Parton Percolation and J/Psi Suppression

The geometric clustering of partons in the transverse plane of nuclear collisions leads for increasing A or sqrt(s) to percolation. In the resulting condensate, the partons are deconfined but not yet in thermal equilibrium. We discuss quarkonium dissociation in this precursor of the quark-gluon plasma, with an onset of dissociation when the saturation scale of the parton condensate reaches that of the given quarkonium state.

Statistical QCD predicts color deconfinement for sufficiently hot strongly interacting systems in full equilibrium. In the resulting quark-gluon plasma, both the momenta and the relative abundances of quarks, anti-quarks and gluons are determined by the temperature of the medium. It is not evident if and at what evolution stage high energy nuclear collisions produce such equilibrium, nor is it evident that color deconfinement is restricted to such ideal thermal systems. It thus seems natural to ask what conditions are necessary in the pre-equilibrium stage to achieve deconfinement and perhaps subsequent quark-gluon plasma formation. In recent years, the occurrence of color deconfinement in nuclear collisions without assuming prior equilibration has therefore been addressed on the basis of two closely related concepts, parton percolation [1,2] and parton saturation [3,4,5].
Both start from the observation that in a central nucleus-nucleus collision at high energy, one finds in the transverse nuclear plane interacting partons 1 of different transverse scales. At low densities, one can define individual partons originating from nucleons of the incident nuclei. Once the density of partons becomes so high that they form a dense interacting cluster, independent parton existences and origins are no longer meaningful: the resulting cluster forms a condensate of deconfined partons. The condensate is formed in the sense of droplets condensing to form a liquid, and the partons which make up this condensate are no longer constrained by any hadronic conditions. Consider a distribution of partons of transverse size πr 2 over the transverse nuclear plane πR 2 A , with r << R A . 2 The fundamental aim of parton percolation studies [1,2,6] is the determination of the transition from a normal hadronic collision situation of disjoint partonic 'discs' to a connected cluster of such discs spanning the nucleus, the parton condensate. Percolation theory predicts this transition to occur when N q πR 2 A πr 2 = n q πr 2 ≡ η → η c = 1.128 (1) i.e., when the parton density η = n q πr 2 measured in terms of parton size reaches the percolation threshold η c . In the 'thermodynamic' limit of infinite spatial size R A → ∞ and infinite parton number N q → ∞, the largest connected cluster first spans the system at this point. The critical value η c = 1.128 is determined by extensive numerical studies [7]. Since the partons overlap, this does not mean that the entire transverse nuclear surface is covered by parton discs. In fact, at the percolation point, only the fraction 1 − exp(−η c ) ≃ 2/3 of the nuclear area is covered by partons. At η = η c , the critical clustering behavior of the system can be specified in the usual way in terms of critical exponents. In particular, the size S(η) of the largest cluster diverges for η → η c as with the critical exponent γ = 43/18. While this holds strictly only for infinite systems, it is verified that even for rather small spatial systems, the transition from very small size to percolating cluster occurs in a very narrow density interval. In other words, even at finite size there is almost critical behavior. This will become quite important for our further considerations.
The essential idea of parton saturation is that the increase in the number of partons for small x, as obtained from deep inelastic scattering experiments, must stop when the density of partons becomes so high that they overlap and form large interacting clusters; fusion and splitting then causes their number to approach a constant. The onset of saturation has been discussed in various ways; making use of their transverse size, it can also be quite naturally determined by the percolation condition (1), which then fixes the saturation scale in terms of A and the c.m.s energy √ s. Saturation sets in when the parton density n q in terms of the partonic interaction cross section σ q approaches the critical value η c = 1.128. The partonic cross section depends on its inverse transverse momentum k T , σ g (k T ) ∼ 1/k 2 T , and the parton density n g (k 2 T ) for a fixed resolution scale k 2 T can be obtained from the gluon distribution function determined in deep inelastic scattering. The novel aspect from the point of view of percolation is that the density of partons n q (k 2 T ) is related to their transverse size σ(k 2 T ), so that with the functional form of these two quantities given, the percolation condition specifies the scale of the percolating partons. Let us consider this in more detail.
