Wilson line correlator in the MV model: relating the glasma to deep inelastic scattering
aa r X i v : . [ h e p - ph ] M a r arXiv:0711.3039 [hep-ph]SPhT-T07/146 Wilson line correlator in the MV model: relating the glasma to deep inelasticscattering
T. Lappi ∗ Institut de Physique Th´eorique, Bˆat. 774, CEA/DSM/Saclay, 91191 Gif-sur-Yvette Cedex, France
In the color glass condensate framework the saturation scale measured in deep inelastic scatteringof high energy hadrons and nuclei can be determined from the correlator of Wilson lines in the hadronwavefunction. These same Wilson lines give the initial condition of the classical field computationof the initial gluon multiplicity and energy density in a heavy ion collision. In this paper the Wilsonline correlator in both adjoint and fundamental representations is computed using exactly the samenumerical procedure that has been used to calculate gluon production in a heavy ion collision.In particular the discretization of the longitudinal coordinate has a large numerical effect on therelation between the color charge density parameter g µ and the saturation scale Q s . Our result forthis relation is Q s ≈ . g µ , which results in the classical Yang-Mills value for the “gluon liberationcoefficient” c ≈ . PACS numbers: 24.85.+p, 25.75.-q, 13.60.Hb
I. INTRODUCTION
A useful description of the hadron or nucleus wave-function at high energy is to view the small x degrees offreedom as classical color fields radiated by classical staticcolor sources formed by the large x degrees of freedom[1, 2, 3]. This description, known as the color glass con-densate (for reviews see e.g. [4, 5]), provides a commonframework for understanding both small x deep inelas-tic scattering (DIS) and the initial stages of relativisticheavy ion collisions, both of which can be understoodin terms of Wilson lines of the classical color field. Thecross section for small x DIS can be expressed in termsof the correlator of two Wilson lines in the fundamentalrepresentation (i.e. the dipole cross section), and the ini-tial condition for the classical fields that dominate thefirst fraction of a fermi of a heavy ion collision is deter-mined by these same Wilson lines. The inverse of thecorrelation length of these Wilson lines is known as the saturation scale Q s . The dipole cross section, can bedetermined from the dipole model fits to DIS data onprotons [6, 7, 8, 9, 10] and nuclei [11, 12] or extensionsfrom the proton to the nucleus using a parametrizationof the nuclear geometry [10, 13, 14, 15, 16, 17]. Onthe other hand there is a large body of both analyti-cal [18, 19, 20, 21, 22, 23, 24] and numerical classicalYang-Mills (CYM) [25, 26, 27, 28, 29, 30, 31, 32, 33]computations of the “Glasma” [34] fields in the initialstages of relativistic heavy ion collisions.The aim of this paper is to relate the parameters ofthese two types applications of color glass condensateideas of the to each other more precisely. This is doneby computing the Wilson line correlator in the McLerran-Venugopalan (MV) model [1, 2, 3] using exactly the samenumerical method that has been used to compute theinitial transverse energy and multiplicity in a heavy ion ∗ Electronic address: [email protected] collision. By doing this we can relate saturation scale Q s , whose numerical value can be determined from fitsto DIS data, to the color charge density g µ that deter-mines the initial conditions for a heavy ion collision. Thecalculation relating these two parameters has been doneanalytically by several authors [35, 36, 37, 38, 39, 40, 41].The procedure used to construct the MV model Wilsonlines in this paper is the same as used in the numericalcomputations of the Glasma fields and differs from theseanalytical computations in two ways. Firstly, as notedalso in Ref. [42], the analytical computation is done byspreading out the color source in rapidity, while in thenumerical computations this has not been done. We shallsee that this introduces a factor of 2 difference in the ac-tual numerical relation between Q s and g µ . Secondlythe analytical result for the relation between Q s and g µ depends logarithmically on an infrared cutoff that mustbe used in an intermediate stage of the computation,whereas in most of the numerical work the only suchcutoff has been the size of the system. We shall alsodiscuss the uncertainty arising from the non-Gaussianfunctional form of the Wilson line correlator and arguethat it introduces an additional ambiguity at the 10%level. While the uncertainty from these aspects is para-metrically unimportant (a constant or a logarithm), theymust still be better understood in order to increase thepredictive power of the calculations.The logic of this paper is that, instead of treating thecolor charge density g µ in the “Glasma” calculations asa free phenomenological parameter, one should be able torelate it exactly, even the constant under the logarithm,to the saturation scale Q s measured in DIS experiments.When DIS measurements are used to determine the valueof the saturation scale, choosing what treatment of therapidity direction to use in the MV model is mostly amatter of convenience as long as the value of g µ usedis consistent with this chosen implementation. Let usnote that our concern here is not as much the effectsof high energy evolution on the wavefunction, but theparametrization of the region x ∼ .
