# Parton Saturation, Production, and Equilibration in High Energy Nuclear Collisions

\addressPhysics Department, BNL, Upton, NY 11973, USA

Deeply inelastic scattering of electrons off nuclei can determine whether parton distributions saturate at HERA energies. If so, this phenomenon will also tell us a great deal about how particles are produced, and whether they equilibrate, in high energy nuclear collisions.

## 1 Introduction

The deeply inelastic scattering (DIS) experiments at HERA revealed that structure functions grow rapidly at small x and large [1]. At some , for a fixed , it is expected that parton distributions will saturate–leading to a weaker growth of the structure functions [2]. Asymptotically, at least, one expects cross–sections to not grow faster than the logarithm squared of the energy [3].

Recently, there have been hints in the DIS data from ZEUS that saturation may be occuring already at HERA energies. The derivative of the structure function , with respect to , is plotted as a function of (each bin in being averaged over the acceptance for that particular ) [4]. The data for this quantity, which at high and relatively large may be simply related to the gluon structure function, show the expected rise for decreasing (and decreasing mean ), but then flattens and decreases for very small (and small ). An interesting feature of the data is that the turnover region lies between 1–10 GeV. The GRV94 DGLAP fit overshoots the data in this region. However, other, more recent parton distribution function sets do fit the data.

One must warn that the data should be approached with some caution since the averaging procedure is performed over a wide range of for each bin. However, ZEUS has collected more data in the low region which will enable them to perform the average over a smaller range of [5]. We will assume here that atleast the qualitative features of the Caldwell plot will remain unchanged.

A possible interpretation of the data–often called the “Caldwell Plot”–is that it demonstrates the onset of parton saturation [6]. The caveat about the data aside, if indeed we are seeing saturation at small x at HERA, this is very relevant for nuclear collisions at LHC, and to a lesser extent for RHIC. A sure-fire way to confirm saturation is to collide electrons with nuclei! [7].

The saturation scale determines the typical intrinsic momenta associated with quanta in the nuclear wavefunction. These quanta go on shell in a collision, and eventually produce a large multiplicity of particles, mostly pions. At very high energies, the saturation scale is the only scale in the problem, and one can estimate that the typical momenta of the particles produced in the collision is at this scale. If this momenta is large enough, one can estimate in perturbation theory whether or not they equilibrate to form a quark gluon plasma [8, 10].

Parton saturation, and its implications for both DIS and heavy ion collisions, can be studied systematically in an effective field theory (EFT) approach to QCD at small x [11, 12, 13, 14]. In the following section, we discuss parton saturation in this model, both for quarks and gluons. In section 3, we apply this model to nuclear collisions. The very early stages of the nuclear collision, where the non–linearities in the fields are large, can be studied in this approach. At late times, the fields linearize. Whether the partons, which have “emerged” by these times, equilibrate is an interesting issue, of great relevance to the quark gluon plasma community. This is discussed briefly in section 4.

## 2 Parton Saturation

In the infinite momentum frame , the effective action for the soft modes of the gluon field with longitudinal momenta (or equivalently ) can be written in light cone gauge as

(1) | |||||

Here is the gluon field strength tensor, are the matrices in the adjoint representation, and is the path ordered exponential in the direction in the adjoint representation of ,

(2) |

The effective action in Eq. 1 above has a remarkable saddle point solution [11, 12, 15]. It is equivalent to solving the Yang–Mills equations in the presence of the source . One finds a solution where , and (for ) is a pure gauge field which satisfies the equation . Here is the covariant derivative , is the space–time rapidity, and and are the space-time rapidity and the Lorentz contracted width, respectively, of the hard partons in the fragmentation region. At small x, the space–time and momentum space notions of rapidity are used interchangeably [16]. The momentum space rapidity is defined to be where is Bjorken and is the rapidity in the fragmentation region. The solution of the above equation is [12]

(3) |

To compute the classical nuclear gluon distribution function, one needs in general to average over the product of the classical fields in Eq. 3 at two space–time points with the weight [11]. For a large nucleus, one may approximate , where is the color charge squared per unit area per unit rapidity. The classical gluon distribution for this Gaussian source is

(4) |

where is the Casimir in the fundamental representation and
is the nuclear cross-section ^{1}^{1}1 Above, we have re-written the expression for the gluon distribution
in Ref. [12], using the leading log gluon distribution to
replace and with the gluon distribution
at the scale ..

For large (but smaller that , the distribution falls like a power law –and has a dependence! For very small , the behavior is the perturbative distribution . The scale which determines the cross–over from a logarithmic to a power law distribution is, to follow the notation of Mueller [9], the saturation scale . Setting and the argument of the exponential above to unity, one obtains the relation,

(5) |

which, for a particular , can be solved self–consistently to determine . The value of is approximately 1 GeV for RHIC energies and 2–3 GeV for LHC energies. To compare with the estimate of Gyulassy–McLerran [17], set , in the notation of Ref. [11, 12] where and for GeV.

The model calculation above shows that a) the gluon distribution saturates at some scale –the momentum distribution of partons grows only logarithmically for , and b) this saturation is seen already at the classical level. The gluon distribution has a tail that goes as , so one expects the typical intrinsic momentum of the gluons to be peaked at .

A similar behavior is seen for the structure function . The quark distribution in the classical gluon field-and , can be computed in the same approach [18, 19], and one obtains

(6) | |||||

where is the electric charge squared of a quark of flavor ,

, are the modified Bessel functions, and is the Casimir in the adjoint representation. The above equation is the well known Glauber expression [20] usually derived in the rest frame of the nucleus. It is heartening that, with the assumption of Gaussian color charges [19], the formalism of Ref. [18] for structure functions in the infinite momentum frame (which, in principle also accomodates non–linear evolution for structure functions) reproduces it. For large , it reduces to the standard DGLAP expression, while at small it goes to zero as . One then recovers, qualitatively, the shape of the Caldwell plot for .