The distribution of partons in an incident nucleon of momentum p is given in terms of their fractional momentum x ≃ k/p and the resolution scale Q 2 , through integration over the partonic transverse momentum k T . The relevant resolution scale in a nucleon-nucleon collisions is the largest transverse momentum k max T = Q for which partons are resolved.
The number of partons at fixed x and integrated over k T ≤ Q is given by the sum of the contributions of gluons plus those of sea quarks and antiquarks, where g(x, Q 2 ) denotes the gluon distribution function, q i (x, Q 2 ) andq i (x, Q 2 ) that of up and down (i = 1, 2) quarks and antiquarks, respectively. The distribution functions are determined from parametrisations of deep inelastic scattering data, and thus eq. (3) provides the number of partons at central rapidity y = 0, with x = Q/ √ s. We shall see shortly that for RHIC and higher energies, the gluon contribution is strongly dominant; for SPS energy, however, the quark and antiquark contributions cannot be neglected.
We further need the parton size. In the simplest percolation approach, which we shall follow here, this is just the geometric cross section π/Q 2 . More dynamical considerations lead to numerical modifications, σ g (Q 2 ) = κα s (Q 2 )π/Q 2 , where α s (Q 2 ) is the running coupling at scale Q 2 and κ a given constant. With the geometric cross section we obtain that in an AA collision at y = 0, the equation determines the onset of percolation. Here n s (A) specifies the density of parton sources in the transverse plane of a central AA collision. At SPS energy, this is essentially the density of wounded nucleons [8], n s (A) ≃ n w (A). For higher energies and harder partons, collision-dependent contributions will play a significant role, and so a more suitable form here is a combination of the two sources, with 0 ≤ x ≤ 1 [9].
As determined by eq. (4), partons with k T ≤ Q s condense to form an overlapping and hence interacting cluster spanning the system, the parton condensate. Within this cluster, they can fuse or split and thus lose their independent existence. We recall that at this point, 2/3 of the nuclear area is covered by the parton condensate.
The relation between saturation and percolation has so far not been much emphasized. For a study of the new percolating medium at very large A or √ s, the 'color glass condensate' of Ref. [5], it is indeed not so important. However, it does become crucial for a detailed picture of the onset of parton condensation. We know from percolation theory that in the large volume limit this is a critical phenomenon and hence even for finite systems takes place in an almost singular way.
To study the onset of parton percolation in AA collisions, it is convenient to rewrite eq. (4) in the form with the density of parton sources n s in fm −2 ; the factor 25 arises when this is converted to GeV 2 . In eq. (6), the hadronic parton distribution (dN q /dy)/Q 2 is compared to the density of parton sources in a nucleus-nucleus collision. For low source densities, i.e., for small A, 8.98/n s (A) remains well above the A-independent l.h.s., which is a function only of Q and √ s: there are not enough partons to form a condensate. For sufficiently large A, however, 8.98/n s (A) intersects (dN q /dy)/Q 2 at some Q s , thus defining the onset of percolation. All partons with k T ≤ Q s merge to form the condensate, in which interactions prevent much further increase in the number of partons, i.e., there is saturation.
To see when that happens, we have to make use of a specific set of parton distribution functions. The kinematic range relevant for our analysis is 0.5 GeV 2 < Q 2 < 2 GeV 2 , with 0.02 ≤ x < 0.1 for SPS and 0.003 < x < 0.02 for RHIC. Among the commonly used PDF parametrizations, only the set GRV94 [10] goes down to such small values of Q 2 . This parametrization describes well the available data on the proton structure function F p 2 from the E665 and NMC collaborations for 0.4 GeV 2 < Q 2 < 2 GeV 2 and x < 0.01 [11,12]. It also reproduces quite well the small x HERA data [10,13] for x > 10 −3 . We have therefore calculated (dN q /dy) using the next-to-leading order GRV94 PDF's in the DIS scheme (GRV94DI) [10]; the resulting (dN q /dy)/Q 2 are shown in Figs. 1a and 1b for SPS (20 GeV) and RHIC (200 GeV) energies, respectively.
