An infrared singularity in the damping rate for longitudinal gluons in hot QCD

Abstract

We calculate γ l (0), the damping rate for longitudinal on-shell gluons with zero momentum in hot QCD using the hard-thermal-loop (htl) scheme. We find it to be divergent in the infrared, which means that in this scheme γ l (0) is different from γ t (0), the corresponding damping rate for transverse gluons which is known to be finite. This result suggests that the htl scheme is infrared sensitive and thus may need to be improved upon in this sector. We discuss this issue after we present our calculation. pacs: 11.10.Wx 12.38.-t 12.38.Bx 12.38.Mh keywords: hard thermal loops. soft gluon damping. ENSK-TP-10Besides their importance regarding the stability of the quark-gluon plasma, gluon damping rates have been crucial in better understanding QCD at high temperature T . In early works, it has been noticed that in this regime, the determination of the dispersion laws for quarks and gluons beyond lowest order using standard perturbation theory is gaugedependent [1]. This problem has been emphasized in further works in which the gluon damping rates have been calculated to one-loop order in various gauges and schemes and different results have been obtained [2]. It was then realized that the problem was related to the way the expansion in powers of g, the perturbative QCD coupling constant, was performed: at high T (the hard scale), when the external momenta are soft, i.e., of magnitude gT, the standard loop expansion is not anymore an expansion in powers of g [3]. It was subsequently developed an effective perturbative expansion in the framework of a resummation scheme of the so-called hard thermal loops (htl) [4]. Using this scheme, the transverse-gluon damping rate γ t (0) with zero momentum was shown to be Coulomb-and-covariant-gaugeinvariant and determined in the strict Coulomb gauge to be finite and positive [5]. Later, a

Advances in perturbative thermal field theory

“…However, as will be discussed further below, the limit k → 0 involves infrared problems, and there are even explicit calculations [322,323] that claim to find obstructions to this equality, which are however contradicted by the recent work of [324].…”

NLO dispersion laws for slow-moving quarks in HTL QCD

“…In a line of works, we have used this formalism and looked into the infrared behavior of fully-HTL-dressed one-loop-order damping rates of slow-moving longitudinal [55,56] and transverse gluons [57], quarks [58][59][60], fermions 2 [61][62][63] and photons [64] in QED, and quasiparticles [65] in scalar QED. 3 In this logic, the natural step forward is to try to calculate the NLO energies of the quasiparticles.…”

Damping of very soft moving quarks in high-temperature QCD

“…For example, the gluon and quark on-shell energies ω(p) are obtained in the literature in the form of a series in powers of soft p [43,15]. The same is true for the residue and cut functions intervening in the spectral decomposition of the effective propagators [35]. It is therefore legitimate to expect the perturbation built on hard thermal loops (these being considered as a zeroth order approximation) to be analytic in very soft p, and hence admit an expansion in powers of such momenta.…”

“…On the one hand, this puts 'some water' into our arguments regarding the acceptability of our early expansion in powers of the very soft external momentum. On the other, recalling that the infrared divergences we find in the gluonic sector start also at order (p/m g ) 2 [34,35,37], it would be interesting to try to understand why this is so. Finally, it is useful to recall that the intensive use of the damping rates as a mean to 3 We already know from [38,39] that the first coefficient a 0 is safe.…”

“…The infrared sensitivity of the on-shell gluon damping rates has been emphasized in [34,35] where the damping rate γ l (0) for longitudinal gluons with zero momentum was determined to lowest order g 2 T and found to be different from γ t (0) and infrared divergent. This is to be contrasted with the fact that at zero momentum, there must be no difference between longitudinal and transverse gluons [16].…”