We report results of our ongoing nonperturbative numerical study of a
classical effective theory describing low-x partons in the central region of a
heavy-ion collision. In particular, we give estimates of the initial transverse
energies and multiplicities for a wide range of collision regimes, including
those at RHIC and at LHC.Comment: Talk given at International Symposium for Multiparticle Dynamics,
  Tihany, Hungary, October 2000. LaTeX using sprocl.sty, 6 pages, figures part
  of the LaTeX fil

Krasnitz, A.

Venugopalan, R.

English

arXiv

We report results of our ongoing nonperturbative numerical study of a classical effective theory describing low-x partons in the central region of a heavy-ion collision. In particular, we give estimates of the initial transverse energies and multiplicities for a wide range of collision regimes, including those at RHIC and at LHC

VENUGOPALAN, R.

UNT Digital Library

., i BNL-68097 MULTIPARTICLE PRODUCTION AT RHIC AND LHC: A CLASSICAL POINT OF VIEW A. KRASNITZ CENTRA and Faculdade de Cie^ncias e Tecnologiu, Universidade do Algarve, Campus de Gambelas, P-8000 Fare, Portugal E-mail: lxasnitz@ualg.pt R. VENUGOPALAN Department of Physics and RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA E-mail: rajuQbnl.gov We report results of Our ongoing nonperturbative numerical study of a classical ef- fective theory describing low-z partons in the central region of a heavy-ion collision. In particular, we give estimates of the initial transverse energies and multiplicities for a wide range of collision regimes, including those at RHIC and at LHC. In the central-rapidity region of hea.vy-ion collisions at RHIC and a.$ LHC a. combination of very high center-of-mass energy wit,11 a. very la,rge number of participating valence quarks is expected to lea,d to a novel regime of QCD, oue chara.cterized by a very high parton density. This regime may not be amenable to analysis by conventional methods. such as multiple scattering or classica. cascade descriptions, which ignore the coherence of the gluon field I. This coherence emerges na.tura.lly from the classical effective field theory approa,cll of McLerran and Venugopah (MV) ‘. If the p&on density in the colliding nuclei is high at. sma.ll X, cla.ssica.1 methods arc’vvalid. It has beeu shown recent*ly that a RG-improved genera.liza.tion of this efl’ective action reproduces several key results in small-a: QCD s. The model can be summa.rized as follows. The high-z a.nd the low-~ pa.r- t,ons a.re treated sepa.rately. The former corresponds t,o valence quarks and hrd sea. partons. These high-x pa,rtons are considered recoilless sources of color charge. For a. large Lorentz-contracted nucleus, this results in a. sta.tic Ga.ussia,n distribution of their color chrge density /I in the tlansverse pla.ne: The va.riance A, of the color chrgc clist.ribut~iou is the only dimensional pa- ra.mcter of the model, a.part from t,lic linear size L of the nucleus. l’br central impa.ct, para.meters, A, ca.n be estimat.ed in t*erms of single-nucleon st.ructure 1 functious 4. It is assumed, in addition, that the nucleus is infinitely thiu in the longitudinal direction. Under this simplifying assumption the resulting gauge fields are boost-invariant. The sma,ll z fields a.re then described by the classical Yang-Mills equations DLaFItv = Jv (1) with the random sources on the ho light coues: J,, = Cl 2 Sv,~S(~~)~~,~(~*~). The two signs correspond to two possible directions of motion along the bea,m axis z. As showu by Kovner, McLerra.n and Weigert (KMW) 5, low-~ fields in the cent& region of the collision obey sourceless Yang-Mills equations (this region is in the forward light cone of both ,nuclei) with the initial conditions in the A, = 0 gauge given by Here the pure gauge fields Ai,,, a.re solutions of (1) for each of the two uuclei in the a.bseuce of the other nucleus. 