One obtains from the above equation, in a manner analogous to Eq. 4, the quark saturation scale by replacing in Eq. 5. The relative size of the two saturation scales, glue to quark, is therefore determined simply by the ratio of the two Casimirs, .

What about quantum corrections to the above quark and gluon distributions? At the one loop level, one gets corrections to the Weizsacker–Williams distribution [21, 22, 9]. However, Mueller has argued recently that beyond the one loop level, the distribution has the same form as the as the above classical gluon distribution. What does change due to small evolution is the dependence of the saturation scale [9]. It will be very interesting to see if detailed studies of parton evolution in the non–linear region confirm this result [23, 14].

## 3 Parton Production

In the previous section, we discussed the parton distribution in a single nucleus in an effective field theory approach to QCD at small x. Kovner, McLerran, and Weigert applied this approach to study nuclear collisions at very high energies [24]. One now has two sources of color charge , one on each light cone, described by the currents corresponding to . The classical field describing the small x modes in the EFT is then obtained by solving the Yang–Mills equations in the presence of these sources. It can be written as . Here have and the transverse gauge fields are, as discussed in the previous section, pure gauges.

A convenient co–ordinate system to study nuclear collisions is , where and

Kovner, McLerran, and Weigert solved the Yang–Mills equations perturbatively to . Diagrammatically, this can be represented as . In Schwinger gauge, , and for , this is the dominant contribution to gluon production at small x. The result has been computed in different ways by several authors [24, 25] and it agrees with the pQCD gluon bremsstrahlung expression of Gunion and Bertsch at small x [26].

To the order , the saturation effects discussed in the previous section are not visible. The result agrees with the pQCD mini–jet result at small x. They become less reliable below the scale , where all orders in become important. Given the distribution in Eq. 4, suggesting that most of the partons in the small x component of the nuclear wavefunction have intrinsic momenta , these higher order effects are important for nuclear collisions. They self-consistently regulate the divergence at small , leading to infrared safe distributions of produced particles. Saturation effects may only be marginal at RHIC, but will be absolutely essential at LHC-where most of the produced particles will be semi–hard.

Recently, Krasnitz and I computed gluon production to all orders in numerically [27, 28]. Gauge fields in the forward light cone were obtained by solving Hamilton’s equations, for each configuration, on a lattice. Observables were computed by averaging over these configurations with a Gaussian weight. Assuming boost invariance, the lattice Hamiltonian is the Kogut–Susskind Hamiltonian in 2+1–dimensions, coupled to an adjoint scalar field.

The lattice initial conditions, in analogy with the continuum case, were obtained by matching the lattice equations of motion in the different light cone regions at . Our study was performed on two dimensional transverse lattices ranging from 1010 to 160160. For simplicity, we studied only the SU(2) case.

For large transverse momenta , the results of our simulation agree very well with lattice perturbation theory. This is the case for both the momentum and time dependence of the fields. The lattice coupling is . For , one observes large non–perturbative corrections at smaller momenta. At very early times, the field strengths are very large–and a parton model description is likely not valid. Since (or ) is the only scale in the problem, one expects fields to linearize on the time scale . This is seen when we compute on the lattice. It rises from zero and goes to a constant at . At late times , this quantity is the energy per unit transverse area per unit rapidity of produced particles. It scales as .

To summarize the above, we now have the technology to systematically compute a range of gauge invariant observables in high energy heavy ion collisions. An interesting quantity to compute would be the number of particles produced per unit rapidity. Several years ago, Blaizot and Mueller [8]–assuming saturation, computed on fairly general grounds the number of produced gluons per unit rapidity to be

(7) |

with . As Mueller has pointed out recently [10], this
coefficient cannot be computed precisely without knowing the details
of the collision. It is now feasible for us to confirm this formula
and compute the value of the coefficient ^{2}^{2}2The author thanks
L. McLerran for discussions on this point. We hope to report on
it in the near future.

If saturation does occur at GeV and 10, as the HERA data suggest, then depending on the x dependence of the gluon density, saturation may occur already for large nuclei at 10–which corresponds to RHIC energies. Modulo what above is, the multiplicity of produced particles at RHIC may tell us something about the saturated gluon density at HERA or vice versa. At LHC, the gluon densities would correspond to lower values of x for a fixed than at HERA (hence the urgent need for an eA collider!).

## 4 Parton Equilibration

At very early times after the nuclear collision, non–linearities in the Yang–Mills equations are extremely important, and the concept of partons is not very meaningful. The non–linearities dissipate-producing on shell partons in a time scale . Since this time scale is much less than , it is likely the produced partons will further interact, and perhaps equilibrate. (It is only at this stage that it is meaningful to think of a parton cascade.) The further interaction of produced partons is beyond the scope of the simulations of Krasnitz and myself. However, the “final” distribution of partons in our model can, in principle, be used as the initial conditions for a parton cascade. We should also point the reader to the Mueller’s recent qualitative study of small angle scattering and the onset of equilibration in heavy ion collisions [10]. From the theoretical viewpoint, the problem of parton equilibration is an extremely difficult one–especially since it is not clear that the problem can be formulated in a gauge invariant way.

## Acknowledgments

I would like to thank Prof. Tran Thanh Van and the other organizers for a very well organized and stimulating conference. I would also like to thank the US Moriond committee for an NSF travel award. This work has been supported by DOE nuclear theory at BNL.

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