July 16, 2002 BITP 2002/14
Parton Percolation and Suppression
S. Digal, S. Fortunato, P. Petreczky and H. Satz
Fakultät für Physik, Universität Bielefeld
D33501 Bielefeld, Germany
Abstract:
The geometric clustering of partons in the transverse plane of nuclear collisions leads for increasing or 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 quarkgluon 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 quarkgluon plasma, both the momenta and the relative abundances of quarks, antiquarks 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 preequilibrium stage to achieve deconfinement and perhaps subsequent quarkgluon 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 nucleusnucleus collision at high energy, one finds in the transverse nuclear plane interacting partons^{1}^{1}1A relation between deconfinement and percolation was suggested quite long ago [6]; but the first work on color percolation in nuclear collisions was given in terms of strings [1], rather than partons. We shall nevertheless restrict ourselves here to a parton picture [2]. 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 over the transverse nuclear plane , with .^{2}^{2}2For simplicity, we assume the partons to have a fixed transverse radius; the extension to a distribution of radii is straightforward and does not change the picture. 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
(1) 
i.e., when the parton density measured in terms of parton size reaches the percolation threshold . In the ‘thermodynamic’ limit of infinite spatial size and infinite parton number , the largest connected cluster first spans the system at this point. The critical value 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 of the nuclear area is covered by partons. At , the critical clustering behavior of the system can be specified in the usual way in terms of critical exponents. In particular, the size of the largest cluster diverges for as
(2) 
with the critical exponent . 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 , 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 and the c.m.s energy . Saturation sets in when the parton density in terms of the partonic interaction cross section approaches the critical value . The partonic cross section depends on its inverse transverse momentum , , and the parton density for a fixed resolution scale 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 is related to their transverse size , 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 is given in terms of their fractional momentum and the resolution scale , through integration over the partonic transverse momentum . The relevant resolution scale in a nucleonnucleon collisions is the largest transverse momentum for which partons are resolved. The number of partons at fixed and integrated over is given by the sum of the contributions of gluons plus those of sea quarks and antiquarks,
(3) 
where denotes the gluon distribution function, and that of up and down () 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 , with . 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 . More dynamical considerations lead to numerical modifications, , where is the running coupling at scale and a given constant. With the geometric cross section we obtain that in an collision at , the equation
(4) 
determines the onset of percolation. Here specifies the density of parton sources in the transverse plane of a central collision. At SPS energy, this is essentially the density of wounded nucleons [8], . For higher energies and harder partons, collisiondependent contributions will play a significant role, and so a more suitable form here is a combination of the two sources,
(5) 
with [9].
As determined by eq. (4), partons with 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 or , 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 collisions, it is convenient to rewrite eq. (4) in the form
(6) 
with the density of parton sources in fm; the factor 25 arises when this is converted to GeV. In eq. (6), the hadronic parton distribution is compared to the density of parton sources in a nucleusnucleus collision. For low source densities, i.e., for small , remains well above the independent l.h.s., which is a function only of and : there are not enough partons to form a condensate. For sufficiently large , however, intersects at some , thus defining the onset of percolation. All partons with 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 GeV, with for SPS and for RHIC. Among the commonly used PDF parametrizations, only the set GRV94 [10] goes down to such small values of . This parametrization describes well the available data on the proton structure function from the E665 and NMC collaborations for 0.4 GeV GeV and [11, 12]. It also reproduces quite well the small HERA data [10, 13] for . We have therefore calculated using the nexttoleading order GRV94 PDF’s in the DIS scheme (GRV94DI) [10]; the resulting are shown in Figs. 1a and 1b for SPS (20 GeV) and RHIC (200 GeV) energies, respectively.
Any uncertainty in is mainly due to the gluon distribution. At SPS energy, this can be estimated by comparing calculations using leading and nexttoleading order GRV94 PDF’s. The resulting uncertainty is below 3 % for GeV and increases to at most 10 % at 0.5 GeV. At RHIC, the nexttoleading 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 nexttoleading order GRV94 parametrization.
We also note that these gluon distributions are not very different from those proposed in more recent phenomenological saturation studies for [15]. This approach is very successful in describing the low HERA data and it can in fact also account for the E665 data on in the kinematical region relevant for RHIC. The gluon distribution of [15] is directly related to ; 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 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 and , and with increasing energy, the condensate becomes more and more gluondominated. Thus the ratio of gluons to quarks and antiquarks in the parton condensate based on the GRV94 PDF’s is at GeV found to be about 1.2 for SPS energy and 4.0 for RHIC; in a chemically equilibrated quarkgluon plasma, it is about 0.5. Hence before any thermalization, the medium is strongly gluondominated [19].
In order to illustrate the effect for different central collisions, we assume a spherical nuclear profile, which gives at SPS energy, with in eq. (5),
(7) 
Inserting this into eq. (6), together with the values of at 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 , with GeV. To obtain the corresponding behavior at GeV, nucleon collisions have to be included as source of partons. With
(8) 
and [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 for central collisions are displayed as function of . At SPS energy, we thus do not obtain parton condensation below ; at RHIC energy, the higher parton density lowers the onset to .
