1 Introduction Recently it was pointed out that elementary particles with masses up to GeV may be produced with measurable cross sections in periphera

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1 Production of Supersymmetric Particles in Ultrarelativistic Heavy-Ion Collisions J. Rau, B. Muller, W. Greiner Institut fur Theoretische Physik, Johann Wolfgang Goethe - Universitat, Postfach , D-6 Frankfurt am Main, West Germany G. So Gesellschaft fur Schwerionenforschung (GSI), Planckstrae 1, Postfach 11 55, D-61 Darmstadt, West Germany 199 Abstract Heavy ions with energies of some TeV per nucleon can be considered as carriers of intense photon elds. In peripheral collisions elementary particles with masses up to a few 1 GeV can be produced via two-photon processes. Here we evaluate total cross sections for the formation of charged supersymmetric particles, namely sleptons and winos, in Pb + Pb collisions. At LHC energy E ion = 3:5 TeV/u and for rest masses around m = 5 GeV we obtain ~W + ~ W? ' 1 nb and ~l +~ l? ' 1 nb. Basic features of an impact parameter-dependent treatment of the production mechanism are discussed. present address: Trinity College, Cambridge CB 1TQ, England 1

2 1 Introduction Recently it was pointed out that elementary particles with masses up to GeV may be produced with measurable cross sections in peripheral collisions of heavy nuclei at TeV per nucleon energies [1]. Future accelerators will allow to probe this energy range, with energies up to 3.5 TeV per nucleon at the CERN Large Hadron Collider (LHC) and up to 8 TeV per nucleon at the Superconducting Supercollider (SSC) in Texas. Very rapidly ( 1) moving ions can be considered as carriers of intense photon elds. A simple estimate for the maximum photon energy yields h! = hc R ' 1 GeV ( GeV) ; (1) assuming a Lorentz contraction factor ' 35 (8) at the LHC (SSC) and a nuclear radius R ' 7 fm for a lead nucleus. Hence exotic particles with masses up to a few 1 GeV can be produced via two-photon processes. Recently cross sections for intermediate vector boson and Higgs boson production in U + U or Pb + Pb collisions were predicted to be in the nanobarn range [1]. Here we would like to draw attention to the fact that also supersymmetric particles may be produced in ultrarelativistic heavy-ion collisions. Supersymmetry [, 3] is one of the most promising attempts to unify all fundamental forces in nature. It involves a symmetry between bosons and fermions which allows for a unied description of matter, built of fermions, and interactions, carried by bosons. Apart from its intrinsic beauty there are three major motivations for supersymmetry: (a) Unication of all interactions: In order to unify the strong, weak and electromagnetic interaction with gravity one has to connect internal symmetries with space-time symmetries in a non-trivial way. Due to the theorem of Coleman and Mandula [4] this is impossible in the framework of symmetries described by standard Lie groups. Supersymmetry is a way of obtaining a non-trivial unication of space-time and internal symmetries [5]. (b) Cancellation of divergencies: As boson and fermion loops contribute to radiative corrections with opposite signs many divergencies cancel pairwise. Hence in supersymmetric gauge theories the problem of divergencies is less severe. In particular, there are no quadratic divergencies. (c) Resolution of the gauge hierarchy problem: The standard model is unnatural in the sense that radiative corrections m H to the Higgs mass m H are much bigger than m H itself. Supersymmetry avoids such big radiative corrections so that the mass hierarchy, once established at tree level, is preserved to higher orders of perturbation theory. An immediate consequence of supersymmetry is the existence of supersymmetric partners to all known elementary particles, the so-called superpartners, which have exactly the same properties except that their spin diers by 1. Since no light charged supersymmetric particles (such as the selectron) have been observed, supersymmetry must be broken. Nonetheless superpartners of all known particles should exist, and their detection is the experimentum crucis for supersymmetry. In our report we evaluate total cross sections for the formation of sleptons and winos in ultrarelativistic heavy-ion collisions. Thereby we restrict our considerations to peripheral collisions of high- nuclei in order to avoid any direct hadronic reaction.

