The initial spectrum of fluctuations in the little bang
aa r X i v : . [ nu c l - t h ] S e p The initial spectrum of fluctuationsin the little bang
Kevin Dusling (1) , Fran¸cois Gelis (2) , Raju Venugopalan (3)
November 13, 2018
1. Physics Department, North Carolina State University,Raleigh, NC 27695, USA2. Institut de Physique Th´eorique (URA 2306 du CNRS), CEA/DSM/Saclay,91191, Gif-sur-Yvette Cedex, France3. Physics Department, Bldg. 510A, Brookhaven National Laboratory,Upton, NY 11973, USA
Abstract
High parton densities in ultra-relativistic nuclear collisions suggest a description of thesecollisions wherein the high energy nuclear wavefunctions and the initial stages of the nuclearcollision are dominated by classical fields. This underlying paradigm can be significantly im-proved by including quantum fluctuations around the classical background fields. One classof these contributes to the energy evolution of multi-parton correlators in the nuclear wave-functions. Another dominant class of unstable quantum fluctuations grow rapidly with propertime τ after the collision. These secular terms appear at each loop order; the leading contri-butions can be resummed to all loop orders to obtain expressions for final state observables.The all-order result can be expressed in terms of the spectrum of fluctuations on the initialproper time surface. We compute, in A τ = 0 gauge, the essential elements in this fluctuationspectrum–the small quantum fluctuation modes in the classical background field. With ourderivation in QCD, we have all the ingredients to compute inclusive quantities in heavy ioncollisions at early times including i) all–order leading logs in Bjorken x , of the two nuclei,ii) all strong multiple scattering contributions, and iii) all–order leading secular terms. In thesimpler analogous formalism for a scalar φ theory, numerical analysis of the behavior of theenergy-momentum tensor is strongly suggestive of early hydrodynamic flow in the system [1].In QCD, in addition to studying the possible early onset of hydrodynamic behavior, additionalimportant applications of our results include a) the computation of sphaleron transitions off-equilibrium, and b) “jet quenching”, or medium modification of parton spectra, in strong colorfields at early times. The large flow measured in heavy ion collisions at RHIC [2–5] and more recently at the LHC [6]can be described in hydrodynamic models that have both a nearly perfect fluid value of the shear1iscosity to entropy ratio of the quark-gluon matter produced and fairly short thermalization timesthat usually range between 0.5 and 2 fermis/c [7–9] (depending on the assumptions made about theinitial conditions and the implementation of the freeze-out). How isotropization and (subsequently)thermalization is achieved in heavy ion collisions is an outstanding problem which requires that oneunderstand the properties of the relevant degrees of freedom in the nuclear wavefunctions and howthese degrees of freedom are released in a collision to produce quark-gluon matter. An ab initio approach to the problem can be formulated within the Color Glass Condensate (CGC) effectivefield theory, which describes the relevant degrees of freedom in the nuclei as dynamical gaugefields coupled to static color sources [10,11]. The computational power of this effective theory isa consequence of the dynamical generation of semi-hard saturation scale [12,13] larger than theintrinsic non-perturbative QCD scale ( Q s ≫ Λ ), which allows for a weak coupling treatmentof the relevant degrees of freedom [14–16] in the high energy nuclear wavefunctions.There has been significant recent progress in applying the CGC effective field theory to studyingthe early time behavior of the quark-gluon matter called Glasma [17] produced in the initial littlebang of a high energy heavy ion collision. Inclusive quantities such as the pressure and the energydensity in the Glasma can be written as expressions that factorize, to leading logarithmic accuracyin the longitudinal momentum fraction x , the universal properties of the nuclear wavefunctions(measurable for instance in proton-nucleus or electron-nucleus collisions) from the final state evo-lution of the matter in the collision [18–20]. Key to this approach are the quantum fluctuationsaround the classical fields. In particular, quantum fluctuations that are invariant under boosts canbe shown to factorize into universal density functionals that encode the multi-parton correlationsin the nuclear wavefunctions. The evolution of these density functionals with energy is describedby the JIMWLK renormalization group equation [21–28].There are however quantum fluctuations that are not boost invariant. It was observed in [29–31], via numerical solutions (see also [32,33] for a semi-analytic discussion of some instabilities inthe solutions Yang-Mills equations) of the classical Yang-Mills equations, that rapidity dependentquantum fluctuations in the expanding Glasma are unstable and grow exponentially as the squareroot of the proper time τ after the collision. In fact, both the existence and the specific timedependence of these instabilities was anticipated based on studies of the Weibel instabilities inexpanding anisotropic Yang-Mills plasmas [34–37]. The unstable quantum fluctuations (initially oforder O (1)) become comparable in size to the classical field (of order O ( g − )) on a very short timescale τ ∼ Q − s . Fortunately, one can isolate and resum these rapidly growing secular divergencesto all orders in perturbation theory. The resulting expressions are free of secular divergences, andcan be rephrased as an average over a spectrum of Gaussian fluctuations of the initial data forthe classical field encountered at leading order . A similar observation was made previously in thecontext of inflationary cosmology [38,39].In a previous paper [1], we developed this formalism for a scalar φ theory which, like QCD, hasa dimensionless coupling in 3+1 dimensions and has unstable modes. We computed the spectrumof fluctuations and showed that the resummed expression for the pressure and energy density cansimilarly be expressed as an ensemble average over quantum fluctuations. The rapid growth ofthe unstable fluctuations has drastic consequences. Without resummation, the relation betweenthe energy density and the pressure is not single valued. For the resummed expressions, while therelation between the pressure and energy density is not single valued at early times, it becomes soafter a finite time evolution. This development of an equation of state therefore allows one to writethe conservation equation for the resummed energy momentum tensor T µν as the closed form set ofequations corresponding to the hydrodynamical evolution of a relativistic fluid. This result can be2nterpreted as arising from a phase decoherence of the classical field configurations with differentinitial conditions given by the ensemble of quantum fluctuations. In this theory, the period of theclassical trajectories is proportional to the amplitude of the field. Anharmonicity occurs in anynon–linear system and we expect the same to hold for QCD. As the different trajectories becomephase shifted for different amplitudes there are cancellations resulting in a single valued relationbetween the pressure and the energy density.This phenomenon shares several common features with Srednicki’s hypothesis of eigenstatethermalization and Berry’s conjecture [40–44]. Berry conjectured in [40] that high lying energyeigenstates of systems whose classical counterpart is chaotic have very complicated wavefunctionsthat for many purposes behave like random Gaussian functions. A system in such an eigen-state would display features reminiscent of thermal equilibrium, despite being in a pure quantumstate [42]. For a system starting initially in a coherent state rather than an energy eigenstate,thermalization would merely amount to losing the initial coherence. Although these ideas whereformulated in much simpler systems, they may have some relevance to QCD since here also theunderlying classical theory is believed to be chaotic [45,46].In this paper, we shall focus on computing the initial spectrum of fluctuations in the Glasmaformed at early times after a heavy ion collision. The classical background field at τ = 0 + inthe Glasma can be expressed, from the continuity of the Yang-Mills equations across the light-cone [47,48], in terms of classical solutions of the Yang-Mills equations for each of the two nucleibefore the collision. For later times, analytical solutions are not known ; however, the Yang-Millsequations have been solved numerically with the initial conditions at τ = 0 + [51–55]; for a nicereview, see [56]. Fortunately, inclusive quantities such as components of the energy-momentumtensor are sensitive only to the initial spectrum of fluctuations about the classical field at τ = 0 + ,which can be calculated analytically. Specifically, we will solve the small fluctuations equations ofmotion in A τ = 0 gauge, in order to obtain a complete orthonormal basis of these fluctuations.There was a first attempt to compute the small fluctuations 2-point correlator in the Glasma [57]which, as we shall discuss, was incomplete because it did not include fully the structure of thebackground field.The paper is organized as follows. In the next section, we will outline the power countingof higher order contributions in the Glasma and emphasize the necessity of resumming secularterms. We isolate the leading contributions and obtain an expression for the resummed leadingsecular divergences. We show that this expression for inclusive quantities can be rewritten as apath integral over a spectrum of fluctuations times the leading order (classical) expression for theinclusive quantity. The only unknown ingredient in this reformulation are the small fluctuationfields on the initial proper time hypersurface. Additional sub-sections discuss gauge invarianceissues and the renormalization of ultraviolet divergences. In section 3, we will show how to computethe small fluctuation fields in the vacuum. We first obtain an inner product for fluctuations ona space-like Cauchy surface that we use to define the orthogonality between a pair of fields. Wewill further prove that the inner product is independent of the chosen surface. We then show thatthe small fluctuation fields can be expressed as a linear combination of modes whose coefficientsare Gaussian-distributed random complex numbers. (This is equivalent to diagonalizing the smallfluctuation correlator on the initial proper time surface.) Because even the computation of the Naturally, the wavefunction of a given eigenstate is not a random function. Berry’s conjecture means thatfor the purpose of computing the expectation value of sufficiently inclusive observables, one can replace the truewavefunction by a random Gaussian function. For some interesting recent attempts, see [49,50]. A τ = 0 gauge is non-trivial, we shall first solve small fluctuation equationsin the vacuum. Then, in the section 4, we shall solve the small fluctuation equations in thebackground classical field of the Glasma to construct the corresponding physical small fluctuationmodes. Section 5 outlines a practical algorithm to compute inclusive quantities (including allleading logarithms in x , and all leading secular contributions) as a function of proper time. Aswe shall demonstrate, the complexity of this space-time evolution is manageable, and amounts todiagonalizing certain matrices on the initial proper time surface. In the final section, we re-stateour key results and discuss some important applications. These include a) a systematic studyof possible thermalization of the quantum system whose evolution with proper time we plan tosimulate numerically just as for the scalar case studied previously. b) The nature and role ofsphaleron transitions in the early time dynamics of the system. c) The medium modification ofhard probes to study ’jet quenching’ at early times. An open question for future work is to exploreuntil what times these analysis are valid and how one can incorporate sub-leading contributionsthat become increasingly important at late times. There are two appendices. The first discussestechnical aspects of the computation of small fluctuation fields in the vacuum. Expressions for theWightman functions for free fields in A τ = 0 gauge are discussed in the second appendix, wheresome connections to previous work on these is also discussed [57,58]. We will begin by first outlining how the power counting for computing inclusive quantities in fieldtheories with strong time dependent sources is modified due to the presence of secular divergences.Following this power counting, we derive an explicit expression for the energy-momentum tensor inheavy ion collisions that resums the leading instabilities to all loop orders in perturbation theory.We will show that the resummed expression for the energy momentum tensor can be expressedas a path integral over the product of two terms. The first is a weight functional that samplesthe spectrum of quantum fluctuations on the initial proper time hypersurface, while the second isthe leading order expression for the energy-momentum tensor. In the latter, the classical field isshifted by the sampled quantum fluctuations. Computing the initial spectrum of fluctuations isour primary goal in this paper, the derivation of which will be discussed at length in sections 3and 4. Before we go there, two further sub-sections will discuss the constraints imposed by gaugeinvariance on the spectrum of fluctuations and the nature of ultraviolet divergences respectively.
