Next-to-leading order Debye mass for the quark-gluon plasma.

The Debye screening mass for a quark-gluon plasma at high temperature is calculated to next-to-leading order in the QCD coupling constant from the correlator of two Polyakov loops. The result agrees with the screening mass defined by the location of the pole in the gluon propagator as calculated by Rebhan. It is logarithmically sensitive to nonperturbative effects associated with the screening of static chromomagnetic fields. One of the fundamental properties of a plasma is the Debye screening mass mD, whose inverse is the screening length for electric fields in the plasma. The definition of the Debye mass is conventionally given in terms of the small-momentum (k → 0) limit of the static (ω = 0) Coulomb propagator [k +Π00(0, k)] −1 = [kǫ(0, k)], where Π00(ω, k) is the longitudinal photon self-energy function and ǫ(ω, k) is the corresponding dielectric function: m2D = lim k→0 Π00(0, k). (1) The Debye mass can alternatively be defined in terms of the location of the pole in the static propagator for complex k: k + Π00(0, k) = 0 at k 2 = −m2D. (2) At leading order in the coupling constant, the static self-energy has the simple form Π00(0, k) = m2D independent of k, and the definitions (1) and (2) are equivalent. Beyond leading order in the coupling constant, they need no longer be equivalent. A fundamental question of plasma physics is then this: what is the correct general definition of the Debye mass? This question applies equally well to a quark-gluon plasma, where the Debye mass describes the screening of chromoelectric fields. Because the coupling constant of quantum chromodynamics (QCD) is relatively large, higher order corrections to the Debye mass are probably not negligible. The question of the correct definition therefore becomes one of practical importance. In the case of QCD, the longitudinal gluon self-energy function Π00(ω, k) is gauge dependent. If the Debye mass is relevant to the screening of chromoelectric fields, then it must be gauge invariant. Thus, gauge invariance can be used as a guide to the correct definition of the Debye mass. Formal arguments due to Kobes, Kunstatter, and Rebhan indicate that, in spite of the gauge-dependence of the gluon propagator, the locations of its poles are gauge invariant [1]. This suggests that (2) is the correct general definition. However, it is important to back up these formal arguments with explicit calculations. There have been a number of calculations of the next-to-leading order correction to the Debye mass using the conventional definition (1) [2, 3]. The results are infrared finite but gauge dependent. The naive application of the pole definition (2) gives a gauge-dependent result, which is also infrared-divergent. However, it was recently shown by Rebhan [4] that, 1 with a careful treatment of infrared effects, the pole definition does in fact give a gaugeinvariant result. Since the calculation of Rebhan involves subtle interchanges of limits, it is desirable to have an independent verification of this result. It is also desirable to calculate the nextto-leading order Debye mass directly from a gauge-invariant quantity, to provide assurance that it is relevant to physical quantities. The simplest such quantity is the correlation function of two Polyakov loops. A previous calculation of this correlator to next-to-leading nontrivial order by Nadkarni [3] gave results that seemed to be incompatible with simple Debye screening. In this Letter, we reexamine this calculation and show that, by careful treatment of infrared effects, it can be used to extract the Debye mass to next-to-leading order. The result agrees with that obtained by Rebhan from the pole in the gluon propagator. These results, together with the general arguments of Ref. [1], provide compelling evidence that the pole definition (2) is the correct definition of the Debye mass beyond leading order in the coupling constant. We begin by reviewing the calculation by Rebhan. At leading order in the QCD coupling constant g, the Debye mass is given by a straightforward perturbative calculation: mD = √ (2Nc +Nf )/6 gT , where Nc = 3, Nf is the number of flavors of light quarks, and T is the temperature. At next-to-leading order, it is necessary to resum perturbation theory by including the Debye mass in the Coulomb propagator. The contribution of order g to the static longitudinal gluon self-energy is δΠ00(0, k) = Ncg T ∫ dp (2π)3 [ 1 p2 +m2D + 2(m2D − k ) p2(q2 +m2D) − ξ (k +m2D)(p 2 + 2p · k) (p2)2(q2 +m2D) ]