The dynamics of low-x partons in the transverse plane of a high-energy
nuclear collision is classical, and therefore admits a fully non--perturbative
numerical treatment. We report results of a recent study estimating the initial
energy density in the central region of a collision. Preliminary estimates of
the number of gluons per unit rapidity, and the initial transverse momentum
distribution of gluons, are also provided.Comment: Latex,9 pages, 4 figures, invited talk at 5th workshop on QCD
  (QCD2000), Villefrance-sur-Mer, Jan.3rd-7th, 2000; minor typo correcte

Krasnitz, Alex

Venugopalan, Raju

English

arXiv

The dynamics of low-x partons in the transverse plane of a high-energy nuclear collision is classical, and therefore admits a fully non-perturbative numerical treatment. The authors report results of a recent study estimating the initial energy density in the central region of a collision. Preliminary estimates of the number of gluons per unit rapidity, and the initial transverse momentum distribution of gluons, are also provided

Krasnitz, A.

Venugopalan, R.

UNT Digital Library

,,, -BNL-67344NON-PERTURBATIVE GLUODYNAMICS OF HIGH ENERGYHEAVY-ION COLLISIONSA. KRASNITZUCEH, Universidade do AlgarveCampus de Gambelas, P-8000 Fare, PortugalE-mail: krasnitz@ualg.ptR. VENUGOPALANPhysics Department, Brookhaven National LaboratorylUpton, NY 11973, USAE-mail: raju (i’bnl.govThe dynamics of low-x partons iri the transverse plane of a high-energy nuclearcollision is classical, and therefore admits a fully non–perturbative numerical treat-ment. We report results of a recent study estimating the initial energy density inthe central region of a collision. Preliminary estimates of the number of giuons perunit rapidity, and the initial transverse momentum distribution of gluons, are alsoprovided.In heavy-ion experiments, planned at RHIC later this year, gold ions areexpected to collide at & = 200 GeV per nucleon. A few years later & = 5.5TeV per nucleon will be attained in heavy ion collisions at LHC. In the centralregion of these collisions, a combination of very high center-of-mass energywith a very large number of participating valence quarks will likely give riseto a novel regime of QCD, one characterized by a very high parton density.This regime does not easily lend itself to a description based on conventionalapproaches. Collisions involving large transverse momenta can be adequatelydescribed in terms of pairwise scattering of individual partons comprising thecolliding systems. Final-state interactions of secondary partons formed thereincan be safely neglected 1. However, as the parton density grows, final-state in-teractions of secondary partons must be taken into account. This requirementis only partially satisfied by multiple scattering or by classicai cascade descrip-tions, which ignore the coherence of the secondary field configuration 2.The coherence of the secondary partons is incorporated naturally into theclassical effective field theory approach of McLerran and Venugopalan (MV) 3.If the parton density in the colliding nuclei is high at small z, classical mctll-ods are applicable. It has been shown recently that a RG-improved general-ization of this effective action reproduces several key results in small-x Q(UD:the leading u, log(l/z) 13FIi L equation, the double log GLR equation and itsextensions, and the srnall-.r DGLA P equation for quark distributions’1.I.-,, ,Briefly, the model is based on the following assumptions. Partons in anucleus are separated into high-z and the low-z components. The formercorresponds to valence quarks and hard sea partons. These high-x partons areconsidered recoilless sources of color charge. For a large Lorentz-contractednucleus, this results in a static Gaussian distribution of their color chargedensity p in the transverse plane:The variance p’ of the color charge distribution is the cnly dimensional pa-rameter of the model, apart from the linear size L of the nucleus. For centralimpact parameters, p is given in terms of single-nucleon structure functions 5:A1/3 1/f=—/(dz)~q(z, Q2) + ~g(q Q2) ,7rr~ ~0 2NC cwith the separation scale Z. s Q/~, r. = 1.12 fm, and NC the number Gfcolors. It is assumed, in addition, that the nucleus is infinitely thin in thelongitudinal direction. Under this simplifying assumption the resulting gaugefields are boost-invariant.The small z fields are then described by the classical Yang-Mills equationsDp Fp. ==J. (1)with the random sources on the two light cones: J. = ~1,26.,AC$(ZT )P1,2(rt).The two signs correspond to two possible directions of motion along the beamaxis z. As shown by Kovner, McLerran and Weigert (KMW), low z fields inthe central region of the collision obey sourceless Yang-Mills equations (thisregion is in the forward light cone of both nuclei) with the initial conditions inthe AT = Ogauge given byAi=Ai+ Ai.