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

Thermal field theory 393
Figure 12. One-loop diagrams in HTL-resummed perturbation theory contributing to the gaugeboson
self-energy in pure-glue QCD. HTL-resummed quantities are marked with a blob.
distinguish between spatially transverse and longitudinal polarizations. However, as will be
discussed further later, the limit k → 0 involves IR problems, and there are even explicit
calculations [337, 338] that claim to find obstructions to this equality, which are however
contradicted by the recent work of [339].
In the simpler φ 4 -theory, the long-wavelength plasmon damping constant has been
calculated in [64,65] and it has been shown in [340–343] that this quantity can also be extracted
from classical field theory after perturbatively matching to the HTL mass.
7.2. NLO correction to the plasma frequency
In [344], Schulz has calculated also the real part of the NLO contribution to the gluon
polarization tensor in the limit of k → 0, which determines the NLO correction to the gluonic
plasma frequency.
The original power-counting arguments of [213] suggested that besides one-loop diagrams
with HTL-resummed propagators and vertices given in figure 12 there could be also
contributions from two-loop diagrams to relative order g. The explicit (and lengthy) calculation
of [344] showed that those contribute only at order g 2 ln(1/g) rather than g, and the NLO plasma
frequency in a pure-glue plasma was obtained as
ω pl =ˆω pl [1 − 0.09 √ Ng]. (7.2)
In this particular result, HTL-resummed perturbation theory turns out to give a moderate
correction to the leading-order HTL value even for g ∼ 1.
7.3. NLO correction to the Debye mass
Poles of the gauge-boson propagator at ω0, the A-branch
of the HTL propagator becomes unscreened in the static limit. In QED, a ‘magnetic mass’ is
forbidden by gauge invariance [215,222], but some sort of entirely non-perturbative magnetic
mass is expected in non-Abelian gauge theories in view of the severe IR problems caused by
the self-interactions of massless magnetostatic gluons [149–151].
Branch B, on the other hand, contains the information about screening of (chromo-)electric
fields as generated by static charges (Debye screening). The Debye mass given by the leadingorder
HTL propagator is ˆm D = √ 3ω pl .
The evaluation of corrections in thermal perturbation theory again requires resummation.
Even though the non-localities of the HTL effective action do not play a role in the static limit,
historically these corrections have been determined for QCD, as well as for (ultrarelativistic)
QED, only after the more complicated cases of gluonic plasmon damping and the NLO plasma
frequency discussed above had been mastered.