In previous works [59,60], it was shown that the problem of computing leading order (LO) and next-to-leading order (NLO) contributions to inclusive quantities –such as components of the energymomentum tensor– in field theories with strong time dependent sources can be formulated as aninitial value problem where a classical field determined on an initial Cauchy surface is evolved upto the time at which the (local) observable is computed. Because one anticipates that a semi-hard scale Q s ≫ Λ QCD is generated by the non-linear QCD dynamics at high energy [12,13], asystematic weak coupling expansion of these inclusive quantities is feasible. One can formallyarrange the perturbative expansion of an observable such as the energy momentum tensor as O [ ρ , ρ ] = 1 g h c + c g + c g + · · · i , (1)4here each term corresponds to a different loop order. Each of the coefficients c n is itself an infiniteseries of terms involving arbitrary orders in ( gρ , ) p . These terms are all of order unity becausethe color charge densities are of order ρ , ∼ O ( g − ) in a large nucleus at high energy. The colorcharge densities correspond to the large x color sources in either nucleus 1 or nucleus 2 respectivelyin a heavy ion collision. Their evolution with the separation scale between sources and fields isdescribed by the JIMWLK equation, which will be stated shortly. The LO contribution comesfrom the first coefficient c , O LO [ ρ , ρ ] ≡ c g . (2)This leading term c /g has been studied extensively for the single inclusive gluon distribution inA+A collisions [51–54] and recently for the double inclusive distribution as well [61].Following this terminology, we denote O NLO [ ρ , ρ ] ≡ c , O NNLO [ ρ , ρ ] ≡ c g , · · · (3)At each order in the loop expansion, there can arise contributions from the loop integrals whichare of the same magnitude as lower orders. One set of such contributions are the increasinglylarge logarithms of the momentum fractions x , of partons in the nuclear wavefunctions as higherenergies, or equivalently smaller values of x , , are achieved in nuclear collisions. The term c n canhave up to n powers of such logarithms, with leading logarithmic terms identified as terms thathave as many logarithms as their order in the loop expansion, O LLog [ ρ , ρ ] ≡ g ∞ X n =0 d n h g ln (cid:18) x , (cid:19) i n , (4)where d n is the coefficient of n -th term in the leading log expansion. We were able to show [18–20]that the leading logarithmic contributions in x , , after averaging over the sources ρ , factorizeinto the expression hOi LLog = Z [ Dρ Dρ ] W x [ ρ ] W x [ ρ ] O LO [ ρ , ρ ] , (5)where W x , [ ρ , ] are the density functionals we alluded to previously. These obey the JIMWLKequation [21–28] ∂W x , [ ρ , ] ∂ ln(1 /x , ) = H , W x , [ ρ , ] . (6)Here H , are the JIMWLK Hamiltonians of the two nuclei; since their explicit form is not essentialto the discussion here, we will refer the interested reader to ref. [18] for explicit expressions in ournotation. Given an initial condition at some initial x value, the JIMWLK equation describes theevolution in the nuclear wavefunctions of the multi-parton correlators that contribute to inclusiveobservables measured in the final state.The resummation of quantum corrections arising from logarithms in x , , as sketched here,takes into account contributions that are essential in describing the energy evolution of inclusiveobservables in heavy ion collisions. These contributions are zero modes in ν , the Fourier conjugateof the space-time rapidity η , and are localized in rapidity around the wave functions of the incomingnuclei. There are also quantum fluctuations that are non-zero modes of ν . Such contributions, that5o not bring leading logs of 1 /x , , cannot be factorized into the evolution of the density functionals W x , in eq. (5). As shown in [29–31,62], these terms can be unstable and grow exponentially withthe square root of the proper time (equal to τ ≡ √ x + x − in light-cone co-ordinates) for a systemundergoing one dimensional longitudinal expansion. Based on these considerations, the expansionwe sketched in eqs. (1) and (4) needs to be modified to keep track also of quantum fluctuationsof amplitude g exp( √ µτ ) (where µ is a growth rate of order Q s ) relative to the leading term.This is necessary because the rapid growth of these unstable modes leads to a break down of theperturbative expansion when τ ∼ τ max ≡ µ − ln (cid:18) g (cid:19) (7)is reached, the proper time at which 1-loop corrections become as large as the leading order term.The breakdown of the expansion can be avoided if one resums these divergent contributions, leadingto a resummed result that is well behaved for τ → + ∞ . Taking into account both the leading logsin 1 /x , and the leading unstable contributions, the new expansion reads O LLogLInst . [ ρ , ρ ] ≡ g ∞ X n =0 g n X p + q = n ˜ d p,q ln p (cid:18) x , (cid:19) e q √ µτ . (8)Thus far, we have only resummed the q = 0 sector of this formula, where the result of the resumma-tion is expressed by the factorized formula (5). The two sources of leading quantum fluctuations atthis accuracy can be resummed independently because a given quantum fluctuation mode cannotbe at the same time a zero mode (that generates logarithms in x , ) and a non-zero mode (thatgenerates a secular divergence in proper time τ ). Naturally, in higher loop corrections, one loop canbring a log of 1 /x , while another loop brings a secular divergence. This is why eq. (8) has termswith both p and q non-zero simultaneously. But the independence of the two types of divergences,based ultimately on considerations of causality, leads us to expect that the double series of eq. (8)can be factorized into a series in p times a series in q . We shall now discuss how resumming the leading secular terms modifies the expression of eq. (5).Albeit our considerations apply to any inclusive quantity, for specificity, we shall consider here theenergy-momentum tensor.
Let us recall first that at leading order in g , the energy-momentum tensor T µν LO is given by T µν LO ( x ) = 14 g µν F αβa F a,αβ − F µαa F νa α , (9)with the field strength tensor defined as F µνa = ∂ µ A νa − ∂ ν A µa + gf abc A µb A νc , (10)6here A µa is the solution of the classical Yang-Mills equations with sources ρ , that vanishes at x → −∞ , D abµ F µνb = δ ν + ρ a + δ ν − ρ a , lim x →−∞ A µa ( x ) = 0 . (11)One can then express the NLO contribution to the energy-momentum tensor, for a given distri-bution of color sources and at an arbitrary space-time point, as the action of a functional operatoracting on the LO result [18,59], T µν NLO ( x ) = h Z Σ d u β · T u + 12 Z Σ d u d v X λ,a Z d k (2 π ) k [ a + k λa · T u ][ a − k λa · T v ] i T µν LO ( x ) , (12)where Σ is a Cauchy surface where the initial values of the classical color field and its derivativesare specified. In this formula, β and a ± λ k a are small corrections to the gauge field A µ . Morespecifically, β ( u ) is the one loop correction to the classical field on the surface Σ (see [18] formore details). In applications to heavy ion collisions, a natural choice of Σ is the surface atproper time τ = 0 + , which corresponds physically to times just after the two nuclei have collided.Though τ = 0 + is the initial surface of choice, we will at the outset consider a generic space-timehypersurface. The only constraint on Σ is that, for the forthcoming resummation to be effective, itmust be located at times before the unstable modes have become too large . In the right hand sideof eq. (12), T u is the generator of shifts of the initial data for the classical field on Σ. Generically,it reads a · T u = a µ ( u ) δδ A µ ( u ) + ( ∂ ν a µ ( u )) δδ ( ∂ ν A µ ( u )) , (13)where A µ ( u ) (in curly font and without a time argument) is the value of the classical field onthe initial time surface. Note that in general, specific gauge conditions and specific choices of thesurface Σ reduce the number of terms this operator contains. In its minimal form, it contains oneterm for each independent field component or field derivative component that one must specify inthe initial value problem on Σ.Let us now explicit a bit more the fields a µ ± k λa that appear in eq. (12). They are smallfluctuation fields about the classical field A µ , that obey the equation of motion D µ ( D µ a ν − D ν a µ ) − ig F νµ a µ = 0 , (14) We have written the Yang-Mills equations in a form that involves the adjoint representation of the covariantderivative, D abµ = ∂ µ δ ab − ig A abµ , where A abµ is the classical gauge potential in the adjoint representation. Itis important to distinguish the A abµ ’s from the A aµ ’s that are the components of the SU(3) element A µ in itsdecomposition over the generators of the algebra, A µ ≡ A aµ t a . The two sets of coefficients are related by A bcµ = − if abc A aµ , since the components of the generators in the adjoint representation are ( t a adj ) bc = − if abc . Unless specified otherwise, the dependence on ρ , ρ , the color charge densities in each of the nuclei, will beimplicit in our discussion. The NLO expression in eq. (12) does not depend on the choice of Σ. However, our resummed result will dependon this choice since it includes only a subset of the higher loop corrections. Provided the surface Σ is locatedin a region where the unstable fluctuations are still small, the difference between various choices of Σ is a smallcorrection. We will discuss this point further later in the paper. It has dimension of (mass) because dim[ δδ A ] = (mass) and dim[ a ± k λa ] = (mass) . To avoid cumbersome notations, we have not written explicitly the color indices of the various objects. Here,and henceforth, the D ’s and F should be understood as objects in the adjoint representation. For instance F νµ a µ means F νabµ a µb . Likewise, D µ D µ a ν is D ab,µ D µbc a νc . x →−∞ a µ ± k λa ( x ) = ε µ k λ T a e ± ik · x . (15)The T a ’s are the SU(3) generators and ε µ k λ is the polarization vector. Thus the labels k , λ, a arerespectively the initial momentum, initial polarization and initial color of the gauge fluctuationrepresented by a ± k λa , and the sign ± specifies whether it is a positive or negative energy wave inthe remote past. At leading order (tree level), the energy momentum tensor is of order Q s /g . In the absence ofsecular divergences, from the power counting described previously, the NLO corrections should beof order Q s . This power counting could be obtained in eq. (12) by noting that a ± k λa ∼ O (1) , (16) β ∼ O ( g ) , (17) T u ∼ δδ A ∼ O ( g ) , (18)since A ∼ O ( g − ). The existence of instabilities implies that we must alter our estimate of theorder of magnitude of the operators T u . Indeed, since T u A ( τ, x ) is the propagator of a smallfluctuation over the background field between a point on the initial proper time surface and thepoint ( τ, x ), it grows at the same pace as the unstable fluctuations. Thus the counting rule for T u should be modified to read T u ∼ O ( g e √ µτ ) . (19)The combination T u T v in eq. (12) then grows as g exp(2 √ µτ ) which leads to a break down of thepower counting at the proper time τ max defined in eq. (7). At τ max , the 1-loop correction becomesas large as the leading order contribution, and one may anticipate that an infinite series of higherloop corrections also become equally important at this time. Our goal is now to collect from higher orders all the terms that are leading at the time τ max .This comprises all the terms where the extra powers of g are compensated by an equal numberof factors of e √ µτ . We presume that a typical higher order correction to the energy momentumtensor can still be written in the form of eq. (12), but with a more general operator acting on T µν LO ( x ) of the form Z Σ d u · · · d u n Γ n ( u , · · · , u n ) · T u · · · T u n . (20)Here Γ n is an n -point function, which may or may not be simply connected. This expression hasnot been proven in general but results from a conjecture that inclusive quantities at all loop orderscan be expressed purely in terms of retarded propagators, thereby generalizing known results at8 T µν ( x ) v Γ ( u,v ) Figure 1: Representation of the 1-loop contribution involving the function Γ ( u , v ). The thick redline is the τ = 0 + surface on which the initial value problem is set up. The open circles representthe initial data. The filled blue circles represent the two operators T u , v , and the U-shaped wavyline represented below the light-cone is the function Γ ( u , v ).LO and NLO. While there are specific examples of loop contributions that have been checked tosatisfy this conjecture, there are in particular nested loops contributions for which the conjectureis difficult to confirm. In the figures 1 and 2, we illustrate this formula by some examples of 1-loopand 2-loop contributions.With the stated assumption implicit in eq. (20), the following power counting can be established.If eq. (20) is a piece of a L -loop correction to the energy-momentum tensor, the order g p and thenumber n of points of the function Γ n are related by n + p = 2 L . (21)This formula can be checked for the examples of graphs given in the figures 1 and 2. Note that p = 0 is the smallest possible value for p . Taking into account the effect of the instabilities (i.e.one power of exp( √ µτ ) for each of the n operators T u i ), the order of magnitude of a contributionobtained from eq. (20) is g p h g e √ µτ i n , (22)relative to the leading order contribution. If we just count naively the powers of g , the powercounting would indicate that this contribution gives a correction of order g L , a decrease by afactor g for each extra loop. However, because of the instability, by the time τ max given ineq. (7), the contribution is instead of order g p and does not depend anymore on the number n of T operators. At the time τ max all the terms with p = 0, regardless of the number of loops, are ofthe same order while all the remaining terms for which p > g . It is therefore natural to resum all the p = 0 terms, and to neglect all those with p > n of T operators must be evenand equal to 2 L . Moreover, since we keep only terms of order p = 0 in Γ L , the only possibilitythat remains is to construct Γ L as a product of L factors Γ (an example of which is the left Note that tadpole contributions such as the term β · T u in eq. (12) are also excluded since β ∼ O ( g ). µν ( x ) T µν ( x ) Γ ( u,v,w ) Figure 2: Representation of two examples of 2-loop contributions. The thick red line is the τ = 0 + surface on which the initial value problem is set up. The open circles represent the initial data.The filled blue circles represent operators T u , v . Left: contribution with a Γ that factorizes intotwo Γ ’s. Right: contribution with a Γ .diagram of figure 2) because any non-factorized contribution to Γ L requires more powers of g . Therefore, the leading operator at L loops in eq. (20) is the L -th power of the 2-point operatorthat appears at 1-loop, 1 L ! " Z Σ d u d v Γ ( u , v ) · T u T v L , (23)where Γ ( u , v ) · T u T v = X λ,a Z d k (2 π ) k [ a + k λa · T u ][ a − k λa · T v ] . (24)The inverse factorial prefactor is a symmetry factor that prevents multiple counting when thevarious factors Γ are permuted. Summing all the contributions from L = 0 (leading order) to L = + ∞ , we obtain T µν resummed ( x ) = exp " Z Σ d u d v Γ ( u , v ) · T u T v T µν LO ( x ) . (25) Until the conjecture in eq. (20) is proved, one may choose to interpret eq. (25) as a well motivated ans¨atzresulting from the exponentiation of the NLO result. at the time τ = τ max .However, this expression is very formal as it is expressed in terms of functional derivatives withrespect to the initial conditions for the classical color fields and their time derivatives on the initialproper time surface Σ. Fortunately, as we shall see, we can rewrite this result in a form that ismuch more transparent both conceptually and for computational purposes. We first recall that the operator T u defined in eq. (13) is the generator of shifts of the initialconditions at all points u on the initial proper time surface τ = 0 + for the classical fields A µ and their time derivatives ∂ τ A µ , the latter either being equal to or simply proportional to thecorresponding electric fields, their canonical conjugate momenta. We can therefore writeexp " Z Σ d u [ α · T u ] F (cid:2) A (cid:3) = F (cid:2) A + α (cid:3) , (26)where A ≡ ( A , E ) denotes collectively all the components of the initial classical field and theircanonically conjugate momenta on the initial time surface. One should similarly understand α todenote small perturbations of both the initial classical field and their canonically conjugate electricfields. We next obtain exp " Z Σ d u d v Γ ( u , v ) · T u T v = Z (cid:2) D α (cid:3) F (cid:2) α (cid:3) exp " Z Σ d u [ α · T u ] , (27)with F (cid:2) α (cid:3) ∝ exp " − Z Σ d u d v α ( u ) Γ − ( u , v ) α ( v ) . (28)In eq. (27), the functional integration [ Dα ] is also a shorthand for integrations over all the com-ponents of the perturbation and of its time derivative on the initial surface. Subleading contributions such as the p = 1 contribution g · g exp( √ µτ ) are no longer sub-leading by times τ ∼ µ − ln ( g − ). This time is only slightly larger than the time τ max at which the p = 0 terms become important.Therefore, including one by one the p = 1 terms on top of an expression that resums the p = 0 terms is bound to fail –these contributions must be included all at once via a resummation. Having this in mind, a more important questionis: do these contributions ever become important after they have been appropriately resummed? A resummationof the p = 1 secular terms is outside the scope of this work, but it is plausible that they can be included in ourframework by a modification of the distribution F [ α ] of the fluctuations at the initial time. If this is the case, thenthe p > An elementary form of this identity, e α ∂ x f ( x ) = Z + ∞−∞ dz e − z / α √ πα f ( x + z ) , can be proved by performing a Taylor expansion of the exponential on the left hand side and of f ( x + z ) on theright hand side of this expression. From this simple example, one sees that a Gaussian operator in derivatives is a smearing operator that convolutes the target function with a Gaussian distribution. The unwritten constant prefactor, proportional to [det(Γ )] − / , is such that the distribution F (cid:2) α (cid:3) has anintegral over α normalized to unity. T µν resummed ( x ) = Z (cid:2) D α (cid:3) F (cid:2) α (cid:3) T µν LO [ A + α ]( x ) , (29)where the weight functional F , corresponding to the initial spectrum of fluctuations, is defined ineq. (28). This result is a central expression of our paper and was sketched previously in [18,63].It was also obtained in previous works for a scalar field theory [39] and a gauge field theory [57]respectively using different methods. The expression is quite remarkable because it demonstratesthat the resummation of loop (quantum) corrections that correspond to the most unstable con-figurations in a single heavy ion collision event can be expressed as an average over a Gaussiandistributed ensemble of classical configurations in the Glasma. Note also that, although this for-mula was derived here for the energy-momentum tensor, the power counting that led us to theexponentiation of the 1-loop result did not depend on the choice of a specific observable. Thuswe expect that the same resummation would also be applicable to other inclusive quantities; thespectrum of fluctuations superposed on the Glasma fields is universal.The essential ingredient in eq. (29) is the small fluctuations correlator Γ ( u , v ), defined ineq. (24), which should be computed with the two endpoints on the initial time surface Σ, with theGlasma field in the background. A first attempt to compute this object is given in [57]. However,the expression obtained there was incomplete because it corresponds to an approximate expressionfor the free propagator in A τ = 0 gauge, with the only dependence on the background field comingfrom Gauss’s law. We will compute here (in A τ = 0 gauge) the spectrum of fluctuations both inthe free case and in the presence of a classical background field. We will show later that the latterhas a non-trivial dependence on the classical fields in the Glasma at Σ. The left hand side of the expression in eq. (29) should be gauge invariant because the energy-momentum tensor is a physical quantity. It should therefore be invariant under a gauge transfor-mation of the classical Glasma field,
A −→ Ω † A µ Ω + ig Ω † ∂ µ Ω . (30)This invariance is also true for its leading order counterpart on the right hand side of the expression,when we apply the same gauge transformation to the total field A + α , A + α −→ Ω † ( A µ + α µ ) Ω + ig Ω † ∂ µ Ω . (31)Thus for the expression in eq. (29) to be manifestly gauge invariant, it is sufficient if the initialspectrum of fluctuations F is invariant under the transformation α −→ Ω † α Ω . (32)If we decompose α on the basis of the generators of SU (3), α ≡ α a t a , the identity¯Ω ab t b = Ω t a Ω † , (33)12where ¯Ω ab here is an adjoint SU(3) matrix) gives the equivalent transformation for the components α a to be α a → ¯Ω ab a b ≡ ˜ α a . (34)For the argument of the exponential in F to be invariant under this transformation, α a ( u ) Γ − ,ab ( u , v ) α b ( v ) −→ ˜ α a ( u ) e Γ − ,ab ( u , v ) ˜ α b ( v ) , (35)the inverse small fluctuations correlator in the Glasma (which is an 8 × e Γ − ( u , v ) = ¯Ω( u ) Γ − ( u , v ) ¯Ω † ( v ) . (36)It is clear from the structure of the expression in eq. (24) that this property will be satisfied. It is also important to address the potential ultraviolet divergences in eq. (29). The leading orderenergy-momentum tensor in the Glasma is ultraviolet finite. However, because one is resummingquantum fluctuations in eq. (29), the energy-momentum tensor should receive a contribution fromthe (infinite) zero point energy. One can regularize ultraviolet divergences by introducing a cutoff Λcorresponding to the largest momentum mode of the fluctuation α . Since the energy-momentumtensor has canonical dimension four, we expect that its dependence on this cutoff can be organizedas T µν resummed ( x ) = c Λ + c Λ + c , (37)where c , , are finite quantities. It is easy to renormalize the energy-momentum tensor by subtrac-tion if one can prove that the divergences are truly a property of the vacuum and do not depend onthe background classical field A in the Glasma. The coefficient c is dimensionless – it is thereforea pure number, that cannot depend on the background field. The case of c is trickier. Indeed, itscanonical dimension 2 allows a priori a dependence on the background field. However, we knowthat the left hand side in eq. (37) is invariant under gauge transformations of the background field;we therefore must conclude that the coefficient c must be a gauge invariant, local (because theleft hand side is a local quantity), dimension 2 quantity. There is no such quantity in Yang-Millstheory, which suggests that c = 0. Thus, on the basis of gauge symmetry and locality, we expectthat the only ultraviolet divergence in our expression for the resummed energy-momentum tensoris a quartic divergence, with a coefficient that does not depend on the background field. It can becomputed in principle once and for all in the absence of the background field and subtracted from T µν resummed to give an ultraviolet finite result for this quantity. We noted in eq. (29) of the previous section that the small fluctuations 2-point correlator Γ ( u , v )is the key ingredient in resumming the contributions of leading instabilities at all loop orders tothe energy-momentum tensor. In practical implementations of this resummation, space is discretized on a lattice, and thus the UV cutoff isthe inverse of the lattice spacing.
13n this section and the following one, we will work in the
