of the individual contributions to M ud For our purposes, it is sufficient to consider only the j W gauge parameter dependence As the total amplitude in Eq Ž 7 must be gauge independent, Ej Muds 0 Also the coun- W terterms d e and d s, as defined in w x W, do not depend on j at the one-loop level On the other Ý ud d d ud/ W uu

101 ( ) P Gambino et alrphysics Letters B hand, the gauge variation of the proper vertex d vert is not trivial; it can be studied using the STI that governs the gauge dependence of the Green functions On the mass shell, and after contracting with e m and with the external quark spinors, one finds the following non-linear identity written in terms of PI Green functions Ý E G s G G qg G T q y q q q X j W W ud xg i W W ud x iuh u X W u d is, qg q X W ud G xhxd 8 i d Here g m " and hu,d are the sources associated to the BRST variation of W " and u,d fields and x are m, the sources of the two independent composite operators generated by the variation of the two gauge fixing parameters j and j Following the W, W, common practice, we have set j Wsj W,sj W, At the one-loop level, Eq Ž 8 reduces to g Ý Ž T Ž E G q s G y q j W ud xg W V W i ud ' is, Ý Ž q G V X x iuh u X ud X u Ý Ž q V X ud GxhXd eu a y, 9 i d X d Ž where the superscript indicates that the proper functions are evaluated in the one-loop approximation and T that only the transverse component is considered All terms are evaluated on the mass-shell of the physical fields In a similar way, one may find STI for the two-point functions of the W boson and of the quarks At one-loop level and adopting the standard tadpole renormalization, which consists in removing the whole tadpole amplitude, one finds for the W boson WFR factor E T Ž q y j W W j W W W MW E dz se G p s E p p Ý x i g W W i s, s G T Ž y q M Ž 0 The treatment of the quark two-point functions is slightly more involved: in the case of the u quarks, A detailed illustration of this kind of STI w5,8x will be given in w7 x for instance, we decompose the unrenormalized self-energy according to S X p ss L X pu a qs R X uu uu y uu pu aq qs S X m a qm X uu Ž u y u a q Ž The one-loop STI for S X Ý uu then reads Ž Ž E S X p sy G pu pu ym X j uu x uh X W i u u is, Ž q pu ym G X u xh u pu i u Ž Splitting G Ž m x uh X u into its left and right-handed i u components, G L, RŽ x iuh u X, we find on the mass shell of the u quark X X L, Ž m E S S,Ž L,Ž R,Ž X qs X sm X u j uu uu u Gx uh ymu G x uh, W i u i u R,Ž S,Ž R,Ž L,Ž m E S X qm XE S X sm X u j G ym G W uu u j W uu u x iuhu X u x iuhu X Ž 3 One can then use the above equations in the definition of the on-shell dz L Žsee for ex Eq Ž 34 of Ref wx 3 and find Ý L) R,Ž E dz X j s G m 4 W uu x iuh u X u is, An analogous result holds in the d sector Inserting Eqs Ž 0 and Ž 4 into Eq Ž 9, we can write the gauge dependence of the vertex as L) V E d sy E V dz q dz X V X Ý ud j vert j ud W u u u d u X W Wž Ý ud d d / X q V X dz L X Ž 5 d We observe that, as a consequence of the unitarity of V, in the last two terms of Eq Ž 5 the dependence on the CKM matrix factorizes; this is consistent with the fact that the STI in Eq Ž 8 would be exactly the same, were the CKM matrix diagonal, and Žfrom a diagrammatical point of view that the one-loop vertices involve one and only one charged quark current The limit for massless fermions of Eq Ž 5 agrees with the expressions reported in w8 x Note also that all the gauge dependence of the vertex is contained in two-point function contributions Using Eqs Ž 7 and Ž 5, Ej Muds0 is reduced to Ej d V W W ud s0 In other words, we have shown that the gauge-

102 0 ( ) P Gambino et alrphysics Letters B parameter dependence introduced by the LSZ external field factors Žthat coincide with the on-shell WFR constants is completely absorbed by the proper vertex On the other hand, Vud is a parameter of the bare Lagrangian and its renormalization condition, in the present framework, should preserve the gauge independence It may therefore seem surprising that the counterterm defined on the basis of the on-shell conditions is gauge-dependent Indeed, an explicit one-loop calculation yields for the j W dependent part of the WFR in n dimensions ig m 4yn d n k L,j W ) ) dz X s V XV X X uu ud ud M X Ž p = W Ý H n d j M ym qm X W W u d k yj M Ž kqp ym X W W d, Ž 6 with p sm and u/u X u ; again an analogous ex- L pression holds for dz X We see from Eq Ž 6 dd that the rž n y 4 pole of the anti-hermitian part of dz L,j W is removed by GIM cancelations, ensuring that the divergence of d V in Eq Ž ud 4 is gauge independent On the contrary, the momentum dependence of the integrand in Eq Ž 6 spoils the GIM cancelations for the finite part of d V in Eq Ž ud 4, which turns then out to be gauge dependent We conclude that the W-decay amplitude calculated in the on-shell framework of Ref wx 3 is gauge dependent This can be understood by noting that the finite part of the WFR factors defined on-shell violates the WTI, in particular it does not satisfy Eq Ž 3 A convenient and natural alternative to the prewx 3, which maintains decoupling and scription of enhances the symmetry among the quark generations, can be obtained by imposing the following conditions on the off-diagonal two-point functions: E L G X 0 s 0; G X 0 ' G X uu uu uu Ž pu ay s 0, E pu pus0 Ž 7 for u/u X, and analogously for the d sector These conditions do not fix the diagonal two-point functions They also do not fix the off-diagonal hermitian part of the right-handed WFR, which however is finite and suppressed by light quark masses We choose to set d Zuu R, H X s0 for u/ux The normalization point, pu s0, is the same for all flavors and all divergences related to the mixing are correctly subtracted, as they are logarithmic and do not depend on masses and momenta In addition, Eqs Ž 7 avoid problems in the treatment of the absorptive parts of the two-point functions whenever any of the quarks is heavy Žtypically, in the SM, this is the case with the top quark At the one-loop level, the antihermitian and hermitian WFR factors d Z L, A X uu and d Zuu L, H X obtained from Eq Ž 7 can be expressed in terms of self-energies evaluated at zero momentum transfer: m qm X L, A u u L S X X X uu uu uu m ym X u u d Z s S Ž 0 q S Ž 0 ; d Z L, H X sys L X Ž 0 Ž 8 uu uu It is straightforward to verify that d Zuu L, H the condition of Eq Ž 3 because it reduces it to the case of no mixing The CKM counterterm obtained L, A by the use of d Z in Eq Ž 4 is, in units of Ž g r 64M p, W ÝÝ ) d V s V X V XV X X ud u d ud u d X X u /u d ½ Ž m qm X m X u u d 3 MW = y3ln m X uymu e m y5y 3Ž yy y y y ln y Ž yy Ž yy 5 qž uld, u X ld X, Ž 9 L, A where ysm X rm and es 4yn r d Z X d W uu and d Vud are gauge independent, as can be directly seen from Eq Ž We stress once more that the use of this counterterm, based on Eq Ž 7 and consequently on the Z L, A factors, is independent of the choice of WRF in the rest of a calculation, and corresponds to just one of the many gauge-invariant definition of the one-loop CKM elements Needless to say, it is always the LSZ procedure Žsee Eqs Ž 6 to dictate the treatment of the external lines d Vud can there- fore be used without modifications in the calculation

103 ( ) P Gambino et alrphysics Letters B of wx 3 : in that case, the results for the W-decay amplitude are gauge independent and differ from wx 3 Ž by gauge dependent OGm m light terms We now consider the consistency of the renormalization conditions of Eqs Ž 7 beyond one-loop First, we can show that the Eqs Ž 7 respect the WTI for the Wud-vertex at all orders; this WTI reads Ž p qp G q p, p ym G q u d m Wˆ udž u d W Gˆ udž p u, pd m ž g X y V X a G Ž p ' Ý ud q dd d X d Ý ud uu u y / y V X G X yp a s0 0 X u At the one-loop level and for on-shell amplitudes, Eq Ž 0 holds even in the case the external Goldstone boson G q and the W q are quantized Differentiating with respect to pu and pd and setting all momenta to zero, one finds that Ý Ý L L V X G X 0 y G X 0 V X ud d d uu u ds Vud Gud 0, X X d u Ž where G Ž 0 is a convergent term induced by G q ud Gˆ ud The second condition of Eqs Ž 7 reduces this constraint to the case of no mixing at all orders This is the crucial requirement Similarly, the first condition of Eqs Ž 7, used in Eq Ž 0 at zero momenta, reduces it to the well-known constraint on the renormalization of the Yukawa coupling in the absence of mixing We also investigate the effect of Eqs Ž 7 on the STI: at the one-loop level, they induce several constraints on the renormalization of x-dependent Green Ž functions which appear in Eq, eg G x uh X; using i u them together with Eqs Ž 7 in the two-loop STI, the latter can be linearized and reduces to its one-loop form at p s 0 This is a non-trivial result, as the gauge dependence of the two-loop off-diagonal Ž S X Ž p uu can now be written only in terms of two-loop Green functions At the one-loop level several sim- Ž plifications occur: for example one has G Ž x uh X 0 a i u y s0, because only the left-handed source h L X u is involved Beyond one-loop, h R X u may also contribute, to the effect that this and analogous simplifications are not granted any longer Consequently, the combi- nation S L XŽ 0 q S XŽ uu uu 0 is not guaranteed to be gauge invariant at two or more loops, although analogous but more complicated gauge-invariant combinations do exist As already mentioned, what is certainly gauge invariant at all orders is the MS pole of d V; we have explicitly verified this property using the STI A related problem concerns S R, which is finite at one-loop just because of the GIM mechanism, but it may require a subtraction beyond one-loop; this would also modify the conditions of Eqs Ž 7 Moreover, we note that a rigorous analysis of all the WTI Žnot only of Eq Ž 0 at higher orders cannot be performed without specifying the whole set of normalization conditions The previous points show that a complete discussion of a non- MS renormalization of the mixing matrix beyond one-loop becomes extremely complex w7 x, and is only partially simplified by Eqs Ž 7 The renormalization conditions of Eqs Ž 7 can be used in any model containing Dirac fermion mixing For example, all has been said applies directly to the case of lepton mixing, which is suggested by recent experiments, if the neutrinos have Dirac masses They can also be generalized to exwx 4 tended models, along the lines of Refs In summary, we have reanalyzed the renormalization of the fermion mixing parameters in the Standard Model We have reviewed several possibilities for the definition of the CKM matrix at higher orders, showing the constraints they have to satisfy in order to respect gauge invariance In particular, we have demonstrated that the prescription based on the on-shell wave function renormalization constants is not consistent with the Ward-Takahashi Identities and leads to gauge dependent physical amplitudes We have therefore proposed a simple scheme that naturally satisfies all theoretical requirements and is very convenient for practical calculations Acknowledgements We thank Bernd Kniehl for suggesting the topic, for a careful reading of the manuscript and for many useful suggestions and remarks We are also grateful to D Maison for interesting discussions and for reading the manuscript, to A Denner and T Mannel

104 04 ( ) P Gambino et alrphysics Letters B wx for communications regarding Ref 3, to M Steinhauser for technical help, and to C Becchi, G Degrassi, and A Sirlin for stimulating conversations References wx W Marciano, A Sirlin, Nucl Phys B 93 Ž wx JF Donoghue, Phys Rev D 9 Ž wx 3 A Denner, T Sack, Nucl Phys B 347 Ž wx 4 BA Kniehl, A Pilaftsis, Nucl Phys B 474 Ž ; S Kiyoura, MM Nojiri, DM Pierce, Y Yamada Phys Rev D 58 Ž wx 5 H Kluberg-Stern, JB Zuber, Phys Rev D Ž ; D Ž wx 6 LF Abbott, Nucl Phys B 85 Ž 98 89; Acta Phys Pol Ž 98 ; A Denner, S Dittmaier, G Weiglein, Nucl Phys B 440 Ž wx 7 PA Grassi, MPI-Thr98-57; Nucl Phys B 46 Ž wx 8 NK Nielsen, Nucl Phys B 0 Ž ; K Sibold, O Piguet, Nucl Phys B 53 Ž ; R Haussling, E Kraus, Z Phys C 75 Ž wx 9 C Becchi, A Rouet, R Stora, Comm Math Phys 4 Ž 975 7; Ann Phys Ž NY 98 Ž w0x C Jarlskog, Phys Rev Lett 55 Ž ; Z Phys C 9 Ž wx A Sirlin, Phys Rev D Ž wx TP Cheng, E Eichten, LF Li, Phys Rev D 9 Ž ; ME Machacek, MT Vaughn, Nucl Phys B Ž w3x DJ Gross, F Wilczek, Phys Rev Lett 30 Ž w4x P Gambino, A Kwiatkowski, N Pott, hep-phr w5x K Sasaki, Z Phys C 3 Ž ; KS Babu, Z Phys C 35 Ž ; C Balzereit, T Mannel, B Plumper, hepphr w6x K Aoki, Z Hioki, R Kawabe, M Konuma, T Muta, Prog Theor Phys Suppl 73 Ž 98 w7x P Gambino, PA Grassi, in preparation w8x G Degrassi, A Sirlin, Nucl Phys B 383 Ž 99 73

105 3 May 999 Physics Letters B Pseudoscalar vertex, Goldstone boson and quark masses on the lattice Jean-Rene Cudell a, Alain Le Yaouanc b, Carlotta Pittori a a Institut de Physique, UniÕersite de Liege ` au Sart Tilman, B-4000 Liege, ` Belgique b LPTHE, UniÕersite de Paris Sud, Centre d Orsay, 9405 Orsay, France Received November 998; received in revised form 8 March 999 Editor: PV Landshoff Abstract We analyse the Structure Function collaboration data on the quark pseudoscalar vertex and extract the Goldstone boson pole contribution, in rp The strength of the pole is found to be quite large at presently accessible scales We draw the important consequences of this finding for the various definitions of quark masses Ž short distance and Georgi-Politzer, and point out problems with the operator product expansion and with the non-perturbative renormalisation method q 999 Published by Elsevier Science BV All rights reserved Continuum model for the quark pseudoscalar vertex It is well known that the quark pseudoscalar Ž PS vertex contains a non-perturbative contribution from the Goldstone boson, in the continuum On the lattice, the use of a non perturbative renormalisation scheme wx makes this contribution manifest, although it should go to zero for large momentum transfers The purpose of this letter is to extract it from lattice simulation data, and to show that it is not negligible for presently accessible scales In particular, it must be subtracted when evaluating the short distance quark masses from the lattice through the non perturbative method of Ref wx This method uses the off-shell axial Ward identity Ž AWI and renormalises the mass in the momentum subtraction Ž MOM scheme of Ref wx, which can afterwards be related to the MS scheme The MOM renormalisation involves the pseudoscalar vertex, whence the necessity of the subtraction of the Goldstone contribution to extract the short distance quantity For physical u, d quarks, the Goldstone contribution becomes very large, larger than the perturbative part; this corresponds to a very large dynamical u,d mass, larger than the usual current mass at the scales accessible to standard lattice calculations The expected behaviour of the pseudoscalar vertex in the continuum has been described as follows in the 70 s in the works of Lane and Pagels w4,5x and others Near the chiral limit, the one-particle-irreducible PS quark vertex L5 can be described through Laboratoire associe au CNRS-URA D00063 The problem would be quite different if other methods are wx used to extract quark masses, see eg Ref r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

106 06 ( ) J-R Cudell et alrphysics Letters B a perturbative contribution plus a non-perturbative Goldstone boson contribution The perturbative contribution is of course =g 5, with QCD radiative corrections, which according to the renormalisation group lead to a logarithmic bewa Ž haviour x s p 4r for NFs 0 As to the non-perturbative contribution, firstly, according to PCAC, the Goldstone boson must dominate other pole contributions in the PS vertex near the chiral limit: the coupling of the pion to the PS Ž vertex indeed gives a pole ;r q qm p, where q is the momentum transferred at the vertex; other poles Ž radial excitations contributions are suppressed in the chiral limit At qs0, in terms of the quark mass m, this gives a rm pole contribution We emphasize that this pole contribution explodes at q s 0 in the chiral limit, ie it is singular on the border of the physical region of momenta This well-known fact seems to have been underestimated, especially in its consequence for the renormalisation of mass on the lattice near the chiral limit Žsee below, Section 3 Secondly, according to the Wilson operator product expansion Ž OPE, the non-perturbative contribution must be power behaved, ie at large p it must drop as rp up to logs This power behavior, as shown by Lane, is dictated by one of the pion Bethe-Salpeter amplitude in the OPE In the latter, the dominant operator is the vacuum to pion matrix element of the pseudoscalar density This leads to rp by the canonical dimensions and further to logs, wa Ž x s p 7r at NFs 0 3 Finally, a very important relation, emphasized by Lane and Pagels and derived from the Ward identity, connects directly the forward pseudoscalar vertex to the scalar part of the propagator Žthey are essentially proportional 4 Hence the study of propagator OPE 3 In the work of Lane, as well as in some subsequent works, w Ž xy4 r one quotes another power of logs: as p This latter power corresponds to the anomalous dimension of the local operator cg5c But one requires a gluon exchange at lowest order, which gives an additional as factor This was shown by Politzer wx 6 for the corresponding cc contribution to the propagator 4 Let us also recall that in the context of the lattice, this relation has been discussed and used by the Rome group to fix ZA on quark states wx 7 by Politzer wx 6 closely parallels the above considerations on the PS vertex, with the dominant non-perturbative Ž power contribution corresponding to the quark condensate, which corresponds to the Goldstone pole contribution in the PS vertex We postpone the discussion of the propagator lattice data, which involves delicate problems of improvement, to another paper wx 8 Nevertheless, in the last part of the letter, using only the vertex lattice data, we draw important consequences for the propagator thanks to this Ward identity In addition, we use the Ward identity to borrow the two-loop corrections to the above logarithmic factors, from those calculated in the case of the propagator by the Rome group wx, and by Pascual and de Rafael wx 9 In the following, we will show that the lattice data behave as expected from the continuum theoretical expectations To relate lattice to physical numbers, we will use the lattice unit a y s9 GeV at bs60 and al MS s074 for the QCD scale parameter in the quenched approximation We must stress that nothing in our discussion depends critically on the precise values of these parameters: this work is not oriented towards accurate numerical determinations, but rather towards questions of principle Fit of lattice data for G as function of the 5 hopping parameter k Recently w0 x, Paul Rakow presented, on the behalf of the Structure Function Ž QCDSF collaboration, the bare vertex G data at bs60 Ž 5 Fig of w0x through the product q 5 q 5 am G p 'C= am G qs0,p Ž as a function of a p at three k values, with: TrŽ g5l5 G 5 ' 4 This is the only contribution that survives at qs0 in the continuum limit By construction, C is a constant with the value w0 x: Cs075, Ž 3

107 ( ) J-R Cudell et alrphysics Letters B Ž Fig a The value of the coefficient of a p AŽ p in Eq Ž 6 from the lattice data extrapolated at k ; Ž b Our fit to the extrapolated data c defined in w0x so that the numbers in rhs of Eq Ž equate approximately the lattice data for the scalar part of the bare improved propagator Žsame Fig of w0x on some range of momenta, increasing with k This approximate equality corresponds to the Ward identity to be discussed in Section 4, but we are not concerned with it, as we rely only on the vertex data The QCDSF data were obtained with the SW imwx at csws 769, NFs 0, proved fermionic action in the Landau gauge, and p denotes the following lattice definition of the momentum squared, ap l ap'4w'4ýsin ž / Ž 4 l where a is the lattice unit This notation will be used herefrom, and identified with the continuum p We y take a s9"0 GeV at bs60 w x We also use the standard lattice definition of the bare mass: amq s y Ž 5 k k c Let us write q 5 q am G p sa p qam B p Ž 6 We shall first perform an extrapolation linear in m q of the three datasets to the chiral limit at kc s035, for each value of p This fit gives us both AŽ p and BŽ p, as shown in Figs and Ž As G5 qs0, p A AŽ p ramq qbž p, it seems reasonable to identify the AŽ p term as the Goldstone contribution Ž ie as the pole in am, and q Fig Ž a The value of BŽp in Eq Ž 6 from our extrapolation at k, compared with Eq Ž 7 ; Ž b c Our fit to the lattice data in the region of high p for B0 s735 The plain curves correspond from top to bottom respectively to amq s048, 0078, and 008, and the dashed one to the Goldstone boson contribution

108 08 ( ) J-R Cudell et alrphysics Letters B q 5 Fig 3 Ž a The value of am G Ž qs0, p for light quarks; Ž b The value of the dynamical u,d masses the second one as the perturbative contribution, if we are sufficiently close to the chiral limit The continuum Ward identity enables us to borrow the two-loop renormalisation group improved factors, from those calculated in the case of the propagator by the Rome group wx for the perturbative part: 4r asž p BŽ p sb 0 = asž p q8 4p Ž 7 wx and by Pascual and de Rafael 9 for the non-perturbative part: 7r s sž a p a p A p sa = q0 ap 0 4p Ž 8 with A0 and B0 some constants These two-loop corrections are valid for the MOM renormalisation scheme, in the Landau gauge, but with as taken as 5 the MS coupling constant As already mentioned, we take a L MS s 074 at NF s 0 from the three gluon coupling measurement with asymmetric mow3,4 x Note that, a priori, the evolution menta formulae properly apply to the renormalised propagator, which is the bare one divided by Z Ž p 6 ; 5 q for which we use the expression 4p ra Ž q slog s ž / L MS ž / 0 q q log log L 6 MS In this letter, we use the convention of Ref wx for the Z s ž / c but, at least theoretically and in the continuum, Z Ž p c should evolve very slowly, since gc s0 at one loop in the Landau gauge, and the two-loop correction seems also to have very little effect w5 x Lattice data w6,7x confirm this perturbative argument Then, we conclude that the two-loop corrections should be obeyed by the bare PS vertex with good accuracy The lattice data turn out to be quite close to the continuum theoretical expectations near the chiral limit, confirming the above interpretation of A and B Indeed one finds the following: Ž Ø A p is behaving remarkably close to rp over a large interval of p, see Fig Ž a From the numerical analysis, the Goldstone contribution appears to be very large Indeed, a p AŽ p,005 Ž 9 from the lowest point a p s06 On the other hand, we do not see the log factors expected from the perturbative calculation, Eq Ž 8, which remains to be understood; we shall discuss further problems in the perturbative evaluation of the Wilson coefficient at the end of the paper Ž Ž Ø Bp is found to evolve closely to ln p r L QCD y4r, more precisely it is evolving in good conformity with the two-loop MOM renormalisation formula quoted above We obtain B s

109 ( ) J-R Cudell et alrphysics Letters B which provides a very good fit to the data for p larger than GeV, see Fig Ž b The Goldstone contribution is felt already at rather large quark masses and, for physical u,d quarks it is in fact very large: AŽ p is larger than the perturbative part am BŽ p q even at rather large p, as shown in Fig 3Ž a and further discussed in Section 4 3 Consequences on the short-distance mass from the lattice Let us then recall that the pseudoscalar vertex is an important ingredient in the method first developed by the Rome group wx to determine the short distance quark masses One starts from the axial MOM renormalised quark mass given by: Z Landau A AWI am m sr Ž Z m P where r is a dimensionless parameter determined from a ratio of matrix elements involving the pion and the bare axial and pseudoscalar bilinear operators ZA is the standard renormalisation of the axial current, determined from the axial Ward identity or approximated through the MOM renormalisation non-perturbative method, or from one-loop perturbation theory Finally Z Ž m P is defined through the MOM renormalisation condition for the vertex and is determined by the same non-perturbative method from the PS quark vertex at qs0, or again approximated from one-loop perturbation theory 7 Then, Landau from m Ž m AWI, one can deduce the standard short distance masses, for instance the MS mass, but this requires to work in the perturbative, short distance, region Then our findings for the PS vertex have important consequences Let us note that, in principle, the fact that at qs0, the PS vertex not only is influenced by a large Goldstone contribution Žas noted in wx, w8 x, but really explodes in the chiral limit, is 7 Of course, using the PS quark vertex for normalisation is a particular way to define the MOM scheme, which, in the non-perturbative regime, is not necessarily compatible with eg the use of the scalar vertex See remarks below crucial for the MOM procedure of renormalisation, since the renormalisation constant Z is defined as: Z m MOM ZP Ž m s Ž G qs0,p sm c 5 This definition ensures that G R qs0,p sat p sm Ž 3 5 and one concludes from it that ZP has a trivial chiral limit at fixed p Indeed, near k c, since m p Amq Ž MOM and G5 qs0, p Arm q, ZP tends to zero, or its inverse goes to infinity 3 Calculation of Z P One can translate the above fit of G5 into an expression for ZP MOM, or rather its inverse which is more directly physical: G5Ž p AZ Ž p s s qbz Ž p MOM Z p Z p am P q c P Ž 4 where A Ž p sažp rwcz Ž p x and B Ž p Z c Z s BŽ p rwcz Ž p x c This is to be contrasted with usual fits, which assume that Z Ž P is linear in amq of Eq 5 8 One Ž notes that since Zc p is weakly dependent on p and on k, and close to, the expression is quite similar to the preceding one: it consists in a first term which is approximately in rm q, and the sec- ond one which is approximately constant 9 Just as for the G5 vertex, the former corresponds to the non perturbative Goldstone contribution while the latter, B Z, corresponds to the short distance contribution, as measured by the lattice numerical simulation, including all the orders of perturbation theory by a non perturbative method To give numbers, we need now 8 D Becirevic w9x has now done a fit along the above lines with the data of the Orsay-Rome group and found roughly similar conclusions 9 One could be worried by the fact that the dependence of Z c on k could generate from A rwž rk yrk rx Z c an additional contribution to Z y P independent of k However, it can be seen that since AZ decreases rapidly, this contribution is not large with respect to the one coming from B Z

110 0 ( ) J-R Cudell et alrphysics Letters B values for Z c, which we borrow from the data of the Rome group w 6 x, with an improvement procedure trying to parallel as much as possible the one followed by Rakow for the scalar part However, we would like to emphasize that all the qualitative conclusions of this paper are independent of precise values of Zc as long as they stay around At a p s which is close to the standard reference point ps GeV, and at ks034, which is the k closest to the chiral limit, we estimate: Z c,085 Ž 5 One has then numerically: A,003, B,88 Ž 6 Z Z We emphasize that this identification of the two contributions does not rely on Boosted Perturbation Theory Ž BPT, but purely on numerical simulations We shall refer to BZ as the short distance contribution not to be confused with its one-loop perturbative estimate Now, we shall compare to BPT estimates here and in the following for illustrative purposes only BZ is indeed very close to the one-loop y standard BPT evaluation Z Ža p s s 7 Ž P from ZP s 059 in the chiral limit, with g s 68, whereas the non-perturbative term contributes around 003ram q ; 08 on a total of 7 Therefore, the departure of ZP from its one-loop perturbative evaluation seems to be essentially due to the Goldstone boson contribution Higher-order radiative corrections do not seem to be very large 0 However, these conclusions could be sensitive to details of the data or of the fits 3 Calculation of the MS mass We can now convert our results for the renor- Landau malised mass mawi into a calculation of the MS mass In Ref w0 x, it is suggested that the use of the MOM non-perturbative determination of Z Ž P with linear extrapolation in k in the AWI method improves the results for the MS mass with respect to previous determinations of ZP by one-loop perturbative calculations However, in principle, to make the conversion to a short distance mass, we must still make sure that we work in the perturbative regime Now, we have isolated the Goldstone boson contribution which is essentially non-perturbative as it does not correspond to higher order contributions but rather to power corrections The fact that the non-perturbative estimate of the full ZP differs sizeably from the short distance BZ already at the measured kappas, is a signal that it is not presently possible to work at p high enough for the Goldstone contribution to be negligible Hence, we must first subtract it from Z y P y AZ Ž p Subtr y ZP Ž p s ZP Ž p y sbzž p am q Ž 7 Numerically, the remaining short distance BZ is in fact close to the one-loop BPT estimate of Z P,aswe have just seen Indeed, at a p s, with this subtraction and using again Z s 085 near the chiral c limit, we find ZP Subtr sr88s053 which corresponds to the fully resummed short-distance contribution determined directly from the lattice data To get an estimate of the consequences on the light quark masses, we use wx Žwhich uses the notation am for r where one finds r;am q, with amu,d s , and we take Z s079 w x We then find: A am Landau u,d ;0007 Ž 8 Converted into the MS scheme through wx / 6 a MS Landau s mq smq ž y Ž 9 3 4p with a Žaps s05 at two loops, this gives: s MS am u,d ; therefore about 46 MeV at NFs 0 One would obtain about 6 MeV if one used the full MOM non-perturbative estimate of ZP linearly extrapolated to the chiral limit and 4 MeV from one-loop BPT Note that these numbers are only indicative; in view 0 wx These two possible explanations were suggested in Ref We take ZP Subtr approximatively independent of mass, except for a small variation of Z c One would otherwise require a fit of mq G5 with one more term in m q

111 ( ) J-R Cudell et alrphysics Letters B of the many uncertainties in the subtraction procedure, we do not try to discuss the other sources of error necessary to give a real determination of the mass Our aim is only to underline the necessity of the subtraction of the Goldstone contribution Despite the fact that in this case the two methods lead to comparable results, the subtraction method just described, which does not rely on perturbation theory but which rather uses the lattice data directly, is in general superior to that based on the BPT estimate, and knowledge of the full ZP MOM is in general necessary even if we aim at measuring short distance quantities Indeed, the BPT method has the following drawbacks Firstly, the unknown higher-order perturbative corrections may be large Secondly, the one-loop perturbative evaluation can be tadpole-improved in many ways, potentially leading to very different estimates Finally, the one-loop estimate, even if tadpole-improved, does not automatically follow the behavior dictated by the renormalisation group; the problem is then cured by taking the one-loop estimate at some momentum, and then imposing the renormalisation group evolution for the other momenta, but this is obviously presenting a rather arbitrary choice of a privileged point On the other hand, the subtraction method used here avoids these problems It amounts in general to a non-perturbative measurement of the Z s, followed by the evaluation and removal of the pole contributions As shown above in Fig Ž b, this procedure leads to a result which evolves per se according to the renormalisation group The evaluation of the pole contribution from the pion is especially easy, because of its particular singular nature at m s 0 Other pole contributions q are expected to be smaller, because they are regular in the chiral limit If one were to subtract them, one could only rely on the expected power behavior of the particle vertex function, and their extraction would thus be more difficult In this section, we show that the Goldstone boson contribution to the PS vertex, which is only parasitical in the calculation of MS masses, and has to be subtracted as shown in the previous sections, retains an important physical meaning, as can be seen through the use of other definitions of renormalised quark masses 4 Physical releõance of the full renormalised axial mass One should remember that the full axial MOM renormalised mass, which is calculated through: Landau y amawi p srza ZP p with Z Ž p P not submitted to the above subtraction, retains a physical significance by itself, since it is the mass defined through the divergence of the axial current, with the corresponding natural renormalisation condition which consists in setting the pseudoscalar vertex G R to on quark states at the 5 renormalisation scale We stress that in contrast to the standard MS current mass, it does not vanish in the chiral limit, because the chiral limit r 0 is compensated by the pole in Z y The meaning of P this last result will be further developed in the next subsection Of course, it is unpleasant to have renormalised quantities with a rather queer behavior in the chiral limit Indeed, for instance, hadronic matrix elements of the renormalised pseudoscalar density which do not have a pion pole should tend to zero in the chiral limit- and therefore also, the ones of the scalar density defined in accordance with the Ward identity But as we hope to have shown, this is an unavoidable consequence of the non-perturbative method as applied to the pseudoscalar density Of course, one could avoid this by preferring the corresponding MOM normalisation condition for the scalar vertex But then, one must remember another aspect of the axial MOM renormalised mass, which gives it another important physical significance, and which we shall now discuss 4 Consequences for the renormalised mass a` la Georgi-Politzer Recall then that the scalar vertex should be consistently normalised through the Ward identities, as repeatedly emphasized by the Rome group, therefrom one sees that ZSrZP must be independent of p, in obvious contradiction with imposing simultaneously a similar MOM condition for the scalar vertex

112 ( ) J-R Cudell et alrphysics Letters B Relation with the mass as defined by the propagator It can indeed be easily shown that the axial renormalised mass is also essentially identical to a standard renormalised mass defined through the scalar part of the propagator, as becomes obvious through the renormalised axial Ward identity at zero transfer Ž this has been recalled in the talk of Rakow ; in fact it is essentially identical to the Georgi-Politzer mass Let us indeed write the Ward identity 3 : Tr S Landau R m m G qs0,p,m s AWI 5 Ž p,m y R 4 Here, it is assumed of course that the propagator is also calculated in the Landau gauge, and that the renormalisation of the vertex and of the propagator are both consistently performed according to the MOM scheme Therefore the propagator is normalised in the Euclidean region according to 4 : S y p,m sipu qm GP m at p sm Ž 3 R R which is nothing else than the Georgi-Politzer renormalisation condition Setting p sm, one obtains from Eqs Ž 3, and Ž 3 : Tr S p,m m m s sm m 4 y Landau R < GP AWI p sm R Ž 4 Landau therefore the announced identity of m Ž m AWI and GP the Georgi-Politzer mass function m Ž m R is derived 3 Here we use Eq as a constraint allowing to calculate the rhs from the lhs We do not require the independent input of the scalar part of the propagator data As a side remark, as a check of the validity of Eq on the lattice QCDSF propagator data, we note that it would imply for the C defined above Cs ZA rram q, which is not far from their Cs075 4 We disregard here the difference between this standard MOM condition for Zc and the derivative MOM condition derived from the vector Ward identity by the Rome group wx, which seems to lead numerically to very small differences Through Eq 4, the rp power contribution in Z y P corresponding to the Goldstone boson is related to a similar contribution in the scalar part of the propagator, which represents a dynamically generated mass for light quarks, though off-shell, gaugew3x signal of the spontaneous breakdown of the chiral dependent and Euclidean This is a well-known symmetry We can then translate our knowledge of the PS vertex into information on this dynamical mass 43 Physical consequences Numerically, for the non-perturbative contribution, which corresponds to the chiral limit of mawi Landau, one finds at aps, from Eq Ž, with, as before, Z from Eq Ž 4, Z s079 and r;am w x, P A q am GP a p s ;008 Ž 5 R therefore around 34 MeV at ps9 GeV We have obtained analogous results directly from the scalar part of the bare improved propagator as given in w0 x Extended considerations and estimates on the propagator will be given in a forthcoming paper wx 8 At large p, we want to stress that the non-perturbative contribution to the mass remains of the order of the u,d perturbative masses and even larger, see Fig 3Ž a It must be emphasized that this result is rather safe, since it does not depend critically on improvement procedures The sign is as expected, but the magnitude is much larger than what one would expect from the estimate by the qq condensate and a perturbative wx calculation of the Wilson coefficient 9 : 4 ² 0< qq< 0: GP mr Ž p sy pas Ž 6 3 p MSŽ Indeed, with a aps ;03 Ž one-loop s, and with a standard MS renormalised evaluation of the qq condensate of yž 5 MeV 3, at m s GeV, rescaled at a p s by a factor 8, one would find an answer lower by a factor ten However, a large value is inevitable if we admit as usual that at moderately low p the mass is of the order of a

113 ( ) J-R Cudell et alrphysics Letters B constituent mass, ie several tens of MeV, and if we take into account that the decrease is only in rp, as found with remarkable accuracy on the lattice data of the QCDSF group That a value of several tens of MeV is needed at low p is confirmed by direct lattice calculations of the propagator, to be compared with our estimate of Fig 3Ž b : Ø in configuration space, the propagator in time St Ž has been found exponential over a large range of t, with a coefficient of the exponential around 300 MeV w7,4 x; Ø in momentum space, at the lowest points, corresponding to p 0, where it should be insensitive to the improvement, the scalar part of the inverse propagator is found to be around similar values of MeV w0,6 x, for the lowest mass k s 034 A large value for the coefficient of rp was also found in a phenomenolw5 x, based ogy of the pion as a Goldstone boson on an assumption: 3 4Ž md GP mr Ž p s Ž 7 p where md is a free parameter The phenomenol- GP ogy seems to require m f300 MeV, m Ž p D R ; 7 MeV at p s GeV, in agreement with the above estimate It must be emphasized that this large non-perturbative contribution does not contradict directly the sum rule calculation of correlators, which uses a perturbative evaluation of the quark propagator Indeed, as in the calculation of propagator by Politzer, in sum rule calculations non-perturbative contributions are consistently added through condensates Moreover, as emphasized by Pascual and de Rafael wx 9, the condensate contributions are quite different for the quark propagator and the correlators, therefore our finding for the full quark propagator does not have a direct impact on sum rules Of course, the discrepancy with the naıve perturbative estimate of the Wilson coefficient is nevertheless worrying and deserves further reflection since the latter is known to work in general One could imagine that the perturbative calculation of the Wilson coefficient is not valid for some particular reason In this direction, one must observe that the two-loop correction is very large, around 50% of the first order at a p s It may be therefore that the perturbative expansion happens to fail 5 Conclusion The lattice numerical calculations can be seen as the triumph of the general theoretical predictions of the 70 s for the quark pseudoscalar vertex Nevertheless, a striking and unexpected feature of the lattice data is the very large size of the Goldstone boson contribution to the pseudoscalar vertex This corresponds, through the Ward identity, to a very large non-perturbative contribution to the renormalised mass function of Georgi and Politzer, by far larger than what is expected from the quark condensate and a perturbative evaluation of the Wilson coefficient, but in agreement with other physical expectations These large non-perturbative contributions then give a warning that there may be a possible problem with the use of lowest order perturbation theory in the estimate of the Wilson coefficient of condensates This question requires more investigation The Goldstone contribution must be subtracted from the pseudoscalar vertex to calculate the short distance mass from a normalisation of this pseudoscalar vertex This has important numerical consequences The short distance ZP to be used should be sizeably larger than the one measured directly on the lattice Finally, it must not be forgotten that lattice artefacts may still be large at bs6, and that the vertex has not been improved with rotations as an off-shell Green function, therefore one can expect large uncertainties on the quantitative estimate of the effect at large p Note added in proof When completing this paper, we became aware of the work of the JLQCD collaboration, presented at Denver conference, w6x where certain parallel conclusions on ZP have been drawn from lattice staggered fermion data Paul Rakow has also drawn our attention to Ref w7 x, where connected observations on PS vertex and the scheme of Pagels are made

114 4 ( ) J-R Cudell et alrphysics Letters B Acknowledgements We are very grateful to Paul Rakow for providing us with the data on the pseudoscalar vertex, as well as for very useful related information and discussions, to Guido Martinelli for invaluable encouragement, in particular for many discussions, precious information and ideas or notes on the current work of the Rome group on quark masses and on the quark propagator, and for providing us with their data on the propagator AL would also like to thank Damir Becirevic with whom much of this paper has been intensively discussed, for many informations, and finally all the lattice group, and the quark model group at Orsay for their constant and friendly help and discussions, as well as J Skullerud CP acknowledges illuminating discussions with GC Rossi, warmly thanks J Cugnon and the Groupe de physique nucleaire theorique de l Universite de Liege ` for kind hospitality and acknowledges the partial support of IISN References wx C Allton et al, Rome group, Nucl Phys B 43 Ž wx G Martinelli et al, Rome group, Nucl Phys B 445 Ž 995 8, e-print hep-latr9400 wx 3 ALPHA Collaboration, S Capitani et al, hep-latr97095, talk given by M Luscher at the International Symposium on Lattice Field Theory, July 6, 997, Edinburgh wx 4 KD Lane, Phys Rev D 0 Ž wx 5 H Pagels, Phys Rev D 9 Ž wx 6 HD Politzer, Nucl Phys B 7 Ž wx 7 G Martinelli, S Petrarca, CT Sachrajda, A Vladikas, Phys Lett B 3 Ž 993 4; B 37 Ž Ž E wx 8 JR Cudell, A Le Yaouanc, C Pittori, Lattice Quark Propagator and Quark Masses: an Extensive Analysis of Available Data, article in preparation wx 9 P Pascual, E de Rafael, Zeit Phys C Ž 98 7 w0x Paul Rakow, for S Capitani et al, QCDSF Collaboration, Nucl Phys B Ž Proc Suppl 63 Ž , e-print heplatr wx B Sheikholeslami, R Wohlert, Nucl Phys B 59 Ž wx GS Bali, K Schilling, Phys Rev D 47 Ž w3x D Henty et al, in: the proceedings of the 995 EPS HEP conference, Brussels, 995, p 39, e-print hep-latr w4x JP Leroy, Orsay Quadrics group, private communication New results are given in hep-phr9803 w5x E Franco, V Lubicz, preprint ROME-I ŽMarch 998, e-print hep-phr w6x G Martinelli, propagator data of the Rome group, private communication w7x J Skullerud, for the UKQCD Collaboration, Nucl Phys B Ž Proc Suppl 4 Ž , e-print hep-latr9404 w8x H Olrich, for M Gockeler et al, QCDSF Collaboration, Nucl Phys B Ž Proc Suppl 63 Ž , e-print heplatr97005 w9x D Becirevic, private communication w0x V Gimenez et al, hep-latr98008v wx QCDSF Collaboration, M Gockeler, Phys Rev D 57 Ž wx M Luscher, S Sint, R Sommer, H Wittig, Nucl Phys B 49 Ž , e-print Archive: hep-latr9605 w3x Y Nambu, G Jona-Lasinio, Phys Rev Ž ; 4 Ž w4x C Bernard, D Murphy, A Soni, K Yee, Nucl Phys B Ž Proc Suppl 7 Ž w5x H Pagels, S Stokar, Phys Rev D 0 Ž w6x N Ishizuka, for the JLQCD Collaboration, e-print heplatr98094 w7x M Gockeler et al, preprint DESY , July 998 and e-print hep-latr

115 3 May 999 Physics Letters B ž / The Goldberger Treiman discrepancy in SU 3 Jose L Goity a,b,, Randy Lewis a,c,, Martin Schvellinger a,b,d,3, Longzhe Zhang a,b,4 a Jefferson Lab, 000 Jefferson AÕenue, Newport News, VA 3606, USA b Department of Physics, Hampton UniÕersity, Hampton, VA 3668, USA c Department of Physics, UniÕersity of Regina, Regina, SK, S4S 0A Canada d Department of Physics, UniÕersidad Nacional de La Plata, CC 67, 900 La Plata, Argentina Received 6 January 999 Editor: H Georgi Abstract The Goldberger Treiman discrepancy in SUŽ 3 is analyzed in the framework of heavy baryon chiral perturbation theory Ž HBChPT It is shown that the discrepancy at leading order is entirely given by counterterms from the OŽ p 3 Lagrangian, and that the first subleading corrections are suppressed by two powers in the HBChPT expansion These subleading corrections include meson-loop contributions as well as counterterms from the OŽ p 5 Lagrangian Some one-loop contributions are calculated and found to be small Using the three discrepancies Ž p NN, KNL and KNS which can be extracted from existing experimental data, we find that the HBChPT calculation favors the smaller gp NN values obtained in recent partial wave analyses q 999 Elsevier Science BV All rights reserved Introduction The Goldberger Treiman relation Ž GTR wx, obtained from matrix elements of the divergence of axial currents between spin r baryons, is an important indicator of explicit chiral symmetry breaking by the quark masses It interrelates baryon masses, axial vector couplings, the baryon-pseudoscalar meson Ž Goldstone boson ' GB couplings and the GB decay constants Explicit chiral symmetry breaking leads to a departure from the GTR Ž defined below which is called the Goldberger Treiman discrepancy Ž GTD goity@jlaborg randylewis@ureginaca 3 martin@venusfisicaunlpeduar 4 lzhang@jlaborg The GTD has been repeatedly discussed over time wx and for several reasons there were difficulties in arriving at a clear understanding On one hand, there was no available effective theory with a systematic expansion to address the problem, and on the other hand the experimental values of the baryon-gb couplings were too poorly known In recent years, progress has been made on both fronts There is now a baryon chiral effective theory that permits a consisw3 6 x There has tent expansion of the discrepancy also been progress in the determinations of the baryon-gb couplings that are the main source of uncertainty in the phenomenological extraction of the discrepancies In fact, the current knowledge of the couplings g p NN, gknl and, to a lesser extent, gkns is good enough to justify a new look at the GTD in SUŽ 3 In this work we study the GTD in the light of heavy baryon chiral perturbation theory Ž HBChPT w4,5 x r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

116 6 ( ) JL Goity et alrphysics Letters B Let us first briefly review the derivation of the GTR wx 7 and the definition of the GTD We consider the matrix elements of the octet axial current A a m a s q x gg l qž x Ž m 5 the Gell Mann matrices are Ž a b ab normalized to Tr ll s d between states of the baryon octet: c ² b, pb NAm Na, pa: subž pb g g abc Ž q m A abc yqm g q g5ua p a, Ž where a,b,cs,,8 and qspbypa is the mo- mentum transfer between baryons a and b From Eq Ž, the matrix elements of the divergence of the axial currents become ² b, p NE m A c Na, p : b m a abc bž b Ž a b A siu p y M qm g q abc qq g Ž q g5uaž p a, where Ma is a baryon mass Crucial to the derivation of the GTR is the GB pole contribution represented in Fig To explicitly expose the pole term, the matrix element in Eq Ž can be rewritten as ² b, p NE m A c Na, p : b m a N abc q siubž pb g U Ž p, Ž 3 5 a a q ym qie c abc Ž Ž c Ž Ž where N q s gcab q P q q q y abc m d Žq, m is a GB mass, and g Žq c c cab the baryon-gb form factor, defined such that in the physical basis of the Gell Mann matrices g Ž M is equal to g c Ž 0, etc P q 3,6qi7,6yi7 p p nn represent the couplings of the pseudoscalar currents to c the GB s, given in the chiral limit by P sm F ŽF c c c Fig Diagrams representing the contact term and the pole term in the matrix elements of the divergence of the axial currents Crosses represent the divergence of axial currents is the decay constant, where F s94 MeV p ; the q dependence of P c Žq starts at OŽp 4 and is abc henceforth disregarded Finally, d Žq denotes contributions not involving the GB pole, and it starts as a quantity of OŽ p This separation of pole and non-pole contributions is not unique Žthe off-shell functions separately are not observables ; for instance, up to higher order terms in q, we can choose to remove the q dependence in g Žq cab around the point q sm c by a simple redefinition of d abc m c In the chiral limit E A s0, and at q s0 Eq Ž m gives: Mg abc Ž 0 s lim q g abc q sf g Ž 0, Ž 4 A c cab q 0 which is the general form of the GTR Here M is the common octet baryon mass in the chiral limit In the real world, chiral symmetry is explicitly broken by the quark masses and the GB s become massive In this case, Eqs Ž and Ž 3 lead to g q P c q abc mc g Ž mc s lim Ž 5 q ym qie q m c cab In order to define the GTD it is also convenient to take the limit q 0 which gives Ž M qm g Ž 0 abc a b A c abc s gcabž 0 P Ž 0 y d Ž 0 Ž 6 m c The discrepancy D abc is then defined by: Ž M qm g Ž 0 abc a b A Ž yd abc c cab c c m c s g m P m Ž 7 c Notice that while the GTR, Eq Ž 4, is defined at Ž q s0, the GTD in Eq 7 is given at q sm c because only at that point is the coupling g cab unambiguously determined At leading order in the quark masses, the GTD can then be expressed as follows: E abc abc D sm log N q N c q sm Ž 8 E q c

117 ( ) JL Goity et alrphysics Letters B Tree level contributions Throughout we are going to use standard definitions, namely: p a l a u'exp yi, Ž 9 F 0 ž / x' B Ž sqip, Ž 0 0 x "'u x u "ux u, Ž i m q m m q v ' Ž u E uyue u, Ž i m mn S Õ ' gs 5 Õ n Ž 3 The HBChPT Lagrangian is ordered in powers of momenta and GB masses, which are small compared to both the chiral scale and the baryon masses, LsL Ž ql Ž ql Ž3 q Ž 4 Although the Lagrangian is written as a single expansion, it will be useful to keep track of the chiral and rm suppression factors separately As will be demonstrated explicitly below, leading order Ž LO contributions to the GTD appear within L Ž3 Subleading contributions are suppressed by at least two suppression factors, so we will refer to any contribution at the order of L Ž5 as a next-to-leading order Ž NLO contribution The tree level contributions to the GTD stem from contact terms in the effective Lagrangian that can contribute to d abc, and also from terms that can give a q dependence to g First we notice that in cab HBChPT such terms must contain the spin operator S m that results from the non-relativistic reduction of Õ the baryon pseudoscalar density There are two types of terms which contribute to the GTD The first type must contain the pseudoscalar source x y The sec- ond type must contain monomials such as w D m, w D, v xx and w D n, w D, v m xx m n m between the Ž m baryon field operators here D is the chiral covariant derivative However, upon using the classical equations of motion satisfied by the GB fields at OŽ p, it turns out that terms of the second type can be recast into terms among which there are terms of the first type In this way, one moves the explicit q dependence from gcab to contact terms, some of which contribute to d abc Such reduction of terms has been implemented for L Ž ql Ž ql Ž3, for instance in Ref wx 8, and in the relativistic effective wx Ž3 Lagrangian as well 3 Some terms in L whose coefficients are determined by reparametrization inwx variance 8 may seem at first glance to give a q dependence to g cab, but a careful calculation shows that this is not so Since x is OŽ p y, and since a factor of the spin operator SÕ m is needed, the LO tree contributions to the GTD must come from L Ž3 One can further argue that there are no contributions from the evenorder Lagrangians, L Ž n The reason is that an even number of derivatives would require factors in the monomial of the form Õ= which, when acting on the baryon field, are in effect replaced by = r M; the other possibility would be factors of ÕSÕ that vanish Ž Ž3 In the case of SU, the Lagrangian L has been given by Ecker and Mojzis ˇˇw8,9 x There are only two terms in L Ž3 that are of interest to us, namely the terms O and O given in Refs w8,0 x 9 0 In the scheme used by Ecker and Mojzis ˇˇ these are finite counterterms We note that although O7 and O8 do contribute to gcab and to g A abc, they are such that no contribution to the GTD results, as noticed in Ref w x Ž Ž3 0 In SU 3 there are instead three L terms that are of interest to us, namely, ž / Ž3 m LGTD syif9tr BS Õ = m x y, B m 9 ž Õ = m x y, B4/ yid Tr BS m y ib0tr BSÕ B Tr Em x y 5 The NLO contributions come from LGTD Ž5 and will not be displayed here There are, for instance, terms quadratic in the quark masses such as Ž m Tr BS x w= x, Bx Õ q m y and others The contribution to d abc from LGTD Ž3 is given by abc d CT abc abc 4MB 0 s s if f qd d cde d bea abe qd s if9 f qd9 d c ab qs 3 D9 qb0 d, 6

118 8 ( ) JL Goity et alrphysics Letters B where s0 s 3 muqmdqm s, 7 a s sda3ž muymd y da8ž msymuym d ' 3 Ž 8 In deriving Eq Ž 6 from Eq Ž 5 we used the Ward identity: d L d L d L m a a abc b ye Am s s0 q 3 s qd s, Ž 9 0 d p d p d p a as well as the following correspondence of operators between the heavy baryon and relativistic theories: m BSE Õ Õ m pbõ lyimbg 5 pb, 0 c where B and BÕ are the relativistic and heavy baryon fields respectively The leading terms in the GTD are therefore of order p There are several relations among the discrepancies that are exact at LO One of them is the Dashen Weinstein relation wx 7 : m / NNp g A NNp K ž D g V ž / ž / / NL K NS K ga g NL K A NSK p ž gv gv s m 3 D y D Ž This particular relation provides useful insight as will be shown in the phenomenological discussion Since the bulk of the contribution to the GTD will result from the counterterms of Eq Ž 5, it is important to consider what physics determines their magnitude It seems likely that a meson dominance model may provide the correct picture In such a model the size of the counterterms would be determined by the lightest excited pseudoscalar mesons that can attach a the pseudoscalar current qg5 lq to the baryons The relevant such states are in the P X octet consisting of p Ž 300, hž 440 and KŽ 460 The next set of pseudoscalar states is in the range of 800 to 000 MeV, and thus, one may expect that they only give corrections at the order of 0 to 30% The meson dominance model can be implemented using an efw x fective Lagrangian in analogy with Ref The coupling of the P X octet to the pseudoscalar current is obtained from the effective Lagrangian: L s Tr = P = P y M P m X X X X X P m P qid XTrŽ P X x q, P y where we display only those terms relevant to our problem Here the P X octet responds to chiral rotations in the same way as the baryon octet The matrix element of the divergence of the axial current is given by B m a X b ² : b a 0NE A NP sy d X m P TrŽ l l, Mq 4 syd ab d X m, Ž 3 and the P X -baryon coupling can be expressed through the effective Lagrangian: X X X X X P B 5 4 5w x P L sd Tr Bg P, B qf Tr Bg P, B Ž 4 a From Eqs Ž 3 and Ž 4 one readily obtains the contribution to d abc : m c m abc abc abc c d X syd Xg X fd Xg X P P P B P P B q ym X M X P P Ž 5 abc F X Ž bw c ax D X Ž b c a Here g X s Tr l l,l q Tr l l,l 4 P B ' 8 ' 8 The current situation is that the couplings of the P X are not known, and there is no estimate in the literature that one could judge reliable As we comment later, the GTD s actually serve to determine Ž abc d X q g X P P B much more precisely than any model calculation available, provided the meson dominance model is realistic 3 Loop contributions There are several one-loop contributions to the GTD that we illustrate in Fig There are also, at the same NLO, two-loop contributions that we do not display here Although we do not perform here a full calculation, we do arrive a some interesting observations about such NLO effects by loops Let

119 ( ) JL Goity et alrphysics Letters B Fig One loop diagrams that give NLO corrections to the GTD In Ž a the cross represents the divergence of the axial current obtained from the OŽ p Lagrangian, and in Ž b the same divergence obtained from the OŽ p 3 baryon Lagrangian us consider the loop diagram in Fig a We can show that in HBChPT this loop effect on the GTD is OŽrM, and must therefore be suppressed by two powers relative to the LO contribution Indeed, in HBChPT the diagram is proportional to the following loop integral: d d k km k mn n yit H d Ž p k ymd qkpõr Mf qo rmf = kõqk r M yd m q kqq Õr Me qo rme = Ž kqq Õq Ž kqq rž M yd m f fa e eb, Ž 6 where d m ab'maym b, and T mn is transverse to the four-velocity Õ For spin r baryons in the loop T mn ASÕ m qpsõsõ n It is also easy to show explicitly that T mn is transverse if one or both lines in the loop are spin 3r baryons From energy-momentum conservation we have q qõs Ž MbyMa y Ž 7 M b Using this and the transversity of T mn, the expansion of Eq 6 shows no q -dependence at OŽ and OŽ rm We conclude that the one-loop diagrams considered here must affect the GTD at OŽr 5 M, and are thus negligible in the large M limit Another type of one-loop contribution is not suppressed by rm These are the diagrams involving the insertion of terms from L Ž3 as shown in Fig b, which correct the GTD at NLO Similarly there are NLO two-loop contributions that are of leading order in rm It is interesting to comment here on a one-loop calculation in the framework of a relativistic baryon effective Lagrangian, as used in Refs w3,3 x It turns out that the relativistic version of the loop diagram in Fig a gives a finite q dependence to the g cab coupling, namely, gcabž q ygcabž 0 3 s ž F p / = 8 afd ebd fec fed Ý g A g A g A J q, M a, M b,m c d,e, fs where the integral J fed Ž a b c J fed q, M, M,m s Ž 4p = is given by: C Ž M a, M b, M e, Mf Ž 8 NŽ x, y d x d y, Ž 9 yx H H 0 0 D x, y Ž a b N x, y s xqyy M qm yq xqy qž yx MaMeq xmamf qž yy MfMbq ymbme q M ym, Ž 30 f DŽ x, y s Ž yxyyž xma qymb ymd e ym xym yqxyq, Ž 3 f 5 w x For a related discussion, see Ref e

120 0 ( ) JL Goity et alrphysics Letters B C Ž M a, M b, M e, Mf s Ž M qm M qm M qm Ž 3 b e a f e f One can readily check that for SUŽ one obtains the result in Ref wx 3 The interesting thing here is that the contribution to the GTD by the loop is not suppressed by rm Actually, it is nearly constant for baryon masses ranging from a few hundred MeV to an arbitrarily large mass This result seems at odds with the one from HBChPT, but the two can be harmonized as follows: in the limit of large M it turns out that in the relativistic calculation there are contributions to the loop integral from momenta that are OŽ M M acts in fact as a regulator scale In HBChPT on the other hand, one is doing a rm expansion of the integrand, which implies that one is assuming a cutoff in the loop integrals given by a QCD scale The relativistic and HBChPT frameworks must each lead to the same physical results; in the present case this implies that in order to lead to the same results for the discrepancies, the coefficients F9 and D9 in L Ž3 must be readjusted when going from one framework to the other In the real world, M;L x and we may use the relativistic calculation as an estimate of this class of loop contributions to the discrepancy For the discrepancies of interest herein, these loop contributions are small, between ten to twenty percent of the discrepancies themselves, and smaller than their current errors The numerical results are Dloop NNp s00043 Ž 33 Dloop NL K sy0044 Ž 34 Dloop NS K s0044, Ž 35 where we use Ds079 and Fs046 for the SUŽ 3 axial vector couplings Of course the calculated loop contribution is not all that there is; the inclusion of decuplet baryons in the loop also gives contributions to the discrepancy ŽRef w4x discusses some DŽ 3 effects with only two quark flavors Using Rarita Schwinger propagators and three quark flavors, we have checked that the q -dependent part does show an UV divergence in the relativistic framework HBChPT also permits two-loop contributions at NLO 4 Results There are only three discrepancies that can be determined from existing data on baryon-pseudoscalar couplings: D NNp, D NL K, and D NS K Due to the smallness of the u and d quark masses, D NNp is necessarily very small, and its determination requires a very precise knowledge of the g coupling Ž p NN g A and Fp are already known to enough precision, leaving most of the uncertainty in the determination of D NNp to the uncertainty in g p NN A recent value for gp NN from NN, NN and p N data is obtained by the Nijmegen group w5 x They analyzed a total of twelve thousand data and arrived at gp NNs 305" 008 Similar results are obtained by the VPI group w6 x There is still some disagreement between determinations of gp NN by different groups Larger values have been obtained, such as gp NNs365"030 by Bugg and Mach- leidt w7 x, and gp NNs35"03 by J Rahm et al w8 x We note that some of the quoted errors do not include systematic errors We will use an error larger by roughly a factor of two to account for them As we find out below, our analysis of the discrepancies strongly favors the smaller gp NN val- g NNp A ues Using Fp s 94 MeV, Ž g V sy67" 0004 w9 x, Eq Ž 7 gives, D NNp s004"00 for g expt p NN s305"008 g p NN s355"07, Ž 36 4p D NNp s0047"00 for g expt p NN s35"03 g p NN s45"06 Ž 37 4p The determination of the gknl and gkns couplings relies on a more sparse data set The Nijmegen group analyzed data from YY production at LEAR, and they obtained w0 x: gknlsy37"04 and g s 39" 07 These values are consistent KNS

121 ( ) JL Goity et alrphysics Letters B with an earlier analysis by Martin w x, where only an upper bound for g is given Using NL K KNS g A FK s Fp and ž / sy078" 005 and g V NS K g ž / A s0340"007 w9 x, Eq Ž 7 gives, g V D NL K s07"003 Ž 38 expt Dexpt NS K s08"05 Ž 39 Disregarding SUŽ breaking, which implies that there is no contribution from the term proportional to b0 to these discrepancies, we can use the three measured discrepancies to determine the two LO parameters in HBChPT: MF9 s00"006 GeV y, MD9 s035"03 GeV y, for gp NNs305"00 Ž 40 MF9 s00"006 GeV y, MD9 s04"03 GeV y, for gp NNs35"00 Ž 4 where M is here the common baryon-octet mass in the chiral limit The x in Ž 40 and Ž 4 is respectively 008 and 36 If one uses the errors quoted in w x w x Ref 5 and 8 the respective x become 03 and 76, and the central values given by the fit have no significant change The LO discrepancies resulting from our fit are: D NNp s008"0005; 009"0005, Ž 4 D NL K s07"004; 09"004, Ž 43 D NS K s07"005; 04"007, Ž 44 where the quoted results correspond respectively to the smaller and larger gp NN couplings As evidenced by the large value of x and by Eq Ž 4, the larger NNp value D s 0047 of Eq Ž 37 Žcorresponding to the larger g coupling p NN cannot come out consis- tently from the fit To understand this one can use the Dashen Weinstein relation, Eq Ž, which holds exactly in our LO calculation For the results of the discrepancies involving the hyperons the term proportional to D NS K in the Dashen Weinstein relation is about one fifth of that proportional to D NL K, and the right hand side of Eq Ž would imply that D NNp must be about 5% The only way to accommodate a larger D NNp would be larger D NL K or D NS K or else a large deviation from the Dashen Weinstein relation The latter seems unlikely because the corrections to the relation must be suppressed by two powers in HBChPT Žthis is so because the corrections to the axial-vector couplings and to the discrepancies are of OŽ p On the other hand the former possibility would require that the magnitudes of gknl and gkns be unrealistically large In fact, either gknl and gkns would need to be increased by many standard deviations This would result in D NL K )03 or D NS K )05, implying a serious failure of the low energy expansion Thus, we conclude that only the smaller values of D NNp, and thus of g p NN, are consistent This shows the importance of the current analysis of the GTD in SUŽ 3 Finally, the coupling constants required in the meson dominance model resulting from our analysis are as follows: d X F X P s"03 GeV Ž 45 d X D X P s7"03 GeV Ž 46 Since here F X and D X are baryon-meson couplings, it is not unreasonable that they should have values similar to those of, say, the pion-nucleon coupling This would imply that the coupling d X P should be a few hundred MeV This makes the meson dominance picture quite plausible In conclusion, we have shown that the GTD in SUŽ 3 is given at leading order by two tree-level contributions, and that the corrections are suppressed by two powers in HBChPT Some of the loop corrections were calculated explicitly and found to be small Our leading order analysis indicates a strong preference for a smaller Goldberger Treiman discrepancy in the pion-nucleon sector, thus favoring the smaller values of the pion-nucleon coupling extracted in recent partial wave analyses Acknowledgements We would like to thank Juerg Gasser for allowing us to use material from an earlier unpublished collaboration and for useful discussions We also thank T Ericson, G Hohler, B Holstein, U Meißner and R Workman for useful comments, and Jan Stern for bringing to our attention the Dashen Weinstein relation This work was supported by the National Science Foundation through grant a HRD

122 ( ) JL Goity et alrphysics Letters B Ž JLG and MS, and a PHY Ž JLG and by the Department of Energy through contract DE- AC05-84ER4050 Ž JLG, RL, and in part by Natural Sciences and Engineering Research Council of Canada Ž RL, the Fundacion Antorchas of Argentina Ž MS and by the grant a PMT-PICT0079 of the ANPCYT of Argentina Ž MS References wx ML Goldberger, SB Treiman, Phys Rev 0 Ž wx CA Dominguez, Riv Nuovo Cim 8 Ž 985, and references therein; BR Holstein, Nucleon axial matrix elements, nucl-thr wx 3 J Gasser, ME Sainio, A Svarc, ˇ Nucl Phys B 307 Ž wx 4 E Jenkins, AV Manohar, Phys Lett B 55 Ž wx 5 U-G Meissner, in Themes in Strong Interactions, Proth ceedings of the HUGS at CEBAF, JL Goity Ž Ed, World Scientific Ž , and references therein wx 6 NH Fuchs, H Sazdjian, J Stern, Phys Lett B 38 Ž wx 7 R Dashen, M Weinstein Phys Rev 88 Ž wx 8 G Ecker, M Mojzis, ˇˇ Phys Lett B 365 Ž wx 9 N Fettes, U-G Meissner, S Steininger, Nucl Phys A640 Ž w0x HW Fearing, R Lewis, N Mobed, S Scherer, Phys Rev D56 Ž wx G Ecker, J Gasser, A Pich, E de Rafael, Nucl Phys B 3 Ž wx JA McGovern, MC Birse, MC-TH-98-3 preprint Ž 998 e-print Archive: hep-phr w3x J Gasser, JL Goity, unpublished w4x V Bernard, HW Fearing, TR Hemmert, U-G Meissner, Nucl Phys A 635 Ž 998 ; Nucl Phys A 64 Ž w5x JJ de Swart, MCM Rentmeester, RGE Timmermans, Proc of MENU97, TRIUMF Report TRI-97- Ž w6x RA Arndt, II Strokovsky, RL Workman, Phys Rev C 5 Ž w7x DV Bugg, R Machleidt, Phys Rev C 5 Ž w8x J Rahm et al, Phys Rev C 57 Ž w9x Particle Data Group, C Caso et al, Eur Phys J C 3 Ž 998 w0x RGE Timmermans, TA Rijken, JJ de Swart, Phys Lett B 57 Ž 99 7 REG Timmermans, ThA Rijken, JJ de Swart, Nucl Phys A 585 Ž c wx AD Martin, Nucl Phys B 79 Ž 98 33

123 3 May 999 Physics Letters B B h X X in the standard model s Xiao-Gang He a,b,, Guey-Lin Lin c, a Department of Physics, National Taiwan UniÕersity, Taipei, Taiwan, 0764, Taiwan b School of Physics, UniÕersity of Melbourne, ParkÕille, Vic 305, Australia c Institute of Physics, National Chiao-Tung UniÕersity, Hsinchu, Taiwan, 300, Taiwan Received December 998; received in revised form 4 March 999 Editor: H Georgi Abstract We study B h X X within the framework of the Standard Model Several mechanisms such as b h X s sg through the X X QCD anomaly, and b h s and B h sq arising from four-quark operators are treated simultaneously Using QCD equations of motion, we relate the effective Hamiltonian for the first mechanism to that for the latter two By incorporating next-to-leading-logarithmicž NLL contributions, the first mechanism is shown to give a significant branching ratio for B h X X s, while the other two mechanisms account for about 5% of the experimental value The Standard Model prediction for B h X X is consistent with the CLEO data q 999 Elsevier Science BV All rights reserved PACS: 35Hw; 340Hq s X The recent observation of B h K wx and B X X h X wx s decays with high momentum h mesons has stimulated many theoretical activities w3 0 x One of the mechanisms proposed to account for this decay is ) X X b sg sgh w3,4x where the h meson is produced via the anomalous h X y g y g coupling According to a previous analysis wx 4, this mechanism within the Standard Model Ž SM can only account for r3 of the measured branching ratio: X q00 BŽ B h X s s 6"6Ž stat "3Ž syst y5 y4 = Ž bkg =0 wx with 0-p X h -7 GeV There are also other calculations of B h X X based on hexg@physntuedutw glin@beautyphysnctuedutw s four-quark operators of the effective weak-hamiltow5,6 x These contributions to the branching ratio, nian typically 0 y4, are also too small to account for B h X X s, although the four-quark-operator contri- bution is capable of explaining the branching ratio X for the exclusive B h K decays w8,9 x These results have inspired proposals for an enhanced b sg and other mechanisms arising from physics beyond the Standard Model w4,6,7 x In order to see if new physics should play any role in B h X X s, one has to have a better understanding on the SM prediction In this letter, we carry out a careful analysis on B h X Xs in the SM using next-to-leading effective Hamiltonian and consider several mechanisms simultaneously We have observed that all earlier calculations on X b sgh were either based upon one-loop result wx r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

124 4 ( ) X-G He, G-L LinrPhysics Letters B which neglects the running of QCD renormalization -scale from MW to Mb or only taking into account part of the running effect wx 3 Since the short-distance QCD effect is generally significant in weak decays, it is therefore crucial to compute b sgh X using the effective Hamiltonian approach As will be shown later, the process b sgh X alone contribute significantly to B h X Xs while contributions from b X X h s and B h sq are suppressed The effective Hamiltonian 3 for the B h X X s decay is given by: G F ) f eff Ý fb fs Ž ' fsu,c H Ž D Bs s V V C Ž m O Ž m with 4 ž / VyAž / qcž m O f Ž m f O s sf i j fb j i, VyA VyA f O s sf i i fb j j, VyA VyA Ý 6 ) ts tb Ý i i ž is3 O s sb qq, 3 i i j j VyA q Ý O s sb qq, 4 i j VyA j i VyA q VyA Ý O s sb qq, 5 i i j j VqA q Ý O s sb qq, yv V C Ž m O Ž m qc8ž m O8 Ž m, Ž / 6 i j VyA j i VqA q g s mn a a 8 i s L b R ij j mn O sy s s Ž m P qm P T b G, 4p where V"A'"g 5 In the above, we have dropped O since its contribution is negligible For 7 numerical analyses, we use the scheme-independent Wilson coefficients discussed in Ref w3,4 x For m s75 GeV, a Žm t s Z s08 and msmbs5 GeV, we have w4x Csy033, Cs50, C3s007, C sy0037, C s000, C sy0045 Ž At the NLL level, the effective Hamiltonian is modified by one-loop matrix elements which effectively Ž change C m is3, PPP,6 into C m qc q,m i i i with C4 q,m sc6 q,m sy3c3 q,m where sy3c5 q,m syps q,m, 4 a s 0 s 9 c Ž P q,m s C Ž m qg m,q,m, Ž 5 8p with G m,q,m Ž c ž / m c yxž yx q s4hxž yx log d x Ž 6 m The coefficient C8 is equal to y044 at m s 5 GeV 5, and mc is taken to be 4 GeV Before we discuss the dominant b sgh X process, let us first work out the four-quark-operator contribution to B h X Xs using the above effective Hamiltonian We follow the approach of Ref w3,5,5x which uses factorization approximation to estimate various hadronic matrix elements The four-quark operators can induce three types of processes repre- X X sented by Ž ² h < qg bb < :² X< sg q < 0 :, s X ² h < q G q < 0 : ² X < sg b < B :, and Ž 3 s X X X ² < < :² < < : Ž h Xs sg3q 0 0qG3bB Here Gi denotes ap- propriate gamma matrices The contribution from Ž X gives a three-body type of decay, B h sq The contribution from Ž gives a two-body type of X decay b sh The contribution from Ž 3 is the annihilation type which is relatively suppressed and will be neglected Note that there are interferences 3 For an extensive review on the subject of effective Hamiltow x, which contains a detailed list of original nian, see Ref literatures 4 The sign of O8 is consistent with the covariant derivative, a a D se yigt A, in the QCD Lagrangian See, w x m m m 5 For an extensive review on the subject of effective Hamiltow x, which contains a detailed list of original nian, see Ref literatures

125 ( ) X-G He, G-L LinrPhysics Letters B between Ž and Ž, so they must be coherently added together wx 5 Several decay constants and form factors needed in the calculations are listed below: ² 0< ug g u< h : s² 0< dg g d< h : sif X p, X X u h X m 5 m 5 h m ² 0< sg g s< h : sif X p, X s h X m 5 h m m X h X ² < < : u s 0 sg s h si f Xyf X 5 h h, m s ž / u f X h s fcosuq f8sinu 8, ' 3 ' s f Xs f cosu y' h Ž f8sinu 8, ' 3 X y X ² < < : ² < < 0 h ug bb s h dg bb : m m sf Bq pm B qp hx m mb ym X Bq Bq h q F yf q, 0 m q ž / Bq Bh 8 Bh F s sinu F qcosu F,0,0,0 Ž 7 ' 3 ' For the h X yh mixing associated with decay constants above, we have used the two-angle -parametrization The numerical values of various parameters are obtained from Ref w6x with f s 57 MeV, f s 68 MeV, and the mixing angles u s y98, 8 u 8 sy8 For the mixing angle associated with form factors, we use the one-angle parametrization with usy548 w6 x, since these form factors were calculated in that formulation w5,5 x In the latter discussion of b sgh X, we shall use the same parametrization in order to compare our results with those of earlier works w3,4 x For form factors, we assume that F Bh sf Bh 8 sf Bp with dipole and monopole q dependence for F and F 0, respec- tively We used the running mass ms f0 MeV at Bp ms5 GeV and F s033 following Ref wx 9 The branching ratios of the above processes also depend on two less well-determined KM matrix elements, Vts and V ub The dependences on Vts arise from the penguin-diagram contributions while the dependences on Vub and its phase g occur through the tree-diagram contributions We will use g s 648 obtained from Ref w7 x, < V < f< V < ts cb s 0038 and < V < r< V < ub cb s 008 for an illustration We find that, for m s 5 GeV, the branching ratio in the signal region p XG 0 GeV Ž m F 35 GeV is h X BŽ b h X Xs f0=0 y4 Ž 8 The branching ratio can reach =0 y4 if all parameters take values in favour of B h X X s Clearly the mechanism by four-quark operator is not sufficient to explain the observed B h X Xs branching ratio We now turn to the major mechanism for B h X X : b h X s sg through the QCD anomaly To see how the effective Hamiltonian in Eq Ž can be applied to calculate this process, we rearrange part of the effective Hamiltonian such that ž / ž / 6 C4 C6 COs C q O q C q O N N Ý i i is3 c c where yž C4yC6 OA qž C4qC6 O V, Ý a m a OA ssgm yg5 T b qg g5t q, q Ý Ž 9 a m a OV ssgm yg5 T b qg T q 0 q Since the light-quark bilinear in OV carries the quantum number of a gluon, one expects wx 3 OV give contribution to the b sg ) form factors In fact, by applying the QCD equation of motion: DG n a mn s m a g Ýqg T q, we have O s Ž rg sg Ž y g s V s m 5 - T a bdn Ga mn 6 In this form, OV is easily seen to give 6 By applying the QCD equation of motion or performing a direct calculation, it was shown that the operator basis of O3yO6 are suitable to describe nonleptonic weak decays although effective vertices such as s dqgluons are encountered Here the operator basis on the rhs of Eq Ž 9 is more suitable for our purpose For detail, see Ref w8 x