‘, A* = +*[A~,A; ]. (2)Here the pure gauge fields A\,2 are solutions of (1) for each of the two nucleiin the absence of the other nucleus.Equation (1) with the initial condition (2) can now be solved, in order toobtain the resulting gluon field configuration at late proper times. Since theinitial condition depends on the random color source, averages over realizationsof the source must be performed. This aspect of the solution resembles theclassical thermal theory, wherein an average is performed over initial conditions2,., -Figure 1: Transverse-plane energy density per unit rapidity versus proper time for the va[ues5.66 (diamonds), 35.36 (plusses), and 297 (squares) of g2pL. Both the energy density andthe proper time are expressed in units of g2 V. The solid lines are fits of the data to the forma + flexp(–-m).drawn from the canonical ensemble. In fact, the analogy between the MVeffective theory and the classical thermal theory goes further. This analogycan be made explicit by considering the perturbative solution of ( 1) obtained byKMW. They showed that in perturbation theory the gluon number distributionby transverse momentum (per unit rapidity) suffers from an infrared divergenceand argued that the distribution must have the form“+(%?41’1(% (3)for k~ >> CX,p. We can now draw a parallel between p and the temperature T ofthe thermal system. In particular, the log term suggests that the perturbativedescription breaks down for kl = a,p. Likewise, the perturbative thermaltheory loses validity at the non-perturbative scale k w g2T.It is therefore clear that a fully non-perturbative study of MV model isnecessary. The model is discretized on a lattice in the transverse plane andthe lattice field equations solved numerically. Boost invariance and periodicboundary col~clitions in the transversc plane are assumed. Technical detailsof the lattice formulation can be found in Ref. 8. The cluantity g2p ancl the3. -., . .linear size L of the nucleus are the only physically interesting dimensionalparameters of the MV mode17. Any dimensional qurmtity q can then be writtenas (g2p)dfq(g2pL), where d is the dimension of g. All the non-trivial physicalinformation is contained in the dimensionless funct.on fqQT21JL).On a lattice,g will generally depend also on the lattice spacing a; we will seek to removethis dependence by taking the continuum limit a + O. Finally, we estimatethe values of the dimensional parameter g2pL which correspond to key colliderexperiments. Assuming Au-Au collisions, we take L = 11.6 fm (for a squarenucleus!) and estimate the standard deviation p to be 0.5 GeV for RHIC and1 GeV for LHC 5. Also, we have approximately g = 2 for energies of interest.The rough estimate is then g2pL w 120 for RHIC and g2pL N 240 for LHC.Clearly, there is some variation in g2pL due to the various uncertainties inthis estimate. The expression we will derive is a non–perturbative formula,from which one can deduce the number or energy of produced gluons for aparticular choice of g2pL.0.50.4*O ~9.2I I i I I Ibo00 0c’ o 0I I I I I I Io 50 100 150 200 250 30092pLFigure 2: Transverse-plane energy density (in units of g2 p) per unit rapidity versus gz LL.The error bats are smaller than the plotting symbols.Results of a numerical investigation (for SU(2) only) are as follows. Wefirst compute the energy per unit transverse area per unit rapidity, deposited inthe central region by the colliding nuclei. As Figure 1 illustrates, this quantitytends to a constant at late proper times. It is this asymptotic value of theenergy density that we wish to determine. If we express the energy density in4“.,’units of g2p and extrapolate our numerical findings to the continuirm limit,we find that the energy density depends on the dimensionless