Fock-Schwinger gauge A τ = 0.Albeit at first sight a natural gauge for describing hadron-hadron collisions (because it is aninterpolation between two light-cone gauges in the forward light-cone), the Fock-Schwinger gaugeis not frequently used in the literature. This is because even expressions for the free correlator arecomplicated in this gauge. Our motivation here is specific to the nature of the CGC description ofheavy ion collisions. The initial conditions for the evolution of classical gauge fields in the forwardlight cone, in this gauge, are simply expressed [47] in terms of the classical fields in the nuclei beforethe collision. This is an important criterion because our resummed result in eq. (29) for the energy-momentum tensor is expressed in terms of solutions of classical Yang-Mills equations with Gaussiandistributed initial conditions. Further, numerical computations are unavoidable because one is ina strong field regime where perturbative computations are invalid; thus analytically cumbersomeexpressions are not a deterrent if efficient numerical algorithms are feasible.Turning to the computation of the small fluctuations correlator,Γ ( u , v ) = X λ,a Z d k (2 π ) k a + k λa ( u ) a − k λa ( v ) , (38)as noted previously in the discussion after eq. (13), the fluctuation fields a µ ± k λa are plane wavefields at x = −∞ that have been evolved in time by interacting with the classical backgroundfield A . Before going further, let us state two properties of this correlator that are true when thetwo points u and v lie on the same Cauchy surface , i. Γ is symmetric: Γ ( u , v ) = Γ ( v , u ), ii. Γ is real valued.Note that for the correlator as defined, the two properties are equivalent since a − k λa = ( a + k λa ) ∗ .These properties are crucial since Γ is the variance of the fluctuations in our resummation formula,eq. (29).From the definition given in eq. (38), one might presume that the calculation of Γ requires oneto follow the entire evolution from x = −∞ to τ = 0 + . We will demonstrate in section 3.3 thatthis is not the case; to compute Γ , it is sufficient to construct an orthonormal basis of fluctuationfields about classical fields at small proper times τ = 0 + . The fluctuations a µ ± k λa start as plane waves in the remote past and evolve about the Glasmaclassical field A according to the equation of motion (14). This expression of the equation ofmotion, as well as the Yang-Mills equation (11), implicitly assume a Cartesian system of co-ordinates. However, the most natural co-ordinate system in the treatment of the post-collisionevolution is the ( τ, η, x ⊥ ) system, where τ ≡ p t − z , η ≡
12 ln (cid:18) t + zt − z (cid:19) , (39) For these properties to hold true, it is important that the two points are not separated by a time-like interval,which ipso facto is guaranteed if they belong to the same locally space-like surface. x ⊥ collectively denotes the two co-ordinates perpendicular to the collision axis. In theseco-ordinates, the metric tensor g µν = diag (cid:0) , − , − , − τ (cid:1) (40)has a proper time dependent determinant, √− g = τ . This implies small changes to the equationsof motion. The classical Yang-Mills equations (11) become √− g D α (cid:0) √− g g αβ g νµ F βµ (cid:1) = δ ν + ρ + δ ν − ρ , (41)and the small fluctuation eqs. (14) become1 √− g D α (cid:0) √− gg αβ g νµ ( D β a µ − D µ a β ) (cid:1) − ig g αβ g νµ F µβ a α = 0 . (42)In Fock-Schwinger gauge A τ = 0 and ( τ, η, x ⊥ ) co-ordinates, these equations can be written outmore explicitly to read, D η ∂ τ a η + τ D i ∂ τ a i − ∂ τ D η a η − τ ∂ τ D i a i = 0 (cid:0) ∂ τ τ − ∂ τ − τ − P ii (cid:1) a η + τ − P iη a i = 0 (cid:0) ∂ τ τ ∂ τ − τ − P ηη − τ P ii (cid:1) a x + τ − P ηx a η + τ P ix a i = 0 (cid:0) ∂ τ τ ∂ τ − τ − P ηη − τ P ii (cid:1) a y + τ − P ηy a η + τ P iy a i = 0 , (43)where we have introduced the shorthand P , which is defined as P ab IJ a b K ≡ D ac I D cb J a b K + gf acb F c IJ a b K . (44)Here the indices I, J, K denote x, y and η , while we use Latin indices i, j to designate the twotransverse coordinates x, y . For example, ∂ i = ∂ x + ∂ y = ∂ ⊥ . Note that the first equation ineq. (43) is Gauss’ law, which can also be written as D η ∂ τ a η + τ D i ∂ τ a i + ig ( ∂ τ A η ) a η + τ ( ∂ τ A i ) a i = 0 . (45) Field equations can be generalized to an arbitrary system of coordinates by trading ordinary derivatives ∂ µ forcovariant derivatives ∇ µ (here, covariant refers to coordinate transformations, not SU (3) gauge transformations)that involve Christoffel symbols. For the Yang-Mills equations, one should thus write( ∇ µ − igA µ ) F µν = J ν , F µν ≡ ∇ µ A ν − ∇ ν A µ − ig [ A µ , A ν ] . However, it turns out that for an antisymmetric tensor such as F µν one has also (see [64], chapter 5) ∇ µ F µν = 1 √− g ∂ µ (cid:0) √− g F µν (cid:1) . Moreover, one can show that ∇ µ A ν − ∇ ν A µ = ∂ µ A ν − ∂ ν A µ , so that the flat-space expression of F µν can stillbe used (provided the two indices are downstairs). In other words, the usual formulas for the field strength shouldbe used only for lower indices, and the metric tensor should be used to raise indices if necessary. For instance, in A τ = A τ = 0 gauge, one has F τη = ∂ τ A η and F τη = g ττ g ηη F τη = − τ − ∂ τ A η , but ∂ τ A η = ∂ τ ( − τ − A η ) is notequal to F τη . τ – η coordinates, we shall at times make use of light-conecoordinates defined as x ± ≡ t ± z √ , (46)with the metric tensor now having the form g + − = g − + = 1 , g xx = g yy = − . (47)The transformation from light-cone to τ, η coordinates is given by x ± = τ e ± η √ , (48)while the Fock-Schwinger gauge condition, expressed in light-cone coordinates, is A τ = A τ = 1 τ (cid:0) x + A − + x − A + (cid:1) = 0 . (49) Since the equations of motion (42) of the small fluctuations are linear, the set of its solutions isa vector space, and it is sufficient to know a basis of this space in order to be able to constructany solution. For a real background field such as the classical field A µ , the evolution in time ofthe small fluctuations is unitary . Therefore, there should be an inner product between pairs ofsolutions of eq. (42) that remains invariant during the evolution of these solutions. To constructthis inner product, rewrite eq. (42) as O νµ a µ = 0 , (50)with O νµ ≡ D α √− g (cid:16) g νµ g αβ − g νβ g µα (cid:17) D β − ig √− gg να g µβ F αβ . (51)Consider now two solutions a µ and b µ of eq. (50), and start from the identity0 = Z Ω d x a ∗ ν ( x ) h −→ O νµ − ←− O νµ ∗ i b µ ( x ) , (52)where Ω is some domain of space-time. This identity is a trivial consequence of the equations ofmotion for a ∗ and b . A remarkable property of the integrand in the right hand side is that it is atotal derivative , a ∗ ν ( x ) h −→ O νµ − ←− O νµ ∗ i b µ ( x ) = ∂ α h √− g (cid:16) g νµ g αβ − g νβ g µα − g να g µβ (cid:17)(cid:16) a ∗ ν ( x ) ↔ D β b µ ( x ) (cid:17)i . (53)Therefore, one can use Stokes theorem, Z Ω d x ∂ α F α = Z ∂ Ω d S u n α F α (54) In event of confusion from the apparent structure of the last term of eq. (42), note that the components of theadjoint generators are purely imaginary, and therefore the function that multiplies a µ in this term is real. For this property to be true, it is crucial that the last term in eq. (51) is real and one should properly take thecomplex conjugate of the covariant derivatives when they act on the left. This property is in fact closely related tothe operator O νµ being Hermitean; the evolution of the fluctuations is unitary. d S u is the measure on the boundary ∂ Ω, and n α is a normal vector to the boundary,oriented outwards. Let us assume that the boundary ∂ Ω is made of two locally space-like surfacesΣ and Σ , and a third boundary located at infinity in the spatial directions on which all the fieldsare vanishing. Then eq. (52) is equivalent to Z Σ d S u √− g (cid:16) g νµ g αβ − g νβ g µα − g να g µβ (cid:17) n α (cid:16) a ∗ ν ( u ) ↔ D β b µ ( u ) (cid:17) = Z Σ d S u √− g (cid:16) g νµ g αβ − g νβ g µα − g να g µβ (cid:17) n α (cid:16) a ∗ ν ( u ) ↔ D β b µ ( u ) (cid:1) . (55)We have thus proved, most generally, that an inner product defined as (cid:0) a (cid:12)(cid:12) b (cid:1) ≡ i Z Σ d S u √− g (cid:16) g νµ g αβ − g νβ g µα − g να g µβ (cid:17) n α (cid:16) a ∗ ν ( u ) ↔ D β b µ ( u ) (cid:17) , (56)is independent of the Cauchy surface Σ used to define it, provided a µ and b µ obey the equation ofmotion of small fluctuations. Note that we have added a factor i to its definition to ensure that itis Hermitean, (cid:0) a (cid:12)(cid:12) b (cid:1) ∗ = (cid:0) b (cid:12)(cid:12) a (cid:1) , (cid:0) a ∗ (cid:12)(cid:12) b ∗ (cid:1) = − (cid:0) b (cid:12)(cid:12) a (cid:1) = − (cid:0) a (cid:12)(cid:12) b (cid:1) ∗ . (57)In the special case where Σ is a surface of constant τ and we work in the Fock-Schwinger gauge A τ = 0, we have n · D = ∂ τ , and n · a = 0, n · b = 0. Therefore the inner product simplifies into (cid:0) a (cid:12)(cid:12) b (cid:1) ≡ i Z τ = const d S u √− g g νµ (cid:16) a ∗ ν ( u ) ↔ ∂ τ b µ ( u ) (cid:17) . (58)Now let us evaluate the inner product for pairs of field fluctuations taken from the set of the a ± k λa . Since the inner product does not depend on the chosen time surface and since we knowthese fields at x → −∞ (because they are defined via their initial condition in the remote past),we can evaluate the inner product by using plane wave initial conditions for these fluctuation fields.This gives (cid:0) a + k λa (cid:12)(cid:12) a − l ρb (cid:1) = 0 (cid:0) a + k λa (cid:12)(cid:12) a + l ρb (cid:1) = δ λρ δ ab (2 π ) kδ ( k − l ) (cid:0) a − k λa (cid:12)(cid:12) a − l ρb (cid:1) = − δ λρ δ ab (2 π ) kδ ( k − l ) . (59)Thus this particular basis of the space of solutions of eq. (14) is orthonormal with respect to theinvariant inner product defined in eq. (56). Note also that the a + k λa ’s represent only one half ofthe basis of the vector space of solutions of eq. (42) –namely the solutions that have a positivefrequency in the remote past. The other half is simply obtained by complex conjugation. It easy tocheck that any unitary transformation of the positive energy solutions (and a concomitant changeto the negative energy ones, that are their complex conjugates) transforms an orthonormal basisinto another orthonormal basis, and leaves the formula eq. (38) unchanged. This remark is usefulbecause it leaves us the freedom to label the elements of the basis by other quantities than theCartesian 3-momentum. This will be true in our specific case where we are interested in a basis ina curvilinear co-ordinate system. 17 .3.1 Normalization of the fields and choice of basis It is important to note that the prefactor in front of the δ functions in eq. (59) exactly cancels thefactors that are included in the integration measure in eq. (38), namely one has Z d k (2 π ) k (cid:0) a + k λa (cid:12)(cid:12) a + l ρb (cid:1) = 1 . (60)This remark in fact defines uniquely how the inner product of the basis elements should be normal-ized given a generic choice for the integration measure in eq. (38); in particular, this rule will become in handy later when we use other labels than the usual 3-momentum to index the elementsof the basis. Moreover, this makes clear that eq. (38) is just one particular representation of thecorrelator Γ ; there exists such a representation for any orthonormal basis of the space of solutionsof eq. (42), as we shall explain now. Thanks to the above inner product, one can spell out a generalprocedure for constructing the correlator Γ : i. Find a complete set of independent positive energy solutions a K of eq. (42), where K denotescollectively (usually a mix of continuous and discrete labels) all the labels necessary to indexthese solutions. ii. This set of solutions should obey the orthogonality condition, (cid:0) a K (cid:12)(cid:12) a K ′ (cid:1) = N K δ KK ′ (61)with N K real and positive definite , iii. The correlator Γ is then given byΓ ( u , v ) = Z dµ K a K ( u ) a ∗ K ( v ) , (62)where the measure dµ K (a mix of integrals and discrete sums) is such that Z dµ K N K δ KK ′ = 1 . (63)It is clear from eqs. (61) and (63) that the Γ given by eq. (62) is independent of how we normalizethe solutions (i.e. on the constants N K ), provided we choose the integration measure accordingly.Moreover, we only need to know the form of the solutions at the time of interest, and we can avoidthe complication of evolving the plane waves from the past through the forward light-cone. Thisis particularly helpful when one uses τ, η coordinates, because this system of coordinates has asingularity at τ = 0.A further simplification is possible because in practice we won’t need to use directly eq. (62)for Γ . Indeed, an ensemble of real-valued field fluctuations a µ that have a 2-point