01 relevant for cen-tral rapidities at RHIC, which would be a reasonable ini-tial condition for solving the BK or JIMWLK equation.The glasma field configurations obtained are the boostinvariant fields that serve as the background for study-ing things like instabilities in the classical [43, 44, 45]field and higher order contributions to particle produc-tion [46, 47, 48].We shall first introduce our notation for the Wilsonline correlators in Sec. II. Then our numerical resultsare presented in Sec. III and their implications for theinterpretation of some of the earlier phenomenologicalwork discussed in Sec. IV.
II. WILSON LINES AND GLASMA FIELDS
Consider a high energy nucleus or a hadron movingalong the x + -axis. Its fast degrees of freedom can beconsidered as a classical color current J + = gρ ( x T , x − ) , (1)which acts as a source to a classical color field represent-ing the slower partons[ D µ , F µν ] = J ν . (2)In the MV model the color charge density is taken to be astochastic random variable with a Gaussian distribution.In covariant gauge Eq. (2) can be solved as A + ( x − , x T ) = − gρ ( x T , x − ) ∇ T . (3)The path ordered exponential of this field gives the Wil-son line in the fundamental representation U ( x T ) = P e i R d x − A + . (4)It is this quantity that will concern us in the following.The cross section for a virtual photon scattering off ahigh energy hadron or nucleus can be expressed in termsof the dipole cross section, which is determined by thecorrelator of two Wilson lines in the fundamental repre-sentation [5, 49] e C ( x T − y T ) = h Tr U † ( x T ) U ( y T ) i , (5)with the expectation value hi evaluated with the distri-bution of the sources.The Wilson line in the adjoint representation is givenby U ab ( x T ) = 2 Tr (cid:2) t a U † ( x T ) t b U ( x T ) (cid:3) . (6)The correlator of adjoint representation Wilson lines C ( x T − y T ) = h U ab ( x T ) U ab ( y T ) i (7)is related to the gluon distribution of a nucleus [35, 37, 50](See Refs. [40, 51, 52] for a discussion on the intricacies µ C ~ ( k ) N y = 1N y = 3N y = 5N y = 10N y = 20 N y = 50 µ C ( k ) N y = 1N y = 3N y = 5N y = 10N y = 20 N y = 50 FIG. 1: The Wilson line correlator for g µL = 100 anddifferent values of N y . Above: fundamental representation˜ k e C (˜ k ), below: adjoint representation ˜ k C (˜ k ). of defining a gluon distribution in this case.) With somealgebra this the adjoint representation correlator can berelated to a higher correlator of fundamental representa-tion Wilson lines C ( x T − y T ) = D(cid:12)(cid:12) Tr (cid:2) U † ( x T ) U ( y T ) (cid:3)(cid:12)(cid:12) − E , (8)which is the form we shall use to evaluate it numerically.The initial conditions for the glasma fields are deter-mined by the pure gauge fields (in light cone gauge) ofthe two colliding nuclei [18, 20]. In terms of the Wilsonline (4) the pure gauge field of one nucleus is A i ( x T ) = ig U ( x T ) ∂ i U † ( x T ) , (9)and the initial conditions for the glasma fields are givenby the sum and commutator of the pure gauge fieldsof the two nuclei. In the numerical computation of theglasma fields [25, 26, 27, 28, 29, 30, 31] there has beenno longitudinal structure in the source, and the Wilsonlines have been constructed simply as U ( x T ) = exp (cid:26) − i gρ ( x T ) ∇ T (cid:27) , (10)with the transverse charge densities depending on a singleparameter µ , independent of x − : h ρ a ( x T ) ρ b ( y T ) i = δ ab δ ( x T − y T ) g µ . (11)The analytical calculation [35, 36, 37, 38, 39, 40, 41]of the Wilson line correlator requires