parameter g2pLas described in Figure 2. Note the very slow variation of tl:is dimensionlessfunction in the entire range of g2,uL values, which iilcludes both our RHIC andLHC estimates. Using this plot, and assuming, in accordance with Ref. 11, the(N: - 1)/Nc dependence of the energy on the number of colors IVc, we arriveat the values of 2700 GeV and of 25000 GeV for the transverse energy per unitrapidity at RHIC and at LHC, respectively 9.I I I I 1 +’0.16 -+ o(g2;L)2 o000.12 - -i-+I I I I I I0 50 100 150 200 250 300g2pLFigure 3: Transverse-plane gl~]on number density per unit rapidity versus g2pL. f,-omCoulomb gauge fixing (diamonds) and from relaxation (plusses).Next, we report our preliminary estimates of low-x gluon multiplicities inthe central region. Determination of the total number of produced gluons isof considerable interest: this quantity may be directly related to the number10 Further, the momentum distribution of gluons in theof produced hadrons .transverse plane can be used as initial data for a Boltzmann-type equationdescribing evolution of the gluon gas towards thermal equilibrium 11.The particle number is a well-defined notion in a free ficid theory whewIIwniitonian in momentum space has thc form(1 ;15.,- .10.1n(~~) _\ I0.010.001 T0.0001 t I I I I Io 2 4 6 8 10~92PFigure 4: Transverse-plane gluon number (per unit rapidity) distribution for g2yL = 35.5(plusses) and for g2.uL = 297 (diamonds). The transverse momentum is expressed in unitsof g2~. The solid line is a fit of the g2pL = 35.5 data to the perturbative expression (3).where ~(k) is k-th momentum component of the field, r(k) is its conjugate mo-mentum, and w(k) is the corresponding eigenfrequency. The average particlenumber of the k-th mode is thenn(k) = w(k)(l~(k)12) = /(l#(k)121n(k) 12), (5)where, in our case, the average () is over the initial conditions. Obviously, anyextension of this notion to interacting theories should reduce to the standardfree-field definition of the particle number in the weak-coupling limit. However,this requirement alone does not define the particle number uniquely outsidea free theory. We therefore use two different generalizations of the particlenumber to an interacting theory, each having the correct free-field limit. Weverify that the two definitions agree in the weak-coupling regime correspondingto late proper times in the central region. We note, however, that the theory inquestion may have low-lying metastable states. If, for a small value of g2pL,the system finds itself in the vicinity of a rnetastability, then the system isfar from linearity, and both our definitions of the number make little sense.We therefore restrict our attention to values of gzp L for which energies ofmetastable minima are much lower than the average energy of a configuration.In all such cases the two definitions give results close to each other.6‘.“..Our first definition is straightf_orwa~d. We impose the Coulomb gaugecondition in the transverse plane: VL . Al = O and substitute the momentumcomponents of the resulting field configuration into (5). At this point, there aretwo possibilities open to us. We can assume u(k) to be the standard massless(lattice) dispersion relation and use the middle expression of (5) to computen(k). Alternatively, we can determine n(k) from the rightmost expression of(5); the middle expression of (5) can then be used to obtain u(k).Our second definition is based on the behavior of a free-field theory underrelaxation. Consider a simple relaxation equation for a field in real space,8,4(2) = –aH/a@(x), (6)where t is the relaxation time (not to be confused with real or proper time)and H is the Hamiltonian. For a free field (H = Hj ) the relaxation equa-tiou has exactly the same form in the momentum space with the solution~(k, t) = #(k, O) exp(–02 (kjt). The potential energy of the relaxed free field isV(t) = (1/2) ~k CJ2(k)14(k, t) Iz. It is then easy to derive the following integralexpression for the total particle number of a free-field system:(7)Now (6) can be solved numerically for interacting fields. Subsequently, V(t) canbe determined, and N can be computed by numerical integration. Note that ina gauge theory the relaxation equations are gauge-covariant, and the relaxedpotential V(t) is gauge-invariant, entailing gauge invariance of this