In the experimental study of suppression, the production is measured at different centralities, so that we now have to determine the parton source density at fixed and varying impact parameter. This is done in a Glauber analysis based on WoodsSaxon nuclear profiles, with a collisiondetermined weight [18]. In Fig. 3a, we show the resulting percolation behavior as function of the effective number of participants in a collision at GeV. The threshold for parton percolation is found to be slightly below . The corresponding calculations for collisions at GeV, with a collisiondependent term in and again 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 is shown; at the onset point, the condensate contains partons of different sizes , with at SPS and at RHIC.
We now turn to the effect of parton condensation on production. It is known from collisions that normal nuclear matter leads to reduced charmonium production. Therefore we first have to consider such ‘normal’ suppression, since for collisions, at least up to RHIC energies, we are well below the threshold for parton condensation^{3}^{3}3A determination of the collision energy for which parton saturation occurs in or collisions is presently difficult, since it involves values much smaller than presently accessible in deep inelastic scattering. Disregarding this small behavior leads to a very early onset of parton percolation [22].. Consider the production of charmonium in the nuclear target rest frame. A gluon from the incident proton fluctuates into a virtual pair; in a collision with one of the target nucleons this is brought onshell, its color is neutralized, and it eventually becomes a physical of size fm. Thus in the restframe of the pair, a time of at least 0.4 fm is needed for formation. During its preresonance stage, for fm, it will travel a distance
(9) 
in the target rest frame; here denotes the mass of the nucleon, that of the pair and the Feynman momentum fraction. For the production of a at rest in the nucleonnucleon c.m.s., this becomes . From this it is immediately seen that at , and , the nascent effectively traverses even heavy nuclear targets in its preresonance stage. The situation is very similar for and production, for which the preresonance lifetimes are if anything even larger. Thus for the mentioned and , the nuclear target sees of all charmonium states only the small preresonance precursor, so that all should suffer the same degree of nuclear suppression [20]. For negative (and perhaps to some extent also for lower ), 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 preresonance dissociation in standard nuclear matter, leading to a breakup 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 preresonance in the rest frame of the nucleus: otherwise, the is wounded several times before it has had a chance to register the first interaction. For collision energies 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 preresonance breakup in normal nuclear matter could change, and on the other hand the coherence length becomes dilated ten times more. Now interference effects of the LandauPomeranchukMigdal 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 collisions at RHIC are absolutely essential for an understanding of whatever suppression in collisions is observed there.
We now want to consider the additional ‘anomalous’ suppression of charmonium production due to the dense partonic medium created in collisions. The virtual partons in the incoming nuclei coalesce to form a condensate in a time 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 quarkgluon plasma; a crucial factor for this is the energy density reached at thermalization. We restrict ourselves here to the parton condensate stage.
In 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 preresonance 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 preresonance absorption in normal nuclear matter, followed by the effect of the parton condensate on the survivors.
In the first attempt to describe suppression in terms of parton percolation [2], it was assumed that different charmonium states define particular scales , 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 scaleindependent function of . Parton saturation in fact specifies as function of the scale , so that we now have (see Figs. 2 and 4) at given collision energy a parameterfree determination of the onset line or of parton condensation.
Within the parton condensate, color fields of a dependent strength will affect the binding of a dipole charmonium state. This effect can be addressed in different ways. One possible way is to consider the 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 , where is the dipole size and is a dimensional parameter determined by the field strength. In the color glass approach, [5]. Thus for small singlet dipoles, , 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 of scale will be dissociated if it finds itself in a parton condensate of scale ; otherwise it will survive. This very simplistic picture allows an analysis of nuclear profile effects (condensed and noncondensed regions of the collision profile) and thus provides some direct predictions for the centralitydependence of anomalous suppression for given and . 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]
(10) 
These have to be compared to the partonic saturation radii shown in Fig. 4 as functions of the number of participants in central collisions. It is seen that at SPS energy, the onset of and suppression effectively coincides with the onset of parton condensation, while the survives up to larger and higher parton densities^{4}^{4}4The fate of the is most likely more complex. It should certainly be dissociated once parton condensation sets in; however, it is very much more weakly bound than and , so that a less dense environment could also lead to its breakup.. Hence in collisions, anomalous suppression should start with the elimination of feeddown from and at ; the further suppression of direct production should set in for . In Fig. 5 we show the suppression pattern observed by the NA50 collaboration at the CERNSPS [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 , all charmonium states should suffer anomalous suppression, so that here there should only be one onset point. Moreover, collisions with correspond to an impact parameter of 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 collisions for much lower , as already noted previously [2].
In conclusion, we have shown that a very simplistic parton percolation approach leads to a conceptually reasonable and effectively parameterfree description of the observed anomalous suppression pattern.
Acknowledgements
It is a pleasure to thank D. Kharzeev and M. Nardi for their contribution to an early version of this study [2] as well as for many helpful discussions and comments.
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