TPI–MINN–95–02/T NUC–MINN–95–8/T HEP–MINN–95–1327 5 hep-ph/9502289 9 9 February 1995 1 b Gluon Production from Non-Abelian e F 3 Weizs¨acker-Williams Fields in 1 1 Nucleus-Nucleus Collisions v 9 8 2 Alex Kovner, Larry McLerran, and Heribert Weigert 2 0 5 School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455 9 / h p Abstract - p e h We consider the collisions of large nuclei using the theory of McLerran and Venu- : v gopalan. Thetwonucleiareultra-relativistic andsourcesofnon-abelianWeizsa¨cker- i X Williams fields. These sources are in the end averaged over all color orientations r a locally with a Gaussian weight. We show that there is a solution of the equations of motion forthetwonucleusscattering problemwherethefieldsaretimeandrapidity independent before the collision. After the collision the solution depends on proper time, but is independent of rapidity. We show how to extract the produced gluons from the classical evolution of the fields. 1 1 Introduction Nucleus-nucleus collisions at ultra-relativistic energy have long been recognized as anenvironmentwherehotdensematterisformed[1–3]. Ithasbeenconjecturedthat in such an environment one might produce and experimentally study a quark-gluon plasma [4]. Theoretical studies of quark-gluon plasma formation have typically assumed some initial conditions at some time after the collision was initiated, and then evolved the matter distributions forwards in time according to the equations of perfect fluid hydrodynamics [3,5]. While such an approach may work well for the late stages of the collision when the particles are not so energetic, it does not work well for the earliest stages of the collision. In the earliest stages, the quarks and gluons emerge from their quantum mechanical wavefunction and cannot be described as a perfect fluid until at least enough time has passed for there to be scattering. Intheearliest stages ofthecollision, thequarkandgluoninteractions shouldbe most energetic. Such scatterings are therefore more easy to experimentally probe as they presumably induce hard experimental signatures which are more easily dis- entangled from backgrounds due to soft final state processes. During the hydrody- namic expansion, typically the scale of energy in the interaction is softer and more difficult to disentangle from backgrounds. There has been recent progress in attempting to describe the early evolution of matter producedinnuclear collisions [6]. Inthepartoncascade modelofGeiger and Mu¨ller,onetakestheexperimentallymeasureddistributionfunctionsforquarksand gluons and assumes that they may be treated as an incoherent beam of particles arising from each nucleus. The scattering of partons from partons is computed making reasonable assumptions aboutquantum coherence and time dilation effects. The system is thereby evolved from very early times in the collision until a later time when hydrodynamics may be applicable. In such a theory, the hard scattering 2 signals are computed and may be compared with experiment. The parton cascade model whileelegant and well motivated, in our opinion still lacks some theoretical underpinning. In particular, the issue of quantum coherence intheinitialstateistreated phenomenologically, andneedsadeeperunderstanding. This problem has at least two important aspects. The first and most glaring problem is that the partons arise from a quantum mechanical state. In such a state the uncertainty in momentum, ∆p, times the uncertainty in position, ∆x, is close to saturated, ∆p∆x 1 (1) ∼ Forexample,inthelongitudinalmomentumdistributionofpartons,theweepartons have a longitudinal momentum of order of 100’s of MeV. This corresponds to a longitudinal size of order of fractions of a Fermi. On the other hand, in the parton cascade one assumes knowledge of both the position and momentum of the partons, sincethepartonsaredescribedbyclassicalphasespacedistributionfunctions. While this should be true later in the collision as the scale of spatial gradients becomes larger, early in the collision it is most certainly violated. Although in the parton cascade, the assumptions on the initial distributions are plausible, they can at best give a qualitative agreement with precise results which include the effects of coherence, and at worst totally ignore some classes of interference phenomena. For example, one obvious problem is that for a single nucleus, the partons will spread out since they are an incoherent distribution of partons with different momentum. After some time, one therefore no longer has a spatially compact nucleus. Another class of phenomena which is not fully treated in the parton cascade model is the problem of coherent addition of the color charges of quarks and gluons. Such coherent addition is for example responsiblefor Debye screening, and presum- ably magnetic screening, which will serve as a cutoff for divergent transport cross 3 sections in parton-parton processes. In the parton cascade, a low momentum cutoff is introduced by hand, and of course results for many processes depend upon this cutoff. While the parton cascade may lack precision in many detailed computations, it nevertheless is outstanding for its qualitative predictions. We nevertheless would like to put this model onto firmer foundations, and understand clearly its limits of applicability. To begin to tackle this problem, one must understand at least some aspects of the quantum mechanical wavefunction of the quarks and gluons in the nuclear wavefunction. In the past, one rarely considered the nuclear wavefunction, and the structure functions for a nucleus were taken as a given quantity. There was no constructive description of how such structure functions arise. Recent work by McLerran and Venugopalan has given rise to a picture of how the structure functions arise at small x for very large nuclei at ultra-relativistic energy. In this description, the