Collisional energy loss above the critical temperature in QCD

We compute the collisional energy loss for a heavy quark above the critical temperature in Quantum ChromoDynamics (QCD). We work in the semi Quark-Gluon Plasma, which assumes that this region is dominated by the non-trivial holonomy of the thermal Wilson line. Relative to the result to leading order in perturbation theory, at a fixed value of the coupling constant we generically we find that collisional energy loss is suppressed by powers of the Polyakov loop, l<1. For small values of the loop, this suppression is linear for the scattering off of light quarks, and quadratic for the scattering off of gluons, or for Compton scattering.

small probability to create particles with nonzero color charge. Nevertheless, numerical simulations on the lattice indicate that this probability is really rather small near the critical temperature, T χ . In any case, particles in the adjoint representation, such as gluons, are more strongly suppressed than quarks, which lie in the fundamental representation. To a good approximation for three colors, when the Polyakov loop is small the density of gluons is proportional to the square of the loop, while the density of quarks is proportional to a single power of the loop.
We work to leading order in perturbation theory in a fixed background field for the nontrivial holonomy. For collisional energy loss, in the limit of a small value of the loop, we find that the scattering off of light quarks is suppressed by a single factor of the loop, while that for gluons, or for Compton scattering in a thermal bath, is quadratically suppressed.
We expect that the suppression of scattering off of quarks and gluons near T χ holds in any effective theory, although surely the details differ. For the time being we defer a detailed comparison to experiment to future study.

A. Introduction
To represent nontrivial holonomy for a SU (N c ) gauge group, we expand about a background, classical gluon field A cl 0 , where Here g is the coupling constant for the gauge field, so as the background field is proportional to 1/g, it is manifestly nonperturbative. Further, the gluon field A 0 is not real, but purely imaginary. We introduce such a mean field to model the effect of nontrivial holonomy, and so it should be understood as arising from an ensemble average over nonperturbative fluctuations. Thus we do not attempt to derive from first principles how this field arises, but simply use results from lattice simulations to determine the Q's.
Since the gauge group is SU (N c ), the vector potential is traceless, The elements Q a are real, and we can assume that they are distributed symmetrically about the origin. (This is equivalent to assuming the expectation value of the Polyakov loop, Eq. (5) below, is real. This is true if there is no net baryon density; otherwise it is necessary to generalize the ansatz.) For three colors, as in QCD, this implies there is only one independent variable, Q: The Wilson line in the temporal direction is T is the temperature, τ is the imaginary time, and P denotes time ordering and is gauge invariant. There are also higher loops, (1/N c )tr L n , which are obviously also gauge invariant. For a general field in SU (N c ), there are N c − 1 independent loops.
For three colors, under the mean field ansatz of Eq. (3) there is one independent loop, which we can take to be the simplest, While the Wilson line is not gauge invariant, its eigenvalues are. To leading order in weak coupling it suffices to deal with the background A cl 0 field. Beyond leading order it is necessary to deal with the eigenvalues of the Wilson line, which are gauge invariant. Typically, lattice simulations do not measure the eigenvalues directly, but only the Polyakov loop, which is a sum over the eigenvalues. The eigenvalues were measured directly in one recent study, [14].
The results, however, agree with measurements of the Polyakov loop.
Physically the background field which generates non-trivial holonomy can be thought of as an imaginary chemical potential for color. The Bose-Einstein/Fermi-Dirac statistical distribution function for a gluons and quarks are given, respectively, by n q a (E) = one color index, through Q a . For gluons in the adjoint representation, a difference of two indices enters, Q ab = Q a − Q b . We also introduce the notation |z| r = sign(Rez) z.
These are the statistical distribution functions for emission into a thermal bath. Those for absorption from a thermal bath are given bȳ n g ab = 1 + n g ab (9) for gluons, andn for quarks. The relative minus sign is because quarks obey the Fermi-Dirac exclusion principle.
To illustrate the physics of non-trivial holonomy, consider the sum over all colors for the quark statistical distribution function: We first compute the number of quarks in the deconfined phase at very high temperature.
At high temperature the theory is essentially perturbative, and we can set all Q's to vanish.
The number of colored quarks is then where the Riemann zeta-function ζ(3) ≈ 1.20206.... The numerical value of the right hand side is not important, what we wish to emphasize is that as expected in the deconfined phase, the quark density is proportional to the number of quarks, N c .
Contrast this with the background field in the confined phase of the pure glue theory, where any Polyakov loop with nonzero Z(N c ) charge vanishes. The explicit Q which produces the confined vacuum is For three colors, Eq. (3), Q = 2πT /3. In the confined phase, all loops with nonzero Z(N c ) charge vanish. The only Polyakov loops which are nonzero are those which wrap around the imaginary time direction by an integral multiple of N c . These correspond to a type of "baryon": In the confined phase, then, For massless quarks, the energy is related to the momentum k by E = |k|. Integrating over the momenta, the total number of colored particles in the confined phase is This computation illustrates several points. In the limit of an infinite number of colors, the number of quarks is ∼ N c in the deconfined phase, and very small, ∼ 1/N 2 c , in the confined phase. This ratio is strictly zero only in the limit of infinite N c . For finite N c , there is a small density of quarks in the "confined" phase. This matters with dynamical quarks, where this density is nonzero. This is the usual observation that there is no strict order parameter for confinement in the presence of dynamical quarks. As mentioned previously, however, in practice the density of quarks in the confined phase is small, at least as measured by numerical simulations on the lattice for three colors and for two or three flavors of quarks.