3 The Method of Equivalent Photons For 1, because of Lorentz contraction, the electromagnetic eld of a rapidly moving nucleus becomes almost transverse, and electric and magnetic eld both have the same strength. Hence an observer in the laboratory frame cannot distinguish the moving Coulomb eld from that of a plane wave. So, eectively, the moving nucleus can be regarded as a carrier of quasi-real photons with a certain frequency spectrum n(!). If two highly relativistic heavy ions collide, processes in the electromagnetic eld of these two nuclei can be described as a sum over incoherent photonphoton scatterings. Within this approximation, which is the method of equivalent photons or Weizsacker-Williams-method [, 6, 7, 8], two-photon cross sections are summed up incoherently. If, for example, the nal state f is produced in the electromagnetic eld of two colliding nuclei A and B, the total production cross section reads (h = c = 1) f AB = d! A d! B n A (! A ) n B (! B ) (! f A ;! B ) = d! A d! B n A (! A ) n B (! B ) ds (4! A! B? s) (s) f ; () f being the elementary two-photon cross section for the production of f. f only depends on the C.M. energy p s, where s is given by s = 4! A! B provided the two photons are massless and move collinearly. This is guaranteed by the -function in (). n A and n B are the photon spectra of the nuclei A and B, respectively. The frequency spectrum of a nucleus with charge e is given by n(!) =! 1!= dk k? (!=) k 3 F (k ) : (3) Here we assumed the nuclear charge distribution to be spherically symmetric. F (k ) then denotes the elastic nuclear form factor, i.e. the exact Fourier transform of the charge density of the nucleus. Especially in the case of a homogeneously charged sphere with radius R we obtain This yields with! F (k ) = 3 j 1(kR) kr : (4) n(!) =! f(!=! ) (5) =R and f(x) = 1 x dz z z? x z 3 sin z? z cos z : (6) z 3 Due to the 4 dependence of the total production cross section heavy ions are more eective than protons or electrons by a factor up to 1 8. For nal states with invariant mass below! this enhancement far outweighs suppression caused by the nuclear form factor. Actually the cross section depends on the range of impact parameters b admitted. As will be shown in the appendix, equation () can be modied to with = 16 4 d b d! A d! B (!; q) = 1! d q () eiqb (! A ; q)(! B ; q) (! A! B ) (7) d k? F (?k ) F [?(k? q) ] () k? : (8) k (k? q) The integration R d b can be performed over an arbitrary domain of impact parameters. 3