equal-timecorrelation given by Γ can be generated by the following formula, a µ ( x ) = Z dµ K h c K a µ K ( x ) + c ∗ K a µ ∗ K ( x ) i , (64) In this light, eq. (38) which represents Γ in terms of the a ± k λa ’s, exploits one possible method of constructingsuch an orthonormal basis. In this case, one starts at x = −∞ with the plane waves, that are known to form anorthonormal basis, and evolves them forward to the time of interest. The time invariance of the inner product thenguarantees us that we get an orthonormal basis on the forward light-cone. This means that the solutions a K will in general be complex solutions. c K are random Gaussian-distributed complex numbers whose variance isgiven by D c K c ∗ K ′ E = N K δ KK ′ (cid:10) c K c K ′ (cid:11) = D c ∗ K c ∗ K ′ E = 0 . (65)This method of generating the field fluctuations offers the advantage that it does not require thatone diagonalizes the correlation function Γ . A τ = 0 gauge Let us start by calculating the fluctuation correlator in A τ = 0 gauge and on the surface τ = 0 + in the free case , namely, in the absence of the classical background field. Given the complicationsintroduced by the choice of gauge and the system of curvilinear coordinates, this is a useful exerciseto pursue before attacking the more general case of the Glasma background field. In this situation,eqs. (43) simplify into ∂ η ∂ τ a η + τ ∂ i ∂ τ a i = 0 (cid:0) ∂ τ τ − ∂ τ − τ − ∂ ⊥ (cid:1) a η + τ − ∂ i ∂ η a i = 0 (cid:0) ∂ τ τ ∂ τ − τ − ∂ η − τ ∂ ⊥ (cid:1) a x + τ − ∂ η ∂ x a η + τ ∂ i ∂ x a i = 0 (cid:0) ∂ τ τ ∂ τ − τ − ∂ η − τ ∂ ⊥ (cid:1) a y + τ − ∂ η ∂ y a η + τ ∂ i ∂ y a i = 0 . (66) A general solution a µ to eq. (66) has a priori four components. However, a massless vector field hasonly two physical degrees of freedom. One of the seemingly independent components of the vectorfield is removed by the gauge condition a τ = a τ = 0 (this is already implemented in eqs. (66)).However, even after imposing this condition, there is a residual gauge symmetry in the equationsof motion, namely, these are invariant under τ independent gauge transformations a µ → a µ + ∂ µ Λ( η, x ⊥ ) , (67)where Λ is an arbitrary τ independent function. As a consequence of this residual gauge freedom,the three remaining components of a µ are not all physical degrees of freedom.In order to find the two physical solutions, we begin by finding the unphysical solution. Thissolution must be a pure gauge , which here means it is a τ independent total derivative. As wewill see, the unphysical solution is not a dynamical variable but is completely constrained by theinitial and boundary conditions. After finding the most general τ independent solution to theequations of motion, the two physical solutions can be determined relatively easily. Their formwill be narrowed down by requiring that the three solutions are mutually orthonormal, and thenthe residual gauge freedom will be fixed by imposing the equations of motion.In the ( τ, η, x ⊥ ) system of coordinates, a convenient set of labels for the solutions of eq. (66)is ν, k ⊥ , λ, a , where ν is the Fourier conjugate of the space-time rapidity η (as used previously, λ denotes the polarization and a the color). We choose λ = 1 , λ = 3 to be the unphysical one. Since in this section we are in the vacuum, all the colors a ( x , x ′ ) = X λ =1 , Z d k ⊥ dν (2 π ) a k λ ( τ = 0 + , x ) a ∗ k λ ( τ = 0 + , x ′ ) , (cid:0) a k λ (cid:12)(cid:12) a k ′ λ ′ (cid:1) = (2 π ) δ λλ ′ δ ( ν − ν ′ ) δ ( k ⊥ − k ′⊥ ) | {z } δ ( k − k ′ ) , (68)where we use the shorthands k ≡ ( ν, k ⊥ ) and x ≡ ( η, x ⊥ ). In a linear system of coordinates, the a µ k λ ( x ) introduced above would have the simple followingparametrization, a µ k λ ( x ) = e i k · x e − iω k x ε µ k λ , (69)where ε µ k λ is a constant polarization vector. Note that the minus sign in the exponential that givesthe time dependence is necessary in order to ensure that a µ k λ is a positive energy solution. Butbecause we work in a curvilinear co-ordinate system, the time dependence of the solutions cannotbe a simple exponential. Let us generalize the previous expression by writing a µ k λ ( τ, x ) = e i k · x α µ k λ ( τ ) , (70)where we have combined the polarization vector and the time dependence in a unique quantity thatwe denote α µ k λ ( τ ). (Since the equations of motion do not have coefficients that depend explicitlyon η or x ⊥ , it is clear that we can still look for solutions whose η and x ⊥ dependence is of theform exp( i k · x ).) Here again, we will have to make sure that the a µ k λ ( τ, x ) constructed in this waycontains only positive energy contributions.Let us consider first the unphysical solution. This is a pure gauge solution independent of τ .The most general τ independent solution is of the form α µ k = k x k y ν α ( ν, k ⊥ ) , (71)where α ( ν, k ⊥ ) is an arbitrary function. The inner product of the unphysical fluctuation a µ k withone of the physical solutions ( a µ k ′ λ with λ = 1 ,
2) is (cid:0) a k (cid:12)(cid:12) a k ′ λ (cid:1) = iτ α ∗ ( ν, k ⊥ ) Z + ∞−∞ dη Z d x ⊥ e i ( k ′ − k ) · x ∂ τ (cid:0) k x α x k ′ λ + k y α y k ′ λ + ντ − α η k ′ λ (cid:1) = iτ (2 π ) δ ( k − k ′ ) α ∗ ( ν, k ⊥ ) ∂ τ (cid:0) k x α x k λ + k y α y k λ + ντ − α η k λ (cid:1) . (72)We can satisfy this orthogonality condition by choosing a second solution of the form α µ k ( τ ) = k y − k x α ( τ, ν, k ⊥ ) . (73)20he functional form of α should be fixed by the equations of motion. Substituting the aboveexpression into the equations of motion (66) yields a differential equation for α , whose generalsolution can be expressed in terms of the Hankel functions H (1) iν and H (2) iν , α ( τ, ν, k ⊥ ) = a k H (1) iν ( k ⊥ τ ) + b k H (2) iν ( k ⊥ τ ) . (74)Recall here the integral representation of the Hankel functions, H (1) iν ( x ) = − iπ e + πν/ Z + ∞−∞ e ix cosh t + iνt dtH (2) iν ( x ) = + iπ e − πν/ Z + ∞−∞ e − ix cosh t − iνt dt , (75)The convention set by eq. (69) implies that only H (2) iν ( k ⊥ τ ) has the appropriate frequency sign tobe one the a k λ ’s. We can therefore set a k = 0 and keep only the second term in eq. (74). Thevalue of b k can then be determined by the orthogonality condition. For this, we need the identity H (2) ∗ iν ( x ) ↔ ∂ x H (2) iν ( x ) = − ie − πν πx , (76)from which we obtain (cid:0) a k (cid:12)(cid:12) a k ′ (cid:1) = (2 π ) δ ( k − k ′ ) | b k | k ⊥ e − πν π . (77)In order to get the same normalization as in eq. (68), we obtain (up to an irrelevant phase) b k = √ πe πν/ k ⊥ , (78)and the first physical solution reads a µ k ( x ) = √ πe πν/ k ⊥ k y − k x e i k · x H (2) iν ( k ⊥ τ ) . (79)The second physical solution can be found by requiring that it be orthogonal to the two solutionswe have so far. This restricts it to be of the form α µ k = ν k x α ⊥ ( τ, ν, k ⊥ ) ν k y α ⊥ ( τ, ν, k ⊥ ) − α η ( τ, ν, k ⊥ ) , (80)provided that k ⊥ τ ∂ τ α ⊥ = ∂ τ α η . This last constraint was derived by substituting the generalform of the second solution eq. (80) into the orthogonality condition eq. (72) with the unphysicalsolution. It turns out that this is the same constraint needed for this solution to fulfill Gauss’s law.Dynamical equations for the functions α ⊥ and α η can be found by substituting eq. (80) intoeqs. (66). One obtains, ∂ τ α η − τ ∂ τ α η + (cid:18) ν + 1 τ + k ⊥ (cid:19) ∂ τ α η = 0 ,∂ τ α ⊥ + 3 τ ∂ τ α ⊥ + (cid:18) ν + 1 τ + k ⊥ (cid:19) ∂ τ α ⊥ = 0 . (81)21he positive energy solutions to the above third order differential equations can be written as α η = const × Z τ dτ ′ τ ′ H (2) iν ( k ⊥ τ ′ ) α ⊥ = const × Z τ dτ ′ τ ′ H (2) iν ( k ⊥ τ ′ ) . (82)Note that the differential equations above imply that the functional form of α ⊥ and α η areonly determined up to an arbitrary τ independent function. This corresponds to a residual gaugefreedom in which we can always add to the second physical solution eq. (80) a pure gauge solutionhaving the form of eq. (71). This residual gauge freedom can be removed by imposing an additionalgauge fixing condition. For example, in simulations of classical Yang–Mills one typically imposestransverse Coulomb gauge.The integrals over the Hankel functions can be written in terms of hypergeometric functionsbut their form is not very enlightening. To streamline our notation, following Makhlin [58], let usdefine R ( a ) b,α ( k ⊥ τ ) ≡ Z τ dx x b H ( a ) α ( k ⊥ x ) . (83)The properly normalized form for the second physical degree of freedom is a µ k ( τ, η, x ⊥ ) = √ πe πν/ k ⊥ ν k x R (2) − ,iν ( k ⊥ τ ) ν k y R (2) − ,iν ( k ⊥ τ ) − R (2)+1 ,iν ( k ⊥ τ ) e i k · x . (84)For k ⊥ τ ≪
1, we can make use of the series expansion of the Hankel functions and rewrite thesolution as a µ k ( x ) ≈ √ πe πν/ k ⊥ k x k y − ( k ⊥ τ ) / ( ν + 2 i ) e i k · x H (2) iν ( k ⊥ τ ) . (85)From these explicit solutions, we will construct in appendix B the correlation function Γ for freefields on the initial surface τ = 0 + . However, for the purposes of generating a Gaussian ensemble offluctuations with the proper variance, the above results along with eqs. (64) and (65) are sufficient. After our extended discussion of the free fluctuations in A τ = 0 gauge, we now turn to the derivationof the small fluctuations spectrum in the Glasma. We shall first write down the small fluctuationequations of motion in the presence of the background classical fields. The main difficulty here isthat the Glasma background fields are not known analytically at arbitrary proper times. They arehowever known in closed form at τ = 0 + in terms of the classical CGC fields before the collision.We will perform a small time expansion, valid at very short proper times τ ≪ Q − s , of both theclassical fields and the small fluctuation fields and show that the fluctuations only depend on theclassical gauge fields immediately after the collision. As our final result, we will obtain explicitexpressions for an orthonormal basis of small fluctuations that generalize eqs. (79) and (84) to thecase of a non-zero background field at small proper times.22 .1 Structure of the Glasma background field In the Fock–Schwinger gauge, the classical gauge field configurations can be expressed as [47,51] A i = θ ( − x + ) θ ( x − ) α i + θ ( x + ) θ ( − x − ) α i + θ ( x + ) θ ( x − ) A i A η = θ ( x + ) θ ( x − ) A η (86)The fields α i , are the color fields of the two nuclei before the collision, that take the form oftransverse pure gauge fields, while A µ denotes the classical fields after the collision.Since we are interested in the spectrum of fluctuations at τ = 0 + we need only the behavior ofthe background fields shortly after the collision. The classical Glasma fields in the forward lightcone can be expanded, in all generality, at early times as A I = ∞ X n =0 A ( n ) I τ n (87)The initial conditions for these background fields at τ = 0 + are obtained by matching the Yang-Mills equations just below and just above the forward light-cone (to ensure a regular behavior ofthe field equations). One obtains [47] for the fields and their time derivatives, A i ( τ = 0 + ) = α i + α i A η ( τ = 0 + ) = 0 E i ( τ = 0 + ) = τ ∂ τ A i | τ =0 = 0 E η ( τ = 0 + ) = 1 τ ∂ τ A η | τ =0 = ig (cid:2) α i , α i (cid:3) . (88)As alluded to previously, explicit analytical solutions are known for the fields α i , . The Taylorexpansions of eqs. (87) begin with A i = A i (0 + ) + O ( τ ) , A η = 12 E η (0 + ) τ + O ( τ ) . (89)At this point it is useful to introduce some extra notation. Based on the Taylor expansion ineq. (87) of the classical background field, it will be convenient to introduce an analogous expansionof the covariant derivatives and projectors as defined in eq. (44), D µ = X n τ n D ( n ) µ , P µν = X n τ n P ( n ) µν . (90)Naturally, the coefficients D ( n ) µ and P ( n ) µν can be written in terms of the Taylor coefficients of theclassical background field, A ( n ) I . For example, if we perform the Taylor expansion of the covariantderivative, D ( n ) µ , the leading coefficients can be expressed in terms of the Taylor coefficients of theclassical background field, D ab (0) i = δ ab ∂ i − ig A ab (0) i , D ab (0) η = δ ab ∂ η , D ab (2) η = − ig A ab (2) η . (91) Of course, the D τ derivative does not need to be expanded, since in the Fock-Schwinger gauge one has D τ ≡ ∂ τ .