that, unlike the nu-merical procedure in [25, 26, 27, 28, 29, 30, 31], the sourcebe extended in the x − direction h ρ a ( x T , x − ) ρ b ( y T , y − ) i = g δ ab δ ( x T − y T ) δ ( x − − y − ) µ ( x − ) . (12)With this longitudinal structure the Wilson line corre-lators can be computed analytically up to a logarithmicinfrared cutoff that must be introduced in solving thePoisson equation (3). The result is e C ( x T ) ≈ d F e C F8 π χ x T ln( m | x T | ) (13) C ( x T ) ≈ d A e C A8 π χ x T ln( m | x T | ) , with χ = g Z d x − µ ( x − ) . (14)The dimensions and Casimirs of the two representationsin Eq. (13) are d A = N c2 − d F = N c , C A = N c and C F = ( N c2 − / N c . It could be argued that the cutoff m should be ∼ Λ QCD . In any case, running coupling andconfinement effects are not included in this treatmentand the cutoff cannot be consistently defined within thiscalculation. When looking at length scales | x T | ≪ /m results depend very weakly on this cutoff; in the latticecalculation it can be replaced by the finite size of thelattice. It would be very tempting to identify µ , thesource strength of the delta function source, appearing inEq. (11), with the integral over the spread distribution µ ( x − ) of Eq. (14), but as we will see in the following,this correspondence is not exact. Note that the form (13) is compatible with the expec-tation that in the large N c limit the four point functionin Eq. (8) factorizes into a product of two point functionsand lim N c →∞ C ( x T ) = e C ( x T ) . (15) III. NUMERICAL PROCEDURE AND RESULTS
The Wilson lines used in the numerical calculation ofthe Glasma fields [25, 26, 27, 28, 29, 30, 31, 32] areSU(3) matrices defined on the sites of a 2 dimensionaldiscrete lattice corresponding to the transverse plane. Asin most of these calculations, we shall consider a square It is relatively easy to see that the identification of R d x − µ ( x T , x − ) with µ ( x T ) of Eq. (11) would be exact inthe Abelian case or in the large N c limit in which the termsresulting from the noncommutative nature of ρ are suppressed. µ a0.50.60.70.80.9 Q s / g µ N y = 1, fundN y = 1, adjN y = 10, fundN y = 10, adj FIG. 2: The lattice spacing dependence of the saturationscales Q s and q C A C F e Q s for g µL = 100 and different N y . Thecontinuum limit is the g µa = 0 axis on the left. µ L0.50.60.70.80.911.11.2 Q s / g µ N y = 1, fundN y = 1, adjN y = 5, fundN y = 5, adjN y = 20, fundN y = 20, adjN y = 100, fundN y = 100, adj FIG. 3: The dependence of the adjoint and fundamentalrepresentation saturation scales Q s and q C A C F e Q s on g µL for g µa = 0 . N y . lattice with periodic boundary conditions and an averagecolor charge density g µ that is constant throughout theplane. The Wilson lines are constructed as follows: Oneach lattice site x T one constructs random color chargeswith a local Gaussian distribution (cid:10) ρ ak ( x T ) ρ bl ( y T ) (cid:11) = δ ab δ kl δ ( x T − y T ) g µ N y , (16)with the indices k, l = 1 , . . . , N y representing a dis-cretized longitudinal coordinate. The numerical calcu-lations so far have been done using N y = 1, whereas thederivation of the analytical expression of the correlator,Eq. (13) are derived with an extended source, correspond-ing to the limit N y → ∞ . Our normalization is chosenso that X k,l (cid:10) ρ ak ( x T ) ρ bl ( y T ) (cid:11) = δ ab δ ( x T − y T ) g µ . (17) y Q s / g µ m = 0, fundm = 0, adjm = 0.1 g µ , fundm = 0.1 g µ , adj FIG. 4: Dependence on N y of the saturation scales Q s and q C A C F e Q s for g µL = 100 and g µa = 0 .