B. Energy loss in the sQGP
Consider a heavy quark of mass M and energy E, where E = p 2 + M 2 . We assume that the heavy quark is moving rapidly, with p M . The energy loss per unit length x is given by Here P = (E, p ) is the four momentum of the incident heavy quark, P = (E , p ) that of the outgoing heavy quark, whilst K = (k, k ) and K = (k , k ) are the four momenta of the particles which the heavy quark scatters off, and ω = E − E . The integrals are over spatial momenta of the virtual particles, and similarly for k and p . The index i represents a label over the different types of particles, including whether they are bosons or fermions/antifermions, as well as color, flavor, and spin. For the squared amplitude, |M i | 2 in Eq. (17), one sums over the initial and final spin states, and divides by the degeneracy factor, d = 2, of the incoming particle.
The computations of energy loss for a heavy quark in hot QCD were first carried out by Braaten and Thoma [15]. They showed that the integral over phase space can be simplified considerably; see also Appendix A of Peigné and Peshier [16]. Using these simplifications, Here s, t, and u are the usual Mandelstam variables, and we introduce the quantity γ, where Since the square root of γ enters into Eq. (19), γ must be positive. Requiring that γ > 0 fixes the limit of integration over t and ω, with t : t min → 0, and ω : ω min → ω max .
In this article we only compute the energy loss to leading logarithmic order. In this instance the distribution functionn(k + ω) can be replaced with 1 [15,16]. Thus we obtain

C. Coulomb scattering
The amplitude for Coulomb scattering, of a heavy quark off of a light quark in the thermal bath, is illustrated in the left hand side of Fig. (1). This amplitude involves the color trace where N f is the number of light quark flavors, and thus Here the subscript on dE/dx refers to the dependence on the background field A cl 0 through Q e .
The integration with respect to ω is Integration with respect to the spatial momentum k is done by expanding the quark distribution function into a series which starts with the Boltzmann term, Keeping only the terms to leading logarithmic order, we find a very simple result: the expression in the semi-QGP is an overall factor, which depends upon the Q's, times that for the perturbative QGP: where the result in the perturbative QGP is (32) N f is the number of light quark flavors. We regulate the infrared logarithmic divergence of the integral over the Mandelstam variable t by the gluon Debye mass.
We note that in the semi-QGP that the gluon Debye mass depends upon the background field through the Q's. We can neglect this dependence because it only enters beyond leading logarithmic order.
The Q-dependent factor in Eq. (31) is given by The superscript in S qk (Q) denotes that it is due to scattering off of light quarks.
In the perturbative regime this suppression factor equals unity, as In the confined phase of the pure glue theory, only loops with n = kN c contribute, so that by Eq. (14), For physically relevant case, N c = 3, and under the mean-field anzatz of Eq. (3), by using the identity ∞ n=1 (−) n n 2 cos(2π n x) = π 2 x 2 − 1 12 the suppression factor S qk (Q) can be calculated analytically, Using Eq. (6), we have for the nearly confining and perturbative background field respectively. u-channel (right). Only the diagram on the right hand side generates a logarithm.

D. Compton scattering
There are three diagrams which contribute to what is termed Compton scattering. There is scattering off of a gluon in the t-channel, which is illustrated by the diagram on the right hand side in Fig. (1). There are also two diagrams for scattering off of a gluon in the s-channel and u-channel, as illustrated in Fig. (2). Among them, only the t-channel and u-channel generate leading logarithm contribution through small angles scattering. Furthermore, all cross terms between different channels do not lead to leading logarithm contribution, thus we will focus on the squared amplitude of t-channel and u-channel diagrams.