23n writing down the above expressions we have used the fact that the transverse components of thebackground field have non–vanishing zero order Taylor coefficients in contrast to the longitudinalcomponent of the background field whose leading behavior starts with τ . Similarly we can expressthe Taylor coefficients of P ( n ) µν in terms of the Taylor coefficients of the classical background field.The terms which will needed later on in our discussion include, P ab (0) ηi = P ab (0) iη = D ab (0) i ∂ η , P ab (0) ij = δ ab ∂ i ∂ j − ig A ab (0) i ∂ j − ig A ab (0) j ∂ i − ig (cid:0) ∂ i A (0) j (cid:1) ab − g A ac (0) i A cb (0) j , P ab (2) ηη = − ig A ab (2) η ∂ η . (92)In deriving the above expressions we have used the fact that the background field is boost invariant, ∂ η A I ≡
0. It is this property of the background field that leads to the simple form of P (0) ηi andyields the factor of two in the expression for P (2) ηη . In the case of the τ, η, x ⊥ system of coordinates and in the Fock–Schwinger gauge, the equationsof motion for the fluctuations propagating over such a background field are written explicitly ineqs. (43). Solving for the full time dependence of the fluctuations is both intractable (since thetime dependence of the background field is only known numerically) and unnecessary (since weonly need the spectrum of fluctuations at early times).A crucial property of the background Glasma fields is that they are invariant under boostsin the longitudinal direction. This implies that the fields A i and A η in the forward light-cone,after the collision, are independent of the space-time rapidity η . Thus the variable ν , the Fourierconjugate of η , is a conserved quantum number for fluctuations propagating over the Glasma fields,that can be used to label the elements of the basis. We can therefore write the elements of thebasis as a µν l λ ( τ, η, x ⊥ ) ≡ e iνη β µν l λ ( τ, x ⊥ ) , (93)where λ = 1 , l collectively represents the remaining quantum numbersnecessary to label the basis. The main difference with the free case (see eq. (70)) is that we cannotassume that the x ⊥ dependence of the fluctuations has the form of a plane wave and insteadrepresent it more generally as the function β µν l λ ( τ, x ⊥ ). This is because the background field has anon-trivial dependence on x ⊥ . A further consequence is that, in contrast to the vacuum case, theremaining quantum numbers encoded in l will not simply be transverse momenta. As in the freecase, we shall keep only solutions that have positive frequencies in this basis.We would now like to motivate one of the main results of this work; a modified form of thelinearized equation of motion eq. (43), which captures the early time behavior of a quantumfluctuation propagating on top of the classical background field. The simplest way to arrive at ourresult is to simply replace all of the projectors appearing in eq. (43) with the corresponding zerothorder Taylor coefficient from the proper time expansion given in eq. (90). Following this proceduregives essentially the right result except for one subtlety. We will argue that we also need to include It may be instructive to work this term out explicitly. Since the field strength tensor is anti–symmetric we have P abηη = D acη D cbη . We can therefore write P ab (2) ηη = D ac (0) η D cb (2) η + D ac (2) η D cb (0) η . From eq. (91) we know that D ab (0) η = δ ab ∂ η and therefore commutes with the boost invariant background field. The final result is then P ab (2) ηη = 2 D ab (2) η ∂ η . τ P (2) ηη , that appears at higher order in the expansion of the projector. The resultingsmall–time linearized equations of motion are, (cid:0) ∂ τ τ − ∂ τ − τ − P (0) ii (cid:1) a η + τ − P (0) iη a i = 0 , (cid:0) ∂ τ τ ∂ τ − τ − P (0) ηη − τ P (2) ηη − τ P (0) ii (cid:1) a x + τ − P (0) ηx a η + τ P (0) ix a i = 0 , (cid:0) ∂ τ τ ∂ τ − τ − P (0) ηη − τ P (2) ηη − τ P (0) ii (cid:1) a y + τ − P (0) ηy a η + τ P (0) iy a i = 0 . (94)The necessity of including the term ( P (2) ηη ) can be seen by looking at the structure of the operatoracting on a x in the second equation (or equivalently acting on a y in the third equation) above; (cid:0) ∂ τ τ ∂ τ − τ − P (0) ηη − τ P (2) ηη − τ P (0) ii (cid:1) a x , (95)By examining the power counting in τ of each term in this operator, it would be inconsistent toignore the τ component of P ηη which is of the same order as P (0) ii . One could argue that both P (0) ii and P (2) ηη are suppressed by τ at early times relative to P (0) ηη and therefore can both beignored. If we drop these terms, when we turn off the background field, we would not recover thevacuum wavefunctions and this is clearly unsatisfactory. We therefore conclude that we need toinclude all projectors that are of the same order in τ as the constants appearing in the vacuumcase. By the argument presented we should also include the term P (2) ηη in order to have thecorrect power counting.A more formal way of arriving at eq. (94) is by considering the series expansion of the smallfluctuations, a I . Using the method of Frobenius one finds that the leading τ behavior of thetransverse components behaves as a i ∼ τ iν while that of the longitudinal component goes as a η ∼ τ iν . These coefficients are exactly those needed to reproduce the small τ expansion of thephysical solutions found in the vacuum case. Furthermore, if we continue to use the Frobeniusmethod we find that only a small subset of the Taylor Coefficients in eq. (90) are needed todetermine the lowest order coefficient in the Frobenius expansion of a I . These are precisely theTaylor coefficients that have been included in eq. (94).Finally, we need to discuss why we have not included Gauss’s law in eq. (94). The reasoningis that Gauss’s law is not a dynamical equation but a constraint, and therefore not amenable to aseries expansion. This can be seen by noting that Gauss’s law, G ≡ τ − D η E η + τ D i E i = 0 (96)is a constant of motion ( ∂ τ G = 0) and therefore if Gauss’s law is obeyed at τ = 0 + it will remainsatisfied for all times. For the first physical solution, we take a ην l = 0 as was done in the vacuum case. With this choice,the first equation in (94) coincides with Gauss’s law. The last two equations control the evolutionof the transverse components a iν l . Since we expect the time dependence in eq. (93) to enter in thesame way as in the vacuum case, we postulate that the solution at early times will be of the form a µν l ( τ, η, x ⊥ ) = √ πe πν/ Q ν l b xν l ( x ⊥ ) b yν l ( x ⊥ )0 e iνη H (2) iν ( Q ν l τ ) . (97)25ext we substitute eq. (97) into the early time linearized equation of motion (94). Requiring thateq. (97) is a solution to the equations of motion leads to, − h D (0) y D (0) y + P ( ν )(2) ηη i b xν l + P (0) yx b yν l = Q ν l b xν l − h D (0) x D (0) x + P ( ν )(2) ηη i b yν l + P (0) xy b xν l = Q ν l b yν l . (98)We have explicitly included a superscript ( ν ) on P ν (2) ηη in order to remind ourselves that thederivative with respect to η should be replaced by iν when acting on the exponential in eq. (97).We should also stress that the equations above only depend on the background fields at τ = 0 + ,which are known analytically.Solving eqs. (98) amounts to finding the eigenvalues Q ν l and eigenfunctions (the doublets( b xν l ( x ⊥ ) , b yν l ( x ⊥ ))) of an Hermitean operator. Since this operator is Hermitean, its spectrum ismade of real eigenvalues, and its eigenfunctions can be chosen to form an orthonormal basis, Z d x ⊥ b i ∗ ν l ( x ⊥ ) b iν l ′ ( x ⊥ ) = δ ll ′ . (99)Note that the choice of normalization in eqs. (97) and (99) is such that the inner product definedin eq. (68) is satisfied (cid:0) a ν l (cid:12)(cid:12) a ν ′ l ′ (cid:1) = 2 πδ ( ν − ν ′ ) δ ll ′ . (100)The operator that we need to diagonalize in eqs. (98) has a spectrum that is twice larger thanthe size expected for the space of solutions with polarization λ = 1. Half of this spectrum isincompatible with Gauss’ law and must be discarded . As was the case for the first physical solution, the second physical solution will maintain the same τ dependence at τ = 0 + but with a modified dispersion relation. We therefore write the mostgeneral form of the second physical solution as a µν l ( τ, η, x ⊥ ) = √ πe πν/ Q ν l b xν l ( x ⊥ ) R (2) − ,iν ( Q ν l τ ) b yν l ( x ⊥ ) R (2) − ,iν ( Q ν l τ ) b ην l ( x ⊥ ) R (2)+1 ,iν ( Q ν l τ ) e iνη . (101)Following the same procedure as for the first solution, we substitute eq. (101) into the linearizedequations of motion given by eq. (94). Requiring that eq. (101) is a solution to the equation ofmotion at lowest order in τ leads to the following equations for b iν l b xν l = iν D (0) x b ην l b yν l = iν D (0) y b ην l −P (0) ii b ην l ( x ⊥ ) = Q ν l b ην l ( x ⊥ ) . (102) In the free case, this operator is O ij ≡ − ∂ ⊥ δ ij + ∂ i ∂ j . It has two types of eigenfunctions: (i) b i = ∂ i χ , witheigenvalue Q = 0, and (ii) b i = ǫ ij ∂ j χ , with eigenvalue Q = k ⊥ (where χ ( x ⊥ ) ≡ exp( i k ⊥ · x ⊥ )). Only the secondeigenfunction is compatible with Gauss’ law ∂ i b i = 0. Interestingly, one of the perturbative solutions of the classicalequations of motion at large transverse momenta in the forward light-cone has an identical structure [47]. b x,yν l in terms of b ην l . The third equation determines b ην l as an eigenfunction of the operator −P (0) ii , with eigenvalue Q ν l . Because this operator isHermitean, these eigenvalues are real, and the eigenfunctions are mutually orthogonal, Z d x ⊥ b η ∗ ν l ( x ⊥ ) b ην l ′ ( x ⊥ ) = δ ll ′ . (103)We can therefore write the final form of the second physical solution in terms of the singleeigenfunction b ην l a µν l ( τ, η, x ⊥ ) = √ πe πν/ Q ν l iνR (2) − ,iν ( Q ν l τ ) D (0) x iνR (2) − ,iν ( Q ν l τ ) D (0) y R (2)+1 ,iν ( Q ν l τ ) b ην l ( x ⊥ ) e iνη , (104)where b ην l ( x ⊥ ) is a solution to the eigenvalue equation (102). Finally, let us rewrite the solutionusing the small time approximation of R (2) ± ,iν , as done in the vacuum case (see eq. (85)) a µ k ( x ) ≈ √ πe πν/ Q ν l D (0) x D (0) y − ( Q ν l τ ) / ( ν + 2 i ) b ην l ( x ⊥ ) e iνη H (2) iν ( Q ν l τ ) . (105) The results of the previous section provide all the ingredients we need in order to evaluate aninclusive quantity such as the energy-momentum tensor, resumming in the calculation both thelarge logs of 1 /x , and the secular terms that plague fixed order calculations. The algorithm forperforming such a calculation can be broken down into several independent steps: i. Solve the JIMWLK equation. i.a
Generate an ensemble of color source densities ρ a ( x ⊥ ) (or, equivalently, of Wilson linesΩ ab ( x ⊥ )) that represent the distribution W [ ρ ] at large x , close to the fragmentationregion of a nucleus. i.b For each of these configurations, evolve it to smaller x by using the Langevin formulationof the JIMWLK equation. This amounts to performing a random walk on the space ofmappings from R to the group SU(3) [65,66]. ii. Pick two elements (one for each nucleus) in the above ensembles for each projectile, evolvedat the values of x relevant to the observable of interest. Compute the gauge fields and theirfirst time derivatives on the initial surface τ = 0 + immediately after the collision [52,56]. iii. Generate fluctuations on top of the classical Glasma fields. iii.a
Solve the eigenvalue equations in eqs. (98) and (102). In the former case, only thesolutions that fulfill Gauss’s law should be kept. It should be noted that in a latticediscretization, this amounts to diagonalizing large but sparse matrices.27 ii.b
Evaluate the Hankel functions H (2) iν and the hypergeometric functions R (2) ± ,iν at theinitial time of interest. (Note that because they oscillate as ln( τ ) → −∞ , this initialtime cannot be exactly zero.) To do this, one can go back to their defining differentialequations, and solve them numerically, keeping only the solution that becomes a positivefrequency plane wave as τ → + ∞ . iii.c Superimpose on top of the classical Glasma fields (obtained in step ii ), a linear combi-nation of the small fluctuations that are obtained by solving the eigenvalue equations ineqs. (97) and (104), multiplied by random Gaussian coefficients that have a variance given by eq. (65). iv. For each initial condition generated in this way, solve numerically the classical Yang-Millsequations forward in time, up to the proper time at which the observable should be evaluated.Repeat steps iii and iv in order to do a Monte-Carlo evaluation of the average over thefluctuations of the initial gauge fields. With this work, we have completed the resummation of all the leading contributions from quantumfluctuations to inclusive quantities at early times, in the Color Glass Condensate effective fieldtheory approach to high energy nucleus-nucleus collisions. These quantum fluctuations can befactorized into ν = 0 and ν = 0 modes, where ν is the Fourier conjugate to the space-time rapidity η . The expression in eq. (5) described the contribution of ν = 0 modes, which correspond tosumming the leading logarithmic α s ln(1 /x , ) contributions. Our final expression for the energy-momentum tensor, extending the expression in eq. (5) to include the leading secular terms, is h T µν i LLx+LInst . = Z [ Dρ Dρ ] W x [ ρ ] W x [ ρ ] × Z (cid:2) D α (cid:3) F (cid:2) α (cid:3) T µν LO [ A [ ρ , ρ ] + α ]( x ) , (106)where the weight functionals W x , [ ρ , ] satisfy the JIMWLK renormalization group equation ineq. (6). The argument A ≡ ( A , E ) denotes collectively the components of the classical fields andtheir canonicallly conjugate momenta on the initial proper time surface. These quantities arefunctionals of ρ , ρ and, as discussed previously, analytical expressions for these are available at τ = 0 + [47,51,55,56]. The initial spectrum of fluctuations F (cid:2) α (cid:3) , defined in eq. (28), requires thatone compute small fluctuations around the classical background fields A as τ → + . The formalexpressions for these were derived in section 4. In section 5, we have outlined a practical algorithmto compute the path integral over small fluctuations. As numerical algorithms for solving theJIMWLK equation are now also available [65] and have been successfully implemented [66], a fullfledged numerical computation of eq. (106) is feasible in the near future. We should emphasizethat this equation is valid for any inclusive quantity, not just the energy-momentum tensor.There are several applications of this formalism to understand key features of early time dy-namics in heavy ion collisions. We shall discuss a few of these here. Note that the (arbitrary) factor N K in this variance is cancelled by the normalization of the eigenfunctions a K and of the integration measure dµ K . Indeed, in eq. (64) one has c K ∼ p N K , a K ∼ p N K and dµ K ∼ /N K . Early thermalization