0.5 I I I I 0.3 ET --Y- A: 0.2 k I i+yre 1: Transverse-plnneenergy clcnsil,v per unit rapiclit.y versus proper time for the values 5.GG (dianmncls), 35.:X (pl~~sscs), and 297 (squares) of ASL.Both the energy density and the proper time are expressed in units of A,. ‘Tlw solid lines are fits of the data to the form a+/3esp(--(T). In order to obtain l.lic rcsultiiig gluon lick1 configuration at. late proper hnes, one needs t.o solve (1) wit.11 the ini1,ia.l couclition (2). Siiicc th latter depends on the random color, source, averages over realizations of the source must be performed. KMW showed that in perturbation theory the gluon munber distribution by transverse momeutum (per unit rapidity) suffers from au infrared divergeuce and argued that the distributiou must have the form for kl >> A,. The log term clearly iudicates tl1a.t the perturbative description brea.ks down for kl - A,. A relia.ble wa.y to go beyond perturbatiou theory is to re-formulate the EFT on a. la.ttice by cliscretizing the transverse plane. The resultiug lattice theory cau then be solved munerically. We sha.11 not dwell here on the details of the lattice formulation, which is described in detail in Ref. ‘. Keeping in mind tl1a.t A, a.ud the 1inea.r size L of the nucleus a,re the only physica.lly in- teresting dimensional parameters of the model 7, we cau writ#e a,ny dimensional quautity Q as hf.f, (A, L) , where d is the dimension of (I. All the non-trivial physical informatiou is couthed in the climeusionless function .fcl (A, L). We can estimate the va.lues of the product A,& which correspond to key collider experiments. Assullling Au-Au collisions, we take L = ll.G fm (for a square nucleus!) a.ud estima.te the standa.rcl devia.tiou A, to be 2 GeV for R.HIC and 4 Ge\T for LHC ‘. Also, we hve a.pproxima.te.ly 11 = 2 for energies of interest. The rough estima.te is then A,qL M 120 for RHIC aad A,?L M 240 for LHC. We now proceed, to describe the results qf our numerical study so far. Our simulst~ions were performed for the SU(2) ga.uge group, in order to keep the comput,a.tioiia.l cost,s low. h,L 17.68 35.36 70.7 106.06 148.49 212.13 296.95 SB .323( 4) .208(4) .300( 5) .211( 1) .332( 1) .234(2) .257(5) Tal~le 1: The function f~, i.~., the energy per unit. tnmsversearea per unit rapidity, expressed in unit.s of A,<, tabulated vs h,L. We first1 compute the energy per unit trausverse arca per uuit rapidity, &posited in the centra.1 region by hlle colliding uuclei. As Figure 1 illustra.tes, this quantity teucls t.o a coushut, at late proper times. We Iiud tl1a.t at, T --+ 1s~ he energy densit,); in units or A, depends on the climensiouless parameter A,L as described in Table 1. Note the very slow variation of this dimeusionless function in the entire range of A, L values, which includes l)oth our RHIC and T,I-1CI cst#ima.tes. \lsing t-his plot9. and assuming. iii accordance with Ref. ‘> the (IV<? - l)/fVc dependeuce of the energy on tlic number of colors IV,, we a.rrive at the values of 2700 GeV and of 25000 GeV for the transverse energy per unit rapidity at RHIC and at LHC, respectively ‘. 1 4kd 0.1 0.01 t 0 0.5 1 1.5 2 2.5 3 k.L/As Figure 2: n(kl) E dN/L’/d’kl as a function of the gluon momentum kl for h,L = 297 (diamonds). The solid line is a fit to Bose-Einstein distribution, corresponding to a gas of free particles al. the inverse temperature /3 = (1.25 zk O.Ol)A, and with the mass m = (0.092 f O.O03)A,. The uurnlxr and distribution of produced gluons are oE considerable in- tere& as initial da.ta. for possible evolution of the gluon ga.s towasds therma. eyuilil~rium8~10~11. Strictly spea,l;ing, a. pa.rticle number is only well-defined in a. free-field theory, and there is no unique est,ension of this notmion to a. gcn- era1 intera.cting case. For this rea,son we use two different generalizations of the paxticle number to an int,era.cting theory, each lmving the correct free-field limit. We verify that the two clehitions agree in the wea.li-coupling regime corresponding to late proper times in the centra.1 region “‘. Our first definition is straiglftforyard. We impwe the Coulomb gauge con- dition in the tramverse phne: VI . Al = 0 a.nd a.ud determine the momentun~ cou~pouents of the resulting field configuration. Our second definition is lx~sed ou he behavior of a free-field theory under rela.xation. Consider a. simple relaxakion equa.tion for a. fielcl in real spa,ce, where t is the relasa.tiou tiinc (not to be conli~secl wit-11 maI or proper t.ime) and ,. 