Compton scattering in the t-channel
For Compton scattering in the t-channel, the relevant diagram is that on the right hand side of Fig. (1). The color structure which enters for this diagram is Again, there is no summation over the color indices e and f , which correspond to those for the gluon in the inital state, which the heavy quark scatters off of.
The matrix element for scattering in the t-channel is The expression for energy loss in the t-channel is Performing the integrals over k, t, and ω as before, again the result is a Q-dependent factor times the perturbative result: The perturbative result is dE dx The modification of the pertubative result in the semi-QGP is given by a factor In the perturbative QGP this factor is unity, as it must be. The superscript in S gl (Q) denotes that it is due to scattering off of a gluon.
We show in the next subsection that to leading logarithmic order, the suppression factor for scattering in the u-channel is the same as in the t-channel, Eq. (54).
In a confining background field, the suppression factor for t-channel scattering is found to vanish, For three colors, N c = 3, using the anzatz Eq. (3) and the identity ∞ n=1 1 n 2 cos(2π n x) = π 2 12 we obtain In the limiting cases of a confining and perturbative background field we get

Compton scattering in the u-channel
The only diagram which generates a logarithm at leading order is that on the right hand side of Fig. (2). The color structure for this diagram is Here C f = (N 2 c − 1)/(2N c ) is the Casimir for the fundamental representation. The matrix element for scattering in the u-channel becomes After integrating over k, t, and ω, once again the result is a Q-dependent factor times the perturbative result: The result in the perturbative limit is We find that the color dependent factor in the semi-QGP is the same in the u-channel as in the t-channel, given by Eq. (45).

II. COMPLETE RESULT
A. Extracting the loop from the lattice Besides non-perturbative contributions in the semi-QGP, the Polyakov loop also receives contributions from ordinary perturbation theory, To order ∼ g 4 [17], Notice that the leading contribution is positive. This implies that the expectation value of the loop exceeds unity. While this cannot be true classically, it occurs because of renormalization. In Eq. (57), and where Λ M S is the renormalization mass scale in the modified minimal substraction scheme.
Lastly, the coefficients for c g and c m are We assume that these perturbative corrections exponentiate, Even with Q = 0, exponentiating the leading order corrections is an assumption about those to higher order. Further, the corrections to ∼ g 3 and ∼ g 4 will certainly change in the semi-QGP, when the Q's are nonzero. We do not include this effect for the time being.
Thus we first compute (Q) from Eq. (63) and by using Eq. (6) determine Q(T ). The results are shown in Fig. (3).

B. Suppression factors in the semi-QGP
Summing up the contributions from Coulomb scattering, Eq. (31), from Compton scattering in the t-channel, Eq. (43) and from Compton scattering in the u-channel, Eq. (54), gives a total result for energy loss which is We can then use the results for the temperature dependence of Q to plot the suppression factors in the semi-QGP, versus the perturbative results. These are illustrated in Fig. (4).
For temperatures near T χ , where the expectation value of the Polyakov loop is small, we find that the suppression of the gluon terms, S gl (Q), is much stronger than for the quark term, S qk (Q). This is obvious from the corresponding expressions, Eq. (45) Physically this is evident. For small values of the loop, the density of quarks is ∼ , while that of gluons is ∼ 2 . This is simply because the quarks are in the fundamental representation, and the gluons, in the adjoint. In another way, in the double line notation (which is useful at large N c , but can be used at any N c ), quarks have one line, and gluons, two lines.
We have only illustrated the suppression factors, and leave it for later analysis to make a detailed comparison to experiment. However, our study shows that for temperatures which are probed at both RHIC and even at the LHC, that the scattering off of light quarks completely dominates over scattering off of gluons. This is directly a manifestation of the "bleaching" of color in the semi-QGP, as the density of colored particles decreases. (3). Notice that the suppression is much greater for gluons than for quarks.

III. CONCLUSIONS AND OUTLOOK
In this paper we computed the suppression of collisional energy loss in the semi-QGP.
We find, on elementary and very general grounds, that scattering off of gluons is strongly suppressed in the semi-QGP, while that of quarks is only moderately suppressed.
It is interesting that when the dust settles, we obtain rather simple expressions for the collisional energy loss, as simple suppression factors times the usual perturbative result. This suggests that the same will be true for other electromagnetic probes. We have computed the effects of the semi-QGP upon both dilepton production, and on the production of real photons, and will present these results shortly.
These computations represent the first attempt to extend perturbative computations of quantities in thermal QCD to phenomenologically relevant temperatures of interest, building crucially upon results from numerical simulations on the lattice.