126 6 ( ) X-G He, G-L LinrPhysics Letters B rise to b sg ) vertex Let us write the effective b sg ) vertex as GF g bsg ) s Gm sy Vts Vtb ' 4p a = ž D Fs Ž qgmyqr qm / LT b n a yifm bssmn qrtb In the above, we define the form factors D F and F according to the convention in Ref wx 4 Inferring from Eq Ž 9, we arrive at 4p D Fs Ž C4Ž m qc6ž m, FsyC8Ž m a s Ž We note that our relative sign between D F and F agree with those in Ref w4,6 x, and shall result in a destructive interference for the rate of b sgh X We stress that this relative sign is fixed by treating the sign of O8 and the convention of QCD covariant derivative consistently 7 To ensure the sign, we also check against the result by Simma and Wyler w9x on b sg ) form factors An agreement on sign is found Finally, we remark that, at the NLL level, D F should be corrected by one-loop matrix elements The dominant contribution arises from the operator O where its charm-quark-pair meets to form a gluon In fact, this contribution, denoted as D F for convenience, has been shown in Eqs ŽŽ 4-6, 4p Ž Ž Ž namely D F s C q,m qc q,m a s 4 6 To proceed further, we recall the distribution of the bž p sžp X q gž k q h X Žk X branching ratio wx 4: ž / d BŽ b sgh g Ž m a Ž m m (0cos u d x d y 4p 4 X s g b ( = < D F < ) c0qre D FF c c q< D F <, Ž 3 y where a Ž m ' N a Ž m rp f X g F s h is the strength of h X ygyg vertex: agcosuemnab q a k b with q and k X the momenta of two gluons; x' Ž p qk rm and 7 We thank A Kagan for pointing out this to us, which helps us to detect a sign error in our earlier calculation b y y' Žkqk X rm b; c 0, c and c are functions of x and y as given by: X X c0 s y x yq Ž yyž yyx Ž xqyyx r, X c s yy yyx, X X c s x y y Ž yyž yyx Ž xyyyqx r, Ž 4 with x X 'm Xrm ; and the h X h b yh mixing angle u is taken to be y548 as noted earlier Finally, in obtaining the normalization factor: 0, we have taken into account the one-loop QCD correction w0x to the semi-leptonic b c decay for consistency In previous one-loop calculations without QCD corrections, it was found D Ffy5 and Ff0 w3,4 x In our approach, we obtain D F sy486 and F s088 from Eqs Ž 3 and Ž However, D F is enhanced significantly by the matrix-element correc- Ž tion D F q,m The latter quantity develops an imaginary part as q passes the charm-pair threshold, and the magnitude of its real part also becomes maximal at this threshold From Eqs ŽŽ 3, 4 and Ž 5, Ž Ž one finds Re D F 4m,m c sy58 at ms5 GeV Ž Including the contribution by D F q,m with ms5 GeV, and using Eq Ž 3, we find BŽb sgh X s y4 X 56 = 0 with the cut m '( X Ž k q p F 35 GeV imposed in the CLEO measurement wx This branching ratio is consistent with CLEO s measure- X ment on the B h X branching ratio wx s Without the kinematic cut, we obtain BŽ b sgh X s0= 0 y3, which is much larger than 43=0 y4 calcuwx 4 We also obtain the spectrum lated previously d BŽ b sgh X rdm X as depicted in Fig The peak of the spectrum corresponds to m X f4 GeV It is interesting to note that the CLEO analysis wx indicates that, without the anomaly-induced contribu- X tion, the recoil-massž m X spectrum of B h Xs can not be well reproduced even if the four-quark operator contributions are normalized to fit the branching ratio of the process On the other hand, if b sg ) sgh X dominates the contributions to B h X X s,as shown here, the m X spectrum can be fitted better as shown in Fig of Ref wx It is also interesting to remark that although the four-quark operator contributions can not fit the branching ratio nor the spectrum, it does play a role in producing a small peak in the spectrum, which corresponds to the B h X K

127 ( ) X-G He, G-L LinrPhysics Letters B Fig The distribution of BŽ b sq g qh as a function of the recoil mass m X mode Specifically, the B h X K mode is accounted for by the b sh X type of decays discussed earlier Based on results obtained so far, one concludes that the Standard Model is not in conflict the experimental data on B h X X s It can produce not only the branching ratio for B h X Xs but also the recoil-mass spectrum when contributions from the anomaly mechanism and the four-quark operators are properly treated X Up to this point, a Ž m g of the h ygyg vertex has been treated as a constant independent of invariant-masses of the gluons, and m is set to be 5 GeV In practice, a Ž m g should behave like a form-factor which becomes suppressed as the gluons attached to it go farther off-shell w3,4,6 x However, it remains unclear how much the form-factor suppression might be It is possible that the branching ratio we just obtained gets reduced significantly by the form-factor effect in h X ygyg vertex Should a large formfactor suppression occur, the additional contribution X X from b h s and B h sq discussed earlier would become crucial We however like to stress that our estimate of b sgh X with as evaluated at ms5 GeV is conservative To illustrate this, let us compare branching ratios for b sgh X obtained at ms5 GeV and ms5 GeV respectively In NDR scheme 8, branching ratios at the above two scales with the cut m F35 GeV are 49=0 y4 X X and 8 In NDR scheme, apart from a different set of Wilson coeffi- 0 cients compared to Eq Ž 3, the constant term: 9 at the rhs of Eq Ž 5 is replaced by For details, see, for example Ref w x 3 9=0 y4 respectively One can clearly see the significant scale-dependence! With the enhancement resulting from lowering the renormalization scale, there seems to be some room for the form-factor suppression in the attempt of explaining B h X X s by b sgh X 9 It should be noted that the above scale-dependence is solely due to the coupling constant a Ž m s appearing in the h X ygyg vertex In fact, the b sg ) vertex is rather insensitive to the renormalization scale Indeed, from Eq Ž, we compute in the NDR scheme the scale-dependence of g Ž s D Fq Ž D F q We find that, as m decreases from 5 GeV to 5 GeV, the peak value of the above quantity increases by only 0% Therefore, to stabilize the scale-dependence, one should include corrections beyond those which simply renormalize the b sg ) vertex We shall leave this to a future investigation It is instructive to compare our results with those of Refs w3,4 x With the kinematic cut, our numerical result for BŽ b sgh X is only slightly smaller than y4 the branching ratio, 8=0, reported in Ref wx 3, X where the a Ž m s coupling of h ygyg vertex is evaluated at m f GeV, and D F receives only short-distance contributions from the Wilson coefficients C4 and C 6 Although we have a much smaller a s, which is evaluated at ms5 GeV, and the inter- ference of D F and F is destructive wx 4 rather than constructive wx 3, there exists a compensating enhancement in D F due to one-loop matrix elements The branching ratio in Ref wx 4 is y3 times smaller than ours since it is given by a D F smaller than ours but comparable to that of Ref wx 3 Concerning the relative importance of D F and F, we find that Ž X y4 D F alone gives B b sgh s65=0 with the kinematic cut m X F35 GeV Hence the inclusion of F lowers down the branching ratio by only 4% Such a small interference effect is quite distinct from results of Refs w3,4x where 0% 50% of inter- 9 Ž We do notice that Bb sgh is suppressed by more than one order of magnitude if a Ž m in Eq Ž 3 is replaced by a m P according to Ref 6 However, this prescrip- g m X h Ž X wx g h Ž X mh y q tion for a stems from the assumption that g ) gh X g form factor behaves in the same way as the QED-anomaly form factor g ) 0 gp It remains unclear as raised in Refs w3,4x that one could make such a connection between two distinct form factors X

128 8 ( ) X-G He, G-L LinrPhysics Letters B ference effects are found We attribute this to the enhancement of D F in our calculation Before closing we would like to comment on the branching ratio for B hx s It is interesting to note that the width of b hsg is suppressed by tan u compared to that of b h X sg Taking usy548, y5 we obtain B B hxs f4=0 The contribu- tion from the four-quark operator can be larger Depending on the choice of parameters, we find that y5 B B hxs is in the range of 6;0 =0 In conclusion, we have calculated the branching ratio of b sgh X by including the NLL correction to the b sg ) vertex By assuming a low-energy h X ygyg vertex, and cutting the recoil-mass m X at 35 GeV, we obtained BŽ b sgh X sž 5y9 = 0 y4 depending on the choice of the QCD renormalization-scale Although the form-factor suppression in the h X ygyg vertex is anticipated, it remains possible that the anomaly-induced process b sgh X could account for the CLEO measurement on BŽB X h X s For the four-quark operator contribution, X y4 we obtain B B h Xs f=0 This accounts for roughly 5% of the experimental central-value and can reach 30% if favourable parameters are used Finally, combining contributions from the anomaly-mechanism and the four-quark operators, the entire range of B h X Xs spectrum can be well reproduced Acknowledgements We thank W-S Hou, A Kagan and A Soni for discussions The work of XGH is supported by Australian Research Council and National Science Council of ROC under the grant numbers NSC 87-8-M and NSC 88--M The work of GLL is supported by National Science Council of ROC under the grant numbers NSC 87--M , NSC 88--M , and National Center for Theoretical Sciences of ROC under the topical program: PQCD, B and CP References wx CLEO Collaboration, BH Behrens et al, Phys Rev Lett 80 Ž wx CLEO Collaboration, TE Browder et al, Phys Rev Lett 8 Ž wx 3 D Atwood, A Soni, Phys Lett B 405 Ž wx 4 WS Hou, B Tseng, Phys Rev Lett 80 Ž wx 5 A Datta, X-G He, S Pakvasa, Phys Lett B 49 Ž wx 6 AL Kagan, A Petrov, hep-phr ; hep-phr wx 7 H Fritzsch, Phys Lett B 45 Ž ; X-G He, W-S Hou, CS Huang, Phys Lett B 49 Ž wx 8 H-Y Cheng, B Tseng, hep-phr ; A Ali, J Chay, C Greub, P Ko, Phys Lett B 44 Ž 998 6; N Deshpande, B Dutta, S Oh, Phys Rev D 57 Ž wx 9 A Ali, G Kramer, C-D Lu, hep-phr w0x I Halperin, A Zhitnitsky, Phys Rev Lett 80 Ž ; F Araki, M Musakonov, H Toki, hep-phr ; DS Du, Y-D Yang, G-H Zhu, hep-phr980545; M Ahmady, E Kou, A Sugamoto, Phys Rev D 58 Ž ; DS Du, CS Kim, Y-D Yang, Phys Lett B 46 Ž ; F Yuan, K-T Chao, Phys Rev D 56 Ž ; A Dighe, M Gronau, J Rosner, Phys Rev Lett 79 Ž wx G Buchalla, AJ Buras, ME Lautenbacher, Review of Modern Physics 68 Ž wx A Lenz, U Nierste and Ostermaier, Phys Rev D 56 Ž w3x A Buras, M Jamin, M Lautenbacher, P Weisz, Nucl Phys B 370 Ž w4x NG Deshpande, X-G He, Phys Lett B 336 Ž w5x TE Browder, A Datta, X-G He, S Pakvasa, Phys Rev D 57 Ž w6x T Feldmann, P Kroll, B Stech, e-print hep-phr w7x F Parodi, P Roudeau, A Stocchi, e-print hep-phr98089 w8x AI Vainshtein et al, JETP Lett Ž ; MA Shifman et al, Nucl Phys B 0 Ž ; MB Wise, E Witten, Phys Rev D 0 Ž w9x H Simma, D Wyler, Nucl Phys B 344 Ž w0x G Corbo, Nucl Phys B Ž ; N Cabibbo, G Corbo, L Maiani, ibid B 55 Ž wx Ali and Greub, Phys Rev D 57 Ž

129 3 May 999 Physics Letters B One-loop QCD interconnection effects in pair production of top quarks W Beenakker a,, FA Berends b, AP Chapovsky b, a Physics Department, UniÕersity of Durham, Durham DH 3LE, UK b Instituut Lorentz, UniÕersity of Leiden, PO Box 9506, 300 RA Leiden, The Netherlands Received 7 February 999; received in revised form 9 February 999 Editor: PV Landshoff Abstract We calculate the one-loop non-factorizable QCD corrections to the production and decay of pairs of top quarks at various collider experiments These non-factorizable corrections interconnect the different production and decay stages of the off-shell top-pair production processes This in particular affects the invariant-mass distributions of the off-shell top quarks, resulting in a shift of the maximum of the distorted Breit Wigner distributions Although the non-factorizable corrections can be large, the actual shift in the mass as determined from the peak position of the corrected Breit Wigner line-shape is below 00 MeV q 999 Published by Elsevier Science BV All rights reserved Introduction At present and future collider experiments, a detailed investigation of the production of top-quark pairs will substantially contribute to our knowledge of the top-quark properties and thereby of the Standard Model An improved measurement of the topquark mass m t, for instance, can serve to obtain improved indirect sensitivity to the mass of the Standard Model Higgs boson This is achieved by combining the high-precision measurements of the electroweak parameters at LEPrSLC with the direct measurements of the top-quark and W-boson masses Pairs of top quarks can be produced in hadron collisions at the Tevatron Ž pp and LHC Ž pp,as well as in e q e y and gg collisions at a future linear collider Since the top quark has a large width as compared to the QCD hadronization scale, G f t 4 GeV 4 LQCD f MeV, it predominantly decays before hadronization takes place Therefore the perturbative approach can be used for describing top quarks The main lowest-order Ž partonic mechanisms for the pair production of top quarks are q y q y e e,gg tt bw bw 6 fermions, Ž q y qq, gg tt bw bw 6 fermions Research supported by a PPARC Research Fellowship Research supported by the Stichting FOM A lot of effort has been put into an adequate theoretical description of these reactions Žsee eg Ref wx r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

130 30 ( ) W Beenakker et alrphysics Letters B for two review papers Most of these studies treat the top quarks as stable particles, which is a reasonable approximation since G rm s OŽ % t t For the reactions qq, gg tt these studies comprise QCD wx and electroweak wx 3 one-loop corrections, as well as the resummation of soft-gluon effects wx 4 Also for q y the reactions e e,gg tt both the QCD wx 5 and electroweak wx 6 one-loop corrections are known Moreover, the tt threshold, with its sizeable QCD wx 7 and Yukawa interactions wx 8, has been analysed in detail One would, however, like to treat the top quark as an unstable particle, with a Breit Wigner distribution describing its line shape The most economic approach for treating processes that involve the production and subsequent decay of unstable particles is the so-called leading-pole approximation Ž LPA wx 9 This approximation is based on an expansion of the complete amplitude around the poles of the unstable particles, which can be viewed as a prescription for performing an effective expansion in powers of GirM i Here Mi and Gi stand for the masses and widths of the various unstable particles The residues in the pole expansion are physically observable and therefore gauge-invariant The actual approximation consists in retaining only the terms with the highest degree of resonance In the case of top-quark pair production only the double-pole residues are hence considered and the LPA becomes a double-pole approximation Ž DPA This approximation will be valid sufficiently far above the tt-threshold If in reactions Ž and Ž also the W bosons are treated as unstable particles, then also for these particles the leading pole residues should be taken In this approach the complete set of corrections to reactions Ž and Ž naturally splits into two groups: factorizable and non-factorizable corrections The factorizable corrections are directly linked to the density matrices for on-shell production and decay of the unstable particles The non-factorizable corrections can be viewed as describing interactions that interconnect different Ž productionrdecay stages of the off-shell process A detailed discussion of this method with all its subw0 x, where the method tleties can be found in Ref has been applied to the complete set of OŽ a radiative corrections to the process e q e y W q W y 4 fermions For tt production partial results along this line exist w x, involving a subset of the factoriz- able corrections to the reaction q y e e tt q y bw bw 6 fermions However, the non-factorizable corrections are needed for a complete OŽ a s calculation In recent years the necessary methods for calculating such non-factorizable corrections have been de- veloped In Ref wx a first complete calculation was q y performed for tt production in e e collisions In Ref w3x the non-factorizable corrections were calculated for W-pair production, revealing differences with the results of Ref w x The results of Ref w3x were confirmed by an independent calculation w4x as well as by a re-analysis of the results of Ref w x In this paper we apply our calculations presented in Ref w3x to the non-factorizable OŽ a s correc- tions to tt production at various colliders We discuss the effect on the invariant-mass distribution of the off-shell top quark and the resulting shift in the maximum of the distorted Breit Wigner distribution Definition of the non-factorizable corrections In the LPA approach reactions like Ž and Ž, which involve unstable particles during intermediate stages, can be viewed as consisting of separate subprocesses, ie the production and decay of the unstable particles Having this picture in mind, the complete set of radiative corrections can be separated naturally into a sum of corrections to these subprocesses, called factorizable corrections, and those corrections that interconnect various subprocesses, called non-factorizable corrections It should be noted, however, that it is often misleading to identify the non-factorizable contributions on the basis of diagrams Such a definition is in general not gauge-invariant Rather one should realize that only realrvir- 3 tual semi-soft gluons with E s OŽ G g i will con- tribute, the contributions of the hard gluons being suppressed by GirE g This is a consequence of the fact that the various subprocesses are typically separated by a big space-time interval of OŽ rg i due to the propagation of the unstable particles The subpro- 3 These gluons will still be perturbative in our case as their typical energy Ž E ; G R4 GeV g t,w largely exceeds the QCD hadronization scale Ž L f MeV QCD

131 ( ) W Beenakker et alrphysics Letters B Fig The generic structure of the complete tt-production process q y qq tt bw bw 6 fermions in the LPA The open circles denote the various production and decay subprocesses As an example also the non-factorizable semi-soft gluon interaction between the two top-quark decay subprocesses is shown cesses can be interconnected only by the radiation of semi-soft gluons with energy of OŽ G i, which in- duce interactions that are sufficiently long range Hard gluons ŽE sož M 4G g i i as well as massive particles induce short-range interactions and therefore contribute exclusively to the factorizable corrections, which are governed by the relatively short time interval ; rmi on which the decay and production subprocesses occur A more detailed discussion of these issues can be found in Refs w0,5,6 x In Fig we show schematically the partonic q y process qq tt bw bw 6 fermions The process consists of five subprocesses, which we will q y denote by tt prod, t dec, t dec, W dec, and W dec In Fig these subprocesses are indicated by the open circles The non-factorizable semi-soft gluon interactions interconnect any two different subprocesses, as is exemplified in Fig for the two top-quark decay subprocesses The coupling of such a gluon to a certain subprocess can be written in terms of semisoft currents In contrast to soft-gluon currents, the effect of the gluon momentum on the unstable-particle propagators cannot be neglected in the semi-soft currents The various non-factorizable corrections to the cross-section are just given by all possible interferences of the semi-soft currents This will be made more explicit in the next section 3 Colour dependence of the non-factorizable corrections We start off by considering the simpler case of stable W bosons At the end of this section we will indicate what happens if the W bosons decay hadronically For stable W bosons one can identify three subprocesses: tt prod, t dec, t dec The non-factoriz- able corrections are given by the semi-soft gluon interferences between these different subprocesses As only semi-soft gluons contribute, the virtual and real matrix elements factorize in terms of lowestorder matrix elements and semi-soft currents In view of the possible presence of coloured particles in the initial state Ž qq, gg, this factorization depends on the colour structure For the reactions Ž, which involve only colourless initial-state particles, the tt pair is produced in a singlet state In contrast, the tt pair is produced in an octet state in the lowest-order annihilation process qq tt, which involves the time-like exchange of a gluon Both singlet and octet states are present in the lowest-order gluon-fusion reaction gg tt, since in that case also space-like top-quark-exchange diagrams contribute Because of these differences in the colour structure of the lowest-order reactions, also the non-factorizable corrections will come out differently, as we will see from the following discussion In order to keep the notation as general as possible, we write the lowest-order partonic reactions in the generic form QŽ q QŽ q tž p tž p q X y X Ž Ž b k W k b k W k, Ž 3 q y 4 where QQs ee,gg, qq, gg The corresponding lowest-order matrix element will be denoted by Ž cc M 0 ij, where i, j indicate the t,t colour indices in the fundamental representation The colour indices c,c belonging to Q,Q depend on the specific initial state: they are absent for the colourless e q e y and gg initial states, and they are in the fundamentalradjoint representation for the qqrgg initial states The momentum, Lorentz index, and colour index of the semi-soft gluon will be denoted by k, m, and a, respectively By using the relation / a a Ž T ijž T kls ž dildkjy dijdkl Ž 4 N

132 3 ( ) W Beenakker et alrphysics Letters B a for the SU N generators T in the fundamental representation Ž with Ns3 for QCD, the virtual and real non-factorizable corrections take the generic form: dg XX XX X X virt 0 ) cc cc ds s Ž M XX XX Ž M X X nf 0 i j 0 i j K s in ½ 5 d 4 k XX X X XX c c ;c c virt = Re ih Ž D XX X X XX nf i i ; jj, 4 Ž p wk qiox Ž 5 dg XX XX X X real 0 ) cc cc ds sy Ž M XX XX Ž M X X nf 0 i j 0 i j K s in ½ 5 d k XX X X XX c c ;c c real = Re H Ž D XX X X XX Ž 6 3 nf i i ; jj Ž p k0 Here the pre-factor consists of the lowest-order phase-space factor in the LPA w d G x 0, the partonic flux factor wrž s x, and the initial-state spin and colour average wrk x in The non-factorizable kernels can be expressed in terms of semi-soft currents according to Ž D c XX c X ;c X c XX virt XX X X XX nf i i ; jj cc cc½ i j j i tž tt tt [/ m m s d XX X d X XX d XX XX d X X J J yj yj ž / m m qjt JttqJtt qj] q Jt J m t,m ž / m qd XX X d X XX N J J qj qj i i j j t tt tt [ m ž / m qn J J yj yj t tt tt ] m m m m y J J qj J qj J N ž / t t,m t tt,m t tt,m 5 a c XX c X ;c X c XX a XX X m X XX in jj t [,m qž Q d T i i J J 4 a X XX m qd XX X i i T jj Jt J ],m, 7 Ž D c XX c X ;c X c XX real XX X X XX nf i i ; jj cc cc½ i j j i t ž tt tt 0/ m ) m s d XX X d X XX d XX XX d X X I I yi yi ž / ) m ) m qit IttqIttqI 0 q It I m t,m ž / ) m qd XX X d X XX N I I qi qi i i j j t tt tt 0 m ž / ) m qn I I yi yi t tt tt 0 m ) m ) m ) m y I I qi I qi I N ž / t t,m t tt,m t tt,m 5 a c XX c X ;c X c XX a XX X ) m X XX in jj t 0,m qž Q d T i ii I a X XX ) m qd XX X i i T jj It I 0,m 8 The terms proportional to d XX XX d X X i j j i project on the lowest-order singlet tt states, whereas the terms proportional to d XX X d X XX i i j j completely factorize the lowest-order cross-section The colour structure 0 for e q e y,gg a a XX X a X XX XX X Q ~ dc cž T c cqdc cž T c X c XX for qq in s a XX X a d X XX F qd XX X F X XX c c c c for gg cc cc 4 Ž 9 depends on the specific initial state and in general does not project on explicit lowest-order tt colour a states Here F are the SUŽ N generators in the adjoint representation, which are defined in terms of the SUŽ N structure constant according to ŽF a bc s yif abc Note that for e q e y and gg initial states the currents J m, J m and I m ] [ 0 completely drop out of Eqs Ž 5 and Ž 6, as it should be for colourless particles in the initial state The semi-soft currents appearing in the virtual non-factorizable corrections are given by m m p k D m Jt sygs y, m Jt sygs kp qio kk qio D qkp m m p k D y ykpqio ykkqio Dykp Ž 0

133 ( ) W Beenakker et alrphysics Letters B for gluon emission from the decay stages of the process, and m m p p m Jtt sgs q, kp qio ykp qio m m p p J m tt sgs y, kp qio ykp qio m m q q m J[ sygs y, kq qio kq qio m m q q J m [ sygs q, kq qio kq qio m m q q m J] sgs y, ykq qio ykq qio m m q q J m ] sgs q Ž ykq qio ykq qio for gluon emission from the production stage of the process Here gs is the QCD gauge coupling and D, sp, ym tqimtgt is a shorthand notation for the inverse top-quark propagators Note the difference in the sign of the io parts appearing in the currents J, J and J, J [ [ ] ] These infinitesimal imaginary parts are needed to ensure a proper incorporation of causality The corresponding semi-soft real-gluon currents read m m p k D m It sygs y, kp kk D qkp m m p k D m It sgs y and kp kk D qkp p p p p I sg y, I sg q, m m m m m m tt s tt s kp kp kp kp m m q q m I0 sygs y, kq kq m m q q I m 0 sygs q Ž 3 kq kq By simple power counting one can explicitly see from the above specified currents that the contributions of hard gluons are suppressed and that effectively only semi-soft gluons with E sk sož G g 0 t contribute In view of the pole structure of the virtual corrections, governed by the infinitesimal imaginary parts io, many of the non-factorizable corrections will vanish when virtual and real-gluon corrections are added up For instance, all initial final state interferences will vanish, leaving behind a very limw0,6 x The ited subset of final-state interferences following holds for the remaining interferences: ) m ) m It Itt,m yit I tt,m, ) m ) m It Itt,m It I tt,m, with similar effective replacements for J tt As a result of these properties of the non-factorizable corrections, a factorization per colour structure emerges: N y dsnf s dnf dsborn,y ds Born,8, Ž 4 N N ½ d 4 k dnf sre ih 4 Ž p wk qiox = Jt m Jt,m qjt m Jtt,m qjt m Jtt,m dk yh 3 Ž p k0 = I ) m I qi ) m I qi ) m I t t,m t tt,m t tt,m 5 Ž 5 Here dsborn, and dsborn,8 are the lowest-order multi-differential cross-sections for producing the intermediate tt pair in a singlet and octet state, respectively For completeness we note that q y e e,gg q y e e,gg qq qq Born Born, Born Born,8 ds sds, ds sds, dsborn gg sdsborn, gg qdsborn,8 gg Ž 6 The non-factorizable factor dnf can be obtained from Ref w3 x The results of Section 4 of that paper should be used, since those allow for massive decay

134 34 ( ) W Beenakker et alrphysics Letters B products from the unstable particles, which is the case for the top-quark decay We conclude by considering the case that also the W bosons are unstable This adds two decay subprocesses, W q and W y, to the three we have considdec dec ered so far If the W bosons decay leptonically, nothing changes as the gluon cannot couple to the W decay subprocesses in that case For a hadronically decaying W boson additional interferences have to be taken into account However, such interferences trivially vanish as a result of the singlet nature of the w Ž a x W-boson decays ie Tr T s 0 4 Numerical results With the help of Eqs Ž 4 Ž 6 we can now in principle evaluate all kinds of multi-differential distributions, with and without non-factorizable corrections Although the factorized structure of the nonfactorizable corrections is very transparent in Eq Ž 4, integration of the multi-differential cross-sections will affect this structure For instance, in Eq Ž 4 the correction to the singlet cross-section differs by a factor y8 with respect to the octet one However, for the calculation of the relative non-factorizable corrections to a one-dimensional distribution, one has to evaluate the ratio of the integrated Eqs Ž 4 and Ž 6 Since dnf depends on the integration variables, the thus-obtained singlet and octet correction factors will not necessarily differ by the factor y8 At this point we stress that any observable that is inclusive in both top-quark invariant masses, such as the total cross-section, will not receive any nonfactorizable corrections This is a typical feature of these interconnection effects w6 x As an example of a distribution that is subject to non-vanishing nonfactorizable corrections we focus on the invariantmass distribution of the top quark, which can be used for the mass determination To this end we determine the non-factorizable correction d Ž M nf for the distri- bution ds ds Born s qdnf Ž M, Ž 7 d M d M where M is the invariant mass of the b-quark and the W q boson The maximum of the Breit Wigner distribution can be used to determine the top-quark mass The linearized shift of this maximum as induced by the non-factorizable corrections is given by ddnf Ž M D Ms 8 G t Ž 8 d M Ms m t The correction d Ž M nf is calculated for the four different mechanisms of tt production, ie initiated q y by e e, gg, qq and gg For the centre-of-mass energies of these Ž partonic reactions we take ' s s 355 GeV and 500 GeV These values exemplify the non-factorizable corrections in the vicinity of the threshold and far above it As mentioned before, the adopted approximation in our calculation Ž LPA forces us to stay sufficiently far above the tt threshold Ž read: a few times G t The numerical values for the input parameters are m s738 GeV, M s806 GeV, t W M s 987 GeV, Ž 9 Z and Gts390 GeV, Ž 0 the latter being the OŽ a S corrected top-quark width The correction dnf is proportional to a S, for which we have to choose the relevant scale For ' s s 355 GeV the main contribution originates from the non-factorizable Coulomb effect present in d nf Its typical momentum is determined by the top-quark width Gt and velocity b: Gtrb; 68 GeV At Fig The relative non-factorizable correction d Ž M nf to the single invariant-mass distribution ds rdm Centre-of-mass energy: ' s s355gev

135 ( ) W Beenakker et alrphysics Letters B GeV softer gluons contribute and therefore the typical gluon momentum is G t;4 GeV Therefore we choose ' a 4 GeV f for s s 500 GeV, S ' a 68 GeV f 0955 for s s 355 GeV, S Ž corresponding to a Ž M S Z s080 at the Z peak It should be noted that choosing another scale in a S will only affect the normalization of the correction In Fig the non-factorizable correction dnf is plotted as a function of the invariant mass M at the centre-of-mass energy of 355 GeV The dnf values for the pure singlet e q e y initial state and the pure octet qq initial state differ approximately by the afore-mentioned factor of y8 For the gg initial state the Born octet part is larger than the singlet one, resulting in a non-factorizable correction that q y lies between the e e and the qq case The correction for the gg initial state is virtually indistinguishable from the e q e y one and is therefore not displayed Evidently the distortion effects from the singlet corrections are very large, which is due to a large non-factorizable Coulomb correction inside d nf The maximum of the Breit Wigner distribution is hardly affected by this large correction One finds for q y the various initial states e e gg, gg and qq D Mfy85, y5 and q0mev respectively The situation at 500 GeV is depicted in Fig 3 The overall correction is small, which is typical for nonfactorizable corrections further away from threshold The shift in the maximum of the Breit Wigner distribution is of the order of 5MeV for the e q e y and gg initial states, and even smaller for the qq and gg initial states In order to obtain hadronic distributions from the partonic ones, the results for the qq and gg initial states should of course be properly folded with the parton densities of the colliding hadrons Ž pp at the Tevatron, pp at the LHC The bulk of the partonic contributions originates from the energy region not far above the tt threshold Ž s Q 8 m, ie ' t s Q 500 GeV, which is exemplified by the partonic energies 355 and 500 GeV used in our analysis 5 Conclusions In this paper we have summarized the gauge-invariant description for calculating the OŽ a S nonfactorizable QCD corrections to pair production of top quarks The formalism is presented in a general way, making it applicable to all relevant initial states The resulting final formula for the non-factorizable corrections involves the same quantity dnf for all reactions This quantity can be numerically calculated using expressions available in the literature Although the formalism can be used for numerical studies of many distributions, the focus of our numerical evaluation has been on the invariant-mass distribution of the top quark, which can be used for extracting the top-quark mass In spite of the possible sizeable deformations of this line-shape distribution, its maximum is shifted by less then 00MeV Therefore, if the top-quark mass is extracted experimentally from the peak position of the line-shape, the non-factorizable corrections can be safely neglected If the precise shape of the Breit Wigner distribution is used in the experimental analysis, the non-factorizable corrections should be taken into account properly In particular if the singlet colour state dominates In addition higher-order nonfactorizable corrections might be needed Fig 3 The relative non-factorizable correction d Ž M nf to the single invariant-mass distribution ds rdm Centre-of-mass energy: ' s s500gev Acknowledgements Discussions with VA Khoze are gratefully acknowledged

136 36 ( ) W Beenakker et alrphysics Letters B References wx JH Kuhn, Lectures presented at the XXIII SLAC Summer Institute on Particle Physics, The Top Quark and the Electroweak Interaction, July 0, 995, SLAC, Stanford, USA, hep-phr97073; E Accomando et al, Phys Rept 99 Ž 998 wx P Nason, S Dawson, RK Ellis, Nucl Phys B 303 Ž ; W Beenakker, H Kuijf, WL van Neerven, J Smith, Phys Rev D 40 Ž ; W Beenakker, WL van Neerven, R Meng, GA Schuler, J Smith, Nucl Phys B 35 Ž wx 3 W Beenakker, A Denner, W Hollik, R Mertig, T Sack, D Wackeroth, Nucl Phys B 4 Ž wx 4 E Laenen, J Smith, WL van Neerven, Nucl Phys B 369 Ž ; E Berger, H Contopanagos, Phys Lett B 36 Ž 995 5; Phys Rev D 54 Ž ; S Catani, M Mangano, P Nason, L Trentadue, Phys Lett B 378 Ž ; Nucl Phys B 478 Ž wx 5 J Jersak, E Laerman, PM Zerwas, Phys Rev D 5 Ž 98 8; B Kamal, Z Merebashvili, AP Contogouris, Phys Rev D 5 Ž wx 6 W Beenakker, SC van der Marck, W Hollik, Nucl Phys B 365 Ž 99 4; AA Akhundov, DY Bardin, A Leike, Phys Lett B 6 Ž 99 3; A Denner, S Dittmaier, M Strobel, Phys Rev D 53 Ž wx 7 VS Fadin, VA Khoze, JETP Lett 46 Ž ; Sov J Nucl Phys 48 Ž ; W Kwong, Phys Rev D 43 Ž ; M Jezabek, JH Kuhn, T Teubner, Z Phys C 56 Ž ; Y Sumino, K Fujii, K Hagiwara, M Murayama, CK Ng, Phys Rev D 47 Ž ; M Peter, Y Sumino, Phys Rev D 57 Ž ; AH Hoang, JH Kuhn, T Teubner, Nucl Phys B 45 Ž ; AH Hoang, Phys Rev D 56 Ž ; A Czarnecki, K Melnikov, Phys Rev Lett 80 Ž ; AH Hoang, T Teubner, Phys Rev D 58 Ž ; K Melnikov, A Yelkhovsky, Nucl Phys B 58 Ž ; O Yakovlev, hep-phr wx 8 H Inazawa, T Morii, Phys Lett B 03 Ž ; K Hagiwara, K Kato, AD Martin, CK Ng, Nucl Phys B 344 Ž 990 ; J Feigenbaum, Phys Rev D 43 Ž 99 64; MJ Strassler, ME Peskin, Phys Rev D 43 Ž ; W Beenakker, W Hollik, Phys Lett B 69 Ž 99 45; VS Fadin, O Yakovlev, Sov J Nucl Phys 53 Ž wx 9 RG Stuart, Phys Lett B 6 Ž 99 3; A Aeppli, GJ van Oldenborgh, D Wyler, Nucl Phys B 48 Ž w0x W Beenakker, AP Chapovsky, FA Berends, hepphr9848 wx CR Schmidt, Phys Rev D 54 Ž wx K Melnikov, O Yakovlev, Nucl Phys B 47 Ž w3x W Beenakker, AP Chapovsky, FA Berends, Phys Lett B 4 Ž ; Nucl Phys B 508 Ž w4x A Denner, S Dittmaier, M Roth, Nucl Phys B 59 Ž ; Phys Lett B 49 Ž w5x Yu L Dokshitzer, VA Khoze, LH Orr, WJ Stirling, Nucl Phys B 403 Ž w6x K Melnikov, O Yakovlev, Phys Lett B 34 Ž 994 7; VS Fadin, VA Khoze, AD Martin, Phys Rev D 49 Ž

137 3 May 999 Physics Letters B Top quark production near threshold and the top quark mass M Beneke a, A Signer b, VA Smirnov c a Theory DiÕision, CERN, CH- GeneÕa 3, Switzerland b Department of Physics, UniÕersity of Durham, Durham DH 3LE, UK c Nuclear Physics Institute, Moscow State UniÕersity, 9889 Moscow, Russia Received 9 March 999 Editor: R Gatto Abstract We consider top-anti-top production near threshold in e q e y collisions, resumming Coulomb-enhanced corrections at next-to-next-to-leading order Ž NNLO We also sum potentially large logarithms of the small top quark velocity at the next-to-leading logarithmic level using the renormalization group The NNLO correction to the cross section is large, and it leads to a significant modification of the peak position and normalization We demonstrate that an accurate top quark mass determination is feasible if one abandons the conventional pole mass scheme and if one uses a subtracted potential and the corresponding mass definition Significant uncertainties in the normalization of the tt cross section, however, remain q 999 Published by Elsevier Science BV All rights reserved Introduction The top quark mass is now known to be around 75 GeV with an accuracy of 5 GeV from the direct measurement at the Fermilab Tevatron Collider wx Accurate mass determinations Žwith errors below GeV are difficult at hadron colliders Despite the fact that orders of magnitudes more top quarks will be produced at the CERN Large Hadron Collider, a precision measurement is reserved for a future lepton collider In this case the method of choice relies on scanning the top quark pair production threshold From an experimental point of view, an error in the 00 MeV range is conceivable wx ; the limiting factor is the accuracy to which the cross section can be predicted theoretically as a function of the top quark mass The literature on top quark physics near threshold q y in e e collisions is substantial w3,4 x Perturbative calculations in the threshold region require that either the toponium Rydberg energy scale mtas 4 L or that the top quark decay width G 4L QCD t QCD Both conditions are satisfied and since G t;mtas ; 5 GeV, narrow toponium resonances do not exist wx 5 In the kinematic region of interest, the top quarks are slow, with typical velocities Õ; r0 As a consequence methods familiar from non-relativistic bound state calculations in QED can be used to compute the tt cross section In particular, the dominant interaction between the t and t can be described by the colour-singlet Coulomb potential, which has to be treated non-perturbatively wx 3 Corrections are computed in the background of this strong Coulomb interaction Recently, the -loop QCD correction to the Coulomb potential wxž 6 correcting an earlier result wx 7 and the -loop relativistic correction to the tt vector coupling to the initial virtual photon or r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

138 38 ( ) M Beneke et alrphysics Letters B Z boson w8,9x have been computed With these two inputs at hand, we can compute the next-to-next-toleading order Ž NNLO QCD correction to the top quark pair production cross section ŽThe precise meaning of NNLO in the present context is given below In the following, we first compute the NNLO QCD correction in the conventional pole mass scheme, ie the threshold cross section is expressed in terms of the top quark pole mass Comparable calculations, with some technical and implementational differences, have already been completed by several groups w0 x We find that the NNLO correction is substantial and leads to an uncomfortably large shift in the top quark mass, raising questions as to the stability of the theoretical prediction In w3 x, one of us suggested that such shifts could occur as a consequence of on-shell mass renormalization, which is particularly Žand in the case at hand, artificially sensitive to non-perturbative, long-distance effects This sensitivity to non-perturbative effects can be removed by using a different mass renormalization scheme Žcalled the potential subtraction Ž PS scheme in w3x with the additional benefit that the new mass definition can also be related more reliably to short-distance masses Žsuch as the MS mass, which are ultimately of more interest for high-energy processes and Yukawa coupling unification relations Ž should they exist The main result of this letter is to demonstrate that this procedure works We show that the mass shifts become small in the PS scheme, and that the PS mass can also be accurately related to the MS mass However, the normalization uncertainty of the cross section remains large, and is larger than could have been anticipated from the earlier next-to-leading order Ž NLO calculations The second improvement which we suggest in this letter is to use renormalization group equations to sum large logarithms, lnõ and lnõ to all orders in perturbation theory The consideration of logarithms is important to identify correctly the scales of the various subprocesses, but not restricted to the scale of the QCD coupling only In the present work, we complete this program Žal- most trivially to the next-to-leading logarithmic Ž NLL order The step to NNLL is substantially more complicated and we will discuss it, together with the details of the present calculation, in a future publicaw4 x tion Before proceeding, let us emphasize that NLO, NNLO etc do not refer to a conventional loop expansion in a s, because the Coulomb interaction cannot be treated perturbatively in the threshold region To be precise, a LO, NLO, approximation to the normalized cross section takes into account all terms of order R's rs q y tt m m a k s Ý ž Õ / k sõ P Ž LO ; a,õ Ž NLO ; a,a Õ,Õ Ž NNLO ;, Ž s s s where logarithms of Õ are suppressed The renormalization group improved treatment extends this to the summation of logarithms such that a LL, NLL, approximation accounts for all terms of order a k ž s / k l Ý Ý s l RsÕ Ž a lnõ P Ž LL ; Õ a,õ Ž NLL ; a,a Õ,Õ Ž NNLL ; s s s The result discussed here is complete at NNLO and NLL Furthermore, near the would-be toponium poles another resummation is necessary, which we discuss below We also emphasize that the NNLO QCD correction is defined as above in the limit G <m a t t s When G t;mtas, as we expect, further corrections arise from so-called non-factorizable contributions w5 x, which have not been calculated so far with the Coulomb interaction treated non-perturbatively Electroweak corrections also enter and the entire concept of a tt cross section has to be revised, since single resonant contributions are expected to contribute at NNLO Žin the above power counting scheme with G ;m a t t s to the WWbb final state These are inter- esting issues to be studied, but we expect them to bear little on the issue of an accurate top mass determination which we address in this letter Hence, we will keep the top quark width only in the form of 4 4

139 ( ) M Beneke et alrphysics Letters B an imaginary mass term igtc c for the non-relati- vistic top quarks; this amounts to evaluating the tt Green function at energy EqiGt as familiar from LO and NLO calculations w3,4 x As a final Ž trivial simplification, we restrict our attention to tt pairs produced through the vector coupling to a virtual photon The vector coupling to a Z boson can be accommodated by a trivial replacement of the electric charge The axial vector coupling is suppressed by a factor of Õ near threshold; no QCD corrections to it are needed at NNLO Derivation of the cross section With the treatment of the top quark width as specified above, and neglecting the axial-vector coupling, the tt production cross section is obtained from the correlation function P q s q q yq g P q mn m n mn H m n 4 iqp si d xe x ² 0< T j Ž x j Ž 0 < 0 :, Ž 3 w x current and s s q the centre-of-mass energy squared Defining the usual R-ratio Rss rs Ž tt 0 s0 s 4pa rž 3s em, where aem is the electromagnetic coupling at the scale m, the relation is m m where j x s tg t x is the top quark vector t 4p et ii Rs Im P Ž sqie, Ž 4 s where etsr3 is the top quark electric charge in units of the positron charge Only the spatial components of the currents contribute up to NNLO In the following, mt refers to the top quark pole mass According to Ž, at NNLO, we have to extract, to all orders in a s, the first three terms of the expan- sion of any Feynman diagram in Õ s ŽŽ' s y m rm r The expansion in Õ is constructed w6x t t by dividing the loop integral into terms related to hard Žl 0 ;m t, l;m t referring to a frame where Ž qs0, soft l 0 ;mtõ, l;mtõ, potential l 0 ;mtõ, Ž l;m Õ and ultrasoft l ;m Õ, l;m Õ t 0 t t momen- tum This split-up requires a regularization to deal with divergent integrals that appear in intermediate expressions and we use dimensional regularization with MS subtractions This procedure has already been used to obtain the cross section at threshold up to order a By expanding the all-order result pres sented below up to order as, we reproduced the result of wx 9, which has been used as a common input to the previous w0 x NNLO tt cross section calculations We begin with integrating out the hard modes, commonly termed relativistic corrections The ef- ) fective g tt coupling seen by the non-relativistic quarks Ždescribed by two-spinor fields c for t and x for t after integrating out the hard modes is given by c i i i tg tsc csxy cs Ž id xq, Ž 5 6m t where the ellipsis refers to terms not needed for P Žq and at NNLO At NNLO, we can use c s, while c is needed at order as The expression for c is asž mh asž m Ž cž m sq Ž c qd yd asž m 4p as Ž qc q Ž 6 Ž 4p where d, related to the -loop anomalous dimension of the non-relativistic current c s i x, is given by d sy560p rž 7b Ž b s y n r3, n s f f and c at the scale m h, at which QCD is matched onto a non-relativistic effective theory, is given by ½ 5 8a Ž m 35p m c m sy q b q ln 3p 3 7 m h s h t Ž h p 4p 5z Ž 3 y y y lny asž mh q n f q Ž 7 7 p The first order result is well known w7 x; the second order contribution is from w8,9 x Eq Ž 6 follows from solving the renormalization group equation for c

140 40 ( ) M Beneke et alrphysics Letters B with the -loop anomalous dimension When evaluated at a scale m of order mõ, this expression sums next-to-leading logarithms of the form a s aslnõ l to all orders This is in fact the only source of next-toleading logarithms in the problem Žthere are no leading logarithms, and Ž 6 is sufficient to obtain a NLL approximation There is an ambiguity in the scale of a in Ž s 6 In our numerics, we actually choose the expression for the NNLL-improved coefficient function, setting the Ž unknown 3-loop anomalous dimension of the current to zero The correlation function Ž 3 is now expressed in terms of the effective current ŽŽ 5 This leaves out a hard correction from the region x; rmt in the integral Ž 3 However, in this region the top quarks are far off-shell and no contribution to the imaginary part of P is obtained The correlation functions of the non-relativistic current are then computed with the non-relativistic effective Lagrangian It is a straightforward matter of counting powers of velocity Ž using the momentum scaling rules given above to show that the following terms in the effective Lagrangian are sufficient at NNLO: ž / t D 0 4 LNRQCD sc id q qigt cq c D c 3 m 8m d gs d g s i i y c spbcq c wd, E xc m 8m t d3 igs ij i j q cs wd, E xc 8m t q antiquark terms ql light Ž 8 Because we use dimensional regularization, some care is needed to define the algebra of Pauli matrices in 3 y e dimensions as well as anti-symmetric ij w i products We use s ' s,s j xrž i Žequal to ijk k e s in three dimensions and s P B must be interpreted as ys ij G ij r in terms of the gluon field strength tensor The last term in Ž 8, L light, denotes the QCD Lagrangian of the massless fields, ie the QCD Lagrangian with the top quark part omitted The coefficient functions d, d, d3 can be set to at NNLO Their leading logarithmic renormalization t t would be required for a NNLL approximation In that case, further operators, notably four-fermion operators Ž of heavy-heavy and heavy-light type would have to be added to the effective Lagrangian We discuss this extension in w4 x The unconventional term involving the top quark width accounts for the fast top quark decay as discussed in the introduction ŽWe should emphasize again that the Lagrangian is not complete to NNLO as far as the treatment of the width is concerned The term we added is the leading order term, but further terms exist which are suppressed by two powers of velocity The loop integrals constructed with the non-relativistic Lagrangian still contain soft, potential and ultrasoft modes Near threshold, where energies are of order mtõ, only potential top quarks and ultrasoft gluons Ž light quarks can appear as external lines of a physical scattering amplitude Hence, we integrate out soft gluons and quarks and potential gluons Žlight quarks and construct the effective Lagrangian for the potential top quarks and ultrasoft gluons Žlight quarks Because the modes that are integrated out have large energy but not large momentum compared to the modes we keep, the resulting Lagrangian contains instantaneous, but spatially non-local interactions In the simplest case, these reduce to what is commonly called the heavy quark potential We refer to this theory as potential non-relativistic QCD Ž PNRQCD, adapting the term PNRQED introw8x to the QCD case duced in The derivation of the potentials in the framework of the threshold expansion w6x will be presented elsewhere w4 x The following result has been obtained by matching the on-shell tt scattering amplitude in NRQCD onto its PNRQCD counterpart We verified that the potential is gauge-independent for a general covariant gauge and the Coulomb gauge ŽTo obtain a gauge invariant result one has to combine the contribution from the soft modes with the one from potential gluons The resulting momentum space potential, after carrying out a colour and spin projection on the components relevant to the calculation of P,is 4p C F a s V Ž p,q sy qdv Ž p,q, Ž 9 q

141 4p CFas q as d V Ž p,q sy ayb0ln q ž m / 4p ž ( ) M Beneke et alrphysics Letters B q q ay Ž abqb 0 ln m q as 0 qb ln m pa < q< / Ž 4p yg q e E ž m / s q 4mt ye G Ž rye G Ž rqe = 3r p G Ž ye ž / = C F y Ž ye qc Ž ye A p q d y7dq0 q q d m m ½ 4Ž dy t t y Ž qd, Ž where ds4ye, CFs4r3, CAs3, dsds in the present NNLOrNLL approximation, and b s0y38n fr3 the two-loop coefficient of the QCD b-function ŽAlways a s a Ž m s s The loop correc- tions to the Coulomb potential are a sž 3CAr9y 0 n r9 w9x and w6x f 4343 z Ž 3 p 4 a sca q q4p y yc n A f 899 8z Ž 3 q n f ycfnf y8z Ž 3 q Ž 6 8 The potential Ž 0 differs from the potential used in w0 x Firstly, we need the potential in d spacetime dimensions, because the terms in the last two lines generate ultraviolet divergent integrals, which we regularize dimensionally The divergences cause a factorization scale dependence which cancels with the factorization scale dependence in the coefficient function Ž 6 of the non-relativistic current ŽThe Coulomb potential does not generate divergences and we can use a in four dimensions, Secondly, our potential contains a C a rž m < q< F s t term, which is absent in w0 x Nevertheless, both potentials Žin four dimensions are in fact equivalent w4 x Having integrated out soft modes and potential gluons, the correlation functions of non-relativistic currents are computed with the Lagrangian L sc ie q qig c PNRQCD 0 t ž m t / 0 t ž m t / qx ie y qig x C a dy F s q d r cc Ž r y r H ž / = xx Ž q c cy x x 3 3 8m 8m H t dy q d r cc r d V r t = xx Ž 0 Ž The terms in the last line are treated as perturbations However, velocity power counting reproduces the well known fact that the leading order Coulomb potential in the second line is not suppressed compared to the free non-relativistic Lagrangian in the first line Consequently perturbation theory in PN- RQCD implies that instead of freely propagating, a tt pair propagates according to the Coulomb Green function, which satisfies p ž / m t ye G p, p ;E X c ž / d dy k y4p C F a s X q G pyk, p ;E dy Ž p k H c dy Ž dy X s p d pyp 3

142 4 with EsEqiGt and Es s ym t Because we use dimensional regularization, all quantities are defined a priori in momentum space; the above equation defines the Coulomb Green function in d dimensions The PNRQCD Lagrangian Ž does not contain any gluon fields any more This is so, because at NNLO the top quarks interact only through potentials Counting powers of velocity for the leading ultrasoft interactions, we find that they are of NNNLO and higher order, ie beyond the accuracy of the present calculation To complete the calculation, we compute with PNRQCD the correlation functions of the non-relativistic currents For the power-suppressed term in Ž 5 the LO Lagrangian suffices For the current x s i c we need the kinetic energy correction to first order, the potentials suppressed by one power of a s or Õ relative to the Coulomb potential to second order, and the potential suppressed by two powers of as or Õ to first order The explicit result for the cross section is lengthy and will be given in w4 x We have checked this result by expanding it to order as, confirming the result of wx 9 in this limit This gives us confidence that the factorization in dimensional regularization has been done correctly The Coulomb Green function contains bound state poles at En LO s ym Ž C a rž4n for positive integer n The locat F s tion and residues of the bound state poles are modified by QCD corrections We computed the bound state energies to order as 4 and residues to order as 5 and find agreement with the results of w0x and w x, respectively The calculation described so far sums correctly all terms at NNLO and NLL, as defined in Ž and Ž However, the result contains terms of the schematic form E LO n a E LO k ' s n 4 y Ž EqiG t ( ) M Beneke et alrphysics Letters B which become large in the vicinity of EsEn LO,if Gt is not much larger than En LO For top quarks G t;e LO and it is necessary to sum singular terms of the form Ž 4 to all orders This is easily done by adding the expression with the exact bound state energy denominator at NNLO and by subtracting the same expression but expanded and truncated at NNLO That is, we add: F LO qf a qf a n Ž s s LO En qeas qe as ye F LO F LO E LO e F LO n n n n f y~ qas y q LO LO LO n En ye n qa E ye E ye LO LO LO LO Fn En e Fn En eqef s y 3 LO LO Ž En ye Ž En ye LO F f n ß q Ž 5 LO E ye n This procedure Žalso discussed in w,x is essentially equivalent Ž near the bound state poles to solving the Schrodinger equation exactly with the potential d VŽ r, rather than treating it perturbatively as we have done so far In practice, we implement this resummation for the first two bound state poles The correction for the remaining ones is tiny, because the residues decrease as rn 3 It is worth noting that even after this resummation, one would not be able to compute the tt cross section in the threshold region, if the width of the top quark were smaller than about two times the error that remains in the location of the S bound state pole at NNLO In the conventional pole mass scheme this requires Gt to be larger than roughly GeV, a constraint which is satisfied but not by a large margin 3 Top quark mass definitions, the PS scheme The top quark cross section at LO, NLO and NNLO Žincluding the summation of logarithms at NLL is shown in Fig a ŽTo be precise, the NLO curves include the second iterations of the NLO potentials The NNLO correction is seen to modify the line shape at the level of 0% It also shifts the position of the peak by approximately 600 MeV This conclusion is in qualitative agreement with the results of w0 x We should note, however, that our result is implemented in a different way: for instance, we do not keep the short-distance coefficient c m as an overall factor, but multiply it out, keeping all terms to NNLO Furthermore, we prefer to choose a different renormalization scale Žequiv-

143 ( ) M Beneke et alrphysics Letters B Fig Žw a upper panel x: The normalized tt cross section Žvirtual photon contribution only in LO Ž short-dashed, NLO Žshort- long-dashed and NNLO Ž solid as function of Es' s ym t Ž pole mass scheme Parameters: mts mhs75 GeV, Gts 40 GeV, a Ž m s Z s08 The three curves for each case refer to ms5ž upper ;30Ž central ;60Ž lower 4GeV Žw b lower panel x: As in Ž a, but in the PS mass scheme with mf s0gev Hence Es' s ym Ž 0GeV t,ps Other parameters as above with mt m Ž 0GeV t,ps alent to the soft scale in w0 x, typically of order 30 GeV, compared to the central value 75 GeV chosen in previous works This choice is motivated by the fact that the typical momentum transfer in the instantaneous interactions is of this order, or, if anything, smaller The significant shift in the location of the peak impacts directly on the top quark mass measurement There is no unique choice of the concept of the top quark mass The top quark pole mass definition has been universally assumed in previous cross section calculations near the threshold; this is indeed an intuitively plausible choice as top quarks do not hadronize However, the top quark pole mass definition is known to be more sensitive to non-perturbaw3x than other mass definitions and tive effects the finite width of the top makes no difference in this respect ŽThe difference is, that the top quark pole mass is irrelevant, because a top quark is always off-shell by an amount ( m t G t In w3x we argued that the shift of the peak position is related to large perturbative corrections which appear only, when the cross section is expressed in terms of the pole mass, and which have their origin in the sensitivity to distances larger than the toponium Bohr radius The point is that the Coulomb potential in coordinate space and the pole mass receive the same large corrections w3,4 x We therefore perform a subtraction on the potential such that the subtracted terms are absorbed into a mass redefinition and at the same time cancel the large corrections to the pole mass This leads to the potential subtraction Ž PS scheme and the corresponding mass definition For further details of the argument we refer to w3 x The PS mass at the subtraction scale mf is defined by m Ž m sm yd m Ž m, Ž 6 t,ps f t t f where 3 d q H d m Ž m sy V Ž q < q < -m f Ž p t f 3 Coulomb ž ž // as m f sdž mf q ayb0 ln y 4p m a s q a Ž 4p ž m f 0 ž / mf mf ž m m // y abqb 4 ln y m qb0 ln y4ln q8, 7 and DŽ m f scfasmfrp In the following, we re- express the tt cross section in terms of the PS mass We suppose that mf scales as mtas and count D mf as order mtõ Then all terms are re-expanded and terms beyond NNLO are dropped Žmodulo the resummation near the bound state poles discussed above Note that DŽ m f is not expanded when it occurs in the combination ' s y Žm Ž m t,ps f q DŽ m f It should also not dominate this combination and this is why we do not use the MS mass directly, which would lead to D;mta s

144 44 ( ) M Beneke et alrphysics Letters B The peak in the tt cross section profile is, roughly speaking, the remnant of the first bound state pole Ž the S pole To understand the effect of the mass redefinition on the cross section qualitatively, it is useful to compute the correction to the S pole in terms of both mass definitions The dominant correction in the pole mass scheme is related to the running coupling in the LO Coulomb potential So let us take ž / y4p CFas b0 as q d Vk Ž q s y ln Ž 8 q 4p m to compute the energy level shift d 3 q d 3 p ) dek sh C Ž pqqr 3 3 S Ž p Ž p = d V Ž q C Ž pyqr k S ž / 3 d q q sh q d V kž q 3 Ž p Ž mc t Fas Ž 9 The mass of the S bound state, including only the leading order Coulomb interaction and the above perturbation, is given by M sm qe LO qde S t k smt,psž mf qe LO,PS Ž mf y q dekqd mtž m f Ž 0 Comparing Ž 9 with Ž 7, we see that the integrands in de qd m Ž m cancel each other for < q< k t f - mc a and < q< t F s -m f At the same time the integral Ž 9 is dominated by the contribution from small q quickly as k increases The result of this exercise for ms30 GeV Žin which case mc a Ž m t F s is also about 30 GeV and mfs0 GeV is shown in Table This shows that the prediction for the mass of the S state is stable in terms of the PS mass, and we expect a qualitatively similar conclusion for the top quark cross section ŽThe same observation can also be used to determine the bottom quark MS mass from the F resonances accurately w5 x To demonstrate that the PS mass can also be related more accurately MS Ž MS to the MS mass m sm m, we assume m s t t t k Table k de rmev de qd m rmev k k t y489 q97 y4 q08 3 y09 q35 4 y78 q0 5 y67 q03 6 y y y GeV and obtain, adding loop corrections consecutively, mts 650q76q6q06 Ž est GeV Ž mt,psž 0 GeV s 650q67qq03 Ž est GeV The -loop correction follows from w6x and Ž 7 and the 3-loop correction is based on an estimate in the so-called large-b -limit w7 x 0 Hence, the present uncertainty in the relation of the PS mass to the MS mass is about 300 MeV 4 Discussion The tt cross section expressed in terms of m Ž m t,ps f with mfs0 GeV is shown in the lower panel of Fig Since the horizontal scale E is now defined as ' s y m Ž 0 GeV t,ps, we observe an overall, but trivial shift, related to the fact that m ym Ž 0 GeV t t,ps s75 GeV for ms30 GeV The important change is that in the PS scheme the location of the peak moves little as we go from LO to NLO to NNLO Furthermore, the scale dependence of the peak location under variations of m between 5 and 60 GeV is reduced by a factor of The transition from the pole to the PS scheme has little effect on the shape and overall normalization of the cross section as expected In particular a significant uncertainty of about "0% in the normalization remains, larger than at NLO The strong enhancement of the peak for the small scale ms5 GeV is a consequence of the fact that the perturbative corrections to the residue of the S pole Žsee Ž 5 become uncontrollable at scales not much smaller than 5 GeV We find that these large

145 ( ) M Beneke et alrphysics Letters B corrections are mainly associated with the logarithms that make the coupling run in the Coulomb potential This could be interpreted either as an indication that higher order corrections are still important Žat such low scales or that the terms associated with b 0 should be treated exactly, because they are numerically Ž but not parametrically large ŽIf the Schrodi- nger equation with the potential Ž 0 is solved exactly by numerical methods, the scale dependence of the peak height is indeed smaller w8 x Inspecting the logarithms in the result for the cross section, Ž Ž choosing an energy-dependent scale m s mt E q G r t r would be most appropriate Although this choice is our preferred one, we refrained from using it for the comparison of the pole and PS scheme, since the peak positions are at different energies in the two schemes The uncertainties due to other parameters turn out to be less than the uncertainty due to the variation of the scale m The dependence on a Ž m is discussed below Varying mh by a factor of two around mt changes the cross section by a few percent near the peak The effect of summing logarithms to NLL is of the same order We have also checked the effect of some NNLL logarithms on the calculation and find a variation of the order of "5% The relatively small effect of renormalization group improvement is a consequence of the absence of leading logarithms We also checked the effect of varying mf between 5 and 40 GeV From a purely pragmatic point of view, values of mf around 40 GeV lead to the most stable result However, since m -m C a Ž m f t F s f30 GeV is required from a conceptual point of view, we have chosen m s0 GeV as our preferred setting Ž f From the point of view of non-perturbative infrared cancellations, it would be sufficient to choose mf larger than the strong interaction scale GeV However, from Ž 9 we see that the cancellation becomes effective and the perturbative correction universal as soon as the integrand is dominated by < q< - mc t Fa s For this reason it is legitimate and advanta- geous to choose m significantly larger than the f strong interaction scale If we take Ž naively the change in the peak position under scale variations as a measure of the uncertainty of the top mass measurement, we conclude that a determination of the PS mass with an error of about MeV is possible Given that s Z the uncertainty in relating the PS mass to the MS mass is about 300 MeV, this accuracy seems to be sufficient We emphasize that it is not sufficient to invent an ad hoc mass definition in terms of which the peak position is stable empirically In addition, such a mass definition needs to have a well-behaved relation order by order in perturbation theory to a mass definition relevant to top quark processes far away from threshold For a realistic assessment of the error in the mass measurement, the theoretical line shape has to be folded with initial state radiation, beamstrahlung and beam energy spread effects Since these effects are well understood, the main question that needs to be addressed is whether the normalization uncertainty leads to a degradation of the mass measurement after these sources of smearing are taken into account This should be studied in a collider design specific setting In the PS scheme the correlation of the top quark mass with as is also strongly reduced, mainly be- cause the perturbative corrections to the S pole are small in this scheme This is indicated by Fig, which shows the dependence of the line shape on the value of a Ž m s Z One can therefore rely less on input from top quark momentum distributions, which have been used in the pole scheme to constrain m t and a Ž m s Z simultaneously Since momentum distri- butions are more sensitive to non-perturbative effects than the inclusive cross section, this is another advantage of the PS scheme Fig Dependence of the NNLO tt cross section on a Ž m s Z in the PS scheme Ž solid and the pole scheme Ž long-short-dashed The three curves in each scheme refer to a Ž m s03 Ž lower s Z, a Ž m s08 Ž middle and a Ž m s03 Ž upper s Z s Z Recall that Es' s ym in the pole scheme but Es' s ym Ž 0GeV t t,ps in the PS scheme Other parameters: mtsmhs75gev, Gts 40 GeV, ms30gev

146 46 ( ) M Beneke et alrphysics Letters B In conclusion, we evaluated the next-to-next-toleading order QCD correction to top quark production near threshold in the conventional pole scheme and in the PS scheme w3 x We employed factorization in dimensional regularization and summed nextto-leading logarithms in the top quark velocity We find that the cross section expressed in the PS scheme allows us to determine the top PS mass more accurately than the top pole mass The PS mass, in turn, can also be related more accurately to the top quark MS mass Acknowledgements We thank AH Hoang for extensive discussions and for comparing numerical results of their calculaw8x prior to publication The work of MB is tion supported in part 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 - MIHT The work of VS has been supported by the Volkswagen Foundation, contract No Ir736, the Russian Foundation for Basic Research, project and by INTAS, grant References wx CDF Collaboration Ž F Abe et al, Phys Rev Lett 80 Ž ; CDF Collaboration Ž F Abe et al, FERMILAB-PUB E, hep-exr98009; D0 Collaboration ŽB Abbott et al, Phys Rev D 58 Ž wx ECFArDESY LC Physics Working Group ŽE Accomando et al, Phys Rep 99 Ž 998 wx 3 VS Fadin, VA Khoze, Pis ma Zh Eksp Teor Fiz 46 Ž wjetp Lett 46 Ž x; Yad Fiz 48 Ž wsov J Nucl Phys 48 Ž x wx 4 MJ Strassler, ME Peskin, Phys Rev D 43 Ž ; W Kwong, Phys Rev D 43 Ž ; M Jezabek, JH Kuhn, T Teubner, Z Phys C 56 Ž ; H Murayama, Y Sumino, Phys Rev D 47 Ž 993 8; M Jezabek, T Teubner, Z Phys C 59 Ž ; Y Sumino, K Fujii, K Hagiwara, H Murayama, C-K Ng, Phys Rev D 47 Ž ; K Fujii, T Matsui, Y Sumino, Phys Rev D 50 Ž ; R Harlander, M Jezabek, JH Kuhn, T Teubner, Phys Lett B 346 Ž ; M Jezabek et al, Phys Rev D 58 Ž wx 5 II Bigi, Phys Lett B 8 Ž wx 6 Y Schroder, hep-phr9805 wx 7 M Peter, Phys Rev Lett 78 Ž ; Nucl Phys B 50 Ž wx 8 M Beneke, A Signer, VA Smirnov, Phys Rev Lett 80 Ž wx 9 A Czarnecki, K Melnikov, Phys Rev Lett 80 Ž w0x AH Hoang, T Teubner, Phys Rev D 58 Ž wx K Melnikov, A Yelkhovsky, Nucl Phys B 58 Ž wx O Yakovlev, hep-phr w3x M Beneke, Phys Lett B 434 Ž w4x M Beneke, A Signer, VA Smirnov, in preparation w5x K Melnikov, O Yakovlev, Phys Lett B 34 Ž 994 7; VS Fadin, VA Khoze, AD Martin, Phys Rev D 49 Ž ; M Peter, Y Sumino, Phys Rev D 57 Ž ; W Beenakker, FA Berends, AP Chapovsky, hepphr w6x M Beneke, VA Smirnov, Nucl Phys B 5 Ž w7x R Barbieri, R Gatto, R Korgerler, Z Kunszt, Phys Lett 57B Ž w8x A Pineda, J Soto, Nucl Phys Proc Suppl 64 Ž , hep-phr970748; Phys Rev D 59 Ž w9x L Susskind, Les Houches lectures 976, in: R Balian et al Ž Eds, Weak and electromagnetic interactions at high energies, North-Holland, Amsterdam, 977; W Fischler, Nucl Phys B 9 Ž ; A Billoire, Phys Lett B 9 Ž w0x A Pineda, FJ Yndurain, Phys Rev D 58 Ž wx K Melnikov, A Yelkhovsky, TTP98-7, hep-phr wx AA Penin, AA Pivovarov, INR , hep-phr w3x M Beneke, VM Braun, Nucl Phys B 46 Ž ; II Bigi, MA Shifman, NG Uraltsev, AI Vainshtein, Phys Rev D 50 Ž ; M Beneke, Phys Lett B 344 Ž ; MC Smith, SS Willenbrock, Phys Rev Lett 79 Ž w4x AH Hoang, MC Smith, T Stelzer, S Willenbrock, UCSDrPTH 98-3, hep-phr98047 w5x M Beneke, A Signer, VA Smirnov, in: Proceedings of RADCOR98, Barcelona, September 998 w6x N Gray, DJ Broadhurst, W Grafe, K Schilcher, Z Phys C 48 Ž w7x M Beneke, VM Braun, Phys Lett 348 Ž ; P Ball, M Beneke, VM Braun, Nucl Phys B 45 Ž w8x AH Hoang, T Teubner, in preparation

147 3 May 999 Physics Letters B Direct instantons and nucleon magnetic moments M Aw a,, MK Banerjee a, H Forkel b a Department of Physics, UniÕersity of Maryland, College Park, MD 074-4, USA b Institut fur Theoretische Physik, UniÕersitat Heidelberg, D-690 Heidelberg, Germany Received 8 February 999 Editor: PV Landshoff Abstract We calculate the leading direct-instanton contributions to the operator product expansion of the nucleon correlator in a magnetic background field and set up improved QCD sum rules for the nucleon magnetic moments Remarkably, the instanton contributions are found to affect only those sum rules which had previously been considered unstable The new sum rules show good stability and reproduce the experimental values of the nucleon magnetic moments with values of x, the quark condensate magnetic susceptibility, consistent with other estimates q 999 Published by Elsevier Science BV All rights reserved For over two decades QCD instantons have been associated with fundamental aspects of strong interaction physics, as, for example, with the u-vacuum wx, the issue of strong CP violation wx, and the X anomalously large h mass wx Unequivocal and quantitative evidence for their role in hadron structure, however, turned out to be much harder to establish This is mainly due to the complexity involved in dealing with interacting instanton ensembles and their coupling to other vacuum fields over large distances Instanton vacuum models wx 3 attack the first part of this problem directly, by approximating the field content of the vacuum as a superposition of solely instantons and anti-instantons This approach has DOErERr , UMD PPa Present address: Department of Physics, Carnegie Mellon University, Pittsburgh, PA 53, USA; mountaga@cmuedu been developed for more than a decade and can describe an impressive amount of hadron phewx 3 More recently, QCD lattice simula- nomenology tions began to complement such vacuum models by isolating instantons in equilibrated lattice configurations and by studying their size distribution and their impact on hadron correlators wx 4 While the results obtained with different, currently developed lattice techniques have not yet reached quantitative agreement, they do confirm the overall importance of instantons and some bulk properties of their distribution in the vacuum Another approach towards linking the instanton component of the vacuum to hadron properties has been developed over the last years by generalizing the nonperturbative operator product expansion Ž OPE and QCD sum rule techniques w5 7 x While its range of applicability is more limited than that of instanton vacuum models and of lattice calculations, it avoids the need for large-scale computer simulations and takes, in contrast to instanton models, all r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

148 48 ( ) M Aw et alrphysics Letters B long-wavelength vacuum fields and also perturbative fluctuations into account Furthermore, the approach is largely model-independent and allows the study of instanton effects in a fully analytic and, therefore, rather transparent fashion In the present paper, we adapt this approach to properties which characterize the response of the hadronic system to a weak external field Specifically, we calculate the direct-instanton contributions to the nucleon correlator in the presence of a constant electromagnetic field With the help of background-field sum-rule techniques due to Ioffe and Smilga wx 8 and Balitsky and Yung wx 9, we then study previously neglected instanton effects in the QCD sum rules for the magnetic moments of the nucleon As already mentioned, our work is based on the nucleon correlation function H 4 ipx i d x e ² 0< Th Ž x hž 0 < 0: mn sp 0 p q 4pa Pmn p F, F ' in the background of a constant electromagnetic field F The interpolating fields hž x mn with proton or neutron quantum numbers w0x are composite operators of massless up and down quark fields: h Ž x s u Ž x Cg u Ž x gg d Ž x e, T a b a c abc p a 5 h sh uld n p For the application in QCD sum rules we need a theoretical description of the correlator Ž at momenta ssyp, GeV, ie at distances xq 0 fm The information on the magnetic moments is contained in the second term on the right-hand side of Eq Ž It characterizes the linear response of the nucleon to the external field and can be decomposed into three independent Lorentz and spinor structures: PmnŽ p s Ž pus mnqsmn pu PŽ p qiž gm pnyg n pm pu P Ž p qsmn P3Ž p Ž 3 Note that the invariant amplitude P corresponds to the chirally-even part of the correlator, while P and P are associated with the chirally-odd part 3 The nonperturbative OPE w,8x of the above correlator at small distances can be generated by splitting each diagram contributing to Ž 3 in all possible ways into a hard and a soft subgraph The hard subgraphs contribute to the Wilson coefficients and are usually calculated perturbatively, with the integration range of each internal momentum restricted 3 to be larger than the OPE scale m;05 GeV The soft subgraphs correspond to hadron-channel independent condensates, renormalized at m In the presence of an external electromagnetic field the OPE Ž up to eight-dimensional operators involves the additional, Lorentz-covariant condensates ' ² 0< qs q< 0: s 4pa xf ² 0< qq< 0 :, Ž 4 mn ' mn g² 0< qg q< 0: s 4pa kf ² 0< qq< 0 :, Ž 5 mn i g² 0< qg G q0 < : s ' 4pa jf ² 0< qq< 0 : 5 mn mn Ž 6 a Ž G s l G, G s G with sy mn a mn mn mnrs rs 03 The parameters x,k, and j play the role of generalized susceptibilities and quantify the vacuum response to weak electromagnetic fields The magnetic susceptibility of the quark condensate, x, for example, originates from the induced spin alignment of quark-antiquark pairs in the vacuum Note also that x is associated with the lowest-dimensional induced condensate, which enhances its role in the OPE and the corresponding sum rules The OPE of P Ž p mn up to operators of dimension eight, with perturbatively calculated Wilson coeffiwx 8 An inherent cients, has been obtained in Ref assumption of this calculation - and of the QCD sum rule program in general - is that the short-distance physics associated with fields of wavelength smaller than m y is predominantly perturbative It is well known, however, that also strong nonperturbative fields of rather small size exist in the QCD vacuum Instantons, ie the finite-action solutions of the clasw x, are sical, Euclidean Yang Mills equation paradigmatic examples of such fields and have a crucial impact on the vacuum structure mn 3 In practice, this restriction is often unnecessary see below