definitionof the particle number. This is an attractive feature of the relaxation method.On the other hand, this technique presently only permits determination of thetotal particle number and cannot be used to find the number distribution.Our findings are summarized in Figures 3 and 4. We consider these re-sults preliminary, since we are yet to perform a careful extrapolation to thecontinuum limit, as we did in the case of the energy. In the case of the en-ergy measurement, the systematic error related to a finite lattice cutoff wasof the order of 1O$ZO.For the particle number, which is better behaved in theultraviolet than the energy, this systematic error should be smaller.As Figure 3 show’s, our t~vo definitions of the particle number agree on a20,% level in a wide range of values of g2/lL, ~~]]ich includes the RHIC andthe LHC regimes. If \ve write the particle number per unit rapidity as Ar =(f\zj(L)2j~(g2pL), then j~(glpL) = 0,14* 0,03 in that range.WC now estimate the number of gluous procluced in oue unit of rapidity,tit central vapidities. at lIIJIC and LH(-~. 11’c extrapolate to S[J(3) in a manner.. . ● ✍analogous to that of the energy estimate. For g2pL s 116 (RHIC) and fIV=0.13-0.15, we obtain Ng = 778-897. For g2pL s 232, (LHC) we obtain Ng =3100-3600. For the same range of f’s, a g2pL = 150 value for RHIC wouldgive IVg = 1300-1500, and g2pL = 300 for LHC would give IJg = 5200-6000.Since Ng depends quadratically on g2pL, and the latter is not known withwith great precision, the range of the prediction is significant. What would bemore interesting though is the slope j of the ratio of the two, for which wehave a prediction up to 20% at present. Varying the energies, and sizes of thenuclei, should enable one to extract this quantity. This point, and comparisonsto predictions from other models, will be discussed further in a forthcomingpaper 12.Finally, Figure 4 shows how N is distributed among various momentummodes, for two extreme cases: g2pL % 300 and g2pL = 35.5. For comparison,we also show a fit of the high-momentum tail (kl >> g2p of the g2pL ~35.5 distribution to the perturbative expression (5). On the low-kl end ofthe spectrum our fully non-perturbative result deviates significantly from theperturbative prediction and remains finite at kl = O. Note that the deviationfirst occurs for kl of the order of the non-perturbative scale g2p.In summary, our numerical implementation of MV model allows one to takeinto account non–perturbative effects at high parton density in the central re-gion. We have derived non–perturbative formulae which relate the energy andnumber of produced gluons to the gluon density and the size of the incomingnuclei. Varying the energy and the size of nuclei should enable us to test thepredictive power of these formulae. Our treatment can be made more accu-rate by switching from the SU(2) to the true physical SU(3) gauge group, byrelaxing the assumption of exact boost invariance, and by replacing periodicboundary conditions by more realistic ones. We plan to address these issuesin future work.AcknowledgementsR.V.’S research was supported by DOE Contract No. DEACO2-98CH1O886.The authors acknowledge support from the Portuguese FCT, under grantsCERN/P/FIS/1203/98 and CERN/P/FIS/15196 /1999.References1. K. Kajantie, P. V. Landshoff, and J. Lindfors, Phys. Reu. Lett. 59 (1987)2527; K. J. Eskola, K. Kajantie, and J. Lindfors, Nucl. Phys. 323 (1989)37; J .-P. Blaizot and A. H. Mueller, Nucl. Phys.; K. J. Eskola, B. Muller,and X.-X. IVang. Phys. Lett. 374 (19!16) 20. B289 (1987) 847.8.+ 4-tion C,n- /-lfi/i-\ ,7 /--:-. – nL. n...2. X.-N. Vv’ang, Phys. Rep. AOU AOI ~IYY i ); n. wager, ritys. acp. 258 ’23?(1995) B. Zhang, Comput. Phys. Commun. 104 (1997) 70.T XN.T-----3. J n x7--.. A---l--u. IVLLIA51 L-W arm ~t. v w UgUpdI WI, PhYs. D-..lLCU. n,40 0.-) ’)9 [l C)C) A\. n.4nu-u /2.400 \lu.YY), u-l=3352 (1994); D50 2225 (1994)1 1-1:1 :,.- X, f.._:.... A 1’- . . ..-.-... nI,. T -.. -.. . ..A u IX7A;--.+4. 0 . !J C1.liuall-lvl .allcml, n. 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Non-Perturbative Gluodynamics of High Energy Heavy-Ion Collisions

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.342.6546

Non-perturbative gluodynamics of high energy heavy-ion collisions

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