Any uncertainty in (dN q /dy) is mainly due to the gluon distribution. At SPS energy, this can be estimated by comparing calculations using leading and next-to-leading order GRV94 PDF's. The resulting uncertainty is below 3 % for Q 2 > 0.6 GeV 2 and increases to at most 10 % at 0.5 GeV 2 . At RHIC, the next-to-leading order GRV94 PDF's are consistent with the gluon distributions determined by the H1 and ZEUS collaborations and with the constraints from charm measurements [14], while the leading order results for the gluon distribution are too large. This gives us some confidence in using the nextto-leading order GRV94 parametrization.
We also note that these gluon distributions are not very different from those proposed in more recent phenomenological saturation studies for x < 0.01 [15]. This approach is very successful in describing the low x HERA data and it can in fact also account for the E665 data on F p 2 in the kinematical region relevant for RHIC. The gluon distribution of [15] is directly related to F p 2 ; the sea quarks are present only virtually as small dipoles from photon splitting [16]. In such an approach, the uncertainty in the gluon distribution can be avoided, and it has been used to predict the √ s-dependence of hadron multiplicities at RHIC [17].
Once the density of the percolating medium is sufficiently high, the PDF approach of Ref. [15] to the gluon distribution is more appropriate to study the resulting condensate; however, it is not suitable to study the onset of condensation. In this connection we note that the effect of the sea quarks cannot be completely neglected even at RHIC energy. In general, the relative weight of the different parton species in the produced condensate will depend on Q and √ s, and with increasing energy, the condensate becomes more and more gluon-dominated. Thus the ratio of gluons to quarks and antiquarks in the parton condensate based on the GRV94 PDF's is at Q = 1 GeV found to be about 1.2 for SPS energy and 4.0 for RHIC; in a chemically equilibrated quark-gluon plasma, it is about 0.5. Hence before any thermalization, the medium is strongly gluon-dominated [19].
In order to illustrate the effect for different central AA collisions, we assume a spherical nuclear profile, which gives at SPS energy, with x = 0 in eq. (5), Inserting this into eq. (6), together with the values of [dN q (x, Q 2 )/dy]/Q 2 at √ s = 20 GeV of the previous section, we obtain the results shown in Fig. 1a. The intersection points determine the onset of percolation. It is seen that in this very simplified picture, parton condensation begins for A ≃ 60, with Q ≃ 0.7 GeV. To obtain the corresponding behavior at √ s = 200 GeV, nucleon collisions have to be included as source of partons.
With n coll = 3 4 and x = 0.09 [9] in eq. (5), we find the percolation points in Fig. 1b. In eq. (8), the factor 3/4 comes from averaging over the nuclear profile. In Fig. 2, the percolation values Q s for central AA collisions are displayed as function of A. At SPS energy, we thus do not obtain parton condensation below A ≃ 60; at RHIC energy, the higher parton density lowers the onset to A ≃ 40.
In the experimental study of J/ψ suppression, the production is measured at different centralities, so that we now have to determine the parton source density at fixed A and varying impact parameter. This is done in a Glauber analysis based on Woods-Saxon nuclear profiles, with a collision-determined weight [18]. In Fig. 3a, we show the resulting percolation behavior as function of the effective number N part of participants in a P b − P b collision at √ s = 20 GeV. The threshold for parton percolation is found to be slightly below N part ≃ 150. The corresponding calculations for Au − Au collisions at √ s = 200 GeV, with a collision-dependent term in n s and again x = 0.09 in eq. (5), lead to the results shown in Fig. 3b. The onset of parton condensation at RHIC is thus shifted to considerably more peripheral collisions. In Fig. 4, the centrality dependence of the percolation scale Q s is shown; at the onset point, the condensate contains partons of different sizes r, with r ≥ 1/(0.7 GeV) at SPS and r ≥ 1/(0.9 GeV) at RHIC.