149 ( ) M Aw et alrphysics Letters B The nonperturbative contributions from shortwavelength fluctuations of quarks and gluons around instantons are thus neglected in the standard treatment of the OPE Their relative importance, and hence the justification for approximately disregarding them, depends both on the instanton size distribution in the vacuum and on the quantum numbers of the hadronic channel under consideration Instantons of smaller Ž average size r are accompanied by fluctuations of smaller wavelength, and those contribute more strongly to the Wilson coefficients The hadron-channel dependence of the instanton contributions originates mainly from the chirality and spin-color coupling of the quark zero-modes in the instanton background Instanton-induced effects are particularly large in the pseudoscalar-isovector and scalar-isoscalar channels, because there the quark zero-modes contribute with maximal strength As a consequence, instanton contributions dominate already at short distances in the pseudoscalar sum rules, and are essential for their stability wx 6 In the vector and axial-vector channels, on the other hand, zero-mode contributions are, to leading order in the instanton density, absent The strength of instanton contributions to the nucleon channel lies about halfway in-between these extreme cases In the chirally-odd amplitudes, it was found to be roughly of the same magnitude as that of the condensate contributions w5,7 x, since the spin-0 diquark operators in the interpolating fields Ž couple strongly to instantons The nucleon channel is therefore well suited for studying the interplay bewx 7 tween instantons and other vacuum fields The calculation of the leading direct-instanton contributions to the background-field correlator Ž proceeds essentially along the lines described in Refs w5 7 x, to which we refer for more details Similar to the condensates, the bulk properties of the instanton size distribution are generated by long-distance vacuum dynamics and have thus to be taken as input for this calculation As before w5 7 x, we will use the standard values w3x r, 3 fm for the average instanton size and R, fm for the average separation between neighboring Ž anti instantons The results of instanton vacuum models w3,4x confirm these scales, while those from the lattice wx 4 are not yet fully consistent but lie in the same range Žwith maximal deviations of about 50% Since the average instanton size is of the order of y the inverse OPE scale, r-m,04 fm, instanton corrections to the Wilson coefficients can be substany tial Moreover, since r 4 L QCD, these corrections are essentially semiclassical, and at the relevant distances x Q 0 fm < R multi-instanton correlations should be negligible The instanton contributions to the OPE coefficients can therefore be calculated in semiclassical approximation, ie by evaluating the correlator Ž in the background of the instanton and anti-instanton field and by then averaging the instanton parameters over their vacuum distributions ŽThe distribution of the instanton s position and color orientation is uniform, due to translational and gauge invariance We treat the zero-mode sector of the quark propagator in the instanton field exactly and approximate the continuum modes, as before wx 5, by plane waves The recently found zero-mode dominance of the ground-state contributions to the pion and r-meson correlators on the lattice w5x supports the validity of this approximation The impact of the remaining vacuum fields Ž including other instantons on the instanton contributions is accounted for in a meanw6x and generates an effective mass field sense m r sy pr ² qq: 3 for the quark zero modes We further use the approximate instanton size distribution nž r s ndž ry r w3x and the self-connž r sistency condition w7x ² qq: syh d r s mž r n y Ž mž r which is numerically satisfied to good accuracy to eliminate the n dependence from the resulting expressions Up to operators of dimension eight, we find the leading instanton contributions to the correlator Ž to arise from just one type of graph, in which two of the quarks Žemitted from the current at x s 0 propagate in zero-modes while the third interacts with the background field through the magnetized quark condensate Graphs in which the background field couples directly to a hard quark in a zero-mode, vanish The same holds for graphs in which the interaction with the photon causes a transition from zero-mode to continuum-mode propagation 4 The 4 Contributions of this type are essential in the pseudoscalar three-point correlator associated with the pion electromagnetic form factor wx 6

150 50 ( ) M Aw et alrphysics Letters B contribution from direct instantons is thus generated by the interplay between the rather localized quark zero modes and more slowly varying, nonperturbative vacuum fields Effects of this type appear naturally in the OPE, whereas they are difficult to acwx 7 count for in, eg, quark models Evaluating the corresponding graphs in the instanton and magnetic background fields as described above, and averaging over the instanton size distribution, we arrive at ² 0< Th Ž x h Ž 0 < 0: F,inst p sp 0 inst p Ž x 3 4 eu r y ² qs q: mn F s 4 mn 3p m 4 = H d x0 Ž r qr x qr Ž 0 Ž r s x y x 0, where x0 specifies the center of the instanton for the proton The corresponding neutron correlator is obtained by replacing eu with e din order to put the instanton contribution Ž 7 to use in the sum rules, we also need its Fourier and Borel transform 5 Again for the proton, it reads e u ˆP Ž M s axr M IŽ z Ž 8 8p Here, M denotes the Borel mass parameter We have also used the standard definitions z' Mr, a' y p ² qq :, and abbreviated the integral da z y IŽ z sh e 4a Ž ya 0 a Ž ya z z z y 0ž / ž / s4e K qk Ž 9 Note that Ž 9 shows the typical exponential Borelwx 5 To- mass dependence of instanton contributions gether with the appearance of the new scale r, this distinguishes them from the logarithms and power terms of the standard OPE An important qualitative property of the instanton contribution has been made explicit in Eq Ž 8 : to 5 For the Borel transform we follow the convention of Ref w0 x leading order, direct instantons contribute almost exclusively 6 to one invariant amplitude, P 3, which is associated with the chirally-odd Dirac structure s mn Exactly this amplitude was singled out in the previous sum rule analysis of Ref wx 8, for two reasons: first, one of its Wilson coefficients contains, in the pragmatic version of the OPE Ž see below, an infrared divergence Secondly, the P 3 sum rule failed to show a fiducial Borel region of stability w8x Ž even after proper subtraction of the divergence, while the other two sum rules provide stable and accurate values for the nucleon magnetic moments even without direct-instanton corrections, and with a value of x consistent with other, independent estiw9 x To understand the first point we note that, since in mates QCD perturbative contributions from soft loop momenta are normally small compared to the corresponding condensate contributions, it is standard procedure not to remove them in sum-rule calculations w0 x This simplification which goes under the name of pragmatic OPE fails, however, if infrared divergences appear in diagrams associated with a Wilson coefficient Such an infrared divergence was encountered in the OPE of P 3, in a graph where a vacuum gluon field and the background photon interact with the same hard quark line Hence the contribution from soft loop momenta has to be cut off explicitly, according to the rules of the exact OPE A similar divergence was found before wx in the vector meson correlator when two soft gluon fields couple to the same quark line The authors of wx 8 conjectured that the appearance of infrared singularities in P 3 and the absence of a stability region in the associated sum rule might be related This seems unlikely, however, in view of the later finding wx of similar infrared divergencies Žwhich always occur when a quark line interacts with multiple soft gauge fields in the other two amplitudes of Ž 3, which nevertheless lead to satisfactory sum rules Direct instantons, on the other hand, contribute almost exclusively to P 3, and their previous neglect could offer a more plausible explanation 6 There is also a small direct instanton contribution of similar structure to the amplitude P This term turns out to be too small to have an appreciable impact on the corresponding sum rules, however, and will not be discussed further

151 ( ) M Aw et alrphysics Letters B for the instability of the P 3 sum rule Support for this conjecture, which we are going to test quantitatively below, comes from two other chirally-odd nucleon sum rules Žfor the nucleon mass wx 5 and its isospin splitting wx 7, where exactly such a selective stabilization due to instantons has been found Our modified P 3 sum rule is obtained by equating the Borel transform of the standard OPE from Ref wx 8 and the instanton contributions Ž 8 to the Borel-transformed double dispersion relation for the correlator Ž, with a spectral function parametrized in terms of the nucleon pole contribution Žcontaining the magnetic moments and a continuum based on local duality Including the infrared-divergent term encountered in Ref wx 8, duly truncated at the OPE renormalization point, the new sum rule for the proton reads 7 ½ am euy 6 ed q4kqj E M m0 M 4 y 9 6 u EM q e ln yg L M m q ed M x EŽ M L 6 y y L 7 8 u c 5 y e xr M I z m a mp m y p s l me 4 N M y qa p, Ž 0 M m where m is the nucleon mass, W the continuum threshold, and l the coupling of the current Ž N to the nucleon state, ² 0 < h < N : s lnu For the mixed quark condensate we use the standard parametrizamn ² : ² : tion qsmn G q sym0 qq with m 0 s 08 GeV, and g EM,0577 is the Euler-Mascheroni constant The additional parameters A p,n determine the strength of electromagnetically induced transitions between the nucleon and its excited states The sum rule for the neutron is obtained from Ž 0 by interchanging eu and ed and by replacing m p,m a p m and A A We have also defined l n p n N s 4 3p l, LslnŽ MrL rlnž mrl Ž Ls0 GeV, N 7 We have corrected some errors which appeared in the expreswx 8 sions of Ref and transferred, using the standard expressions j W n W y n M Ý ž j! M / E Ž M sye q, Ž the continuum contributions to the OPE-side of the sum rules The appropriate form of these contribuw3 x In principle, several alternative options are avail- tions has recently been clarified in Ref able for the quantitative analysis of background-field sum rules In practice, however, one is limited by the fact that their fiducial domain Žie the Borel-mass region in which the neglect of higher-order terms in the short-distance expansion is justified while the nucleon pole still dominates over the continuum is generally not large enough to determine all the unknown parameters from a stable fit The authors of Ref wx 8 succeeded, however, in eliminating the susceptibilities and other constants by combining the two sum rules for P and P and their M -derivatives The magnetic moments can then be fitted and are in good agreement with experiment, although taking derivatives of the sum rules generally reduces their reliability Unfortunately, the above procedure also eliminates the continuum contributions, which makes it impossible to determine whether a fiducial stability domain exists In any case, this procedure is ineffective for our new sum rules Ž 0, as it cannot eliminate the additional x-dependence introduced by the direct-instanton contribution Similarly, working with the ratio of background-field and mass sum rules, as done in Ref w4x for the P sum rules, offers no advantage in our case since already the leading terms in the OPE of the chirally-odd mass and background-field sum rules differ Therefore, we resort to a direct minimization of the relative deviations between the two sides of Eq Ž 0 The coupling l s93 GeV 6 and the continuum threshold Ws66 GeV are, folwx 8, obtained by fitting lowing the procedure of Ref the instanton-improved nucleon mass sum rule wx 5 to the experimental nucleon mass 8 The values of the 8 We have included anomalous-dimension corrections in the mass sum rule of Ref wx 5 and restored the full four-quark condensates, since we are dealing exclusively with Ioffe s interpolating field Ž

152 5 ( ) M Aw et alrphysics Letters B two susceptibilities k sy034 " 0 and j s y074" 0 were estimated in independent work by Kogan and Wyler w5 x This enables us to fit both sides of the sum rules Ž 0 by varying x and A Ž p or A, respectively n while keeping the magnetic mo- ments fixed at their experimental values The fits are performed in the fiducial Borel mass domain, which is bounded from below by requiring the highest-dimensional operators to contribute at most 0% to the OPE and from above by restricting the continuum contribution to maximally 50% The resulting fiducial domains of both the proton and neutron sum rules, 08 GeV Q M Q 5 GeV, are larger than those of the sum rules based on P and P The fit results are shown in Fig, where for both the proton and the neutron sum rules the direct-instanton contributions, the remaining OPE including the continuum contributions, their sum Žwhich makes up the left-hand side of Eq Ž 0, and the right-hand sides are plotted The fit quality of both the proton and neutron sum rules is excellent As previously in the instanton-improved, chirally-odd nucleon mass sum rule wx 5, the theoretical side of the sum rules, including the instanton-induced part, is almost indistinguishable from the phenomenological side Moreover, Fig shows that the direct-instanton contributions can reach about half the magnitude of the remaining terms in the OPE, which makes it evident why their previous neglect had a detrimental impact on the stability properties An alternative way of evaluating the optimized sum rules consists in solving them for mn and plotting the result as a function of the Borel mass, as shown in Fig The resulting functions m Ž M p,n specify the value of the magnetic moment which is required to make both sides of the sum rule Ž 0 match exactly at each value of the Borel mass The instanton-corrected sum rules render mž M practically M-independent, thereby again indicating an almost perfect fit for both proton and neutron This fit predicts x,y496 GeV y for the proton and x,y473 GeV y for the neutron sum rule These values correspond to the OPE scale m s 05 GeV adopted for our sum rules and lie inside the range obtained from other estimates w8,9, x They are somewhat smaller in magnitude than the value x,y57 GeV y found in the twow9 x ŽOur predicted and three-pole models of Ref values for the excited-state transition parameters are A p, 08 GeV and A, y07 GeV In conclusion, we have recovered a third reliable sum rule for the nucleon magnetic moments In contrast to the other two, it receives previously y n Fig The OPE Ž dashed line and direct instanton Ž dotted line contributions to the new s sum rules for the proton Ž positive range mn and neutron Their sum Ž dot-dashed line is compared to the RHS Ž solid line

153 ( ) M Aw et alrphysics Letters B Fig The Borel mass dependence of the magnetic moments of the proton upper and neutron calculated from the optimal fit of LHS and RHS neglected direct-instanton contributions which arise from the interplay with long-wavelength vacuum fields Our new sum rule is built on the chirally-odd amplitude P 3 of the nucleon correlator in an electromagnetic background field and found to be at least as stable as the other two, although it had previously been regarded as flawed The new sum rule adds to the predictive power of the backgroundfield sum rules and strengthens their mutual consistency Furthermore, our results reinforce a systematic pattern which emerged from previous studies of direct-instanton effects both in the nucleon and pion channels: those sum rules which worked satisfactorily without instanton corrections receive little or no direct instanton contributions, and previously less reliable or completely unstable sum rules are stabilized by large instanton contributions This pattern points not only towards the importance of direct instantons in particular sum rules, but also supports the adequacy of their semiclassical implementation into the OPE Our results show that these conclusions continue to hold in the presence of a magnetized vacuum Acknowledgements MA and MKB acknowledge support from the US Dept of Energy under grant number DE-FG0-93ER-4076 HF acknowledges support from Deutsche Forschungsgemeinschaft under habilitation grant Fo 56r- References wx R Jackiw, C Rebbi, Phys Rev Lett 37 Ž 976 7; CG Callan Jr, RF Dashen, DJ Gross, Phys Lett B 63 Ž wx G thooft, Phys Rev Lett 37 Ž wx 3 T Schafer, E Shuryak, Rev Mod Phys 70 Ž ; DI Diakonov, VYu Petrov, Nucl Phys B 45 Ž ; Phys Lett B 47 Ž ; Nucl Phys B 7 Ž wx 4 See for example MC Chu, S Huang, Phys Rev D 45 Ž ; M-C Chu, JM Grandy, S Huang, J Negele, Phys Rev D 49 Ž ; P van Baal, Nucl Phys Proc Suppl 63 Ž 998 6; T DeGrand, A Hasenfratz, T Kovacs, talk given at the 997 Yukawa Seminar, COLO-HEP- 396, hep-latr980037; DA Smith, MJ Teper, Oxford preprint OUTP-97-76P, hep-latr980008; A Hasenfratz, C Nieter, hep-latr wx 5 H Forkel, MK Banerjee, Phys Rev Lett 7 Ž wx 6 H Forkel, M Nielsen, Phys Lett B 345 Ž

154 54 ( ) M Aw et alrphysics Letters B wx 7 H Forkel, M Nielsen, Phys Rev D 55 Ž wx 8 BL Ioffe, AV Smilga, Nucl Phys B 3 Ž wx 9 II Balitsky, Phys Lett B 4 Ž 98 53; II Balitsky, AV Yung, Phys Lett B 9 Ž w0x BL Ioffe, Nucl Phys B 88 Ž 98 37; Nucl Phys B 9 Ž 98 59; VM Belyaev, BL Ioffe, Sov Phys JETP 56 Ž wx MA Shifman, AI Vainshtein, VI Zakharov, Nucl Phys B 47 Ž , 448 wx AA Belavin, AM Polyakov, AS Schwartz, YuS Tyupkin, Phys Lett B 59 Ž w3x EV Shuryak, Nucl Phys B 03 Ž 98 93, 6 w4x EV Shuryak, JJM Verbaarschot, Nucl Phys B 34 Ž 990 w5x TL Ivanenko, JW Negele, Nucl Phys Ž Proc Suppl 63 Ž w6x MA Shifman, AI Vainshtein, VI Zakharov, Nucl Phys B 63 Ž w7x CG Callen Jr, R Dashen, DJ Gross, Phys Rev D 7 Ž , 763; DG Caldi, Phys Rev Lett 39 Ž 977 w8x B Ioffe, A Smilga, private communication w9x VM Belyaev, YaI Kogan, Yad Fiz 40 Ž wsov J Nucl Phys 40 Ž x; Ya Balitsky, AV Kolesnichenko, AV Yung, Yad Fiz 4 Ž wsov J Nucl Phys 4 Ž x w0x VA Novikov, MA Shifman, AI Vainshtein, VI Zakharov, Nucl Phys B 49 Ž wx AV Smilga, Yad Fiz 35 Ž wsov J Nucl Phys 35 Ž 98 7x wx SL Wilson, J Pasupathy, CB Chiu, Phys Rev D 36 Ž ; CB Chiu, SL Wilson, J Pasupathy, JP Singh, Phys Rev D 36 Ž w3x B Ioffe, Phys Atom Nucl 58 Ž w4x CB Chiu, J Pasupathy, SL Wilson, Phys Rev D 33 Ž w5x II Kogan, D Wyler, Phys Lett B 74 Ž 99 00

155 3 May 999 Physics Letters B Large Coulomb corrections to the e q e y pair production at relativistic heavy ion colliders DYu Ivanov a,, A Schiller b,, VG Serbo c,3 a Institute of Mathematics, NoÕosibirsk, , Russia b Institut fur Theoretische Physik and NTZ, UniÕersitat Leipzig, D-0409 Leipzig, Germany c NoÕosibirsk State UniÕersity, NoÕosibirsk, , Russia Received 5 January 999; received in revised form 0 March 999 Editor: PV Landshoff Abstract q y We consider the Coulomb correction CC to the e e pair production related to multiphoton exchange of the produced e " with nuclei The contribution of CC to the energy distribution of e q and e y as well as to the total pair production cross section is calculated with an accuracy of the order of % The found correction to the total Born cross section is negative and equals y5% at the RHIC for Au-Au and y4% at the LHC for Pb Pb collisions q 999 Elsevier Science BV All rights reserved PACS: 0ym; 575yq; 3450ys Introduction Two new large colliders with relativistic heavy nuclei, the RHIC and the LHC, are scheduled to be in operation in the nearest future The charge number of the nuclei Z sz sz with masses M sm sm d-ivanov@mathnscru schiller@tph04physikuni-leipzigde 3 serbo@mathnscru and their Lorentz factors g s g s g s ErM are the following Zs79, gs08 for RHIC Ž Au Au collisions, Zs8, gs3000 for LHC Ž Pb Pb collisions Ž Here E is the heavy ion energy in the cms One of the important processes at these colliders is ZZ Z Z e q e y Its cross section is huge In the Born approximation X Žsee Fig with nsn s the total cross section according to the Racah formula wx is equal to s Born s 36 kbarn for the RHIC and 7 kbarn for the LHC Therefore it will contribute as a serious back r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

156 56 ( ) DYu IÕanoÕ et alrphysics Letters B Fig The amplitude M X of the process Ž with n Žn X nn virtual photon emitted by the first Ž second nucleus ground to a number of experiments Žfor example, at the LHC it will lead to about 50 pairs per single beam collision Besides, this process is also important to determine the luminosity and the beam lifetime The pair production, accompanied by the capture of an electron on an atomic orbit, is the leading beam loss mechanism Žfor details see review wx It means that the differential and total cross sections of reaction Ž should be known with a good accuracy The cross sections of the process Ž in the Born approximation are known with accuracy ;rg Žsee, for example, w,3x and more recent calculations reviewed in Refs w,4 x However, besides of the Born amplitude MBorn s M, also other amplitudes M X Ž see Fig nn have to be taken into account for heavy nuclei since in this case the parameter of the perturbation series Za is of the order of unity For example, Za f 06 for colliders Ž Therefore, the whole series in Za has to be summed to obtain the cross section with sufficient accuracy Following Ref wx 5, we call the Coulomb correction Ž CC the difference dscoul between the whole sum ds and the Born approximation dssds qds Ž 3 Born Coul Such kind of CC is well known in the photoproq y duction of e e pairs on atoms Žsee wx 5 and 98 of wx 6 The Coulomb correction to the total cross section of that process decreases the Born contribution by about 0% for a Pb target For the pair production of reaction Ž with Za< and Za; CC has been obtained in w7, x Recently this correction has been calculated for the pair production in the colliwx 8 The results of sions of muons with heavy nuclei Refs w7,,8x agree with each other in the corresponding kinematic regions and noticeably change the Born cross sections Formulae of the CC for two heavy ions were suggested ad hoc in Section 73 of wx However, our calculations presented here do show that this suggestion is incorrect A large CC was also found for the positronium production in collisions of photons, electrons and nuclei with heavy nuclei wx 9 In the present paper we calculate the Coulomb correction for process Ž omitting terms of the order of % compared with the main term of the Born cross section We find that these corrections are negative and quite important: scoulrsborn sy5% for RHIC, s rs sy4% for LHC Ž 4 Coul Born This means that at the RHIC the process with the largest cross section will have a production rate 5% smaller than expected Recently in Refs w0 x the Coulomb effects were studied in the frame of a light-cone or an eikonal approach However, the approximations used there fail to reproduce the classical results of Bethe and Maximon wx 5 The most striking result of Refs w0 x is that the CC is absent, dscouls0, both for the case Za;Za; and Za<, Za; This clearly contradicts not only the results of the present paper but also those obtained in Refs w7,,8,9 x An application to SPS fixed-target experiw3 x, given in Ref w0 x, is ments with heavy ions ungrounded Ž for details see the Discussion Our main notations are given in Eq Ž and Fig, besides, Ž P q P s 4E s 4g M, qi s X Ž v, q sp yp, s q and i i i i q y a 4 Z Z PP s0 s, Lsln slng, Ž 5 p m MM where m is the electron mass The quantities q i H and p "H denote the transverse part of the corresponding three-momenta Throughout the paper we use the well known function wx 5 ` Ý ns n n qz a fž Z sz a, Ž 6 its particular values for the colliders under discussion are f 79 s033 and f 8 s033

157 ( ) DYu IÕanoÕ et alrphysics Letters B Selection of the leading diagrams and the structure of the amplitude Let M be the sum of the amplitudes M X nn of Fig It can be presented in the form Ms M XsM qm qm qm, Ž 7 Ý nn Born X nn G M s M X, M s M, Ý Ý n n X n G ng Ý M s M X nn X nn G The Born amplitude MBorn contains the one-photon exchange both with the first and the second nucleus, whereas the amplitude M Ž M contains the onephoton exchange only with the upper Ž lower nucleus In the last amplitude M we have no one-photon exchange According to this classification we write the total cross section as sss qs qs qs, Ž 8 where Born ds A< M <, Born Born ) ds ARe M M q< M <, Born ds ARe M M ) q< M <, Born Ž ds ARe M M qm M qm M ) Born ) ) ) < < qm M q M In the amplitude M X nn the e q e y pair is produced by nqn X virtual photons, therefore its charge parity is CsŽ y nqnx Due to C conservation, in the total cross section sah < M < dg only those pieces of the form HM X M ) X nn kk dg contribute to the cross section for which Ž y nqnx sy kqkx Therefore, the sum nqn X qkqk X has to be even and the ratio sirsborn is a function of Ž Za only but not of Za itself Additionally we estimate the leading logarithms appearing in the cross sections s i The integration over the transferred momentum squared q and q results in two large Weizsacker Williams Ž WW logarithms ;L for s Born, in one large WW logarithm ;L for s and s The cross section s contains no large WW logarithm Therefore, the relative contribution of the cross sections s is s rs ss rs ; i Born Born Ž Za rl and s rs Born; Za rl -04% for the colliders Ž As a result, with an accuracy of the order of % we can neglect s in the total cross section and use the equation sss qs qs Ž 9 Born With that accuracy it is sufficient to calculate s and s in the leading logarithmic approximation Ž LLA only since the next to leading log terms are of the order of Ž ZarL This fact greatly simplifies the calculations The calculation in the LLA can be performed using the equivalent photon or WW approximation The main contribution to s and s is given by the region Ž v rg < yq < m and v rg < yq <m, respectively In the first region the main contribution arises from the amplitudes MBornq M, in the second region from M qm Born Let us consider the first region for definiteness The virtual photon with four-momentum q is almost real in this region and the amplitude can be expressed via the amplitude Mg for the real photo- q production g Z Z e e y Ž see, for example, 99 of wx 6 < q < H E M qm f' Born 4pa Z M g Ž 0 yq v Ž The amplitude M has been calculated in Ref wx g 5 We use the convenient form of that amplitude dew4x and w5 x: rived in the works M s f M Born qi f DM e if, Ž g g g where Mg Born is the Born amplitude for the g Z Z e q e y process This Born amplitude depends on the transverse momenta p "H only via the two combinations Asjqyj y and BsjqpqHqj ypyh Ž where j sm r m qp " "H The quantity DMg is obtained from Mg Born replacing A jqqj yy and B jqpqhyj yp yh All the nontrivial dependence on the parameter Z a' n is accumulated in the Bethe Maximon phase Ž pqp jq Fsn ln Ž Ž p P j y y

158 58 ( ) DYu IÕanoÕ et alrphysics Letters B Ž Ž and in the two functions with z s y yq r m j j q y FŽ in,yin ;; z yz X fs, fs fž z Ž 3 FŽ in,yin ;; n The function f Ž z and its derivative f X Ž z are given with the help of the Gauss hypergeometric function Fa,b;c; Ž z It can be clearly seen that in the region p "H;m the amplitude M differs considerably from the g Mg Born amplitude and, therefore, the whole amplitude M differs from its Born limit M Let us stress Born that just this transverse momentum region p ;m "H gives the main contribution into the total Born cross section s and into s Born Outside this region the CC vanishes Indeed, for p <m or p 4m the variable zf, there- "H "H fore, f f, f f0 and MBorn qm smborne if Ž 4 The same is also true for the region yq <m Asa result, the CC is absent in the region q <m, i H is, Note that the region p 4m gives a "H negligible contribution to the total cross section s, however, this region might be of interest for some experiments 3 CC to the energy distribution and to the total cross section As it was explained in the previous section, the basic expression for the cross section ds in the LLA can be directly obtained using the WW approximation To show clearly the terms omitted in the LLA, we start with a more exact expression for ds derived for the case of mz collisions considered in Ref wx 8 The reason is that for the most interesting Ž " region when the energy of relativistic e pairs is much smaller than the nucleus energy the muon in the mz scattering as well as the upper nucleus of the ion-ion collision can be equally well treated as spinless and pointlike particles Using Eqs Ž 4 and Ž 7 from Ref wxž 8 given in the lab frame of the muon projectile on a nucleus target and the invariant variables x sž p P " " r Ž qp, ysž qp rž PP we obtain ds for the pair production in ZZ form Ž and at y< 4 dssy3s0f Z collisions in the invariant qj 4ya = ½ Ž qj ay ln yaq j qj 5 = dy dxqdxyd Ž xqqx yy Ž 5 y Ž with as qx qx q y, js Ž M yrm xqx y The main contribution to s is given by the region M M Ž PP <j< Ž 6 The corresponding expression for ds can be ob- tained by making the replacements ds sds Ž q q, P lp,z lz Ž 7 Below we consider only the experimentally most interesting case when in the collider system Žg s q y ErM ;g serm both e and e are ultrarelativistic Ž 4 m " We assume that the z-axis is directed along the initial three-momentum of the first nucleus P To obtain the energy distribution of e q and e y in the LLA we have to take into account two regions p 4m and Ž yp " z " z 4m where the lepton pair is produced either in forward or backward direction In the first region we have x " s " r, ys re, and from Eqs Ž 5 and Ž 6 we obtain in the LLA 4 q Ž y ds sy4s0fž Z y 3 ž / Ž mg d qd y = ln, q y m< " <mg Ž 8 In the second region we have x " f r, y f m rž 4E and q y y ž / 4 q Ž ds sy4s0fž Z y 3 = g q y d qd y ln, m m< " <mg Ž 9

159 ( ) DYu IÕanoÕ et alrphysics Letters B Summing up these two contributions, we find 4 q y d qd y dssy8 s0fž Z ž y lng / 3 Ž 0 To obtain s we have to integrate the expressions Ž 8 and Ž 9 over Ž y with logarithmic accu- racy ds sy s f Z ln, mg d Ž 8 q 9 0 q q m< q<mg, Ž Ž g q d Ž 8 q ds sy 9 s0fž Z ln, m m< q<mg, from which it follows that 8 d q 9 0 q ds sy s fž Z lng Ž 3 The further integration of Eqs and over q results in PP 8 ssy 9 s0fž Z ln Ž 4 MM This expression is in agreement with the similar result for the mz scattering Žsee Eq Ž 3 from wx 8 for Z s, Z sz The corresponding formulae for s can be ob- tained from Eqs Ž 0, Ž 3 and Ž 4 by replacing gl g, Zl Z The whole CC contribution ds sdž s qs Coul for the symmetric case Z sz sz and gsg sg takes the following form 4 q y d qd y dscoulsy6 s0 fž Z y L, 3 ž / m< " <mg, Ž 5 d q Coul 9 0 q q ds sy s f Z L, m< <mg and finally q Ž 6 56 scoulsy 9 s0 f Z L 7 Fig The total cross section of the process ZZ ZZe q e y with Ž solid line and without Ž dashed line Coulomb correction as function of the Lorentz factor g of Pb nuclei Ž Zs8 The size of this correction for the two colliders was given before in Eq Ž 4 The total cross section with and without Coulomb correction as function of the Lorentz factor g is illustrated in Fig for Pb nuclei 4 Discussion We have calculated the Coulomb corrections to e q e y pair production in relativistic heavy ion collisions for the case of colliding beams Our main results are given in Eqs Ž 5 Ž 7 We have restricted ourselves to the Coulomb corrections for the energy distribution of electrons and positrons and for the total cross section In our analysis we neglected the contributions which are of the relative order of ; Ž Za rl The CC to the angular distribution of e q e y can be easily obtained in a similar way, however only with an accuracy ;ZarL Since our basic formulae Ž 5, Ž 7 are given in invariant form, a similar calculation can be easily repeated for fixed-target experiments This interesting question will be considered in a future work And let us make a last remark about Refs w0 x The main object in these papers is the amplitude of

160 60 ( ) DYu IÕanoÕ et alrphysics Letters B the process Ž in the impact-parameter representation AŽ b given, for example, by Eq Ž 7 from Ref X wxwith the natural replacement p yp q, p py Žthe corresponding formula in Ref wx is given by Eq Ž 43 If we transform this amplitude to the amplitude in the momentum representation, used here, we immediately obtain H Ms A b e yi q H b d bse iž byc M, Ž 8 CsZ a ln q r4 qz a ln q r4, H H Born G Ž yiza G Ž yiza i b e s Ž 9 G Ž iz a G Ž iz a From this equation it follows that < M < s< M < Born or dscouls 0, which obviously contradicts the results of Refs w7,,8,9x and of the present work Moreover, it is clearly seen that Eq Ž 8 does not reproduce Eqs Ž 0 Ž 4 in any kinematical region We remind that the results of Bethe and Maximon wx 5, which are used here in the form of Eqs Ž 0 Ž 4, are the basis for our investigation Those results were confirmed in a number of papers Žsee, for example, Refs w6,5 x, using various approaches In the paper w0x it was claimed that in the differential cross section the CC is absent everywhere except of the region of small transverse momentum-transfer q i H;aZmrg It was further sug- gested there that just this region is responsible for the excess cross section over the Born contribution observed in the SPS experiments w3 x We would like to stress that this suggestion is completely ungrounded Indeed, as it was explained in Section, in the region q H< m and q H< m the main contribution to the cross section is given by the amplitude with one-photon exchange both with the first and with the second nucleus, ie by M Born Therefore, the CC is absent in the region q ; i H azmrg Acknowledgements We are very grateful to G Baur, Yu Dokshitzer, U Eichmann, V Fadin, I Ginzburg and V Telnov for useful discussions AS and VGS acknowledge support from the Volkswagen Stiftung ŽAz No Ir7 30 DYuI and VGS are partially supported by the Russian Foundation for Basic Research Žcode References wx G Racah, Nuovo Cim 4 Ž wx CA Bertulani, G Baur, Phys Rep 63 Ž wx 3 VN Baier, VS Fadin, ZhETF 6 Ž ; EA Kuraev, VG Lasurik-Elzufin, Pis ma ZhETF 3 Ž 97 39; VM Budnev, IF Ginzburg, GV Meledin, VG Serbo, Nucl Phys B 63 Ž wx 4 G Baur, K Henken, D Trautman, J Phys G 4 Ž wx 5 H Bethe, LC Maximon, Phys Rev 93 Ž ; H Davies, H Bethe, LC Maximon, Phys Rev 93 Ž wx 6 VB Berestetskii, EM Lifshitz, LB Pitaevskii, Quantum Electrodynamics Ž Nauka, Moscow, 989 wx 7 AI Nikishov, NV Pichkurov, Sov J Nucl Phys 35 Ž wx 8 DYu Ivanov, EA Kuraev, A Schiller, VG Serbo, Phys Lett B 44 Ž wx 9 SR Gevorkyan, EA Kuraev, A Schiller, VG Serbo, AV Tarasov, Phys Rev A 58 Ž ; GL Kotkin, EA Kuraev, A Schiller, VG Serbo, hep-phr98494 and Phys Rev C Ž in print w0x B Segev, JC Wells, Phys Rev A 57 Ž ; physicsr wx AJ Baltz, L McLerran, Phys Rev C 58 Ž wx U Eichmann, J Reinhardt, W Greiner, nucl-thr w3x CR Vane et al Phys Rev Lett 69 Ž 99 9; Phys Rev A 50 Ž 994 ; A 56 Ž w4x H Olsen, LC Maximon, Phys Rev 4 Ž w5x D Ivanov, K Melnikov, Phys Rev D 57 Ž w6x H Olsen, LC Maximon, H Wergeland, Phys Rev 06 Ž 957 7; VN Baier, VM Katkov, ZhETF 55 Ž

161 3 May 999 Physics Letters B Experimental verification of the doughnut scattering mechanism of a high-energy beam deflection by a bent crystal AA Greenenko a, NF Shul ga a,b,) a National Science Center KharkoÕ Institute of Physics and Technology, KharkoÕ, 3008 Ukraine b Belgorod State UniÕersity, Belgorod, Russia Received 9 July 998; received in revised form 5 March 999 PV Landshoff Abstract An analysis of the recent CERN experimental data dealt with new mechanisms of beam deflection by a bent crystal is presented The modifications of the experiment conditions necessary for observation of the beam deflection connected with the doughnut scattering are specified q 999 Published by Elsevier Science BV All rights reserved PACS: 0545qb; 97Ac Keywords: High energy particle; Beam deflection; Bent crystal; Channelling New possibilities of deflection of a particle beam while passing through a bent crystal were investigated recently at CERN wx These possibilities deal with the doughnut-scattering effect, which takes place for particles incident close to a crystallographic axis direction The experiment wx resulted in no bending effect for negatively charged particles Žp y The experiment showed also that some fraction of positively charged particles Žp q of the beam may follow along the bent crystal s axis In the present work it is shown that in the condiwx the bending effect caused tions of the experiment by the doughnut scattering does not manifest itself It is noted, that positively particle beam bending, which ) Corresponding author shulga@kiptkharkovua was detected in this experiment, may be caused by a hyperchannelling effect We show below how one has to modify the experimental conditions in order to be able to observe the beam deflection connected with the doughnut scattering Elsewhere w,3 x, the possibility of positively and negatively charged particles deflection under their multiple scattering by atomic strings was emphasized This beam-deflection mechanism differs from another one proposed by Tsyganov wx 4 The mechanism of wx 4 is based on plane channelling of positively charged particles in a bent crystal Contrary to wx 4, the deflection effect, which was considered in Refs w,3 x, takes place for over-barrier particles This result was obtained in Ref wx 3 by computer simulation of multiple scattering of charged relativistic particles by bent atomic strings The anawx 5 leads to the following lytical treatment condition r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S X

162 6 ( ) AA Greenenko, NF Shul garphysics Letters B for the realization of the bending effect Žsee also Eq Ž 8 of wx : L l H as P -, Ž Rc Rc c c where L is the crystal length, R is the crystal bending curvature, cc is the critical angle of axial channelling, and l H is the length of beam transverse momentum equalization due to multiple scattering by atomic strings For c F c one may estimate wx c 6 l Hf ccnda TF y, where n is the atomic density, d is the distance between atoms along the passage direction Ž along z-axis, and atf is the screening radius of atomic potential An attempt for the experimental detection of this effect was made in wx The passage of a proton beam with energy E s 450 GeV and a negatively pion beam with energy Es00 GeV close to the -0 ) axis of a bent Si crystal was studied The length of a bent part of the crystal was Lf3 cm in this experiment The radius of curvature was Rf0 m The beam size was = mm, and the experiment measurement base was 4 m In wx it was used the following condition for beam doughnut deflection Žsee Eq Ž 9 of wx l H - Rc c The parameters of the experiment wx satisfied the inequality Ž Note, however, that the condition Eq Ž in comparison to Eq Ž contains an additional factor Ž LrRc c, which invalidates the bending condition Ž in the case of the experiment wx Namely, the parameter a values for wx are: a q p f 83 for protons and apyf83 for pions The results of the experiment wx have demonstrated that in the case of positively charged particles a small part of the beam follows the bent axis, while in the case of negatively charged particles the bending was absent 3 For analysis of the above situation we have improved the simulation code, which was used in wx 3 to calculate the particle near-axis over-barrier passage through a bent crystal We have taken into account both the finite and infinite motion of particles in a periodic field of bent strings, as well as the noncoherent effects in scattering The last effects lead to the dechannelling and rechannelling Continuous potential of an atomic row is computed on the basis of the Moliere ` approximation for the potential of a separate atom Equipotential surfaces for this potential for the -0) crystal axis in silicon are given in Refs w7,8x Žsee Fig 67 in wx 7, Fig in wx 8 The passage of particles through a crystal is considered as a step-by-step two-dimensional movement in a plane orthogonal to the direction of the crystallographic axis Calculations of both coherent scattering due to averaged continuous potential and incoherent scattering of particles are made at each step Incoherent scattering is caused by thermal displacements of lattice atoms from the equilibrium positions and by electrons of the crystal Incoherent scattering results in a change of the transverse energy of a particle The account of the incoherent scattering is made using the assumption of Gaussian distribution of calculated values Fig presents simulation results for the condiwx These results seem in tions of the experiment good qualitative agreement with experimental data both for positively and for negatively charged particles ŽSee Fig 5 and Fig from wx The simulation statistics here is 000 particles The horizontal profile of a proton beam leaving the crystal is depicted in Fig Ž c The numerical analysis of the particles parameters shows, that for conditions of the experiment wx several mechanisms of beam deflection are realized Namely, the small peak in the vicinity of 3 mrad is due to protons, hyperchannelling along the axis - 0) The peaks in the vicinity of mrad, mrad and 05 mrad are due to protons captured and channelled in plane channels, formed by atomic strings The dependence of the relative number of axially deflected particles Žie particles within the angle range cc relative to the current direction of the crystallographic axis, u y c F u F u q c R c R c on thickness of the crystal Ž u s LrR R is shown in Fig Ž e The top curve represents a fraction of all particles deflected in the specified interval of angles The bottom curve represents the relative number of hyperchannelling protons Žaccording to Refs w7,8x the hyper channelling potential barrier in this case is about 5 6 ev The top curve is seen to merge

163 ( ) AA Greenenko, NF Shul garphysics Letters B rapidly with bottom one, ie, the axially deflected protons are hyperchannelling The horizontal profile and the relative number of axially deflected particles for a p y beam are shown respectively in Fig Ž d and Fig Ž f One can see, that for the negatively charged particles the effect of axial bending is absent under the given conditions For the p y ybeam, effects of axial and planar channelling, responsible for the bending of proton beam, are suppressed The reason is that the negatively charged particles pass close to the sites of the lattice, ie they are influenced by the atomic thermal fluctuations, which cause strong noncoherent scattering and dechannelling Therefore, the effect of beam bending caused by the doughnut scattering of particles by atomic strings was absent in experiment wx, in complete agreement with the violation of the inequality Ž for conditions of the experiment wx The requirement Ž prompts how one has to change the conditions of the experiment to make the examined effect observable 4 Fig presents simulation results for protons and pions with the same energy as in wx, but for a crystal with the thickness Ls0 cm and the curvature radius R s 00 m Length and curvature values were chosen to satisfy the condition in Eq Ž The values of parameter a for both particles in this case are: apqfr3 for protons and apyf0 for pions The beam size in the calculations is assumed to be the same as in wx and the observation base is 0 m now Fig Beam passage through a bent silicon crystal with length Ls3 cm and curvature radius Rs0 m near the -0) axis: Ž a,b angular particle distributions; Ž c,d horizontal beam profile; Ž e,f relative number of all deflected particles Župper curve and particles executing finite motion in the field of atomic strings as functions of the distance passed Ž q s LrR ; Ž a,c,e R y 450 GeV protons; b,d,f 00 GeV p -pions; dashed line in Ž c,d corresponds to the beam profile after passage through 3 cm layer of amorphous silicon; coordinates of initial beam center are -6 q X,qY s 0,0, beam divergence is 3=0 rad; coordinates of end axis direction are Ž q,q sž 3,0 X Y mrad; beam size is mm = mm; observation base is 4 m Fig The same as in Fig, but for 0 cm crystal length, 00 m curvature radius, and 0 m observation base; filled circles in Ž d correspond to the horizontal profile of p y -pion beam from Fig d Ž

164 64 ( ) AA Greenenko, NF Shul garphysics Letters B Simulation results show that under these conditions the deflection of positively as well as negatively charged particles takes place The horizontal profiles of a proton and pions beams leaving the crystal are depicted respectively in Fig Ž c and Fig Ž d One can see that the significant fraction of beam particles appears deflected on the whole angle of crystal bending about 05 mrad Besides, Fig Ž e and Fig Ž f testify, that the deflection of a large fraction of particles is caused by the doughnut mechanism of scattering Namely, for protons the fraction of particles which execute an infinite motion relative to atomic strings is of the order 65% of all particles axially deflected on the whole angle, and for p y -pions this fraction amounts to about 95% The profile of the deflected beam of p y is given in Fig Ž d The comparison of this profile with the profile from Fig Ž d made on the same scale shows, that the implementation of the condition Ž results not only in relative, but also in absolute increase of the efficiency of beam deflection by a crystal At the end we note, that due to crystal axis bending, the doughnut scattering results in the increase of the beam angular divergence with thickness This fact is accounted for in Eq ŽŽ see Ref wx 5 Therefore, the inequality Ž is the sufficient condition of the beam deflection due to the doughnut scattering The inequality Ž does not depend on the crystal thickness and can be considered only as necessary Ž but not sufficient condition of the realization of the above mechanism of the beam deflection The authors hope that the above analysis will promote the experimental detecting of positively and negatively charged particles deflection caused by the doughnut scattering mechanism in a bent crystal Acknowledgements This work was carried out with partial financial support of the Ukrainian Fundamental Research Fund Ž project LPM-effect and of the RFFR Žproject a References wx A Baurichter et al, Nucl Instr and Meth B 5 Ž wx JF Bak et al, Nucl Phys B 4 Ž 984 wx 3 AA Greenenko, NF Shul ga, Nucl Instr and Meth B 90 Ž wx 4 EN Tsyganov, Fermilab TM-68, TM-684, Ž Batavia, 976 wx 5 NF Shul ga, AA Greenenko, Phys Lett B 353 Ž wx 6 J Lindhard, Dansk Vid Selsk Math Phys Medd 34 Ž 965 wx 7 AIAkhiezer and NF Shul ga, High-Energy Electrodynamics in matter, Gordon and Breach, Amsterdam Ž 996 wx 8 JF Bak et al, Nucl Phys B 30 Ž

165 3 May 999 Physics Letters B Flavour-symmetry correction to the naıve Zweig rule for the scalar-meson flavour singlet Eef van Beveren a,, George Rupp b, a Departamento de Fısica, UniÕersidade de Coimbra, P-3000 Coimbra, Portugal b Centro de Fısica das Interacçoes Fundamentais, Instituto Superior Tecnico, Edifıcio Ciencia, ˆ P-096 Lisboa Codex, Portugal Received February 999 Editor: L Montanet Abstract We present flavour-symmetric couplings for the OZI-allowed three-meson vertices of effective meson theories, which, for the case of the two-meson channels to which the flavour-singlet scalar meson couples, are endowed with a correction factor with respect to the standard formula q 999 Elsevier Science BV All rights reserved PACS: 440Cs; 39Pn; 375Lb Introduction Three-meson vertices give rise to the most important strong interactions which are considered by effective meson theories, as they reflect the simple fact that, by quark-pair creation, mesons couple to pairs of mesons according to processes of the form C l AqB Ž Within multiplets of SU3-flavour or U3-flavour symmetry, the relative magnitudes of the transition amplitudes for processes Ž are given by Ž A B C B A C ltr M M M T "M M M T, where MX is the 3=3 flavour matrix for meson X It is understood in formula Ž that either the symmetric or the antisymmetric trace is to be taken, depending on the sign of the product of the three eef@malapostafisucpt george@ajaxistutlpt charge-conjugation quantum numbers This way, charge conjugation and G-parity are automatically preserved Relation Ž is often referred to as the Zweig rule, since it fully suppresses exactly those two-meson decay modes which do not meet Zweig s criteria for meson-pair decay as given in Ref wx, and, moreover, agrees with the quark-line rules for Quantum Chromodynamics in the limit of large Nc as devel- oped in Ref wx The constant l in front of expression Ž can be adjusted to experiment for each different set of three U -flavour nonets Ž 3 or octets and singlets in the case of SU -flavour 3, such that the interaction Lagrangian for the theory may contain a long sum of all possible three-meson vertices, each with a different coupling constant However, in the absence of a prescription for the relative intensities among the thus occurring terms in the interaction Lagrangian, one easily overlooks Žsee eg Ref wx 3 an inconsistency in the procedure for the scalar flavour-singlet two-meson r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

166 66 ( ) E Õan BeÕeren, G RupprPhysics Letters B transition modes if a unified coupling is introduced, as we will explain in the following Note that, especially in the case of the scalar mesons, the employment of truly flavour-independent couplings may be crucial for the obtainment of reliable predictions in unitarised meson models, due to the large and highly nonlinear coupled-channel effects in these systems wxž 4 see also Ref wx 5 All three-meson vertices and specific examples In Ref wx 6 we describe how the various values for the coupling constants l can be unified Our strategy becomes manageable, if we assume equal effective quark masses in the harmonic-oscillator expansion, resulting in a finite, albeit large, number of possible meson-pair channels for each type of meson, which are all just given by the recoupling of the four involved valence Ž anti quarks As described in Ref wx 6, the coupling constants boil down to the following expression g Tr MA MB MC T =² J, L,S, N; Ž j, l,s,n; A, Ž j, l,s,n;b = P Ž J, l,s,n;c: qmb MA MC T = ² J, L,S, N; Ž j, l,s,n; A, Ž j, l,s,n;b : 4 =P Ž J, l,s,n;c, Ž 3 where g is universal, ie, the same for all possible three-meson vertices The quantum numbers J, L, S, and N in formula Ž 3 represent the total angular momentum, the relative orbital angular momentum, the total spin, and the relative radial excitation of the A q B two-meson channel, respectively, whereas Ž j, l, s, n; X represent the corresponding quantum numbers for the qq-system that describes meson X P represents the exchange operator for quarks and P for antiquarks The matrix elements ² J, L,S, N; Ž j, l,s,n; A, Ž j, l,s,n;b = P Ž J, l,s,n;c: Ž 4a and ² J, L,S, N; Ž j, l,s,n;b, Ž j, l,s,n; A : = P Ž J, l,s,n;c Ž 4b determine the relative coupling constants for the various OZI-allowed three-meson vertices They result from Fermi statistics applied to the valence Ž anti quarks and Bose statistics to the meson pair in the four-particle harmonic-oscillator expansion, which is not be confused with either quark dynamics or quark wave functions Expressions Ž 4a and Ž 4b are equal, up to a factor ", depending on the sign of the product of the three charge-conjugation quantum numbers, which is equivalent to the choice of sign in formula Ž Total spin J, parity, and charge conjugation are conserved, and the OZI-rule ŽRefs w,7x is respected by formula Ž 3 The recoupling scheme is outlined in Ref wx 6, whereas more details on the evaluation of the recoupling matrix elements Ž 4 can be found in Ref wx 8 In order to make our point, instead of exhibiting all details of the calculation, we just give the results for three cases: the two-meson transitions of pseudoscalar, vector, and scalar mesons Table shows the nomenclature we use for the relevant mesons in this paper The squares of the transition amplitudes to all channels which couple within our procedure are given in Table for pseudoscalar mesons, in Table 3 for vector mesons, and in Table 4 for scalar mesons In order to keep the tables as condensed as possible, and since we assume that isospin is indeed a perfect symmetry, we may represent all members of an isomultiplet by the same symbol Žt for isotriplet, d for isodoublet, 8 for the isosinglet flavour-octet member, and for the flavour singlet Now, let us just analyse one horizontal line of one of the three tables, to make sure that the reader understands what the numbers stand for Let us take Table Nomenclature of mesonic qq systems relevant to this paper The columns respectively contain our notation for the mesons, the qq quantum numbers Žn is radial quantum number, s is total spin, l is orbital and J is total angular momentum, and the more common quantum numbers ŽJ, parity P sy lq, and charge conjugation CsŽ y lq s Symbol Ž nq l J sq PC J yq X 0 yq 3 0 yy 0 X 3 yy 0 3 yy 3 qq 3 0 qq qy P S 0 P S 0 V S V S V D S P 0 T P U P

167 ( ) E Õan BeÕeren, G RupprPhysics Letters B Table Transition intensities for the coupling of pseudoscalar mesons to meson pairs The interpretation of the content of the table is explained in the text Decay products Flavour channels and totals for P A B rel SU -octet members SU singlets 3 3 LSN isotriplets isodoublets isoscalars Ž Ž Ž t d 8 tt dd t8 t T td d8 d T tt dd 88 8 T tt dd 88 T P S V0 U V0 T P V V0 V the fourth line of Table In the first column, under A, we find P for meson A, which hence characterises a meson out of the lowest-lying pseudoscalar nonet In the second column, under B, we similarly find that meson B represents a meson out of the lowest vector nonet In the third column, we find the quantum numbers for the relative motion of A and B, ie, P-wave Ž Ls with total spin one Ž Ss, in the lowest radial excitation Ž N s 0 Since the table refers to two-meson transitions of the lowest pseudoscalar meson nonet Ž P, indicated in the top of the table, the next four columns refer to its isotriplet member, which is the pion We then find that the pion couples with a strength ' r6 to the tt Žisotri- plet-isotriplet channel, which, following the abovediscussed particle assignments for A and B, ie, pseudoscalar and vector respectively, represents the pr channel Following a similar reasoning, we find that the pion couples with a strength ' r to KK ) The total coupling of a pion to pseudoscalar- Table 3 Transition intensities for the coupling of vector mesons to meson pairs Decay products Flavour channels and totals for V 0 A B rel SU -octet members SU singlets 3 3 LSN isotriplets isodoublets isoscalars Ž Ž Ž t d 8 tt dd t8 t T td d8 d T tt dd 88 8 T tt dd 88 T P U P T V0 U V0 T S V P P V0 V P V V0 V

168 68 ( ) E Õan BeÕeren, G RupprPhysics Letters B Table 4 Transition intensities for the coupling of scalar mesons to meson pairs Decay products Flavour channels and totals for S A B rel SU -octet members SU singlets 3 3 LSN isotriplets isodoublets isoscalars Ž Ž Ž t d 8 tt dd t8 t T td d8 d T tt dd 88 8 T tt dd 88 T P P X P P V0 V X V V V0 V U U S S T T P T V U V V 0 vector channels is given in the column under T by 'r4, which is the square root of the quadratic sum of the two previous couplings, ie, ' r6qr The next set of coupling constants refer to the two-meson transitions of a kaon We find ' r8 to td, which represents both of the possibilities pseu- ) doscalar isotriplet q vector isodoublet, ie, p K, and pseudoscalar Ž isodoublet q vector Ž isotriplet, ie, K r, each with half the intensity given in the table Next, we find in the table that the kaon couples with ' r8 tod8, which represents both of the possibilities pseudoscalar Ž isodoublet q vector Ž SU -octet isoscalar 3, ie, K q octet mixture of v and f, and pseudoscalar Ž SU -octet isoscalar 3 q X vector Ž isodoublet, ie, octet mixture of h and h q K ), each with half the intensity given in the table The kaon does not couple to the d channels in pseudoscalar q vector, which represent the channels with one isodoublet and one SU3 singlet The total coupling for the kaon to its pseudoscalar q vector channels sums up to ' r4, as one verifies in the column under T The next two sets of coupling constants similarly refer to the two-meson transition modes of the isoscalar, either SU3-octet or SU3-sing- let, partners of the pseudoscalar nonet A remark is here in place: when we identify the flavour-isotriplet members of the lowest-lying harmonic-oscillator state with pions, the isodoublet states with kaons, and so on, then we actually have in mind that the corresponding coupling constants of their three-meson vertices are to be folded in a coupledchannel model where the real mesons come out as bound states and resonances Žsee eg Refs w3,9,0 x Hence, the above particle identification should not be taken too literally At best, one may identify the harmonic-oscillator states with objects that do not really exist in Nature, the so-called bare hadrons, ie, valence qq-systems which are forbidden to couple to two-meson channels by means of valencequark-pair creation Nevertheless, the unification of the coupling constants can only be achieved by taking into account the internal structure of the three mesons involved, for which we have chosen here harmonic oscillators 3 A closer look at the results From the three tables we may notice the following: The intensities Ž couplings squared for all strong two-meson transition modes of the pion Žcolumns 4 to 7 in Table add up to Žnumber at the very bottom of the eighth column ; and the same result

169 ( ) E Õan BeÕeren, G RupprPhysics Letters B holds for the couplings to the two-meson channels of all other pseudoscalar nonet members, as well as for vector and scalar mesons Ž Tables 3 and 4 The reason for this property is the wave-function normalisation for the recoupling matrix elements of formula Ž 3, which this way translates the flavour independence of strong interactions, very recently reconw x The subtotals Ž columns under T for the strong firmed by experiment two-meson transition intensities of the octet members are equal for each different mode Žone horizontal line in each of the tables This translates SU3- flavour independence of the strong interactions 3 The subtotals of the flavour-singlet pseudoscalar and vector mesons are either twice as large as those of the flavour-octet members, or zero, in such a way that in both cases the total intensity adds up to Unfortunately, the tables for axial vectors, tensors, etc are too long to be shown here in a manageable form Nevertheless, let us just mention that for all higher quantum numbers we find similar factors two and zero for the flavour-singlet couplings, with only one exception: the scalar mesons Ž Table 4 4 If one uses formula Ž, one similarly obtains these factors two and zero However, in this case the same would hold for scalar-meson transitions, contrary to our findings In our procedure Žformula Ž 3, we thus find full flavour independence for all strong two-meson transitions of all mesons, whereas with formula Ž the flavour-singlet scalar couples twice as strongly to its two-meson channels This feature of the three-meson couplings can only be fully appreciated once all two-meson channels Ž open and closed are taken into account, which is much easier when their number is finite, and which takes a particularly manageable form in the harmonic-oscillator expansion for equal effective quark masses 4 Conclusions The flavour singlet of scalar mesons has the quantum numbers of the vacuum Ž< 0 :, which we also believe to be the quantum numbers of the valence qq-pair created in OZI-allowed strong twomeson transitions Now, in general, the normalised sum of two orthonormal states < f: and < 0: is given by Ž< f: q< 0: r' However, when < f: s< 0: Žin which case they are not orthogonal, then the correctly normalised sum is given by Ž< f: q< 0: r This is exactly the reason for the extra factor r' which we find with our procedure Hence, we propose to modify formula Ž into ² : Tr MA MB MC T "MB MA MC T l, Ž 5 ( q C flavour-singlet scalar meson in order to restore universal flavour independence for the three-meson vertices of effective theories for strong interactions 5 Epilogue Finally, we should mention that the Zweig rule for strong decays does not necessarily imply that singlets couple twice as strongly as octet members On the contrary, the couplings for three-meson vertices for both subsets of the flavour nonet may be chosen independently in SU3-flavour-symmetric theories Moreover, we do neither assume here that the numbers of the three tables Ž, 3, and 4 are the rigorously correct relative intensities for three-meson vertices, nor that the number of two-meson channels must be finite For that, both the limit of equal effective valence quark masses and the harmonic oscillator expansion are probably too crude approximations Those numbers are principally meant to pinpoint the scalar flavour-singlet problem Nevertheless, in the light of the promising results of Ref w0 x, in which works a unified coupling for pseudoscalars, vectors, and scalars has been applied, the tables gain some credibility Furthermore, the anti-de Sitter geometry w x, which has recently revived w3x as a possible candidate for quark confinement w4 x, is well approximated by a harmonic oscillator of universal frequency This might provide an additional justification for the here employed harmonic-oscillator expansion, as not just a purely mathematical tool

170 70 ( ) E Õan BeÕeren, G RupprPhysics Letters B Acknowledgements We wish to thank H Walliser and A Blin for useful discussions and suggestions References wx G Zweig, An SU3 model for strong interaction symmetry and its breaking, CERN Reports TH-40 and TH-4 Ž 964 wx G t Hooft, Nucl Phys B 7 Ž ; G Veneziano, Nucl Phys B 7 Ž ; E Witten, Nucl Phys B 56 Ž wx 3 Nils A Tornqvist, Zeit Phys C 68 Ž wx 4 Eef van Beveren and George Rupp, Comment on Understanding the scalar meson qq nonet, hep-phr Ž 998, submitted for publication wx 5 Deirdre Black, Amir H Fariborz, Francesco Sannino, and Joseph Schechter, Phys Rev D 58 Ž ; Putative light scalar nonet, hep-phr Ž 998, submitted for publication; Amir H Fariborz and Joseph Schechter, h X hpp decay as a probe of a possible lowest-lying scalar nonet, hep-phr99038 Ž 999 ; Shin Ishida, Muneyuki Ishida, Taku Ishida, Kunio Takamatsu, and Tsuneaki Tsuru, Prog Theor Phys 95 Ž ; M Napsuciale, Scalar meson masses and mixing angle in a U 3=U3 linear sigma model, hep-phr Ž 998 wx 6 Eef van Beveren and George Rupp, Flavour symmetry of mesonic decay couplings, hep-phr Ž 998, submitted for publication wx 7 S Okubo, Phys Lett 5 Ž ; J Iizuka, K Okada, O Shito, Prog Theor Phys 35 Ž wx 8 E van Beveren, Zeit Phys C 7 Ž ; C Ž wx 9 E Eichten et al, Phys Rev Lett 36 Ž ; Phys Rev D Ž ; Nils A Tornqvist, Phys Rev Lett 49 Ž 98 64; Nils A Tornqvist and Matts Roos, Phys Rev Lett 76 Ž ; Veronique Bernard, Ulf G Meissner, Alex Blin, and Brigitte Hiller, Phys Lett B 53 Ž w0x E van Beveren, C Dullemond, G Rupp, Phys Rev D, 77; Ž E D Ž ; E van Beveren, G Rupp, TA Rijken, C Dullemond, Phys Rev D 7 Ž ; E van Beveren et al, Zeit Phys C 30 Ž wx K Abe, Phys Rev D 59 Ž wx E van Beveren, C Dullemond, TA Rijken, Phys Rev D 30 Ž ; E van Beveren, TA Rijken, C Dullemond, G Rupp, Geometric quark confinement and hadronic resonances, in: S Albeverio, LS Ferreira, L Streit Ž Eds, Resonances - Models and Phenomena, volume Lecture Notes in Physics, pages , Bielefeld Ž 984 ; C Dullemond, TA Rijken, E van Beveren, Il Nuovo Cim 80A Ž ; E van Beveren, TA Rijken, C Dullemond, J Math Phys 7 Ž w3x Juan Martın Maldacena, Adv Theor Math Phys Ž w4x IV Volovich, Large-N gauge theories and the anti-de Sitter bag model, hep-thr Ž 998 ; Nicholas Dorey, Timothy J Hollowood, Valentin V Khoze, Michael P Mattis, and Stefan Vandoren, Multi-instanton calculus and the adsrcft correspondence in Ns4 superconformal field theory, hepthr9908 Ž 999

171 0 May 999 Physics Letters B Crossing and anticrossing of energies and widths for unbound levels P von Brentano ), M Philipp Institut fur Kernphysik, UniÕersitat zu Koln, Zulpicher Str 77, Koln, Germany Received 9 November 998; received in revised form 8 January 999 Editor: J-P Blaizot Abstract The crossing and anticrossing properties of the energies and widths of two unbound levels under the influence of a symmetrical complex interaction are investigated It is found that a sufficiently large variation of the difference of the unperturbed energies or of the widths leads always to a crossing of either the energies or the widths of the perturbed system A particularly interesting result is that for a real off diagonal interaction there is a joint crossing of the unperturbed energies and of either the perturbed energies or the perturbed widths q 999 Elsevier Science BV All rights reserved PACS: 0365-w; 0380qr; 380Bx; 0-k The two level system is a fruitful tool in physics and has many applications w 5 x One usually considers, the properties of a system of two bound states It is of interest to extend this study from bound states to unbound states w6 5 x Interesting examples of unbound two level systems are eg: p q 8 The I s, Ts, Ts0 doublet in Be w6,7 x The r v T s, T s 0 doublet of mesons w9,8,9 x 3 Doublets of resonance s in cavities w0 x ) brentano@ikpuni-koelnde Present address: SAP AG, Neurottstr 6, 6990 Waldorf, Germany This paper discusses the crossing and anticrossing of energies and widths of the two level system for unbound levels For the system of two bound states it is known that the difference D EsEyE of the energies E and E can not vanish if the off diagonal matrix element of the interaction does not vanish w 6 x In short: the energies of two bound states, anticross for a non vanishing offdiagonal interaction: n / 0 This statement is a special case of a theorem of Wigner and von Neumann wx In the present paper the crossing and anticrossing of unbound levels is studied The energy of an unbound level is in general a complex number: seyir G Ž Here EsRe is the real energy and GsyIm is the width of the unbound state The complex r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

172 7 ( ) P Õon Brentano, M PhilipprPhysics Letters B energy difference y of two unbound states is thus also in general a complex number y sdeyir DG There are different possible cases of crossing and of anticrossing in the unbound two level system depending on the vanishing or non vanishing of the real part D E or of the imaginary part yrd G of the complex energy difference y We mention that there have been several works in which the concept of energy repulsion and width attraction was discussed for a system of two unw6,7 x This problem is related to bound states but different from the crossing and anticrossing relation which will be discussed here Furthermore the crossing and anticrossing relations are derived here for the general case of a = matrix with a symmetrical complex off diagonal interaction whereas the energy repulsion width attraction relations were derived previously only for an off diagonal interaction which was either real or purely imaginary Before the crossing problem is discussed in detail, the meaning of the effective Hamiltonian for unbound states will be clarified A proper description of the unbound system is done in the frame of an S-matrix We can define an effective Hamiltonian H from the propagator of the S-matrix A convenient form of an unitary S-matrix which exhibits the propagator has been given by Mahaux and Weidenmuller in their book wx and was also used by other authors eg: w3 5, 5 x Time reversal invariance is assumed and therefore the S-matrix is symmetric For a system with two unbound states this leads to the following representation of the S-matrix w x: 4 y t q S E su yiw D E W U, 3 DŽ E nm sed nm yh nm with sed yh qriž W t W, Ž 4 nm nm ) ) h sh,w sw and n,mg w,x Ž 5 nm nm cn cn Here Wcn is the M= matrix of the decay ampli- tudes which couple the M channels to the levels U is a unitary matrix, which describes the background nm From time reversal invariance one obtains further Ž t that the width matrix G s WW nm nm and the en- ergy matrix hnm are real and symmetric = matri- ces The width matrix GsW t W is a positive semidefinite matrix Combining the energy matrix hnm with the width matrix Gnm one obtains the effective Hamiltonian matrix: Hshyir G One can write the effective symmetric Hamilto- Ž nian H in the form of Eq 6 Hsž H H / H H ž / E 0 yir G 0 nyirv s Ž nyirv DE yir G It is further assumed for simplicity that the effective Hamiltonian H is energy independent This is a reasonable assumption far from thresholds This form allows the standard decomposition of H into an unperturbed effective Hamiltonian H 0 and a complex off diagonal interaction V: ž / sž H nyirv / 0 s 0 0 nyirv 0 Ž 7 The special form n y ir v of writing the off diagonal complex interaction matrix element is used in order to simplify the relations below The poles and of the S-matrix are identical with the complex eigenenergies of the effective Hamiltonian and are given by the solutions of Eq Ž 8 : det Ž d yh s0 Ž 8 nm nm from which the well-known expressions for the complex energies of the two level system are obtained w 6 x:,srž HqH r Ž "r H yh q4h H 9

173 ( ) P Õon Brentano, M PhilipprPhysics Letters B From Eq Ž 9 one obtains the square of the difference of the complex energies Ž y : 0 0 Ž y s Ž y q4ž nyirv s Ayi B Ž 0 In order to discuss Eq Ž 0 it is useful to consider the differences of the energies D E and of the widths D G of the two perturbed states and of D E 0 and D G 0 of the unperturbed states By decomposing Eq Ž 0 into its real part: A and its imaginary part: yb one obtains Eqs Ž a and Ž b : 0 0 Ž DE y r4ž DG s Ž DE y r4ž DG q4 n yr4v sa Ž a Ž DEDG s Ž DE 0 DG 0 q4nvsb Ž b These equations are the basis of the following discussions of the crossing and anticrossing of the levels One notes that the functional dependence of the quantities A and B on the parameters of the Hamiltonian H of Eqs Ž and Ž 6 are given by Eqs Ž a and Ž b To begin the discussion we first consider the full complex energy crossing ie the case: s One finds the Eq Ž : s mas0 and Bs0 Ž Eq Ž gives the conditions for full complex energy crossing Such crossing is possible in the case of unbound levels also for a nonvanishing interaction nyirv/0 This was noted before w8 x The reason is, that a complex symmetrical = matrix has more parameters than the real symmetrical = matrix and these many parameters make it possible to fulfil the two relations As0 and Bs0 also for a non vanishing off diagonal interaction The complex energy crossing has been discussed in great detail by Mondragon and Hernandez w x We now discuss the partial crossing which is particularly interesting for the unbound system Namely we consider that either the energy difference D E, or the width difference D G vanish One finds in both cases that the parameter B must vanish One obtains Eq 3 : Bs0m Ž DEDG s0mdes0ordgs0 Ž 3 Eq Ž 3 contains the logical or, which includes of course the possibility that D E and D G vanish simultaneously as was discussed above Eq Ž 3 is a crossing anticrossing relation One finds further that the sign of A specifies whether there is energy crossing Ž D Es0 or width crossing Ž D Gs0 as is shown in Eqs Ž 4a and Ž 4b : A)0 and Bs0mDE/0 and DGs0 Ž 4a A-0 and Bs0mDEs0 and DG/0 Ž 4b Relation Ž 4a is interesting It states that for B s 0 energy anticrossing implies width crossing For bound states the relation is trivial because the widths vanish everywhere The relation is nontrivial for unbound states, however Eq Ž b implies that by varying D E 0 or D G 0 in a sufficiently large range while keeping the other parameters of H constant one can make Bs0 Thus in cases where D E 0 or D G 0 can be varied in the experiment in a sufficiently large range one finds either energy or width crossing The width crossing relation is a rather general, somewhat surprising and interesting result The conditions under which it holds can be realized in experiments Particularly simple and strong results are obtained for a special off diagonal interaction for which nvs 0 That is for either a real or a purely imaginary off diagonal interaction One finds: nvs0and Bs0m Ž DEDG s Ž DE 0 DG 0 s0 Ž 5 Thus for this special interaction the perturbed widths or energies will cross at the crossing point of the unperturbed widths or energies The question whether the perturbed widths or energies cross depends again on the sign of A as is shown in Eqs Ž a and Ž b One finds Eqs Ž 6a and Ž 6b : < < < 0 < 0 n ) r DG and DE s0 and vs0 EDE DGs0 and and DE/0 Ž 6a EDE 0 < < < 0 < 0 n - r DG and DE s0 and vs0 EDG DEs0 and and DG/0 Ž 6b EDE 0

174 74 ( ) P Õon Brentano, M PhilipprPhysics Letters B In Eqs Ž 6a and Ž 6b all quantities of the perturbed system as eg D E, D G, treated as functions of the parameter D E0 are to be taken at the value D E 0 s0 One can also obtain corresponding relations for a variation of the parameter D G 0 The 0 0 derivatives EDErEDE and EDGrEDE in Eqs Ž 6a and Ž 6b are obtained from Eqs Ž a and Ž b by keeping all parameters Ž D G,n, v 0 constant except D E The relation Ž 6a 0 implies that for a sufficiently large real interaction with < n < ) < r D G 0 < the three quantities D E 0, D G and EDGrEDE 0 will vanish jointly wheras D E does not vanish for D E 0 s0 Eq Ž 6b gives the conditions for a joint vanishing of the three quantities D E 0, D E and EDGrEDE 0 whereas D G does not vanish for D E 0 s0 One notes that Eq Ž 6a is well known for boundstates It shows that the difference of the energies Ž D E sže ye of the two states has an extremal value at the Ž 0 crossing point of the unperturbed energies D E s0 It should be stressed that the joint crossing of the three quantities is found for purely real or imaginary interactions For a general complex interaction the three quantities will not vanish jointly in general It is useful to derive the width crossing relation directly in a simple model The electromagnetic decay of a system of two interacting bound states c and c to the groundstate cg is considered in perturbation theory The system is described by the Hamiltonian of Eq Ž 6 with v s 0 and vanishing unperturbed widths: G0 sg 0 s0 One assumes that the two states c, c decay by electromagnetic E-transitions to the unperturbed groundstate < c 0 : s< c : g g Thus one obtains the transition decay widths G t and G t in perturbation theory by Eq Ž 7 : t G <² 0 < < :< t <² 0 < < :< gs cg E c a, Ggs cg E c a Ž 7 where a is a constant We use the same relations also for the unperturbed case One assumes in this model for simplicity that in the unperturbed system H 0 there is only one nonvanishing transition width G t g 0 /0, G t g 0 s0 With these assumptions one can calculate the difference of the widths: D G t s Ž G t g yg t g as a function of the parameters D E 0 and n D G t is directly related to the difference of the amplitudes a and b of the expansion of the perturbed states into the unperturbed states which is given in Eq Ž 8 : csac 0 qbc 0, csybc 0 qac 0 Ž 8 t Ž One finds DG s a y b This shows the origin of width crossing Namely the amplitudes a and b cross at the parameter value D E 0 s0 This model gives an intuitive explanation for width crossing in terms of a complete mixing of the two states This model is valid of course only for very small widths, when perturbation theory applies We do not know a similarly intuitive explanation for energy crossing Summing up the anticrossing relation for the energies in a system of two bound states is extended to crossing anticrossing relations for the energies and widths of a system of two unbound states Since the energies of the unbound states are complex numbers there is a rich physical scenario for crossing and anticrossing of the real or imaginary parts of the complex energies A particularly interesting result is that in a system of two unbound states with an off diagonal interaction there is a value of the difference of the real unperturbed energies D E0 for which the perturbed energies or the perturbed widths cross An even stronger result is obtained for a sufficiently strong real interaction For such an interaction one finds that the three quantities D E 0, D G and EDEr EDE 0 will vanish jointly This is a rather general result because the conditions for which it holds are rather weak and can be realized in experiments Finally we give an experimental example for which the scenario of joint unperturbed energy crossing: D E 0 s0 and perturbed width crossing: D Gs0 is at least approximately fullfilled This example is the famous doublet of I p s q Ts0 and Ts resonances in 8 Be These resonances were studied both in experiment and in analysis in great detail, eg by Hinterberger etal w6 x They have become one of the most completely studied exam-