5, shown for m = 0and m = 0 . g µ . The Wilson lines are then constructed from the sources(16) by solving a Poisson equation and exponentiating: U ( x T ) = N y Y k =1 exp (cid:26) − ig ρ k ( x T ) ∇ T + m (cid:27) . (18)Here we have introduced an infrared regulator m for in-verting the Laplace operator. This is the same regulatoras the one appearing in the analytical expression Eq. (13).For large N y the charge densities ρ k in Eq. (16) becomesmall, and the individual elements in the product (18)approach identity. This is precisely the procedure thatleads in the N y → ∞ limit to the continuum path orderedexponential (4).To summarize, our calculation depends on the follow-ing parameters: • g µ , determining the color charge density. • N y , the number of points in the discretization ofthe longitudinal ( x − or rapidity) direction. • The infrared regulator m . When m = 0, as inmost of the results presented, the Poisson equa-tion is solved by leaving out the zero transversemomentum mode. This procedure corresponds toan infrared cutoff given by the size of the system. • The lattice spacing a . • The number of transverse lattice sites N ⊥ , givingthe size of the lattice L = N ⊥ a .Of the parameters a , g µ and m , only the dimension-less combinations g µa and ma appear in the numericalcalculation, and the continuum limit a → N ⊥ → ∞ so that g µa → g µL = g µaN ⊥ remains constant. What we are looking at is relatively in-frared quantity and thus should converge very well in the continuum limit. Based on the analytical calculation wemay expect a logarithmic dependence of the saturationscale on g µ/m or, for m = 0, on g µL .By Fourier transforming the Wilson lines we can thanconstruct the momentum space correlators in the adjointand fundamental representations, C ( k T ) and e C ( k T ) re-spectively. These correlators, averaged over the polarangle, for different values of N y are plotted in Fig. 1 asa function of ˜ k = 2 a vuut X i =1 sin ( k i a/ . (19)For small momenta the correlators look like Gaussians,which is the form used in the “GBW” fit of DIS datain Refs. [6, 7, 8]. For large momenta there is a powerlaw tail 1 / k T that differs from the original GBW fits,but resembles more closely the form required to matchsmoothly to DGLAP evolution for large Q [53].We define the numerically measured saturation scalesas follows. The scale Q s is determined by the adjointrepresentation Wilson line correlator as the momentum˜ k max corresponding to the maximum of ˜ k C ( k T ). Thisnormalization in terms of the adjoint representation cor-responds to that of Refs. [37, 50]. Similarly, from themaximum of the fundamental representation correlator˜ k e C ( k T ) we define the fundamental representation sat-uration scale e Q s as e Q = ˜ k . Our definition of thesaturation scale is not sensitive to the exact shape of thecorrelator for very large or small transverse momenta,and for a Gaussian correlator it reproduces the GBWsaturation scale as 1 /R = e Q . The saturation scale isexpected to scale according to the Casimir of the repre-sentation, meaning e Q ≈ C F C A Q s2 . In the plots (Figs. 2, 3,4 and 6) we shall rescale e Q s by this color factor to makethe validity of this scaling clearer.We first check the lattice spacing dependence of our re-sult. Figure 2 shows that, as expected, the ratio Q s /g µ depends in fact so little on the lattice spacing that wewill in the following not perform any continuum extrap-olation for this quantity. The dependence of Q s on thelattice size through the combination g µL (without theadditional infrared cutoff m ) is shown in Fig. 3. Thevalues used in the numerical computations of the glasmafields [25, 26, 27, 28, 29, 30, 31, 32, 33] correspond to N y = 1 and g µL ∼
100 in Fig. 3, with Q s ≈ . g µ .Figure 4 shows the dependence of g µ/Q s on the num-ber of points used to discretize the longitudinal direction, N y . When m = 0, i.e. the infrared singularity is regu-lated only by leaving out the zero mode, there is approx-imately a factor of two difference between Q s = 0 . g µ for N y = 1 (the numerical CYM prescription) and Q s ≈ . g µ for N y → ∞ (the analytical computation of the s C ~ ( k ) N y = 1N y = 3N y = 5N y = 10N y = 20 N y = 50 s C ( k ) N y = 1N y = 3N y = 5N y = 10N y = 20 N y = 50 FIG. 5: The same fundamental representation Wilson linecorrelator as in Fig. 1 plotted as a function of the scalingvariable ˜ k/Q s . Above: fundamental representation ˜ k e C (˜ k )vs. ˜ k/ e Q s , below: adjoint representation ˜ k C (˜ k ) vs. ˜ k/Q s . dipole cross section) . When a regulator m = 0 . g µ isintroduced the dependence on N y is weaker, which canalso be seen in Fig. 6. In Fig. 5 we show the same corre-lators as in Fig. 1 as a function of ˜ k/Q s instead of ˜ k/g µ .One sees that the correlator has a scaling form indepen-dent of N y ; from which only the N y = 1 result deviatesslightly. This suggests that, as argued in Sec. I, oncethe appropriate relation between Q s and g µ is used, thephysical results depend very little on N y . Thus no sig-nificant change to the numerical CYM results should beexpected if the calculations were repeated using a differ-ent treatment of the longitudinal coordinate in the source ρ . Explicitly regulating the infrared behavior with a massscale m makes it possible to compare the numerical re-sult to the analytical one of Eq. (13). If one introducesan infrared scale m as in Eq. (10) and replaces ln ( m | x T | )with − ln (cid:0) g µ/m (cid:1) in the coordinate space correlator itbecomes a Gaussian. Fourier transforming this one ob- Because the initial energy density ǫ of the glasma is proportionalto Q s4 /g , this factor of 2 could be an explanation of the factorof 16 difference in ǫ/ ( g µ ) observed in Ref. [42]. µ Q s / g µ N y = 1, adjN y = 10, adjN y = 50, adjC = 0.616C = 0 FIG. 6: Dependence on the regulator m of the saturationscale Q s for g µL = 100. The solid line is the expected log-arithmic dependence, Eq. (21) with the constant C = 0 . C = 0. µ Q s , c oo r d . / g µ N y = 1, adjN y = 10, adjN y = 50, adjC = 0.616C = 0 FIG. 7: Dependence on the regulator m of the coordinatespace saturation scale Q s , coord . for g µL = 100. The solidline is the expected logarithmic dependence, Eq. (21) withthe constant C = 0 .