1 V is the potential. It is then easy to derive the following integral expression for the total particle number of a free-field system: N= s ss m dt 7r 0 - V(f), 4 (5) where V(t) is the potential for the relaxed field. Now (4) can be solved numer- ically for interacting fields. Subsequent.ly, V(t) can be determined, and N can be computed by numerical integration. This technique presently only permits determination of the total particle number. We now present our results using both the techniques discussed. Our results for the number distribution, computed in Coulomb gauge, axe as follows. We have verified that for la.rge kl our numerically obtained multiplicity a.grees wit,h the lattice analogue of the perturbakive expression (3). At smaller kl, the distribution softens and converges to a. constant, value, unlike its perturba.tive counterpa.rt. Notably, this qua.lita.tive change of the distribution occurs a.t kl N A,. We tried to quantify this non-perturbative beha.vior by fitting the distribution to a variety of physica.lly motivated functional forms. Surprisingly, we find tha.t the sha.pe of the distribution is closely reproduced by the Bose- Einstein form n(k) = A/(esp(@k) - l), wit,h t,lie inverse temperature fi of the order of 1 in units of As, and with wk corresponding to a free massive dispersion relation, with the ma.ss of the order of O.lR,. This is an unexpected result, for a purely classical theory, whose mea,ning, beyond providing us with a convenient para.metrization, is not, yet c1ea.r. Table 2 summaxizes our findings for the tot,al gluon number, written a.s N = (A8L/s)2f~~(A8L). Note t,he good agreement between the two methods used to determine N (a, more deta.iled a.na.lysis shows tha.t the small difference between the two is likely an a.rtifact, of the Coulomb gauge). Note that, simi- larly to f~ (&L), .f,v (A,L) is a. slowly-va.rying function. Ba,sed on Table 2, and making a naive estrapola.tion from the SU(2) to the SU(3) gauge group, we estima.te the initia,l gluon number per unit. ra.pidity as N 950 for central Au-Au collisions at RHIC energies, a.nd as - 4300 for central Au-Au collisions at LHC). Let us a.gain emphasize tha.t, even though our figure for RHIC is not far from the recent experimental result,, both t.he dist.ribut,ion and the total multiplicity are likely to be cha.nged by furtkr evolution of the field configura.tion. A,L 35.36 7O.il 106.1 148.5 '13.1 '97.0 fj~ (cooling) .llG(l) .119( 1) .137(l) .138(l) .lG( 1) .151(l) ]A’ (Coulollll~) .127(Z) .135(") .1X( 1) .142( 1) .1.45( 1) .153(l) Tal~lc 2: Values of fp vs AsL fur t.lk two definitions of t.he total pallick number. At the current stage of our project we are only able to make qualitative predictions. One obvious way to improve the accuracy is by replacing the SU(2) color group by the physical SU(3) one. Another is by allowing deviations from the strict boost invariance. These issues will be a.ddressed in the future. Acknowledgments We would like to tha.nk Larry McLerra.n and Al X/Iueller for very useful dis- cussions. R. V.‘s research was supported by DOE Contract No. DEACOB- 98CH10886. The authors acknowledge support from the Portuguese FCT, under grants CERN/P/FIS/1203/98 and CERN/P/FIS/15196/1999. References 1. S.-N. Wa.ng, Phys. Rep. 280 287 (1997); I<. Geiger, Phys.Rep. 258 237 (1995) B. Zhang, Cornput. Phys. Comn~zm. 104 (1997) 70. 2. L. McLerran and R. Venugopalan, Phys. Rev. D49 2233 (1994); D49 3352 (1994); D50 2225 (1994). 3. J. Jalilian-Marian, A. Kovuer, L. McLerran, and Ii. Weigert, Phys. Rev. D55 (1997) 5414; J. Jalilian-Ma.rian, A. Kovner, A. Leonidov, and H. Weigert, A&l. Phys. B504 415 (1997); Phys. Rev. D59 034007 (1999); Erratum-ibid. D59 099903 (1999); J. Jalilian-hIarian, A. Kovner, and I-I. Weigert, Phys. Rev. D59 014015 (1999); L. McLerran and R. Venu- gopalan, Phys. Rev. D59 094002 (1999); 1’u.V. Kovchegov, Phys. Rev. D 54 (199G) 54G3, hep-ph/9005446. 4. M. Gyulassy and L. hfcLerran, Phys. Rev. C5G (1997) 2219. 5. A. Kovner, L. McLerran and I-I. Weigert, Phys. Rc,u D52 3809 (1995); D52 6231 (1995). 6. A. Krasnitz and R.. Venugopalan, hcp-ph/970G329, hep-ph/9808332; Nzd Phys. B557 237 (1999). 7. R. V. Ga.vai and R. Venugopalan, Phys. Reu. D54 5795 (1990). 8. A. H. Mueller, Nucl. Phys. B572 (2000) 227, hep-ph/9906322; Phys. Lett. B475 (2000) 220, hep-ph/9909388. 9. A. Krasnitz and R. Venugopakm, Phys. Rev. Lett. 84 (2000) 4309, liep-pli/9909203; hep-pli/9910391. 10. J. Bjoraker and R.. Venugopaiau, Phys. Rev. CG3 (2001) 024609, hep- ph/OOO8294; A. Dumit,ru and hl. C:yulassy, hep-ph/OOOG257. 11. R,. Ba.ier, A.H. hilueller, D. Schiff, D.T. Son, hep-ph/0009237. 12. A. Krasnitz and R.. Venugopalan. hep-ph/OU07108. 

Multiparticle Production at Rhic and Lhc: A Classical Point of View.

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Multiparticle production at RHIC and LHC: a classical point of view

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