175 ( ) P Õon Brentano, M PhilipprPhysics Letters B ples of resonance doublets in nuclear physics Žsee also references in w6 x We follow here an analysis w7x which used the same form of the S-matrix as is used in the present paper The parameters, and 0, 0 and n ob- 8 tained for the Be doublet in Ref w7x are: s Ž 67y ir P 08 kev s Ž 700y ir P 74 kev 0 s Ž 6838yirP8 kev 0 s6893 kev ns48 kev From these parameters one can find the parameters D, D 0 and n One finds: D 0 sž y55yirp8 kev D sž y88yirp35 kev As the interaction n s 48 kev is real, Eq Ž 5 holds This is indeed true: ( 88=35s55=8 We note that although both D E0 and D G both do not vanish they are both rather small compared to < D E< and < D G < respectively: 0 < < DE s55 kev<88 kevsde 0 < < < < and DG s35 kev<8 kevs DG 0 Thus although there is no true perturbed width crossing in 8 Be the system is near to such a crossing We also note that the interaction < n < is large: < n < s 98 kev)9 kevs< r D G < and thus Eq Ž 6a 0 implies near perturbed width crossing as found in the experiment The two resonances in 8 Be have also been used as an example of a near complex energy crossing w x Other applications may be found in atomic physics or in microwave cavities One can consider eg that the energies of an atom are changed by a magnetic field or one can consider a set of nearly identical coupled microwave cavities in which the energies widths and the interaction n of the individual caviw0,6 x ties can be separately varied Acknowledgements We thank Dr G Pascovici for many discussions References wx J von Neumann, EP Wigner, Z Phys 30 Ž wx LD Landau and EM Lifschitz, Quantenmechanik 8 Auflage, Akademie-Verlag Berlin, 988, Ch wx 3 The Feynman lectures, Vol 3 Quantum mechanics, RP Feynman, RB Leighton and M Sands, Addision-Wesley Publishing Co, London, 970, Chs 9 wx 4 MV Berry, M Wilkinson, Proc Roy Soc London A 39 Ž wx 5 MV Berry, Proc Roy Soc London A 39 Ž wx 6 C Cohen-Tannoudji, B Diu, F Laloe, Mecanique quantique, Herman, Paris, 973, Ch 4 wx 7 H Friedrich, D Wintgen, Phys Rev A 3 Ž wx 8 P von Brentano, Phys Lett B 38 Ž 990 wx 9 P von Brentano, Physics Report 64 Ž w0x E Hernandez, ` A Jauregui, A Mondragon, Rev Mex Fis 38 Ž S Ž 99 8 wx A Mondragon, E Hernandez, ` J Phys A 6 Ž wx E Hernandez, ` A Mondragon, Phys Lett B 36 Ž 994 w3x VV Sokolov, VG Zelevinski, Phys Lett B 0 Ž w4x VV Sokolov, VG Zelevinski, Nucl Phys A 504 Ž w5x VV Sokolov, VG Zelevinski, Ann Phys Ž NY 6 Ž w6x F Hinterberger et al, Nucl Phys A 99 Ž and references therein w7x P von Brentano, Phys Lett B46 Ž w8x LM Barkov et al, Nucl Phys B 56 Ž and references therein w9x RP Feynman, Photon Hadron Interactions, WA Benjamin, 97 w0x H Alt et al, Nucl Phys A 560 Ž and references therein wx C Mahaux and HA Weidenmuller, Shell-Model approach to nuclear reactions, North-Holland, Amsterdam, 969, p 5 wx H Feshbach, Theoretical Nuclear Physics, nuclear reactions, Wiley, New York, 99 w3x JJM Verbaarschot, HA Weidenmuller, MR Zirnbauer, Phys Rep 9 Ž w4x VD Kirilyuk, NN Nikolaev, LB Okun, Yad Fiz 0 Ž w5x VD Kirilyuk, NN Nikolaev, LB Okun, Sov J Nucl Phys 0 Ž w6x H Alt et al, Phys Rev Lett 8 Ž

176 0 May 999 Physics Letters B On the treatment of NN interaction effects in meson production in NN collisions C Hanhart a,b,c,, K Nakayama b,d, a Institut fur Theoretische Kernphysik, UniÕersitat Bonn, D-535 Bonn, Germany b Institut fur Kernphysik, Forschungszentrum Julich GmbH, D-545 Julich, Germany c Department of Physics and INT, UniÕersity of Washington, Seattle, WA 9895, USA 3 d Department of Physics and Astronomy, UniÕersity of Georgia, Athens, GA 3060, USA Received 4 September 998; received in revised form March 999 Editor: J-P Blaizot Abstract We clarify under what circumstances the nucleon nucleon final state interaction fixes the energy dependence of the total cross-section for the reaction NN NNx close to production threshold, where x can be any meson whose interaction with the nucleon is not too strong It is shown that the results obtained from the procedure used recently by several authors to include the final state interaction in the reactions under discussion should be interpreted with caution In addition, we give a formula that allows one to estimate the effect of the initial state interaction for the production of heavy mesons q 999 Published by Elsevier Science BV All rights reserved PACS: 540yh; 375Cs Keywords: Final state interaction; Initial state interaction; Meson production As early as 95 K Watson pointed out under what circumstances one expects the final state interaction to strongly modify the energy dependence of the total production cross-section NN NNx wx given by H m x h X X s Ž h A dr Ž q < AŽ E, p < NN NNx 0 hanhart@physwashingtonedu knakayama@fz-juelichde 3 Present address In the above equation q X is the momentum of the outgoing meson and AE, Ž p X is the NN NNx transition amplitude, which, for future convenience, is expressed as a function of the total energy E and the relative momentum of the two nucleons in the final X X X state p Žnote, that p and q are related to each other via energy conservation The phase space is denoted by držq X and h denotes the maximum momentum of the emitted meson in units of its mass Watson wx argues that if there is a strong and attractive force between two of the outgoing particles, as is the case for the reactions under consideration, the energy dependence of the total cross-section is determined r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

177 ( ) C Hanhart, K NakayamarPhysics Letters B Fig Decomposition of the production amplitude in the final state NN interaction and the production part Here C denotes the nucleon nucleon wave function and T stands for the NN T-matrix by the phase space and the energy dependence of the relevant attractive interaction, ie, H m x h X X X s Ž h A dr Ž q < TŽ p, p < NN NNx 0 H X X ž p / m xh sind p X A dr Ž q, Ž 0 X In the above equation TŽ p, p X is the on-shell NN T-matrix, dž p X denotes the NN phase shifts at the energy, EŽ p X, of the final NN subsystem Žhere Ž X X restricted to s-waves, where E p 'p rm, with m denoting the nucleon mass When data for the reaction pp ppp 0 close to threshold became available wx Eq Ž indeed turned out to give the correct energy dependence of the total cross-section wx 3 Several authors w4 9x concluded from this observation that it is appropriate to calculate the transition NN NNx to lowest order in perturbation theory and just include the final state interaction Ž FSI by using a formula of the type in Eq Ž ; they implement the FSI by use of just the on-shell NN T-matrix, not only to get the right energy dependence of the cross-section, but also to get the strength of the matrix elements In this letter we criticize this procedure We shall demonstrate that the observation that the energy dependence of the cross-section is given by the on-shell FSI does not necessarily imply that the strength of the matrix elements is also determined by the on-shell NN interaction We also show that Watson s requirement that the FSI be attractive in order to obtain the energy dependence of the cross-section given by Eq Ž is unnecessary Finally we give an expression that allows one to estimate the effect of the initial state interaction Ž ISI on the reaction NN NNx, with x any meson heavier than the pion, in terms of the Ž on-shell NN scattering phase shifts and inelasticities The starting point of the present investigation is the decomposition of the total transition amplitude into a production amplitude, hereafter called M, and the NN FSI Ž see also Fig The decomposition is to be done in such a way that all the NN interactions taking place after the meson is produced are regarded as part of the FSI Ža more formal definition of the FSI can be given based on the last cut lemma w0 x As M is not specified, no approximation is involved in this decomposition Schematically we can write AsMqTGM An integration over the intermediate momenta is needed to evaluate the second term on the right hand side This is actually the term where the off-shell information of both the NN T-matrix and M enters, as will become clear below To be concrete, we use non-relativistic kinematics for simplicity The generalization to a fully relativistic treatment is straightforward and does not provide any new insights In addition, since we only want to investigate effects of the FSI on the energy dependence of the total crosssection, overall constant factors are dropped Using wx GŽ E,k sp yi pdž Ey EŽ k, Ey EŽ k

178 78 ( ) C Hanhart, K NakayamarPhysics Letters B where P denotes the principal value, we write the total transition amplitude A in the form ½ AŽ E, p X smž E, p X yik Ž p X TŽ p X, p X = i X q PŽ E, p ap 5 X Ž X X where k p sp pmr is the phase space density; the factor of rap X, with a denoting the low-energy NN scattering length, has been introduced for further convenience Also, for convenience, we display only those arguments of M that are relevant for the present discussion, that is the total energy E and the relative momentum p X of the two nucleons in the final state As pointed out in Ref wx, M depends weakly on E if the production mechanism is short ranged In the above equation, all the off-shell effects are contained in the function PŽ E, p X, whose explicit form is X ap ` k f Ž E, k X PŽ E, p s X PH dk k Ž p 0 Ey EŽ k X a ` k fž E,k yp s H dk, Ž 3 X p 0 p yk with the function f defined as TŽ p X,k MŽ E,k fž E,k s X X X TŽ p, p MŽ E, p KŽ p X,k MŽ E,k s X X X Ž 4 KŽ p, p MŽ E, p The last equality in the above equation follows from the half-off-shell unitarity relation of the NN T-matrix, namely X X id X Ž p X T p,k s h p e q K p,k with the K-matrix real by definition Therefore, all of the imaginary part of f and therefore of P is introduced by the production amplitude M In the latter formula use has been made of the fact that the on-shell T-matrix and the phase shifts are related by i X X X X id X Ž p k Ž p TŽ p, p s Ž hž p e y, Ž 5 where hž p X denotes the inelasticity Substituting Eq Ž 5 into Eq Ž, we get for the transition amplitude AŽ E, p s MŽ E, p e id Ž p X X X = X X X id Ž p yid Ž p Ž hž p e qe X X X id Ž p yid Ž p y hž p e ye i = X X PŽ E, p Ž 6 ap For energies near the production threshold energy one has hž p X s, so that, A E, p X sm E, p X e id Ž px = X Ž Ž p cos d X Ž Ž p X X sin d y PŽ E, p Ž 7 ap Using the effective range expansion X nq ` p X X Ý a n ž L / ns0 p cot d Ž p sy q L r Ž 8 Eq Ž 7 can be further reduced to sin d Ž p X X AŽ E, p X symž E, p X e id Ž p X ap = ž / X o P E, p qy ar p y X Ž 9 This is the central formula of the present discussion It reveals a number of important features First of all, it shows that the energy dependence of the total cross-section is, indeed, given by Eq Ž as has been shown by Watson, provided the production amplitude MŽ E, p X and the function PŽE, p X have

179 ( ) C Hanhart, K NakayamarPhysics Letters B a weak energy dependence compared to that due to the FSI Secondly, it is not necessary that the FSI be attractive in order for the total cross section to have the energy dependence given by Eq Ž : as long as MŽ E, p X and PŽE, p X have a weak energy dependence, the energy dependence of the total cross-section will be given by the on-shell FSI times phase X space for p < Ž ar 0 y Thirdly, and most relevant to the present discussion, the above formula also shows that the strength of the amplitude AE, Ž p X depends on the function PŽ E, p X As has been mentioned before, the function PŽ E, p X summarizes all the off-shell effects of the FSI and production amplitude As such, it is an unmeasurable and model-dependent quantity In particular, it depends on the particular regularization scheme used For example, in conventional calculations based on meson-exchange models, where the regularization is done by introducing form factors, the function Ž X 4 P E, p is very large and cannot be neglected Other regularization schemes, however, may yield a vanishing function PŽ E, p X Since the total amplitude AE, Ž p X should not depend on the particular regularization scheme, the production amplitude MŽ E, p X in Eq Ž 9 must depend on the regularization scheme in such a way to compensate for the regularization dependence of PŽ E, p X The above consideration shows that results from calculations aimed at quantitatiõe predictions, such as those using the procedure of evaluating M in the on-shell tree level approximation and multiplying it with the on-shell NN T-matrix w4 9 x, without consistency between the NN scattering and production amplitudes should be interpreted cautiously All the above considerations are not restricted to the NN final states; whenever there is a strong two-particle correlation in the final state, the energy dependence of a total production cross-section is given by the on-shell phase shifts of two of the outgoing particles This condition is for example also met in the reaction pp pk L, as demonstrated in Ref w x The situation is very different for the effect of the NN interaction, responsible for the initial state distor- 4 We checked this numerically tions Since the kinetic energy of the initial state has to be large enough to produce a meson, the NN ISI is evaluated at large energies Therefore, in this regime we expect the variation with energy of the ISI to be small 5 At least in the case of meson-exchange models this implies a flat off-shell behavior of the NN T-matrix at a given energy, in which case the principal value integral is expected to be small, as can be seen from Eq Ž 3 It is this observation that allows us to use Eq Ž 6 to estimate the effect of the ISI on the total production cross-section for the production of heavier mesons The ISI therefore leads to a reduction of the total cross-section of the order of ls e h Ž p e qe id LŽ p id LŽ p yid LŽ p L Ž shl p cos dl p q 4 yhl p F 4 qh p, 0 L where p denotes the relative momentum of the two nucleons in the initial state with the total energy E The index L indicates the quantum numbers of the corresponding initial state Note that, for production reactions close to threshold, selection rules strongly restrict the number of allowed initial states In the literature there is one example that quantifies the effect of the ISI for meson production reactions, namely Ref w3 x, where the reaction pp pph is studied The inclusion of the ISI in this work leads to a reduction of the total cross-section by roughly a factor of 03 At threshold only the Ls 3 P0 state contributes to the ISI The phase shifts and inelasticiw3x for the ISI are ties given by the model used in d Ž p sy6078 and h Ž p s 057 w4x L L at TLabs 50 MeV These values agree with the phase shift analysis given by the SAID program w5 x Both phase shifts and the inelasticity vary by 0% only over an energy range of 500 MeV w5 x Using the above mentioned values for d Ž p and h Ž p we get 5 In case of pion production the phase shifts of the 3 P0 partial wave, which is the initial state for the s-wave p 0 production, still vary reasonably rapidly with energy Therefore we do not expect the principal value integral to be small L L

180 80 ( ) C Hanhart, K NakayamarPhysics Letters B for the reduction factor l, defined in Eq Ž 0, a value of 0 Therefore, in the case of the kinematics of the ISI for the h production, the principal value integral within the meson-exchange model used indeed turns out to be a correction of the order of 0% compared to the leading on-shell contribution In summary, our primary point has been to demonstrate that, for the purpose of achieving quantitative predictions, FSI must be treated cautiously and in a way which is consistent with the corresponding production amplitude Our criticism is not to the result of Ref wx In fact, our function PŽ E, p X appearing in Eq Ž is related to the factor ŽŽ f r, R in Eq Ž 3 of Ref wx, where fž r accounts for the short-range behavior of the strongly interacting partiwx does not cles in the final state Note that Watson give a prescription how to calculate the overlap integral ŽfŽ r, R, which would be required to fix the overall normalization We emphasize that we do not claim that off-shell effects are measurable w6 x The result of this paper is the demonstration of the necessity to properly account for loop effects of the FSI in situations where the latter strongly influences the energy dependence of the total cross-section as in meson production in NN collisions In addition, we have given a compact formula that allows one to estimate the effect of the ISI in terms of the phase shifts and inelasticities of NN scattering This formula should prove to be useful for theoretical investigations of the production of heavy mesons close to their production threshold Acknowledgements We thank J Durso, J Haidenbauer, Th Hemmert and N Kaiser for useful discussions and W Melnitchouk for careful reading of the manuscript One of the authors Ž CH is grateful for the financial support by COSY FFE Project Nr References wx K Watson, Phys Rev 88 Ž wx HO Meyer et al, Phys Rev Lett 65 Ž ; Nucl Phys A 539 Ž wx 3 G Miller, P Sauer, Phys Rev C 44 Ž wx 4 A Moalem, Nucl Phys A 589 Ž wx 5 R Shyam, U Mosel, Phys Lett B 46 Ž 998 wx 6 A Sibirtsev, W Cassing, nucl-thr98005 wx 7 A Sibirtsev, W Cassing, Eur Phys J A Ž wx 8 E Gedalin et al, nucl-thr wx 9 V Bernard, N Kaiser, U-G Meißner, Eur Phys J A, accepted for publication, and nucl-thr w0x JG Taylor, Phys Rev 50 Ž wx M Goldberger, K Watson, Collision Theory, Wiley, New York, 964 wx JT Balewski, Eur Phys J A Ž w3x M Batinic, A Svarc, T-SH Lee, Phys Scripta 56 Ž w4x T-SH Lee, private communication w5x Extracted from the VIRGINIA TECH PARTIAL-WAVE ANALYSES ON-LINE Ž CAPSrsaid_branchhtml w6x For a recent discussion, see HW Fearing, Phys Rev Lett 8 Ž

181 0 May 999 Physics Letters B Proton scattering on the radioactive nucleus 0 O and the 0 q q transition in the neutron-rich oxygen isotopes gs JK Jewell a, LA Riley a, PD Cottle a, KW Kemper a, T Glasmacher b, RW Ibbotson b, H Scheit b, M Chromik b, Y Blumenfeld c, SE Hirzebruch c, F Marechal c, T Suomijarvi c a Department of Physics, Florida State UniÕersity, Tallahassee, FL , USA b National Superconducting Cyclotron Laboratory, Michigan State UniÕersity, East Lansing, MI 4884, USA c Institut de Physique Nucleaire, IN P -CNRS, 9406 Orsay, France 3 Received 0 October 998; received in revised form 3 March 999 Editor: RH Siemssen Abstract Elastic and inelastic proton scattering to the q state of the single closed shell radioactive nucleus 0 O have been measured in inverse kinematics with a beam energy of 30 MeVru The matrix element determined for the 0 q gs q transition in this work is compared with the corresponding electromagnetic matrix element to determine the ratio of the neutron and proton multipole matrix elements for this transition, M rm s9ž n p 4, which is quite different from the value MnrMpsNrZs5 that would be expected for a purely isoscalar transition A comparison of this result to a corresponding result for 8 O suggests that the strength of the core polarizing interactions between the valence neutrons and the core protons and neutrons is changing significantly with mass q 999 Published by Elsevier Science BV All rights reserved The measurement of proton and neutron contributions to transitions between nuclear states provides one of the most important tools for understanding the relative importance of valence and core contributions to these transitions The competition between valence and core contributions is of particular interest in single-closed-shell nuclei, where the low-lying excitations would be composed exclusively of the valence neutrons or protons if the closed core were truly inert Methods for determining proton and neutron matrix elements, defined as ² << l M s J S r Y Ž V << J :, Ž pž n f pž n i l i i generally involve the comparison of measurements of a transition using two experimental probes with different sensitivities to proton and neutron contributions While studies of 0 q gs q transitions in stable nuclei have been performed using a variety of combinations of experimental probes Žfor example, see w, x, it has until recently been impossible to examine neutron and proton contributions in this way in short-lived radioactive nuclei Such studies would be of particular interest because of the relatively small binding energies of the valence nucleons Data on the electromagnetic matrix elements for 0 q gs q transitions have been available for some short-lived even-even nuclei for some time wx 3, and recent advances in intermediate energy Coulomb exw4,5x have made even more information of citation this type available These electromagnetic data pro r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

182 8 ( ) JK Jewell et alrphysics Letters B vide information on the proton contributions to the 0 q gs q matrix elements With recent advances in techniques for providing intense beams of radioactive nuclei, inverse kinematics proton scattering provides a way to determine the neutron contributions to these matrix elements At center-of-mass energies less than 50 MeV - corresponding to radioactive beam energies of less than 50 MeVrnucleon inelastic proton scattering is much more sensitive to the neutron contributions in transitions than those of the protons, and therefore can be used together with electromagnetic data to understand the relationship between proton and neutron contributions w,6 x Here, we report the results of an inverse kinematics proton scattering study of the 0 q gs q transition in the single-closed-shell radioactive nucleus 0 O, which is two neutrons from stability but only four neutrons from the heaviest bound oxygen isotope Ž 4 O w7,8 x Reliable electromagnetic data are already available for this transition from measurements of q the lifetime of the state Žsee wx 3 for a compilation, so the proton scattering data provide a determination of MnrM p, the ratio of the neutron and proton multipole matrix elements wx The trend of this ratio in the oxygen isotopes provides insight about the role of core polarization in the 0 q gs q transition Both the experiment and interpretation described here for 0 O are similar in approach to those used in the inverse kinematics proton scattering study of the radioactive nucleus 38 S by Kelley et al wx 9 Angular distributions of protons scattered from the ground state and 67 MeV q state of 0 Ointhe pž 0 O,p reaction were measured The 30 MeVrnucleon 0 O beam was produced in the A00 fragment separator w0x at the National Superconducting Cyclotron Laboratory via fragmentation of a 65 MeVrnucleon Ne primary beam on a water cooled 360 mgrcm 9 Be target A 00 mgrcm C achromatic degrading wedge subtending an angle of 9 mrad and placed at the second dispersive image of the A00 was used to reduce the number of beam species transmitted, and the momentum acceptance of the A00 was % The secondary beam, with a maximum intensity of 30,000 particlesrs, was approximately 99% pure A 36 mgrcm polypropylene foil was used as the hydrogen target for the secondary beam This foil was mounted at 558 with respect to the beam axis to minimize the angular straggling of scattered protons leaving the target The beam was tracked by two position sensitive parallel plate avalanche counters Ž PPAC wx upstream of the target chamber and stopped in a fastrslow plastic phoswich telescope, allowing for beam tracking and particle identification on an event by event basis Data were also taken with a 3 mgrcm C target to determine the background caused by the carbon in the polypropylene The arrangement for detecting the scattered protons was similar to that used by Kelley et al for the measurement of the pž 38 S,p reaction wx 9 Scattered protons were detected using the FSU-MSU array of 8 Si strip-si PIN-CsI particle telescopes The telescopes have an active area of 5 cm x 5 cm and were mounted 8 cm from the target position, yielding a total laboratory-frame angular range of 08 for each telescope The 300 mm-thick strip detectors consisted of 6 3 mm-wide strips The 470 mm PIN diode and cm CsI layers stopped protons which passed through the strip detectors Protons stopped in the strip detectors were identified by time-of-flight, while higher energy protons were identified in gates on the D EyE spectra The telescopes were oriented such that the Si strips were tangent to circles of constant scattering angle with respect to the beamline alignment axis Due to finite strip width and to the fact that the strips were not curved to follow lines of constant scattering angle, the geometric angular range of each strip, corresponding to an uncertainty in scattering angle measurements, was approximately 0858 Three telescopes were mounted with their centers at laboratory-frame angles of 758 and three others were centered at 708 Hence, we detected protons in the laboratory scattering angle range , corresponding to a center-of-mass angular range of approximately The laboratory energy range of detected protons was MeV, sufficient to detect protons which were elastically and inelastically scattered to the q state in the desired angular range Fig illustrates the data taken by the telescopes Fig Ž a is a scatter plot of laboratory frame proton energies versus laboratory frame angles for the telescopes centered at 708, and Fig Ž b gives a calculation of the kinematics for the reaction The data shown are in coincidence with both beam and proton

183 ( ) JK Jewell et alrphysics Letters B shown in the form of a lab-frame kinetic energy versus scattering angle spectrum in Fig Ž c Scattering angles have been determined from Si strip positions and beam tracking information; the distribution in the directions of the incident beam particles results in a distribution in the scattering angles even for protons detected in a single strip As can be seen in Fig Ž c, clear separation between elastic and inelastic events was obtained The absence of a significant background is also apparent The data taken with a C target yielded no evidence of any contributions to the proton data from the carbon in the polypropylene target The natural binning of the data by the Si strips in the laboratory was used to generate the proton angular distributions in Fig It should be noted that this leads to a difference in scattering angles for elastic and inelastic points in the center-of-mass for each bin The horizontal bars in Fig correspond to the geometric angular ranges of the individual Si strips Due to lower yields, the inelastic points have been generated using two strips each To provide absolute Fig Ž a Plot of proton laboratory energy versus scattering angle data for pž 0 O,p for the telescopes centered at 708 in the laboratory frame Ž b Calculated kinematics for the pž 0 O,p reaction Ž c Proton laboratory energy versus scattering angle data for pž 0 O,p for the strip centered at 78 in the telescopes centered at 758 The spread in the data arises from the variation in the directions of the incident 0 O beam particles identification gates The scattered proton data from the 78 strips in the telescopes centered at 758 are Fig Angular distributions of protons scattered from the ground state and q state of 0 O The smooth curves are coupled channels calculations described in the text The shaded band indicates the experimental uncertainty for the coupling parameter b s050"004 Data points indicated as squares were detected in the telescopes centered at 708 in the laboratory frame; circles are points detected in the telescopes centered at 758 in the lab frame

184 84 ( ) JK Jewell et alrphysics Letters B normalization of the data and thus absolute differential cross sections the elastic scattering data were compared to an angular distribution calculated using optical model parameters from the 30 MeV 0 NeŽ p,p study of de Swiniarski et al w x The elastic scattering data were normalized on a telescope-by-telescope basis, so that the relationships between data points in the same telescope were not altered The computer code CHUCK w3x was used to perform coupled channels calculations from which the strength of the 0 q gs q transition in the present Ž p,p reaction was extracted A standard vibrational form factor was used for the calculation of the inelastic cross section The curve through the inelastic data points in Fig is given by b s050 We assign an experimental uncertainty of D b s 004, which corresponds to an uncertainty of 6% in the magnitude of the differential cross section, on the basis of a visual comparison of the data with the calculation The range of differential cross sections given by this experimental uncertainty is indicated in Fig by the shaded band Inelastic proton scattering at low energies ŽF 50 MeV is much more sensitive to neutron contributions to a transition than proton contributions If Ž p, p X b is the external-field neutron Ž proton nž p interac- tion strength for low-energy proton scattering, then Ž p, p X Ž p, p X the ratio b rb is approximately 3 wx n p In contrast, the electromagnetic matrix element Ž q B E;0 q gs measures the proton matrix element M p To relate the microscopic picture represented in Eq Ž to the collective model parameters deter- mined in the present analysis of the Ž p,p data, we use an equation given in Ref wx : Mn Ndn s, M Zd p p where dnž p is the deformation length d s br for the neutrons Ž protons and R is the nuclear radius Bernstein et al wx use Eq Ž and distorted wave theory to obtain the approximate equation d F q bn F rbp F MnrMp f, Ž 3 d F F p q Ž bnrbp Ž NrZ where dž F is the deformation length extracted with an experimental probe F and bn F rbp F is the interac- tion strength ratio for that probe The denominator assumes that the ratio of neutron and proton ground state densities is NrZ Solving Eq Ž 3 for MnrMp with dp being given by the deformation length from an electromagnetic measurement and the probe F being Ž p,p, we obtain ž ž / / M b d X n p Ž p, p bn N M s b d q b Z y Ž 4 p n em p The compilation of Raman et al wx 3 gives d em s Ž 9 fm for O To calculate d X Ž p, p, we use the radius parameter from the real part of the optical model potential, rr s0 fm, so that Rs99 fm This gives d X sb Rs49Ž Ž p, p fm From a simple comparison of these two deformation lengths, it is clear that there is an isovector component to this transition and that the neutrons are playing a disproportionately large role, as is expected for a nucleus with a closed proton shell Eq Ž 4 yields MnrMps 9Ž 4, which can be compared to the value MnrMp snrzs5 that would be expected for a purely isoscalar transition It is important to note that Bernw,x used this method to analyze stable stein et al nuclei as light as 8 O, so its use in 0 O is appropriate One of the primary motivations for the present efforts in nuclear structure physics with radioactive beams is to observe the evolution of nuclear structure effects as the drip lines are approached The transition from stability to particle instability occurs over a relatively small change in neutron number in the oxygen isotopes While 8 O is stable, it appears from an exhaustive search for 6 O that 4 Oisthe heaviest bound isotope w7,8 x Hence, the oxygen isotopes provide an excellent opportunity to track changes that occur in the structure of the q state as the neutron drip line is approached In this context, it is interesting to compare the values of M rm for the 0 q q transitions in 8 n p gs O and 0 O to look for suggestions of a trend Several methods have been employed to extract the MnrMp value for the 0 q q transition in 8 gs O, and a significant range of results has been obtained However, we begin by comparing the present MnrMp result for 0 O with a value for 8 O extracted using an identical analysis of low energy proton scattering and electromagnetic results From the compilation of

185 ( ) JK Jewell et alrphysics Letters B wx 3, we obtain an electromagnetic deformation length 8 of d s Ž em 3 fm for O Proton inelastic scatter- ing data for 8 O were taken at 45 MeV by Escudie et al w4 x, but were reanalyzed by Grabmayr et al w5x using an optical model potential formulated to systematically reproduce the available neutron and proton scattering data for 6,8 O at energies between 4 and 5 MeV Hence, the result of the distortedwave Born approximation Ž DWBA analysis of the 8 OŽ p,p data in Ref w5x Žb s 045Ž 4 is adopted, giving d X s 30Ž Ž p, p fm Using these values with 8 Eq Ž 4, we obtain M rm s 50Ž 7 n p for O, a value which is significantly smaller than the result for 0 O and is not so far from the result we would expect for an isoscalar transition, MnrMpsNrZs 5 This comparison of results extracted using the same experimental probes in both nuclei suggests a trend in which MnrMp is increasing as a function of mass Values of MnrMp for the 0 q gs q transition in 8 O have been extracted using a variety of methods, including the comparison of intermediate energy prow6 x, the com- ton scattering with electron scattering parison of the electromagnetic result for 8 O with the corresponding electromagnetic result in the mirror 8 nucleus Ne w7 x, the comparison of low energy proton scattering with low energy neutron scattering w x q 5, and the comparison of the scattering of p and p y w8,9 x Results using these methods, as well as the values for 8 O and 0 O extracted using the comparison of low energy proton scattering and electromagnetic data, are shown in Fig 3 ŽThe mirror nucleus result of w7x has been updated with matrix elements compiled in Ref wx 3 ; the result shown for the pion scattering data of w8x is that cited in Ref w7 x; and the value shown for the Ž p,p versus Ž n,n analysis of w5x is that cited in the compilation of wx The weighted mean of all the results shown for 8 O in Fig 3 is M rm s03ž n p 3 Once again, the results suggest that M rm is larger in 0 O than in 8 O n We can examine the role of core polarization in the 0 q gs q transition to understand the physical significance of the difference between MnrMp val- ues in 8 O and 0 O A simple understanding of the roles of the valence nucleons and core polarization in the 0 q gs q transition can be achieved by writing Mn and Mp in terms of valence-space matrix ele- p Fig 3 M rm values for the 0 q q transitions in 8 n p gs O and 0 O The results illustrated here are described in the text The dashed lines correspond to MnrMps NrZ ments M X and M X n p and core-polarization contribu- tions as wx M sm X qd nn qm X n n pd np, Ž 5 and M sm X d pn qm X p n pž qd pp, Ž 6 where d xy is the core-polarization parameter corresponding to core x s polarization by valence y s That is, d xy reflects the amount of core polarization per unit of contribution from the Õalence nucleons The connection of the core-polarization parameters and the usual electromagnetic effective charges is given by ensd pn and epsqd pp The oxygen isotopes have no valence proton contribution because of the closed proton shell, so that M X p s0 Therefore, the ratio M rm is given by n p M rm s Ž qd nn rd pn Ž 7 n p That is, MnrMp depends only on the core polariza- tion parameters and not on the number of valence neutrons If these parameters and, therefore, the effective charges are constant, then MnrMpshould be constant as well Conversely, any change in MnrMp would be due to changes in the polarization parameters As discussed above, such an increase would not be a simple consequence of the increase in the number of valence neutrons, since the increase in the valence neutron contribution M X cancels out of n MnrM p Instead, it appears that either the interaction

186 86 ( ) JK Jewell et alrphysics Letters B of the valence neutrons with the core neutrons Žas nn 0 8 reflected in d is stronger in O than in O or that the interaction of the valence neutrons with the Ž pn core protons d is becoming weaker with increasing mass Confirmation of this trend will require inverse kinematics proton scattering measurements of the corresponding transitions in the heavier even-a isotopes,4 O These measurements will become possible as new radioactive beam facilities come on line Furthermore, a microscopic study of the trends suggested by the macroscopic analysis presented here would likely lend new insights regarding the core polarization mechanisms at work in the oxygen isotopes In summary, we have measured inverse kinematics proton scattering with a 30 MeVrnucleon beam of radioactive 0 O nuclei to study the 0 q gs q tran- sition A result for MnrMp for this transition was obtained from a comparison of the transition strength extracted from the present data with the strength obtained from electromagnetic data A comparison of the 0 O result with corresponding values in 8 O suggests that the relative strengths of the interactions between the valence neutrons and the core protons and neutrons are changing with mass; however, confirmation of this trend will require similar measurements for,4 O Acknowledgements This work was supported by National Science Foundation grants PHY , PHY , and PHY References wx AM Bernstein, VR Brown, VA Madsen, Phys Lett B 03 Ž wx AM Bernstein, VR Brown, VA Madsen, Comments Nucl Part Phys Ž wx 3 S Raman, CH Malarkey, WT Milner, CW Nestor Jr, PH Stelson, At Data, Nucl Data Tables 36 Ž 987 wx 4 T Motobayashi, Y Ikeda, Y Ando, K Ieki, M Inoue, N Iwasa, T Kikuchi, M Kurokawa, S Moriya, S Ogawa, H Murakami, S Shimoura, Y Yanagisawa, T Nakamura, Y Watanabe, M Ishihara, T Teranishi, H Okuno, RF Casten, Phys Lett B 346 Ž wx 5 T Glasmacher, Ann Rev Nucl Part Sci 48 Ž 998 wx 6 VA Madsen, VR Brown, JD Anderson, Phys Rev C Ž wx 7 D Guillemaud-Mueller, JC Jacmart, E Kashy, A Latimier, AC Mueller, F Pougheon, A Richard, YuE Penionzhkevich, AG Artuhk, AV Belozyorov, SM Lukyanov, R Anne, P Bricault, C Detraz, M Lewitowicz, Y Zhang, YuS Lyutostansky, MV Zverev, D Bazin, WD Schmidt- Ott, Phys Rev C 4 Ž wx 8 M Fauerbach, DJ Morrissey, W Benenson, BA Brown, M Hellstrom, JH Kelley, RA Kryger, R Pfaff, CF Powell, BM Sherrill, Phys Rev C 53 Ž wx 9 JH Kelley, T Suomijarvi, SE Hirzebruch, A Azhari, D Bazin, Y Blumenfeld, JA Brown, PD Cottle, S Danczyk, M Fauerbach, T Glasmacher, JK Jewell, KW Kemper, F Marechal, DJ Morrissey, S Ottini, JA Scarpaci, P Thirolf, Phys Rev C 56 Ž 997 R06 w0x BM Sherrill, DJ Morrissey, JA Nolen Jr, N Orr, JA Winger, Nucl Inst Meth B 56r57 Ž wx D Swan, J Yurkon, DJ Morrissey, Nucl Inst Meth A 348 Ž wx R de Swiniarski, A Genoux-Lubain, G Bagieu, JF Cavaignac, Can J Phys 5 Ž w3x PD Kunz, University of Colorado report, unpublished w4x JL Escudie, R Lombard, M Pignanelli, F Resmini, A Tarrats, Phys Rev C 0 Ž w5x P Grabmayr, J Rapaport, RW Finlay, Nucl Phys A 350 Ž w6x J Kelly, W Bertozzi, TN Buti, JM Finn, FW Hersman, MV Hynes, C Hyde-Wright, BE Norum, AD Bacher, GT Emery, CC Foster, WP Jones, DW Miller, BL Berman, JA Carr, F Petrovich, Phys Lett B 69 Ž w7x AM Bernstein, VR Brown, VA Madsen, Phys Rev Lett 4 Ž w8x S Iversen, H Nann, A Obst, KK Seth, N Tanaka, CL Morris, HA Thiessen, K Boyer, W Cottingame, CF Moore, RL Boudrie, D Dehnhard, Phys Lett B 8 Ž w9x SJ Seestrom-Morris, D Dehnhard, MA Franey, DB Holtkamp, CL Blilie, CL Morris, JD Zumbro, HT Fortune, Phys Rev C 37 Ž

187 0 May 999 Physics Letters B Black holes and Calogero models GW Gibbons, PK Townsend DAMTP, UniÕersity of Cambridge, SilÕer St, Cambridge CB3 9EW, UK Received 8 February 999; received in revised form 4 February 999 Editor: PV Landshoff Abstract We argue that the large n limit of the n-particle SUŽ,< superconformal Calogero model provides a microscopic description of the extreme Reissner Nordstrom black hole in the near-horizon limit q 999 Elsevier Science BV All rights reserved An amusing feature of the matrix model approach to M-theory is that apparently intractable problems of quantum Ds supergravity are resolved by a return to non-relatiõistic quantum mechanics wx The MŽ atrix model Hamiltonian, which can be viewed as that of an SUŽ ` Ds super-yang Mills Ž SYM theory, was originally found from a light-front gauge-fixed version of the D s supermembrane wx, but the interpretation given to it in Ref wx was inspired by the observation wx 3 that there is a close similarity between this supermembrane Hamiltonian and that of n IIA D0-branes wx 4 in the large n limit However, because the D0-brane Hamiltonian differs from that of the MŽ atrix model by the inclusion of relativistic corrections, the precise connection has only recently become clear By viewing the limit in which relativistic corrections are suppressed as one in which the spacelike circle of S -compactified M-theory becomes null, it was shown in Ref wx 5 that all degrees of freedom other than D0-branes are also suppressed, thus establishing that the MŽ atrix model Hamiltonian captures the full dynamics of Žuncom- pactified M-theory Here we wish to observe, because it will provide a useful perspective on our later results, that this limit can be understood as a nearwx 6, the horizon limit In the dual-frame metric D0-brane solution of IIA supergravity approaches ads =S 8 wx 7 Near the ads Killing horizon, and for sufficiently large n, there is a dual description of the D0-brane dynamics wx 8 in terms of IIA supergravity on ads =S 8 wxž 7 which is the null reduction of the M-wave wx 9 Because ads has an SLŽ ;R isometry group one might expect the MŽ atrix model Hamiltonian to exhibit this symmetry as a worldline conformal symmetry, in analogy to the dual adsrcft descriptions of D3-brane dynamics w0x in a similar near-horiw x The SYM interpretation of the zon limit MŽ atrix model makes it clear that there can be no such symmetry because D-dimensional SYM theories are conformally invariant only for D s 4 In accord with this observation, the SLŽ ;R symmetry of the ads =S 8 metric does not extend to the complete supergravity solution because it is broken by the dilaton field Nevertheless, since the value of the dilaton is simply related to the matrix model r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S X

188 88 ( ) GW Gibbons, PK TownsendrPhysics Letters B coupling constant one might expect there to be a generalized conformal symmetry taking a matrix model at one value of the coupling constant to the same model at a different value of the coupling constant Just such a generalized conformal symw x, but the above metry was exhibited in Ref explanation of this result indicates that it should be a general phenomenon applicable not only to D0-branes but to all 0-brane intersections for which the dualframe supergravity metric has an ads factor The generalized SLŽ ;R conformal invariance will become a genuine conformal invariance in those cases for which the dilaton is constant These genuinely conformal cases are the focus of this paper A case in point is the four intersecting D3-brane configuration D3: 3 y y y y y y D3: y y 4 5 y y y y D3: y y 4 y 6 y y y D3: y y 3 y 5 6 y y y As explained in Ref w3 x, this corresponds to a black hole solution of the T 6 reduction of IIB supergravity for which all scalar fields, including the dilaton, approach constant values near the horizon The special case for which the scalar fields are everywhere constant is the extreme Reissner Nordstrom Ž RN black hole In all cases the near horizon geometry is ads =S and the isometry supergroup is SUŽ,< Assuming that ads =S is equally a solution of the full, non-perturbative, IIB superstring theory Žwhich seems likely in view of the results of w4x we may conclude that SUŽ,< acts as a superconformal group on the quantum mechanical model governing the fluctuation of the branes in the region of the intersection But what is this model? Whatever it is, we expect that it has a dual description as Ns Ds4 supergravity on ads =S in a limit that involves interpreting each of the four supergravity D3-branes as a large number of coincident microscopic D3-branes Equivalently, we expect some limit of the superconformal mechanics model to provide a microscopic description of the extreme RN black hole, at least near the horizon The determination of this model is therefore likely to be an important step in our understanding of the quantum mechanics of black holes The field theory on the string intersection of any two of the four D3-branes of the above configuration is a Ž 4,4 supersymmetric D s SYM theory The intersection with the remaining two D3-branes must reduce this to a Ds Ns4 superconformal quantum mechanics, so we need some n-particle SUŽ, < -invariant superconformal mechanics that is related, in the large n limit, to a reduction of a Ds SYM theory One such model, omitting fermions, is the n particle Calogero model w5x with Hamiltonian l Hs Ýpi q Ý i i-j Ž qiyqj where Ž p, q Ž is,,n i i are the n-particle phase space coordinates, and l is a coupling constant This model, like a number of variants of it, is integrable and has been studied extensively in this context Žsee eg w7 x A distinguishing feature of the particular model defined by the above Hamiltonian is that its action is invariant under an SLŽ ;R group acting as a worldline conformal group w8 x, so we shall call it the Conformal Calogero Ž CC model The conformal symmetry arises from an action of SLŽ ;R on phase space, generated by Ž H, K, D Ž where H is the Hamiltonian, Ks Ýiqi gener- ates conformal boosts and D sy Ýipi qi generates dilatations The generators H and K are lightlike with respect to the Killing form on slž ;R while D is spacelike In our conventions timelike generators are compact We propose that the model describing the 0-brane intersection of the above four D3-brane configuration is the Ns4 supersymmetric extension of the CC model, and hence that this model provides a microscopic description of the extreme RN black hole in the near-horizon limit The Ns superconformal Calogero model is described by the superpotential lý log< q yq < i- j i j It is a special case of the N s supersymmetric, but generically non-conformal, Calogero Moser Ž CM model studied in Ref Calogero models have arisen previously in the context of the matrix model describing D0-brane dynamics w6x but since the latter cannot be conformal invariant, for the reasons given above, they are necessarily different from those considered here

189 ( ) GW Gibbons, PK TownsendrPhysics Letters B w9 x; this fact will play a part in our proposal Unfortunately, the N s 4 supersymmetric extensions of the CC and CM models have not yet been constructed but the main features are clear and we shall discuss them later First we wish to explain the motivation for our proposal in the context of the bosonic model The n-particle CC model was shown in Ref w0x to be equivalent in the large n limit to a D s SUŽ n gauge theory on a cylinder A related observation that we shall elaborate on here is that the CC model can be obtained by symplectic reduction of a class of matrix models Consider, for example, the space of hermitian n = n matrices X with the flat metric trž dx Ž dx The corresponding free particle mechanics model is manifestly conformal invariant It is also invariant under SUŽ n transformations acting by conjugation on X The corresponding conserved angular momenta are encoded in the conserved traceless hermitian matrix msiwx, P x, where P s X is the momentum canonically conjugate to X The idea now is to work at some fixed values of the angular momenta This gives constraints, and one quotients phase space by the action generated by these constraints to get a reduced Hamiltonian system on the quotient In order to obtain the CC system this way one must choose the angular momenta such that m has ny equal eigenvalues l The stability group of the matrix m is then SU Ž =UŽ ny ;UŽ n, and the action of this group may be used to bring X to the diagonal form XsdiagŽ q,,q n and P to a form with diagonal entries p and off-diagonal entries P s i lrž i ij qi y q The reduced Hamiltonian, trp, is just Ž j It will prove instructive to consider the ns case in more detail In this case the configuration space is E 4 We can write X as XsUDU y, Ž 3 where DsdiagŽ q,q and U is the SU matrix ž / u Õ Us Ž < u< q< Õ< s Ž 4 yõ u There is a UŽ redundancy in this description bei a cause we can take u,õ e Ž u,õ without changing X We thus have a parametrization of X in terms of Ž q,q and the coordinates of SU ruž (S Introducing the centre of mass coordinate QsŽ q qq r and the relative position coordinate qsž q yq we find that ds s dq q dq qq dv 5 Note that the 3-metric describing the relative motion on E 3 is flat The angular momentum matrix m has eigenvalues "l where l is the length of the angular momentum 3-vector Lassociated with the SOŽ 3 isometry of S At fixed l, the Hamiltonian is g Hs p q, Ž 6 q where gsl Ž 7 This is the conformal mechanics Hamiltonian of de Alfaro, Fubini and Furlan Ž DFF wx with coupling constant g The CC models thus provide a natural generalization of DFF conformal mechanics The Ns supersymmetric extension of the DFF model, with SUŽ,< superconformal symmetry was constructed in Ref w x This model is clearly related to the -particle N s superconformal Calogero model in the same way as above The superpotential in the latter case is llog< q yq < sllog< q <, Ž 8 but this is precisely the superpotential of N s superconformal mechanics Conversely, by simply adding in a trivial centre of mass motion one can obtain the -particle N s superconformal Calogero model by a simple change of variables, and the n-particle model is a straightforward generalization w9 x The Ns4 extension of the DFF model, with SUŽ,< superconformal symmetry, was conw3 x; the construction is not com- structed in Ref pletely obvious because there is still only one physical boson variable, q, despite the N s 4 supersymmetry However, it should be clear from the above discussion that the N s 4 -particle superconformal Calogero model is already implicit in the N s 4 superconformal mechanics We expect that the n- particle generalization will again be straightforward, but the complete construction will not be attempted here

190 90 ( ) GW Gibbons, PK TownsendrPhysics Letters B We now come to the main motivation for our proposal It was shown in Ref w4x that the Ns4 superconformal mechanics model describes the dynamics of a superparticle of unit mass Žand charge to mass ratio equal to that of the black hole in the near-horizon geometry of an extreme RN black hole in the limit of large black hole mass The coupling constant was found to be gs4 l, Ž 9 where l is the particle s orbital angular momentum operator quantum number Actually, the result of w x 4 is that gs4l where L is the operator with eigenvalues l Ž lq, but the shift of l to l Ž lq can be interpreted as a consequence of integrating out the fermions w,5 x, so one finds gs4 l if the fermions are simply omitted Even so, there is an apparent discrepancy with Ž 7 because it would be natural to suppose from our derivation of conformal mechanics from the = hermitian matrix model that, in the quantum theory, l should take on integer values, but this is consistent with Ž 9, and integer l, only if l is an eõen integer The resolution of this discrepancy is that in arriving at Ž 6 we implicitly assumed that q was positive If one allows q to be negative, then the parametrization Ž 3 of X has an additional redundancy which can be removed by identification of antipodal points on the -sphere Only those harmonics with even angular momentum quantum number are well defined on S rz (R P, so allowing q to be negative leads to the restriction ls l for integer l The derivation of conformal mechanics from a charged particle on ads =S makes crucial use of a coordinate system for which the ads metric is ds syr 4 q y4 dt q4r q y dq, Ž 0 where R is the radius of curvature The singularity at Solutions F of the covariant Klein Gordan equation on ads ˆ ˆ Ž y in satisfy ietf s HF where Hs qp q p q g rq, and the y inner product is Hdrr < F < It follows that Hˆ provides the correct resolution of the operator ordering ambiguity inherent in the classical hamiltonian Ž 6 This ordering differs from the naive one used in studies of quantum conformal mechanics but the change of ordering does not affect the qualitative properties of the model needed here q s ` is just a coordinate singularity at a Killing horizon of hse The vector fields t 4 dstetq qe q, ks t qq rr EtqtqE q are also Killing, and Ž h,d,k have the slž ;R commutation relations wd,hxsyh, wd,kxsk, wh,kxs d Ž The key result of w4x is that the particle trajectory qt Ž is described by a relativistic conformal mechanics, and that the Hamiltonian describing this evolution in t reduces in the large R limit to the non-relativistic Hamiltonian Ž 6 Just as this Hamiltonian, H, is associated with the Killing vector field h, so there are two other functions on phase space, D and K associated to d and k Their Poisson bracket algebra is isomorphic to Ž In the quantum theory, the DFF Hamiltonian has a continuous spectrum with E) 0 but no ground state at Es0 w x This feature is a reflection of the incompleteness of the classical dynamics because a state of zero energy would be time-independent and hence associated with a fixed set of Et on ads, ie its Killing horizon The classical cure for the incompleteness due to the Killing horizon is simply to choose a global coordinate system on ads, eg sint Rcos r ts, q s, Ž 3 costysin r costysin r where < r < - pr, and t is periodically identified with period p The metric is now ds s Ž Rsec r ydt qdr Ž 4 We see that E is Killing It is a compact generator of t SL ;R ; in fact Et s hqk 5 Classical evolution in t is complete because Et has no horizon The corresponding Hamiltonian H Ž r, p r can be found by solving the mass-shell constraint in the new coordinates, but the quantum states it evolves belong to a new Hilbert space It is also no longer obvious how to take the non-relativistic limit and hence unclear how to generalize to the multi-particle case For these reasons we shall not pursue this approach here Instead, we retain the original Hilbert space but evolve the states via a new, non-conformal, Hamiltonian H X shqk, as suggested by the

191 ( ) GW Gibbons, PK TownsendrPhysics Letters B formula Ž 5 This was also done in Ref w x, where it was found that H X has a unique ground state with a series of evenly spaced excited states, as expected for evolution in a periodically identified time parameter Curiously, despite the fact that orbits of Et pass through qs`, a particle in the ground state of H X has zero probability of being at qs` because the ground state wave-function is wx a yq r 0 ž ' / c sq e, as q q4 g Ž 6 Two particles do not make a black hole, so we now wish to generalize the above ns discussion to n) A complication of the n) case is that there are many possibilities for the angular-momentum matrix m Only the choice described above, n y equal eigenvalues, leads to the n-particle conformal Calogero model A partial motivation for this choice comes from the observation w5x that the one independent eigenvalue l of the ns case is an almost-central charge in the superconformal algebra; it is not truly central because it can be removed by a redefinition of the slž ;R charges but, for a given definition, the anticommutator of the odd charges is l-dependent The equal eigenvalue condition for n ) is thus analogous to a BPS condition In any case, having made this choice we arrive at the CC Ž Hamiltonian, which we now rewrite as l ny y HsHny q pnq Ý Ž yqirq n Ž 7 q n is where Hny contains all terms independent of Ž q, p n n If we now suppose that all qi are small except q Žso that we may omit OŽ q rq terms n i n, and take n large Žso that we can ignore reduced-mass effects from factoring out the centre of mass motion then the Hamiltonian governing the motion of the nth particle is the DFF model with gsž ny l s8ž ny l, Ž n4 Ž 8 Recalling that in the quantum theory l is an integer multiple of Planck s constant, we see that the large n limit is one in which the particle orbiting the cluster of n y particles acquires a macroscopic angular momentum We thus arrive at a picture of an extreme black hole as a composite of a large number of particles interacting via a repulsive inverse cube force law However, this picture is rather misleading because, as mentioned above, the variable q is actually an inõerse radial variable in the sense that qs` corresponds to the black hole horizon The cluster of ny particles near qs0 is actually much further from the horizon than the one at large q, although they are still in the near-horizon region of the black hole One can view them as living at the ads boundary, and the large n CC model as the boundary conformal field theory in the sense of the adsrcft correspondence w0 x It is then natural to interpret these Calogerons as the microscopic degrees of freedom of the black hole The CC model describes the dynamics of nordered particles in the sense that if we choose the q i such that qiq yq i)0 then the CC dynamics imple- mented by H preserves this ordering As for conformal mechanics, the full dynamics in which we allow qiqy qif 0 will be described by some other Hamiltonian If we follow the DFF approach described above we would replace the CC Hamiltonian by the new Hamiltonian HCM shqv K for some constant v Now n K' Ýqi s Ý Ž qiyqj q Q Ž 9 n i i-j where QsŽ rn Ýiqi is the centre of mass position We may set Q s 0 without loss of generality, in which case l v HCM s Ýpi qý q Ý Ž qiyq j i i-j Ž q yq n i j i-j Ž 0 This is the Calogero Moser Ž CM Hamiltonian; it has a unique ground state and towers of excited states with energy spacings v,3v,,nv w5,9 x It is therefore natural to associate HCM with a time parameter that is periodically identified with period 3 pv y 3 In the ns case the periodicity could be taken to be pv y but the inclusion of fermions will require a periodicity of pv y even in this case One way to see this is to note that ads can be embedded in E Ž,, in which case its Killing spinors are the restriction of constant spinors on E Ž, It follows that the Killing spinors of ads are antiperiodic in t when written in a spin basis associated to the one-forms dt,dr, where t and r are the ads coordinates introduced previously

192 9 ( ) GW Gibbons, PK TownsendrPhysics Letters B The ground state wave-function of HCM is similar to Ž 6 in that particles in the ground state have zero probability of being at any of the boundaries between the ny regions with different signs of the relative coordinates qiq y q i Because classical tra- jectories connect these regions it is natural to consider them as distinct, but in the quantum theory we must choose to put the particles in one region It is thus natural to assign the black hole an entropy S sž n y log Žcf w6 x On the other hand, the n-particle Calogero Moser Hamiltonian describes a system of size L; ' n as n ` for fixed v and l w9 x, so the identification of this system with the RN black hole implies that the area A of the black hole horizon is A;L It follows that S;A, in qualitative agreement with the Bekenstein Hawking entropy References wx T Banks, W Fischler, S Shenker, L Susskind, M-theory as a matrix model: a conjecture, Phys Rev D 55, 5 wx B de Wit, J Hoppe, H Nicolai, Nucl Phys B wfs3x 305 Ž wx 3 PK Townsend, Phys Lett B 373 Ž wx 4 E Witten, Nucl Phys B 460 Ž wx 5 N Seiberg, Phys Rev Lett 79 Ž ; A Sen, Adv Theor Math Phys Ž wx 6 MJ Duff, GW Gibbons, PK Townsend, Phys Lett B 33 Ž wx 7 HJ Boonstra, K Skenderis, PK Townsend, JHEP 0 Ž wx 8 N Itzhaki, J Maldacena, J Sonnenschein, S Yankielowicz, Phys Rev D 58 Ž wx 9 K Becker, M Becker, J Polchinski, A Tseytlin, Phys Rev D 56 Ž w0x J Maldacena, Adv Theor Math Phys Ž 998 3; SS Gubser, IR Klebanov, AM Polyakov, Phys Lett B 48 Ž ; E Witten, Adv Theor Math Phys Ž wx GW Gibbons, PK Townsend, Phys Rev Lett 7 Ž wx A Jevicki, T Yoneya, Nucl Phys B 535 Ž w3x IR Klebanov, AA Tseytlin, Nucl Phys B 475 Ž ; V Balasubramanian, F Larsen, Nucl Phys B 495 Ž w4x R Kallosh, A Rajaraman, Phys Rev D 58 Ž w5x F Calogero, J Math Phys Ž w6x AP Polychronakos, Phys Lett 408B Ž w7x MA Olhanetsky, AM Perelomov, Phys Rep 7 Ž 98 33; AM Perelomov, Integrable systems of classical mechanics and Lie algebras, Birkhauser, 990 w8x G Barucchi, T Regge, J Math Phys 8 Ž ; S Wojciechowski, Phys Lett A 64 Ž w9x DZ Freedman, A Mende, Nucl Phys B 344 Ž w0x A Gorskii, N Nekrasov, Nucl Phys B 44 Ž wx V de Alfaro, S Fubini, G Furlan, Nuovo Cim A 34 Ž wx VP Akulov, IA Pashnev, Theor Math Phys 56 Ž ; S Fubini, E Rabinovici, Nucl Phys B 54 Ž w3x E Ivanov, S Krivonos, V Leviant, J Phys Ž w4x P Claus, M Derix, R Kallosh, J Kumar, PK Townsend, A Van Proeyen, Phys Rev Lett 8 Ž w5x JA de Azcarraga, JM Izquierdo, JC Perez-Bueno, PK Townsend, Superconformal mechanics, black holes and nonlinear realizations, Phys Rev D, in press, hep-thr98030 w6x JD Bekenstein, VF Mukhanov, Phys Lett B 360 Ž 995 7

193 0 May 999 Physics Letters B On the thermodynamics of a gas of AdS black holes and the quark-hadron phase transition John Ellis a, A Ghosh a, NE Mavromatos a,b, a CERN, Theory DiÕision, GeneÕa CH-, GeneÕa 3, Switzerland b UniÕersity of Oxford, Department of Physics, Theoretical Physics, Keble Road, Oxford OX 3NP, UK Received 8 March 999 Editor: L Alvarez-Gaumé Abstract We discuss the thermodynamics of a gas of black holes in five-dimensional anti-de-sitter AdS space, showing that they are described by a van der Waals equation of state Motivated by the Maldacena conjecture, we relate the energy density and pressure of this non-ideal AdS black-hole gas to those of four-dimensional gauge theory in the unconfined phase We find that the energy density rises rapidly above the deconfinement transition temperature, whilst the pressure rises more slowly towards its asymptotic high-temperature value, in qualitative agreement with lattice simulations q 999 Elsevier Science BV All rights reserved Introduction The striking conjecture of Maldacena wx on the equivalence of large-nc superconformal quantum gauge theories on d-dimensional Minkowski space M considered as the boundary of Ž dq d -dimen- sional anti-de Sitter space AdS dq to classical supergravity in the AdS bulk, has opened a new dialogue between students of non-perturbative gauge theories and string theorists Quantities in the strong-coupling limit of gauge theory may be calculable using classical correlators in AdS Ž super gravity, andror non-perturbative aspects of string theory may be related to correlators in gauge theories w,3 x, PPARC Advanced Fellow in a holographic spirit wx 4 In particular, the AdS approach was used in wx 5 to relax the assumption of four-dimensional supersymmetry by starting from a supersymmetric theory in six dimensions, which was one of the cases for which the conjecture was thought to be valid, and compactifying appropriately two of the dimensions The resulting compactification led to a high-temperature regime for the four-dimensional boundary theory, which had broken supersymmetry In this way, confinement at low temperatures and deconfinement at high temperatures could be demonstrated However, the gauge theory was still conformal, and asymptotic freedom was therefore not present Nevertheless, this approach has motivated inwx 6, the quark- triguing estimates of glueball masses antiquark potential wx 7 and QCD vacuum condensates wx 8 that agree surprisingly well with lattice and other phenomenological estimates r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

194 94 ( ) J Ellis et alrphysics Letters B Two of us Ž JE and NEM have proposed wx 9 a generalization of this holographic approach to the AdSdqrMd correspondence which is based on Li- ouville string theory w0, x, in which conformal symmetry and supersymmetry need not be assumed, provided world-sheet defects wx are taken into account properly in the Liouville-dressed theory The Liouville field itself provides an extra bulk dimension, and the AdS structure is induced by the recoil of the world-sheet defect, when considered in interw3 x Within this action with a closed-string loop approach, it was possible to demonstrate the formation of a condensate of world-sheet defects at low temperatures, which was related to the condensation of magnetic monopoles in target space and induced confinement wx 9 It was also possible to demonstrate the logarithmic running of the gauge coupling strength Although this Liouville approach is somewhat heuristic, it opens up a new way to discuss non-supersymmetric QCD in the strong-coupling regime and at finite temperature In particular, the target-space quark-hadron deconfinement-confinement transition may be viewed as a Berezinskii- Kosterlitz-Thouless phase transition of world-sheet vortices w4, x, which can also be related to the phase transition of black holes in AdS w5 x In this paper we embark on a heuristic attempt to use this approach to model aspects of quark-hadron phase transition Lattice analyses w6x indicate that the free energy of pure QCD rises relatively rapidly above the critical temperature to approach the asymptotic ideal-gas value as predicted in perturbative QCD On the other hand, the pressure is calcuw6x to rise much more slowly towards its lated asymptotic ideal-gas value, and one possible interpretation is that massive effective degrees of freedom are important close to the transition, causing a larger departure from the ideal-gas picture for the pressure than is the case for the free energy Calculations close to the phase transition necessarily require non-perturbative techniques, such as the lattice, and our hope is that the M4rAdS5 correspondence may also prove useful in this region Specifically, we model aspects of the gluon plasma using a non-ideal gas of black holes in AdS, inter- 5 acting via forces of van der Waals type, and described by an effective van der Waals equation of state that we derive in this paper We discuss both a high-energy limit related to wx 5 and an approach to the phase-transition region related to Liouville string theory w,7,9 x In both QCD w8x and AdS gravity w5 x, three distinct transition temperatures have been identified: T, below which only the confined phase 0 of the gauge theory exists and black holes condense leaving a residual gas of radiation; T, at which the free energies of the confined and deconfined phases Ž or black-hole and radiation phases are equal; and T, beyond which only the deconfined and stable- black-hole phases exist We relate T0 and T to the temperatures of Berezinskii-Kosterlitz-Thouless tran- sitions for vortex and spike condensation wx 9, and construct an interpolation to describe qualitatively the intermediate region T;T Thermodynamics of a gas of AdS black holes We consider a homogeneous gas of AdS black holes, each of mass M, and restrict ourselves to the case where the characteristic velocity of a generic black hole in the ensemble is either zero or very small: < u < <, corresponding to the case of very i massive black holes We consider first the static case: uis0 This will help in gaining insight into the u i/0 case, which we treat later as a perturba- tion of the static case We consider an ensemble of N indistinguishable Schwarzschild black holes in a five-dimensional AdS space-time with radius b, which is related to the critical temperature T above which massive black holes are stable: T s rž p b The invariant line element is be taken to be Ž in Minkowskian signature w5,5 x: ž / r v M ž b r / r v 4 M ds sy q y dt b r y 4 q q y dr qr dv, Ž where the AdS radius b is related to the negative cosmological constant L by bs' y3rl, and v 4 '8GNr3p, where GN is the five-dimensional Newton constant that is related to the Planck length l P via G sl 3 N P, and M is the ADM mass of the black 3

195 ( ) J Ellis et alrphysics Letters B hole The outer horizon of the black hole is defined to be the larger positive root r of the equation v M q y s0, namely rq 4 b r q r rqsb yq q4v4 Mrb 3 q ž ( / For the purposes of calculating the partition function and other thermodynamic quantities of the ensemble, a Wick rotation to a Euclidean AdS geometry: t it, will always be understood The Euclideanized AdS- Schwarzschild space-time has been found to be smooth w5 x, provided the Euclidean time direction is perodic at a particular radius b 0: 4p b r y q b 0 'TH s Ž 4 4r qb q where TH is the Hawking temperature of the black hole In subsequent sections of this paper, we consider two limits of Ž 4 that correspond to high Hawking Ž temperatures T R T, namely i b < r and Ž ii H 0 q b 4r, where we assume in each case that l q P < r,b It is easy to check using Ž q 3 that in the limit Ž i we also have v4 Mrr q;rqrb, whereas in the limit ii we also have r q,v4 M4l P, so that we are consistent with the static limit in both cases According to the analysis of w5 x, the thermodynamic ensemble of black holes is stable in the limit Ž i: l P <b <r q <v4 M, which was considered in wx 5 This corresponds to the region T;T The limit ii : l P <r q ;v4 M<b, where the radius of the AdS space-time is large compared to the outer horizon, was studied perturbatively in wx 9, using the Liouville approach where b;d y `, with d 0 q a parameter appropriately defined to regulate the recoil operators According to w5 x, this limit corresponds to a temperature TRT 0 However, the intermediate regime of the phase transition where T;T, lying between the regions Ž i and Ž ii, cannot be studied reliably by analytic methods, and we resort later to continuity arguments in order to describe the energy and pressure curves in this region 3 Is there a phase transition? To answer this question and to investigate its order, one should examine the equation of state for an ensemble of AdS black holes in appropriate regimes of the parameters We consider first the limit Ž i above Using standard General Relativity, the effective static potential Ur between two black holes in the ensemble with a radial separation r is given by the temporal metric component g 00, that can be obtained from Ž : r v 4 M g00 syyuž r, UŽ r s y Ž 5 b r provided the potential is weak We note that the potential Ž 5 vanishes at a radius r : r4 U r0 s0 r0s v4mb 6 b Using Ž 3, we see that r, r q 0 q 4r q Therefore, if we restrict ourselves to a thin shell outside the horizon: rqf r F rqq, the potential is indeed small, and one can take s r0 yr q,b r4rq to a good approximation Since the potential varies very little over the range rqfrfr qq, we make the second approximation Ur,U, where U is a constant, justified because d U; The fact that the absolute value of the constant U is a small number also justifies our weak-field approximation These statements can be checked more precisely using the following formulae: UŽ r,usconst'knrdv, H rqq 4 d xu r r q ks Ž rn 5 p r q, yv4 M r q, Ž 7 4N b where the effective volume DVs V4y V 0 is the size of the four-volume of the shell rqfrfr qq and N is the number of black holes in the ensemble We denote by V 0 the four-volume inside the horizon From the expression Ž 7 for k, one can easily check that U- We shall see that the above 0

196 96 ( ) J Ellis et alrphysics Letters B approximations lead to analytic expressions for thermodynamic quantities of our AdS black-hole system We first evaluate the classical partition function Z of the system, assuming that it is in thermal equilibrium at a Hawking temperature T'b y : ž / 4 4 d xd p yb wp r MqMUŽ r x H 4 Zs e N! Ž p ž / ž pb / M s N! pb ž Hd xe / N N 4 yb UŽ r M N M N yb UMN s DV e Ž 8 N! Some remarks are in order at this point First, in the static case which we are considering now, the kinetic term of the black holes has no physical meaning and serves only as a cut-off for the momentum integrals Secondly, the spatial integral is understood to be taken over the prescribed shell rqf r F rqq, which give rise to the volume factor DV Finally, we underline that we have made explicit use of the approximations mentioned above Before proceeding to compute other thermodynamic quantities, we also examine the low-velocity non-static case As we shall see, the difference between the static and the non static cases can be described effectively by a renormalisation shift of the mass term M MqT in the partition function above Ž 8 To prove this, we note that in the non-static case, where the heavy black holes move with a small velocity < u < i < relative to each other, we may employ both the small-velocity and weak-potential approximations simultaneously To obtain the velocity corrections to the potentials, we use the Minkowski-signature Lagrangian: ds A B LsyM, ds sygabdx dx, dt A, Bs04, Ž 9 where ds is given by Ž We then write ( ( i j ds dx dx i j s yg00 ygij s yg00ygiju u dt dt dt N Ž 0 and set 4 00 ij ij ij i g syyu r, g sd qh, r ' x Ý is from which we see that dsr dt s quy < u < i j ( i yhiju u Expanding in powers of both U and u i to leading non-trivial order, we find: < < i j ž i ij L,yM quy u y uh u q Uu < < q Ž i / Parametrising with a generalized velocity-dependent potential U in the Lagrangian: LsyMq Mu < i < ymu Ž x,u, we get U x,u suq Uu < < y uh u, 3 i j i ij which clearly reduces to U in the static limit u 0 i The partition function for slowly-moving black holes can be computed in a straightforward manner First, we note that the generalized momenta are given by E L j pis smž uiyuiuqhiju, Ž 4 E u i giving rise to the Hamiltonian: i i HspiuyLsM qu q pu i 5 Taking into account the facts that hijsuxi x jrr pj and u, wd Ž qu yh x i M ij ij, we can re-express the Hamiltonian as: Hs pikijp jqmu, Kijs dijž qu yh ij, Ž 6 M which resembles the Hamiltonian of a particle with momenta pi in a curved space whose metric Kij depends solely on the potential U and not on the

197 ( ) J Ellis et alrphysics Letters B velocity u Ž to this order i The resulting partition function is 4 4 N d xd p yb w p ik p jqmuž r Zs e ij x ž H 4 N! Ž p / N N 4 yb UŽ r M ž / ž H / M s d x N! pb (det KM e, Ž 7 where Ž ij ij (detž KM s Det Ž qu d yh r U,exp Tr Ud ijyhij q,e Ž 8 Ž Thus, one has, upon approximating U, U cf 7 : ž / N M N ynž b MqU Z, DV e Ž 9 N! pb which, when compared with Ž 8, demonstrates the aforementioned renormalisation shift in the mass by T In view of this simple change, from now on we shall deal with the general velocity-dependent case The energy of the system is defined as E Es Ž ylnzqbmn, Ž 0 Eb where the ground-state energy due to the chemical potential m s E Ln ZrE N has been appropriately added The pressure of the system is defined as: E LnZ P', Ž b EDV Ž which upon using 8 yields the following equation of state: ž / N NTs PykM Ž V yv, Ž 4 0 Ž V yv 4 0 where the Boltzmann constant has been put to unity and M represents the shifted mass appearing in the partition function Ž 9 The relation is nothing other than a Õan der Waals equation of state In our view, this leads to the prima facie expectation that a first-order phase transition takes place in the bulk, though this remains to be verified For the non-static case Ž 9, the quantity E is EsyNLn NyN pb E M M y N y Ž yb M Eb pb pb q N Ln ž / qn Ln DVq ž / M N EDV pb DV Eb qž yb MNUyNU Ž 3 Ž In the limit under consideration, we may use 4 to relate the black-hole mass to the temperature: p M; bt Ž 4 v 4 for a five-dimensional AdS-Schwarzschild black hole We assume Ž see later that the AdS radius b scales with the temperature as b;c0t y, Ž 5 where c0 is taken sufficiently large to ensure that v M4b 4 Thus E is easily evaluated: Esconstq NTy6N LnT XX y XX y3 qc UNT yc NUT, 6 where U-0, and is assumed constant, and we have used Ln N!,N Ln N for large N The constant in Ž 6 can be set to zero by an appropriate normalization of the energy, since only energy differences matter The energy density r for the four-dimensional system on the boundary of AdS5 is obtained by dividing E by the three-volume which, in view of the above discussion, scales like T y3 Thus rrt 4 A Ny6NT y LnT XX y3 XX y4 qc UNT yc NUT 7 As for the energy density, the pressure in four dimensions Žthree space dimensions, one periodic temperature dimension, denoted by p, is computed In the case of an Ž nq -dimensional AdS-Schwarzschild black hole of large mass M, the temperature scales as T n

198 98 ( ) J Ellis et alrphysics Letters B from the equation of state after writing it in the form p'bpsconst= NT, Ž 8 DV 3 Ž using 7 and assuming that the variations of the potential with the volume are small The quantity bp simply represents the fact that the three-space pressure should be computed with respect to a threevolume shell, DV, and not a four-volume shell, 3 DV 4 The former scales with one length dimension less compared to the latter, and thus the quantity bp in has the right scaling with T With in this mind, we remark that DV3 scales like T y3 in the very-high-temperature regime, beyond the upper phase transition, and hence that prt 4 ;const Ž 9 The constant term in 9 may be fixed by the fact that in the very-high-temperature regime the system is supposed to represent a gas of massless gluons, and hence, from the classical statistical mechanics of a ideal gas of massless Bose particles, the energy density is three times the pressure The energy density curve is plotted in Fig We observe that the qualitative features of QCD are Fig The scaled energy density rrt 4 Ž dashed line and pres- 4 sure 3 prt Ž continuous line in a gauge theory, plotted as functions of the temperature T, as calculated in the high-temperature limit Ž i b< r wx q 5 using a typical set of parameters for N indistinguishable AdS black holes The bump in the energy density is reminiscent of the transition from a gas of pions to a deconfined quark-gluon plasma in the QCD case, but the approximations made in the limit Ž i are not reliable in the regions where the lines are dotted correctly captured by our classical statistical system of AdS black holes The energy density drops sharply as we approach low temperatures, and it is tempting to identify this region might with the deconfined region of QCD, approached from the high-temperature unconfined phase Our approximate calculation exhibits a bump in the energy-density curve before the confined region is reached, due to the y yt LnT term in Ž 7 However, the limit Ž i that we have used above is valid only for high temperatures, and should not be trusted quantitatively in this region On the other hand, the appearance of this bump may indicate the existence of a thermodynamic instability, given that the bump region is followed by a sharp drop in the energy density as the confined region is approached 4 Another view of the phase transition In this section we shall look at the phase transition from the opposite viewpoint, described by the limit ii defined above: l P < r q, v4 M < b The approximations made in the analysis of the previous section are also valid in this parameter regime, though in a different way When there is a large separation between any pair of black holes in the ensemble: r q<r<b, it is again a good approximation to take Ur,U, where U is a positive constant, because the potential varies very little over the range r q<r <b Not only that, but the constant U is also a small number and hence one can again make a weak-field approximation The analogues of the formulae Ž 7 are in this case: UŽ r,usconst'knrv, H b 4 ks Ž rn d xuž r r q 6 p r 4 q s b y y3v4 MŽ b y r q, Ž 30 4N b where, as before, the effective volume VsV4yV 0 is the size of the four-volume of the shell r q<r< b, and N is the number of black holes in the ensemble Given that V;b 4, and using the above formula Ž 30 for k, one can easily check that U- Notice that the potential is attractiõe in the region r q<r-r 0 and repulsiõe in the region r0-r<b