616 and the dashed one with C = 0. tains the estimate Q s2 ( g µ ) ≈ C A C F e Q ( g µ ) ≈ C A π (cid:20) ln g µm + 12 + ln 2 − γ E (cid:21) . (20)Because of the replacement | x T | ∼ /g µ there is stillan uncertainty in the constant term. In Fig. 6 we plotthe numerical result for Q s /g µ as a function of m/g µ compared to the estimate Q s2 ( g µ ) = C A π (cid:20) ln g µm + C (cid:21) , (21)with values C = + ln 2 − γ E ≈ .
616 and C = 0. Inan intermediate range of m/g µ and for a large enoughvalue of N y (recall that the analytical result correspondsto N y → ∞ ) the behavior of Q s /g µ is similar, but thenormalization different.Another common way to define the saturation scaleis in terms of the coordinate space correlator C ( x T ),because this is the object appearing in the calculation µ L1.041.061.081.11.121.14 Q s , c oo r d . / Q s N y = 1N y = 5N y = 20N y = 100 FIG. 8: The ratio of the two definitions of the saturationscale, the coordinate space definition ( Q s , coord . ) and the mo-mentum space definition used in most of this paper ( Q s ). of most observables in DIS. Kowalski and Teaney [10]define the saturation scale Q s , coord . from the conditionthat C ( x T ) = d A e − / at x T = 2 /Q , coord . . Notethat the definition in Ref. [17] where the same IPsatmodel is used differs slightly: C ( x T ) = d A e − / at x T = 1 /Q , coord . . This definition can also be used in thenumerical CYM computation, most straightforwardly byHankel-transforming the correlator C ( | k T | ) back into co-ordinate space. As shown in Fig. 7, using this definitionis closer to the analytical estimate Eq. (21). The ratioof Q s , coord . to our original definition of Q s for differentvalues of g µL and N y is plotted in Fig. 8. The differ-ence between the two definitions is of the order of 10%with small variations. One must emphasize here that foran exactly Gaussian Wilson line correlator (the GBWform) the two definitions would be equal. They differ inthe MV model, because the correlator is not Gaussian.Thus if one tries to determine g µ from a comparison tothe experimental DIS data using GBW-type fits, whichis one of the alternatives we consider in the next sec-tion, the ambiguity in the definition of Q s leads to a 10%uncertainty in the value of g µ . IV. DISCUSSION
Let us finally use the results of the previous section for Q s /g µ and studies of DIS data to estimate the relevantvalue of g µ for RHIC physics. In deep inelastic scatter-ing the variables x and Q are precisely defined, and thesaturation scale is a function of x , typically Q s2 ∼ x − λ with λ ≈ . . In the context of a heavy ion collision oneis in fact, at a fixed energy and rapidity, summing up glu-ons produced at different transverse momenta and thusrelated to partons of different x in the nuclear wavefunc-tion. The value of x at which to evaluate the satura-tion scale must therefore be some kind of effective x eff , Q sRHIC g µ Naive A / . . A / [11] 1 . . CA / [12] 1 . . ∼ CA / ln A [17] 1 . . depending on the typical transverse momentum of theproduced gluons, x eff ∼ h p T i√ s ∼ Q s √ s . (22)This introduces an additional uncertainty into our at-tempt to determine the color charge density based onthe deep inelastic scattering data; by varying Q s / √ s 20 GBW = Q ( x/x ) λ . (23)The result of the fit including charm quarks gives λ =0 . 