It is important to emphasize that a numerical simulation of eq. (106)would describe the real time evolution of a quantum field theory that goes far beyond purelyclassical contributions but includes as well important quantum effects to all orders in per-turbation theory. (For similar considerations of the relative roles of classical versus quantumeffects within the 2PI framework, we refer readers to refs. [67–69] and references therein.) Inref. [1], we developed the corresponding formalism for a scalar φ theory, and demonstratedthat the system developed an equation of state, allowing one to write the energy-momentumconservation equation for the resummed energy momentum tensor T µν as the closed form setof equations corresponding to the ideal hydrodynamics satisfied by a perfect relativistic fluid.We interpreted this result as arising from a phase decoherence of the classical trajectoriesin eq. (106), with the different initial conditions given by the spectrum of initial quantumfluctuations. As discussed previously, we expect the same to occur in QCD as well.It is widely believed that the strong hydrodynamic behavior seen in heavy ion collisionsrequires early thermalization. However, as shown for the scalar theory, and may likely alsobe true for gauge theories, this is not a necessary condition for (nearly) perfect fluidity.It is interesting to ask what is the proper condition for thermalization in a quantum fieldtheory. One such criterion is based on Berry’s conjecture [40,41,43,44], that states that in aquantum system whose classical counterpart is chaotic, high lying energy eigenfunctions looklike Gaussian random functions. It was later argued by Srednicki [42] that such eigenstateswould appear to be thermal, for example, if one measures the single particle distribution.Since the underlying classical Yang-Mills theory is believed to be chaotic [45,46], the quantumsystem described by eq. (106) is a good candidate to explore these ideas in a quantum fieldtheory. In our approach, the initial state is not an energy eigenstate, but rather a coherentstate. It is formulated as a sum of plane waves with random Gaussian coefficients (seeeqs. (64) and (65)) but there is no constraint that restrict the Gaussian random eigenstatesto states of a given energy, in contrast to the states postulated by Berry. The self-interactionsof the fields lead to a loss of this initial coherence, and it would be very interesting to see ifthermalization occurs on the same time-scales as decoherence.Alternatively, one can look at the spectral function, defined in terms of the imaginary partof the retarded Green function, for the appearance of quasi-particle behavior, which allowsfor a kinetic theory description in terms of single particle distributions. It is interesting toexplore whether there is a region of overlap between a description in terms of high occupationnumber fields and a kinetic theory description in terms of quasi-particles [70,71]. Such aregime may equivalently be described by a coupled set of equations for the classical fields andquasi-particle modes [72] which is reminiscent of the description of superfluids in condensedmatter physics. Indeed, it is conceivable that the best description of such an overpopulatedsystem, for intermediate times, might be as a Bose-Einstein superfluid [73], before the inelasticprocesses in the Glasma begin to dominate [74,75] and lead eventually to a conventionalkinetic description.While the mechanism for thermalization is non-perturbative, it is still unclear whether themechanism is weak coupling or strong coupling. One might anticipate that the former is morelikely at higher energies due to the increasing dominance of semi-hard scales, but where such atransition occurs is unknown. It is intriguing that while the the mechanism of thermalizationin AdS/CFT inspired strong coupling approaches appears very different (more “top down”than “bottom up”), there appear to be technical similarities between aspects of our weak29oupling approach and these approaches [76,77]. • Sphaleron transitions
As emphasized, our master formula in eq. (106) is valid not only for the expectation valueof the energy-momentum tensor but also for other inclusive quantities. One such exampleis the sphaleron transition rate Γ sphal . , which controls the mean squared change in the axialcharge in thermal equilibrium through the relation h (∆ Q A,q ) i therm . = 4 V t Γ sphal . , (107)where V is the spatial volume and q denotes quark flavor. In ref. [78], it was noted thatsphaleron transitions are not allowed in the boost invariant classical Glasma. Because theclassical dynamics is effectively 2+1-dimensional, the second homotopy group of SU(3) gaugetheory is zero, disallowing integer valued topological transitions. The quantum fluctuationswe have been describing are no longer boost invariant, thereby allowing sphaleron transitionsto go. In the non-equilibrium Glasma, the relevant quantity for computing the mean squarechange in the axial charge is the Wightman function [79] G >F ˜ F ( X, Y ) = (cid:28) g π F aµν ˜ F µνb ( X ) g π F aαβ ˜ F αβb ( Y ) (cid:29) , (108)with ˜ F µν ≡ ǫ µναβ F αβ . The sphaleron transition rate is defined in terms of this quantity asΓ sphal = Z d X G >F ˜ F ( X, . (109)This quantity may be computed by a formula similar to eq. (106). It has aroused muchinterest recently because in semi-peripheral heavy ion collisions, the combined effect of largeexternal electromagnetic ~B fields and a large rate for topological transitions can lead to an in-duced charge separation phenomenon, called the Chiral Magnetic effect [80], with observableconsequences. Our approach allows in principle an ab initio computation of this effect. • Jet quenching
Besides the large flow observed in heavy ion collisions, the apparent strong modificationof rare final states (such as jets, high p ⊥ hadrons and heavy flavors) by the medium isadduced as confirmation of the high degree of opacity in the medium consistent with astrongly correlated fluid. The standard energy loss mechanism is radiative energy loss, whichis dominated by collinear splittings that are primarily influenced by late time dynamics in astrongly interacting quark-gluon plasma. For a sampling of reviews, see refs. [81–84]. Thereare however potentially large modifications of the spectra of hard final states from energyloss at early times, that are are not included in this energy loss scenario [85]; for a recenttreatment of large angle contributions in a framework similar to ours, see ref. [86]. However,no computation has thus far included next-to-leading order corrections to, for example, thesingle inclusive parton spectrum at early times taking into account both small x evolutionand multiple scattering effects. This is done in our formalism when T µν in eq. (106) isreplaced by E p dNd p ; in fact, eq. (106) sums up a class of all order graphs that correspondto coherent multiple emissions where the momenta of all but one of the final state gluons30 Figure 3: Example of graph included via the resummation of eq. (106). The green and red dotsrepresent the color charges that describe the gluon content of the colliding nuclei in the CGCframework.is integrated over. A typical contribution is illustrated in the figure 3. Note that thisdiagram, corresponding to the resummed case, looks like a parton shower interacting withthe background Glasma fields. However, unlike vacuum showers which can in space-timebe visualized as being logarithmically divergent in the proper time, these “showers” are aconsequence of the exponentially growing contributions from leading instabilities at eachorder in perturbation theory.We note that the single inclusive gluon spectrum also potentially suffers from collinear andinfrared singularities. In the Glasma, it is possible that these could be regulated by strongmultiple scattering and/or screening effects, but that remains to be proved. To avoid suchcomplications, as in usual jet physics [87,88], one can look instead at correlators of the energy-momentum tensor that correspond to energy flow and are manifestly infrared and collinearsafe. Clearly, there are a number of issues that need to be resolved here; the promising aspectof our formalism is that parton evolution, radiation and re-scattering can likely be treated,without ad hoc assumptions, in a consistent manner at early times.
Acknowledgements
We would like to thank Miklos Gyulassy for providing the encouragement to initiate this work.We would also like to thank T. Epelbaum, K. Fukushima, Y. Hatta, C. Jarzynski, T. Lappi, J.Liao, L. McLerran, A. Mueller, S. Srednyak, D. Teaney and G. Torrieri for useful discussions. R.V.is supported by the US Department of Energy under DOE Contract DE-AC02-98CH10886. K.D.is supported by the US Department of Energy under DOE Contracts DE-FG02-03ER41260 andDE-AC02-98CH10886. F. G. would like to thank the Nuclear Theory group at BNL for hospitalityand support during the completion of this work.31
Alternative basis for the free field solution
We will show in this appendix, that the form of the vacuum fluctuations can be written in termsof simple analytic functions if we trade the index ν for an index θ introduced via the followingtransformation a µ k ( x ) ≡ Z dν π e − iνθ a µ k ( x ) , (110)where we denote k ≡ ( θ, k ⊥ ).To see that we are free to make this change in the quantum numbers, let us start from thereal–valued field fluctuation as written in terms of the ν coordinate a µ ( x ) = X λ =1 , Z d k ⊥ dν (2 π ) h c k λ a µ k λ ( x ) + c ∗ k λ a µ ∗ k λ ( x ) i . Then, making the trade from ν to θ we obtain a µ ( x ) = X λ =1 , Z d k ⊥ dν (2 π ) h c k λ Z dθe iνθ a µ k λ ( x ) + c ∗ k λ Z dθe iνθ a µ ∗ k λ ( x ) i = X λ =1 , Z d k ⊥ dθ (2 π ) h Z dνe iνθ c k λ a µ k λ ( x ) + Z dνe iνθ c ∗ k λ a µ ∗ k λ ( x ) i = X λ =1 , Z d k ⊥ dθ (2 π ) h d k λ a µ k λ ( x ) + d ∗ k λ a µ ∗ k λ ( x ) i , where we defined d k λ ≡ Z dνc k λ e iνθ and in keeping with the notation employed throughout the text we have used k ≡ ( ν, k ⊥ ) and k ≡ ( θ, k ⊥ ). Since c k λ is a Gaussian random variable so is its Fourier Transform, d k λ , and we arefree to generate random fluctuations using either basis.Using this new basis, the first physical solutions transforms to a µ k ( x ) ≡ Z dν π e − iνθ a µ k ( x ) = i √ πk ⊥ k y − k x e − ik ⊥ τ cosh( θ − η )+ i k ⊥ · x ⊥ , (111)After the transformation to the θ variable, eqs. (82) can be integrated over time α ⊥ = i √ πk ⊥ tanh( θ − η ) e − ik ⊥ τ cosh( θ − η ) α η = i √ πk ⊥ (cid:20) ik ⊥ τ cosh( θ − η )cosh ( θ − η ) (cid:21) e − ik ⊥ τ cosh( θ − η ) . (112)Note that eqs. (81) only specify α ⊥ and α η up to a τ independent function. The τ independentfunctions that can be added to α ⊥ and α η are not independent, because this modification mustcorrespond to a residual gauge transformation. Thus the allowed modifications are α ⊥ → α ⊥ + f , α η → α η + i ( ∂ θ f ) , (113)32here f is an arbitrary τ independent function. We can choose to fix this residual gauge freedomby choosing the function f such that a µ k ( x ) vanishes at τ = 0 + . After fixing the residual gaugefreedom, we have the following expression for the second physical solution a µ k ( x ) = i √ πk ⊥ k x g ⊥ k y g ⊥ g η e − ik ⊥ τ cosh( θ − η )+ i k ⊥ · x ⊥ , (114)where we denote, g ⊥ = i tanh( θ − η ) h − e ik ⊥ τ cosh( θ − η ) i g η = − − e ik ⊥ τ cosh( θ − η ) + ik ⊥ τ cosh( θ − η )cosh ( θ − η ) . (115) B Wightman functions for free fields
In this appendix, we shall construct the equal τ Wightman function for free fields. The equal-timeWightman function corresponding to the first physical solution a µ k is given by G µν ≡ Z d k ⊥ (2 π ) dθ a µ ∗ k ( τ, η, x ⊥ ) a ν k ( τ, η ′ , x ′⊥ ) . (116)Using the explicit expression in eq. (110), G µν can be written in the following formal way, G ij = 1(2 π ) (cid:18) − ∂ y ∂ y ∂ x ∂ x ∂ y − ∂ x (cid:19) F (cid:18) i∂ η , q − τ ∂ ⊥ (cid:19) δ ( η − η ′ ) ln (cid:18) | x ⊥ − x ′⊥ | (cid:19) , (117)where Λ is an infrared cutoff and we have defined F ( ν, d ⊥ ) ≡ Z dx √ x dη e iνη e − ixd ⊥ sinh( η/ . (118)Likewise, the Wightman function corresponding to the second physical solution is given by G µν ≡ Z d k ⊥ (2 π ) dθ a µ ∗ k ( τ, η, x ⊥ ) a ν k ( τ, η ′ , x ′⊥ ) . (119)From the discussion following eq. (81), we noted that there was a residual gauge freedom remainingeven after finding the two physical solutions. In the appendix A, we chose to fix this gauge byrequiring that α η → τ = 0 + in order that Hamilton’s equations are regular at τ = 0. Clearly,for the problem at hand, this is the correct choice. However, computing the Wightman functionwill be made much easier by choosing a different gauge, g ⊥ = i tanh( θ − η ) , g η = − ik ⊥ τ cosh( θ − η )cosh ( θ − η ) . The final result does not depend on this cutoff, thanks to the derivatives acting on the logarithm.
