Gluon saturation effects on the color singlet J/Psi production in high energy dA and AA collisions

We derive the formulae for the cross section of J/Psi production in high energy pA and AA collisions taking into account the gluon saturation/color glass condensate effects. We then perform the numerical calculations of the corresponding nuclear modification factors and find a good agreement between our calculations and the experimental data on J/Psi production in dA collisions. We also observe that cold nuclear modification effects alone cannot describe the data on J/Psi production in AA collisions. Additional final state suppression (at RHIC) and enhancement (at LHC) mechanisms are required to explain the experimental observations.


I. INTRODUCTION
The goal of this paper is to provide an improved analysis of the gluon saturation effects on the color singlet mechanism of J/ψ production in dA and AA collisions at RHIC and LHC. In our recent publications [1][2][3] we argued that a mechanism responsible for J/ψ production in central nuclear collisions is different from the one in pp collisions. This is because the symmetry properties of J/ψ under the parity and charge conjugation transformations dictate that there must be an odd Obviously, such contribution breaks the perturbative QCD factorization already at the leading order in α s .
In [1] we assumed that the cc pair propagates through the nucleus in the color octet state and becomes color singlet only after the last interaction with the nucleus. In this paper we drop this assumption by taking into account a possibility that the cc pair converts from the color octet to the color singlet state already inside the nucleus. In the large N c approximation further color conversions of the cc state are suppressed and thus can be neglected. Therefore, in this case the cc experiences the last inelastic interaction inside the nucleus after which it rescatters only elastically.
As a result, the last inelastic interaction does not exponentiate with the rest of the scatterings and -as we will show -automatically selects an odd number of inelastic scatterings as required by the parity of J/ψ. This is different from our approach in [1][2][3] where we had to select the odd number of inelastic scatterings in the scattering amplitude. Additionally, we give a more accurate treatment of J/ψ wave function with parameters taken from a fit to the exclusive J/ψ production in deep inelastic scattering.
Our paper is structured as follows. In Sec. II we derive the cross section for J/ψ production in pA collisions; our main result is given by Eq. (15). In Sec. III we propose a generalization of this result to the AA collisions. The derived cross section is given by (17), (18) and satisfies the constraints imposed by the symmetry of the J/ψ wave function. The results in Sec. II and Sec. III are derived in the quasi-classical approximation, i.e. assuming that the coherence length for J/ψ production is much larger than the nuclear radius, but neglecting the low-x evolution. In Sec. IV we derive expression for the scattering amplitude (26) that includes the low-x evolution and thus gives a dependence on energy and rapidity. Sec. V is dedicated to the description of the numerical calculations performed with different models for the dipole scattering amplitudes. Our main results are exhibited in Figs. 3,4. We discuss them and conclude in Sec. VI.

II. PRODUCTION OF J/ψ IN PA COLLISIONS
The cross section for J/ψ production in pA collisions can be written in the factorized form In order to set normalizations for it is convenient to compare the gA scattering process in (1) with that of γA where there is a well developed phenomenology. Start with γ-proton scattering where dσ γp→J/ψp dt and t is given in terms of the momentum transfer by t = −∆ 2 . Call and where N T = 1.23, R 2 T = 6.5 GeV −2 [8]. Except for a factor of z(1 − z) in (6) our notation, and choice of J/ψ wave function exactly matches that of Ref. [9]. Because (1) is a collinear factorized expression the gluon projectile on the right hand side of (1) is on-shell and so only transverse polarizations appear. We have taken the photon in (2) also on-shell so that the relationship between the photon and gluon induced processes will involve only a normalization change in (5) and a change of the 1 − S factor in (3).
We can get (2) in a more convenient form by using (3) and integrating over ∆. Thus where we have suppressed the energy and impact parameter dependence in the 1 − S factors in (7). The S factors are given, in the McLerran-Venugopalan model [4], by and the cross section in (7) allows nuclear breakup but is elastic at the dipole-nucleon scattering level. Q s in (8) is the gluon saturation momentum with impact parameter dependence again suppressed.
The main change necessary to convert (7) to a cross section for gA → J/ψX is the way the cc dipole scatters off nucleons in the nucleus. In (7) the scatterings are purely elastic, and such Sample interactions are illustrated in Fig. 1.
The interaction at ξ gives the factor The interactions occurring before ξ give the factor while those occurring after give In going from γA → J/ψA to gA → J/ψX the [1 − S * (r )] [1 − S(r)] factor in (7) gets replaced by the product of the factors in (9)- (11).
In addition there is a color factor. In γ induced J/ψ production there is a factor of N c in the amplitude and a factor of N c in the complex conjugate amplitude. This is the factor of N c explicit where we have used the large-N c limit in the right hand side of (12). The C F /(N 2 c − 1) 2 factors go into making up part of the two factors of Q 2 s that come from the graphs. Explicit calculation confirms that the remaining factor, after taking out the factor in (9) and the factor linear in Q 2 s when expanding (10) is just the factor 1/2N c on the right hand side of (12).
Putting all this together gives where, finally, in (14) we have introduced the replacement 2 3 e → g. Doing the integral over ξ and using (1) we get