277 and x = 0 . · − , with the one redundant pa-rameter chosen as Q = 1 GeV, while the fit withoutcharm makes Q s approximately 30% larger.This result must then be extended to finite nuclei.Let us denote the nuclear modification of Q s by g ( A ) ≡ Q s2 A /Q s2 p . The most straightforward theoretical expec-tation for the nuclear dependence would be g ( A ) = A / . Freund et al. [11] perform a fit of the form g ( A ) = A δ to the available nuclear DIS data and ob-tain δ = 1 / 4. Taking into account modifications to the A / behavior of the nuclear radius leads Armesto etal. [12] to consider a fit of the form g ( A ) ∼ AR p /R A = C (cid:2) A/ ( A / − . A − / ) (cid:3) δ with the result δ ≈ . C ≈ . 5. Although for asymptotically large nu-clei this would imply g ( A ) ∼ A . , for the physical case A < ∼ 200 the nuclear modification factor g ( A ) obtainedin Ref. [12] is actually less than A / .A more detailed description of the saturation scale in anucleus can be obtained by the IPsat model [10, 16, 17].HERA data and the DGLAP equations are used toparametrize the dipole cross section for a proton. Tak-ing into account the fluctuations in the positions of thenucleons in the nucleus within a realistic nuclear geom-etry leads to a nuclear dipole cross section, from whichalso the saturation scale can be determined. As shownin Ref. [17] this picture leads to a good parameter freedescription of all the existing small x eA data. The re-sult is a more realistic picture of an impact parameterdependent saturation scale also influenced by DGLAPevolution, where the nuclear geometry leads to a g ( A )that can roughly be understood as a CA / -like depen-dence (with C < 1) enhanced by a logarithmic increase in A resulting from the DGLAP evolution. Because scatter-ing off nuclei is less dominated by the dilute edge than inthe proton, the typical Q s (conveniently taken as corre-sponding to b med. , the median impact parameter in deepinelastic scattering) is closer to the maximal Q s in thenucleus than in the proton. We shall use here for goldthe value at b med. ≈ . x = 0 . ≈ Q s / √ s .Table I summarizes the estimated saturation scales forcalculating the classical field at central rapidity in RHICbased on the different fits explained above. The tablealso shows the corresponding values of g µ = Q s / . N d η = ( g µ ) πR A g f N (24)d E T d η = ( g µ ) πR A g f E , (25)where the numerical result is f E ≈ . 25 and f N ≈ . . < ∼ g µ < ∼ . p d V work thus decreasing the energy atcentral rapidities, decreasing d E T / d y . It is very hardto imagine a process that would increase the energy atmidrapidity and thus the measured final transverse en-ergy gives a lower limit to the initial energy and to g µ .The upper limit follows from the requirement that thenumber of gluons in the initial state should be less orequal to that of hadrons in the final state. In ideal hy-drodynamical flow the two are related by entropy con-servation, and nonequilibrium processes should increaseentropy and consequently the multiplicity during the evo-lution, not decrease it. The measured hadron multiplic-ity thus gives an upper limit on the initial multiplicityand g µ . Quark pair production [54, 55] or in generalhigher order processes would generically increase the ini-tial multiplicity for a given g µ and thus decrease theupper limit for g µ below 2 GeV. The only overlap re-gion between these estimates and the DIS based ones inTable I is around g µ ≈ x parton distribution functions,of Ref. [20] that was used in the CYM calculations ofRefs. [26, 27, 28]. On the DIS side the value g µ ≈ c , introduced by A.Mueller [57], is defined by writing the produced gluonmultiplicity as d N d x T d y = c C F Q s2 π α s . (26)With Eq. (24) this leads to c = πf N C F (cid:18) g µQ s (cid:19) . (27)The original expectation was that c should be of orderunity. The analytical calculation by Y. Kovchegov [50]gave the estimate c ≈ ≈ . 