Advances in perturbative thermal field theory - Ultra-relativistic ...

392 U Kraemmer and A Rebhan
perturbation theory leads to next-to-leading order corrections of quasi-particle dispersion laws
that are typically suppressed by a single power of g rather than g 2 .
In the following, we shall review what is at present known about next-to-leading order
corrections to the HTL quasi-particles in gauge theories. It will turn out that some corrections,
namely screening lengths for frequencies below the plasma frequency and damping rates for
moving excitations are even more enhanced by IR effects to an extent that they cannot be
determined beyond the leading logarithmic term without non-perturbative input.
7.1. Long-wavelength plasmon damping
Historically, the first full-fledged application of HTL resummation techniques was the
calculation of the damping constant of gluonic plasmons in the long-wavelength limit.
In fact, the development of the HTL resummation methods was stimulated by the failure
of conventional thermal perturbation theory to determine this quantity.
In bare perturbation theory, the one-loop long-wavelength plasmon damping constant
had been found to be gauge-parameter-dependent and negative definite in covariant gauges
[216]. Since even the gauge-independent frameworks of the Vilkovisky–DeWitt effective
action [328, 329] and the gauge-independent pinch technique [330] led to negative one-loop
damping constants, this was sometimes interpreted as a signal of an instability of the quark–
gluon plasma [331,332]. Other authors instead argued in favour of ghost-free ‘physical’ gauges
such as temporal and Coulomb gauges, where the result turned out to be positive and seemingly
gauge-independent in this class of gauges [196, 197].
It was pointed out in particular by Pisarski [333] that all these results at bare one-loop order
were incomplete, as also implied by the arguments for gauge-independence of the singularities
of the gluon propagator of [191]. The appropriate resummation scheme was finally developed
by Braaten and Pisarski in 1990 [213, 281], who first obtained the complete leading term in
the plasmon damping constant by evaluating the diagrams displayed in figure 12 with the
result [334]
γ(k = 0) = 1 Im δ A (k 0 =ˆω pl , k = 0) = +6.635 ... g2 NT
2 ˆω pl 24π = 0.264√ Ng ˆω pl . (7.1)
For g ≪ 1, this implies the existence of weakly damped gluonic plasmons. In QCD
(N = 3), where for all temperatures of interest g 1, the existence of plasmons as identifiable
quasi-particles requires that g is significantly less than about 2.2, so that the situation is
somewhat marginal.
The corresponding quantity for fermionic quasi-particles has been calculated in [335,336]
with a comparable result: weakly damped long-wavelength fermionic quasi-particles in two- or
three-flavour QCD require that g is significantly less than about 2.7.
These results have been obtained in Coulomb gauge with formal verification of their
gauge independence. Actual calculations, however, later revealed that in covariant gauges
HTL-resummed perturbation theory still leads to explicit gauge-dependent contributions to
the damping of fermionic [210] as well as gluonic [211] quasi-particles. But, as was pointed
out subsequently in [212], these apparent gauge dependences are avoided if the quasi-particle
mass-shell is approached with a general IR cutoff (such as finite volume) in place, and this cutoff
lifted only in the end. This procedure defines gauge-independent dispersion laws; the gaugedependent
parts are found to pertain to the residue, which at finite temperature happens to be
linearly IR singular in covariant gauges rather than only logarithmically as at zero temperature,
due to Bose enhancement.
For kinematical reasons, the result (7.1), which has been derived for branch A of the
gluon propagator, should equally hold for branch B, since with k → 0 one can no longer