III. J/ψ CROSS SECTION IN AA COLLISIONS
Generalization of the result of the previous section to nucleus-nucleus collisions is achieved by letting the initial gluon be emitted from either nucleus and taking into account cc dipole scattering in both nuclei. The scattering amplitudes and the saturation scales for the two nuclei depend on their respective impact parameters b 1 and b 2 . To make our notations more compact we will not indicate the impact parameter dependence explicitly. Introducing the relative impact parameter B = b 1 − b 2 and using the relation we can write the cross section as where Expanding (18) at small Q 2 s1 we recover Eq. (15). The first few terms in the expansion of (18) in nuclear density read Averaging over the relative angle between r and r yields This is the leading contribution to the J/ψ production; it is easily seen that it breaks the factorization. We believe that (18) is a reasonable starting point for phenomenology of J/ψ production in AA collisions. Nevertheless a better theoretical understanding of the AA production amplitude

IV. RAPIDITY AND ENERGY DEPENDENCE
Eqs. (17), (18) can be readily generalized to include quantum evolution effects. To that end we recall that the initial condition for the BK [5,10] where subscript F indicates the fundamental representation. Evolution of the gluon dipole scattering amplitude (adjoint representation) obeys the equation and its initial condition is Accordingly, we can incorporate evolution effects in (18) by the following replacements [12] e − 1 8 Q 2 Omitting the impact parameter dependence as before, we thus obtain The experimental data is expressed in terms of the nuclear modification factor (NMF) defined as where S stands for the overall area of two colliding nuclei. Since the mechanism of J/ψ production in pp collisions remains elusive, we follow our approach in the previous publications and approximate with C = const. We fix the constant to provide the best description of the pp and dA data. It is reassuring that the numerical calculations described in the next section indicate that C is close to unity.
The results of our calculations are exhibited in Fig. 3 and Fig. 4; we have used two different models for the dipole scattering amplitude: DHJ [13] and bCGC [9] models (see Appendix A for the description of these models). Comparison of the results of the two models gives an idea about the model dependence of the numerical results. We observe a reasonable agreement with the experimental data on J/ψ production in dA collisions.
Concerning the J/ψ production in AA collisions all models underestimate the suppression at RHIC both at mid-rapidity and in the forward rapidity. Moreover, it appears that the gluon saturation effects on NMF show very little rapidity dependence at RHIC which contradicts the experimental data. We also find that there is almost no change between the NMF at LHC √ s = 2.76 TeV and 5.5 TeV. We note that our calculation overestimates the NMF at √ s = 2.76 TeV.  Fig. 3 using the bCGC model [9].

VI. DISCUSSION AND CONCLUSIONS
Our calculations indicate that the nuclear modification of J/ψ production in dA collisions at RHIC is dominated by the cold nuclear matter effects. It would be important to study J/ψ production in pA collisions at LHC; Fig. 3 and Fig. 4 provide our predictions. In contrast, the cold nuclear matter effects alone cannot provide neither quantitative nor even a qualitative description of the AA data. Additional mechanisms beyond the initial state effects are required to explain the experimental data. It is remarkable that at RHIC these additional mechanisms must provide extra suppression of the NMF, perhaps via the Matsui-Satz color screening mechanism [20] or the gluon-induced dissociation [24,25], whereas at LHC they must produce enhancement.
Our successful description of the J/ψ NMF in pA collisions with the normalization factor C = 1 in (28) may be an evidence that the J/ψ production mechanism in pp collisions is similar to that in pA implying that it is perhaps dominated by the higher twist effects.
To summarize, we derived the formulae for the cross sections of J/ψ production in pA and AA collisions taking into account the gluon saturation/color glass condensate effects. Our numerical results provide an estimate of the color nuclear matter effects on J/ψ production in heavy-ion collisions.
kinematic region. In this model, the dipole scattering amplitude is parameterized as follows The gluon saturation scale is given by where the parameters Λ = 0.6 GeV and λ = 0.3 are fixed by DIS data [22].
where γ s = 0.628 is implied by theoretical arguments [15] and d = 1.2 is fixed by fitting to the hadron production data in dA collisions at RHIC. Y = ln(1/x), with x = me −y / √ s. The quark dipole scattering amplitude is given by which follows from (22).
We used the bCGC model [9] with a modification: we treat the nuclei and proton profiles as step-functions; the saturation scales are assumed to scale with A as Q 2 s ∝ A 1/3 . The advantage of this model -besides its compliance with the known analytical approximations to the BK equation [23] -is that its parameters are fitted to the low x DIS data. The explicit form of the scattering amplitude N is given by where Q 2 s is the the quark saturation scale related to the gluon saturation scale Q 2 s -which we have called simply the 'saturation scale' throughout the paper -by Q 2 s = (4/9)Q 2 s . Its functional form is where s is the square of the center-of-mass energy and y is rapidity with respect to the central rapidity. The anomalous dimension is γ = γ s + 1 c λ (ln √ s + y) ln 2 rQ s .