4. Using the formula(20) for the ratio Q s /g µ led to the interpretation [56,58] that the CYM result would be c ≈ . 5. We nowsee that when Q s /g µ is computed consistently with thenumerical calculation the resulting CYM value for theliberation coefficient is c ≈ . 1. We must emphasize that,because c is defined in terms of the physical multiplicityand the physical correlation length Q s , there is no largelogarithmic or N y uncertainty in the result c ≈ . 1. Thenon-Gaussianity of the MV model correlator, as seen inthe differing coordinate and momentum space results for Q s , does introduce an ambiguity at the 10% level.Let us summarize the major sources of error in estimat-ing the relevant value of the saturation scale for RHICphysics from the DIS data. We have already mentionedthe questions of the Wilson line correlator not being ex-actly of the GBW form, the exact value of x to use andthe considerable variance in the estimates of A depen-dence of Q s . It is also possible that including a more real-istic description of the transverse coordinate dependenceof the saturation scale [10, 14, 17, 30, 59] in the CYMcalculation will have an impact on the gluon multiplicityand energy in an ion-ion collision, modifying our previousdiscussion. The solution to the problems related to theshape of the correlator can be solved by using the actualsolution of the BK or JIMWLK equations to understandboth DIS data (as is done in Ref. [9]) and to calculatethe Glasma fields. Confirming the calculations like thatof Ref. [17] relating the saturation descriptions of the pro-ton and a nucleus will require more experimental input inthe form of more data on small x DIS on nuclei. Finallyand perhaps most importantly, the influence of quantumcorrections and instabilities of small rapidity-dependentfluctuations is not yet understood quantitatively. Acknowledgments The author would like to thank R. Venugopalan fornumerous discussions and comments on the manuscript. [1] L. D. McLerran and R. Venugopalan, Phys. Rev. D49 ,2233 (1994), [arXiv:hep-ph/9309289].[2] L. D. McLerran and R. Venugopalan, Phys. Rev. D49 ,3352 (1994), [arXiv:hep-ph/9311205].[3] L. D. McLerran and R. Venugopalan, Phys. Rev. D50 ,2225 (1994), [arXiv:hep-ph/9402335].[4] E. Iancu and R. Venugopalan, arXiv:hep-ph/0303204.[5] H. Weigert, Prog. Part. Nucl. Phys. , 461 (2005),[arXiv:hep-ph/0501087].[6] K. Golec-Biernat and M. Wusthoff, Phys. Rev. D59 ,014017 (1999), [arXiv:hep-ph/9807513].[7] K. Golec-Biernat and M. Wusthoff, Phys. Rev. D60 ,114023 (1999), [arXiv:hep-ph/9903358].[8] A. M. Stasto, K. Golec-Biernat and J. Kwiecinski, Phys.Rev. Lett. , 596 (2001), [arXiv:hep-ph/0007192].[9] E. Iancu, K. Itakura and S. Munier, Phys. Lett. B590 ,199 (2004), [arXiv:hep-ph/0310338].[10] H. Kowalski and D. Teaney, Phys. Rev. D68 , 114005(2003), [arXiv:hep-ph/0304189].[11] A. Freund, K. Rummukainen, H. Weigert and A. Schafer,Phys. Rev. Lett. , 222002 (2003), [arXiv:hep-ph/0210139].[12] N. Armesto, C. A. Salgado and U. A. Wiedemann, Phys.Rev. Lett. , 022002 (2005), [arXiv:hep-ph/0407018].[13] E. Levin and M. Lublinsky, Nucl. Phys. A696 , 833(2001), [arXiv:hep-ph/0104108].[14] E. Gotsman, E. Levin, M. Lublinsky and U. Maor, Eur.Phys. J. C27 , 411 (2003), [arXiv:hep-ph/0209074].[15] E. Levin and M. Lublinsky, Nucl. Phys. A712 , 95 (2002),[arXiv:hep-ph/0207374].[16] H. Kowalski, L. Motyka and G. Watt, Phys. Rev. D74 ,074016 (2006), [arXiv:hep-ph/0606272].[17] H. Kowalski, T. Lappi and R. Venugopalan, Phys. Rev.Lett. , 022303 (2008), [arXiv:0705.3047 [hep-ph]].[18] A. Kovner, L. D. McLerran and H. Weigert, Phys. Rev. D52 , 3809 (1995), [arXiv:hep-ph/9505320].