4 3 Production of Scalars and Dirac Fermions The two-photon cross section for the production of a pair f + f? of Dirac fermions reads [, 8] " f+ f? (s) = 4 (1 + 1 s x? 1 x ) ln(p x + p x? 1)? r1? 1x (1 + 1x # ) (x? 1) (9) with the abbreviation x = s ; 4m where m denotes the particle mass. Similarly one nds for the two-photon production cross section of a pair b + b? of scalar bosons "r b+ b? (s) = 1? 1 s x (1 + 1 x )? 1 x (? 1 x ) ln(p x + p # x? 1) (x? 1) :(11) Given the same mass m, the production cross section of Dirac fermions is generally larger than that of scalar bosons, partly due to the contribution of dierent spin orientations. Inserting (9) and (11) into () it is now easy to evaluate total cross sections for pair creation in relativistic collisions of heavy nuclei [7]. In order to illustrate the mass dependence of the total production cross section we computed for pair production of charged Dirac fermions in Pb + Pb collisions as a function of the fermion mass m. Figure 1 shows the results for SSC energy E ion = 8 TeV/u, where we used the form factor of a homogeneously charged sphere with radius R = 7:17 fm. (1) 4 Production of Supersymmetric Particles So far our considerations have been very general, and no assumptions were made concerning the particular type of particles produced. Only now we concentrate on supersymmetric particles (sparticles). Formally one distinguishes them from ordinary particles by introducing a quantum number R (R parity) which is +1 for particles and?1 for sparticles. In most supersymmetric theories this quantum number is multiplicatively conserved. Immediate consequences are: (a) sparticles have to be produced in pairs, (b) heavy sparticles decay into lighter sparticles and (c) the lightest supersymmetric particle (LSP) is stable, because there are no decay channels available. Cosmological bounds require that the LSP should be neutral [9]. Possible candidates are the photino ~, the sneutrino ~ or the Higgsino h. ~ In any case, the LSP is very weakly interacting and, once produced, escapes undetected. As we consider production of sparticles in heavy-ion collisions, i.e. by virtue of electromagnetic interaction, it is obvious to focus on charged sparticles. In particular, we compute the production cross section for sleptons ~ l and winos W ~. Since global supersymmetry and conventional internal symmetries decouple, internal quantum numbers like charge, weak isospin etc. remain unaected by supersymmetry transformations. Therefore the interaction of charged sparticles with the electromagnetic eld can be described by minimal coupling as usual. Especially sleptons behave like ordinary charged scalars and winos like ordinary charged Dirac fermions in the electromagnetic eld. So in order to evaluate total production cross sections in ultrarelativistic heavy-ion collisions one simply has to insert (9) and (11) into (), m now being a parameter for the unknown slepton and wino mass, respectively. Lower mass limits are provided by existing high-energy accelerators where sparticles have not yet been observed. Actually one can exclude certain regions in the m ~ l m ~ - and m W m ~ ~ - plane [1]. From the TOPA collaboration [11] we know m ~e > 8 GeV, m ~ > 4 GeV, m ~ > 1 GeV and 4

5 m W ~ > 5 GeV, assuming the photino to be massless. So far upper limits are not available, but from cosmological constraints m ~ l and m W ~ are expected to be not larger than some 1 GeV [1]. Hence these masses are supposed to lie in the energy region which the planned supercolliders can explore. Figure shows the lowest-order two-photon processes for the formation of sleptons ~ l and winos ~W, whereas possible decay channels are depicted in gure 5 assuming that either photinos or Higgsinos (left) or sneutrinos (right) are the LSP. As a common signature, one always nds a high-energetic lepton together with some missing energy and momentum, because the neutrino and the LSP cannot be detected. 5 Results and Conclusions Using the Weizsacker-Williams method we computed the total production cross section of sleptons and winos in Pb + Pb collisions as a function of the beam energy, varying the sparticle mass between 5 GeV and GeV. Figures 3 and 4 show the results. For rest masses m = 5 (1) GeV and LHC energy E ion = 3.5 TeV/u we obtained W ~ + W ~ = 1 (.15) nb and? ~ l+~ l =? (.3) nb. Considerations of accelerator-technological aspects indicate that a heavy-ion luminosity L ' 1 8 cm? s?1 can be achieved for Pb ions at 3.5 TeV/u [13]. Requiring a signal of at least 1 events/yr leads to the condition > :3 nb for the cross section to be measurable. At LHC energy sleptons and winos with masses up to 1 GeV could be detected. Given the same luminosity, at SSC energy the accessible mass region might be extended to GeV. Corresponding results for proton - proton collisions at E = 4 TeV are W ~ + W ~.1 nb and? ~ l+~ l.1 nb for masses around m = 1 GeV [14]. The disadvantage of lower luminosity? of a heavy-ion beam may well be compensated by the low-multiplicity topology of the events predicted for peripheral heavy-ion collisions without direct hadronic interaction. In order to determine the eect caused by excluding collisions with impact parameters b < R the analytical expression (7) has to be evaluated numerically. This is currently under investigation. Appendix: Impact Parameter-Dependence of the Cross Section Although we restricted our considerations to peripheral collisions with impact parameters b > R we employed the equivalent photon approximation which involves an integration over all impact parameters. As long as the mass m of the produced particles is far below the threshold energy h! this approximation is well justied. However, as soon as m ' h!, excluding impact parameters b < R might cause a considerable eect on the production cross section. In order to calculate this suppression one has to investigate the production mechanism within the framework of QED. In this section we derive a modication of the Weizsacker-Williams formula () so that a minimum impact parameter b m can be introduced []. First we re-write the frequency spectrum (3) in a slightly dierent way: n(!) = 4 1! d k? () k? k 4 F (?k ) (1) 5