We now turn to the effect of parton condensation on J/ψ production. It is known from pA collisions that normal nuclear matter leads to reduced charmonium production. Therefore we first have to consider such 'normal' suppression, since for pA collisions, at least up to RHIC energies, we are well below the threshold for parton condensation 3 . Consider the production of charmonium in the nuclear target rest frame. A gluon from the incident proton fluctuates into a virtual cc pair; in a collision with one of the target nucleons this is brought on-shell, its color is neutralized, and it eventually becomes a physical J/ψ of size d J/ψ ≃ 0.4 fm. Thus in the rest-frame of the cc pair, a time of at least 0.4 fm is needed for J/ψ formation. During its pre-resonance stage, for τ cc ≤ 0.4 fm, it will travel a distance 3 A determination of the collision energy for which parton saturation occurs in pp/pp or pA collisions is presently difficult, since it involves x-values much smaller than presently accessible in deep inelastic scattering. Disregarding this small x-behavior leads to a very early onset of parton percolation [22].
in the target rest frame; here m denotes the mass of the nucleon, M that of the cc pair and x F the cc Feynman momentum fraction. For the production of a cc at rest in the nucleon-nucleon c.m.s., this becomes d cc = τ cc ( √ s/2m). From this it is immediately seen that at √ s ≃ 40, and x F ≥ 0, the nascent J/ψ effectively traverses even heavy nuclear targets in its pre-resonance stage. The situation is very similar for χ c and ψ ′ production, for which the pre-resonance life-times are if anything even larger. Thus for the mentioned √ s and x F , the nuclear target sees of all charmonium states only the small pre-resonance precursor, so that all should suffer the same degree of nuclear suppression [20]. For negative x F (and perhaps to some extent also for lower √ s), the nucleus should begin to see physical resonances, and as a result the suppression should become larger for the higher excited states with their larger radii [21].
On the basis of this information, normal charmonium suppression has generally been studied in terms of pre-resonance dissociation in standard nuclear matter, leading to a break-up cross section around 5 -6 mb [18,23]. This implies a mean free path of about 12 fm. For a multiple collision analysis of Glauber type, the mean free path has to exceed the coherence length (essentially the size) of the pre-resonance in the rest frame of the nucleus: otherwise, the cc is wounded several times before it has had a chance to register the first interaction. For collision energies √ s ≤ 40 GeV, i.e., in the range of fixed target experiments, this condition is satisfied, with coherence lengths below 8 fm. At RHIC energy, on the one hand the cross section for pre-resonance break-up in normal nuclear matter could change, and on the other hand the coherence length becomes dilated ten times more. Now interference effects of the Landau-Pomeranchuk-Migdal type may have to be taken into account, which could lead to a reduction of 'normal' nuclear suppression [24]. Hence measurements of charmonium production in pA collisions at RHIC are absolutely essential for an understanding of whatever J/ψ suppression in AA collisions is observed there.
We now want to consider the additional 'anomalous' suppression of charmonium production due to the dense partonic medium created in AA collisions. The virtual partons in the incoming nuclei coalesce to form a condensate in a time t c ≃ 1/Q s determined by the saturation scale; in the color glass picture [5], this is the time needed to melt the frozen glass. The interacting and expanding parton condensate can subsequently lead to the formation of a thermalized quark-gluon plasma; a crucial factor for this is the energy density reached at thermalization. We restrict ourselves here to the parton condensate stage.
In AA collisions at RHIC energy, the colliding nuclei are in the overall c.m.s Lorentzcontracted to about 0.1 fm. They will therefore sweep past the nascent charmonium in its pre-resonance state and before parton condensation sets in, resulting in some form of normal nuclear absorption. After about 0.2 fm, the nuclei are well out of the way and the parton condensate is formed. Any produced and surviving charmonium states now encounter this new medium, either as fully formed resonances or in the late preresonance stage. For SPS energy, a similar discussion is more complex, since the nuclei are contracted to only about 1 fm diameter; this will introduce a smearing in the comparison of the different time scales. We shall here neglect this and assume that also at the SPS there is first pre-resonance absorption in normal nuclear matter, followed by the effect of the parton condensate on the survivors.