33e should stress that there is a residual gauge freedom remaining in any correlator of gauge fields.With this new gauge choice we find G ij = − π ) (cid:18) ∂ x ∂ y ∂ x ∂ x ∂ y ∂ y (cid:19) F (cid:18) i∂ η , q − τ ∂ ⊥ (cid:19) δ ( η − η ′ ) ln (cid:18) | x ⊥ − x ′⊥ | (cid:19) , (120)with F ( ν, d ⊥ ) ≡ Z dx √ x dη (cid:20) − η x + cosh η (cid:21) e iνη e − ixd ⊥ sinh( η/ . (121)We can also calculate the correlation function of the canonical momenta defined by e i = τ ∂ τ a i .The free Wightman function for the transverse components of the electric field is defined as H ij = X λ =1 , τ Z d k ⊥ (2 π ) dθ h ∂ τ a i ∗ k λ ( τ, η, x ⊥ ) i h ∂ τ a j k λ ( τ, η ′ , x ′⊥ ) i . (122)Note that the τ derivatives remove the residual gauge freedom that still remained in the expressionfor G ij as written above. This is expected since we don’t expect any residual gauge freedom toremain when computing correlators of physical quantities such as th electric field. The Wightmanfunction for the first physical solution is H ij = 1(2 π ) (cid:18) − ∂ y ∂ y ∂ x ∂ x ∂ y − ∂ x (cid:19) F (cid:18) i∂ η , q − τ ∂ ⊥ (cid:19) δ ( η − η ′ ) ln (cid:18) | x ⊥ − x ′⊥ | (cid:19) , (123)and similarly for the second physical solution H ij = − π ) (cid:18) ∂ x ∂ y ∂ x ∂ x ∂ y ∂ y (cid:19) F (cid:18) i∂ η , q − τ ∂ ⊥ (cid:19) δ ( η − η ′ ) ln (cid:18) | x ⊥ − x ′⊥ | (cid:19) , (124)where the functions F , are defined as F ( ν, d ⊥ ) ≡ Z dx √ x dη (cid:2) x + cosh η (cid:3) e iνη e − ixd ⊥ sinh( η/ , (125) F ( ν, d ⊥ ) ≡ Z dx √ x dη (cid:2) x − cosh η (cid:3) e iνη e − ixd ⊥ sinh( η/ . (126) References [1] K. Dusling, T. Epelbaum, F. Gelis, R. Venugopalan, Nucl. Phys.
A 850 , 69 (2011).[2] J. Adams, et al., [STAR Collaboration] Nucl. Phys.
A 757 , 102 (2005).[3] K. Adcox, et al., [PHENIX Collaboration] Nucl. Phys.
A 757 , 184 (2005).[4] I. Arsene, et al., [BRAHMS collaboration] Nucl. Phys.
A 757 , 1 (2005).[5] B.B. Back, et al., [PHOBOS collaboration] Nucl. Phys.
A 757 , 28 (2005).[6] K. Aamodt, et al, [ALICE Collaboration] arXiv:1011.3914 [nucl-ex].347] M. Luzum, P. Romatschke, Phys. Rev.
C 78 , 034915 (2008), Erratum-ibid.C , 039903(2009).[8] P. Romatschke, Int. J. Mod. Phys. E , 1 (2010).[9] D. Teaney, Prog. Part. Nucl. Phys. , 451 (2009).[10] F. Gelis, E. Iancu, J. Jalilian-Marian, R. Venugopalan, Ann. Rev. Part. Nucl. Sci. , 463(2010).[11] E. Iancu, R. Venugopalan, Quark Gluon Plasma 3, Eds. R.C. Hwa and X.N. Wang, WorldScientific, hep-ph/0303204.[12] L.V. Gribov, E.M. Levin, M.G. Ryskin, Phys. Rept. , 1 (1983).[13] A.H. Mueller, J-W. Qiu, Nucl. Phys. B 268 , 427 (1986).[14] L.D. McLerran, R. Venugopalan, Phys. Rev.
D 49 , 2233 (1994).[15] L.D. McLerran, R. Venugopalan, Phys. Rev.
D 49 , 3352 (1994).[16] L.D. McLerran, R. Venugopalan, Phys. Rev.
D 50 , 2225 (1994).[17] T. Lappi, L.D. McLerran, Nucl. Phys.
A 772 , 200 (2006).[18] F. Gelis, T. Lappi, R. Venugopalan, Phys. Rev.
D 78 , 054019 (2008).[19] F. Gelis, T. Lappi, R. Venugopalan, Phys. Rev.
D 78 , 054020 (2008).[20] F. Gelis, T. Lappi, R. Venugopalan, Phys. Rev.
D 79 , 094017 (2009).[21] J. Jalilian-Marian, A. Kovner, L.D. McLerran, H. Weigert, Phys. Rev.
D 55 , 5414 (1997).[22] J. Jalilian-Marian, A. Kovner, A. Leonidov, H. Weigert, Nucl. Phys.
B 504 , 415 (1997).[23] J. Jalilian-Marian, A. Kovner, A. Leonidov, H. Weigert, Phys. Rev.
D 59 , 014014 (1999).[24] J. Jalilian-Marian, A. Kovner, A. Leonidov, H. Weigert, Phys. Rev.
D 59 , 034007 (1999).[25] J. Jalilian-Marian, A. Kovner, A. Leonidov, H. Weigert, Phys. Rev.
D 59 , 099903 (1999).[26] E. Iancu, A. Leonidov, L.D. McLerran, Nucl. Phys.
A 692 , 583 (2001).[27] E. Iancu, A. Leonidov, L.D. McLerran, Phys. Lett.
B 510 , 133 (2001).[28] E. Ferreiro, E. Iancu, A. Leonidov, L.D. McLerran, Nucl. Phys.
A 703 , 489 (2002).[29] P. Romatschke, R. Venugopalan, Phys. Rev. Lett. , 062302 (2006).[30] P. Romatschke, R. Venugopalan, Eur. Phys. J. A 29 , 71 (2006).[31] P. Romatschke, R. Venugopalan, Phys. Rev.
D 74 , 045011 (2006).[32] H. Fujii, K. Itakura, Nucl. Phys.
A 809 , 88 (2008).3533] H. Fujii, K. Itakura, A. Iwazaki, Nucl. Phys.
A 828 , 178 (2009).[34] S. Mrowczynski, Phys. Lett.
B 314 , 118 (1993).[35] A.K. Rebhan, P. Romatschke, M. Strickland, Phys. Rev. Lett. , 102303 (2005).[36] P. Arnold, G.D. Moore, L.G. Yaffe, Phys. Rev. D 72 , 054003 (2005).[37] S. Mrowczynski, M.H. Thoma, Phys. Rev.D , 036011 (2000).[38] D. Polarski, A.A. Starobinsky, Class. Quant. Grav. , 377 (1996).[39] D.T. Son, hep-ph/9601377.[40] M.V. Berry, J. Phys. A: Math. Gen. , 2083 (1977).[41] J.M. Deutsch, Phys. Rev. A 43 , 2046 (1991).[42] M. Srednicki, Phys. Rev. E , 888 (1994).[43] C. Jarzynski, Phys. Rev. E , 2254 (1997).[44] M. Rigol, V. Dunjko, M. Olshanii, Nature , 854 (2008).[45] T.S. Biro, C. Gong, B. Muller, A. Trayanov, Int. J. Mod. Phys. C 5 , 113 (1994).[46] U.W. Heinz, C.R. Hu, S. Leupold, S.G. Matinyan, B. Muller, Phys. Rev.
D 55 , 2464 (1997).[47] A. Kovner, L.D. McLerran, H. Weigert, Phys. Rev.
D 52 , 3809 (1995).[48] Yu.V. Kovchegov, D.H. Rischke, Phys. Rev.
C 56 , 1084 (1997).[49] J.P. Blaizot, Y. Mehtar-Tani, Nucl. Phys.
A 818 , 97 (2009).[50] J.P. Blaizot, T. Lappi, Y. Mehtar-Tani, Nucl. Phys.
A 846 , 63 (2010).[51] A. Krasnitz, R. Venugopalan, Nucl. Phys.
B 557 , 237 (1999).[52] A. Krasnitz, R. Venugopalan, Phys. Rev. Lett. , 4309 (2000).[53] A. Krasnitz, Y. Nara, R. Venugopalan, Phys. Rev. Lett. , 192302 (2001).[54] T. Lappi, Phys. Rev. C 67 , 054903 (2003).[55] T. Lappi, Eur. Phys. J.
C 55 , 285 (2008).[56] T. Lappi, Int. J. Mod. Phys.
E20 , 1 (2011).[57] K. Fukushima, F. Gelis, L. McLerran, Nucl. Phys.
A 786 , 107 (2007).[58] A. Makhlin, hep-ph/9608261.[59] F. Gelis, R. Venugopalan, Nucl. Phys.
A 776 , 135 (2006).[60] F. Gelis, R. Venugopalan, Nucl. Phys.
A 779 , 177 (2006).3661] T. Lappi, S. Srednyak, R. Venugopalan, JHEP , 066 (2010).[62] K. Fukushima, F. Gelis, arXiv:1106.1396 [hep-ph].[63] F. Gelis, T. Lappi, R. Venugopalan, Int. J. Mod. Phys. E , 2595 (2007).[64] S. Weinberg, Gravitation and Cosmology , John Wiley & Sons (1972).[65] J.P. Blaizot, E. Iancu, H. Weigert, Nucl. Phys.
A 713 , 441 (2003).[66] K. Rummukainen, H. Weigert, Nucl. Phys.
A 739 , 183 (2004).[67] J. Berges, S. Roth, Nucl. Phys.
B 847 , 197 (2011).[68] J. Berges, J. Serreau, hep-ph/0208070.[69] J. Berges, S. Bors´anyi, J. Serreau, Nucl. Phys.
B 660 , 51 (2003).[70] A.H. Mueller, D.T. Son, Phys. Lett.
B 582 , 279 (2004).[71] S. Jeon, Phys. Rev.
C 72 , 014907 (2005).[72] F. Gelis, S. Jeon, R. Venugopalan, Nucl. Phys.
A 817 , 61 (2009).[73] J.P. Blaizot, F. Gelis, J. Liao, L. McLerran, R. Venugopalan, Work in progress.[74] R. Baier, A.H. Mueller, D. Schiff, D. Son, Phys. Lett.
B 502 , 51 (2001).[75] A.H. Mueller, A.I. Shoshi, S.M.H. Wong, Nucl. Phys.
B 760 , 145 (2007).[76] S. Caron-Huot, P.M. Chesler, D. Teaney, arXiv:1102.1073 [hep-th].[77] V. Balasubramanian, A. Bernamonti, J. de Boer, N.B. Copland, B. Craps, E. Keski-Vakkuri,B. Muller, A. Schafer, M. Shigemori, W. Staessens, arXiv:1103.2683 [hep-th].[78] D. Kharzeev, A. Krasnitz, R. Venugopalan, Phys. Lett.
B 545 , 298 (2002).[79] G.D. Moore, M. Tassler, JHEP , 105 (2011).[80] D.E. Kharzeev, L.D. McLerran, H.J. Warringa, Nucl. Phys.