199 ( ) J Ellis et alrphysics Letters B Ž In this region, 4 tells us that the black-hole mass is related to the temperature by: b M; Ž 3 4pv 4 One can perform calculations for the partition function, energy and pressure of the ensemble that are similar to those described in the previous section, which we do not reproduce in detail here As in the limit Ž i, we find again in the limit Ž ii a Õan der Waals equation of state, asin To understand qualitatively the physics in the lower end of the transition region T;T, we recall that, as we approach the transition region from above, we enter a regime where the Liouville theory takes over In this theory the radius b may be assumed to be independent of temperature wx 9, and large In this limit of large b and T-independence, a different approximation is needed to capture correctly the features of the transition region, since the classical description of a gas of stable black-hole particles breaks down in this case However, we can still obtain qualitative ideas of the dynamics by applying the above statistical-mechanical approach to this case Notice that the smallness of v4 M compared to b implies that the AdS space is regular for large r This is the regime discussed in the Liouville ap- proach of wx 9 We have calculated the energy density r and the pressure 3 p, with the results shown in Fig We observe that the pressure is almost constant near the q Fig As in Fig, but in the limit ii : r < b wx 9, conjectured to represent the start of the phase transition regime The pressure curve lies below the energy density curve and is almost constant: the bump in the energy density is less marked than indicated in the limit Ž i transition region, whilst the energy increases and exhibits a bump As compared to our results in limit Ž i, this bump is rather smoother The constant value of the pressure is again fixed by the fact that at low temperatures the system again enters an ideal-gas regime, where in this case the physical degrees of freedom are the bound states, ie, massless pions in the case of QCD, so that the relation rs3 p should again be valid 5 Relating the two descriptions The regime Ž i which describes the high-temperature tail of the phase transition must connect smoothly with the regime Ž ii as the scales cross: rql b Since one expects that the temperature should rise after the phase transition, we assume that T Žii <T Ži At the boundary of the two regions, one has rq B ;( v 4 M B ; b B, and the temperature T B ; rb B should therefore lie in the range T Žii <T B <T Ži A natural way to arrange this crossover is to keep r q fixed at the boundary value and study the variations of the other parameters as we go from one regime to the other Clearly this puts the following bounds on the other parameters: b Ži <b B <b Žii, M Žii ;M B <M Ži Ž 3 These bounds seem natural and consistent with the definitions of the limits Ž i and Ž ii In particular, one outcome of this crossover, namely that the black-hole degrees of freedom are more massive in region Ž i and hence decouple from infrared physics, seems consistent for describing the region just above the phase transition, which, according to Ž 3, is associated with lower-mass black holes: M Žii ;M B <MŽi in region Ž ii We would also like to comment on the behaviour of the AdS radius b in the transition region The scaling Ž 5 is justified in the Liouville approach of wx 9, in which the recoil-induced AdS radius b is proportional to a homotopic time variable In the analysis of wx 9, this homotopic time was identified with the target time X 0 in a real-time formulation of Liouville QCD In this real-time formalism, the time X 0 should not be confused with the temperature However, from the equivalence of the real-time and Matsubara formalisms, where one identifies the

200 00 ( ) J Ellis et alrphysics Letters B temperature with the inverse radius of a compactified Euclidean time, it is natural to assume that, at least in the high-temperature regime where one assumes thermal equilibrium with a heat bath of temperature T, the scaling Ž 5 should be valid An alternative way to justify the scaling Ž 5 is to notice that, in order to arrive at the regime where the analysis of wx 9 is valid, one needs to go to very low temperatures, where b is huge This result is not in contradiction with our above procedure of identifying b with rt However, in the low-temperature regime b is almost constant wx 9, and not scaling with temperature We conjecture that there are in general competing contributions to b, so that b;d y qo Ž rt, Ž 33 and that the d y term dominates in the low-temperature regime, whilst d is comparatively large in the high-temperature regime, and the rt term domiwx 9, d was identified with the area of the nates In Wilson loop that generated the world sheet of the string This is consistent with the above picture: for low temperatures in the confining regime, the dominant degrees of freedom are related to large worldsheet areas, in the sense that the Žtemporal Polyakov or spatial Wilson loops that define the order parameters relevant for confinement are large It is these quark-antiquark loops that can be described by weakly-coupled string theory, for which the analysis of wx 9 is valid At high temperatures, on the other hand, the areas defined by the dominant order parameters Ž Wilson andror Polyakov loops are relawx 9 This tively small or microscopic, as remarked in corresponds to the pure stringy limit d y 0 In that limit the perturbative string theory approach of wx 9 is invalid, and should be replaced by the above semi-classical picture wx 5 of a gas of very massive black holes We now remark that, in our approximate treatment of near-horizon distances r, where the potential is weak, one obtains the typical order of magnitude estimate ž ( / 4 4 y r, v4mb q O b v4mb 34 Combining Ž 3, Ž 4 and Ž 5, we find that the volume V Ar 4 varies with T as T y4, and hence 0 q DV'V yv Ar 4 yr 4,c X 4 0 q T y4 Ž 35 where c X is a constant As commented above and in w5 x, the phase transition of the five-dimensional black-hole system is expected to be of first order Moreover, it is here identified with a deconfining transition in gauge theory However, at present our analysis cannot determine the precise order of these associated transitions, and this remains an open issue A related issue is whether holography wx 4 survives the first-order phase transitions associated with the boundary and bulk dynamics Based on the Liouville renormalization argument given above, we would expect so, but this issue is also open 6 Comparison with QCD Here we comment on the temperature-dependence of the pressure, and relate it to what is known for QCD from lattice simulations We recall from the discussion of Section 4 that in low-temperature limit Ž i of the phase transition region Ž see Fig 3, where b is roughly T-independent and the mass of the black hole M;T 4, there is no difference in scaling between the four- and five-dimensional pressures, and 4 hence 3 prt in Ž 8 is initially approximately constant and then increases slightly Ždue to the small- Fig 3 Interpolations of the scaled energy density rrt dashed line and pressure 3 prt 4 Ž continuous line, including the transition region between the limits Ž i and Ž ii shown in Figs and, respectively The behaviours of the energy density and pressure in the intermediate-temperature region are reminiscent of lattice calw6 x: in particular, the pressure curve rises more slowly culations than that of the energy density 4 Ž

201 ( ) J Ellis et alrphysics Letters B ness of d as the temperature increases Thus the pressure curve does not increase as abruptly as the energy density, and always lies beneath it as long as it can be calculated reliably 3 As the temperature increases towards the upper end of the phase transition, the increase in the pressure may be obtained from terms that have been ignored so far in deriving Ž These include terms that express fluctuations of the potential U with the volume V4y V 0 These are required by continuity between the two asymptotic regimes for the pressure computed above The generic Ž approximate form of such terms may be found by representing the potential fluctuations as N X U,Uqe, Ž 36 V4yV 0 where e X is small and positive Such a dependence of the potential on V4 results in extra terms on the right-hand side of the equation of state Thus, for example, in the high-temperature phase we expect the the boundary pressure to have the form: 4 prt ;const 4 DV 3 T = X NTqe N Ž MqT rž V4yV 0 Ž 37 We now recall that, on the high-temperature side of y3 the phase transition, DV scales like T, Ž V y V scales like T y4, and the mass of the black hole scales like T y The mass is sufficiently large that the M-dependent term is still dominant Hence, from 4 36 one obtains a linear increase for 3 prt : 3 prt 4 ;constqož const X =e X N T Ž 38 As the temperature increases, the e X term becomes smaller and smaller, and one recovers a constant temperature at the end of the of the transition region The proportionality constants are again fixed by requiring that this scaling should describe at high 3 w x It has been argued 9 that this feature should be expected on generic grounds based on thermodynamics However, it is highly non trivial that it is reproduced here in the context of statistical physics of a gas of five-dimensional AdS black-holes temperature an almost ideal gas of massless particles, in which case we have the relation rs3 p in three space dimensions We display in Fig 3 heuristic interpolations of r and 3 p between the high- and low-temperature limits Ž i and Ž ii These curves can be compared with those calculated for QCD on the lattice w6 x In both cases, there appears to be a sharp jump of the energy density at the onset of the deconfining phase transition, from the value where the system is equivalent to a gas of pions, towards that where the system is described by an almost ideal gas of quarks and gluons On the other hand, the increase in the pressure is much smoother in both the lattice and our AdS calculations This is related in our approach to the weak-field assumption for the potential: < U < <, which is valid for near-horizon AdS geometries in the high-temperature phase We should repeat that our analysis in the limit Ž ii is not yet quantitative at low temperatures However, the Liouville world-sheet approach of wx 9, which describes the dynamics of world-sheet defects via D particles, describes this regime qualitatively correctly, leading in particular to confinement as a low-temperature property In this case, the space-time obtained from D-particle recoil is indeed of AdS type with M 0 in Žw 3,9 x We have shown in this paper that this approach has, moreover, a plausible regular continuation to the high-temperature limit Ž i explored in wx 5 This gives us further reason to hope that the Liouville string approach may be suitable for development into a reliable tool for describing non-perturbative gauge dynamics, and therefore may contribute to the new avenue for non-perturbaw,5 8 x tive gauge-theory calculations opened up in Acknowledgements One of us Ž AG thanks the World Laboratory for a John Bell Scholarship Another Ž NEM thanks Mike Teper for useful discussions The results of this paper were presented by JE at the memorial meeting for Klaus Geiger: RHIC Physics and Beyond Kay-Kay-Gee Day, BNL, October 3rd, 998 w0 x We dedicate this paper to our friend Klaus s memory

202 0 ( ) J Ellis et alrphysics Letters B References wx J Maldacena, Adv Theor Math Phys Ž wx SS Gubser, IR Klebanov, AM Polyakov, Phys Lett B48 Ž wx 3 E Witten, Adv Theor Math Phys Ž wx 4 G t Hooft, gr-qcr93006; L Susskind, J Math Phys 36 Ž wx 5 E Witten, Adv Theor Math Phys Ž wx 6 C Csaki, H Ooguri, Y Oz, J Terning, preprint UCB-PTH , hep-thr98060; R de Mello Koch, A Jevicki, M Mihailescu, JP Nunes, Phys Rev D58 Ž ; M Zyskin, Phys Lett B439 Ž ; H Ooguri, H Robins, J Tannenhauser, Phys Lett B437 Ž wx 7 J Greensite, P Olesen, JHEP 9808 Ž and hepthr wx 8 A Hashimoto, Y Oz, hep-thr wx 9 J Ellis, NE Mavromatos, hep-thr98087; see also NE Mavromatos, hep-thr w0x F David, Mod Phys Lett A3 Ž ; J Distler, H Kawai, Nucl Phys B3 Ž wx J Ellis, NE Mavromatos, DV Nanopoulos, Erice Subnu- clear Series Ž World Sci, Singapore 3 Ž 993 ; hepthr wx J Cardy, Nucl Phys B05 Ž 98 7; J Cardy, E Rabinovici, Nucl Phys B05 Ž 98 ; B Ovrut, S Thomas, Phys Lett B37 Ž 99 9; J Ellis, NE Mavromatos, DV Nanopoulos, Phys Lett B89 Ž 99 5 and Mod Phys Lett A Ž w3x J Ellis, P Kanti, NE Mavromatos, DV Nanopoulos, E Winstanley, Mod Phys A3 Ž w4x I Kogan, JETP Lett Ž 987, 709; B Sathiapalan, Phys Rev D35 Ž ; J Atick, E Witten, Nucl Phys B30 Ž 988 9; AA Abrikosov Jr, I Kogan, Int J Mod Phys A6 Ž w5x S Hawking, D Page, Comm Math Phys 87 Ž w6x For a recent review, see: H Satz, hep-phr9789 w7x J Polchinski, Phys Rev Lett 75 Ž ; J Polchinski, S Chaudhuri, C Johnson, hep-thr96005 and references therein; J Polchinski, TASI lectures on D-branes, hepthr96050, and references therein w8x BA Campbell, J Ellis, KA Olive, Phys Lett B35 Ž and Nucl Phys B345 Ž w9x M Asakawa, T Hatsuda, Phys Rev D55 Ž w0x J Ellis, nucl-thr99006

203 0 May 999 Physics Letters B Black holes, string theory and quantum coherence Daniele Amati ( ) International School for AdÕanced Studies SISSA, and INFN - Sezione di Trieste, Trieste, Italy Received 8 January 999 Editor: L Alvarez-Gaumé Abstract On the basis of recently discovered connections between D-branes and black holes, I show how the information puzzle is solved by superstring theory as the fundamental theory of quantum gravity The picture that emerges is that a well-defined quantum state does not give rise to a black hole even if the apparent distribution of energy, momenta, charges, etc would predict one on classical grounds Indeed, geometry general relativistic space time description is unwarranted at the quantum microstate level It is the decoherence leading to macrostates Ž average over degenerate microstates that provides on the same token the loss of quantum coherence, the emergence of a space time description with causal properties and, thus, the formation of a black hole and its Hawking evaporation q 999 Elsevier Science BV All rights reserved Many extraordinary coincidences between string theory and black hole physics have been recently uncovered and different opinions have been advanced on how these coincidences may explain or solve the well-known information loss puzzle The idea wx that very massive string excitations should represent black holes has been better substantiated by analysing massive BPS states that, as known, have properties that are not renormalized, ie do not depend on the coupling strength In the weak string coupling Ž g regime, D-branes in four wx and five wx 3 dimensions with a convenient number of charges have been studied BPS states have been counted as well as nearly BPS states for certain regions of moduli space where perturbative computations are feasible wx 4 Decay rates have also been computed wx 5 by averaging over the many initial states and shown to have, a typical thermal amati@sissait distribution The moduli independence of these rewx 6 of their validity beyond sults allow the conjecture the moduli region where they were computed And their g independence Žalso suggested by non-renorwx 7 may imply that they could malization arguments be continued beyond the weak coupling regime An independent treatment on totally different grounds of the strong coupling regime substantiates that impression The large coupling description of the 4 and 5 dimensional systems just discussed is found by solving the 0-d supergravity equations after reduction on the same compact manifold used for the D-brane description The solution generates a metric wx 8 that depends on parameters that are related to the charges through the moduli of the compact manifold The metric shows an event horizon even in the extreme limit; its area in this limit gives the Beckenstein Hawking entropy of extremal bh This entropy and the ADM mass coincide exactly with the mass and entropy Žgiven by the log of the state multiplicity of the BPS state with the same charges r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

204 04 ( ) D AmatirPhysics Letters B as computed from D-branes in the small coupling regime For nearly extremal bh the entropy, the rate and the spectrum of evaporation wx 9 obtained by solving wave equations in the corresponding metric background coincide again wx 5 with those computed for small g And, even more remarkably, also deviations from black body spectrum agree w0 x These magic coincidences between such different calculations gave confidence to the g continuation between a unitary D-brane description and the conventional black hole with its information loss Different interpretations of this apparent contradiction have been advanced Hawking questioned the D-brane formalism because the causal properties of space time are not properly taken into account And even more so having shown wx that the inconsistency cannot be assigned to the lack of identificability of D-brane states due to their prompt decay String theorists wx favour the attitude that black body radiation is only an approximation and that the connection with unitary D-brane formalism guarantees information retrieval in bh evaporation This even more so since duality indicates situations that have flat unitary realizations for g< as well as for g 4 with a bh intermediate region with quite different space time geometry but with an allegedly common spectrum Let me share the consensus of a smooth g behaviour by considering an S-matrix approach where a continuation has a clear meaning and let me start in the small coupling regime where a perturbative string approach is granted If over a well-defined BPS state impinges from far away where for small g the space is flat a well-defined quantum state Ž say a graviton the S matrix, calculable by perturbative string theory, will be unitary It describes the absorption of the graviton generating open strings on the D-brane then decaying, through an arbitrary number of steps, to some BPS state plus outgoing closed string ground states Ž as gravitons and scalars The physical content of this large set of S matrix elements is better analysed by computing final state correlation functions that give semi-inclusive quantities as multiplicities, spectra or multiparticle correlations Old string theorists will remember the techniques used to directly compute these correlators from multiparticle amplitudes, order by order in g They will also recall, however, the need to recur to the full set of final state correlators in order to disentangle the high degeneracy of initial states This means that if we change the choice of the initial BPS state Ž among the very many degenerate ones or if we consider two gravitons impinging instead of a single one, or if we change g, all these correlators will be modified in a non-trivial and coherent way The whole set of correlators represent the complete memory of the original state The result we discussed before concerning the Hawking spectrum, tells us that if we now perform an average over the very many possible initial Žde- generate states, all multiparticle correlators will average to zero apart from the single particle spectrum that by energy conservation averages to a thermal one And this for each g, thus order by order in the string loop expansion This is perhaps not too surprising: the average over the very many degenerate microstates washes out all information over the initial identity leading to the informationless black body radiation Let me stress that this average over degenerate microstates is implied in any classical limit In increasing g towards a black hole regime, it is the quantum S matrix that should be analytically continued Its analytic structure may be very complex, with new singularities being eventually formed, but with unitarity preserved with the concurrence as before of the very many phases that depend on the initial Ž black holish microstate These are essential in order to determine the many non-trivial correlation functions The fact that these averaged to zero independently from g for small g giving the same Hawking spectrum of the large g regime shows that no new physics Žnew singularity in the g continuation has to be invoked in order to understand the spectrum and entropy of macroscopic black holes Ž statistical collection of microscopic states This same reasoning, however, shows that the decay spectrum of a single microstate differs crucially from a black body one This was the case for all small g, thus for all g by continuation in a context in which no new physics is evoked We thus expect for a would be black holish microstate, far from vanishing correlators that encode the whole informa- tion about the coherent formation process

205 ( ) D AmatirPhysics Letters B It is apparent that a single microstate even in the large g regime has not much to do with a black hole and that it is only the decoherence implied by the macroscopic description Žie average over microscopic states that generates the black hole physics Such a statement needs, however, a parallel understanding of why it is only at the macroscopic level that the geometrical interpretation of general relativity emerges with its causal properties, singularities and event horizons Superstring theory contains gravity in the infrared limit; for frequencies much smaller than the string scale, the Einstein classical equations appear as the non renormalization Ž b s 0 condition At the quantum level, however, fluctuations at the string scale will generate all other Ž massive background fields which will appear Ž thanks to the bs0 condition in a large system of coupled equations together with the metric field In a more common language, this implies a very large number of quantum hairs These many non-metric coupled fields, that inhibit a geometrical space time description, are expected to have quickly varying phases so as to be averaged out in the decoherence procedure implied by classical Žor mesoscopic physics This is perhaps not surprising, superstrings are pregeometric quantum theories in which even a space time description is not warranted: Xm are operators and it is only at the mesoscopic level Žin which quantum string fluctuations are averaged away that they appear as coordinates parametrizing a metric space It is thus this decoherence that generates a geometrical space time description Ž ie general relativity and with it, causal properties event horizons and the paraphernalia of black hole physics The consideration up to now of charged extremal or near extremal black holes or, in the string language, solitonic D branes which are BPS or quasi BPS states, was an essential step in order to identify and count stable or nearly stable states This allowed the consideration of S matrices with well-defined quantum initial states, excited in the process and evaporating back to stable final states For unstable states Žhigh string excitations or Schwarzschild bh a consistent quantum treatment has to comprehend both formation and decay In the small g regime this is anyhow the conventional approach of perturbative string amplitudes In the bh regime it implies the consideration at a consistent quantum level of both the formation and evaporation of a bh, thus avoiding, ultimately, the hybrid theoretically procedure of quantizing in the presence of a purely classical solution It is obvious that one might unambiguously prepare imploding states Žspherical waves or even high energy low mass particles colliding at very small impact parameter at large separations where a flat metric is granted At a classical level and even at a semiclassical one w3x Ž ie with radiation black holes would be formed in these conditions losing memory of the state they originated from This is not the case at the quantum level Again, at small g where everything is in principle calculable, final state correlators are far from vanishing and contain all the information needed to disentangle the initial state Ž unitarity No new physics has to be invoked in order to continue to the large g black hole regime: it is the presence of non metric fields of arbitrary high tensorial rank that avoids the Schwarzschild singularity of the usual general relativistic Ž metric solution And, again, it is the mesoscopic decoherence implied by averaging over microstates, that on one hand averages out the correlators that carry all the microscopic information thus leading to a black body Ž or approximate black body spectrum and, on the other hand, washes away the non metric fields thus leading to a geometric space time general relativistic picture with causal properties, horizons, black holes, etc That the decoherence procedure is the cause of the black body spectrum in the small coupling regime is actually shown in an accompanying paper w4 x Inclusive spectra of massless particles Žphotons for open strings, gravitons for closed emitted in the decay of high string excitations, are computed at the tree level A thermal distribution with Hagedorn temperature is shown to emerge when averaging over initial degenerate string states At this level it may be sound to ask if this decoherence may be avoided in a gedanken experiment so as to show the full quantum structure of the fundamental gravity theory Could, for instance, a classically expected black hole be avoided by preparing well-defined coherent imploding states? In my opinion the infrared properties of gravitation may jeopardize this possibility Indeed, it seems even

206 06 ( ) D AmatirPhysics Letters B conceptually hard to avoid incoherent arbitrary soft gravitons and, with them, their high decoherence power due to the very large density of microstates Let me stress the important role that the high degeneracy of states had in the smooth merging of a unitary microstate description into a black hole macrostate one Qualitatively, it is apparent that a degeneracy that grows exponentially with the mass is a border-line between a tendency of states, interacting through vertex operators, to split as in particle physics or join as for bh due to the final states multiplicity One would be tempted to think that consistent fundamental theories of quantum gravity have to have such a degeneracy in order to lead to macroscopic general relativity No wonder, in this sense, that superstring theory is a good candidate while supergravity is not It would be interesting to understand how other proposals that have been advanced, such as topological gravity, may solve this problem in their attempt to qualify as possible consistent theories of quantum gravity Acknowledgements This work was partially supported by EC Contract no ERBFMRXCT and by the research grant on Theoretical Physics of Fundamental Interaction of the Italian Ministry for Universities and Scientific and Tecnological Research References wx MJ Bowick, L Smolin, LCR Wijewardhana, Phys Rev Lett 56 Ž ; L Susskind, hep-thr wx J Maldacena, A Strominger, Phys Rev Lett 77 Ž , hep-thr ; C Johnson, R Khuri, R Myers, Phys Lett B 378 Ž , hep-thr wx 3 A Strominger, C Vafa, Phys Lett B 379 Ž , hepthr96009 wx 4 C Callan, J Maldacena, Nucl Phys B 47 Ž , hep-thr960043; G Horowitz, A Strominger, Phys Rev Lett 77 Ž , hep-thr96005 wx 5 S Das, S Mathur, Nucl Phys B 478 Ž hepthr960685; S Gubser, I Klebanov, Nucl Phys B 48 Ž , hep-thr wx 6 G Horowitz, J Maldacena, A Strominger, Phys Lett B 383 Ž 996 5, hep-thr wx 7 J Maldacena, hep-th965 wx 8 M Cvetic, D Youm, hep-thr ; hep-thr957; G Horowitz, D Lowe, J Maldacena, Phys Rev Lett 77 Ž , hep-thr wx 9 SW Hawking, Nature 48 Ž , Com Math Phys 43 Ž w0x S Gubser, I Klebanov, Phys Rev Lett 77 Ž , hep-thr ; J Maldacena, A Strominger, Phys Rev D 55 Ž , hep-thr wx SW Hawking, MM Taylor-Robinson, hep-thr wx G t Hooft, Int J Mod Phys A Ž gr-qc 96070; L Susskind, Scientific American, April 997; G Horowitz gr-qc ; JR David, A Dar, G Mandal, SR Wadia, hep-thr9600 w3x PD D Eath, PN Payne, Phys Rev D 46 Ž , 675, 694 w4x D Amati, J Russo hep-thr99009, Phys Lett B

207 0 May 999 Physics Letters B Fundamental strings as black bodies D Amati a,b,, JG Russo c, a ( ) International School for AdÕanced Studies SISSA, I-3403 Trieste, Italy b INFN, Sezione di Trieste, Italy c Departamento de Fısica, UniÕersidad de Buenos Aires, Ciudad UniÕersitaria, Pabellon I, 48 Buenos Aires, Brazil Received 8 January 999 Editor: L Alvarez-Gaumé Abstract We show that the decay spectrum of massive excitations in perturbative string theories is thermal when averaged over the Ž many initial degenerate states We first compute the inclusive photon spectrum for open strings at the tree level showing that a black body spectrum with the Hagedorn temperature emerges in the averaging A similar calculation for a massive closed string state with winding and Kaluza-Klein charges shows that the emitted graviton spectrum is thermal with a grey-body factor, which approaches one near extremality These results uncover a simple physical meaning of the Hagedorn temperature and provide an explicit microscopic derivation of the black body spectrum from a unitary S matrix q 999 Elsevier Science BV All rights reserved Introduction High string excitations have a large degeneracy; the number of string states of a given mass increases exponentially in the excitation mass measured in string units Ž string tension or inverse string length This degeneracy is consistent with the unitarity of the perturbative S matrix Unitarity guarantees that the analysis of final states that arise by the decay of a massive state must allow to retrieve the information on how the original massive state has been formed, ie, on which is the coherent superposition of the many degenerate microstates that have been formed and subsequently decayed into the final states under analysis A consistent and complete way of analysing final states is through semi-inclusive quan- amati@sissait russo@dfubaar tities, such as spectra, two particle correlations, three particle correlations, etc The total set of these quantities is complete in the sense that it characterizes the initial state or, equivalently, the coherent superposition that created the final quantities in question Semi-inclusive quantities are calculable in perturbative string theory Žorder by order in the string coupling with a well-defined algorithm wx, which was extensively used in the past when string theory was being applied to strong interactions In this paper a remarkable property of the decay spectrum of a massive string state will be found: when averaging over all string excitations of mass M Ž M 4 string tension, the decay spectrum exactly averages to a black body spectrum with Hagedorn temperature T H That the average procedure should wash out all the information stored in spectra and correlations leading to a total informationless therwx mal distribution was conjectured in order to r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

208 08 ( ) D Amati, JG RussorPhysics Letters B understand recently uncovered connections between string the ory and black holes For completeness we will briefly mention in Section 4 the possible relevance for black hole physics of this non-trivial property of a unitary microscopic quantum theory generating thermal properties in a decoherence procedure The present results also unveil an important physical meaning of the Hagedorn temperature, as the radiation temperature of a macroscopic Ž averaged string Decay rates in open string theory The inclusive spectrum for the decay of a particular massive state < F : of momentum p sž M,0 m into a particle with momentum k Žk s 0 m is represented by the modulus squared of the amplitude for the decay of < F : into a state < F : m< k : X m, summed over all < F : In string theory, the initial state < F : X will be described by a specific excitation of the string oscillators Žfor simplicity, we restrict the discussion to bosonic string theory If Nn is the occupation number of the n-th mode Ž with frequency n, a state < F : of mass M will be characterized by a partition 4 ` N of NsÝ nn, with n ns n ` ˆ < : < : ˆ N F N 4 sn F N 4, Ns Ý nanpa n, n n ns a M sny, a,a sd h, X m n mn n m nm where m,ns,,d For large N, there are w3,4x Dy NN sconst N e, asp, 6 Dq y a' N 4 ( possible partitions, thus representing the degeneracy of the level N associated with excitations of mass M The total momentum p X of F will be p X m X mspm yk At the tree level, < F : m X will also be a superpo- sition of string excitations, satisfying Nˆ < F : X s N X < F :, X X X X X 0 ( N sa p qsnyk a Ž Ny, 3 Let us momentarily ignore twists or non-planar efw3,4 x They will fects of open string theories be incorporated later The inclusive photon spectrum for the decay of a state F is then given by Ý Nn4 dg Ž k s <² F < VŽ k < F :< F N 4 0 X Nn4 n F < X X N =VolŽ S Dy k0 Dy 3 dk 0, Dy p Dy VolŽ S s, 4 Dy G ž / where ÝF < X means the sum over all states F X N X satisfying the mass condition 3, and VŽ k is the photon vertex operator VŽ k sj E X m Ž 0 e ik X Ž0, j k m s0, k s0, m t ` m m m m Ý n n ' ns n ˆ X 0 sx qi a ya, ` m m ' m m t Ý n n ns ˆ E X 0 sp q n a qa, m m x ˆ, pˆn sid n, 5 m where jm is the photon polarization vector, and we have set a X sr The d-function of energymomentum conservation, coming from the zero-mode part of the amplitude, has been factored out from 4 Let us define a projector Pˆ N over states of level N as dz ˆ NyN ˆ P s z, Pˆ < F X: s d X< F X: N E N N NN N, 6 z C where C is a small contour around the origin Introducing Pˆ X in 4 N we can convert the sum into a sum over all physical states < F : X, and use completeness to write dz X X XyN X Nˆ F F se X z ² F N 4 < V Ž yk z VŽ k < F N 4 :, Nn4 n n C z 7 3yD k dg 0 F N n 4 FF Ž k ' 8 Dy Nn4 VolŽ S dk0

209 ( ) D Amati, JG RussorPhysics Letters B Let us now introduce a coherent set basis, ` < : 4 Ł < : n n n m ns l s exp l Pa 0;p, 9 where l, ns, `, is a set of arbitrary Ž complex n parameters, and define dz dz X X yn XyN F l ) l se z E X z z X z C C X N N = ² l 4< W Ž yk z WŽ k z < l 4:, n ˆ ˆ n 0 where WŽ k 'expw jpe X m xexpwik X x t From stan- dard properties of coherent states w3,4 x, we find dz dz X X yn XyN ppž j qj F l ) l se z E X z e z X z C = C ` ) n X n Ł exp K nqlnpln z z ns ) ) qln PJnqlnPJ n, m m m X n m X n J s' n n Ž j qj z y k Ž yz, ' n ) m m n X n m n m n X n J s' n n Ž j z z qj z y k z Ž yz, ' n K sj Pj nz X n n The spectrum F Ž k F N 4 may be directly computed n from 7 using standard operator techniques or from the explicit expression,, and Žwe omit Lorentz indices ` Nn Nn E E F k s F ) F Ł l l, Nn4 ž / ž ) ns El / n Eln linear in jj ) lnslns0 3 which follows after inverting 9 and noticing that VŽ k is represented by the linear term in j of WŽ k For every < F :, the F Ž k N 4 F are easily obtained n Nn4 and are polynomials in k m The spectrum obtained by averaging over string states with mass M is F Ž k s F Ž k 4 Ý N 0 F N 4 0 N n N F < N One can introduce again the projector 6 to convert the above sum into a sum over all states, so that dz dz X X yn XyN F N Ž k 0 s E z E X z N z X z N C C = Tr z VŽ yk z VŽ k 5 Nˆ X Nˆ The trace in 5 can then be computed from by integrating over l n,l ) n, ) m ) m yl n Pln HŁ ) N 0 n n l l NN n,m F k s dl dl e F, j j 6 where < j j means the term linear in j m,j n, and then setting j s j Using we get FN Ž k0 s m e ppž j qj E N N m E = dw dõ yd X yn NyN w fž w Õ w Õ C C ) n y = exp J PJ Ž yw qk, n n n jj ` Ł Ž n X X ns f w s yw, wszz, Õsz 7 C and C are two small contours around zero The extra factor f Ž w in the integrand arises as usual by incorporating the ghost contribution Žthe D y power corresponds to the D y transverse coordinates Thus we obtain F Ž k s j j N 0 m n N N E E = dw dõ yd X yn NyN w fž w Õ w Õ C C Ž m n mn = p p qh VŽ Õ,w, 8 ž / w Ž Õ qõ V Ž Õ,w s n Õ q 9 ` n n yn n Ý n ns yw

210 0 ( ) D Amati, JG RussorPhysics Letters B Ž X The integral over Õ gives N -N X X yn yd j Ž NyN dw w fž w F k s X N Ž 0 E NyN NN w yw 0 X For large N, the integral in 0 can be computed by a saddle point approximation, with the main contribution coming from w; This is similar to the familiar calculation of the number of states N N of level N The behavior of fž w at w( is well known and given by p yr y 6yw f w (const yw e Therefore the saddle point is at lnw(yar N, X where a was defined in Eq Using N sny X X k ' a N we obtain for large N 0 a Ny k 0' ax N ya' N e X F k (j k ' N Ž 0 0 a N yak0' a ye X X We have set NrN Dq 4 (, since in the regime of X interest, k <M, NrN ( Ž 0 the emission of a photon with energy k0 of order M and higher is supy ' N X pressed by a factor of order e Thus ' N X y' N (yk ' a and 0 yak0' a e X X FN k 0 (j a k0m yak a ye 0' X, 3 or ' k 0 y e TH Dy N 0 k y TH dg Ž k (const k dk, 4 ye where T is the Hagedorn temperature T s H H a a X ' Thus the radiation spectrum from a macroscopic Žie averaged over all degenerate microscopic states string of mass M is exactly thermal Ždespite the spectrum for each microstate being absolutely nonthermal The way the exact Planck formula arises in the final result is striking, since the exponents in the numerator and denominator have different origins Let us now show that the non-planar contribution does not change the above result It arises in open ' X string theory due to the fact that the photon may be emitted by any of the two ends of the string So, strictly speaking, we should have written dg Ž k F N 4 0 n s Ý F < Ž VŽ k qq VŽ k Q < X F N;n 4 ' F < X N X =VolŽ S Dy k0 Dy 3 dk 0, 5 at the place of 4, where Q is the twist operator wx 3 The modulus squared involving VŽ k and that with Q VŽ k Q leads each to half of the planar result 4 obtained before The cross product gives rise to the non-planar contribution that may be computed analogously to the planar loop diagram, of which our spectrum represents the absorptive part Instead of 5 for the non-planar contribution one has dz dz X X yn XyN FN Ž k0 np s E z E X z N z X z N C C = Tr z Q V Ž yk Q z VŽ k 6 Nˆ X Nˆ Using the explicit expression for Q wx 3 and computwx 5 one finally obtains ing the trace F Ž k N 0 np s jm jn N N E E X C = dw dzx X yn yd X NyN w Ž yw fž w z w X z C Ž m n mn = p p qh VŽ Õ,w, 7 where V is given by Eq 9 with Õs X The contour integral in z X now selects the pole at z X sw The remaining integral over w is dominated by the wy z X y z same saddle point that allowed the evaluation of 0 The final result is j a yak 0 ' a X N 0 np ' F Ž k ( e 8 N which is smaller by a factor -rn with respect to the planar result 3 or 4 that dominates the sum 5

211 ( ) D Amati, JG RussorPhysics Letters B Spectrum of a charged closed string state Let us now consider closed bosonic string theory with one dimension compactified on a circle of radius R Let m0 and w0 be the integers representing the Kaluza-Klein momentum and winding number of the string state along the circle Žwe will assume m,w )0 The mass spectrum is given by 0 0 X X ˆ X ˆ a M s4 NR y qa Qq s4 NL y qa Q y, Ž 3 m0 w0r where Q" s ", Consider the decay rate of a X R a state of level N L, NR of given charges m 0,w 0, with NLy NRs m0w, 0 by averaging over all physical states of charge m,w and level N, N The calcu- 0 0 L R lation factorizes in a left and right part, and it is similar as in the previous section We assume that N X and N X R L are sufficiently large for the saddle-point approximation to apply The final result is k0 k0 y y e T e L T R N NŽ 0 R L k0 k0 y y ye T ye L TR dg k sconst j M =VolŽ S Dy k0 Dy dk 0, Ž 3 ( ( M yqq M yqy TRs X, TLs X Ž 33 a' a M a' a M Eq Ž 3 is remarkably similar to the analogous result derived for D-branes in wxž 6 see also wx 7, despite the two calculations being rather different, and describing different physical systems Indeed, the D-brane calculation involves only a single oscillator a,a that excites modes with one unit of Kaluza-Klein momentum wx 6 ; the present elementary string calculation involves all excitations a n,a n, ns,,`, in a very specific way dictated by the vertex insertion By analogy with black holes wx 7, Eq Ž 3 may be written as k 0 y T e Dy dgn NŽ k0 sconst s Ž k0 k0 dk 0, R L k 0 y ye T Ž 34 y y y where T s T q T and s Ž k R L 0 is the grey body factor k 0 y T ž TL/ž TR/ ye s Ž k0 sk 0 Ž 35 k k 0 0 y y ye ye The radiation vanishes if the initial string state saturates the BPS condition MGQ q, ie when Ms Q q For a near BPS state, M(Q q, NL4N R, one gets (M yqq T(TR s X Ž 36 a' a M In the superstring theory, owing to the supersymmetry of the BPS state, corrections to the free string picture are expected to be smaller for near-bps configurations, which might allow to extrapolate Eq Ž 36 from the weak to the strong coupling regime wx 8 4 Discussion The appearance of the Hagedorn temperature characterizing the radiation in the decay of a massive macrostate in string theory at zero temperature sheds light on an old problem concerning the physical interpretation of the Hagedorn temperature The Hagedorn temperature was traditionally interpreted as the temperature at which the canonical thermal ensemble of a string gas breaks down The technical reason is simple at higher temperatures the integral defining the thermal partition function diverges and it is related to the fast growing of the level density with mass The question is what is the physical picture behind this instability The present results lead to the following interpretation Žthis seems conw9,0 x When an open sistent with other suggestions string in a macrostate with N 4 is placed in a thermal bath of temperature Tbath - T H, the string will decay by emitting massless Ž as well as massive

212 ( ) D Amati, JG RussorPhysics Letters B particles, as a black body of temperature T H, until the level decreases to a point where the saddle-point prediction Tstring s TH ceases to apply, and the real temperature of the string is of order T bath A situation of equilibrium occurs immediately if one places the string in a thermal bath with Tbath st H When Tbath ) T H, the energy of the string increase endlessly, since it can only emit at a fixed rate determined by the temperature T The system becomes highly H unstable, and there cannot be any thermal equilibrium The canonical thermal ensemble must therefore break down precisely at TsT H It is interesting to witness how these statistical or classical concepts stem from microstate computa- tions through average procedures Ž decoherence The string scale is of course present in the spectrum and in the correlations of single microstates, but it is only after averaging that appears as the temperature characterizing the emission The emergence of classical properties in a computable weak coupling string regime was indeed one of the motivation for this investigation It substantiates the conjecture wx that a similar mechanism is at work at large couplings in order to generate from string theory classical concepts such as space-time geometry, causal relations and black hole physics Finally, it would be interesting to investigate possible cosmological implications related to the decay spectrum and lifetime of cosmic strings w9, x Acknowledgements JR would like to thank SISSA for hospitality This work was partially supported by EC Contract ERBFMRXCT960090, by the research grant on Theoretical Physics of Fundamental Interactions of the Italian Ministry for Universities and Scientific Research, and by Conicet References wx AH Mueller, Phys Rev D Ž wx D Amati, Black holes, string theory and quantum coherence, hep-thr wx 3 V Alessandrini, D Amati, M Le Bellac, D Olive, Phys Reports C Ž wx 4 M Green, J Schwarz, E Witten, Superstring Theory, vols IrII, Cambridge, 987 wx 5 D Gross, A Neveu, J Sherk, J Schwarz, Phys Rev D Ž wx 6 C Callan, J Maldacena, Nucl Phys B 47 Ž wx 7 JM Maldacena, A Strominger, Phys Rev D 55 Ž wx 8 A Sen, Mod Phys Lett A 0 Ž wx 9 D Mitchell, N Turok, Phys Rev Lett 58 Ž w0x M Bowick, S Giddings, Nucl Phys B 35 Ž ; S Giddings, Phys Lett B 6 Ž ; D Mitchell, B Sundborg, N Turok, Nucl Phys B 335 Ž 990 6; N Deo, S Jain, CI Tan, Phys Rev D 40 Ž ; M Laucelli Meana, M Osorio, J Puente Penalba, Phys Lett B 400 Ž wx A Vilenkin, Phys Rep Ž

213 0 May 999 Physics Letters B Conformal description of horizon s states Sergey N Solodukhin Spinoza Institute, UniÕersity of Utrecht, LeuÕenlaan 4, 3584 CE Utrecht, the Netherlands Received 6 December 998; received in revised form 4 February 999 Editor: PV Landshoff Abstract The existence of black hole horizon is considered as a boundary condition to be imposed on the fluctuating metrics The coordinate invariant form of the condition for class of spherically symmetric metrics is formulated The diffeomorphisms preserving this condition act in Ž arbitrary small vicinity of the horizon and form the group of conformal transformations of two-dimensional space Ž ryt sector of the total space-time The corresponding algebra recovered at the horizon is one copy of the Virasoro algebra For general relativity in d dimensions we find an effective two-dimensional theory which governs the conformal dynamics at the horizon universally for any dg3 The corresponding Virasoro algebra has central charge c proportional to the Bekenstein-Hawking entropy Identifying the zero-mode configuration we calculate L 0 The counting of states of this horizon s conformal field theory by means of Cardy s formula is in complete agreement with the Bekenstein-Hawking expression for the entropy of black hole in d dimensions q 999 Elsevier Science BV All rights reserved Introduction Since the remarkable discovery wx that a black hole has entropy proportional to the area of horizon A h SBH s, Ž 4G it remains a mystery as what states are counted by this formula A number of approaches was proposed wx to answer this question within a conventional field theory However, reproducing correctly the proportionality to the area most of these approaches normally lead to divergent expression for the entropy ssolodukhin@physuunl On the other hand, there is a number of indications that two-dimensional conformal symmetry may provide us with relevant description of black hole s states The Hilbert space of a conformal field theory realizes a representation of the Ž quantum Virasoro algebra c wl, L xs Ž nym L q nž n y d, n m nqm nqm,0 Ž with infinite set of generators Ln and central charge c The number of states in the conformal field theory at the level L0 grows exponentially and the corresponding entropy is given by ( cl 0 Sconfsp Ž r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

214 4 ( ) SN SolodukhinrPhysics Letters B In the current literature, this formula is known as Cardy s formula That entropy Ž 3 of an appropriately defined conformal field theory may fit the expression Ž was demonstrated in number of examples found within string theory wx 3 However, the relevant conformal field theory lives in flat space-time and additional arguments should be given to relate its states to the ones living in the black hole phase Another example which is inspiring for the present consideration is three-dimensional BTZ black hole wx 4 The feature of general relativity in three dimensions is that it can be re-formulated as SLŽ, R = SLŽ, R Chern-Simons theory with only dynamical degrees of freedom living on the boundary It was argued by Carlip wx 5 some time ago that states of the boundary theory realizing a representation of Ž two copies of Virasoro algebra are responsible for the entropy of BTZ black hole An alternative elewx 6 He gant calculation was proposed by Strominger uses the fact proven in wx 7 that imposing boundary condition that 3d metric is asymptotically AdS3 there is group of diffeomorphisms preserving this condition This group is generated by two copies of the Virasoro algebra with central charge c s 3lrG ŽG is Newton s constant and l is AdS radius 3 The relevant conformal theory is the Liouville theory wx 8 living on the Ž two-dimensional boundary and described by the action ' L H Ž yf W s d z yg =f qqfrqle, Ž 4 where Q is related to the central charge cs3lrg as cs48p Q The BTZ metric obeys the boundary condition and the black hole, thus, is in the Hilbert space of the conformal field theory The counting of the degeneracy using the formula Ž 3 then exactly reproduces the Bekenstein-Hawking result This calculation, however, uses features specific for the three-dimensional gravity and it is not seen how it can be extended to higher dimensions A goal of the present paper is to follow a similar line of reasoning but not restricting to specific features of three dimensions The key idea is to formulate a boundary condition describing metrics with black hole horizon Indeed, the existence of the horizon is a requirement which essentially restricts class of possible metrics The coordinate invariant form of this condition is formulated in the next section and is appropriate to describe both dynamical and static black holes Remarkably, we find diffeomorphisms that preserve this condition They act in Ž arbitrary small vicinity of the horizon and form the infinite-dimensional group of conformal transformations in two dimensions The corresponding algebra recovered at the horizon is one copy of the Virasoro algebra The physics at the horizon is, thus, conformal that could be anticipated since it is known that fields becomes effectively massless at the horizon For general relativity the conformal dynamics at the horizon is governed by an effective two-dimensional field theory It is constructed in Section 3 and is shown in Section 5 to be universal theory at the horizon of black hole in any space-time dimensions The counting states of this horizon s conformal field theory by Cardy s formula Ž 3 exactly reproduces the result Ž Note, that in our approach the states responsible for the entropy Ž are states of the horizon itself This differs from the picture present in wx 6 where the relevant states live at infinity Horizon boundary condition and d conformal group In a theory of quantum gravity dealing with fluctuating space-time geometry one should be able to formulate conditions which fix the class of possible metrics One of the conditions one usually considers is the behavior of metric at infinity The space-time is then supposed to be asymptotically flat or asymptotically Ž Anti- Sitter dependent on the physical situation However, the fixing of the asymptotic behavior not completely specifies the topology of the space-time This should be considered as an additional requirement restricting the class of metrics under consideration The presence of the black hole horizon is such topological feature which should be traced in the conditions which one should impose on the space-time metrics There are different definitions of horizon, some of them, in particular, require knowledge of global Ž all-time behavior of spacetime More appropriate for our goals is the notion of

215 ( ) SN SolodukhinrPhysics Letters B apparenthorizon which can be defined locally as the boundary of trapped region wx 9 Consider four-dimensional spherically symmetric metric of the general form ds sgabž x 0, x dx a dx b qr Ž x 0, x Ž du qsin u df, Ž 0 where g x, x ab can be considered as metric on effective d space-time M with coordinates x 0, x ; Ž 0 the radius r x, x is then scalar function on M For this class of metrics the apparent horizon can be w x defined 0 as a curve H on M such that the Ž 0 gradient of the radius r x, x g = r= r s0 ab < a b H vanishes along H This condition is invariant under conformal transformation gab e r gab where r is a regular on H function Therefore, the d metric g ab at the horizon is determined by the condition only up to a Ž regular conformal factor It is convenient to use conformal coordinates x, x in which the d part of the metric q y takes Ž 0 a b s Ž x q, x y the form gab z, z dz dz sye dxqdx y Then locally we may choose coordinates x q, xy in such a way that the equation for the curve H becomes xys 0 Assuming that the function Ž 0 s x, x is regular at x s0, we find that Eq y reads E r x, x < q Ž q y x y s0 s0 3 Note, that the condition is appropriate to describe dynamical black hole The horizon, in general, may consist on different components Then the condition locally defines each component For example, in static case the horizon has two intersecting components H Ž x s0, x )0 and H Ž q y q y xq s0, x -0 The later is defined as y E rž x, x < s0 4 y q y x q s0 In what follows we consider only one component of the horizon defined by 3 Expanding the function rž x, x q y near xqs0 we find rž x, x sr qlž x x qo x 5 q y h q y y that is consistent with the condition 3 ; l is an arbitrary function of x, r is constant Ž q h radius of horizon It follows from 5 that E rse lž x x qo x 6 q q q y y in the vicinity of H Now it is not difficult to find group of diffeomorphisms which preserve the condition, 3, 5 Indeed, consider vector j q s Ž q y j sg x,j s0,0,0 Then using 6 we have q q q L Ž E r se Ž j q E r jq q q q q se gž x E lž x x qo x q q q q y y for the Lie derivative along the vector j q Thus, the condition 3, 5, 6 is preserved under diffeomorphisms generated by jq provided that the func- tion lž x changes as follows y L lsgž x E lž x jq q q q This is standard transformation law for a scalar under diffeomorphisms Note, that lž x in 5 q is indeed scalar since it can be represented in the covariant form as n E rslž x, m < m H where n Žn s 0 is normal to H and lž x m m is function along H Diffeomorphisms generated by vector jq are tan- gential to H and thus preserve H In the case of horizon with two bifurcating components Hq and Hy vector jq generates symmetry of Hq while the component Hy is invariant under diffeomor- phisms generated by vector j Ž ys jy q s 0,jy y s fž x,0,0 y Note, that both vectors jq and jy sat- isfy the equation =j q= js g = j c a b b a ab c for two-dimensional conformal Killing vectors and thus generate the infinite-dimensional group of conformal transformations of the space M The corresponding generators l " se ınx " n E " form two copies of the Virasoro algebra " " " l n,lm sı myn l nqm, 7 X with respect to the Lie bracket wj,j xsž jjy j j X E However, near each component Ž x Hq or H of H there is only one copy of the Virasoro y

216 6 ( ) SN SolodukhinrPhysics Letters B algebra which leaves the horizon invariant As in the case of d space with boundary wx the presence of horizon breaks one of the conformal symmetries Note, that the Virasoro algebra 7 is algebra of diffeomorphisms of non-compact space R In particular, this means that n and m in 7 are arbitrary numbers and not necessarily integers Usually, one considers the compact version of the Virasoro algebra which is algebra of diffeomorphisms of circle S For the further purposes we need the compact version of the algebra Therefore, we consider an arbitrary large interval L of R and impose periodic boundary conditions At the end we take L to be infinite It should be noted that we did not use so far any gravitational field equations and defined a general class of Ž spheri-symmetric metrics with black hole horizon We can also see from the present analysis that the condition defying the horizon H is Ž 0 essentially a condition on the function r x, x in the 4d metric while the form of the d metric Ž 0 g x, x ab remains undetermined Another indica- tion of this is the fact that the condition does not change under the conformal transformation of the d metric Thus, the condition defines a class of metrics modulo this conformal transformation Therefore, the only gravitational dynamics of the fluctuating Ž of-shell 4d metric arising on the horizon is the dynamics of the radial function Ž 0 r x, x while for the d part one can take any metric from the same conformal class In what follows we consider the static case and choose the representative metric on d space-time M in the form dx ds sygž x dt q, 8 gž x with the function gž x vanishing at xsx h, where x h is location of the horizon H in the Schwarzschild coordinates Ž t, x In assumption that we deal with a non-extreme black hole we have that h h b H gž x s Ž xyx qo Ž xyx, 9 where bh is constant related to the surface gravity of the horizon 3 Bekenstein-Hawking entropy as central charge of the Virasoro algebra It follows from the consideration of the previous section that any theory of quantum gravity describing black hole should provide us with a realization of the Virasoro algebra in the region close to horizon In this section we demonstrate how it works for general relativity We start with four-dimensional Einstein-Hilbert action W 4 EH sy H d z' yg R Ž4 Ž 3 6p G M 4 and consider it on the class of spherically symmetric metrics We arrive at an effective two-dimensional theory described by the action Wsy d x' yg Ž =F q F Rq, M 4 G Ž 3 H ž / where FsrG yr and R is d scalar curvature This action takes the form of dilaton gravity Žthe radius r playing the role of dilaton field in two dimensions and can be transformed to the form similar to that of the Liouville theory Ž 4 H Ž 4 h M Wsy Ž =f q qf frquž f Ž 33 by applying the transformation wx ž / h f qf h f F gab s e g ab, f s, Ž 34 f q F h where F hsrhg yr is the classical value of the field F on the horizon, ie the horizon radius r h measured in the Planck units The classical value of the field f is respectively fhsq y F h Since we are interested in the region very close to horizon, where the d metric gab is determined up to conformal factor, we obtain an equivalent system provided that the conformal transformation Ž 34 is regular at the horizon The action Ž 33 depends on an arbitrary constant q So does the central charge which we will calculate in a moment However, the final result Žthe

217 ( ) SN SolodukhinrPhysics Letters B statistical entropy counted in the next section is independent of q The potential UŽ f in Ž 33 is f f h qf U f s e h G f ž / but its form is not important for our consideration Varying the action Ž 33 with respect to the dilaton f and metric gab we obtain the equation of motion for f X Ifs 4 qf h RqU Ž f Ž 35 as well as constraints ab a b 4 ab T ' E fe fy g =f q 4qFhŽ gabify== a bf y gabuž f s0 Ž 36 The theory of the scalar field f described by the action Ž 33 is not conformal Indeed, we find that the trace of Ž 36 ab g Tabs 4 qf hifyuž f Ž 37 does not vanish However, the theory becomes conformal being considered in infinitely small vicinity of the horizon The d metric there takes the form 8 9 Operating with this metric it is convenient to use the coordinate H x dx b H zs s lnž xyx h gž x so that the vicinity of the horizon ŽŽ xyx is small h looks as region of infinite negative z The metric function 9 z ye fqe fs qf Rg Ž z qgž z U Ž f Ž 38 X t z 4 h gž z sg e b H 0 exponentially vanishes at the horizon In the coordinates Ž t, z Eq Ž 35 reads Note, that the d scalar curvature R may be non-zero at the horizon due to terms ; Ž xyx h present in the metric and neglected in 9 We see that due to the exponentially decaying factor gž z the rhs of Eq Ž 38 exponentially vanishes for large negative z Therefore, in the region very close to horizon Žin- finite z the rhs of Ž 38 becomes negligible and we obtain the equation E fye fs0 Ž 39 t z describing free field propagating in flat space-time with coordinates Ž t, z Expressing the constraints Ž 36 in terms of the coordinates Ž t, z we find that qf g X h x T00 s 4 Ž Ž Etf q Ž Ezf y ž Ezfy Ezf 4 / q gž z UŽ f, qf g X h x T0 zs EtfEzfy ž EEfy z t Etf /, 4 T s Ž E f q Ž E f zz 4 t z g ž X / qf h x q yetfq Ezf y gž z UŽ f 4 Ž 30 The trace is yt qt s qf ye fqe f ygž z UŽ f 00 zz 4 h t z In the region of large z this quantity vanishes on the equation of motion Ž 39 This, in particular, guarantees that the Poisson algebra of constraints Ž 30 closes and and in the region of infinite z they form the Virasoro algebra We conclude that the theory of the scalar field f described by the action ( 33) is conformal being considered at the horizon ( actually, in arbitrary small Õicinity of the horizon ) The conformal transformations are generated by charges H Lr T wj xs dzt j Ž z, Ž 3 ylr qq where TqqsT 00qT0 z s Ž E fqe f 4 t z y 4 qf h EzŽ EzqEt fy Ž EzqEt f ž / b H Ž 3 A general solution of Eq Ž 39 is sum of right- and left-moving plane waves fsf Ž tqz qf Ž q y ty z However, only the right-moving part contributes to T qq It is worth noting that the present analysis can be done in terms of the original scalar field F Ž 3 Then in the limit of large z the trace of the corresponding stress-energy tensor vanishes under additional condition that yž E tf q EzF s 0 which indicates that one should consider only a part of modes This is also consistent with our discussion in Section

218 8 ( ) SN SolodukhinrPhysics Letters B In the region of large z we may use the translation invariance z z q Z, Z s const in order to L L adjust z to lie in the interval y FzF After all we take the limit of infinite L Considering field f on this interval we assume the periodic boundary condition to be imposed The vector field j Ž z is also periodic, j Ž zql sj Ž z From Ž 33 we find the Poisson algebra 4 f Ž z,t,e f Ž z X,t sd Ž zyz X t and, as a consequence, we have Ž E qe f z,t, E qe f z X t z Ž t z Ž,t 4 se d Ž zyz X ye X d Ž zyz X z z We now in a position to compute the Poisson algebra of the charges Ž 3 The result is T wj x,t wj x st wj,j x 4 ž / h Lr H w 4 ylr qf q C j,j xdz, X X where wj,j x sjjyjj and X y X y Cwj,j xs j qb j j qb j X X y X X y yž j qbh j Ž jqbh j H H Ž 33 X XX Žjj is a deformation of the well known two-cycle X yj j XX of the algebra of DiffŽS That Ž 33 is identical to the Virasoro algebra 7 is easy to recognize by expanding j Ž z in Fourier series j n s p ı L nz e and introducing the Virasoro generators L Lr p ı nz L s dze L T Ž 34 n H p ylr They form the algebra ıl, L 4 s Ž kyn L k n nqk qq ž / ž / c L q k k q d pb nqk,0 H Ž 35 with the central charge cs3p q F h Since the Bekenstein-Hawking entropy is SBH spfh we thus obtain that the central charge of the Virasoro algebra Ž 35 cs3q SBH Ž 36 is proportional to the Bekenstein-Hawking entropy 4 Zero-mode configuration and the counting of the states In order to use Cardy s formula Ž 3 and count the number of states we need to know value of L 0It is typically determined by zero mode configuration which is, in fact, classical configuration In our case the classical configuration Žup to exponentially small terms ;OŽe p r b H is just a constant fsf h But L0 vanishes for this configuration To resolve this problem note that a more general zero-mode configuration is allowed to exist near the horizon f0 saqpz, Ž 4 which is obviously a solution of the field equations This configuration infinitely grows close to horizon and should be excluded in the region of infinite z But it may present if we consider our system in a box In order to make Ž 4 periodic with the period L we first consider this function on the interval 0 F z F Lr and then continue it to the interval ylrf z F 0 as f Ž yz s f Ž z 0 0 Our condition for the function f0 is that it becomes fh at the right end of the interval f < sf 0 zs L h At the point zs0 we impose the condition Ž y < z 0 H 0 zs0 E f qb f s0, Ž 4 which means that for the zero-mode the boundary Ž a Ž a term n Eaq k f where n is normal and k is extrinsic curvature which appears in the Ž on-shell gravitational action vanishes on the inner boundary zs0 Both conditions result in the following form of the configuration Ž 4 zyb H f0 sf Ž 43 hž / Lyb H in the interval 0FzFLr, that gives us PsEzf 0, f h for large L We see, in particular, that P van- L ishes when L becomes infinitely large that is in accordance with our wish to have constant as the classical configuration in the infinite region Provided that we are happy with the imposed conditions the value of L is calculated as follows 0 L P fh F h L0 s s s Ž 44 8p p p q

219 ( ) SN SolodukhinrPhysics Letters B y In the case of extreme black hole b s0 in Ž 4 H and the zero-mode configuration Ž 4 is constant f0 sf h We find then that L0 s0 Applying now the general formula Ž 3 for the entropy of states in a conformal field theory we find from Ž 36 and Ž 44 that for a non-extreme black hole the corresponding entropy SconfspFh ssbh Ž 45 exactly reproduces the Bekenstein-Hawking expression Note that we need only one copy of the Virasoro algebra to get the correct answer This is in agreement with the discussion in Section 5 Generalization for d)4 and ds3 The analysis we present above can be extended to dimensions other than four The spherically symmetric metric in space-time with d dimensions is ds sg x 0, x dx a dx b qr x 0, x dv dy ab S, Ž 5 where dv dy is metric on Ž d y S -dimensional sphere S dy of unit radius General relativity in d dimensions is described by the action ( d Wsy H dz ygž d R Žd, Ž 5 6p G M d d where GŽ d is Newton s constant Being considered on the class of metrics Ž 5 the action Ž 5 reduces to the effective two-dimensional theory Žomitting the total derivative term S dy dy WŽ d sy H Ž r R 6p G M d dy4 qž dy3ž dy r Ž = r qž dy3ž dy r dy4, Ž 53 where S dy s p rg dy is area of the sphere dy S dy Re-defying the dilaton field r as ž / Sdy dy3 dy F scr, Cs, Ž 54 p G dy d the action Ž 53 takes the form similar to Ž 3 dy W sy H ž Ž d Ž =F q ž / F R M 8 dy3 dy4 dy ž dy / 8 / q Ž dy C F Ž 55 In d dimensions the horizon is Ž d y -dimensional sphere and the Bekenstein-Hawking entropy is proportional to the area of this sphere Sdy dy p Ž d dy SBH s rh s F h, Ž 56 4G dy3 d ž / where the value of the field F on the horizon is related to the horizon radius r according to Ž 54 h After applying the transformation ž / ž / dy3 dy3 f f h dy qf h h ab ab F s qf f, g s e g dy f Ž 57 the action 55 takes the Liouville type form 33 Ž d H ž 4 h Ž d / M W sy =f q qf frqu f Ž 58 As in Eq Ž 33, q here is an arbitrary parameter Note, that the transformation Ž 57 for the metric is independent of d The classical value of the field f on the horizon is given by dy fhs ž / F h Ž 59 q dy3 Only the potential term U Ž f in Ž 58 Ž d depends on the dimension d Its form can be found explicitly but is not important for further consideration As we explained in Section 3, in the region near horizon the potential term effectively disappears Žbeing multiplied on the function gž z exponentially decaying at the horizon in the field equation for f and the constraints So that the action Ž 58 defines at the horizon a conformal theory the form of which is universal for any dimension d The corresponding Virasoro algebra was analyzed in Sections 3 and 4 It has fh cs3p q F h and L0 s 50 p

220 0 ( ) SN SolodukhinrPhysics Letters B From Ž 59 and Ž 54 we find that the central charge Ž 50 ž / dy3 Ž d cs6q SBH Ž 5 dy is proportional to the Bekenstein-Hawking entropy 56 On the other hand, we have / dy L0 s F, Ž 5 ž h 8p q dy3 where Eq Ž 59 was used Substituting Ž 50 Ž 5 into Cardy s formula Ž 3 ž / p dy Ž d Sconfs F h ssbh Ž 53 dy3 we find the precise agreement with the Bekenstein- Hawking expression Ž 56 for the entropy of black hole in d dimensions It is seen from Eqs Ž 53 Ž 58 that the case ds3 is special and should be considered separately In this case the kinetic and potential terms are absent in Ž 53 In order to obtain a non-trivial solution of the gravitational equations we have to add l-term to the Einstein-Hilbert action in three dimensions The resultant effective d theory is WŽ3 sy H Ž rž Rql Ž 54 8G M 3 After applying the transformation f qf h 3 h ab ab rs qg F f, g se g Ž 55 it takes the form 58 with the potential f qf U s lqf f e H Ž3 4 h Note that in this case F h is an arbitrary parameter like q We keep it only in order to illustrate the universality of the action Ž 58 governing the conformal dynamics at the horizon for any d G 3 The classical value of new dilaton field f on the horizon now is f hs qgf h y r h The corresponding con- formal field theory has c and L as in Ž 50 0 and, as we can see, the combination Ž qf indeed drops out h 3 rh 0 8 G in the product cl s Ž 3 we then obtain Applying the formula p r h Sconfs Ž 56 G 3 in agreement with the Bekenstein-Hawking formula in three dimensions Note, that we did not use the Chern-Simons form of the 3d gravity when obtained the correct answer for the entropy 6 Remarks 6 Classical central charge It should be noted that the central charge c s 3p q F h of the Virasoro algebra considered in this paper is classical It appears before quantization on the level of the Poisson bracket ŽOn the quantum level the central charge is Ž qc and is dominated by the classical value for large c Therefore, it is a reasonable question if namely this value of c describes the degrees of freedom of the theory and should be used in the computation of the entropy by formula Ž 3 This question also arises w3x in Strominger s calculation of the entropy of BTZ black hole and is directly related to the problem of degrees of freedom in the Liouville model Ž 4 Indeed, the 3 l large value of the classical central charge cs G seems to be in contradiction with the fact that the Liouville model is a theory of just one scalar field w4x and, hence, the effective central charge ceffs For Strominger s calculation this problem is not yet resolved in the literature In our approach, however, there is a hope to overcome this problem As we have seen, both the central charge c and L Ž 50 0 depend on an arbitrary parameter q and do not have an absolute meaning Only the combination cl 0, which should be substituted into Cardy s formula, has the absolute meaning not being dependent on q Note, that it is an important difference between our approach and the one which uses the Liouville model where the classical central charge is fixed and absolute Žsee, however, w5 x It should be noted that most of the complications of dealing with the Liouville model are due to the exponential potential term The conformal field theory Žwith stress tensor Ž 3

221 ( ) SN SolodukhinrPhysics Letters B appearing in our approach is simpler for analysis since the potential term effectively vanishes at the horizon 6 Non-spherical graõitational excitations and matter fields In a quantum theory of gravity we should take into account all possible fluctuations of the metric Therefore, we should be able to incorporate in our analysis the gravitational excitations which are not spherical A way of doing this is to consider all non-spherical excitations as a set of fields propagating on the spherically symmetric background In this respect they are similar to the quantum matter fields and should be considered in the same way Then, expanding all fields in terms of the spherical harmonics Yl,m we obtain an infinite set of fields la- beled by l,m which are functions on d space M The corresponding d theory can be analyzed and shown to be conformal at the horizon by the same reasons as in Section 3 The contribution SQ of this infinite set of fields to the entropy, though proportional Ž in the leading order to the horizon area, is expected to diverge Žeither due to the infinite number of the fields or when one takes the limit of infinite L Žsee, for example, w6 x Presumable, SQ is what in the literature known as the quantum correction w7x to the Bekenstein-Hawking entropy The spherically symmetric excitations of the gravitational field, thus, are responsible for the classical Bekenstein- Hawking entropy while all other excitations produce a correction It was suggested in w7x that the divergence of SQ can be absorbed in the renormalization of Newton s constant G so that the total entropy, SBH qs Q, remains finite It would be interesting to see how this renormalization works in terms of the d conformal field theory at the horizon 63 The Hilbert space and unitary eõolution The conformal description we present in this paper may help to understand the quantum evolution of a system including a black hole as a part In this description one should assign with black hole horizon elements < H: of the Hilbert space realizing a representation of the Virasoro algebra In combination with states < c) at asymptotic infinity they form the complete Hilbert space Considering the evolution in the space of elements < H : = < c : there are no reasons why it should not be unitary Note in this respect the useful analogy between horizon and boundary In a field theory w8x on space-time with boundary B one should take into account the socalled boundary states < B: which live on the boundary Only then an unitary S-matrix can be constructed While this paper was in preparation there apw9x which overlaps peared an interesting preprint with our consideration In particular, both c and L 0 of the Virasoro algebra found in w9x are proportional to the Bekenstein-Hawking entropy though depend on the choice of period Only the combination cl0 is unambiguous This is in agreement with our result Note added It is interesting to note that in Section the equation xys 0 determining the location of the horizon H can be replaced by a more general one: x sfž x y q, Ž f X Ž x / 0 q Introducing then new coordinates zq sx qfž x, z sx yfž x y q y y q the horizon is now located at zys 0 The condition re-formulates as Ž = r slž z z qožz q y y which is invariant un- der diffeomorphisms generated by vector jˆ s ˆ ınz j Ž z E The basis vectors j s e q q z n Ez again form q q the Virasoro algebra Acknowledgements I thank Steve Carlip for correspondence and suggestion to generalize my analysis to higher dimensions I also would like to thank Gerard t Hooft for interesting discussions References wx JD Bekenstein, Phys Rev D 7 Ž ; SW Hawking, Comm Math Phys 43 Ž wx JW York, Phys Rev D 8 Ž ; WH Zurek, KS Thorne, Phys Rev Lett 54 Ž 985 7; G t Hooft, Nucl Phys B 56 Ž ; V Frolov, I Novikov, Phys Rev D 48 Ž wx 3 A Strominger, C Vafa, Phys Lett B 379 Ž wx 4 M Banados, C Teitelboim, J Zanelli, Phys Rev Lett 69 Ž

222 ( ) SN SolodukhinrPhysics Letters B wx 5 S Carlip, Phys Rev D 5 Ž wx 6 A Strominger, J High Energy Phys 0 Ž wx 7 JD Brown, M Henneaux, Comm Math Phys 04 Ž wx 8 O Coussaert, M Henneaux, P van Driel, Class Quant Grav Ž wx 9 RM Wald, General Relativity, University of Chicago Press, Chicago, 984 w0x See, for example, appendix A in JG Russo, Phys Lett B 359 Ž wx JG Russo, A Tseytlin, Nucl Phys B 38 Ž wx JJ Cardy, Nucl Phys B 34 Ž w3x S Carlip, Class Quant Grav 5 Ž w4x D Kutasov, N Seiberg, Nucl Phys B 358 Ž w5x M Banados, Embeddings of the Virasoro algebra and black hole entropy, hep-thr986 w6x DV Fursaev, A note on entanglement entropy and conformal field theory, hep-thr98 w7x L Susskind, J Uglum, Phys Rev D 50 Ž ; AO Barvinskii, VP Frolov, AI Zelnikov, Phys Rev D 5 Ž ; SN Solodukhin, Phys Rev D 5 Ž ; DV Fursaev, SN Solodukhin, Phys Lett B 365 Ž 996 5; J-G Demers, R Lafrance, RC Myers, Phys Rev D 5 Ž w8x S Ghoshal, AB Zamolodchikov, Int J Mod Phys A 9 Ž w9x S Carlip, Black hole entropy from conformal field theory in any dimensions, hep-thr9803

223 0 May 999 Physics Letters B D-term inflation in type I string theory Edi Halyo Department of Physics, Stanford UniÕersity, Stanford, CA 94305, USA Received February 999; received in revised form 30 March 999 Editor: M Cvetič Abstract D-term inflation realized in heterotic string theory has two problems: the scale of the anomalous D-term is too large for accounting for COBE data and the coupling constant of the anomalous UŽ is too large for supergravity to be valid We show that both of these problems can be easily solved in D-term inflation based on type I string theory or orientifolds of type IIB strings q 999 Published by Elsevier Science BV All rights reserved Introduction Early attempts to incorporate inflationary cosmology in string theory Ž or supergravity were not very successful due to the inflaton mass problem, ie the mass of the inflaton is generically as large as the Hubble constant when the vacuum energy arises from F-terms wx In this case, inflation cannot take place since the slow-roll condition is violated An elegant solution to the inflaton mass problem in string theory Ž or supergravity is D-term inflation wx In this scenario, the vacuum energy needed for inflation is dominated by D-terms rather than F-terms Thus, the inflaton mass problem is trivially solved since the dangerous contribution to the inflaton mass due to the F-terms is vanishing Ž or negligible This can be easily realized in heterotic string theories since generically there is an anomalous D-term arising from an anomalous UŽ A which can contribute to the vacuum energy wx 4 In this scenario, the infla- halyo@dormousestanfordedu wx For earlier work on D-term inflation see 3 ton s is neutral under the anomalous UŽ A but has tree level couplings to other fields f,f in the superpotential, ie Wslsff The fields f,f have " charges under UŽ A and behave as the trigger fields in hybrid inflation models wx 5 The scalar potential including the anomalous D-term is wx Vs< ls < < f< q< f < q< lff< g q < f < y< f < qm Ž where l; OŽ is a Yukawa coupling, g is the gauge coupling of UŽ A and M is the scale of the anomalous D-term For heterotic strings it is given by wx 6 M s g Ž TrQA MP 9p Here M P ;=0 8 GeV is the reduced Planck scale, g; r from gauge coupling unification and the trace is over the whole massless spectrum of the string theory giving generically TrQ A ; 00 Hybrid inflation occurs for large values of s, s;m<m P which gives a positive mass squared to f,f and r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

224 4 ( ) E HalyorPhysics Letters B forces them to have vanishing VEVs Then there is a constant nonzero vacuum energy V0 sg M rresulting in a period of inflation Supersymmetry is broken by V0 and as a result a one-loop potential for s is generated wx / g ls 4 Vone- loopž s s g M q log Ž 3 ž 8p L above L is the renormalization scale which does not affect the physics Due to this potential the inflaton rolls slowly to its minimum during inflation There is a critical value scr s gmrl after which the mass squared of f becomes negative and it rapidly falls to its new minimum at fsm This ends inflation and restores supersymmetry A potentially dangerous problem from D-term inflation is the fact that the dilaton gets a big mass from the D-term during inflation and thus can be displaced or even be overthrown to the runaway branch Thus one has to assume that the dilaton is stabilized by some potential with curvature larger than the Hubble parameter during inflation Such a potential cannot come from the zero-temperature supersymmetry breaking however we assume that it exists in this scenario 3 It has been noted that the above D-term inflation scenario has two problems when it is realized in heterotic string theory w7,8 x 4 The first problem arises from the magnitude of density fluctuations obtained from COBE data which requires w8,0x Ž V0re s67=0 6 GeV Ž 4 where e is one of the slow-roll parameters of infla- X tion; es M V rv For inflation with N Ž ;60 P e-folds one needs ž / r M;85=0 GeV= Ž 5 N However in heterotic string models with an anomalous D-term Eq Ž above gives a scale too large to account for COBE data The second problem arises from the fact that inflation can come to an end before the inflaton reaches its critical value if the second slow-roll XX parameter h s M < V rv < P becomes of order one Ž since slow-roll requires h < This means that w8,0x a M P hž s s( Ž 6 p s where asg r4p We see that h; when ( a s f; MP Ž 7 p This gives s f;mpr0 for the final value of s at the end of inflation which is much larger than s cr Moreover, it can be shown that the initial value of the inflaton s needs to be ( i an s i; MP Ž 8 p which gives s i; 08 M P This is problematic be- cause for Planckian values of the inflaton one cannot use the effective low-energy supergravity approximation Actually, this problem is probably less severe than it seems because the criterion should not be s- MP but rather that the one-loop corrections should be smaller than the tree level results Žwhen the D-term is the source of vacuum energy and the corrections to the Kahler potential are not relevant For example, for Voneyloop we see that even for l;, s can be much larger than MP due to the factor g r8p 5 In this letter, we show that both of these problems can be naturally solved in D-term inflation in the framework of type I string theory rather than heterotic string theory The first problem is solved because in type I string theory the scale of the anomalous D-term is not fixed; it is given by the VEV of a modulus which can be of the required order of magnitude Moreover, unlike the heterotic string case in type I theory there are gauge groups which can have rather small coupling constants As a result, the coupling of UŽ A can be small enough to solve the second problem In the next section we briefly review the features type I strings Ž orientifolds of IIB string theory which 3 I thank the referee for pointing out this assumption of the scenario 4 For D-term inflation in explicit string models see w3,9 x 5 We thank Andrei Linde for clarifying this point