[19] A. Kovner, L. D. McLerran and H. Weigert, Phys. Rev. D52 , 6231 (1995), [arXiv:hep-ph/9502289].[20] M. Gyulassy and L. D. McLerran, Phys. Rev. C56 , 2219(1997), [arXiv:nucl-th/9704034].[21] A. Dumitru and L. D. McLerran, Nucl. Phys. A700 , 492(2002), [arXiv:hep-ph/0105268].[22] Y. V. Kovchegov and D. H. Rischke, Phys. Rev. C56 ,1084 (1997), [arXiv:hep-ph/9704201].[23] R. J. Fries, J. I. Kapusta and Y. Li, arXiv:nucl-th/0604054.[24] K. Fukushima, Phys. Rev. C76 , 021902 (2007),[arXiv:0704.3625 [hep-ph]].[25] A. Krasnitz and R. Venugopalan, Nucl. Phys. B557 , 237(1999), [arXiv:hep-ph/9809433].[26] A. Krasnitz and R. Venugopalan, Phys. Rev. Lett. ,4309 (2000), [arXiv:hep-ph/9909203].[27] A. Krasnitz and R. Venugopalan, Phys. Rev. Lett. ,1717 (2001), [arXiv:hep-ph/0007108].[28] A. Krasnitz, Y. Nara and R. Venugopalan, Phys. Rev.Lett. , 192302 (2001), [arXiv:hep-ph/0108092].[29] A. Krasnitz, Y. Nara and R. Venugopalan, Nucl. Phys. A727 , 427 (2003), [arXiv:hep-ph/0305112].[30] A. Krasnitz, Y. Nara and R. Venugopalan, Nucl. Phys. A717 , 268 (2003), [arXiv:hep-ph/0209269]. [31] T. Lappi, Phys. Rev. C67 , 054903 (2003), [arXiv:hep-ph/0303076].[32] T. Lappi, Phys. Rev. C70 , 054905 (2004), [arXiv:hep-ph/0409328].[33] T. Lappi, Phys. Lett. B643 , 11 (2006), [arXiv:hep-ph/0606207].[34] T. Lappi and L. McLerran, Nucl. Phys. A772 , 200(2006), [arXiv:hep-ph/0602189].[35] J. Jalilian-Marian, A. Kovner, L. D. McLerran andH. Weigert, Phys. Rev. D55 , 5414 (1997), [arXiv:hep-ph/9606337].[36] Y. V. Kovchegov, Phys. Rev. D54 , 5463 (1996),[arXiv:hep-ph/9605446].[37] Y. V. Kovchegov and A. H. Mueller, Nucl. Phys. B529 ,451 (1998), [arXiv:hep-ph/9802440].[38] L. D. McLerran and R. Venugopalan, Phys. Rev. D59 ,094002 (1999), [arXiv:hep-ph/9809427].[39] F. Gelis and A. Peshier, Nucl. Phys. A697 , 879 (2002),[arXiv:hep-ph/0107142].[40] J. P. Blaizot, F. Gelis and R. Venugopalan, Nucl. Phys. A743 , 13 (2004), [arXiv:hep-ph/0402256].[41] J. P. Blaizot, F. Gelis and R. Venugopalan, Nucl. Phys. A743 , 57 (2004), [arXiv:hep-ph/0402257].[42] K. Fukushima, arXiv:0711.2364 [hep-ph].[43] P. Romatschke and R. Venugopalan, Phys. Rev. Lett. , 062302 (2006), [arXiv:hep-ph/0510121].[44] P. Romatschke and R. Venugopalan, Phys. Rev. D74 ,045011 (2006), [arXiv:hep-ph/0605045].[45] K. Fukushima, F. Gelis and L. McLerran, Nucl. Phys. A786 , 107 (2007), [arXiv:hep-ph/0610416].[46] F. Gelis and R. Venugopalan, Nucl. Phys. A776 , 135(2006), [arXiv:hep-ph/0601209].[47] F. Gelis and R. Venugopalan, Nucl. Phys. A779 , 177(2006), [arXiv:hep-ph/0605246].[48] F. Gelis, T. Lappi and R. Venugopalan, Int. J. Mod.Phys. E16 , 2595 (2007), [arXiv:0708.0047 [hep-ph]].[49] K. Rummukainen and H. Weigert, Nucl. Phys. A739 ,183 (2004), [arXiv:hep-ph/0309306].[50] Y. V. Kovchegov, Nucl. Phys. A692 , 557 (2001),[arXiv:hep-ph/0011252].[51] D. Kharzeev, Y. V. Kovchegov and K. Tuchin, Phys.Rev. D68 , 094013 (2003), [arXiv:hep-ph/0307037].[52] F. Gelis, A. M. Stasto and R. Venugopalan, arXiv:hep-ph/0605087.[53] J. Bartels, K. Golec-Biernat and H. Kowalski, Phys. Rev. D66 , 014001 (2002), [arXiv:hep-ph/0203258].[54] F. Gelis, K. Kajantie and T. Lappi, Phys. Rev. C71 ,024904 (2005), [arXiv:hep-ph/0409058].[55] F. Gelis, K. Kajantie and T. Lappi, Phys. Rev. Lett. ,032304 (2006), [arXiv:hep-ph/0508229].[56] A. H. Mueller, Nucl. Phys. A715 , 20 (2003), [arXiv:hep-ph/0208278].[57] A. H. Mueller, Nucl. Phys. B572 , 227 (2000), [arXiv:hep-ph/9906322].[58] R. Baier, A. H. Mueller, D. Schiff and D. T. Son, Phys.Lett. B539 , 46 (2002), [arXiv:hep-ph/0204211].[59] T. Lappi and R. Venugopalan, Phys. Rev.