effects of quantum and charge coherence of the partons in the nuclear wavefunction are properly included. The gluons arise from the non-abelian Weizsa¨cker-Williams fields generated by the color charges of the valence quarks. In this paper, we will extend the treatment of McLerran and Venugopalan from a description of a single nucleus to the collision of two nuclei. This work is in some sense an extension of early effort which were somewhat ad hoc to describe such collisions by classical fields [8,9]. We will see that in the region where most of the parton density sits, the gluon distribution function can initially be described by a classical field. These classical fields are to be interpreted as resulting from coherently superimposing large numbers of gluonic quanta. This way the classical description, wherever applicable will automatically incorporate coherence effects. The gluon field for a single nucleus arising in this way is a non-abelian Weiz- sa¨cker-Williams field. At the initiation of the collision, the non-abelian Weizsa¨cker- 4 Williams fields of the two nuclei play the role of boundary conditions for the time evolution ofthegluonfield. Thisclassical fieldeventually evolves intogluonquanta. The picture we have of the collision is therefore the following. Before the nuclei collide,theyaredescribedbyvalencequarksandtheircoherentWeizsa¨cker-Williams fields. These fields are classical in the sense of classical electromagnetic fields, but of course can not be thought of as composed of particle with classical phase space distributions. Duringthecollision, thefieldsarestillclassical, butsufficientlystrong so that the equations of motion evolve the fields non-linearly with time. As time evolves, the field weakens. When the strength of the gluon field is sufficiently low, the field equations linearize, and the gluon field describes the evolution of weakly interacting classical gluon waves. At this time, the coherent addition of the fields is no longer important, and they should be described by an incoherent distribution of gluons. The parton cascade model may therefore be used. Prior to this time however, the coherence in the gluon field is essential. The simple fact that the evolution of the gluons is described by a classical field is a con- sequenceofthefactthatthegluonsareinsomelocally coherentstate. Adescription in terms of incoherent classical particles is simply not possible. In the second section, we review the relevant results of computation of the small x structure functions for a single large nucleus. We will attempt to describe the kinematic limits of applicability of this description. We will argue that the Weizsa¨cker-Williams fields should describe the distribution of gluons in the region of transverse momenta which gives the dominant contribution after integrating over transverse momenta. In the third section, we set up the problem of nucleus-nucleus scattering. We derive an equation for the time evolution of the gluon field. We relate the results of such a computation to the phase space density of gluon radiation. In the fourth section, we summarize our results and speculate on their region of validity. 5 2 Review of the McLerran-Venugopalan Model In the work of McLerran and Venugopalan [7], it was argued that for very large nuclei, A1/3 at small values of Bjorken x, x << A−1/3, the quark and gluon → ∞ distribution functions are computable in a weak coupling limit. This is because the density of partons per unit area defines a dimensionful scale and when 1 dN µ2 >> Λ2 (2) ∼ πR2 dy QCD the strong coupling parameter α (µ2) should become small. Here y ln(1/x). S ∼ In lowest order ina naive weak coupling expansion, it was shown that thegluon distribution function was of the Weizsa¨cker-Williams form, that is proportional to 1/x. It was also shown that the p dependencewas also of the Weizsa¨cker-Williams ⊥ form dN/d2p 1/p2 for α µ << p << µ where µ Λ A1/6 ⊥ ∼ ⊥ S ⊥ ∼ QCD Of course the naive weak coupling expansion may not be strictly valid, since there is the well known Lipatov enhancement of the low x structure functions [10]. Thisenhancementinvolves quantumcorrections to thelowest ordernaive weak cou- pling result, and changes the small x distribution to 1/x1+Cαs. While this behavior iscomputableintheMcLerran-Venugopalan model,itsnatureisnotyetfullyunder- stood. We expect however that as far as the local effects on the parton distribution at fixed rapidity, y ln(1/x), the main effect is to renormalize the charge which ∼ generates the Weizsa¨cker-Williams field. The charge which generates this field in lowest order in the naive weak cou- pling expansion is the charge of the valence quarks which are treated in a no recoil approximation. While it may be true that the Lipatov correction might involve new physics, and the picture might change, we will ignore its effects here except to state thatwebelieve itwilleffectively renormalize thevalencequarkcharge through some x dependent source of charge. To see how this might occur, recall that the strength of the Weizsa¨cker-Williams distribution is proportional to the amount of 6 charge present at a value of x larger than that of the distribution. We are there- fore assuming the main effect of the quantum fluctuations is to generate an excess amount of charge at values of x larger than that at which we measure the parton. In the work which follow, we will not treat the problems generated by the Li- patov effect. We will instead concentrate on the naive lowest order approximation to the McLerran-Venugopalan model. This will be sufficient to understand many qualitative aspects of nucleus-nucleus collisions, and we hope in the end with small modifications can also be extended to include the effects of the Lipatov enhance- ment. In Refs. [7], it was found that to compute the structure functions one simply treated the valence quarks as a source of