225 ( ) E HalyorPhysics Letters B are relevant for D-term inflation In Section 3, we show how these features can be used to solve the two problems that arise in heterotic string theories We discuss our results and conclude in the last section Type I string theory or orientifolds of type IIB strings Type I string compactifications with N s supersymmetry can be obtained by orientifolds of type II strings w x We start with a type IIB string theory in Ds0 and mode it out by the world-sheet parity transformation V This gives a type I string theory in D s 0 with gauge group SOŽ 3 The gauge group arises from the 3 D9 branes required for tadpole cancellation This type I string theory is further compactified on an orbifold of T 6 Žie on 6 T rg where G is a discrete group such as Z n resulting in a Ds4 theory with Ns supersymmetry and chiral matter content The above construction has only D9 branes but by considering more elaborate orientifolds one can obtain models with two types of branes in the theory There can be two sets of D-branes: either D9 and D5 branes or D3 and D7 branes w x The number of each kind of brane is fixed again by tadpole cancellation On the two different kinds of branes ŽD p and D p X there are gauge fields from strings with both ends on the same X X kind of brane Žie pp or pp strings, giving two gauge groups, Gp and Gp X There are also matter fields which arise from strings with ends on different X kinds of branes Žie pp strings For our purposes it is enough to consider the simplest case, a type IIB orientifold with D3 branes and one set of D7 branes compactified on an orbifold of T 6 with radius R c The D3 branes are along X and the D7 branes are along X,,X Ž,,3 7 Our results apply equally well to the cases with more than one set of D7 branes or to the case with D9 and D5 branes These models have two gauge groups; G3 arising from D3 branes and G7 from D7 branes with couplings wx gi gi a3s a 7s Ž M R I c Here gi is the type I string coupling constant and MIsa yr str is the string scale Newton s constant is given by gi GN s s Ž M 8 M R P I c We see that contrary to the heterotic string case M P and MI do not need to be of the same order of magnitude From Eqs Ž 9 and Ž 0 we get a pmp 7 s s35=0 GeV Ž Ž py7 Ž py6 ' M R I c assuming for one set of branes a p;a U;r5, the unified value of the Standard Model coupling constants For the other set of branes we get gi a p s Ž py3 Ž MR I c Note that there is freedom in the scales of MI and R subject to Eq Ž c This should be compared to the heterotic case for which Mhs( aur8 MP is fixed to be close to MP and independent of the compactification radii The other important feature of type I string theory is related to the anomalous D-term In these models the scale of the anomalous D-term is not fixed Žcompare to the heterotic case with Eq but is given by the VEV of some twisted moduli with a Žw x 6 coefficient of O 3 These moduli VEVs are related to the blowing up of the orbifold which smoothes out the singularities of the compact space They are fixed only after supersymmetry is broken but it is safe to assume that the same mechanism that fixes the radii of the compactification torus also fixes them ŽNote that these radii are given by VEVs of the untwisted moduli Thus, we assume that the twisted moduli VEVs are of order R y c The matter content of these models arises from strings stretched between different branes, ie 33, 37, 73 and 77 strings We denote these fields by 6 Here the situation is different from the one considered in Ref w4x since there are D branes in the vacuum

226 6 ( ) E HalyorPhysics Letters B M 33, M 37, M 73, M 77 M 33 and M 77 are in the adjoint representation of the respective gauge groups, G 3 and G, whereas M 37 and M 73 are in the bifunda- 7 mental representation In realistic models the gauge groups will be broken down by Wilson lines and there will be gauge singlets coming from either M or M Ž or both The tree level superpotential generically contains the terms wx Wsg3Ž M 33 M 73 M 37 qg7ž M 77 M 37 M 73 Ž 3 Note that the Yukawa couplings are given by the Ž two gauge couplings in Eq 9 3 D-term inflation in type I string theory relatively small coupling given by Eq Ž 9 With the above values for MI and R c we get for the coupling of the UŽŽ which comes from the D7 branes or G 7 a 7 ;a;5=0 y4 which is a rather small value Substituting this into Eq Ž 7 we find that the initial value for the inflaton should be s i;0 M P This is still much larger than the critical value scr but small enough for the supergravity approximation to string theory to be valid In this case, the final value of the inflaton is close to the critical value, s f;s cr More complicated models with more than two sets of D-branes and unisotropic tori can give smaller a and therefore smaller s In these cases, if UŽ i A arises from a set of D7 branes with two large Žie larger y than M dimensions from Eq Ž 9 I we see that a can be quite small resulting in si as small as s cr We now consider a type I string model such as the one above This model has a gauge singlet M 33 Ž after symmetry breaking by Wilson lines which we identify with the inflaton field s M 33 has tree level couplings to other gauge nonsinglets such as M, M given by Eq Ž 3 These play the role of the trigger fields f,f Note that they are charged under G so we assume that the anomalous UŽ 7 A comes from this sector Also M 37 and M 73 are conjugates so their UŽ charges are opposite Ž A which we take to be " g 3 ; r gives the Yukawa coupling, ie l in Eq Ž Thus, this simple model has all the ingredients for D-term inflation such as the inflaton and trigger fields, an anomalous D-term and the correct superpotential As mentioned above, in type I models the scale of the anomalous D-term is not fixed but given by the VEV of a twisted modulus M t This VEV should be of the order of magnitude of other moduli VEVs such as compactification radii On the other hand, we saw that in type I compactifications there is some freedom in M and R subject to Eq Ž I c If we embed the Standard Model inside the D3 branes Žie 6 inside G3 we get from Eq M I ;=0 GeV and R y c ;8=0 5 GeV Then, M t;r y c is of the correct order of magnitude to account for COBE data This is the solution of the first problem mentioned in the introduction In addition, we saw that the gauge group arising from one set of D branes Ž in our case G can have a 7 4 Discussion and conclusions In this letter, we showed that the two problems which are generic to D-term inflation in heterotic string models are absent in type I string models The first problem related to the magnitude of the density fluctuations is solved by the low scale of the anomalous D-term in these models The string scale and the compactification radii of type I strings are not fixed and may be much smaller than those of the heterotic ones The scale of the anomalous D-term is of the same order of magnitude which is about the scale needed to accommodate the COBE data The second problem related to the very large field values of the inflaton is also absent due to the very small gauge coupling of UŽ A This is possible in type I models since the gauge group arises from two types of D branes independently One set of branes gives the Standard Model group Ž with a ; r5 U whereas UŽ A can arise from the other set of D branes In y this case, for R )M, the UŽ C I A gauge coupling can be much smaller than a Then from Eq Ž 7 we find that the initial value of the inflaton is at most MPr0 which is small enough for the supergravity approximation to string theory to be valid In this letter, we considered the simplest possible type I string model with two sets of D branes on an isotropic T 6 This can be easily generalized to more Ž complicated models with four sets of D branes one U

227 ( ) E HalyorPhysics Letters B set of D3 and three sets of D7 branes or D9 and D5 branes and a torus with different compactification radii All of our results will also hold in these cases However, due to the extra fields and gauge symmetries present in these cases some other requirements such as reheating may be more easily met We also mentioned that a smaller initial value for the inflaton is possible in these cases In discussing the D-term inflation scenario above we made a few generic assumptions such as the presence of gauge singlet fields and the properties of UŽ A It would be interesting to build realistic D s 4 type I string models and see if these are in fact realized We think it is quite encouraging to find that the generic problems of D-term inflation in heterotic string theory are easily solved in type I string models Acknowledgements We would like to thank Ignatios Antoniadis for drawing our attention to Refs w-3x and Andrei Linde for reading the manuscript References wx E Copeland, A Liddle, D Lyth, E Stewart, D Wands, Phys Rev D 49 Ž ; M Dine, L Randall, S Thomas, Phys Rev Lett 75 Ž wx E Halyo, Phys Lett B 387 Ž , hep-phr960643; P Binetruy, G Dvali, Phys Lett B 388 Ž 996 4, hepphr wx 3 JA Casas, C Munoz, Phys Lett B 6 Ž ; JA Casas, J Moreno, C Munoz, M Quiros, Nucl Phys B 38 Ž wx 4 M Dine, N Seiberg, E Witten, Nucl Phys B 89 Ž wx 5 AD Linde, Phys Lett B 59 Ž 99 38; Phys Rev D 49 Ž wx 6 J Atick, L Dixon, A Sen, Nucl Phys B 9 Ž wx 7 C Kolda, J March Russell, hep-phr wx 8 C Kolda, D Lyth, hep-phr9834 wx 9 J Espinoza, A Riotto, G Ross, Nucl Phys B 53 Ž ; hep-phr98044 w0x D Lyth, A Riotto, hep-phr wx G Aldazabal, A Font, L Ibanez, G Violero, hepthr wx L Ibanez, C Munoz, S Rigolin, hep-phr98397 w3x L Ibanez, R Rabadan, A Uranga, hep-thr w4x J March Russell, Phys Lett B 437 Ž , hepphr980646

228 0 May 999 Physics Letters B Ward identity for membranes A Ghosh a,, J Maharana b,c, a CERN, Theory DiÕision, CH- GeneÕa 3, Switzerland b Max-Planck Institute for GraÕitation, Albert-Einstein Institute, Potsdam, Germany c Institute of Physics, Bhubaneswar 75005, India 3 Received 3 August 998; received in revised form 6 February 999 Editor: L Alvarez-Gaumé Abstract Ward identities in the case of scattering of antisymmetric three form RR gauge fields off a D-brane target has been studied in type-iia theory q 999 Published by Elsevier Science BV All rights reserved Recent progress in understanding the nonperturbaw,x has provided a tive aspects of string theory unified description of the dynamics of the five Žper- turbatively consistent superstring theories The p- branes and D-branes, which appear naturally as solutions of the string effective action, have a crucial role to play in these developments wx 3 It is believed that there is an underlying fundamental theory and the five string theories are various phases of this theory w4,5x and the low energy effective action of the fundamental theory, named M-theory, can be identified with that of D s supergravity There is mounting evidence that M-theory encompasses and unifies string theories and governs string dynamics in diverse dimensions The bosonic sector of the low energy effective action of this theory contains the graviton and the antisymmetric 3-rank tensor and consequently, the membrane that couples to the three index antisymmetric tensor field is expected to be amitghosh@cernch maharana@nxth04cernch 3 Permanent address the natural extended fundamental object w6,7x in eleven dimensions It is natural to explore whether the fundamental Ž super membrane can provide the degrees of freedoms of M-theory It is not evident that there exists a consistent quantum mechanical theory of the Ž super membrane wx 8 This issue is very closely related with the large N behaviour of the UN matrix model wx 9 The proposal of the MŽ atrix model w0x has led to very interesting developments w x The MŽ atrix theory reveals various salient features of the M-theory In this approach, the dynamics of the eleven dimensional M-theory is described by the many body quantum mechanics of N D0-branes of the type IIA theory in the limit N ` The compactified MŽ atrix theory gets related to super Yang-Mills theories through dualities Furthermore, it provides a theoretical basis to the understanding of the microscopic dynamics of M-theory and holds the promise of exploring various aspects of string theories nonperturbatively w x Since the membrane theory provides an intimate connection with M-theory, there have been attempts to study the supermembrane action in curved space with antisymmetric tensor field background w3x in r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

229 ( ) A Ghosh, J MaharanarPhysics Letters B spite of the above mentioned shortcomings It is interesting to investigate how much the world volw4 x One of ume theories know about the spacetime us w5x has investigated how the bosonic membrane theory is encoded with the target space local symmetries such as general coordinate transformation and gauge transformation; the former being associated with the graviton and the latter with antisymmetric tensor field The strategy followed to expose these symmetries was based on the one proposed by Veneziano w6 x, in the context of string theory, to derive gravitational Ward identities, and by Veneziano and one of the present author w7,8x to derive Ward identities for various massless excitations of strings: both compactified and noncompactiw9 x Let us recall the technique adopted in Refs fied w6 8 x A Hamiltonian phase space framework is adopted to obtain a Hamiltonian form of the action Next, a set of generating functionals associated with the local Ž target space symmetries of the theory are introduced It was explicitly demonstrated that the variation of the action, under these canonical transformations, can be compensated by appropriate Ž gauge transformation of the massless backgrounds The Ward identities are derived with the argument that the Hamiltonian path integral measure remains invariant under canonical transformations, at the classical level at least A similar approach was adopted in the case of the membrane and it was shown that it is possible to introduce canonical transformations associated with general coordinate transformation and gauge transformation in the target space of the M-theory Of course, the results for the membrane are to be understood as classical one in view of the preceding remarks regarding the quantum theory of membranes It is worthwhile to mention that the the Ward identities derived for membrane w5x are not easy to verify explicitly, unlike the Ward identities in string theory where one can obtain explicit expressions for the vertex operators for some simple backgrounds and the conformal invariance imposes constraints on the form of various vertex operators In the covariant formulation, the vertex operators are required to be BRST invariant The purpose of this note is to derive Ward identities for scattering of the antisymmetric tensor gauge field from a membrane, specifically D-brane, that appear in the type IIA string theory In fact, the type IIA theory is of special significance since the strong coupling limit of this theory in intimately related to the D s supergravity w0 x Thus the membrane appearing in type IIA theory is very closely connected with the membrane of the eleven dimensional theory We shall see that our test of Ward identity will be at the level of scattering amplitudes similar to the works in w6 8 x, although the techniques will be slightly different Let us recall how the Ward identities are derived in the string theory w6 8 x: H ZwJxs wd X d P xwd Gxexp Ž yis wx, P, G, J x, H Ž here, ZwJxis the generating functional, X represents the string coordinate, P represents the conjugate momentum, G are the ghosts and SH is the Hamiltonian action which is a functional of coordinates, momenta, ghosts and the massless background field J, which corresponds to graviton, antisymmetric tensor or gauge potential according to the case at hand As mentioned earlier, one introduces a suitable genwx, P x erator of the canonical transformation, F J Then it was shown explicitly that for the above backgrounds, the following relation holds: d S sd J S F J H Gauge H In other words, first one computes the variations of X and P under FJ to obtain the variation of S H Then, one checks from the lhs of the above equation that it is the same as implementing general coordinate transformation or Abelian gauge transformation associated with graviton or antisymmetric tensor field In the case of compactified string the nonabelian gauge fields are also permissible backgrounds and in that case the corresponding nonabelian gauge transformation is to be considered in the rhs of the equation Now the argument is that the Hamiltonian phase space measure remains invariant under FJ and the variation of SH under FJ can be compensated by appropriate gauge transformation of background J leading to the relation 0sd J Gauge ZwJx ds D H ² J s Hd x yi d J x : Gauge J Ž 3 bg d JŽ x

230 30 ( ) A Ghosh, J MaharanarPhysics Letters B Here ² : means that the expression is averaged with the path integral factor HwdXdPd GxexpŽ yis H The interpretation of the above equation is as fol- ZwJq J lows: from the preceding arguments ZwJxs d J x leading to Eq Ž Gauge 3 Here Jbg means that the massless fields G mn, Bmn or Am takes their back- ground values after the functional derivative of the Hamiltonian action is taken Notice that dsh d L H shd sd Ž xyx Ž s d JŽ x d JŽ X H s d sd xyx Ž s V Ž X, P, Ž 4 where, V Ž X, P J is the corresponding vertex operator for G mn, Bmn or Am depending on what type of WI one is interested in As an example, let us consider a quick derivation of the gravitational WI, note that d GCT GmnsyGmlj l, nygnlj l, myg mn, lj l Ž 5 Ž Ž Using 3 and 4 we arrive at ² mn l Hd s VG Ž X, P GmlŽ X j Ž X, n l l 4 : nl m mn l G bg qg X j X, qg X, j X s0, Ž 6 here j l Ž X is the local parameter associated with infinitesimal general coordinate transformation Since it is an arbitrary parameter, if we differentiate the above equation with respect to j l and set js0 the rhs will be still zero Furthermore, we can take functional derivatives of the whole expression with respect to a string of G s and set eventually the metric to be flat space metric Ž for simplicity to arrive at d n d G Ž y dg Ž y mn mn n n n mn =² H G ml n d s V G E d Ž xyx 4 Gs h qg nl E m d xyx qg mn, l d xyx : s0 Ž 7 Note that the functional derivatives of metric act in three ways: first when it acts on the path integral J factor buried in ² : it brings down the vertex mn operator V Ž X G, in this case; second, if there is any G dependence in the vertex operator in the above expression, it removes that G and introduces a factor of dž y y X i and thirdly it kills the factor of G which exists inside the curly bracket The WI is rather transparent if one takes the Fourier transform The derivatives of delta function will give factors of momenta Thus Ž n q -graviton amplitude gets related to lower point amplitudes We also know that BRST invariance will impose constraints on the vertex operators We shall adopt following prescription to derive WI for the scattering of three index antisymmetric tensor field of type IIA theory from the D-brane Notice that the three-index potential Žwhose field strength is four index antisymmetric tensor comes from the RR sector First, we obtain the vertex operator for the three index antisymmetric tensor field in the type IIA theory The amplitude for scattering of the gauge boson from the D-brane is obtained using the techniques of conformal field theory As is well known the vertex operators must be BRST invariant and the prescription of deriving them in the covariant formulation of superstring was given by Friedan, Martinec and Shenker w x Finally, one can explicitly check that the WI are satisfied when the scattering amplitude, after separating out the kinematical factors, is contracted with the momenta of the incoming or outgoing gauge bosons Let us recall that the massless excitations of the type II theory arise from the product of the left and right moving sectors involving NS NS and RR oscillators We can represent this as < ma : = < nb :, Ž 8 R L here m,n refer to the NS NS sector and a,b correspond to the RR sector; the former being spacetime indices take values 0,,, 9 and the latter are spinor indices The familiar bosonic fields are graviton, dilaton and the antisymmetric tensor coming from the NS NS sector Furthermore, the bosonic fields originating from RR sector appear as bispinors in the vertex operator V sf U ab Ž 9 RR ab

231 ( ) A Ghosh, J MaharanarPhysics Letters B the bispinor Fab can be expanded in terms of a complete basis of the ten dimensional gamma matrices Ž antisymmetric products as follows 0 k i m m Fabs Ý Fm m m Ž g k ab, Ž 0 k k! ks0 where the k-dimensional tensor g m m k is constructed from the ten dimensional gamma matrices and it is antisymmetric with respect to all its indices Therefore, the tensors Fm m appearing the above k equation are antisymmetric in their indices and one concludes that the massless RR fields are antisymmetric Lorentz tensors As the bispinors have definite chirality projections, thus all the components of the F s are not independent As a consequence, type IIA theory has only field strengths corresponding to even integers of k and type IIB contains field strengths with odd integers of k The former admits only even branes and the latter only odd ones, as is well known In the covariant formulation of superstring wx the vertex operators involving RR fields contain the spin field S a, the bosonized ghost f and of course the plane wave piece e ik X Their combinations have to be such that the vertex operator commutes with the BRST charge Generically, we can write b b Ua svžy a z VŽy z, with y f Ž z ikxž z V z se S z e Žy a a Note that the bar on the argument of the vertex operator refers to complex conjugation here and everywhere Moreover, the g matrices are 3 = 3 dimensional and have the representation ž / 0 g mab m g s m Ž 3 g 0 ab m satisfying the anticommutation relation g,g n 4 s mn mn yh with h sdiag Ž y,,,; furthermore, g sg 0 g 9 sdiagž,y The standard method a for the construction of S is wx to bosonize the worldsheet fermions, and introduce a cocycle operator In the case of D-brane some of the coordinates Ž and therefore also worldsheet fermions satisfy Dirichlet or Neumann boundary condition Thus SŽ z and SŽ z get related depending on the Dp-brane one is considering In our case, IIA, a ab S z sm Sb z, 4 with Msg 0 g p for Dp-brane which are even Let us consider scattering of massless RR 3-form states in Type-IIA off a D-brane target In components form the 4-form field-strength is given by F se C ye C qe C ye C, mnrl m nrl n rlm r lmn l mnr where Cmrl is the 3-form potential Now, in the absence of the antisymmetric tensor field Bmn the Chern-Simons like term Ž FnFnB, where B is the NS NS -form potential, is not present in the low energy effective action; therefore, the equation of motion for the RR field strength is given by E m Fmnrls0 Ž 5 Now we choose a plane wave ansatz for the gauge potential C se a b ce ikp X, a,b,cs,,3; mnl abc m n l e 3 sq, a b c a b c Fmnrlsieabc k m n r l yk n r l m a b c a b c ikp X qk r l m n yk l m n r e, 6 where, k is the momentum, satisfying the on-shell Ž a condition k s 0 ; and em are polarization vectors subject to constraints e a Pks0 as a consequence of Eq Ž 5 The amplitude for scattering of the three index gauge field off the D-brane involves computation of the correlation functions involving two of the V operators: RR dzd zdzd z AsH ² V z, z V z, z : Ž Ž, Ž 7 Ž Vol conformal where Ž Vol conformal represents the conformal group volume factor that has to be factored out and V,V correspond to the vertex operators of asymptotically

232 3 ( ) A Ghosh, J MaharanarPhysics Letters B incoming and outgoing massless RR states The precise form of these vertex operators for this case are given below ab w m m m3 m4x Vs F U g g g g Ž 8 mmmm Ž ab 3 4 Ž 4! The computation of the correlation function in Ž 7 involve the correlators of four spin-fields V Ž z y a, Vy b Ž z, Vy g Ž w and Vy d Ž w, defined through Ž, with the indices a,b,g and d are appropriately contracted with the products of Fmnrl and the gamma matrices Žsee Eq Ž 0 This is known and has been calculated in w x The amplitude can be computed using the techniques of wx as was considered by w3 x When the gauge potential is taken to be of plane wave form, then the amplitude is given by the following expression G Ž s G Ž t As Ž sqt PqsP, Ž 9 G Ž sqtq where, ssk5 sk 5 and tskpk, k and k are incoming and outgoing momenta respectively of the massless plane wave states and k i 5 are the com- ponents of k i parallel to D-brane Furthermore, P and P appearing in Ž 7 are given by the traces of gamma matrices P s e k yk 44!4! a b c a b c abc m n r l n r l m X X X X X a b c a b c mnrls m n r l s r l m n l m n r qk yk A A X X X X X X a b c a b c X X X X X X X X X X X X ss abc m n r l n r l m =h e k yk X X X X X X a b c a b c iž k qk P X X X X X X X X X r l m n l m n r qk yk e, mnrls w m n r l 0 s x where, A s Tr g g g g g g g g and P s e k yk 4!4! a b c a b c abc m n r l n r l m X X X X X a b c a b c mnrlsm n r l s r l m n l m n r qk yk C X X X X X X a b c a b c X X X X X X X X X X X X ss abc m n r l n r l m =h e k yk X X X X X X a b c a b c iž k qk P X X X X X X X X X r l m n l m n r qk yk e, Ž 0 Ž m nrlsm X n X r X l X s X w m n r l 0 where, C s Tr g g g g g s m X n X r X l X 0 s X Ž ggg g g g g gggg qg x Now let us look at the expressions for P and P Each one can be written as a product of a tensor involving only trace of the gamma matrices times another piece which contains polarization tensors and momenta Let us denote them as T Ž X Xand T Ž X mn mn mnmn Xfor P and P respectively It is easy to check that when the tensors T Ž and T Ž are contracted with km or k m X, the corresponding momenta of incoming and outgoing gauge bosons, then both P and P vanish separately and therefore the scattering amplitude, A, given by Eq Ž 7, vanishes This is the gauge invariance of the scattering amplitude reflected through the Ward identity We recall that when the BRST invariance condition is imposed in the first quantized version of string theory on its massless backgrounds one obtains the equations of motion for those backgrounds For example, this requirement, in the NS NS massless sector of the closed string, imposes constraints on the polarization tensors of graviton and antisymmetric tensor field in addition to the mass-shell condition Similarly, if we consider the scattering of massless RR fields the BRST invariance restricts the form of the vertex operators as has been investigated by Polyakov w4 x For Abelian UŽ gauge field he derived the Maxwell equation by imposing BRST invariance on the corresponding RR vertex operator for vector bosons in type II theory For the case at hand, the BRST invariance gives rise to constraints on polarization tensor and mass-shell condition; in other words k m Fmnrls0 is a sufficient condition for the BRST-invariance of the vertex operator Ž 9 Furw4x that one can calcu- thermore, it has been shown late three point function, using the conformal field theory techniques, involving the dilaton and RR gauge fields This interaction does not show up in the string effective action expressed in the string frame metric However, if one first goes over to the Einstein frame by a conformal transformation Žin- volving the dilaton and the redefines the RR gauge fields, the interaction terms involving the dilaton and RR gauge fields can be exhibited and the correspondence with the three point function mentioned above can be established In the light of our investigation we can conclude that the derivation of the Ward identities for the scattering amplitude of D-brane

233 ( ) A Ghosh, J MaharanarPhysics Letters B and 3-form potential is a consistency check of the current conservation as is also reflected in the Ward identities associated with conventional gauge theories We note that in the conformal field theoretic calculation of the scattering process involving D- brane and the three-index antisymmetric gauge fields, we do not see the effects of the Ž FnFnB -like term at this order The presence of this CS-like term, at the tree level calculations, can be seen if one looks 4 at the ByFyF -vertex We mention in passing that one might envisage our results from the perspectives of M theory It is well known that if we start with the M-theory membrane and compactify one of the transverse directions on a circle we obtain the type-iia D-brane with equal tension Furthermore, the antisymmetric 3-form in ds supergravity gives rise to the RR 3-form under Kaluza-Klein reduction Therefore, the symmetries uncovered by the Ward identities in the scattering of RR 3-form from D-brane in IIA theory should also be obeyed in the ds theory when one considers 3-form membrane scattering amplitude We interpret it, at the present level of our understanding, that this is an indirect evidence for the abelian gauge invariance in quantized M-theory Another check would be to calculate directly the scattering amplitude of the 3-form field in eleven dimensional supergravity limit off the M-theory membrane as the target Similar calculations have been done in ten dimensional supergravity using extreme black p-branes as target and the amplitude has been compared with the string calculation when the probe energy is small or the impact parameter is large compared to the string scale They seem to agree perfectly in this limit w3 x At this stage it is not possible to compute the scattering of three form gauge field from the membrane in the -dimensional theory in a reliable manner, as compared to the scattering of 3-form gauge field from D-brane in the case of type IIA theory using the conformal field theory techniques It will be interesting to see whether such perfect matchings come out of a MŽ atrix theory computation We may mention that the type-iia D-brane being a dynamical object asks 4 We thank S Mukhi for discussions related to this issue for a consistent effective quantum description of the M-theory membrane by which one can hope to provide direct checks of these symmetries in ds, most possibly using similar techniques discussed at the beginning of this paper and in w5 x In view of these results, it will be possible to derive Ward identities for the entire massless supergravity multiplets of the ten dimensional type II theories and supergravity and super Yang-Mills theories obtained from other string theories, although partial results were derived by Veneziano and JM a few years ago w5,6 x We hope to report our results in the future Acknowledgements We would like to thank Gabriele Veneziano for valuable suggestions during the course of this work One of us Ž JM would like to acknowledge useful discussions with H Nicolai and would like to thank Max-Planck Institute for Gravitation for warm hospitality References wx A Sen, Int J Mod Phys A 9 Ž , hep-thr0000; A Sen, An Introduction to Non-perturbative String Theory hep-thr98005; A Giveon, M Porrati, E Ravinovici, Phys Rep C 44 Ž ; MJ Duff, RR Khuri, JX Lu, Phys Rep C 59 Ž 995 3, hep-thr9484; E Alvarez, L Alvarez-Gaume, Y Lozano, An Introduction to T-duality in String Theory, hep-thr94037 These are some of the review articles wx E Witten, Some comments on String Theory Dynamics, in: Proc String 95, USC, March 995, hep-thr9507; Nucl Phys B 443 Ž , hep-thr95034; PK Townsend, Phys Lett B 350 Ž ; CM Hull, PK Townsend, Nucl Phys B 438 Ž , hep-thr94067 wx 3 J Polchinski, Lectures on D-branes, in TASI 996, hep-thr 96050; C Bachas, Lectures on D-branes, hep-thr wx 4 JH Schwarz, Lectures on Superstring and M-theory Dualities, ICTP and TASI Lectures, hep-thr96070 wx 5 PK Townsend, Four Lectures on M-theory, Trieste Summer School, 996, hep-thr96; M-Theory from its Superalgebra, 997, Cargese ` Lectures, hep-thr97004; H Nicolai, On M-Theory, hep-thr wx 6 E Bergshoeff, E Sezgin, PK Townsend, Phys Lett B 89 Ž ; Ann Phys 85 Ž wx 7 MJ Duff, Supermembranes, hep-thr9603

234 34 ( ) A Ghosh, J MaharanarPhysics Letters B wx 8 B de Wit, J Hoppe, H Nicolai, Nucl Phys B 305 Ž wx 9 J Goldstone, unpublished; J Hoppe, in: G Longhi, L Lusanna Ž Eds, Proc Int Workshop on Constraint s Theory and Relativistic Dynamics, World Scientific, 987 w0x T Banks, W Fischler, SH Shenker, L Susskind, Phys Rev D 55 Ž 997 5, hep-thr wx T Banks, Matrix Theory, hep-thr9703; D Bigatti, L Susskind, Review of Matrix Theory, hep-thr9707 wx R Dijkgraaf, E Verlinde, H Verinde, Notes on Matrix and Microstrings, hep-thr w3x B de Wit, K Peters, JC Plefka, Supermembranes and Supermatrix Models, Talk Presented at the Valencia Workshop Beyond the Standard Model; from Theory to Experiment, October 997, hep-thr9708 w4x PK Townsend, talk at the Workshop on Nonperturbative Aspects of Strings, Branes and Field Theory, CERN, December 997 w5x J Maharana, Hidden Symmetries of M theory, hep-thr 9808, Phys Lett B, in press w6x G Veneziano, Phys Lett B 67 Ž w7x J Maharana, G Veneziano, Phys Lett B 69 Ž w8x J Maharana, G Veneziano, Nucl Phys B 83 Ž w9x J Maharana, Phys Lett B Ž w0x E Witten, Nucl Phys B 443 Ž 995, hep-thr95034 wx D Friedan, E Martinec, S Shenker, Nucl Phys B 7 Ž wx VA Kostelecky, O Lechtenfeld, W Lerche, S Samuel, S Watamura, Nucl Phys B 88 Ž w3x SS Gubser, A Hashimoto, IR Klebanov, JM Maldacena, Nucl Phys B 47 Ž 996 3; A Hashimoto, IR Klebanov, Scattering of Strings from D-branes, hep-thr964 w4x D Polyakov, RR Dilaton Interaction in a Type IIB Superstring, hep-thr9508 w5x J Maharana, G Veneziano, Unpublished results on Ward identities for Supergravity particles from string theory, January 986 w6x Some of the related works are: M Evans, B Ovrut, Phys Rev D 39 Ž ; B Ovrut, S Kalyana Rama, Phys Rev D 45 Ž ; H Gasparakis, hep-thr930856

235 0 May 999 Physics Letters B A comment on the Zamolodchikov c-function and the black string entropy Akikazu Hashimoto a,, N Itzhaki b, a Institute for Theoretical Physics, UniÕersity of California, Santa Barbara, CA 9306, USA b Department of Physics, UniÕersity of California, Santa Barbara, CA 9306, USA Received March 999 Editor: M Cvetič Abstract Using the spectral representation approach to the Zamolodchikov s c-function and the Maldacena conjecture for the D-branes, we compute the entropy of type IIB strings An agreement, up to a numerical constant which cannot be determined using this approach, with the Bekenstein Hawking entropy is found q 999 Published by Elsevier Science BV All rights reserved In two dimensional field theories, the two-point function of the energy-momentum tensor is a useful concept in studying the relation between the energy and the entropy For conformal theories, they comwx This fact, together pletely determine the entropy with the observation of Brown and Henneaux wx that the asymptotic group of AdS3 yields a two dimensional conformal theory, was used by Strominger to derive the Bekenstein Hawking entropy for black holes in AdS wx 3 3 In this note we study the SYM theory in q dimensions with gauge group SUŽ N and sixteen supercharges which is a non-conformal theory This theory can be thought of as the theory living on a collection of N D-branes in the low energy de- aki@itpucsbedu sunny@physicsucsbedu coupling limit In the extreme UV the theory is free and conformal with central charge c UV ;N Ž Perturbation theory in SYM can be trusted in the UV as long as the effective coupling constant is small, that is 4geffsgYM Nx x<, g ' YM N where x is the scale being probed by the two point function In the deep IR region, the physical energy scale determined by the coupling constant becomes irrelevant and the theory flows to a conformal thew4,5x this theory was shown to be a ory In Refs conformal s-model with the target space Ž R 8 N rs N whose central charge is c IR ;N Ž 3 Note that c UV ) cir as expected from the Zamolodchikov s c-theorem wx 7 The first correction to the orbifold CFT is given by the twist operator g YM V ij r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

236 36 ( ) A Hashimoto, N ItzhakirPhysics Letters B where i, j label the fields on which the twist operator is acting wx 6 There are various ways to show that the perturbation theory with respect to the twist opera- 'N tors breaks down at x- w8 0 x A simple g YM argument which rests on the c-theorem is the following Perturbation theory around the conformal point will break down when the difference between the Zamolodchikov c-function and cir is of the order of c That is, when ² T Ž x T Ž 0: IR zz zz is of the order of ² T Ž x T Ž 0 : zz zz Conservation of the energy-momen- tum tensor implies that Žsee eg wx Ý ij g YM p V T sy Ž 4 zz Therefore, perturbation theory around the conformal point can be trusted when c N ' IR N 4 4 6, x4, Ž 5 x gym x g YM where we have used the fact that the weight of the twist operators is Ž 3r,3r wx 6 In the large N limit there is a large region, ' N <x<, Ž 6 ' Ng g YM YM for which neither the perturbative SYM nor the orbifold CFT description can be trusted In Ref wx 9 it was shown that this region is best described by the type IIB string theory on a background associated with the near horizon geometry of DrF strings Exactly at the points in UV and the IR regions where the perturbative field theory descriptions break down, the curvature Ž in string units is small so that the supergravity approximation becomes reliable The transitions between the perturbative conformal theories Ž at the UV and IR and the supergravity descripw 4x to find a match tion were studied in Refs from both sides up to a numerical coefficient which cannot be determined using current methods This agreement Žfor the entropy w,4x and for the Wilw3,4x supports the Maldacena conjecture son line for this non-conformal theory but it does not give us much information about the way in which the supergravity description interpolates between the UV and the IR perturbative field theories descriptions In this article, we elaborate on the interpolation of the Zamolodchikov s c-function between c and IR cuv and its relation to the entropy of the near-extremal DrF string To make contact with entropy it is useful to use the Kallen Lehmann spectral representation of the correlator of two energyw5 x, ² T Ž x T Ž 0 momentum tensors : mn rs p ` d p s H d mcž m H e ipx 3 0 Ž p Ž mn m nž rs r s = g p yp p g p yp p p qm Ž 7 The fact that the Zamolodchikov s c-function is monotonically decreasing along the RG flow follows from cž m G 0 which must hold for any unitary theory w5 x In two dimensions, covariant quantity with four indices subject to the constraint following from the conservation of energy momentum-tensor is characterized by a single invariant Thus, there is only one possible function of the intermediate mass scale, cž m, which is known as the spectral density The quantity H L c Ž L s dm cž m, Ž 8 eff 0 interpolates between c s c Ž 0 and c s c Ž ` IR eff UV eff Since cž m d m measures the density of degrees of freedom which couple to the energy-momentum tensor, and since all fields couple to the energymomentum tensor, the spectral representation of the correlator of two energy-momentum tensors measures the density of degrees of freedom Using the methods developed in Refs w6,7x to compute field theory correlation functions via the bulk propagation of supergravity modes, we can calculate the two point function of the energymomentum tensor Suppressing numerical factors and Lorentz indices, we find N 3r ² TŽ x TŽ 0 : s Ž 9 g x 5 YM Before substituting this into Eq Ž 7 and discussing the spectral density, it is worth while to make a few comment about this result Eq Ž 9 is obtained by repeating the procedure of w6,7x for the minimally

237 ( ) A Hashimoto, N ItzhakirPhysics Letters B coupled scalar in the near horizon geometry of the D-brane In non-conformal theories it is harder to identify the correspondence between the supergravity modes and the field theory operators since the symmetry group is smaller Žsee however w8 x General covariance indicates that the energy momentum tensor must correspond to the metric fluctuation h mn It is therefore more appropriate to analyze the field equations for metric fluctuations in this background We expect nonetheless for the generic components of the metric fluctuations to behave essentially like a minimal scalar The reason is that hm m mixes with a linear combination of a minimal and fixed scalar 3 w9 x In the supergravity region the fixed scalar contribution is suppressed and we are left with Eq Ž 9 for ² T Ž x T Ž: zz zz 0 All other components are determined in two-dimensions by the conservation of the energy-momentum tensor There are corrections to Ž 9 suppressed by gym x' N xg and YM which can be thought of respectively as 'N curvature and quantum corrections from the point of view of type IIB string theory in the near horizon geometry of the D-brane These corrections are very small and can be ignored in the region given by Eq Ž 6 where supergravity approximation can be trusted Conversely, the point in x-space where these corrections become significant mark the transition point to the UV and IR conformal fixed points At these transition points, between supergravity and perturbative SYM and between supergravity and the orbifold CFT, Eq Ž 9 agrees Žup to a numerical factor with the conformal results c ² TŽ x TŽ 0 : s, Ž 0 x 4 for the central charge appropriate for the UV and IR fixed points given in Eqs Ž and Ž 3 In order to fix the numerical coefficient of Eq Ž 9 unambiguously, it may be necessary to understand the supergravityperturbative SYM crossover in detail so that the normalization in the supergravity region can be matched to the normalization in the perturbative 3 Strictly speaking, these fields are scalars in the 9 dimensional supergravity obtained by dimensionally reducing along the spatial direction of the D-brane SYM region For our purpose, however, there is no need to fix the numerical constant since the relation between the spectral density and the entropy can be determined only up to a numerical factor as we discuss below Combining Eq Ž 9 with Eq Ž 7 we find that 3r c m sn rg YM Therefore, for a given temperature T, ˆ the number of light degrees of freedom with p -Tˆ is N T N Ž T ˆ ;c Ž Tˆ s Ž eff eff 3r ˆ g YM Here we encounter a fundamental ambiguity: the relative numerical coefficient between c Ž Tˆ eff and N Ž Tˆ eff cannot be determined for non-conformal the- 4 ories Žas opposed to conformal theories where N sc eff eff In fact, one can construct two theories with the same ceff whose Neff agrees only up to a numerical factor of order one The contribution to the free energy is F;N Tˆ LT ˆ eff Ž Hence the energy density and entropy density are 3r S N Tˆ E N 3r Tˆ3 ss s, es s Ž 3 L g L g YM We wish to compare this with the black hole thermodynamics In the Einstein frame the near horizon metric of N near-extremal D-branes is, d s U 9r U0 6 s y y dt q d x X 5r 3r4 6 a g N U q YM YM ž ž / / N r4 du 6 U 3r ( 0 g YM U y 6 r4 ' ž U / N U q dv 6, Ž 4 g ( YM where U 6 sg 4 sity w9, x, 0 YM ' e This yields for the entropy den- yr3 r3 ssgym Ne, 5 which is in agreement, up to a numerical factor, with Eq 3 4 w x For an attempt to go off criticality see 0

238 38 ( ) A Hashimoto, N ItzhakirPhysics Letters B Note that unlike in the near horizon geometry of D5 q D branes, the field theory describing string theory in the near horizon geometry of D-branes is known in details: it is the SYM in two dimensions However, our calculation did not rely on the detailed properties of the SYM action at all What we did instead was to use the supergravity dual to compute a field theory quantity, the two-point function of the energy-momentum tensor Then, using rather general field theory arguments which are valid for any unitary field theory in two dimensions, we compute the entropy to find an agreement with the Bekenstein Hawking formula This agreement serves as a check of Maldacena conjecture for D-branes wx 9 It is interesting to note that in the supergravity region the entropy energy relation is similar to that of a gas of a free massless scalar field in Ž q dimensions In the extreme UV and IR, on the other hand, this theory behaves like a free gas in Ž q dimensions Žwith a different number of field, since c ) c UV IR Amusingly, this general behavior is mimicked by the following simple statistical mechanical model Consider N free fields in Ž q dimensions which propagate on a semi-lattice By semi-lattice we mean a chain of Nˆ continuous strings with lattice spacing a So the size of the system is LL where L is an IR cutoff which we can take to infinity and L s Na ˆ The dispersion relation for this system is 4 v s k q sin Ž k ar, a k snprl, ns0,,n ˆ Ž 6 In the IR where the thermal wave-length is larger than L, the contributions from the extra dimension are negligible and we have the entropy of a free massless gas in two dimensions In the UV where the thermal wave-length is smaller than a, v can grow only due to k as k is restricted to a single Brillouin zone Therefore, the entropy is that of a free massless gas in two dimensions where the number of massless fields is NN X as can be read from Eq Ž 6 5 To have an agreement with Eq Ž we set ˆN s N In the intermediate region the system behaves like a gas of N free particles in Ž q dimensions To match with the number of degrees of freedom of q dimensional SYM in the deep UV and IR, we set as 'N g YM Acknowledgements We would like to thank Ilya Gruzberg, Victor Gurarie and Joe Polchinski for helpful discussions AH is supported in part by the National Science Foundation under Grant No PHY NI is supported in part by the NSF grant PHY97-0 References wx JL Cardy, Nucl Phys B 70 Ž wx JD Brown, M Henneaux, Comm Math Phys 04 Ž wx 3 A Strominger, JHEP 0 Ž , hep-thr975 wx 4 JA Harvey, G Moore, A Strominger, Phys Rev D 5 Ž , hep-thr9500 wx 5 M Bershadsky, A Johansen, V Sadov, C Vafa, Nucl Phys B 448 Ž , hep-thr wx 6 R Dijkgraaf, E Verlinde, H Verlinde, Nucl Phys B 500 Ž 997 6, hep-thr wx 7 AB Zamolodchikov, JETP Lett 43 Ž ; Sov J Nucl Phys 46 Ž wx 8 R Dijkgraaf, E Verlinde, H Verlinde, Nucl Phys ŽProc Suppl 6 Ž , hep-thr wx 9 N Itzhaki, JM Maldacena, J Sonnenschein, S Yankielowicz, Phys Rev D 58 Ž , hep-thr98004 w0x AW Peet, J Polchinski, hep-thr98090 wx J Polchinski, String Theory, vol II, Cambridge University Press, 998, p 63 wx GT Horowitz, J Polchinski, Phys Rev D 55 Ž , hep-thr9646 w3x J Maldacena, Phys Rev Lett 80 Ž , hep-thr w4x A Brandhuber, N Itzhaki, J Sonnenschein, S Yankielowicz, JHEP 06 Ž , hep-thr w5x A Cappelli, D Friedan, J Latorre, Nucl Phys B 35 Ž w6x SS Gubser, IR Klebanov, AM Polyakov, Phys Lett B 48 Ž , hep-thr The fields are not exactly massless as can be seen from Eq Ž 6 Rather their masses are bounded by ra which is much smaller then the temperature so their contribution to the entropy at large temperature is similar to the massless fields contributions

239 ( ) A Hashimoto, N ItzhakirPhysics Letters B w7x E Witten, Adv Theor Math Phys Ž , hep-thr w8x A Jevicki, Y Kazama, T Yoneya, Phys Rev D 59 Ž , hep-thr98046 w9x MR Krasnitz, IR Klebanov, Phys Rev D 56 Ž , hep-thr97036; SS Gubser, A Hashimoto, IR Klebanov, M Krasnitz, Nucl Phys B 56 Ž , hep-thr w0x AH Castro Neto, E Fradkin, Nucl Phys B 400 Ž , cond-matr wx IR Klebanov, AA Tseytlin, Nucl Phys B 475 Ž

240 0 May 999 Physics Letters B Non-perturbative structure in heterotic strings from dual F-theory models Donal O Driscoll Department of Physics, UniÕersity of Wales Swansea, Singleton Park, Swansea, SA 8PP, UK Received 6 January 999; received in revised form 9 March 999 Editor: M Cvetič Abstract We examine how to construct explicit heterotic string models dual to F-theory in eight dimensions In doing so we learn about where the moduli spaces of the two theories overlap, and how non-perturbative features leave their trace on a purely perturbative level We also briefly look at the relationship with NS9-branes q 999 Elsevier Science BV All rights reserved PACS: 5Mj; 5Sq Keywords: F-theory; Heterotic string Introduction Over the last few years the idea of string duality has lead to much greater understanding of the nonperturbative features of string theory, to the extent that we can now visualize the various string theories as being different points in a larger moduli space Most notably we have learned about the interplay of geometric features in compactification, especially those based on the rich area of Calabi Yau manifolds In particular we have learned to relate strongly coupled type IIB superstrings in a background of 4 7-branes to heterotic string theory through elliptiw 3 x However, to date cally fibered K3 surfaces the pydan@swanseaacuk majority of this work has been very mathematical wx 4 in nature with little attention being paid to the explicit duality map In this letter we will address this issue There are several well known and phenomenologically interesting methods of constructing heterotic theories in less than ten dimensions that have been known for some time, viz: covariant lattices wx 5, free fermionic constructions wx 6, and asymmetric orbifolds wx 7 It can be shown that these are all essentially equivalent wx 8, with each method having its own benefits and drawbacks Nevertheless, it is not trivial to determine a map between F-theory and the heterotic string compactifications directly because it is not known to what extent the moduli spaces overlap For example, F- theory on K3 has a fixed supersymmetry wx 9, however it is relatively easy to construct heterotic models in eight dimensions with less supersymmetry Since F-theory is non-perturbative understanding the map r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

241 ( ) D O DriscollrPhysics Letters B should provide an interesting relationship between perturbative and non-perturbative aspects of the heterotic string In Section, the basic results needed from F-theory on how to read off gauge groups with the necessary substructure indicating non-perturbative contributions are outlined In Section 3, we look at the prescription to recover the purely perturbative heterotic theory and discuss how to construct the dictionary of the two via the moduli space of Type I theory In Section 4, we conclude the paper with a brief look at how we can relate the work of this paper to the recent work being done on NS9-branes, which are problematic in string theory but appear to be required by duality arguments F-theory F-theory wx is not a string theory per se, though attempts have been made to define it as a dimensional theory with two time dimensions A much more satisfactory approach is to consider it as Type IIB superstring compacted on a sphere Žthe complex projective surface in the background of twenty four 7-branes Type IIB superstring theory in ten dimensions has an SLŽ,Z self-duality and hence has an associated torus When this torus is fibered over the sphere in the brane background an elliptically fibered K3 surface is formed, the properties of which are wx well known The moduli space is 9 Ms SOŽ,8;Z _ SOŽ,8 rsož = SOŽ 8 Ž and is the same as that of a heterotic string compactified on T using a Narain lattice Since the K3 surface has an elliptic structure its singularity structure can be easily read off from the Weierstrass equation These singularities have been classified by Kodaira in a way corresponding to the A D E classification of Lie algebras w0 x It is standard to accept this correspondence as being exact, ie the singularity type corresponds to a gauge group of the same Lie algebra type in the physical theory However, Witten wx has shown that this is not necessarily true, though how this works from a heterotic point of view is not yet fully understood Other algebras can also be constructed using various configurations of mutually non-local 7-branes but as they do not coalesce to a single point they are not of interest here Recently there has been much work done in understanding how the gauge groups arise from the K3 surface through the theory of string networks w x The singularities of the K3 surface correspond to the positions of the twenty four 7-branes forming the background in F-theory We know from work on the related theories of D-branes w3x that perturbatively there should be only sixteen D7-branes By using Seiberg Witten theory, Sen w4x showed using F- theory, that the orientifold planes could be formed out of 7-branes which have different charges with respect to the Ramond and Neveu Schwarz sectors In references w5,6x it is shown how to combine these mutually non-local branes to provide the states needed to fill out the gauge groups corresponding to the respective singularities on the K3 surface The 7-branes are classified by the RR and NS NS charges, w p, q x, they carry; 7-branes with different values of wp, qx are said to be mutually non-local From w6x we will be only concerned with three types of mutually non-local 7-branes, denoted A, B, C If we have n 7-branes of the same type at a singularity then the associated gauge algebra is suž n If there are different types of branes at a singularity, n a n b n A B C c, the associated algebra is suž n msuž n a b msuž n c This is actually a maximal subalgebra of a larger simply laced algebra since extra massless BPS states also appear in representations of the subalgebra and fill out the adjoint of the larger group in a manner analogous to free fermion constructions w7 x A D type singularity corresponds to a 7-brane configuration of the form A n BC; the subalgebra is suž n m už which is enhanced to a Dn algebra Similarly an E type singularity has a 7-brane confign uration of the form ABC; the subalgebra is suž n muž msuž which is enhanced to a En algebra It is the maximal subalgebras that we are most interested in since they obviously encode non-perturbative features and point out where BPS states should be in the heterotic spectrum The orientifold limit Since the 4 7-branes are non-perturbative they will not feature directly in string models so we need

242 4 ( ) D O DriscollrPhysics Letters B to find a limit which relates them to a perturbative regime A method in going between F-theory on K3 and Heterotic on T is to use Type I and I X models as an intermediate step w4 x From this point of view the gauge groups in the Dn series are formed by placing n D-branes on an orientifold 7-plane, O That is in going from F-theory to Type I theory we have the 7-branes behaving as: A n BC A n O In collapsing the BC branes to an orientifold, the NS NS charges cancel whilst the RR charges combine to give the correct value for orientifold planes in eight dimensions The maximal subalgebra enlarges as suž n muž sož n Ž 3 The effect of this limit on an E singularity is A n BC A n OqC Ž 4 with the maximal subalgebra reorganizing itself as suž n muž msu sož n muž Ž 5 Thus dual models to F-theory constructions should have enhanced gauge groups built from these maximal subgroups The extra states should also be BPS A corollary is that the corresponding heterotic string we are going to be interested in is HSO since it is this theory which is S-dual to Type I w8 x In the following we will use HSO to denote the heterotic string compactified on the Narain lattice G m G while HE8 denotes G m G m G, 6, 8 8 Though they are the same theories on compactification, they have different Wilson lines when it comes to embedding other gauge groups 3 Heterotic string on T We now turn to building heterotic string models From duality there are conditions to be satisfied; as already pointed out there can be no supersymmetry breaking The moduli space is equivalent to that of compactification on a Narain lattice, prompting the restriction to gauge preserving compactifications, ie total rank is 8, and switching off background antisymmetric tensor fields We will assume that all rank 8 gauge groups appearing on the F-theory side are acceptable, ie have a heterotic dual; and that we are embedding our Wilson lines in a lattice of the form G mg as opposed to G mg ; a priori this is, 6, 8 due to the Dn structure of the maximal subalgebra in the orientifold limit We compactify on the two dimensional torus w9x T sr rpl Ž 6 where L is a lattice with basis vectors e, < e < i i s R i for i s, where R i are the radii of the circles Generically R / R Winding number is denoted v sn i e, n i gz; while momentum is given by p s i i i me i w i where e w i is a basis vector of the dual lattice L w In the lattice frame, the background gauge fields I are A sa I Že w i m i m with Is, 6 labelling coordi- nates in G6 and m the spacetime dimensions V is a vector in G The momentum, Ž p ;p defined as 6 L R K K K K L ž 4 / p s VqAPv, py V A y A A Pv qv Ž 7 K K K K R ž 4 / p s py V A y A A Pv yv Ž 8 form a self-dual Lorentzian lattice The mass of a state is given by M s N q p y q N q p yc Ž 9 4 L L R R N L, NR are the left and right moving oscillator num- bers and cs0, depending on the periodicity of the right moving fermions The level matching condition is NL q p L ysnr q p R yc 0 Applying this to Eq Ž 9 and then imposing the condition NR s c gives the mass formula for BPS states M sp Ž 4 R The massless vectors belonging to the roots of the underlying gauge group have NL s 0 along with p Rs0,p Ls When the winding number is zero this gives the subgroup of the SOŽ 3 surviving breaking by the Wilson lines However, for certain values of R i then further massless gauge bosons can

243 ( ) D O DriscollrPhysics Letters B appear so as to enhance the gauge group Writing out p in component form we get R p se m y V a y a a n yne Ž w i K K K K j i R i i 4 i j i Note, that i is now a label and not a component as far as the ai K are concerned The third term in the expansion looks problematic as it has the potential to cause coupling between the Wilson lines However, our choices of values for the ai K will actually give zero for the expression ai K aj K,i/j and allow us to decouple the two cases With this choice p se m y V a y Ž a n yne Ž 3 w i K K i i R i i 4 i i where there is no summing over i and Ž a i is the length of the shortest vector of the form aiq l, where lgg 6 If the radius of compactification is, for each i, R i s y ai r then extra massless modes appear allowing an enhancement What actually has occurred here is that the two dimensional case has been split up into to two copies of a single dimensional compactification They are also automatically BPS There are two mechanisms of gauge enhancement: Ž i the standard D-brane approach w3x of clustering branes on top of each other; in the above notation this means identifying coordinates within the bulk of the fundamental cell so that the generic group UŽ 8 becomes G Ž Ž 6 mu with the U dependent only on the structure of L; Ž ii when the relationship R s yž a r is satisfied for Ž a i i i the shortest length of the Wilson line relative to the cluster of D-branes we wish to enhance However in the Type I and I X dual models the second mechanism is non-trivial and requires the x string discussed in Ref w x Its position in the moduli space is arbitrary except for gauge enhancing points when its position satisfies the above relation relating the radii to the length of the Wilson lines The x string can be related to the string junctions as its origin in nine dimensions is from the presence of a D0-brane which can couple to D8-branes and orientifold planes It satisfies the condition that the number of Neumann Dirichlet boundaries on the string is eight, ie NDs8 w3 x When we compact down to eight dimensions the D0yD8 system becomes DyD7 which still satisfies ND s 8 and is similar to the string junction system used in the F-theory duals In the work of wx an investigation of the D0y D8 system was made in nine dimensions where they started off with an arbitrary number, n of D0-branes in the presence of D8-branes and O8-planes It was then shown ns was required for stable configurations as in gauge enhancement When there is further compactification down to eight dimensions we have ns in the decoupled case Decoupling basically allows us to take two copies of the nine dimensional case since we can treat the axes as independent except at the non-trivial point of the origin where they intersect which is only significant if there are branes placed at that point In taking the orientifold limit of the En series there was a C-brane left over Nevertheless, it contributes states necessary for the gauge enhancement and does so in a manner analogous to the states contributed by the x string We now make the tentative identification that the string junctions states related to the C-branes are dual to the states due to the x string and hence the C-brane is itself dual to a DyD7-brane set up in Type I theory ŽT-dual to a D0 y D8-brane set up in nine dimensions Note, that there appears to be a choice between which C-brane we should identify with the orientifold and which with the x string However, in the D0yD8 set up gauge enhancement occurs when the D0-brane is attached to an orientifold, and likewise here the left over C-brane is still at the position of the associated orientifold so it is not possible do separate their overall effects in this picture and the choice does not have to be made We will return to the C-brane later when we discuss NS branes in heterotic theory For gauge groups of rank 8 there are only a finite number of ways of combining the En groups for ng6 The only one with three exceptional groups 3 is E which has been handled already in Ref w0 x 6 It also does not satisfy the decoupling feature but we will return to it later The rest of the possibilities we can cluster together as ENm EMm G or ENm G It remains to be verified if this will still be the case when the Wilson lines are coupled

244 44 ( ) D O DriscollrPhysics Letters B where G is of sufficient rank to make the total 8 3 For simplicity we make it a SOŽ n Lie group with no further breaking Looking at the first case with two exceptional groups, we can make the decomposition in the orientifold limit: Enq memqmd6ynym D md muž md Ž 4 n m 6ynym We can now see that we can associate the two UŽ components to the two circles making up the compactification torus, each dimension being made responsible for the enhancement of a particular Dn or D m The single dimensional case has already been dealt with in Ref w x For the sake of convenience we associate the Dn group with is and Dm with is Then we can give the Wilson lines as n m 6ynym n m Ž 6ynym a s 0 0, a s 0 0 Ž 5 These trivially satisfy the condition that they decouple the Ž p R i as their product is always zero Generalizations to G/ D6ynym are straightforward When the gauge group is of the form EN m G then one merely has to move the appropriate radius away from the critical point of enhancement in the previous case or alter one of the ai depending on the form desired for G The other cases of particular interest with regard to enhancement, SOŽ 36 and SUŽ 9 follow as in the one dimensional case with one of the Wilson lines set entirely to zero 3 Coupled solutions We can use the duality of HSO with Type I to learn more about the structure of the moduli space of heterotic Wilson lines First examine the the group E6mE6mE 6 This is somewhat anomalous as E D muž 6 5 requires three UŽ s The solution is of the form given in Eq Ž 5 with nsms5 However, in order to get the third UŽ for the enhancement the Wilson 3 In the former case the rank of G will always be less than or equal to 6 For it equal to 6 we ignore the possibility it could be E 6 lines have the components a sa sy 0 This violates the decoupling argument above but provides a solution nevertheless Hence there exist other solutions where the Wilson lines do not decouple 6 6 w x This is the generic case though the E6 3 one is the only one with enhancing to an exceptional group that cannot be made to decouple In this notation the Wilson lines act as the coordinates of a square moduli space of axes a, a such that 0 F aif Each pair of components of the Wilson lines, Ž a,a K now forms the coordinate of the Kth D- brane when it is mapped to a dual Type I model in eight dimensions The orientifold planes are repre- sented by the corners Ž 0,0, Ž 0,, Ž,0, Ž, though they only have an effect if there are D-branes on them; decoupled solutions lie purely on the axes However, this space is only a fundamental cell of a larger lattice and extra massless states can arise, as in E6 3, when D-branes are located at special points outside the fundamental cell when other winding modes become massless These situations will break the Z4 symmetry of the Wilson lines that exist when the D-branes all lie within the fundamental cell Coupled solutions lie within the bulk and represent the relative difficulty of finding the solution as the positions here are arbitrary, the solutions giving rise to gauge groups lying on loci as opposed to particular points For many cases the loci of solutions will intersect with the boundary and the decoupled form of the Wilson lines can be recovered A final set of gauge groups of interest are those of the form Dn x mdm y with nxqmys8 It can be shown that if xqyg4 then there will always be more than 4 branes required on the F-theory side That is if xqy)4 then some of the gauge groups would have to have rank less than 4 and thus are in the A series as opposed to the D, in line with the fact that in the Type I picture there are only four orientifold planes In the case x q y s 4 then the gauge group is D4 md5 which from the F-theory side is not acceptable as it would require 6 7-branes On the HSO side it would be constructed by placing 4 D-branes on each orientifold plane and choosing the appropriate radii to enhance two of the D to E 4 5 which would give us back the D gauge groups The 5 Wilson lines are: ž / ž / a s 0 0, a s 0 0 Ž 6

245 ( ) D O DriscollrPhysics Letters B The resolution lies in the fact that is not possible to single out only two D4 we want to enhance and the construction from the heterotic point of view breaks down as well There are no other cases such as this so we are justified in the assumption that all rank 8 gauge groups which can be constructed from F-theory on K3 have an appropriate heterotic dual So far we have been concentrating purely on the HSO type models In theory we should be able to construct a dictionary for embedding in HE8 since it has the same moduli space and is related to HSO by T-duality However, there is no simple orientifold limit as for HSO and the Wilson lines are non-trivial A case in hand is E6 3 which, to build up in G,mG8 the third E Ž 6, we require a su 3 3 maximal subalgebra, the brane structure for which is not apparent The generic SLŽ,Z transformation to convert the standard A n a B n bc n c configuration to a form where the maximal subalgebra reflects the embedding in E8mE8 has not been constructed yet If the T-duality is modified as discussed in the next section, then it may be the case that this transformation does not exist 4 Comparison with NS9-branes and conclusion In recent papers Hull w4x has discussed the existence of NS9-branes in non-perturbative HSO theory NS9-branes are the S-duals of the D9-branes in Type I theory and can also be deduced from M9-branes in M-theory w5 x Their discovery, implied by duality, gives the same brane structure in the heterotic string that has proved to be so rich in Type I theory In particular it is not hard to see that the heterotic Wilson lines should now correspond to the position of the 6 NS9-branes This in turn provides evidence that the map between the Wilson lines of Type I and HSO is exact under S-duality; the RR charge of the orientifolds will change to NS NS in HSO models In deriving the map between F-theory on K3 and HSO on T we have implicitly assumed that the A-branes of F-theory have the same charge as D7- branes This is not strictly necessary as what mattered in the above construction was the correct cancellation and summing of overall charge along with the maximal subalgebra This is an intrinsic feature of the F-theory construction as various brane configurations are considered equivalent if they can be related by an SLŽ,Z transformation w5 x The SLŽ,Z self-duality of the parent Type IIB theory includes an S-duality Thus the standard A n a B n bc n c configuration can be related to another configuration of charged 7-branes, A n ab n bc nc, so that the gauge structure is exactly the same but the NS NS and RR charges are interchanged Hence, when examining the perturbative string models we can only tell the difference between HSO or Type I from the charges of the F-theory configuration we started with In terms of the moduli space, it provides evidence that HSO and Type I with all their permitted Wilson line configurations are equivalent up to gauge groupr Kodaira classification, and that the subgroup structure discussed above persists under an S-duality transformation This ties in nicely with the use of truncation techniques on the parent Type IIB spectrum in ten dimensions to obtain the Type I and HSO theories performed in Ref w6x In this paper we have constructed an explicit map taking us from the Kodaira classification of singularities in F-theory compactified on elliptically fibered K3 surfaces to the Wilson lines in the Heterotic SO 3 string compactified on T, and discussed some issues arising out of gauge enhancement Acknowledgements We would like to extend our thanks to DC Dunbar and MR Gaberdiel for explanation of their work We are also grateful to P Aspinwall, L Bonora, M Gross, C Johnson, H Skarke and Swansea theory group for a series of useful conversions and communications This work was supported by PPARC As this manuscript was in preparation Ref w7x appeared, the results of which overlap with this paper References wx C Vafa, Nucl Phys B 469 Ž wx B Andreas, Ns heteroticrf-theory duality, hep-thr wx 3 L Bonora, C Reina, A Enhan, Enhanced gauge symmetries on elliptic K3, hep-thr

246 46 ( ) D O DriscollrPhysics Letters B wx 4 R Friedman, J Morgan, E Witten, Commun Math Phys 87 Ž ; R Donagi, HeteroticrF-theory duality, hepthr wx 5 W Lerche, D Lust, AN Schellekens, Nucl Phys B 87 Ž wx 6 H Kawai, DC Lewellen, S-HH Tye, Phys Rev Lett 57 Ž ; Nucl Phys B 88 Ž 987 ; I Antoniadis, CP Bachas, C Kounnas, Nucl Phys B 89 Ž wx 7 KS Narain, MH Sarmadi, C Vafa, Nucl Phys B 88 Ž wx 8 D Bailin, DC Dunbar, A Love, Nucl Phys B 330 Ž 990 4; Int J Mod Phys A 5 Ž wx 9 P Aspinwall, hep-thr9637; B Greene, hep-thr97055 w0x M Bershadsky, K Intriligator, S Kachru, DR Morrison, V Sadov, C Vafa, Nucl Phys B 48 Ž wx E Witten, J High Energy Phys 0 Ž wx JH Schwarz, Phys Lett B 360 Ž 995 3, K Dasgupta, S Mukhi, Phys Lett B 43 Ž 998 6; B 385 Ž 996 5, A Sen, JHEP 03 Ž w3x S Chaudhuri, C Johnson, J Polchinski, Notes on D-branes, hep-thr96005 w4x A Sen, Nucl Phys B 475 Ž ; A Dabholkar, Lectures on orientifolds and duality, hep-thr w5x MR Gaberdiel, B Zwiebach, Nucl Phys B 58 Ž 998 5; MR Gaberdiel, T Hauer, B Zweibach, Nucl Phys B 55 Ž 998 7; O DeWolfe, B Zwiebach, Nucl Phys B 54 Ž ; O DeWolfe, Affine Lie algebras, string junctions and 7-branes, hep-thr980906; O DeWolfe, T Hauer, A Iqbal, B Zwiebach, Uncovering the symmetries on wp,qx 7-branes: beyond the Kodaira classification, hep-thr9808 w6x A Johansen, Phys Lett B 395 Ž w7x H Kawai, DC Lewellen, S-HH Tye, Phys Rev Lett 57 Ž w8x J Polchinski, E Witten, Nucl Phys B 460 Ž w9x P Ginsparg, Phys Rev D 35 Ž ; KS Narain, Phys Lett B 69 Ž w0x Z Kakushadze, G Shiu, S-HH Tye, Phys Rev D 54 Ž wx O Bergman, MR Gaberdiel, G Lifschytz, Nucl Phys B 54 Ž wx C Bachas, MB Green, A Schwimmer, J High Energy Phys 0 Ž w3x O Bergman, MR Gaberdiel, G Lifschytz, Nucl Phys B 509 Ž w4x C Hull, The non-perturbative SOŽ 3 heterotic string, hepthr980; C Hull, Nucl Phys B 509 Ž w5x E Bergshoeff, E Eyras, R Halbersma, C Hull, Y Lozano, JP van der Schaar, Spacetime-filling branes and strings with sixteen supercharges, hep-thr984; E Bergshoeff, JP van der Schaar, On M nine-branes, hep-thr w6x E Bergshoeff, M de Roo, B Janssen, T Ortin, The super D9-brane and its truncations, hep-thr w7x Y Imamura, String junctions and their duals in heterotic string theory, hep-thr99000

247 0 May 999 Physics Letters B A note on the superstring BRST operator Jose N Acosta, Nathan Berkovits, Osvaldo Chandıa 3 Instituto de Fısica Teorica, UniÕersidade Estadual Paulista, Rua Pamplona 45, Sao Paulo, SP, Brazil Received March 999 Editor: M Cvetič Abstract zation ghosts This provides a trivial proof that Q is nilpotent q 999 Published by Elsevier Science BV All rights reserved yr We write the BRST operator of the Ns superstring as Qse Edz g b e R p i where g and b are super-reparameteri- Introduction Superstring theory in ten dimensions is a critical N s superconformal theory It can be quantized using a nilpotent BRST operator Qs dz c T q T qg G q G, E m g m g Ž Ž p i Ž where wt,g x m m are the cs5, Ns superconformal generators and w T g,gg x are the csy5, Ns su- perconformal generators constructed from a pair of fermionic ghosts wb,cx and a pair of bosonic ghosts wb,g x Physical states are described by vertex operators in the cohomology of Q and, in order to construct vertex operators for the spacetime fermions, it is convenient to fermionize the bosonic ghosts as bsej e yf, gshe f, where h and j are free fermions and f is a chiral wx boson jose@iftunespbr nberkovi@iftunespbr 3 chandia@iftunespbr Because the above fermionization involves Ej rather than j, it is not possible to write the zero mode of j in terms of the wb,g x ghosts The Hilbert space without the j zero mode is called the small Hilbert space, while the Hilbert space including the j zero mode is called the large Hilbert space The small Hilbert space can be defined as operators annihilated by Edz h, and one can show that any such operator can be constructed out of the original wb,g x ghosts Since physical vertex operators should be in the small Hilbert space, they must be annihilated by both Q and Edz h For consistency, this requires that Q should not only be nilpotent, but should also anti-commute with Edz h In this letter, we will construct a similarity trans- yr formation R such that Q s e Ž Edz g b e R p i ŽNote that the first term of R was constructed in wx Since g b is nilpotent, this trivially proves that Q is nilpotent Furthermore, since Edzg b has trivial cohomology, it proves that Q has trivial cohomology in the large Hilbert space However, R does not commute with Edz h, soq has non-trivial cohomology in the small Hilbert space yr as expected Also, e Ž Edz g b e R p i only anticommutes with Edz h in the critical dimension, so r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

248 48 ( ) JN Acosta et alrphysics Letters B quantize the superstring when D/ 0 yr one cannot use the nilpotent e Edz g b e R p i to Similarity transformation After fermionizing the wb,g x ghosts as in Eq Ž and bosonizing jse x and hse yx, the BRST charge of Eq Ž can be written as Qs p i Edz j BRST where jbrst sc Tm ybe cye fy Ef q E xq Ž Ex qe fyx Gm ybe Ž fyx qe cqe Ž cex Ž 3 We will now show that jbrst se yr je 0 R Ž 4 where j0 sybe Ž fyx, Ž 5 yf x yf x Rs Edz cgm e e y 4 E Ž e e ce c p i Ž 6 Note that j was used in wx BRST 3 as the fermionic generator G q of a twisted Ns superconformal algebra Using Eq Ž 4, jbrst is trivially nilpotent since j0 has no poles with itself To prove Eq Ž 4, we use the expansion ` Ý yr R e j0 e s j n, jns w j ny, R x, Ž 7 n! ns0 where, for Rs Edz rž z p i, the commutator is computed following the rule jnyž y, R s Edz jnyž y r Ž z Ž 8 p i If D is the spacetime dimension of the superstring Žie G Ž y G Ž z DŽ yyz y3 q m m, the ns term of Eq Ž 7 is given by j se e G qbce cy E cqe cž 5Efy4Ex f yx 3 m 3 qc E fy3 Ef y Ex ye xq5efex, Ž 9 the ns term is given by j sct q m D E cqe cž ExyEf Ž qc E xye fq Ef q Ex yefex yf x 5 yf x ye e Gm ce cq 4 e e ce ce c, 0 the ns3 term is given by 3D yf x yf x j3 s3e e Gm ce cy e e ce ce c, Ž 4 and the ns4 term is given by 3D y f x j 4 s e e ce ce c Ž The terms for n)4 in the expansion vanish identically since the OPE between j4 and R has no single poles It is straightforward to check that j of Eq Ž BRST 3 is equal to j0qjq jq j3q j4 Ž 3! 3! 4! when Ds0, so we have proven Eq Ž 4 Note that when D/ 0, the integral of Eq Ž 3 contains the Ž Ž xy f term 0 y D Edz cefex q e ce ce c 6 Since this term does not anti-commute with Edz hs yx Edz e, the integral of Eq Ž 3 can only be used as a BRST charge when Ds0 for the reasons stated in the introduction Acknowledgements JNA would like to acknowledge financial support from FAPESP grant number 98r36-, NB would like to acknowledge partial financial support from CNPq grant number 30056r94-9, and OCh would like to acknowledge financial support from FAPESP grant number 98r References wx D Friedan, E Martinec, S Shenker, Nucl Phys B 7 Ž wx N Berkovits, Phys Rev Lett 77 Ž , hepthr9604 wx 3 N Berkovits, Nucl Phys B 40 Ž , hep-thr93089