In the first attempt to describe J/ψ suppression in terms of parton percolation [2], it was assumed that different charmonium states i define particular scales Q i , and the onset of percolation for partons of that scale leads to the dissociation of all charmonia of that species within the percolating cluster. The number of partons was taken as a scale-independent function of √ s. Parton saturation in fact specifies [dN q /dy](Q 2 s ) as function of the scale Q s , so that we now have (see Figs. 2 and 4) at given collision energy a parameter-free determination of the onset line Q s (A) or Q s (N part ) of parton condensation.
Within the parton condensate, color fields of a Q s -dependent strength will affect the binding of a cc dipole charmonium state. This effect can be addressed in different ways. One possible way is to consider the cc propagation in a classical field, which can represent the gluon field of the condensate or that of the nucleus; the information on specific medium is encoded in the corresponding field correlator [24]. For a small singlet dipole the probability to remain singlet is proportional to exp(−κr 2 ), where r is the dipole size and κ is a dimensional parameter determined by the field strength. In the color glass approach, κ ∼ Q 2 s [5]. Thus for small singlet dipoles, Q 2 s r 2 << 1, the probability to remain in a singlet state is close to unity (color transparency). Motivated by this fact we shall here adopt the simple model of Ref. [2], assuming that a charmonium state i of scale Q i will be dissociated if it finds itself in a parton condensate of scale Q s ≥ Q i ; otherwise it will survive. This very simplistic picture allows an analysis of nuclear profile effects (condensed and non-condensed regions of the collision profile) and thus provides some direct predictions for the centrality-dependence of anomalous J/ψ suppression for given A and √ s. Other approaches that have been suggested include the study of the time evolution of the screening masses in the parton cascade [25] and in the color glass condensate [26].
The radii of the observable charmonium states as obtained from the solution of the Schrödinger equation with Cornell potential are [27] r J/ψ ≃ (0.9 GeV) −1 , r ξ ≃ (0.6 GeV) −1 , r ψ ′ ≃ (0.45 GeV) −1 (10) These have to be compared to the partonic saturation radii Q −1 s shown in Fig. 4 as functions of the number of participants in central AA collisions. It is seen that at SPS energy, the onset of χ c and ψ ′ suppression effectively coincides with the onset of parton condensation, while the J/ψ survives up to larger Q s and higher parton densities 4 . Hence in P b−P b collisions, anomalous suppression should start with the elimination of J/ψ feeddown from χ c and ψ ′ at N part ≃ 150; the further suppression of direct J/ψ production should set in for N part ≃ 250. In Fig. 5 we show the suppression pattern observed by the NA50 collaboration at the CERN-SPS [28]; the two 'steps' observed in the anomalous suppression pattern agree fairly well with the expected onsets of parton condensation. Note that at the percolation points, the percolating cluster covers only a fraction of the transverse area; with increasing centrality, this fraction increases. Hence more detailed studies are needed to determine the actual amount of suppression as function of centrality (see [2]); the present study only indicates the onset points.
Finally we want to consider the suppression pattern expected for RHIC experiments. From Fig. 4, it is seen that for N part ≥ 80, all charmonium states should suffer anomalous suppression, so that here there should only be one onset point. Moreover, collisions with N part = 80 correspond to an impact parameter of b ≃ 10 fm, and this may be too peripheral for meaningful measurements. For a study of the onset of anomalous suppression at RHIC, it will thus most likely be necessary to study AA collisions for much lower A, as already noted previously [2].
In conclusion, we have shown that a very simplistic parton percolation approach leads to a conceptually reasonable and effectively parameter-free description of the observed anomalous J/ψ suppression pattern.   Figure 5: The J/ψ survival probability determined at SPS energy as function of the number of participant nucleons, from [28].