6 Thus in () with 1 n A (! A )n B (! B ) = 16 4 d k A?! A! B () I = F (?k A ) F (?kb ) ka kb d k B? () k A? k A F (?ka) k B? F (?k k B)I (13) B : (14) Now our aim is to derive a similar expression from ordinary Feynman rules in which the factor I can be identied as an integral over some range of impact parameters. For this purpose we make the following assumptions: The ions move on classical trajectories. We consider particle production in the external eld of the colliding nuclei. The matrix element M fi is calculated to lowest (i.e. second) order. Transversal momenta of the virtual photons do not contribute. Hence M fi only depends on the frequencies: M fi = M fi (! A ;! B ). The virtual photons are approximately on mass-shell, i.e. quasi real ("equivalent photons"). Consequently, for M fi we can substitute the matrix element for pair production by two real photons; apart from eventual normalization factors we thus have M fi = M fi (! A;! B ). The electromagnetic eld of a moving nucleus with charge number, elastic nuclear form factor F, four-velocity u and impact parameter b reads where A (k) = e ikb r(k)u ; (15) r(k) = e (k u) F (?k ) (16) k is a real quantity provided the nuclear charge distribution is spherically symmetric. Employing our assumptions we obtain the matrix element for pair production in the electromagnetic eld of two colliding nuclei, S fi d4 k A () 4 d4 k B () 4 eik Ab r A (k A )r B (k B ) 4 (k A + k B + p? p 1 )M fi (! A;! B ) ;(17) where b is the (purely transversal) impact parameter. Figure 6 shows the related graph. Now we nd: js fi j d4 k A d4 k B d4 k A d4 kb () 4 () 4 () 4 () 4 ei(k A?kA )b The two 4 -functions imply r A (k A )r A (ka)r B (k B )r B (kb)m fi (! A;! B )M fi (! A;! B) 4 (k A + k B + p? p 1 ) 4 (ka + kb + p? p 1 ) (18) k A + k B = k A + k B : (19) Furthermore, because of the -functions contained in r(k) (cf. (16)), only the transversal components of k A=B and k A=B dier. In particular! =! and hence fi (! A;! B )M fi (! A ;! B) = jm fi (! A;! B )j : () M 6

7 Integrating jm fi j 4 (k A + k B + p? p 1 ) over all nal momenta p 1 ; p yields (again without eventual normalization factors) the production cross section (! A! B ) for the two-photon process. After performing also the kb -integration it remains: d3 p 1 d3 p () 3 () js fij d4 k A d4 k B d4 ka 3 () 4 () 4 () 4 ei(k A?kA )b r A (k A )r B (k B )r A (k A)r B (k B + k A? k A) (! A! B ) (1) If we integrated this expression over all impact parameters, the phase factor would lead to () (k A?? k A?), and we would recover the Weizsacker-Williams formula (). However, here we do not perform this integration. Let us consider A := d4 k A () 4 ei(k A??k A? )b r A (k A) r B (k B + k A? k A) : () Integrating out all the -functions contained in r(k) yields A d ka? () ei(k A??k )b A? F (?k A ) F [?(k B + k A?? k A?) ] ka (k B + k A?? k : (3) A?) Dening the purely transversal quantity we can now write A q := k A? k A = k A?? k A? (4) d q () eiqb F [?(k A? q) ] F [?(k B + q) ] : (5) (k A? q) (k B + q) Integrating A over all impact parameters leads to d b A = F (?k A ) k A F (?k B ) k B ; (6) i.e. exactly the quantity I we introduced in (14). Now in (13) replace I by I! d b d q () eiqb F [?(k A? q) ] F [?(k B + q) ] : (7) (k A? q) (k B + q) Here the integration R d b can be performed over an arbitrary domain of impact parameters. Using () we obtain the nal result (7). It is easy to see that after integration over all impact parameters the production cross section of the Weizsacker-Williams approximation is reproduced. If the integration domain includes only impact parameters b b m, i.e. a circle with radius b m, (7) can be written as with (b m ) = (!; q) = 1 1 d! A d! B dq b m J 1 (qb m )(! A ; q)(! B ; q) (! A! B ) (8) 1 1 dk k F 3 [(!=) + k ] d F [(!=) + k + q? kq cos ] ()! (!=) + k (!=) + k + q? kq cos :(9) 7