249 0 May 999 Physics Letters B Light-cone formulation and spin spectrum of non-critical fermionic string Marcin Daszkiewicz a,, Zbigniew Hasiewicz b,, Zbigniew Jaskolski a,3 a Institute of Theoretical Physics, UniÕersity of Wrocław, pl Maxa Borna 9, Wrocław, Poland b Institute of Mathematics, UniÕersity in Białystok, ul Akademicka, 5-67 Białystok, Poland Received March 999; received in revised form 6 March 999 Editor: PV Landshoff Abstract A free fermionic string quantum model is constructed directly in the light-cone variables in the range of dimensions -d-0 It is shown that after the GSO projection this model is equivalent to the fermionic massive string and to the non-critical Ramond-Neveu-Schwarz string The spin spectrum of the model is analysed For ds4 the character generating functions is obtained and the particle content of first few levels is numerically calculated q 999 Published by Elsevier Science BV All rights reserved PACS: 5Pm Keywords: Non-critical string; Massive string; Fermionic string; GSO projection; No-ghost theorem Introduction One of the main physical motivations for string theory in four dimensions is an old and compelling idea that the low energy QCD should be equivalent to some effective string theory w, x In spite of numerous efforts the problem remains open One of possible approaches is to look for a consistent free string model with a physically acceptable spectrum Although very far from a complete answer a well developed free theory can help in posing the problem of interactions or at least in analysing kinematical requirements an interacting theory should satisfy It marcin@iftuniwrocpl zhas@alphafuwbedupl 3 jaskolsk@iftuniwrocpl may also shed new light on the structure of longitudinal excitations Their dynamics seems to be the central problem of non-critical strings In the case of the non-critical Nambu-Goto bosonic string Ž - d- 6 wx 3, and of the noncritical Ramond-Neveu-Schwarz fermionic string Ž -d-0 w4,5x the covariant quantization leads to consistent quantum models with longitudinal excitations which are not present in the classical theory The relevance of the Liouville theory for a proper description of these extra degrees of freedom was first pointed out by Polyakov in his celebrated papers on conformal anomaly in string theories w6,7 x In the context of the free string model the role of the Liouville excitations was analysed by Marnelius wx 8 He considered the standard string modified by adding the Liouville sector Under some general assumpwx tions about the Liouville dynamics he showed r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S X

250 50 ( ) M Daszkiewicz et alrphysics Letters B that the non-zero Liouville modes can be identified with the longitudinal Brower excitations of the noncritical Nambu-Goto string He also conjectured that the model is equivalent with the massive string constructed by Chodos and Thorn w0x which was latter justified within the covariant functional apw x The modification of the bosonic free string by proach Liouville sector was recently reconsidered under the assumption that the bulk and the boundary cosmow x It was shown that the logical constants vanish covariant quantization leads to a family of new non-critical string models One of them characterised by the largest subspace of null states Žand called for this property the critical massiõe string turned out to be equivalent to the non-critical Nambu-Goto string and to the massive Chodos-Thorn string This equivalence provided a new description of the longitudinal excitations It was further used in constructing a light-cone formulation which allows for instance to calculate the spin spectrum of the model w3 x In all the non-critical bosonic strings analysed in Ref wx the ground state is tachyonic One may expect that this problem can be overcome in fermionic counterparts of these models Indeed it was recently shown that the covariant quantization of the Ramond-Neveu-Schwarz string modified by adding the supersymmetric Liouville sector with vanishing cosmological constants leads in the dimensions -d-0 to a family of tachyon free unitary models w4 x For every member of this family the Neveu-Schwarz sector does not contain any massless states which justifies the name fermionic massiõe string introduced in w4 x One of the quantum models characterised by the largest subspace of null states is equivalent to the non-critical RNS spinning string truncated in the Neveu-Schwarz sector to a tachyon free subspace of the fermion parity operator It is called the critical fermionic massiõe string The aim of the present paper is to develop the light-cone formulation of this model Our motivation is twofold First, the light-cone formulation can be used to calculate the spin spectrum of the string Secondly, since the dual model recipe for amplitudes breaks down Ž the c) barrier such formulation seems to be the only available way to introduce and analyse the splitting-joining interaction of non-critical strings The paper is organised as follows In Section we define a quantum free string model directly in the light-cone variables We shall call it the fermionic non-critical light-cone string or simply the light-cone string In this model the longitudinal degrees of freedom are described by a background charge Fock space realisation w5x of the superconformal Verma module with the central charge cs0yd, ˆ and the highest weight h s A similar construction was first discussed by Marnelius in the context of the non-critical Polyakov fermionic string w6 x The GSO projection is introduced as a projection on a suitably chosen eigenspace of the Ž world-sheet fermion parity operator In Section 3 we show that after the GSO projection the fermionic non-critical light-cone string is equivalent to the critical massive string and therefore to the suitably projected Ramond-Neveu-Schwarz non-critical string In Section 4 the spectrum of the light-cone string is analysed For ds4 the expansion of the character generating function in terms of irreducible characters is derived It is illustrated by numerical calculations of spin content of first few levels The spectrum of the GSO projected tachyon free model is also calculated The corresponding results for the closed noncritical light-cone string are presented in Section 5 An interesting feature of the model is that the closed string spectrum does not contain space-time fermions The spectra of the open and the closed critical massive strings derived in this paper exclude the fundamental string interpretation of the model It might be however a good candidate for an effective low energy description of strong interactions It shares two important features of the critical fermionic string the light-cone formulation and the absence of tachyons Whether it is enough for a consistent interaction is an interesting open problem Fermionic non-critical light-cone string Let us fix a light-cone frame e,e,,e 4 " dy in d-dimensional Minkowski space normalised by e " s 0, eqpe ysy, and eipejsdij for i, js,,dy We shall use the following notation for the lightcone components of a vector V " i i V se PV, V se PV, VsV e " i i

251 ( ) M Daszkiewicz et alrphysics Letters B The fermionic non-critical light-cone string is defined as a representation of the algebra i j ij q y a 0,q0 syid, a 0,q0 si, L c 0,q0 syi, i j ij a m,an smd d m,yn, wc m,cnxsmdm,yn 4 i j ij b,b sd d, d,d 4 sd, Ž r s r,ys r s r,ys supplemented by the conjugation properties i i i i L L q q y y i i i i m ym r yr m ym r yr a sa, q sq, c sc, q sq, a sa, q sq, a sa, b sb, Ž c sc, Ž d sd, e q where m,ngz;r,sgzq The operators P s ' q i ' i y y i i a 0, P s a 0, and x s q 0, x s q0 are ' a ' a interpreted as components of the total momentum of the string, and the barycentric coordinates, respectively Ž q Let us denote by F p, p e the Fock space gener- ated by the algebra of non-zero modes Žwith negative labels out of the unique vacuum state V satisfying P i V sp i V, P q V sp q V, c V slv e e e e 0 e e The space of states is a direct integral of Hilbert spaces over the spectrum of momentum operators H dp q dy q e H e s d ph p, p < p q < In the Neveu-Schwarz sector Ž e s q q H p, p sf p, p In the Ramond sector Ž e s 0 the fermionic zero modes b0 i,d0 form the real Euclidean Clifford algebra CŽ d y,0 If one requires a well defined fermion parity operator the zero mode sector of Ž q H p, p 0 must carry a representation of the Clifford algebra CŽ d,0 We assume that this sector is described by an irreducible representation Dd of the complexified Clifford algebra C C Ž d s CŽ d,0 m C, and q q H0 p, p sf0 p, p md d e Ž Ž q 0 The representation of the algebra on H given by p, p is a i msa i mm, cmsc mm, m/0, i i F F brsbrmg, drsdrmg, r/0, i i L b0sm G, d0sm G, ' ' i i Ž q where a,c,b,d denote the operators on F p, p m m r r 0 representing the non-zero bosonic and fermionic modes, and G,,G dy,g L,G F are the gamma matrices of the Dd representation In order to construct generators of a unitary representation of the Poincare group we introduce the operators Lm s Ý :aynpa nqm:q Ý r :byrpb rqm: e ngz rgzq dq qž ye dm,0q Ý :cync nqm: 6 ngz q r :d d : Ý yr rqm e rgzq ' qi b mc qbd, Ý m Ý G s a Pb q c d q4i b rd, r yn nqr yn nqr r ngz ngz forming an N s superconformal algebra with the central charge ˆ csdyq3b wl, L xs Ž myn L q Ž dyq3b w m0 m n mqn 8 x = Ž m 3 ym d m,yn, L m,gr s myr G mqr, G,G 4 s L r s rqs ž / q Ž dyq3b r y d r,ys 4 The generators of the translations in the longitudinal and the transverse directions are given by the operators P q and P i, respectively The generator of the translation in the x q -direction is defined by a y P s Ž L ya P q 0 0 '

252 5 ( ) M Daszkiewicz et alrphysics Letters B The x q coordinate is regarded as an evolution parameter and P y plays the role of the string Hamiltonian The generators of the Lorentz group are defined by M sx P yx Pyi a a ya a n ij i j j i i j j i Ý yn n yn n n)0 0 0 Ý Ž yr r yr r q ye ib i b j yi b i b j yb j b i, r)0 iq i q qy q y y q M sx P, M s P x qx P, M s Ž P x qx P yx P iy y i i y y i i i y a L yl a q a n i Ý Ž yn n yn n 0 n)0 i i qž ye bg 0 0 q a 0 i i i y b G yg b, q Ý yr r yr r a0 r)0 The algebra of the generators P q, P y, P i, M qy, M iq, M iy, M ij closes to the Lie algebra of the Poincare group up to some anomalous terms They appear only in the commutators / iy jy w M, M xs Ý DnyD q ž n 8a 0 n)0 = Ž ayn i an j yayn j a i n ž / q Ý DyD q a r 0 r)0 = Ž byr i br j ybyr j br i, where Dsdy9q3b, Ds6a 0 ydqy3b, and vanish if and only if bs Ž 9yd 3, and a0 s The first condition implies that the operators P y, M iy are self-adjoint only in the range FdF9 The second leads to the following expression for the mass square operator l dy Me sa Re q ye, ž 6 / where R s Ý Ž a P a q c c e m ) 0 y m m y m m q Ý rž b Pb qd d r ) 0 yr rqm yr rqm is the level operator Note that in the covariant massive string model the eigenvalue l of the bosonic Liouville zero mode c 0 is restricted by the constraint c0 s0 In the present construction it is regarded as a free real parameter Ž It follows from that for l small enough the ground states in the Neveu-Schwarz sector are tachyonic One can try to solve this problem by introduc- Pb e r ) 0 yr r ing the GSO projection w7 x Let F sý b qý d d r ) 0 yr r be the fermion number operator on Ž q F p, p e We introduce the fermion parity operators on the total Hilbert space HsH 0[H : F F F F 0 q y s y mg [ y The GSO projection is defined as the projection on the q eigenspace of Ž y F In the case of even dimensions there exists another operator Q with all the properties of the fermion parity operator, and anticommuting with Ž y F One can show that the GSO projections with respect to Q, and Ž y F lead to equivalent models 3 Equivalence to other models In this section we shall show that the light-cone string is equivalent to the critical fermionic massive string recently introduced in w4 x In the covariantly quantized fermionic massive string the conditions for physical states can be solved in terms of the transverse A i m, Br i, the super-liouville C m, D r, and the shifted longitudinal A L m, Br L DDF operators For details concerning the DDF construction and the notation used in this section we refer to w4 x The critical fermionic massive string corresponds 9y d to a special choice of the parameters bs 3,m 0 s 0 In this case all states containing the shifted longitudinal excitations are null The space of physical states can be identified with all states generated by the transverse and the super-liouville DDF operai i tors A, B, C, D They have the same Ž anti m r m r com- mutation relations and the conjugation properties, as the light-cone excitations a i m,br i, c m,d r Also the continuous spectra of the bosonic zero modes in both

253 models are identical In the critical massive string the representation of the transverse, the Liouville, and the fermion parity gamma matrices on the on-massshell physical states coincides with the representation Dd The only difference is that in the covariant model one gets a neutral, while in the light-cone string a positive definite scalar product This discrepancy is not essential - the subspaces with definite products in the covariant model are eigenspaces of the the fermion parity operator w4 x Note that the neutral product of the covariant model is a consequence of the assumption that the zero and the non-zero fermionic modes have the same conjugation properties One way to show the equivalence of the Poincaré group representations is to calculate the commutators of the Poincare generators with the DDF operators This calculations can be facilitated by the technique of the leading terms w0 x It is based on the observation that the DDF operators expressed in terms of elementary excitations are uniquely determined by they leading terms ie parts of such expressions which do not contain any a q m,br q ;m/0 excitations The most tedious calculations are involved in the commutators: M iy j, A m q q 0 a0 i q0 i i j j ij L sm A ym A y d A ql q q q q a a a a ( ) M Daszkiewicz et alrphysics Letters B m m m m iž ye i j ym B 0 B m q a0 im q A A ya A q a n i j j i Ý Ž yn mqn myn n 0 n)0 im q B B qb B i j j i q Ý Ž yr mqr myr r a0 r)0 q a q M, B sr B yr B q i i iy j 0 0 j 0 j r q q r q r a0 a0 a0 i ij L y d B qg q a 0 Ž r r iž ye i j y B 0 A r q a0 ž i r q A B yb A q a n i j j i Ý Ž yn rqn ryn n 0 n) i j j i Ž yn rqn ryn n / q A B qb A i y B A qa B i j j i q Ý Ž yr rqs rys r a0 s)0 q a q iy M,C sm C ym C q i i m q q m q m a0 a0 a0 ' b iž ye i i ym A ym B D q q a a m 0 m 0 0 im i q A C yc A q a n i Ý Ž yn mqn myn n 0 n)0 im i q B D qd B i q Ý Ž yr mqr myr r a0 r)0 q i i q0 a0 q0 4' b iy i M, Dr sr Dryr Drqr B q q q q r a0 a0 a0 a0 iž ye i y q a0 ž BC 0 r i r i q A D yd A q a n i Ý Ž yn rqn ryn n 0 n) i i Ž yn rqn ryn n / q A D qd A i i i y B C qc B q Ý yr rqs rys r a0 s)0 Setting in these formulae the evolution parameter q x q 0 q s s 0, and neglecting the shifted longitudi- 0 'a nal DDF operators A L m, Br L one reproduces the corre- sponding light-cone commutators One can easily check that this is also true for all the other generators We have shown that the GSO projected fermionic light-cone string is isomorphic to the GSO projected critical massive string The latter model is equivalent to the non-critical Ramond-Neveu-Schwarz string truncated in the Neveu-Schwartz sector to the tachion free eigenspace of the fermion parity operator w4 x One thus has three equivalent descriptions of the same fermionic non-critical string model

254 54 ( ) M Daszkiewicz et alrphysics Letters B Spin spectrum The problem of the spin spectrum is to decompose the unitary representation of the Poincare group on the Hilbert space of string into irreducible representation It follows from formula Ž that the decomposition of He into representations of a fixed mass coincides with the level structure dp H q dy N q e N G 0 e < p q < H s[ d ph p, p, N q N q Re He p, p snhe p, p dy For l in the range 0Fl - 8 the lowest level subspace H 0 in the Neveu-Schwarz sector carries an dy irreducible tachyonic representation For l s, 8 0 dy H is a massless, and for l ) 8, a massive scalar representation In the Ramond sector the 0-level subspaces 0Ž q H p, p 0 are by construction isomorphic with the irreducible representation Dd of the complex Clif- C Ž ford algebra C d For a massive momentum l ) 0 the representation of the little group SpinŽ d y on Dd is a direct sum of two isomorphic fundamental irreducible representations Sdy Ž of SpinŽ dy Ž In the massless case l s0 the maximal compact subgroup of the little group is SpinŽ dy In the odd dimensions Dd is a direct sum of two fundamental irreducible representations of SpinŽd y while in the even dimensions it is a direct sum of four such representations In particular for ds4 and Ž Fig Spin spectrum of the open massive fermionic string in four dimensions, before GSO projection ds4,l s0

255 ( ) M Daszkiewicz et alrphysics Letters B Ž Fig Spin spectrum of the open massive fermionic string in four dimensions, after GSO projection ds4,l s0 l s0 the zero level in the Ramond sector contains two pairs of the left, and the right Weyl spinors Since all higher levels are massive, the spaces NŽ q H p, p e should be decomposed into irreducible representations of the little group SpinŽ dy For Ž every momentum p with m s a N q l y dy 8, N) 0, one can choose a light-cone frame q such that p s ' a, p s 0, and the little group is generated by m j jy jq ji G sm y M, M a We shall use the method developed in the case of the bosonic light-cone string w3 x It relies on the observation that, as far as the character generating function is concerned, the vector representation of Spin- Ž dy formed by the transverse excitation can be extended to a vector representation of SpinŽ dy by means of the Liouville excitations The only novelty is that in the present case we have two vector representations Vym B and Vyr F spanned by the creation operators km c ym, as0 a aym s ½ a, a ym, FaFdy kr d yr, as0 a byr s ½ a b yr, FaFdy The normalisation constants lyi' b k r ( l q4b k ( k s, k s m ' ly4i b r l q6b r

256 56 ( ) M Daszkiewicz et alrphysics Letters B Ž Fig 3 Spin spectrum of the closed massive fermionic string in four dimensions, before GSO projection ds4,l s0 are chosen in order to obtain the canonical antisym- Ži metric matrix generators D of SpinŽ d y vector representation: ( ( i a Ži a b G, Aym si l q4m b Db Aym q, i a Ži a b G, Byr si l q6r b Db Byr q The dots in the formulae above denote all terms of higher order in the excitation operators Such terms do not contribute to the character functions N The subspace He Ž ' a,0 decomposes into a direct sum of tensor products of the symmetric tensor powers of Vym B, the antisymmetric tensor powers of F V, and Dd Then using the method of w3x one yr can write the character of the SpinŽ dy representa- N tion on H Ž ' a,0 as e Ý Ý Ý Ł Ł xe N s x S m kxa m rx e 0, N qn sn p gpž N p gpž N mkgpb mrgp F B F B B F F where the sum runs over all partitions p sm 4 B k, p sm 4 F r of the bosonic N B, and the fermionic NF level number The symbols x m k, and x m r S A stand for the characters of the mk-th symmetric, and the mr-th antisymmetric tensor power of the vector representa- 0 tion of Spin d y, respectively Finally, xe is given by x 0 s x, x 0 s, 3 0 SŽ dy

257 ( ) M Daszkiewicz et alrphysics Letters B Ž Fig 4 Spin spectrum of the closed massive fermionic string in four dimensions, after GSO projection ds4,l s0 where xsž dy is the character of the fundamental irreducible representation of SpinŽ dy Using the formulae for characters of tensor products w8x one gets the character generating function N N xe Ž t, g s Ý t x Ž g s Ł k det yt D Ž g NG0 kgn Ł Ž v e e rgny Ž v = det qt r D Ž g x 0 Ž g, where Dv denotes the vector representation of SpinŽ dy, and N is the set of all positive integers The expansions of x Ž t, g e in terms of irreducible characters can be found using the techniques develw9x for strings oped by Curtright and Thorn with only transverse excitations In the case of ds4 one gets: Ž y ye 8 4 e e Ý l ey lgnq x Ž t,w s t p Ž t p Ž t x Ž w = = ky k Ý Ž y Ž yt kgn ŽkŽ ky qm mq Ý t ž yt / ey mgnq = Ž t k < lym< yt kžlqmq,

258 58 ( ) M Daszkiewicz et alrphysics Letters B where Ł n y p t s yt, ngn Ł p Ž t s Ž qt r, e e rgny and x Ž f l is the character of the spin l irreducible representation of Spin d y For l s 0 the spin spectrum up to 6-th level is presented in Fig The doubling of the spectrum in the Ramond sector Ž 3 is related to the presence of the fermionic zero mode in the Liouville sector For all dimensions in the range -d-0 the GSO projection removes this doubling without any extra conditions for the parameters of the model In the Neveu-Schwarz sector it simply removes the integer levels The GSO truncated spectrum is presented in Fig 5 Closed string The closed string Hilbert space Hc can be con- structed as the subspace of the tensor product of two copies of the open string Hilbert spaces Ž H [H 0 m Ž H [ H determined by the conditions 0 Pc i Pc q i i q q a sa s, a sa s, c sc sl, ' a ' a and anihillated by the twist operator TsŽ R [R 0 mymž R [R 0 Since the mass levels of different sectors of the open non-critical string never coincide Ž the mixed sectors H mh, H mh 0 0 are excluded In consequence the spectrum of the closed fermionic string does not contain space-time fermions This is in fact a common feature of all the covariant closed string models corresponding to the family of non-critical open strings considered in w4 x The representation of the Poincare generators are constructed in a standard manner In particular, the Hamiltonian P y generating the x q -evolution, and the mass square operator M e are given by a y P s L ql y, c 0 0 P q / dy Me s4a ž RqRql ye 8 The character generating function can be calculated as the diagonal part Žie all terms of the form N X t t N of the product of two open string generating functions Žclosed X x t, g sdiag x t, g x t, g < X e Ž e e tst For ds4, and l s0 the results of the numerical calculations of first few levels, before and after GSO projection are presented in Fig 3 and in Fig 4, respectively Acknowledgements The authors would like to thank Andrzej Ostrowski for many stimulating discussions This work is supported by the Polish Committee of Scientific Research Ž Grant Nr PB 337rPO3r97r References wx J Polchinski, Strings and QCD?, hep-thr90045 wx AM Polyakov, The wall of the cave, hep-thr wx 3 RC Brower, Phys Rev D 6 Ž wx 4 JH Schwarz, Nucl Phys B 46 Ž 97 6 wx 5 R Brower, K Friedman, Phys Rev D 3 Ž wx 6 AM Polyakov, Phys Lett B 03 Ž wx 7 AM Polyakov, Phys Lett B 03 Ž 98 wx 8 R Marnelius, Nucl Phys B Ž wx 9 R Marnelius, Phys Lett B 7 Ž w0x A Chodos, CB Thorn, Nucl Phys B 7 Ž wx Z Jaskolski, K Meissner, Nucl Phys B 48 Ž wx Z Hasiewicz, Z Jaskolski, Nucl Phys B 464 Ž w3x M Daszkiewicz, Z Hasiewicz, Z Jaskolski, Nucl Phys B 54 Ž w4x Z Hasiewicz, Z Jaskolski, A Ostrowski, Spectrum generating algebra and no-ghost theorem for fermionic massive string, Preprint IFT UWr 9r99, hep-thr9906 w5x M Bershadsky, V Knizhnik, M Teitelman, Phys Lett B 5 Ž 985 3; G Mussardo, G Sotkov, M Stanishkov, Phys Lett B 95 Ž w6x R Marnelius, Nucl Phys B Ž w7x F Gliozzi, J Scherk, D Olive, Nucl Phys B Ž w8x H Weyl, The classical groups, Princeton University Press, 946 w9x TL Curtright, CB Thorn, Nucl Phys B 74 Ž w0x P Goddard, C Rebbi, CB Thorn, Nuovo Cimento A Ž 97 45

259 0 May 999 Physics Letters B Vortex strings and nonabelian sine-gordon theories Q-Han Park, HJ Shin Department of Physics and Research Institute of Basic Science, Kyung Hee UniÕersity, Seoul 30-70, South Korea Received 8 February 999 Editor: M Cvetič Abstract We generalize the Lund-Regge model for vortex string dynamics in 4-dimensional Minkowski space to the arbitrary n-dimensional case The n-dimensional vortex equation is identified with a nonabelian sine-gordon equation and its integrability is proven by finding the associated linear equations of the inverse scattering An explicit expression of vortex coordinates in terms of the variables of the nonabelian sine-gordon system is derived In particular, we obtain the n-dimensional vortex soliton solution of the Hasimoto-type from the one soliton solution of the nonabelian sine-gordon equation q 999 Elsevier Science BV All rights reserved The relativistic motion of vortex strings in a superfluid was first modeled by Lund and Regge in 976 wx Among many vortex models, their model is distinguished in that it is an exactly integrable model and it becomes the Nambu string model in the no-coupling limit In the context of string theory, the nonvanishing coupling term also received an interpretation as describing the interaction of a string with background antisymmetric tensor fields wx Lund and Regge has proven the integrability of the model by recognizing the vortex equation as the integrability equation of Gauss and Codazzi 3 w,3 x They have identified the vortex equation with the qpark@nmskyungheeackr hjshin@nmskyungheeackr 3 Many integrable equations arise from the study of the surface embedding problem in differential geometry which provides a clear geometrical meaning to integrable equations For the modern formulation of surface embedding problem, see for example in wx 4 complex sine-gordon equation wx 5 and found the associated linear equations of the inverse scattering Since then, the complex sine-gordon theory has been studied intensively w6 0 x, with applications to nonw x Extensions to more general cases linear optics of the nonabelian sine-gordon theories have been also made by associating them with symmetric spaces wx and their properties were investigated in detail w3,4 x However, the vortex model by Lund and Regge is defined only in the 3 q -dimensional Minkowski space and the higher-dimensional model which generalizes the n q -dimensional Nambu string model is not known Even in the 3q-dimensional case, the identification of the vortex equation with the complex sine-gordon equation is not complete The variables of the complex sine-gordon equation has been given in terms of the vortex string coordinates, but the expression of the vortex string coordinates in terms of the complex sine-gordon variables is not known Since exact solutions, eg solitons and r99r$ - see front matter q 999 Elsevier Science BV All rights reserved PII: S

260 60 ( ) Q-H Park, HJ ShinrPhysics Letters B breathers, have been constructed only in the context of the complex sine-gordon equation, the explicit correspondence to the vortex string coordinates is critical in obtaining exact solutions of the vortex equation systematically using the inverse scattering method In this letter, we resolve these two problems We first present an n q -dimensional generalization of the vortex equation which reduces to the Nambu string in the no-coupling limit We identify the nq -dimensional vortex equation with the nonabelian sine-gordon equation and prove the integrability by finding the associated linear equations of the inverse scattering In doing so, we obtain an expression of vortex coordinates in terms of the variables of the nonabelian sine-gordon system Using this relation, we obtain explicitly an n-dimensional Hasimoto-type vortex soliton from the one soliton solution of the nonabelian sine-gordon equation We begin with a review of the vortex model by Lund and Regge The relativistic motion of vortices in a uniform static field is governed by the equation Ž 0 of motion in a Lorentz frame in which X st : E ye X i qce E X j E X k s0; is,,3 Ž t s ijk t s and also by the quadratic constraints: i i Ž t Ž s Ž s i Ž t i E X q E X s, E X E X s 0 Here, X m Ž s,t ;m s 0,,,3 are the vortex coordinates and s, t are local coordinates on the string world-sheet In the no-coupling limit Ž c s 0, this equation describes the transverse modes of the 4-dimensional Nambu-Goto string in the orthonormal gauge The critical step leading to the integration of the vortex equation Ž and Ž was to interpret the equation as the Gauss-Codazzi integrability condition for the embedding of a surface, ie the embedding of the string world-sheet projected down to the X 0 st hypersurface into the 3-dimensional Euclidean space, X 0 st The induced metric on the projected world-sheet is given by ds s Ž E X ds qž E XPE X ds dt s s t Ž t q E X dt, 3 or ds scos f ds qsin f dt Ž 4 by parameterizing Ž E s X s cos f, Et X s sin f according to Eq Ž The unit tangent vectors, N and N, spanning the plane tangent to the surface, and the unit normal vector N3 consisting a moving frame are given by N s Es X, N s Et X, < E X < < E X < s N3 s Es X=Et X Ž 5 < E X=E X < s t The vectors Ž N ; is,,3 i, given coordinates u,u on the surface, satisfy the equation of Gauss and Weingarten: E Ni E N l 3 ij sgiknlqlikn 3, syg LkjN i, Ž 6 E u E u k where Gik l are the Christoffel symbols and the Lij are the components of the extrinsic curvature tensor They are a set of overdetermined linear equations and the consistency of which requires the Gauss- Codazzi equation: R ijklslikljlylill jk, Lij;ksL ik; j, Ž 7 where the semicolon denotes covariant differentiation on the surface and R ijkl are the components of its Riemann tensor From Eq Ž 7, it follows that there exists a field h such that Eh Eh L scotf, Ž LqL scotf Ž 8 E u E u t k We introduce the light-cone coordinates z sžs q t r, zsž syt r and make the coordinate transformation: z zrl, z l z under which the Gauss-Codazzi equation is invariant due to its Lorentz invariance In this case, the Gauss-Weingarten equation in the spin-r representation changes into the linear equation of the inverse scattering wx 3 : y EFsy U0qlU F, EFsy V0ql V F, Ž 9

261 ( ) Q-H Park, HJ ShinrPhysics Letters B where ž / iclr4 q iehcosfrsin f yef q iehcotf U0q lusy Ef q iehcotf yiclr4 y iehcosfrsin f, ž / i yccosfrl y Ehrsin f ycsinfrl y V0 q l Vy sy 4 ycsinfrl ccosfrl q Ehrsin f The integrability equation: Ž 0 y EqU0qlU, EqV0ql Vy s0, then becomes the complex sine-gordon equation: c cosf EEfy sinfq EhEhs0, 3 sin f E cot f Eh qe cot f Eh s0 This reduces to the well-known sine-gordon equation when hs0 Even though the vortex equation as in Eqs Ž and Ž has been identified with the complex sine-gordon equation as in Eq Ž, the explicit correspondence between variables of each equations is not well understood In particular, it is not known how to write the vortex coordinates X i from the variables of the complex sine-gordon system In order to resolve this problem, and also to extend the vortex equation to the higher-dimensional case, we first consider the linear equation in Eq Ž 9 and assume that matrices U 0,U and V 0,Vy are valued in a certain Lie algebra g but otherwise arbitrary Define E y F'F Ž z, z,l l FŽ z, z,l Ž 3 El Then, using Eq Ž 9, we have y y E FsylF UF, E Fs F VyF Ž 4 l Ž Also, using Eqs 9 and, we obtain y EEFsF wu, VyxFs we F, E F x Ž 5 i i i Let FsÝa XT is,,nsdim g where T are generators of the Lie algebra g normalized by Tr i j TT sd and a is some constant Then, Eq Ž 5 ij becomes i ijk j k EE X sa f E X E X, Ž 6 where f ijk are structure constants of the Lie algebra g Note that for gfsož 3 this becomes precisely the vortex equation in Eq Ž The constraints as in Eq Ž, after the coordinate transformation z zrl, z lz, are equivalent to the condition: Tr Ž E F sl Tr Ž U sla, Tr Ž E F s Tr Ž V s a Ž 7 y l l Thus, we define the n-dimensional generalization of the vortex equation in terms of Eqs Ž 5 Ž 7 This equation is integrable in the sense that it arises from the linear Eq Ž 9 of the inverse scattering In order to better understand the model defined by Eq Ž 9 and the constraint in Eq Ž 7, we first solve the constraint by fixing U and V y By a gauge transformation, we can always set U st for some constant element Tgg satisfying Tr T sa The remaining constraint, Tr Ž V y s a, may be y solved for Vy s g Tg for some constant element T satisfying Tr T s a and an arbitrary group variable gž z, z The zero curvature condition in Eq Ž should hold for any l, that is, each coefficients of the polynomial in l should vanish Thus, the coefficients of the l and the l y terms give rise to respectively y EqU 0, g Tg s0 and EqV 0, T s0, Ž 8 which we solve for U0sg y E gqg y Ag and V0s A for some fields A and A satisfying the relation wa, T xs0 and wa, T xs0 Finally, the zeroth-order term results in y y y Eqg E gqg Ag, EqA q T, g Tg s0 Ž 9 This is precisely the nonabelian sine-gordon equawx 9 We emphasize that, at tion introduced in the Ref this stage, A and A are regarded only as background fields which commute with arbitrary constant elements T and T respectively Further specifications of these variables and their physical meanings are given below As for the field theory formulation, one can readily check that the nonabelian sine-gordon Eq

262 6 ( ) Q-H Park, HJ ShinrPhysics Letters B Ž 9 arises from the gauged Wess-Zumino-Novikov- Witten action plus a potential term: SsSWZNW Ž g qsgaugeyspot SWZNW Ž g y y sy H dzd ztr g E gg E g 4p S y y y y H Tr ž g dgng dgng dg /, p B y y Sgauges HTrŽyAE gg qag E g p y qagag yaa, y Spot s Hdzd z TrŽ gtg T, Ž 0 p where S Ž g WZNW is the usual Wess-Zumino-Novikov-Witten action The nonabelian sine-gordon model in Eq Ž 9 associated with a Lie algebra g is rather general for the practical purpose of obtaining exact solutions Thus, we make further restrictions by specifying T and T as follows; we assume that T and T belong to the Cartan subalgebra of g and the subalgebra h;g is the common centralizer of T and T, ie hsh; g: wt, hxs0, wt,hxs0 4 We also assume A and A to be valued in h so that the gauged Wess-Zumino- Novikov-Witten action, S Ž g WZNW q S gauge, becomes the GrH-WZNW action for the coset conforw5 x Note that the potential term mal field theory Spot is invariant under the H-group action so that the whole action S possesses the group H-vector gauge invariance if we treat A and A as gauge connections Moreover, we could further restrict the model by treating A and A as Lagrangian multipliers which give rise to the constraint equations, Ž Ž y y ye gg qgag ya h s0, h g y E gqg y AgyA s0 Ž Here, the subscript h denotes the projection to the subalgebra h This restricted nonabelian sine-gordon model corresponding to the coset GrH has been named as the symmetric space sine-gordonž SSSG model for the type-ii symmetric spaces w x It has been also shown that the field strength of A, A vanishes, ie F sweqa, EqA x zz Thus, we may fix the vector gauge invariance by taking A s A s 0 Note that the vortex string coordinates in Eq Ž 3 are also invariant under the vector gauge transformation: F hf for an element hz, Ž z g H This means that the vortex solution X i can be obtained from the solution of the gauge fixed Ž A s A s 0 SSSG equation: y y ye g E g q T, g Tg s0, and the constraints, y y g E g h s0, E gg h s0 3 Other types of symmetries of the vortex and the SSSG models are also interconnected For example, the symmetry of the SSSG equation under the parity transformation: g Pg, z yz, for P a constant element which anti-commutes with T and T, induces a symmetry of the vortex equation under the exchangež t l s of string world-sheet coordinates On the other hand, the transformation: FŽ l, z, z FŽ l, z, z F Ž l for a unitary element F Ž l, which leaves the linear Eq Ž 9 invariant, induces the rotational and the translational transformation a vortex such that df Ž l y y F F Ž l FF Ž l qlf Ž l Ž 4 dl If F Ž l is not unitary, eg F Ž l sfž l for some function fž l, the trace of F changes Thus we can always set the trace to zero by choosing an appropriate fž l Next, we derive a vortex solution from the one soliton solution of the SSSG equation Instead of applying the method of inverse scattering, we adopt the following Backlund transformation to derive the one soliton solution; let Ž f,f f is a solution of the linear Eq Ž 9 Then, Ž g,f g is another solution provided that ž / l ib F s q gy g f F f Ž 5 lyib l and y y y g E gyf E fyib g f, T s0, y y y ibe g f qg Tgyf Tfs0, 6

263 ( ) Q-H Park, HJ ShinrPhysics Letters B where b is a parameter of the Backlund transformation For simplicity, we take a trivial solution for Ž f, F such that f y fs, Ffsexp yltzyl Tz 7 One can easily see that this corresponds to the y straight vortex line Ff syltzql Tz In order to solve Eq Ž 6 with the trivial solution in Eq Ž 7, we use the fact that g y E g is anti-hermitian so that w y g y g, T x s 0 due to Eq Ž 6 This may be solved in terms of a Hermitian projection matrix P satisfying P sp, P sp by gscosu Pye iu Ž 8 where u is some constant parameter The linear equation now changes into Ž yp Ž Eyibe iu T Ps0, iu Ž yp ibe EyT Ps0 Ž 9 If we consider only the -dimensional projection, we may write n ) ) ij i j Ý k k ks P ss s r ss Ž 30 so that Eq 9 becomes iu iu Ž Eyibe T ss0, ibe EyT ss0 Ž 3 This can be integrated immediately to yield n iu yiu sis Ý expž ibe Tzyie T zrb u k, 3 ik ks where ui are constants of integration Finally, using ib Fg sf Ž l ž q Ž cosu Pye yi u l / exp yltzyl Tz Ž 33 = y where fž l is chosen to make F traceless, we obtain the n-dimensional vortex soliton solution of the Hasimoto-type w6 x: iblcosu Fs blsinuyl yb y y = expž ltzql Tz PexpŽ yltzyl Tz y yr yltzql Tz, Ž 34 where P is defined by Eqs 30 and 3 Now, we restrict to the case of Lund and Regge Ž i by taking gfso 3 and T ssi where si are Pauli matrices c and a as in Eqs Ž and Ž 6 are related by c sy4ia Choosing T and T by T syts yics3r4 and also with an appropriate parametrization of an SUŽ element g, one can readily see that the SSSG equation becomes the complex sine-gordon i equation in Eq The vortex coordinates X in Eq Ž are given by the components of F with the following scaling; 4i l Xisy Fiž zs Ž sqt, zs Ž syt / c l Ž 35 Then, the vortex soliton in Eq Ž 34 becomes XsRsechS cosq, XsRsechS sinq, X 3 srtanhsqs, Ž 36 where 4blcosu Rs, cž blsinuyl yb ž / ž / c b l Ss cosu Ž sqt q Ž syt, 4 l b c c l b Qs tq sinu Ž syt y Ž sqt 4 b l Ž 37 In this paper, we have extended the vortex equation by Lund and Regge to the n-dimensional case and proved its integrability by mapping the vortex equation into the nonabelian sine-gordon equation defined in association with a Lie algebra g of dimension n Through the identification, we have obtained explicit correspondence between vortex coordinates and the variables of the nonabelian sine-gordon system, and also the Hasimoto-type one soliton solution for the vortex equation Other explicit solutions can be also found through this correspondence with interesting physical implications This will appear elsewhere w7 x Acknowledgements We are grateful to K Lee for many helpful discussions This work was supported in part by the program of Basic Science Research, Ministry of

264 64 ( ) Q-H Park, HJ ShinrPhysics Letters B Education D00073, and by Korea Science and Engineering Foundation, References wx F Lund, T Regge, Phys Rev D 4 Ž wx A Zee, Nucl Phys B 4 Ž 994 wx 3 F Lund, Phys Rev Lett 38 Ž ; Ann of Phys 5 Ž wx 4 AI Bobenko, in Harmonic Maps and Integrable Systems, edited by AP Fordy, JC Wood Ž Vieweg, 993 wx 5 K Pohlmeyer, Commun Math Phys 46 Ž wx 6 BS Getmanov, JETP Lett 5 Ž wx 7 HJ de Vega, JM Maillet, Phys Lett B 0 Ž 98 30; Phys Rev D 8 Ž wx 8 I Bakas, Int J Mod Phys A 9 Ž wx 9 Q-H Park, Phys Lett B 38 Ž w0x Q-H Park, HJ Shin, Phys Lett B 359 Ž wx Q-H Park, HJ Shin, J Korean Phys Soc 30 Ž ; Phys Rev A 57 Ž ; ibid A 57 Ž wx I Bakas, Q-H Park, HJ Shin, Phys Lett B 37 Ž w3x Q-H Park, HJ Shin, Phys Lett B 347 Ž ; Nucl Phys B 458 Ž ; TJ Hollowood, JL Miramontes and Q-H Park, Nucl Phys B 445 Ž w4x CR Fernandez-Pousa, MV Gallas, TJ Hollowood, JL Miramontes, Nucl Phys B 484 Ž ; Nucl Phys B 499 Ž w5x D Karabali, Q-H Park, HJ Schnitzer, Z Yang, Phys Lett B 6 Ž ; K Gawedski, A Kupiainen, Nucl Phys B 30 Ž w6x H Hasimoto, J Fluid Mech 5 Ž w7x K Lee, Q-H Park, HJ Shin, in preparation

265 0 May 999 Physics Letters B New N s superconformal field theories and their supergravity description Andreas Karch a,, Dieter Lust b,, Andre Miemiec b,3 a Center for Theoretical Physics, MIT, Cambridge, MA 039, USA b Humboldt-UniÕersitat, Institut fur Physik, D-05 Berlin, Germany Received 5 March 999 Editor: L Alvarez-Gaumé Abstract In this note we construct a new class of superconformal field theories as mass deformed Ns4 super Yang Mills theories We will argue that these theories correspond to the fixed points which were recently found wx studying the deformations of the dual IIB string theory on AdS 5=S 5 q 999 Published by Elsevier Science BV All rights reserved Introduction The investigation of superconformal field theories has already a long history One appealing feature of superconformal models is that they often exhibit a strong-weak coupling duality symmetry Ž S-duality In a remarkable recent development it became clear that in a large class of superconformal field theories a new type of duality symmetry arises, namely they can be equivalently described by supergravity in anti-de-sitter Ž AdS spaces wx In particular, there is a correspondence between four-dimensional superconformal field theories and supergravity on AdS 5 = M 5, where M 5 is a certain five-dimensional Einstein space In the simplest case, M 5 is given by S 5, and the corresponding superconformal field theory is just Ns4 super Yang Mills with SU n gauge symme- karch@ctpmitedu luest@physikhu-berlinde 3 miemiec@physikhu-berlinde try This is nothing else as the superconformal theory which lives on the world volume of n parallel D3 branes Another well studied example for M 5 is the coset space T, which leads to a Ns superconformal gauge theory, which is the celebrated superconformal theory of D3 branes probing a conifold singularity wx 3 The prescription w4 7x of the holographic map allows for several non-trivial checks of the conjectured AdSrCFT correspondence For example, the central charge of the conformal field theory is inversely proportional to the volume of M 5 This check works very nicely for the correspondence between the coset space T, and the superconformal field theory from D3 branes at the conifold singularity Recently, deformations of the usual IIB string 5 theory on AdS = S has been studied w8,9,x 5 by analyzing critical points of N s 8 gauged supergravwx a new fixed point was found, leaving ity In Ns unbroken in the bulk Žcorresponding to Ns on the brane The information about this fixed point r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

266 66 ( ) A Karch et alrphysics Letters B that has been extracted from the supergravity description are the global symmetries and the ratio of the central charges of the undeformed UV theory and the interacting IR theory: cirrcuv s 7r3 The aim of this letter is to show that this new fixed point corresponds to a particular mass deformation of the Ns4 super-yang Mills theory We will show that our new fixed point field theories obtained by mass deforming the N s 4 theory indeed reproduce the global symmetry as well as the ratio cirrcuv from the supergravity side In section we will first briefly review the method w0x of deforming a supersymmetric field theory with a marginal operator to obtain a new class of superconformal models Then we recall the existence of a family of N s superconformal theories as mass deformed N s theories Deforming the finite N s SUŽ n theory with n flavors by a mass for the adjoint chiral multiplet leaves an N s theory with a quartic superpotential By the method of w0x it can be established that this quartic superpotential is a marginal deformation of the IR physics The superconformal theory along the fixed line parametrized by the marginal operator is precisely the superconformal theory of n D3 branes probing a conifold Then we will argue that the same arguments will basically establish that deforming the N s 4 SUSY gauge theory by a mass term to one of the adjoint chiral multiplets will lead to a one parameter family of Ns superconformal theories They can be expressed as Ns theories with two massless adjoints A and B deformed by a quartic superpotential W; Ž AB This has to be contrasted with the mass deformation of the Ns4 theories by a mass for a full hypermultiplet studied eg in w x While latter one leaves an N s theory, our deformation leaves only Ns unbroken In section 3 we will turn to the dual supergravity description provided by the supersymmetric fixed point found in wx from the deformation of the AdS 5 =S 5 supergravity While the field theory considerations presented up to this point are actually valid for an arbitrary gauge group, only the SUŽ n theories will be realized on D3 branes probes in IIB 4 4 Allowing for orientifolds, the SO and Sp examples are also accessible The new conformal theories Let us first briefly recall the construction of w0x to establish the existence of a family of new Ns superconformal theories by mass deforming a given superconformal theory The mass deformation causes a flow from the original theory in the UV to the deformed theory in the IR Specifically, start with a theory at a fixed point with a marginal operator provided by the superpotential WsgXff X, Ž and add the following mass term to the superpotential Wmass smx Via its equation of motion the heavy field can be integrated out, and one obtains the new, non-renormalizable superpotential g X Wnew sy Ž ff Ž 3 m As shown in w0x this is again a marginal operator The flow from finite Ns theories to superconformal Ns theories In this section be briefly recall how this method establishes the existence and S-duality of a family of N s superconformal theories as mass deformed finite Ns theories In the simplest case the Ns theory is SUŽ n gauge theory with n fundamental hypermultiplets, whose b-function is zero In N s language the superpotential has the form WsgQQX, 4 where X is an adjoint scalar multiplet and the n fundamental fields Q and Q originate from the hypermultiplets Now giving mass to X via Wmass s mx breaks the supersymmetry to Ns with, after integrating out X, the marginal operator g W sh QQ new, hs Ž 5 m

267 ( ) A Karch et alrphysics Letters B As discussed in w0x there is a second way to flow to the curve parametrized by this operator They consider supersymmetric quantum chromesodynamics Ž SQCMD, that is SUŽ n gauge theory with n flavors, a singlet meson field N and: m 0 WslNQQq N 5 Again there is the marginal operator Ž QQ, since b AŽ nyng gauge Q Abl and hence vanishing of the b functions only imposes one constraint on the two couplings Integrating out N from the superpotential, we generate the marginal operator with a coupling lrm 0 One interesting point is that the S-duality of the finite Ns theory, gl, translates directly into a g N s S-duality, h l, on the fixed line h parametrized by h The special point where this operator is turned off corresponds to Ns with no superpotential In SQCMD this can be achieved at m s`, in Ns bygs0 At this point the global 0 symmetry is enhanced In SQCMD there is another special point with this enhanced symmetry, m0 s 0 Here we have Seiberg s dual SUŽ n with n flavors and W s Nqq In the N s language this corre- sponds to the free magnetic theory at gs` Therefore the Ns S-duality means from the Ns point of view that the theory is selfdual under Seiberg duality This type of N s gauge theory with gauge group SUŽ n = SUŽ n, bifundamental chiral matter fields and with quartic superpotential precisely appears as the superconformal theory living on n D3 branes probing a conifold, respectively as the dual, Ž, supergravity on AdS = T T s ŽSUŽ 5 = SUŽ ruž In a T-dual brane picture w3,4x a` la Hanany Witten w5 x, the mass deformations corresponds to the rotation of one of the two NS branes by a certain angle On the other hand, the flow from Ns to Ns corresponds in the supergravity context to deforming the Ns orbifold space S 5 rz a blow up to the coset space T, 5 Keeping track of the index structure the second term in the s r r s superpotential should really read NNy r s N NN r s for n colors Similarly the Ž QQ operator will read Ž Q r Q a Ž Q s Q b a s b r Ž r a Ž s b y Q Q Q Q N a r b s Here and in the rest of the paper we will use the compact and sloppy notation Ž QQ by The flow from Ns4 theories to superconformal Ns theories By the same reasoning as in the finite Ns case we can also study the Ns4 theory This is really just a special case of a finite N s theory The matter content is just one adjoint hypermultiplet So the analysis from above applies to this case as well This has however some interesting implications So let us spell out this result that is implicit in the analysis of w0 x From the Ns point of view, the Ns4 theory provides us three adjoint chiral fields A, B and X The superpotential is just the cubic expression W s gf abc A B X Now we add the mass term for the a b c chiral field X As a result we get that any Ns theory with two adjoint matter fields A and B allows for a marginal deformation by adding the quartic superpotential g abc dec Ws f f AaBb AdB e Ž 6 m Not all values of this marginal coupling are distinct There exists an S-duality inherited from the N s 4 theory, mapping strong coupling to weak coupling This is difficult to see from the field theory point of view, but it is a direct consequence of type IIB S-duality and the AdSrCFT correspondence once the dual supergravity description is established As in the case of the finite N s theory this S-duality seems to imply a selfduality of the adjoint theories under Seiberg duality Let us add a few comments about this model: Ž i The self-duality under Seiberg duality is not in apparent conflict with the models with two adjoints discussed by w6x where always an ADE-type of superpotential is present Ž ii In contrast to the previous case we cannot reach the marginal operator Eq Ž 6 from an SQCMD description with four singlet meson fields 6 Four massive singlets would yield Ws Ž tr AB Ž 7 which can only be identified with Ž 6 if the product of two f abc symbols can be written as a product of 6 We are grateful to M Strassler for correcting us on this point

268 68 ( ) A Karch et alrphysics Letters B d ab symbols This is however only possible for SUŽ Ž iii This deformation is not the same as the deformation of the Ns4 theory by a mass term for a full hypermultiplet as eg studied in w x Latter one leaves an Ns SUSY unbroken and the mass deformation is given by one complex parameter In our case it is just one real mass parameter Ž iv As in the previous case of deforming N s models there will be again a description in terms of deformed brane configurations, now in terms of a brane box In contrast to the realization of the mass deformed Ns4 theory on the interval studied by wx where one gives a complex mass to a hypermultiplet leaving N s unbroken, the brane box naturally give the possibility to incorporate a real mass for a chiral multiplet Ž breaking down to Ns by a very similar mechanism A more detailed description of this duality will be given in w7 x 3 The dual supergravity description Following the ideas of wx one would expect these conformal field theories to have a dual supergravity description Since the field theory arises as a mass deformation of N s 4 SYM, the dual supergravity description should be a deformation of the usual IIB 5 string theory on AdS = S In w8,9,x 5 such deformations where studied by analyzing critical points of Ns8 gauged supergravity In wx a new fixed point was found, leaving N s unbroken in the bulk Ž corresponding to N s on the brane We will argue that this deformation indeed corresponds to the dual of the superconformal field theories we were studying in this paper As a first piece of evidence for our identification of the conformal N s theory obtained by mass deforming the Ns4 theory with the SUGRA soluwx let us compare the global symmetries tion of According to wx the subgroup of SOŽ 5 unbroken by the solution is SUŽ =UŽ The SUŽ in the field theory rotates the adjoints into each other The ABAB super potential is invariant The UŽ is the UŽ R symmetry of the Ns theory under which A and B both have charge r A more quantitative test is to compare the ratio of the central charges c of the undeformed UV and the deformed IR theory The UV central charge will be given just by the free field contributions The central charge of the IR conformal theory can be calculated from the anomaly of the R-charge, since they sit in the same supermultiplet This calculation can be found in great detail eg in wx 7 On the supergravity side the ratio can be calculated by comparing the volume of the undeformed SUGRA solution with that of the deformed The prediction is 7r3 Let us show that this value is reproduced by our proposed dual field theory As in wx 7 we calculate the central charge c and the axial charge a computing w8x the correlators among the energy momentum tensor T and the R-current R: E ² TTR : ; ayc Ž 8 m and, with the same proportionality factor, 9 E ² RRR : m ; 5ay3c Ž 9 6 First consider the UV theory The UV theory is Ž the unbroken Ns4 theory It has csr4p N c y To see this use the above relations We have Ž Ž N y gauginos with rs and 3P N y c c mat- ter fermions with r syr3 Žthe superpotential has to have rs, so the scalars have rsr3 and the fermions r syr3 E ² TTR: is given by the sum of all r-charges, m ² : Em TTR s Ž Nc y Pwx q3p Ž Nc y P y 3 s0 Ž 0 hence aycs0 orasc Moreover E ² : m RRR s Ž Nc y Pwx 6 6 ½ 3 c 3 5 q3p N y P y s P Ž Nc y Ž Ž hence 5ay3cs P N y and with asc we get c cuv s P Ž Nc y Ž 4

269 ( ) A Karch et alrphysics Letters B In the IR we see the mass deformed Ns the- ory, so W s ABX q X produces W sž AB, and Ž we are left with N y c gauginos with r-charge Ž rs plus P N y c matter fermions A, B with r-charge r syr Therefore ² : Em TTR s Ž Nc y Pwx qp Ž Nc y P y s0, Ž 3 and hence still asc In addition we have E ² : m RRR s Ž Nc y Pwx 6 6 ½ 3 c 5 qp N y P y 7 s P Ž Nc y, Ž Ž and hence 5ay3cscs P N y or 7 cir s P Ž Nc y Ž 5 8 Comparing with above we find cirrcuv s 7r3 as predicted by supergravity! It would be interesting to compare also the chiral spectrum of the superconformal field theory with the spectrum of the scalar Laplacian of the deformed S 5 manifold Note that the numerical value 7r3 is precisely the same as the one obtained in the related setup of the conifold as viewed as a mass deformation of the Z orbifold w7, x This lead wx to the speculation that these two theories are indeed related Here we see that they are quite distinct The reason for the matching of the numerical values is just due to the mechanism by which we deform: a finite theory with a cubic superpotential Žthe only choice in a finite theory is deformed by a mass term, giving rise to quartic superpotential while killing r3 of the fields The superpotential uniquely fixes the r-charge which in turn determines the central charge Having identified the deformation of wx leading to an Ns superconformal field theory in the dual language, one might hope that we can understand the 64 c deformations leading to the similar spirit Acknowledgements N s 0 theories in a Work partially supported by the EC project ERBFMRXCT and the Deutsche Forschungs Gemeinschaft We like to thank M Aganagic for useful discussions We are grateful to M Strassler for correcting an error in an earlier version References wx A Khavaev, K Pilch, NP Warner, New vacua of gauged Ns8 supergravity in five dimensions, hep-thr98035 wx J Maldacena, Adv Theor Math Phys Ž 998 3, hepthr9700 wx 3 IR Klebanov, E Witten, Nucl Phys 536 Ž , hep-thr wx 4 SS Gubser, IR Klebanov, AM Polyakov, Phys Lett 48 Ž , hep-thr98009 wx 5 E Witten, Adv Theor Math Phys Ž , hepthr98033 wx 6 M Henningson, K Skenderis, J High Energy Phys 9807 Ž , hep-thr w7x SS Gubser, Phys Rev D59 Ž , hepthr wx 8 L Girardello, M Petrini, M Porrati, A Zaffaroni, Novel local CFT and exact results on perturbations of Ns4 super Yang Mills from AdS dynamics, hep-thr9806 wx 9 J Distler, F Zamora, Nonsupersymmetric conformal field theory from stable anti-de Sitter spaces, hep-thr98006 w0x RG Leigh, MJ Strassler, Nucl Phys 447 Ž , hep-thr9503 wx R Donagi, E Witten, Nucl Phys 460 Ž , hepthr9500 wx E Witten, Nucl Phys 500 Ž 997 3, hep-thr w3x AM Uranga, Brane configurations for branes at conifolds, hep-thr98004 w4x K Dasgupta, S Mukhi, Brane constructions, conifolds and M theory, hep-thr9839 w5x A Hanany, E Witten, Nucl Phys 49 Ž 997 5, hepthr9630 w6x JH Brodie, MJ Strassler, Nucl Phys 54 Ž 998 4, hep-thr9697 w7x M Aganagic, A Karch, D Lust, A Miemiec, work in progress w8x D Anselmi, DZ Freedman, MT Grisaru, AA Johansen, Nucl Phys 56 Ž , hep-thr970804

270 0 May 999 Physics Letters B On running couplings in gauge theories from type-iib supergravity A Kehagias, K Sfetsos Theory DiÕision, CERN, CH- GeneÕa 3, Switzerland Received 6 February 999 Editor: L Alvarez-Gaumé Abstract We construct an explicit solution of type-iib supergravity describing the strong coupling regime of a non-supersymmetric gauge theory The latter has a running coupling with an ultraviolet stable fixed point corresponding to the Ns4 SUŽ N super-yang Mills theory at large N The running coupling has a power law behaviour, argued to be universal, that is consistent with holography Around the critical point, our solution defines an asymptotic expansion for the gauge coupling beta-function We also calculate the first correction to the Coulombic quark antiquark potential q 999 Published by Elsevier Science BV All rights reserved Introduction and computations One of the well-known vacua of the type-iib supergravity theory is the AdS 5 =S 5 one, first described in wx The non-vanishing fields are the metric and the anti-self-dual five-form F 5 The latter is given by the Freund Rubin-type ansatz, which is explicitly written as ' L Fmnrklsy e mnrkl, m,n,s0,,,4, ' L Fijkpqs e ijkpq, i, j,s5,,9, Ž and is clearly anti-self-dual This background has received a lot of attention recently because of its kehagias@mailcernch sfetsos@mailcernch conjectured connection to Ns 4 SUŽ N super- Yang Mills Ž SYM theory at large N w,3 x The SYM coupling g YM is given, in terms of the dilaton F, as g YM s4p e F, and the t Hooft coupling is g sg N, where NsH 5 F is the flux of the H YM S 5 five-form through the S 5 The dilaton is constant in this background, which is related to the finiteness of the Ns4 SYM theory In order to make contact with QCD, it is important to investigate deformations of the SYM theory that break conformal invariance and supersymmetry In this case, the couplings are running corresponding to a non-constant dilaton in the supergravity side It is then clear that the background we are after is a perturbation of AdS 5 =S 5 Attempts to find supergravity backgrounds that allow a non-constant dilaton, and hence a running coupling of the YM theory, have been exploited within type-0 theories w4 6 x r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

271 ( ) A Kehagias, K SfetsosrPhysics Letters B Deformations of the Ns4 theory, which flow to an interacting conformal fixed point, have been considwx 7 ered in The purpose of the present paper is to show that running couplings are also possible within the type- IIB string theory We will study the minimal case, that is we will keep the same F form as in Ž 5 and turn on a non-constant dilaton We will show that such a solution, which breaks supersymmetry and conformal invariance, exists We will assume for the metric four-dimensional Poincare invariance ISOŽ,3, since we would like a gauge theory defined on Minkowski space time In addition, we will preserve the original SOŽ 6 symmetry of the AdS 5 = 5 S As a result, the ISOŽ,3 =SOŽ 6 invariant tendimensional metric is of the form 3 ds sgmn dx m dx n qgijdx i dx j, where Ž a g dx m dx n sv r dr qdx dx a, mn as0,,,3, and gij is the metric on S 5 The dilaton, by ISOŽ,3 =SOŽ 6 invariance, can only be a function of r The supergravity equations turn out to be RmnsyLgmnq EmFE nf, ' mn Em Ž ygg En F s0, Ž 3 ' yg and R s Lg Ž 4 ij ij The above equation is automatically solved for a five-sphere of radius r' L and a first integral of the dilaton equation in Ž 3 is ErFsAV y3, Ž 5 where A is a dimensionful integration constant 3 Supersymmetric solutions to type-iib supergravity which, however, do not preserve the Poincare invariance in the brane world-volume have been found in wx 8 Moreover, the non-zero components of the Ricci Ž tensor for the metric are Rrrsy4Er lnv, ab abž r Ž r / R syh E lnvq3 E lnv, Ž 6 Ž and the first equation in 3 reduces to solving A L y6 Er lnv s V q V Ž The solution of the above equation for V as a function of r is given implicitly, in terms of a hypergeometric function, by 6L V F,, ;y V s" < A< 8 8 ( 8 Ž ryr 0, ž A / Ž 8 where r0 is another constant of integration The different overall signs in the right-hand side of Ž 8 arise from taking the square root in Ž 7 We impose the boundary condition that the space described by Ž q becomes AdS5 when r 0 That means that the conformal factor should assume the form V, Rrr for small r, where R' Ž 4p g s N r4 Using well-known formulae for the hypergeometric functions, we see that this naturally leads to the choice of Ž the minus sign in 8 and also fixes Ls4rR In addition, the constants A and r are related by h r 0 ( 3r8 < A< sr, 3 G Ž 3r8 G Ž r8 h',87 Ž 9 8p 4 Ž Then 8 can be written as ž / Vr 3 4 Vr ž F,, ;y R / 8 8 h 8 ž R / / r 3 4 s( 8 h ž y Ž 0 We also find that r 0 ž / r3 R r r6 4r3 y V, Ž 3r8 h y r0 r0, as r r 0 Ž 0

272 7 ( ) A Kehagias, K SfetsosrPhysics Letters B We may solve the dilaton equation in Ž 5 close to rs0 and rsr 0 The result for the string coupling is h r h r F e sgs qs ž q ž 4 r0 3 r0 / ž / r ž / / h q qs q, as r 0, 3456 r 0 Ž where ss"ssignž A, 4 ž / while on the other side r r y FsF 0ys Ž 8r3 ln y, as r r 0 r 0 Ž 3 The form of the above solution is dictated by the fact that, in the limit r0 `, the dilaton should be F F e s g ' e 0 and Ž and Ž 3 s should coincide Now, at rsr0 there is a singularity that may be easily seen by computing the Ricci scalar using Ž 3 m The latter is Rs EmFE F, which at r,r0 be- haves as R; Ž r 0 yr y8r3 Hence, we may consider our solution only as an asymptotic expansion around the AdS5 geometry at rs0 of the form ` 8 n r ž Ý n ž / / 0 ns R VŽ r s q a, r-r, Ž 4 r r 0 where the coefficients an are computed using the series representation for the hypergeometric function in Ž 0 for large V The first coefficient of the expansion turns out to be a syh 8 r43,y035 Hence, VŽ r is given by 8 8 ž ž / / R h r VŽ r, y Ž 5 r 43 r 0 to a very good approximation, except when r takes values very close to r Then we may use, to a very 0 4 The solutions corresponding to the two possible choices for s are related by an S-duality transformation and correspond to different gauge theories This reflects the fact that, except for r s`, corresponding to the Ns4 SUŽ N 0 SYM at large N, our solution describes gauge theories that are not S-duality-invariant Also in Ž we have used Ž 4 below Fig Plot of VŽ r rr in units where r s Curves Ž and Ž 0 were plotted using Ž 5 and Ž respectively The curve corresponding to VŽ r rr, obtained by numerically solving Ž 8, coincides with the union of these curves good approximation as well, Ž instead of Ž 5, and the results are plotted in Fig Note that our analysis was done in the Einstein frame and it is not difficult to translate everything into the string frame by multiplying the Einstein metric by e F r Our solution is singular at r s r0 in both frames and can, indeed, be trusted away from that point Note that, the metric Ž with the conformal factor V specified by Ž 0 can be written in horospherical coordinates 5 Žr, x a determined by VŽ r d r s dr, as g dx m dx n sdr qk Ž r dx dx a, Ž 6 mn where 4Ž r0 yr ' y r 0 r R r KŽ r s Re sinh Ž 7 R In the same coordinate system the string coupling takes the form ž // a ž / sa Ž r0 yr F e sg coth a' ' s 6 r4 Ž 8 ž R In the rest of the paper we prefer to work in the Ž r, x a coordinate system 5 We thank AA Tseytlin for comments on this point

273 ( ) A Kehagias, K SfetsosrPhysics Letters B Running coupling In the AdSrCFT scheme, the dependence of the bulk fields on the radial coordinate r may be interpreted as energy dependence In fact, it is a general feature in the AdSrCFT scheme that long Ž short distances in the AdS space correspond to high Ž low energies in the CFT w,9 x In particular, if the dilaton in the supergravity side is a function of r, then the t Hooft coupling of the boundary CFT has an energy dependence and can be interpreted as the running coupling of the CFT Running coupling means of course that we are away from conformality; thus, backgrounds that admit non-constant dilaton correspond to non-conformal field theories As long as supersymmetry is unbroken, spin-sum rules for the AdS supersymmetry are expected to protect the t Hooft coupling g H of the boundary Ns 4 YM theory against running However, if supersymmetry is broken, there are then no more cancellations between fermionic and bosonic contributions leading to the running of g H The specific background we found here clearly breaks supersymmetry, and A M ) r ) dls g EMFe s ge, dcmsdme 3 V Ž 9 are the associated fermionic zero modes If we now follow the correspondence between longdistancesrhigh-energies in the AdSrCFT scheme, we find that the dual theory of the supergravity solution we obtained has a coupling with power-law running Indeed, by changing the variable rsr ru and interpreting U as the energy of the boundary field theory, we find from Ž the running of g H : h 4 R 8 h 8 R 6 ) g Hsg H qs q ž r0u 8 r0u 7h R 4 qs q, Ž r U / 0 where g H ) sr is the UV value of the t Hooft coupling Ž the result in plotted in Fig From this expression, it may easily be found that the behaviour ) Fig Plot of g r g as a function of U using Ž 0 H H of the beta-function for the t Hooft coupling around g ) is H dg H g Hyg H ) ) U sy4ž g Hyg H y ) du g H 3 H H ) 4 ) Ž H H 4 g yg ) y qo g yg 7 g H Ž However, the above equation does not specify the beta-function, but rather its derivative at the g Hsg H ) point The reason is that our solution breaks down at energies U;R rr From Ž we see that 0 b X g ) sy4, H which means that g YM ) is a UV-stable fixed point 6 We believe that is universal, namely, that it is valid for all models that approach AdS 5 =S 5 at some boundary This can be seen by recalling that, 5 near AdS = S, the dilaton always satisfies Ž 5 3 with VsRrr As a result, F will behave as e FyF 0 ;r 4, where F 0 is the value of F at rs0 We also see that Ž determines the second and third derivatives of the coupling beta-function at the fixed point, 6 Using a radiousrenergy relation in horospherical coordinates of the form Us R e yr r R we find that the running of g is ž ž / ž / / H q y H a H a ) H ) ) H H dg g g U sy ag y Ž 3 du g g However, the above expression is trustable only around the fixed point g ) H H

274 74 ( ) A Kehagias, K SfetsosrPhysics Letters B which, however, are not expected to be model-independent Let us also note that there is no known perturbative field theory with UV-stable fixed points A behaviour of the form Ž 0, namely power-law running of the couplings, was also found in type-0 theories Žsee second ref in wx 6, in gauge theories in higher dimensions w0x and extensively discussed in gauge-coupling unification in theories with large inw 4 x In this scenario, the inter- ternal dimensions nal dimensions are shown up in the four-dimensional theory as the massive KK modes These modes can run in the loops of the four-dimensional theory, giving rise to a power law running of the couplings In particular, for d large extra dimensions and for energies E above the infrared cutoff, which is specified by the mass scale m of the extra dimensions, we find, just for dimensional reasons, that the running coupling constant of the effective four-dimensional Ž4 Ž4 theory is of the form g yg 0 ; mre d r, where g0 Ž4 is the bare coupling Thus, in our case, since we have a four-dimensional theory coming from ten dimensions, we should expect the coupling to run in the sixth power of E Instead, we find here that the coupling depends on the fourth power of U s E, indicating that when holography is involved, we get a softer running of the couplings It is possible to identify those operators that are responsible for the running of the coupling in the boundary field theory Since the dilaton approaches a constant value at r s 0 and the asymptotic background is an AdS5 space, the corresponding boundary field theory is expected to be a deformation of the Ns 4 SUŽ N supersymmetric YM theory The explicit form of the deformation may be specified by recalling that our solution still has an SOŽ 6 symmetry There are not many SOŽ 6 singlets in the spectrum of the S 5 compactification In fact, from the results in w5x we see that the only scalar singlets are the complex scalar of type-iib theory B, aijkl and h i i, with masses m Bs0, m asm hs3 in AdS-mass units Since we have not perturbed the five-form F 5, a s0 and thus the perturbations we have turned ijkl on are the real part of B and h i From their masses i we find that the former corresponds to marginal deformations of the type F, while the latter corremn sponds, to the dimension-eight operator F 4, which mn is irrelevant However, it gives contributions to the boundary field theory since we have an IR cutoff specified by r 0 Sending r0 ` all bulk perturbations disappear and, similarly, the boundary field theory turns out to be the Ns 4 large-n SUŽ N SYM theory 3 The quark antiquark potential The breaking of the superconformal invariance of the Ns4 theory by our solution should be apparent in the expression for the quark antiquark potential, which we now compute along the lines of w6,7 x We will find corrections to the purely Coulombic behaviour, which, on purely dimensional grounds, we expect to be in powers of AL 4, where L is the quark antiquark distance We are eventually interested in the first such correction, which, as is apparent when comparing Ž 4 with Ž, is due to the dilaton, but for the moment we keep the formalism general As usual, we have to minimize the Nambu Goto action M N S s Hdt ds( detž GMNEa X Eb X, Ž 4 p where GMN is the target-space metric in the string frame For the static configuration x 0 st, x 'xs s, and x, x 3 as well as the coordinates of S 5 held fixed, we find that Ž 4 becomes Žwe use the notation of w6x ( T Q r 4 4 H x Q F 4 ` R ž / 8 n ns n r 0 U Ss dx e Ž E U qu rr, Ž 5 p where e 'e S and SsqÝ a is the function multiplying Rrr in Ž 4 Žrewritten using rsr ru It is clear that any background that approaches AdS5 will always have a S of the form Ž 4, so that our analysis is quite general at this point It is easy to see that the solution is expressed as U R du xs H, Ž 6 QyQ U e U ru y U 0 ( 0 where U0 is the smallest distance of the trajectory to the center and Q 0 is the value of the function Q evaluated at U 0 We assume that one of the branes is taken out to Us` and that the string configuration starts and ends at this brane The rest of the branes are located at UsR rr 0 Setting xslr corre-

275 ( ) A Kehagias, K SfetsosrPhysics Letters B sponds to Us` In turn, this gives a condition that relates U to L as 0 RdU Ls Ž 7 U e U ru y ` H U ( QyQ Proceeding in a standard fashion, we substitute back our solution Ž 6 into the action Ž 5 and obtain an integral that is infinity This is because we have included into the potential energy the Ž equal masses of the infinitely heavy Žin the supergravity approximation quark and antiquark In order to compute these masses, we assume that N-branes are at Us R rr 0 and at UsUmax that is assumed to be large but finite The mass of a single quark is computed if 0 in 5 we consider a configuration with x s t, U s s and with fixed spatial world-volume coordinates x a, as,,3, as well as S 5 coordinates Then the self-energy of the quark is U max Q r H E s due Ž 8 self p R rr 0 Subtracting off this energy twice and letting Umax `, we obtain a finite result for the quark antiquark potential given by 4 Q r ` U ru0 e H p U 4 Q 0yQ 0 (U ru0 ye E s du qq ` Q r y H due Ž 9 p R rr 0 At this point we have to solve Ž 7 for U0 as a function of L and substitute the result back into Ž 9 to obtain Eqq as a function of L only This can be done perturbatively in powers of Lrr 0, and we are interested in the first correction to the Coulombic law behaviour of the potential As explained, the correction due to the non-constant dilaton is dominant and we will therefore use V s Rrr, which corresponds to AdS =S 5 for the string metric, 5 whereas for the string coupling we will keep the first two terms in Then we find that 4 R h s h L L 8 h / ž r / 0 U, q, 0 ž ž / p r G Ž 3r4 h s,0599, Ž 30 G Ž r4 where the value of the numerical constant h has been given in Ž 9 and Umax R h 4 E self, y ys Ž 3 ž p pr 4 0 / The result for the quark antiquark potential is 4 4 h R s h L p L 8 h / ž r / 0 E,y y qq ž ž / ž / R h 4 q ys Ž 3 pr 4 0 We see that the Coulombic potential receives a correction proportional to L 3 due to the breaking of conformal invariance 7 This is remarkably similar to the potential obtained in w8x for the quark antiquark pair for Ns 4, at finite temperature, using the near-horizon supergravity solution for N coincident D3-branes However, in that case supersymmetry is broken by thermal effects, whereas in our case it is broken, by the presence of a non-trivial dilaton, even at zero temperature The last term in Ž 3 does not depend on L and represents a constant shift of the potential energy As a final remark we note that the computation of the potential for the monopole antimonopole pair proceeds along the same lines as that for the quark antiquark pair, with the only difference that we start, similarly to w9 x, with the action for a D-string This means that the integrand in Ž 5 should be multiplied yf Q by e Hence, the function e entering into Ž 5 is defined as e Q se yf S 4 Consequently, the first correction to the Coulombic behaviour of the monopole antimonopole potential is given by 4 4 / ž / s 0 ž / h R s h L E mm,y q p g L 8ž h r ž / R h 4 q qs, Ž 33 pr 4 0 which is the same as Ž 3 after we use the fact that under S-duality g rg and s ys Hence we s 7 Supergravity is valid when r < r 0, which means that U G U 4 R r r Using the leading term in Ž 7 0 0, we deduce that Lr r <, which is indeed the condition for the validity of Ž 3 0 s

276 76 ( ) A Kehagias, K SfetsosrPhysics Letters B see a screening Ž antiscreening of the quark antiquark pair for ssqž y and exactly the opposite behaviour for the monopole antimonopole pair Acknowledgements We would like to thank K Dienes, E Dudas, T Gherghetta, AA Tseytlin and A Zaffaroni for discussions References wx JH Schwarz, Nucl Phys B 6 Ž wx J Maldacena, Adv Theor Math Phys Ž 998 3, hepthr9700 wx 3 SS Gubser, IR Klebanov, AM Polyakov, Phys Lett B 48 Ž , hep-thr98009; E Witten, Adv Theor Math Phys Ž , hep-thr98050 wx 4 IR Klebanov, AA Tseytlin, D-branes and dual gauge theories in type 0 strings, hep-thr98035, Asymptotic freedom and infrared behaviour in the type 0 string approach to gauge theories, hep-thr98089; A non-supersymmetric large-n CFT from type 0 string theory, hep-thr9900 wx 5 JA Minahan, Glueball mass spectra and other issues for supergravity duals of QCD models, hep-thr9856; Asymptotic freedom and confinement from type 0 string theory, hep-thr wx 6 G Ferretti, D Martelli, On the construction of gauge theories from non-critical type 0 strings, hep-thr9808; E Alvarez, C Gomez, Non-critical confining strings and the renormalization group, hep-thr9900 wx 7 J Distler, F Zamora, Non-supersymmetric conformal field theories from stable Anti-de sitter spaces, hep-thr98006; L Girardello, M Petrini, M Porrati, A Zaffaroni, Novel local CFT and exact results on perturbations of Ns4 super Yang-Mills from AdS dynamics, hep-thr9806; A Karch, D Lust, A Miemiec, New Ns superconformal field theories and their supergravity description, hep-thr99004 wx 8 M Cederwall, U Gran, M Holm, BEW Nilsson, Finite tensor deformations of supergravity solitons, hep-thr 9844 wx 9 L Susskind, E Witten, The holographic bound in Anti-de Sitter space, hep-thr98054; AW Peet, J Polchinski, UVrIR relations in AdS dynamics, hep-thr98090 w0x TR Taylor, G Veneziano, Phys Lett B Ž wx I Antoniadis, Phys Lett B 46 Ž wx E Witten, Nucl Phys B 47 Ž , hep-thr960070; JD Lykken, Phys Rev D 54 Ž , hep-thr w3x KR Dienes, E Dudas, T Gherghetta, Phys Lett B 436 Ž , hep-phr ; Nucl Phys B 537 Ž , hep-phr98069 w4x C Bachas, JHEP 3 Ž , hep-phr w5x M Gunaydin, N Marcus, Class Quant Grav Ž 985 L; HJ Kim, LJ Romans, P van Nieuwenhuizen, Phys Rev D 3 Ž w6x J Maldacena, Phys Rev Lett 80 Ž , hep-thr w7x S-J Rey, J Yee, Macroscopic strings as heavy quarks in large N gauge theory and Anti-de-Sitter supergravity, hep-thr w8x A Brandhuber, N Itzhaki, J Sonnenschein, S Yankielowicz, Phys Lett B 434 Ž , hep-thr w9x DJ Gross, H Ooguri, Phys Rev D 58 Ž , hep-thr98059