light cone charge. Here the valence quarks are being treated as a source of charge moving at the speed of light along the light cone x− = 0. The source of charge is being treated classically. This approximation as justified so long as the typical transverse momentum scale is p << µ (3) ⊥ where µ is proportional to the number of valence quarks per unit area. At the same time the number of gluon quanta at resolutions with p << µ ⊥ will be sufficiently high to allow for a description of the gluonic degrees of freedom through a classical field. Within these limits, all one has to do is to formulate and solve the Yang-Mills equations in the presence of the classical current induced by the valence quarks: [D ,Fµν] = U[A](x,z(x))Jν(z(x))U[A](z(x),x) . (4) µ Here z(x) = x serves as a reference point used to define “initial values” for |x+=0 the color distribution of the valence quarks Jν(z(x)) = δν+δ(x−)ρ(x ) (5) ⊥ 7 which then, due to covariant current conservation [D ,Jν(x)] = 0 evolve along ν the particles trajectory via parallel transport or link operators U[A](x,z(x)) := Pexp ig x dx′+A−(x′+,x ,x− = 0) connecting the points x and z(x) along the − z(x) ⊥ R particles trajectory. Given a solution to the equations of motion, charge density ρ is to be treated as a stochastic variable, and to compute ground state expectation values one must average over all sources with a local Gaussian weight 1 [dρ] exp d2x ρ(x )2 (6) Z (cid:26)−Z ⊥2µ2 ⊥ (cid:27) This Gaussian distribution arose from the approximations used in Ref. [7]. It was argued there that on the transverse resolution scales corresponding to p << µ T that the valence quark charges may be treated classically. The exponential factor is the contribution to the phase space density associated with counting the number of states of valence charges for a fixed value of the classical charge. It can be thought of as arising from the following classical picture. Supposewe look in a tubethrough the nucleus. This tube has a transverse size much less than a Fermi but large enough so that it intersects many nucleons. In this case, there will typically be many valence quarks inside the tube each coming from a different nucleon. The color charge of each quark is therefore uncorrelated with that of any other quark and the color charges will add together in a random walk. This will lead to the above Gaussian distribution. Physically, thepictureonehasisthefollowing: Thevalencequarksarerecoilless sources of color charge propagating along the light cone. Their charge can fluctuate from process to process and the averaging over charges corresponds to this fluctu- ation. The local charge density is therefore a random variable. The reason why such a stochastic source of charge arises is because the transverse resolution scales whichweareinterested in aresmallcompared toafermi. Onsuchascale, whenone looks at the nucleus, one sees uncorrelated quarks coming from different nucleons. 8 The source of color charge therefore random walks in color space. The criteria that p << µ is the criteria that within each transverse resolution scale, there are many ⊥ quarks so that the color charge is typically large and can be treated classically. Thesolutions oftheabove Yang-Mills equations canbechosentobeoftheform A+ = 0 A− = 0 Ai = θ(x−)α (x ) (7) i ⊥ Here the first line may be interpreted gauge choice. Using light cone gauge A+ = 0 one has direct access to the gluon distribution functions of the parton model. The requirement to have A− = 0 then could still be implemented as a gauge choice at leastalongthetrajectories oftheparticles,makinguseoftheresidualgaugefreedom presentinanyaxialgauge. Inthiscaseitturnsoutthatthereisa particular solution to the equations of motion which has A− vanishing everywhere. On such a solution the link operators on the right hand side of the Yang-Mills equations drop out entirely and the equations become Fij = 0 α = ρ(x ) (8) ⊥ ∇· The solution to these equations is that 1 α = U(x ) U†(x ) (9) i ⊥ i ⊥ −ig ∇ that is a pure two dimensional gauge transform of a the vacuum. Physically, this solution is also easy to understand. The solution is a gauge transform of vacuum on one side of the sheet of valence charge, and another gauge transform of vacuum on the other side of the sheet. We have chosen the field to be zero on one side of the sheet as an overall gauge choice. (This could be relaxed by an overall gauge transformation). Because of the discontinuity in the fields at the 9 sheet of valence charge, the solution is not a gauge transform of the vacuum fields. Its discontinuity gives the source of valence charge. Although we have not been successful in explicitly finding the solution to this equation, it is in principle possible to do numerically. Several generic features of the averaging over different sources of charge are possible to infer nevertheless. For p α µ, the typical value of the external charge is so large that it is a ⊥ S ≤ bad approximation to linearize the gauge transformation and directly compute the field in a naive weak coupling expansion. In this region, the non-linearities of the field equation become important. In this kinematic region, the shape of the Weizsa¨cker-Williams distribution changes form, as is shown in Fig. 1. dN/d2 p Fig. 1: Weizs¨acker-Williams T distributionforasinglenucleus 1 p2 T Λ α µ µ QCD s The 1/p2 behavior turns over and goes to a constant. This provides a low ⊥ momentum cutoff in the number of gluons generated by the distribution. At high momentum, we can compute no further than p µ. The distribution should nev- ⊥ ≤ ertheless extend beyond this region. In fact the upper momentum cutoff should be determined only by the kinematic limit of the process considered. Strictly speaking the number of Weizsa¨cker-Williams gluons is infinite, but only logarithmically, and the cutoff will be determined by the process of physical interest. 10