8 References [1] M. Grabiak, B. Muller, W. Greiner, G. So, P. Koch, J. Phys. G15 (1989) L5 E. Papageorgiu, Phys. Rev. D4 (1989) 9; Nucl. Phys. A498 (1989) 593c M. Drees, J. Ellis, D. eppenfeld, Phys. Lett. 3B (1989) 454 G. So, J. Rau, M. Grabiak, B. Muller, W. Greiner, preprint GSI{89{55 (1989) [] J. Rau, Supersymmetrie, GSI Report 89{ (1989) [3] see e.g. M. F. Sohnius, Phys. Rep. 18 (1985) 39 H.J.W. Muller-Kirsten, A. Wiedemann, Supersymmetry, (World Scientic, Singapore, 1987) [4] S. Coleman, J. Mandula, Phys. Rev. 159 (1967) 151 [5] R. Haag, J. T. Lopuszanski, M. F. Sohnius, Nucl. Phys. B88 (1975) 57 [6] E. Fermi,. Physik 9 (194) 315 E.J. Williams, Proc. Roy. Soc. A139 (1933) 163 C. Weizsacker,. Physik 88 (1934) 61 [7] C.A. Bertulani, G. Baur, Phys. Rep. 161 (1988) 99 [8] V.M. Budnev, I.F. Ginzburg, G.V. Meledin, V.G. Serbo, Phys. Rep. 15 (1975) 181 [9] J. Ellis, J. S. Hagelin, D. V. Nanopoulos, K. Olive, M. Srednicki, Nucl. Phys. B38 (1984) 453 [1] MARK-J Collab., B. Adeva et al., Phys. Lett. B194 (1987) 167 [11] I. Adachi et al., Phys. Lett. B18 (1989) 15 [1] H. Goldberg, Phys. Rev. Lett. 5 (1983) 1419 J. Ellis, J. S. Hagelin, D. V. Nanopoulos, Phys. Lett. B159 (1985) 6 [13] D. Brandt, Relativistic heavy ions in the LHC, CERN, SPS/AMS/Note/89{6, LHC/Note No. 87 (1989) [14] S. Dawson, E. Eichten, C. Quigg, Phys. Rev. D31 (1985) 1581 E. Eichten, I. Hinchlie, K. Lane, C. Quigg, Rev. Mod. Phys. 56 (1984) 579 8

9 Figure captions: Figure 1 Total cross section for pair production of charged Dirac fermions in Pb + Pb collisions with E ion = 8 TeV/u. is plotted versus the rest mass m of the fermion. Figure Feynman diagrams for the two-photon production of sleptons ~ l and winos ~ W. Figure 3 Production cross section of winos W ~ as a function of the ion energy E for Pb + Pb collisions. For the rest mass of the produced sparticles we assumed 5 GeV (solid line), 1 GeV (dashed-dotted line), 15 GeV (dashed line) and GeV (dotted line). Figure 4 The same as in gure 3 for the production of sleptons ~ l. Figure 5 Possible decay modes of sleptons ~ l and winos W ~. Left: The photino or the Higgsino is LSP. Right: The sneutrino is LSP. Figure 6 Graph related to pair production in the electromagnetic eld of two colliding nuclei (lowest order in perturbation theory). 9