277 0 May 999 Physics Letters B Surviving on the slope: Supersymmetric vacuum in the theories where it is not supposed to be G Dvali a,b, M Shifman c a Physics Department, New York UniÕersity, New York, NY 0003, USA b International Center for Theoretical Physics, Trieste, I-3404, Italy c Theoretical Physics Institute, UniÕersity of Minnesota, Minneapolis, MN 55455, USA Received February 999; received in revised form April 999 Editor: M Cvetič Abstract In supersymmetric models with the run-away vacua or with the stable but non-supersymmetric ground state there exist stable field configurations which restore one half of supersymmetry and are characterized by constant positive energy density We call these solutions vacua since they are stable with respect to local deformations of the fields The energy-momentum tensor for these configurations is proportional to diag,y,y,04, ie they break Ž spontaneously the Lorentz and rotational invariances The formal foundation for such vacua is provided by the central extension of the Ns superalgebra with the infinite central charge q 999 Published by Elsevier Science BV All rights reserved In Ref wx we found a class of unconventional solutions, which exist in supersymmetric theories with a vacuum moduli space, and are characterized by Ž i constant energy density; Ž ii topological stability They can be considered as a limiting case of the domain walls Žsometimes we deal with the so-called constant phase configurations, sometimes with the winding phase configurations, see Section 5 of wx One half of supersymmetry may or may not be preserved on these solutions Because of their topological stability, they can become vacua of a theory breaking a part of the Lorentz invariance and supersymmetry Thus, this is a particular realization of the dynamical compactification, the idea central in Ref wx The existence of the topologically stable field configurations Žthey will be referred to as vacua in what follows with the purely gradient Ž constant energy density is intuitively clear in the examples considered in Ref wx since in these examples there exist moduli forming a continuous manifold of supersymmetric vacuum states Being physically inequivalent, these vacua are degenerate, the vacuum energy density vanishes The vacuum moduli spaces occur frequently in supersymmetric theories Here we discuss another class of supersymmetric theories, which, within the standard understanding, have no supersymmetric vacua at all: either there is no vacuum whatsoever Žthe scalar potential has a run-away behavior, or the vacuum state exists but is non-supersymmetric In other words, the conventional vacuum manifold is the empty one Our solutions are stable and restore a part of supersymmetry The pattern we discover is quite unusual r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

278 78 ( ) G DÕali, M ShifmanrPhysics Letters B normally, the stable field configurations with higher energy density have less supersymmetry than the ground state Our analysis shows that the opposite situation is also possible First, we will consider the so-called run-away theories The most well-known example of this type is SUŽ SQCD with one massless flavor wxž in general, SUŽ N SQCD with Ny flavors At any finite values of fields, the minimal energy is not achieved One approaches the vanishing energy density at infinitely distant points in the space of fields So, there is no vacuum in the conventional sense of this word Then, we will consider models with the spontaneous breaking of supersymmetry of the O Raifeartaigh type wx 3 Both phenomena are quite common in the zoo of supersymmetric theories The O Raifeartaigh models appear as a low-energy limit of various gauge models producing the dynamical supersymmetry breaking Žfor a recent review see wx 4 We will show that in both cases BPS saturated solutions exist; they are stable under all localized perturbations, preserve one half of the original supersymmetry and, thus, present supersymmetric vacuum states It may well happen that such solutions in the future will become a component of a phenomenologically successful scenario Žeg wx 5 Let us start from the run-away vacua Many models with the run-away vacuum were considered in the literature For definiteness we focus on models with the logarithmic superpotential for the moduli, H ž H / Ls 4 d u d uffq d u Wqhc where WsyiM 3 lnf The parameter M can be always chosen to be real The scalar potential V is proportional to < f < y Ža mountain peak centered at the origin in the space of fields The run-away behavior is obvious For the wall-like solutions Žie the static field configurations depending only on one coordinate z the condition of the BPS saturation takes the form wx Ef E W s Ž 3 E z Ef It is quite obvious that, given the superpotential Ž, the solution of Eq Ž 3 of the winding-phase type is M 3 i a Ž z f0 z sme, a z s z 4 m For convenience we assumed m to be real; its absolute value is arbitrary Žone can always pass to a complex m by a phase rotation of F The solution Ž 3 preserves two out of four supercharges The energy functional can be written as 3 ½ Ef M Ef Ef 3 EsH d x qi q q E z f E x E y ž /5 E W q qhc Ž 5 E z The corresponding energy-momentum tensor is M 6 umn se diag Ž,y,y,0, Es, Ž 6 m implying a constant vacuum energy density E Needless to say that the solution we present breaks Ž spontaneously the Lorentz and rotational symmetries Eq Ž 5 explicitly demonstrates that the system is stable under the spatially localized perturbations Indeed, if fsf0 qdf, and df vanishes at infinity, ½ 3 E Ž df M E Ž df 3 desh d x yyi df q E z f E x 5 E Ž df q Ž 7 E y There are no negative modes Thus, we get a continuous family of vacua with a constant energy density labeled by the parameter m It is worth noting that as m ` the vacuum energy density E tends to zero the BPS solutions we present allow one to approach the vanishing vacuum energy arbitrarily closely Another way to understand the stability is by compactifying the coordinate z on a circle of the radius R and then taking the limit R ` For finite R only a discrete number of solutions is allowed 3 3 M rm snrr ns,, Thus, M rm is a topologically conserved winding number density which

279 ( ) G DÕali, M ShifmanrPhysics Letters B guarantees the stability of the configuration Now, taking the limit R,n ` with nrr fixed we recover Eq Ž 4 Now, let us discuss a model presenting a classic example of the spontaneous supersymmetry breaking Žthe O Raifeartaigh mechanism wx 3 It includes three chiral superfields, F, with the superpotential,,3 WslF F ym qmf F Ž Again, it is convenient to choose the parameters l, M and m real Superpotential of the type Ž 8 appear in the low-energy limit of various gauge field theories with matter If M -m ržl the minimum of energy is achieved at fsf3s0 and f undetermined At the minimum the F term does not vanish, F sl M, so that supersymmetry is bro- ken, and the vacuum energy density Esl M 4 Note that the flat direction along f is lifted by the Z factor arising as a Ž perturbative quantum correction to the kinetic term Thus, in the flat vacuum SUSY is totally broken Instead, one can try to find a BPS saturated wall-like solution preserving one half of SUSY The BPS saturation conditions now take the form Efi E W s, is,,3 Ž 9 E z Ef i They have an obvious solution fsylm z, fs0, f3s0 Ž 0 The vacuum energy density for this field configuration is Es l M 4, Ž ie twice higher than in the Lorentz-invariant nonsupersymmetric vacuum Nevertheless, the configuration Ž 0 is absolutely stable under all localized deformations, much in the same way as in the case of the winding phase configuration of the previous example In both cases the residual one half of SUSY guarantees that the fermion-boson degeneracy persists for the excitation modes in the given backgrounds In the latter case, Eq Ž 0, the excitation modes from F 3 are localized in the z direction The total vacuum energy gets no quantum corrections due to the BPS-saturated nature of the wall-like solutions considered However, the z-independence of the vacuum energy density is lifted, generally speaking, by quantum corrections to the kinetic term In the weak coupling regime these quantum corrections are small, however The mathematical foundation for the existence of the spatially delocalized vacuum configurations with the residual supersymmetry and a Ž classically constant energy density, which we present here, is the central extension of the Ns superalgebra with an infinite central charge, Qa Qb4sSab Z Ž where Sab is proportional to the area tensor in the plane perpendicular to the z direction, and 4 Zs W Ž zsl yw Ž zsyl Aconst L ` Ž 3 in both models considered This is a natural generalization of the central extensions of the Ns superalgebra with a finite value of the central charge found and discussed previously wxž 6 Note that when we speak of the finiterinfinite central charge we do not include in Z a trivial area factor Sab A A, which is, of course, infinite since the wall area A ` For a recent discussion of a general theory of the tensorial central charges in various superalgebras in three and four dimensions see wx 7 In the examwx 6 the walls interpolate be- ples discussed in Ref tween a discrete set of vacua related to each other by phase transformations Therefore, the central charge can take one of several possible Ž finite values from a discrete finite set Whereas in the present case there exists a symmetry of the model per se, orof the vacuum state, under which the superpotential W gets a shift This explains why the central charge is infinite In the non-supersymmetric context stable solitonlike vacua in the theories without the Lorentz-inwx 8 and, more variant vacua were discussed in Ref recently, in Ref wx 9, where the question was raised as to the relevance of such configurations in the cosmological setting In supersymmetric world the models with no supersymmetric vacuum are abundant The vacua of the type we discuss here may play an important role in the description of the cosmology emerging, in particular, in the context of

280 80 ( ) G DÕali, M ShifmanrPhysics Letters B the TeV Planck scale scenario w0 x First ideas in this direction will be presented in Ref wx 5 Acknowledgements This work was supported in part by DOE under the grant number DE-FG0-94ER4083 References wx G Dvali, M Shifman, Nucl Phys B 504 Ž wx I Affleck, M Dine, N Seiberg, Nucl Phys B 56 Ž wx 3 L O Raifeartaigh, Nucl Phys B 96 Ž wx 4 E Poppitz, S Trivedi, Dynamical Supersymmetry Breaking, hep-thr wx 5 G Dvali, M Shifman, to be published wx 6 G Dvali, M Shifman, Phys Lett B 396 Ž ; B 407 Ž Ž E ; A Kovner, M Shifman, A Smilga, Phys Rev D 56 Ž ; B Chibisov, M Shifman, Phys Rev D 56 Ž wx 7 S Ferrara, M Porrati, Phys Lett B 43 Ž wx 8 E D Hoker, R Jackiw, Phys Rev D 6 Ž ; Phys Rev Lett 50 Ž ; E D Hoker, D Freedman, R Jackiw, Phys Rev D 8 Ž wx 9 I Cho, A Vilenkin, hep-thr w0x N Arkani-Hamed, S Dimopoulos, G Dvali, Phys Lett B 49 Ž ; I Antoniadis, N Arkani-Hamed, S Dimopoulos, G Dvali, Phys Lett B 436 Ž

281 0 May 999 Physics Letters B k-fractional spin through quantum algebras and quantum superalgebras M Mansour a, M Daoud a,b, Y Hassouni a a Laboratoire de Physique Theorique, Faculte des Sciences, BP 04, Rabat, Morocco b Departement de Physique, Faculte des Sciences, BP 8rS, UniÕersite Ibnou Zohr, Agadir, Morocco Received 9 December 998; received in revised form 9 March 999 Editor: R Gatto Abstract The decomposition of a Q-deformed boson, in the Q qse p i r k limit, is discussed The equivalence between a Q-fermion and a conventional one is given The properties of quantum algebras A n, B n, Cn and Dn and quantum superalgebras Am,n, Bm,n, Cnq and Dn,m in the limit when their deformation parameter Q goes to a root of unity, are investigated These properties are seen to be related to the fractional supersymmetry and k-fermionic spin q 999 Published by Elsevier Science BV All rights reserved Introduction Due to the intensive developments of the quantum inverse problem method and the Yang-Baxter equation, a new algebraical objects have been discovered, which were called quantum groups, Žor quantum algebras The quantum analogous of simple Lie algebras have been worked out in the works of Drinfeld wx and Jimbo wx The quantum analogue of Lie superalgebras has been constructed in wx 3 The representation theory of quantum Ž super algebras has been also an object of intensive studies Available are the results for the oscillator representation of quantum algebras and quantum superalgebras the latters are obtained w4,5x through consistent realizations involving deformed bose and Fermi operators wx 6 Recently, in connection with quantum group theory, a new geometrical foundations and a braided interpretation of the fractional supersymmetry has been developed in Refs wx 7 In these works, the authors show that the one-dimensional superspace is isomorphic to the braided line when the deformation parameter goes to a root of unity Fractional supersymmetry is identified as translational invariance along this line In the limit qse p i r k the braided line algebra wx 8 can be separated into two parts, one described by a generalized grassmann variable and the other by an ordinary even variable Similar techwx 9, to show how internal niques are used, in the Ref spin arises naturally in certain limit of the Q-deformed angular momentum algebra U ŽslŽ Q Indeed, using Q-Schwinger realization, it is shown that the decomposition of the U Žsl Q into a direct product of undeformed Usl Ž and U Žsl q which is the same version of U Ž sl Q at Q s q Since there exists Q-oscillator realization of all quantum r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

282 8 ( ) M Mansour et alrphysics Letters B enveloping Ž super algebras U Ž g Q, it is reasonable to expect these to establish analogous decomposition Ž or splitting when Q q The aim of this paper is to investigate the property of splitting of quantum algebras A n, B n, Cn and D and quantum superalgebras AŽ m, n, BŽ m, n n, Cnq and Dn,m in the Q q limit The case of deformed Virasoro algebra and some other particular quantum Ž Super -algebras is given in w0 x As a first step we wish to present in Section a number of results concerning the property of Q-boson decomposition in the Q q limit We shall first of all discuss the way in which one obtains two independent objects Ž an ordinary boson and a k-fermion from one Q-deformed boson when Q goes to a root of unity We establish also the equivalence between a Q-deformed fermion and a conventional Žordinary or undeformed one Using these results, we analyse the Q q limit of quantum algebras of type A n, B n, C and D Ž Section 3 n n and quantum superalgebras Am,n, Bm,n, Cnq and Dn,m Ž Section 4 Concluding remarks are given in Section 5 Fractional spin through Q-bosons Let us start with Q-deformed bosonic algebra Ý Q The algebra Ý Ž n Q is generated by annihilation operators a y i, creation operators a q i and the operators Q N i and Q yn i satisfying the following commutation relations Q N i Q yn isq yn iq N is, Q N iq N jsq N jq N i, Q N i a y Q yn i sq yd ija y, Q N i a q Q yn i sq qd j j j ija q j, y y q q a i,ai s a i,ai s0, y q a i,aj s0 for i/j a y a q yqa q a y sq yn i ; i i i i a y a q yq y a q a y sq N i i i i i Ž Where Fi, jfn The algebra Ý Ž n Q is called the Q-analogue of the Weyl algebra Ž or Q-weyl algebra, and the operators Ni are taken to be hermitians One can verify that the relations Ž are coherent with the expressions: y q q y a a s wn q x ; a a s wn x, Ž 3 i i i Q i i i Q with Q X yq yx wx xq s, Ž 4 y QyQ X may be an operator or a number and Q is an Ž arbitrary complex number From Eq we obtain a Ž a s wx l Q Ž a qq Ž a a, y q l yn q ly l q l i y i i i i i Ž a a s wx l Ž a Q qq a Ž a Ž 5 y l q q ly yn l q y l i i i i i i where the symbol ww xx is defined by yq n wnx s yq Now, we define the deformation parameter q to be k-th root of unity; q k s, where kg is a positive integer In the particular case when one approaches the root of unity on the unit circle Q q Eqs Ž are amenable to the form: y q k q k y y k q q y k i i i i i i i i a a s a a ; a a sa a Ž In addition, Eqs leads to: Ž 6 N q k q k N N y k y k i i i Ni i i i i Q a s a Q ; Q a s a Q Ž 7 We point out that Ža q k Ž q i and ai k are elements of the centre of Ý Ž n Q algebra So, if one deals with a k-dimensional irreducible representation, we obtain: q k y k Ž i Ž i a sai, a sbi Ž 8 where I is the k = k matrix identity The extra possibilities parametrized by Ž as0, b/0; a/0, bs0; Ž 3 a/0, b/0 are not of interest in this work The case Ž 3 correspond to the periodic representation and in cases Ž and Ž, we have the so-called semi-periodic Žsemi- cyclic representation In this paper we shall deal with a representation of the algebra Ý Ž n such that y k q k Ž i Ž i a s0, a s0 Ž 9 Q

283 ( ) M Mansour et alrphysics Letters B are satisfied we note that the algebra Ý Ž n y ob- tained for k s, corresponds to ordinary fermion operators with Ža q Ž y i s 0, ai s 0, a relation that reflects the Pauli exclusion principle In the limiting case k ` we have the algebra Ý Ž n which correspond to ordinary boson operators For k arbitrary, the algebra Ý corresponds to k-fermions Ž Q or anyons with fractional spin in the sense of Majid wx 8 operators that interpolate between fermion and boson It follows from the definition Ž that: yn i y yni y Q a i, Q a i, yn y q k i i Ž i k Q 4 Q Q Q a, a k ky sq wk x! Ž 0 where the Q-deformed factorial is defined by: wk x!s wkxwkyxwky x; w x w0!s x Ž In the limit Q q, Eq Ž 0 becomes: yn i y yni y lim Q a i, Q a i, Q q wk x! yn y q k i i Ž i k Q 4 Q Q Q a, a k ky Q yn y k q k Ž i Ž i Q q s lim Q a, a wk x! k ky sq, which can be written as follows: kni kni y k q k Ž i Ž i Q a a Q lim, s Ž 3 Q q wk x! wk x! ( ( We remark that since q is a root of unity, it is possible to change the sign on the exponent of q kn i terms in the above and in the following definitions wx In the spirit of the work 9 we define "k N i y k Ž ai Q y bi s lim, Q q ( wk x! "k N i q k Ž ai Q q bi s lim Q q 4 ( wk x! So, y q b i,bj sd ij, 5 which is just the defining relation of non-commuting ordinary bosons The number operators of these new bosonic oscillators, are defined in the usual way as Nb sb q i b y i i This idea, concerning the Q q limit of Q-bow0,x in son, was introduced initially in references order to investigate the fractional supersymmetry and to show that there is an isomorphism between the braided line and the one dimensional superspace In the following we will introduce the new set of generators given by: yk Nb i yknb y y q q i i i i A sa q ; A sa q ; N sn ykn, Ž 6 Ai i bi and satisfying the following commutation relations: y q NA y q yn A, A y sq i; A, A sq Ai i i q i i q ; " " N A, Ai s"ai 7 i i which are the defining relations of a k-fermion w x The two algebras b q, b y, N,FiFn4 i i b i and q y A, A, N,FiFn4 i i A i are mutually commutative We conclude that in the limit Q q, the Q-deformed bosonic oscillator decomposes into two independents oscillators, an Ž undeformed boson and a k-fermion An appealing question is to ask whether is possible to find Q-deformed fermionic operators exhibit a similar property of splitting to Q-deformed bosons, when the deformation parameter Q reduces to a root of unity q To answer this question, we consider the y q yn i N set of operators c i,c i,q,q i4 satisfying the following relations: Q N i Q yn isq yn iq N is; Q N iq yn jsq N jq N i ; Q N i c y Q yn i sq yd j ijc y j ; Q N i c q Q yn i sq qd j ijc q j ;

284 84 ( ) M Mansour et alrphysics Letters B c y i,c y i 4 s c q i,c q i 4 s0; 4 c y,c q s0 for i/j Ž 8 i j c y c q qqc q c y sq yn i ; i i i i c y c q qq y c q c y sq N i i i i i Ž 9 We denote this algebra by A Ž n Q If we define the new operators yn i yni y y q q i i i i F sq c ; F sc Q, Ž 0 we obtain by a direct calculation the following anticommutation relation 4 F y,f q sd Ž i j ij Moreover, we have " Fi s0 Thus we see that the Q-deformed fermion reproduces the conventional Ž ordinary fermion Although there exists a freedom to define a Q-deformed fermion as constant multiple of a conventional one and regard the two objects as equivalents It should be noted that the Q-deformed fermions was used by Hayachi wx 4 to give Q-fermionic representation of quantum groups U Ž X Q where X is a finite Lie algebra of type A n, Bn or D n However the Q-ferm- ions are nothing but the conventional ones So, the spinor representation of these quantum groups can be given in terms of a set of fermionic oscillator algebras 3 Quantum algebras at Q a root of unity Let Xswa x Ž Fi, jfn ij be a symmetrisable generalized Cartan matrix and Let d Ž FiFn i be the non-zero integers such that da i ijsaijdi and the greatest common divisor of this is Let Q/0 bea complex number For Q generic the quantum enveloping algebra U Ž X corresponding to X is a Hopf algebra with Q " "d i h i " h and generators e, f,k sq sq i,fifn4 i i i i satisfying the following relations: k iyk y i e i, fj sd ij ; y Q yq i kek y sq a ij e,k f k y sq ya ij f, i j i i j i j i i j i kk i y i sk y i k is, kksk i j jki Ž 3 and some well-known quantum Serre relations The Cartan matrices of classical type A n, B n, C n, Dn and the corresponding non-zero integers are given in w x We shall give now the Q-oscillator representation of quantum algebras A n, B n, Cn and D n we shall provide explicit expressions for corresponding generators as linears and bilinears in Q-deformed bosonic and fermionic oscillators We consider successively the quantum algebras U Ž A, U Ž B, U Ž C Q n Q n Q n and U Ž D Q n Firstly, the quantum algebra U Ž A Q n admits the two following representations e sa y a q ; f sa q a y ; k sq yn iqn iq i i iq i i iq i, Ž 4 for FiFn, in terms of Ž nq Q-deformed bosons and can also be realized by means of Ž nq q-deformed fermions: e sc y c q, f sc q c y, k sq N iyn iq i i iq i i iq i Ž 5 The representation of U Ž B Q n requires only Q-fermions e sc y c q ; f sc q c y ; i i iq i i iq k sq N iyn iq, i FiFny; e sc y, f sc q, k sqq N n n n n n n Ž 6 The U Ž C Q n representation, involving nq-bosons, is given as follows e sa y a q ; f sa q a y ; i i iq i i iq k sq yn iqn iq, i FiFny; y q Ž an Ž an ens, fnsy, y y QqQ QqQ yž N q n k sq Ž 7 n

285 ( ) M Mansour et alrphysics Letters B Finally, the only way to realize U Ž D Q n is based on the use of nq-fermionic operators eisc y i c q iq; fisc q i c y iq; k sq N iyn iq i, FiFny; ensc y nyc y n, fnsc q n c q ny, k sq N nqn q ny Ž 8 n Now, we are in position to investigate the limit Q q of the quantum algebras U Ž A, U Ž B Q n Q n, U Ž C and U Ž D Q n Q n We remark that the investigation presented here is based on the Q-oscillator representations of the above algebras So, all results obtained are specific to the use of Q-Schwinger realization In the Q q, the splitting of Q-deformed bosons leads to undeformed bosons b q i, b y i, N 4 given by Eq Ž 4 b i and k-fermionic operators q y A, A, N 4 defined by Eqs Ž 6 i i A i From the undeformed bosons, we define the operators e sb y b q ; f sb q b y ; k syn qn, i i iq i i iq i i iq Ž 9 Ž F i F n which generates the classical Žunde- formed algebra UŽ A n From the remaining opera- q y tors A, A, N 4, one can realize the U Ž A i i A q n alge- i bra Indeed, the generators defined by EisA y i A q iq; FisA q i A y iq; K sq yn A i qn A iq i Ž 30 generate the U Ž A q n algebra which is the same ver- sion of U Ž A Q n obtained by simply setting Q s q, rather than by taking the limit as above The element of U Ž A and UŽ A q n n algebras are mutually commutative So, we obtain the following decomposition: lim U Ž A su Ž A muž A Q q Q n q n n The analysis of Q q limit of the quantum algebra U Ž C Q n leads to similar decomposition as in the An case In fact, the undeformed bosons b q, b y, N 4 i i b i permits to build up the generators: eisb y i b q iq; fisb q i b y iq; k sq yn b i qn biq, i FiFny; y q Ž bn Ž bn ens, fnsy, k nsy Nnq, 3 which are nothing but those generating the classical UC n algebra Through the k-fermionic operators q y A, A, N 4, we can realize the U Ž C i i A q n by genera- i tors defined by: E sa y A q ; F sa q A y ; i i iq i i iq K sq yn A i qn A iq, i FiFny; y q Ž An Ž An Ens, Fnsy, y y qqq QqQ yž N q A K sq n n Ž 3 Since the two UC and U Ž C n q n are mutually com- mutative, we have the following decomposition lim U Ž C su Ž C muž C Q q Q n q n n which is similar to the one obtained for A n The algebra U Ž A Q n possess a Q-fermion realization beside to Q-bosonic one introduced above While the algebras U Ž B and U Ž D Q n Q n admits only a Q- fermionic representation We have discussed in the first section how one can identify the conventional fermions with Q-deformed fermions, there have in fact an equivalence between these two objects Consequently, due to this equivalence, it is possible to construct Q-deformed algebras of type A, B and Dn using ordinary fermions It is also possible to construct the undeformed algebras of type A n, Bn and Dn by considering Q-deformed fermions So, in the fermionic realization we have equivalence be- tween UŽ A, UB and UD and U Ž A, U Ž B n n n Q n Q n and U Ž D Q n respectively To be more clear, we consider the U Ž A Q n case in the Q-fermionic representation The generators are given by: e sc y c q ; f sc q c y ; k sq N iyn iq i i iq i i iq i, Ž 33 due to equivalence fermion Q-fermion, the operators c q i,c y i are defined as a constant multiple of conventional fermion operators, ie, Ni Ni y y y q q y i i i i F sq c ; F sc Q Ž 34 from which we can realize the generators: EisF y i F q iq; FisF q i F y iq; HisNiyN iq Ž 35 n n

286 86 ( ) M Mansour et alrphysics Letters B The set E, F, H, FiFn4 i i i are elements of UŽ A n The analogous analysis can be performed in the U Ž B and U Ž D q n q n cases The equivalence between fermions and Q-fermions seems to be a class of invertible maps between the classical algebras and their Q-fermionic analogous 4 Quantum superalgebras at Q a root of unity A general description of the quantum Lie superalwx 3 ; this description comprises the gebras is given in Q-oscillator realization of these quantum superalgebras A superalgebra g of rank r can be characterized by a Cartan matrix Ž a ij and a subset t;is,,n4 that identifies the odd generators Unless g is an ordinary Lie algebra, in which case tsf, the set t can actually be taken to consist to only one element Let wx, stands for the graded product deg x deg y wx, yxsxyyž y yx Let QgCy04 be the deformation parameter we shall use also Q sq d i i with di are numbers, that symmetrise the Cartan matrix Ž a ij The quantum superalgebra U Ž g Q is generated by 3r elements e i, fi and h i,igi, which satisfy: Q d i h iyq yd i h i e i, fj sd ij y Q yq, i i h i,h j s0, h i,ej saije j, h i, fj syaijf j, Ž 36 with deg his 0, deg eis deg fis 0, i f t and deg eisdeg fis, igt and further obey certain generalized Serre relations It is convenient to introduce the quantities k sq h i i i in terms of which the defining relations Ž 36 become k iyk y i e i, fj sd ij, y Q yq i kek y sq a ij e, k f k y sq ya ij f, i j i i j i j i i j i kk i y i sk y i k is, kksk i j jki Ž 37 The quantum serre relations are most simply prew3x sented in in terms of the following rescaled generators: The Cartan matrix for quantum superalgebras of type Am,n, Bm,n, Cnq and Dm,n are given in w4,5x Now we give the Q-oscillator representations of the quantum superalgebras We shall present the explicit expressions for the corresponding generators as linears and bilinears in Q-deformed bosonic and fermionic oscillator operators After we investigate their Q q limit, we consider successively the quantum superalgebras U Ž AŽ m, n, U Ž BŽ m, n Q Q, U ŽCŽ nq and U Ž DŽ m, n Q Q associated respectively to Am,n, Bm,n, Cnq and Dm,n Lie superalgebras series described above The quantum superalgebra U ŽAŽ m, n Q can be realized simply by Ž m q Q-deformed fermions and Ž n q Q-bosons Explicitly the generators of A Ž m,n are given by: Q e sc q c y, f sc y c q, i i iq i i iq k sq d iž M i ym iq, i FiFm, e sc q a y, f sc y a q, mq mq mq mq k sq d mq Ž M mq qn, mq e sa q a y, f sa y a q, FjFn, mq j jy j mqj jy j k sq d mq jž N jy yn j mq j, FjFn Ž 38 Due to the property of Q-boson decomposition in the Q q limit, each Q-boson a q, a y, N 4 i i i reproduces an ordinary boson b q, b y, N 4 i i b i and a k-fermion q y operator A, A, N 4 i i A i In the limit the Q-fermions become q-fermions which are objects equivalents to q y conventional fermions F, F, M 4 i i F i The algebra UŽAŽ m, n is generated by: E sf q F y, F sf y F q, i i iq i i iq H sm ym, FiFm, i F i F iq E sf q b y, F sf y b q, mq mq mq mq H sm qn, mq F b mq E sb q b y, F sb y b q, mq j jy j mqj jy j H sn yn, FjFn Ž 39 mq j bjy bj

287 ( ) M Mansour et alrphysics Letters B q y From the operators A, A, N 4 i i A i we construct the generators e sa y A q ; f sa q A y ; k sq yn A i qn A iq, i i iq i i iq i Ž 40 for F i F n q, which generates the algebra U Ž A It is easy to verify that U Ž A and Am,n q n q n are mutually commutative As results, we have the following decomposition of quantum superalgebra A Ž m,n in the Q q limit Q lim U AŽ m,n su AŽ m,n mu Ž A Q q Q q n The quantum superalgebra U ŽBŽ m, n Q, described by the set of elements e, f, h,fifmqn 4 i i i The Z -grading on U ŽBŽ m,n Q is defined by the requirement that the only odd generators are fmq n and e The superalgebra U ŽBŽ m, n nqm Q can be realized in terms of m Q-deformed fermions and n Q-deformed bosons as follows: e sa q a y, f sa y a q, i i iq i i iq k sq d iž N i yn iq, i FiFny, e sc y a q, f sc q a y, k sq d nž M qn n, n n n n n e sa q c y, f sc q a y, nqj j jq mqj jq j k sq d nq jž M j yn jq, FjFmy, mq j M q y M nqm m nqm m e s y c, f s c y, k sq d mq nž M y F m mq n, Ž 4 where MF i are the fermionic operators number, and Ž y M is the so called Klein operator Now we examine the Q q limit of quantum superalgebra U Ž Q B m, n The Q-deformed bosons a q i, a y i, N i, Ž FiFn4 give in the Q q limit two sets of non-commuting operators b q, b y, N Ž F i F n4 i i b i y y and A, A, N Ž F i F n 4 i i A i They correspond respectively to classical boson operators and k- fermionic ones The Q-deformed fermions are equivalents to ordinary one Using these properties, one can investigate the behaviour of U ŽBŽ m, n Q when Ž k the deformation parameter Q goes to q q s Indeed, using the classical bosons b q, b y, N i i b i 4 q y FiFn and the conventional fermions F i, F i, M Ž FiFm4 F i one realizes the operators Eisb q i b y iq, Fisb y i b q iq, H sn yn, FiFny, i bi biq E sf y b q, F sf q b y, H sm qn, n n n n n F b n E sb q F y, F sf q b y, nqj j jq mqj jq j H sm yn, FjFmy, mq j F j b jq MF q y MF nqm m nqm m E s y F, F s F y, H s M y, Ž 4 mq n F m where MF s Ý m is M F i The set of operators E,F, H Ž FiFmqn4 i i i generates the undeformed algebra UBm, Ž Ž n From the remaining generators q y A, A, N Ž F i F n 4 i i A i, we can realize the U ŽBŽ 0, n q generated by the set of operators j,z,t,is,, n 4 The elements of U ŽBŽ 0,n i i i q are given in terms of k-fermionic operators, by j sa q A y, z sa y A q, i i iq i i iq T sn yn, FiFny, i Ai Aiq jnsa q n, znsa y n, Tns NA n q Ž 43 The algebras UBm,n Ž and U Ž BŽ 0,n q are mutu- ally commutative Consequently the following decomposition lim U BŽ m,n su BŽ m,n mu BŽ 0,n Q q Q q Ž 44 holds Repeating the similar analysis, we shall discuss the Q q limit of quantum superalgebra U ŽCŽ q nq The Q-Schwinger realization of generators of this algebra is given by e sc q a y, f sc y a q, k sq d iž MqN i, i i i i i e sa q a y, f sa q a y, iq i iq iq iq i k sq d iq Ž N i yn iq, is,,ny, iq q y n i an y nq y i a enq s, f s, QqQ QqQ k sq d n Ž N n q Ž 45 nq

288 88 ( ) M Mansour et alrphysics Letters B in terms of n Q-deformed bosons and one Q-deformed fermion Using the properties of splitting of Q-deformed bosons and equivalence between Q-deformed fermions and ordinary fermions, we construct in the Q q limit the following operators EisF q b y i, FisF y b q i, HisMF qn b i, E sb q b y, F sb q b y, iq i iq iq iq i H sn yn, is,,ny, iq bi biq q y n i bn nq i b Enq s, F s, Hnq snn q Ž 46 The elements E, F, H 4 i i i generate the well known UCnq Ž Ž algebra From the remaining operators one can realize U Ž AŽ ny or U Ž BŽ 0,n q q which commutes with the superalgebra UCnq Ž Ž In this case, we have two possibilities of decompositions lim U CŽ nq su CŽ nq mu Ž A, Q q Q q ny lim UQŽ CŽ nq Q q su CŽ nq mu BŽ 0,n Ž 47 q Finally, we discuss the splitting property of Q-deformed superalgebra U ŽDŽ m, n Q The Q-realization of Q-deformed superalgebra, involving n Q-deformed bosons and mq-deformed fermions is given by e sa q a y, f sa y a q, i i iq i i iq k sq d iž N i yn iq, i is,,ny, e sc y a q, f sc q a y, k sq d nž N n qm c, n n n n n e sc q c y, f sc y c q, nqj j jq nqj j jq k sq d jq nž M cj ym cjq, js,,my, nqj e sc q c q, f sc y c y, mq n m my mqn m my k sq d mq nž M cmy qm y cm mq n, Ž 48 As was the case above, in the limit Q q, from the classical bosons and ordinary fermions we construct the classical superalgebra UDm, Ž Ž n and from the k-fermionic operators, we realize U Ž A q ny or U Ž BŽ o,n thus in the U Ž DŽ m,n case, we have q lim U DŽ m,n su DŽ m,n mu Ž A, Q q Q Q q ny lim U DŽ m,n su DŽ m,n mu BŽ 0,n Q q Q q Ž 49 two possibilities of decomposition which are similar to the result obtained in the U ŽCŽ nq case 5 Conclusion We have presented the general method leading to the investigation the Q qse p i r k limit of all quantum algebras and quantum superalgebras based on the decomposition of Q-bosons in this limit We note that Q-oscillator realization is crucial in this decomposition of these algebras The relation between our approach to the representation theory and some other approaches will be discussed elswhere We believe that the techniques and formulae, used here, will be useful foundation to extend this study to all Q-deformed affine Lie algebras and superalgebras This matter will be treated in a forthcoming paper w4 x Acknowledgements The authors would like to thank the referee for useful comments References wx VG Drinfeld, Quantum groups, in: Proc Intern Congr Math, Berkeley, 986, vol, p 798 wx M Jimbo, Lett Math Phys Ž wx 3 M Chaichan, P Kulich, Phys Lett B 34 Ž wx 4 T Hayashi, Commun Math Phys 7 Ž wx 5 R Floreanini, P Spiridinov, L Vinet, Phys Lett B 4 Ž ; R Floreanini, P Spiridinov, L Vinet, Commun Math Phys 37 Ž wx 6 LC Biedenharn, J Phys A Ž 989 L873; AJ Macfarlane, J Phys A Ž Q

289 ( ) M Mansour et alrphysics Letters B wx 7 RS Dunne, AJ Macfarlane, JA de Azcaraga, JC Perez Bueno, Phys Lett B 387 Ž ; hep-thr960087; Czech J Phys 46 Ž ; JA de Azcaraga, RS Dunne, AJ Macfarlane, JC Perez Bueno, Czech J Phys 46 Ž ; RS Dunne, hep-thr9703 wx 8 S Majid, Anyonic Quantum groups, in: Spinors, Twistors, Clifford algebras and quantum deformations ŽProc of nd Max Born Symp, Wroclaw, Poland, 99, Z Oziewicz et al Ž Eds, Kluwer, Dordrecht; S Majid, hep-thr9404 wx 9 RS Dunne, Intrinsic anyonic spin through deformed geometry, hep-thr w0x M Mansour, M Daoud, Y Hassouni, AS-ICTP preprint ICr98r64 wx M Daoud, Y Hassouni, M Kibler, Symmetries in Science X, B Gruber, M Ramek Ž Eds, Plenum Press, New York, 998 wx V Chary, A Presely, Guide to quantum groups, Cambridge University Press, Cambridge, 994 w3x M Rosso, Commun Math Phys 4 Ž w4x M Mansour, M Daoud, Y Hassouni, in preparation

290 0 May 999 Physics Letters B Exact results for non-holomorphic masses in softly broken supersymmetric gauge theories Nima Arkani-Hamed a, Riccardo Rattazzi b a Stanford Linear Accelerator Center, Stanford UniÕersity, Stanford, CA 94309, USA b Theory DiÕision, CERN, CH- GeneÕe 3, Switzerland ` Received 9 April 998; received in revised form 3 June 998 Editor: M Cvetič Abstract We consider strongly coupled supersymmetric gauge theories softly broken by the addition of gaugino masses ml and non-holomorphic scalar masses m, taken to be small relative to the dynamical scale L For theories with a weakly coupled dual description in the infrared, we compute exactly the leading soft masses for the magnetic degrees of freedom, with uncalculable corrections suppressed by powers of Ž m rl, Ž mrl l The exact relations hold between the infrared fixed point magnetic soft masses and the ultraviolet fixed point electric soft masses, and correspond to a duality mapping for soft terms We briefly discuss implications of these results for the vacuum structure of these theories q 999 Published by Elsevier Science BV All rights reserved Recent years have seen enormous progress in our understanding of strongly coupled supersymmetric gauge theories w, x In particular, a large class of models have dual descriptions which are weakly coupled in terms of magnetic degrees of freedom in the deep infrared It is natural to attempt to extrapolate these supersymmetric results to non-supersymmetric theories by adding soft masses for the superpartners in order to decouple them w3 8 x As a modest first step towards the decoupling limit, one can study the response of the theory to soft masses m much smaller than the dynamical scale of the theory L This is also of considerable interest to models where some of the Standard Model fields arise as composites of elementary preons If the preon soft masses are known, what are the soft masses of the composite states? The difficulty in addressing these simple questions is that scalar soft terms are given by manifestly non-holomorphic terms in the bare Lagrangian H 4 V L soft>y d uu u m f e f and so the usually powerful constraint of holomorphy can not be used to analyse this type of soft breaking in theories with N s supersymmetry Nevertheless, in this letter we will show that the leading contribution to the soft masses for the magnetic fields can be computed exactly in terms of the soft masses for the original fields, with uncalculable corrections suppressed by powers of Ž mrl These results are possible due to an interpretation of scalar soft masses as auxiliary components of the vector field of an anomalous background UŽ gauge symwx 9 In Ref wx 9, this symmetry was exploited metry in r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

291 ( ) N Arkani-Hamed, R RattazzirPhysics Letters B perturbation theory, allowing high-loop supersymmetry breaking results to be obtained from lower loop supersymmetric computations Here we show that the same symmetry can be useful for computing soft masses in strongly coupled theories For simplicity, we will consider Žasymptotically free SUSY gauge theories with a simple gauge group, softly broken by a gaugino mass ml and universal scalar masses m The extension of our methods to include several group factors as well as arbitrary scalar masses will be obvious Begin by considering the exactly supersymmetric limit The bare lagrangian with ultraviolet cutoff m is H d usž m W qhc UV H UV UV q d 4 u F SŽ m qs Ž m Q Q where Q Q stands for Q e V Q The muv dependence of S,F is dictated by the Wilsonian renormalization group, which requires that the low energy physics stay fixed as the ultraviolet cutoff muv is varied It is well-known that S only changes at -loop w0 x, whereas F runs at all orders in perturbation theory: ds b s Ž 3 dlnm 8p UV dln F sg S 4 dlnm UV where g is the supersymmetric anomalous dimension Now consider the reparametrization Q ' Z Q By the rescaling anomaly w,x the new lagrangian is ž / T d u SŽ m q lnz W qhc 8p H UV H UV UV q d 4 uzf SŽ m qs Ž m Q Q Ž 5 Where T is the total Dynkin index of the matter fields Q After relabeling SŽ m qžtr8p lnz UV UV SŽ m UV, we can start over again with the following bare lagrangian H d usž m W qhc UV ž H UV UV q d 4 uzf SŽ m qs Ž m T y lnz Q Q Ž 6 4p / and treat S and Z as independent parameters The theory defined by Eq Ž 6 is invariant under the transformation Z Zxx, Q Qrx, SŽ m SŽ m q Tr4p ln x Ž 7 UV UV Notice that the physical coupling S q S y ŽTr4p ln Z is invariant We can now consider the situation where SŽ m UV and Z are respectively promoted to chiral and real vector superfields For us this just means that these quantities have non vanishing u and uu components, corresponding to soft gaugino and squark masses Notice that, when S and Z are promoted to superfields, the bare lagrangian of Eq Ž 6 still defines a cut-off independent low energy theory also including insertions of soft terms This follows from simple power counting Apart from a cosmological constant term, no new divergences are generated Indeed, these would have to involve covariant derivatives acting on S and Z, but there is no local counterterm of this type also involving the physical fields Now that S and Z are superfields the above invariance becomes an abelian background UŽ A gauge symmetry Physical quantities have to be UŽ A and RG invariant The parameter space of the theory is described by the only UŽ A and RG invariant object that can be formed with S and Z I'L Z T r b L Ž 8 h h where L sm e y8p Sr b h UV is the holomorphic dy- namical scale The u 0 component of I gives the w x physical strong scale I s L w3 x As we usus0

292 9 ( ) N Arkani-Hamed, R RattazzirPhysics Letters B show immediately below, the u and uu components are related to the UV fixed point limits of the gaugino mass ml and the squark mass m Q respec- tively: ž / 6p 6p m l wln I x u ' m g s lim, b m ` b g UV T T wln I x s wlnz x u u u u 'y m b b T sy lim m Ž 9 b muv ` As a first step in making these identifications, we show that in the deep UV muv `, the u and uu components of F vanish and hence make no contribution to soft terms By dimensional analysis and invariance under the anomalous symmetry the Ž wave function has the form F s F m ri Ž UV note that ybr8p ln m UVrIsSqS ytr4p ln Z is just the argument of F in Eq Ž 6 Therefore we have dln F wln F xu sy wln I x dlnm UV u 8p sy gž muv m g, Ž 0 b dln F wln F x syy wln I x u u u u dlnm UV d ln F q < wln I x u < 4 dlnmuv Q ž / T 8p s gž muv m q gž muv m g b b Ž UV UV where g s dgr dlnm As m `, the theory becomes free, g,g 0, and wln F x,lnf w x u u u both vanish We now establish the first of Eq Ž 9 Defining the anomalous UŽ invariant quantity R via w0x A A Ry ln RsSqS y Ž Tr4p lnzf Ž 8p Žwhere A is the Dynkin index of the adjoint representation the physical gauge coupling in any given scheme has the form wx 9 ` yn R phys srq Ý cn R 3 ns0 where the c are scheme dependent constants Ž n the coefficient of the leading term is fixed by the equality of Wilsonian and physical coupling at tree level At any scale m w x UV, we have that mlrg s R phys u However, as muv `, R y 0, so that only the first term in Eq Ž 3 matters, and the first of Eq Ž 9 follows trivially The same result was discussed in Refs w4,9 x Consider now the running squark mass given by the matter kinetic term in Eq Ž 6 w x m m sy ln Z Q UV u u y ln F m UV u u, Ž 4 where we have used the well-known fact that the soft scalar mass for any field f is the uu component of the logarithm of the coefficient of f f, which in our case is ZF Again, as muv `, the second term in Eq Ž 4 vanishes and we recover the second of Eq Ž 9 Having established the physical interpretation of the various components of the UŽ A and RG invariant superfield I, we discuss the computation of magnetic soft masses This will be possible since the anomalous UŽ symmetry of Eq Ž A 7 provides a powerful constraint on the way in which S, Z Žand hence the soft masses enter into the theory As an example, consider SUŽ N SUSY QCD with Ž Nq flavors Q i,q i, for the moment in the supersymmetric limit In the deep infrared and at the origin in moduli space, this theory has a weakly coupled description in terms of the composite mesons Miis QQ i i i Ž N i i Ž N i and baryons B s Q, B s Q wx We expect that, as long as the soft masses are much smaller than the strong scale L, the mesons and the baryons still give a good description of the low energy theory In particular, we expect the Kahler potential for these fields to be smooth everywhere on moduli space Therefore we can expand it in a power series in M, B, B around the origin By using invariance

293 ( ) N Arkani-Hamed, R RattazzirPhysics Letters B under the flavor symmetries, under Eq Ž 7, and under the RG, the Kahler potential must depend on S,Z as N N MZM BZ B BZ B Ksc qc qc q PPP M B Ny B I I I Ny Ž 5 The effective Kahler potential K is also associated with a coarse-graining scale mir -L, and the wave function coefficients c Ž which depend on m M, B, B IR play a role similar to F in the UV theory At any m IR, the soft terms for the composites are given by eg Z mm Ž mir sy ln y ln cm Ž m IR u u I u u Ž 6 By invariance under the ultraviolet RG and the anomalous UŽ, the wave functions have the form Ž c 'c m ri M, B, B M, B, B IR As for the UV wave function F, the dependence of the c s on the soft terms is determined by the RG T ln cm, B, B s g m m u u M, B, B Ž IR b ž / 8p q M g Ž m m b, B, B IR g Ž 7 where, similarly as before, g s dln cr dln mir and gs d lncr dln m IR Now, the effective theory of mesons and baryons is free in the IR In fact it involves one marginal Yukawa interaction W > BMB which goes to zero for mir 0 More precisely as mir 0, cc, B, B `, so that the effective coupling l eff m IR ;rcmcbcb 0 and the anomalous di- mensions gm, B, B 0 We conclude that at mirs0 the c s do not affect the soft terms in Eq Ž 6 We emphasize that this argument is completely analogous to the one given above for the irrelevance of F to the soft terms in the deep UV Eq Ž 7 determines a relation between soft terms and RG which is somewhat similar to the one discussed in Ref w5 x In that case the role of the invariant I was played by the messenger threshold superfield XX By the above discussion, the IR fixed point value of the composites are determined by the uu components of Z, I which are in turn related to the UV fixed point value of the squark masses as in Eq Ž 9 We therefore find a purely algebraic relationship between the composite soft masses in the deep IR and the squark masses in the deep UV Using the anomalous UŽ symmetry, we can seemingly control the exact soft masses for the composites, at least at the origin of moduli space! This is however true only in the limit in which the soft masses m g and m are much smaller than the strong scale L Indeed there are UŽ A invariant terms in K which can Ž a a involve the U A field strength WA sd D lnz which is non-vanishing when ln Z has a non-vanish- ing uu component One such term is DW a A a Z 4 d u M M Ž 8 I I H ž / Since WA a has positive mass dimension, however, this and all other such operators make contributions to the composite soft masses which are suppressed by powers of Ž mrl It is these uncontrollable operators which prevent us from taking the decoupling limit Ž mrl `, however, their effects are power suppressed for Ž mrl < We have now all the ingredients to determine the mapping of soft terms between the microscopic and macroscopic theories, up to corrections suppressed Ž by powers of m rl : Z mm Ž mirs0 sy ln I u u Ny4 s mqž muvs` Ž 9 Ny Z N mb, B Ž mirs0 sy ln Ny I u u yn s mqž muvs` Ž 0 Ny Note that the IR soft masses only depend on the deep UV value of the squark mass but not on the deep UV value of mlrg sm g Of course, for any given gaugino and squark masses m Ž m,m Ž m l UV Q UV at a finite value of m UV, the deep UV soft scalar mass obtained by running to muv s` will be a function of both m Ž m,m Ž m l UV Q UV

294 94 ( ) N Arkani-Hamed, R RattazzirPhysics Letters B The IR soft masses given in Eq Ž 0 satisfy the relation mmqmbqmbs0 This sum rule can be inferred from the low energy theory due to the RG focusing effect of the Yukawa interaction BMB, and could have been established without using the anomalous symmetry wx 7 The symmetry is however crucial to fix the value of each mass Notice that for N) and for positive squark masses the baryons are tachyonic The implications of this result for the symmetry properties of the vacuum will be discussed below Notice also that for the special case of SUŽ the baryons and the mesons coincide and have vanishing soft mass While this result holds in the deep IR, by Eqs Ž 6 and Ž 7 we can establish how this limit is approached This is a pure Yukawa theory for which both gm and gm are negative in the perturbative domain Therefore we conclude that m M is positive at finite mir and approaches zero as mir 0 The result in Eq Ž 0 has a nice interpretation in terms of the anomalous UŽ charges of the canonically normalized fields Mˆ s MrL and Bˆ h s Ny BrL, Ž L has charge Trb h h In terms of the canonical fields Eq Ž 5 reads q ˆ q ˆ ˆ q ˆ Ksc ˆ ˆ ˆ ˆ ˆ MM Z M MqcBB Z B BqcBB Z B Bq PPP Ž The charges of the composites qmˆ and qbˆ coincide with the corresponding IRrUV mass ratios of Eq Ž 0 Notice that, without the contribution to the soft terms from the powers of I in Eq Ž 5, the soft masses of the composites would just be determined in the naive way, by adding the masses of the constituents We stress that the existence of a relationship between deep UV and IR quantities is just a consequence of RG invariance For instance, in QCD one may ask for the expression of the pion mass in terms of the fundamental parameters It will have the form m pscmˆ q L QCD, where c is a constant and mˆ q is an RG invariant combination of the running quark mass m Ž m and gauge coupling g Ž m q In practice, as for m in Eq Ž g 9, one can define mˆ q just by using the -loop RG in the deep UV mˆ q s lim g p Ž m m Ž m m ` q, where p is determined by the -loop b function and mass anomalous dimension The striking feature of our case, with respect to QCD, is that we can calculate the analogue of the coefficient c Another example is given by SpŽ k gauge theory with kq4s NF chiral multiplets Li in the fundamental representation ŽSU with 3 flavors is just the special case ks The low energy description involves the antisymmetric meson field VijsLieLj kq with a superpotential W s Pf VrL w6 x conf h By adding soft terms the resulting mass for the meson is Žfrom now on it is understood that the LHS and RHS soft masses are the IR and UV fixed point values respectively ky mvs ml kq Notice that Pf V;V kq is an irrelevant operator in the low energy theory for k) In this case Ž g m ri V IR goes to zero with a power law when m 0 and the running mass m Ž m IR V IR approaches Eq equally fast Finally consider SUŽ N gauge theory with Nq -NF -3Nr for which the low-energy description is in terms of a dual magnetic theory with gauge group SUŽ N yn F The magnetic theory contains an elementary meson Mij and NF flavors of dual quarks i i q,q in the fundamental representation of SUŽ N F y N The UŽ A charge of the canonically normalized meson Mˆ s M rl is just q s qqqrl ij ij h Mˆ i j h s 3Ny Ž N rž 3NyN F F The charge of the dual quarks simply follows from the invariance of the tree i j level magnetic superpotential Wmagn s qmijqrlh Žor by matching the baryons in the two theories N F yn N NyN F NyN bsq sq rlh sbrl h F We thus obtaining the soft masses of the magnetic theory 3Ny NF 3Ny N F mms m, mq,q sy m 3NyNF 3NyNF Ž 3 Again the baryons q,q are tachyonic for all theories in the free magnetic phase NF - 3 Nr For 3 Nr- NF - 3 N the theory is in an interacting non-abelian Coulomb phase Here we cannot apply our method in an obvious way since there are no points where the theory is free It is interesting that all the magnetic soft masses vanish at the boundary between the free magnetic and conformal windows, NF s3nr

295 ( ) N Arkani-Hamed, R RattazzirPhysics Letters B Notice that the normalization of the magnetic quarks q,q is arbitrary, and that in Ref wx a scale m was introduced to give proper dimension to the i j superpotential: Wmagn s Mijq q rm Correspondingly the holomorphic scales in the electric and magnetic b b N theory are related by wx L L s m F Ž h h where the tilded quantities refer to the magnetic theory In our derivation we have fixed msl h, but our results do not depend on that choice Indeed one could have argued as follows With the normalization of Ref wx the dual quarks have charge y under the anomalous UŽ Ž m is neutral and M sq Q A ij i j Aswe did for the electric theory in Eq Ž 6, we can define a dual wave function Z multiplying the kinetic term of the dual quarks q, q as well as an invariant scale T r b I s L Z L h h However, since the electric and magnetic theory describe the same physics, it must be IsI Therefore we deduce the following duality relation b b Z s Z 4 which holds up to a gauge transformation which does not affect the mapping of soft terms Eq Ž 4 correctly gives the magnetic quark masses in Eq Ž 3 By Eq Ž 4 the opposite sign of electric and magnetic squark masses is a reflection of duality between IR and UV free theories Finally, by consid- x w x ering wln I us ln I u, a similar duality is obtained for gaugino masses More precisely one gets Žnotice again the flip in sign ml ml lim s lim Ž 5 m ` ž bg ž / m IR 0 bg magn UV / Our results for the soft masses of composite and magnetic fields have obvious implications for composite model-building in theories where supersymmetry breaking is communicated to the preon fields at a scale higher than the dynamical scale L Ž for instance by supergravity mediation Clearly one must check that eg none of the composite squarks obtain negative soft masses Next we consider the vacuum structure of these theories, beginning with the SUŽ N theories with NqFNF F3r N flavors and N) In the supersymmetric limit, these theories have a moduli space of vacua, and we are interested in how the soft breaking effects lift this vacuum degeneracy In all el these theories, for positive squark masses in the deep ultraviolet, the meson fields get positive soft masses while the baryonic fields get negative soft masses The origin of moduli space is therefore unstable, and some of the mesons or baryons must have non-vanishing vevs in the true vacuum Note that our method only gives us information on the form of the potential close to the origin, since far from the origin operators with higher powers of meson and baryon fields Žwhich we have no control over are unsuppressed, and therefore we can not determine the location of the true vacuum even for small soft breakings Nevertheless, establishing the instability of the origin has important consequences, since in these theories all points on the moduli space away from the origin break vector-like symmetries If any baryonic fields obtain vevs baryon number is broken, and if all the baryons vevs vanish, there is no point on the quantum moduli space where M ij A d and so SUŽ N is broken This is to be conij F V trasted with the non-supersymmetric theory obtained by decoupling the scalars, where a general theorem w7x shows that vector-like symmetries are never broken It is easy to argue that the broken vector-like symmetries are restored for squark masses larger than a finite critical value Squarks of mass m 4 L can be integrated out of the theory, generating higher dimension operators suppressed by rm in the non-supersymmetric low energy theory These operators can at most correct the spectrum of states Ž in the low energy theory by O L rm Since all the scalar states in the non-supersymmetric theory get masses of OŽ L Žwith the exception of Goldstone bosons associated with chiral symmetry breaking, there are no scalars which can be brought down Ž to zero mass due to the O L rm corrections and there is therefore no candidate for the Goldstone boson of a broken vector-like symmetry Therefore, the vector-like symmetries must be exactly restored above a finite critical squark mass m ) ;L, and a phase transition must separate the nearly supersymmetric and non-supersymmetric theories For SpŽ k theories with kq4 chiral multiplets, the soft mass of the mesons is positive and the origin of moduli space is at least a local vacuum At this point, the fermionic mesons are massless bound states of a massless quark and a massive squark, the binding energy exactly canceling the squark mass This

296 96 ( ) N Arkani-Hamed, R RattazzirPhysics Letters B provides a rigorous counter-example to the the perw8,9 x In conclusion we remark that, while we have sistent mass condition of illustrated our ideas with specific examples, our technique for computing soft masses can clearly be applied in any asymptotically free supersymmetric theory where the theory in the deep infrared is known and is weakly coupled, as in all syconfining w0x or magnetic free theories Acknowledgements NAH would like to thank S Thomas and M Peskin for very useful discussions RR would like to thank Alberto Zaffaroni for useful conversations RR is also indebted to Erich Poppitz for discussions and for suggesting the analogy with the pion mass Both authors acknowledge very useful comments and criticism from Markus Luty, and useful correspondence with H Nishino References wx N Seiberg, Nucl Phys B 435 Ž 995 9; K Intriligator, N Seiberg, Nucl Phys Ž Proc Suppl 45BC Ž 996 wx N Seiberg, E Witten, Nucl Phys B 46 Ž 994 9; B 43 Ž wx 3 N Evans, SDH Hsu, M Schwetz, Phys Lett B 355 Ž ; N Evans, SDH Hsu, M Schwetz, SB Selipsky, Nucl Phys B 456 Ž ; N Evans, SDH Hsu, M Schwetz, Phys Lett B 404 Ž wx 4 O Aharony, J Sonnenschein, ME Peskin, S Yankielowicz, Phys Rev D 5 Ž wx 5 E D Hoker, Y Mimura, N Sakai, Phys Rev D 54 Ž wx 6 L Alvarez-Gaume, ` J Distler, C Kounnas, M Marino, Int Jour Mod Phys Ž ; L Alvarez-Gaume, ` M Marino, F Zamora, Int Jour Mod Phys 3 Ž and hep-thr wx 7 H-C Cheng, Y Shadmi, hep-thr98046 wx 8 M Chaichian, W-F Chen, T Kobayashi, hep-thr wx 9 N Arkani-Hamed, GF Giudice, MA Luty, R Rattazzi, hep-phr w0x MA Shifman, AI Vainshtein, Nucl Phys B 77 Ž wx K Konishi, Phys Lett B 35 Ž ; K Konishi, K Shizuya, Nuovo Cim A 90 Ž 985 wx N Arkani-Hamed, H Murayama, hep-thr w3x N Arkani-Hamed, H Murayama, hep-thr w4x J Hisano, M Shifman, Phys Rev D 56 Ž w5x GF Giudice, R Rattazzi, Nucl Phys B 5 Ž w6x K Intriligator, P Pouliot, Phys Lett B 353 Ž w7x C Vafa, E Witten, Nucl Phys B 34 Ž w8x J Preskill, S Weinberg, Phys Rev D 4 Ž w9x S Dimopoulos, J Preskill, Nucl Phys B 99 Ž w0x C Csaki, M Schmaltz, W Skiba, Phys Rev D 55 Ž

297 0 May 999 Physics Letters B Probing the MSSM Higgs sector via weak boson fusion at the LHC Tilman Plehn, David Rainwater, Dieter Zeppenfeld Department of Physics, UniÕersity of Wisconsin, Madison, WI 53706, USA Received 4 February 999 Editor: M Cvetič Abstract In the MSSM weak boson fusion produces the two CP even Higgs bosons with a combined strength equivalent to the production of the Standard Model Higgs boson The tt decay mode supplemented by gg provides a highly significant signal for at least one of the CP even Higgs bosons at the LHC with reasonable luminosity The accessible parameter space covers the entire physical range which will be left unexplored by LEP q 999 Published by Elsevier Science BV All rights reserved Introduction The search for the Higgs boson and the origin of spontaneous breaking of the electroweak gauge symmetry is one of the main tasks of the CERN Large Hadron Collider Ž LHC Within the Standard Model Ž SM, a combination of search strategies will allow a positive identification of the Higgs signal wx : for small masses Ž m Q 40 GeV H the Higgs boson can be seen as a narrow resonance in inclusive two-photon events and in associated production in the tth, bbh and WH channels with subsequent decay H gg w 4 x For large Higgs masses Ž m R30 GeV H, the search in H ZZ Ž) 4 l events is promising Additional modes have been suggested recently: the ) inclusive search for H WW l l pu wx T 5, and the search for H gg or tt in weak boson fusion events w6,7 x With its two forward quark jets, the weak boson fusion possesses unique characteristics which allow identification with a very low level of background at the LHC At the same time, reconstruction of the tt invariant mass is possible; modest luminosity, of order of 30 fb y, should suffice for a 5s signal In the minimal supersymmetric extension of the SM the situation is less clear wx The search is open for two CP even mass eigenstates, h and H, for a CP odd A, and for a charged Higgs boson H " For large tanb, the light neutral Higgs boson may couple much more strongly to the T3 syr members of the weak isospin doublets than its SM analogue As a result, the total width can increase significantly compared to a SM Higgs boson of the same mass This comes at the expense of the branching ratio Bh Ž gg, the cleanest Higgs discovery mode, possibly rendering it unobservable and forcing the considera r99r$ - see front matter q 999 Published by Elsevier Science BV All rights reserved PII: S

298 98 ( ) T Plehn et alrphysics Letters B tion of alternative search channels Even when discovery in the inclusive gg channel is possible, observation in alternative production and decay channels is needed to measure the various couplings of the Higgs resonance and thus identify the structure of the Higgs sector wx 8 In this Letter we explore the reach of weak boson fusion with subsequent decay to tt for Higgs bosons in the MSSM framework We will show that, except for the low tanb region which is being excluded by LEP, the weak boson fusion channels are most likely to produce significant h andror H signals Neutral Higgs bosons in the MSSM Some relevant features of the minimal supersymmetric Higgs sector can be illustrated in a particuwx 9 : including the leading larly simple approximation contributions with respect to GF and the top flavor Yukawa coupling, h s m rž Õs t t b The qualitative features remain unchanged in a more detailed description All our numerical evaluations make use of a renormalization group improved next-to-leading order calculation w0, x The inclusion of two loop effects is not expected to change the results dramatiw x Including the leading contributions with cally respect to GF and h t, the mass matrix for the neutral CP even Higgs bosons is given by ž / ž ys c / ys c s s b ysbcb M sm A ysb cb cb c b b b 0 0 Z ž 0 / b b b qm q, 3m 4 tgf s ' p sb = ž / M A A log q y SUSY t t mt MSUSY MSUSY Ž Here s b,cb denote sinb,cos b The bottom Yukawa coupling as well as the higgsino mass parameter have been neglected Ž m < M SUSY The orthogonal diagonalization of this mass matrix defines the CP even mixing angle a Only three parameters govern the Higgs sector: the pseudo-scalar Higgs mass, m, A tanb, and, which describes the corrections arising from the supersymmetric top sector For the scan of SUSY parameter space we will concentrate on two particular values of the trilinear mixing term, Ats 0 and A s' 6 M, which commonly are referred t SUSY to as no mixing and maximal mixing Varying the pseudoscalar Higgs boson mass, one finds saturation for very large and very small values of m either m or m approach a plateau: A h H h Z b b b A m,m c ys qs for m `, m H,m Zqs b for ma 0 For large values of tan b these plateaus meet at m h, Hfm Zq Smaller tanb values decrease the asymptotic mass values and soften the transition region between the plateau behavior and the linear dependence of the scalar Higgs masses on m These A effects are shown in Fig, where the variation of mh and mh with ma is shown for tanbs4,30 The small tan b region will be constrained by the LEP analysis of Zh, ZH associated production, essentially imposing lower bounds on tan b if no signal is observed The theoretical upper limit on the light Higgs boson mass, to two loop order, depends predominantly on the mixing parameter A t, the higgsino mass parameter m and the soft-breaking stop mass parameters, which we treat as being identical to a supersymmetry breaking mass scale: mqs mus M w0 x SUSY As shown in Fig, the plateau mass value hardly exceeds ; 30 GeV, even for large values of tanb, MSUSY s TeV, and maximal mixing w x Theoretical limits arising from the current Although the search for MSSM Higgs bosons at the Tevatron is promising w3x we only quote the Zh,ZH analysis of LEP w4x which is complementary to the LHC processes under consideration The LEP reach is estimated by scaling the current limits for y ' w x y L s58 pb and s s89 GeV 4 to L s00 pb and 's s00 GeV

299 ( ) T Plehn et alrphysics Letters B Fig Variation of Higgs boson masses, couplings to gauge bosons, and signal rate, spbž tt, for the CP even MSSM Higgs bosons as a function of the pseudoscalar Higgs mass The complementarity of the search for the lighter h Ž upper row and heavier H Ž lower row is shown for tanbs4,30 Ž dashed, solid lines Other MSSM parameters are fixed to ms00 GeV, M s TeV, and maximal mixing SUSY LEP and Tevatron squark search as well as the expected results from Zh, ZH production at LEP assure that the lowest plateau masses are well separated from the Z mass peak The production of the CP even Higgs bosons in weak boson fusion is governed by the hww, HWW couplings, which, compared to the SM case, are suppressed by factors sinž b y a,cosž b y a, rew5 x In the m plateau region Ž large m spectively h A, the mixing angle approaches asbypr, whereas in the m plateau region Ž small m H A one finds a fyb This yields asymptotic MSSM coupling factors of unity for h production and < cosž b < R08 for the H channel, assuming tanbr3 As a result, the production cross section of the plateau states in weak boson fusion is essentially of SM strength In Fig the SUSY suppression factors for s Žqq qqhrh, as compared to a SM Higgs boson of equal mass, are shown as a function of m A The weak boson fusion cross section is sizable mainly in the plateau regions, and here the h or H masses are in q y the interesting range where decays into bb and t t are expected to dominate Crucial for the observability of a Higgs boson are the tt or bb couplings of the two resonances Splitting the couplings into the SM prediction and a SUSY factor, they can be written as mb sina hbbhs y Õ ž cosb h bbh m b / s Ž sinž bya ytanb cosž bya, Õ mb cosa s Õ cosb m b s Ž cosž bya qtanb sinž bya Ž 3 Õ and analogously for the t couplings Since for effective production of h and H by weak boson fusion we need sin Ž bya f and cos Ž bya f, respectively, the coupling of the observable resonance to bb and tt is essentially of SM strength The SUSY factors for the top and charm couplings are obtained by replacing tanb yrtanb in the final expressions above They are not enhanced for tan b ) This leads to bb and tt branching ratios very similar to the SM results In fact, in the plateau regions they somewhat exceed the SM branching ratios for a given mass

300 300 ( ) T Plehn et alrphysics Letters B The tt h and tth couplings vanish for sinas0 and cosas0, respectively, or sinž a s0 In leading order, as well as in the simple -approximation given in Eq Ž, this only happens in the unphysical limits tan b s 0,` Including further off-diagonal contributions to the Higgs mass matrix might introduce a new parameter region for the mixing angle a: the off-diagonal element of the Higgs mass matrix and thereby sinž a can pass zero at finite ma and tan b Indeed, by also considering the dominant contribution with respect to Ž mrm, one finds w0x Ž M sym A s b c b SUSY 4 3 tanb ht m A t ymz sbcb y, 4 8p g M SUSY Ž M sinž a s, Ž 4 m ym H h and sinž a may vanish in the physical region The exact trajectory sinž a s0 in parameter space depends strongly on the approximation made in perturbative expansion; we observe this behavior for large A R 3 M, iein part of the non-msugra pat SUSY Fig Mass of the CP even Higgs bosons and weak boson fusion rates spbž tt,gg as a function of the trilinear mixing term, A t Curves are shown for MSUSY s TeV and ms400 GeV with m A s30 GeV, tanb s30 Ž h: upper row, and m A s05 GeV, tanb s Ž H: lower row rameter space If the observed Higgs sector turns out to be located in this parameter region, the vanishing coupling to bb,tt would render the total widths small This can dramatically increase the hrh gg branching ratio, even though G Ž hrh gg may be suppressed compared to the SM case This situation is shown in Fig, where the scalar masses and the tt and gg rates are shown as a function of A t: the vanishing of the tt rate is associated with a very large increase of s BŽ gg Note that the variation of Higgs masses and decay properties with At is quite mild in general, apart from this sinž a s0 effect 3 Higgs search in weak boson fusion Methods for the isolation of a SM Higgs boson signal in the weak boson fusion process Žqq qqh, qqh and crossing related processes have been analyzed for the H gg channel wx 6 and for H tt wx 7 The analysis for the MSSM is completely analogous: backgrounds are identical to the SM case and the changes for the signal, given by the SUSY factors for production cross sections and decay rates, have been discussed in the previous section For the h, H gg signal, the backgrounds considered are gg jj production from QCD and electroweak processes, and via double parton scattering wx 6 It was found that the backgrounds can be reduced to a level well below that of the signal, by tagging the two forward jets arising from the scattered Ž anti quarks in weak boson scattering, and by exploiting the excellent gg invariant mass resolution expected for the LHC detectors w6,7 x, of order GeV For h, H tt decays, only the semileptonic decay channel of the t leptons, tt l " h pu T is considered, assuming the t-identification efficiencies and procedures described by ATLAS for the incluw7,7 x According to the AT- sive H, A tt search LAS study, hadronic t decays, producing a t jet of E T )40 GeV, can be identified with an acceptance of 6% while rejecting hadronic jets with an efficiency of 9975% In weak boson fusion, and with the t identification requirements of Refs w7,7x which ask for substantial transverse momenta of the charged t decay products Žp Ž l " ) 0 GeV and T

301 ( ) T Plehn et alrphysics Letters B p Ž h )40 GeV T, the Higgs boson is produced at high p T In the collinear t decay approximation, this allows reconstruction of the t " momenta from the directions of the decay products and the two measured components of the missing transverse momenw7,8 x Thus, the Higgs boson mass can tum vector be reconstructed in the tt mode, with a mass resolution of order 0%, which provides for substantial background reduction as long as the Higgs resonance is not too close to the Z tt peak With these t-identification criteria, and by using double forward jet tagging cuts similar to the h, H gg study, the backgrounds can be reduced below the signal level, for SM Higgs boson masses between 05 to 50 GeV and within a 0 GeV invariant mass bin Here, irreducible backgrounds from Zjj events with subsequent decay of the Ž virtual Z,g into t pairs, as well as reducible backgrounds with isolated hard leptons from Wjqjj and bbjj events, have been considered Moreover, it was shown that a further background reduction, to a level of about 0% of the signal, can be achieved by a veto on additional central jets of E T )0 GeV between the two tagging jets This final cut makes use of the different gluon radiation patterns in the signal, which proceeds via color singlet exchange in the t-channel, and in the QCD backgrounds, which prefer to emit additional partons in the central region w9,0 x Using the SUSY factors of the last section for production cross sections and decay rates, one can directly translate the SM results into a discovery reach for supersymmetric Higgs bosons The expected signal rates, s BŽ hrh tt,gg are shown in Figs and They can be compared to SM rates, within cuts, of s BŽ H tt s035 fb and s BŽH gg s fb for mh s0 GeV Except for the small parameter region where the tt signal vanishes, and for very large values of m Ž A the decoupling limit, the gg channel is not expected to be useful for the MSSM Higgs search in weak boson fusion The tt signal, on the other hand, compares favorably with the SM expectation over wide regions of parameter space The SUSY factors for the production process determine the structure of spbžhrh tt Apart from the typical flat behavior in the asymptotic plateau regions they strongly depend on b, in particular in the transition region, where all three neutral Higgs bosons have similar masses and where mixing effects are most pronounced Given the background rates determined in Ref wx 7, which are of order 003 fb in a 0 GeV mass bin, except in the vicinity of the Z-peak, the expected significance of the hrh tt signal can be determined 5 s contours for an integrated luminosity of 00 fb y are shown in Fig 3, as a function of tanb and m A Here the significances are determined from the Poisson probabilities of background fluctuations wx 7 Weak boson fusion, followed by decay to t-pairs, provides for a highly significant signal of at least one of the CP even Higgs bosons Even in the low tanb Fig 3 5s discovery contours for h tt and H tt in weak boson fusion at the LHC, with 00 fb y Also shown are the projected LEP exclusion limits Ž see text Results are shown for SUSY parameters as in Fig, for maximal mixing Ž left and no mixing Ž right The marked point is illustrated in Fig 4