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

380 U Kraemmer and A Rebhan
Figure 9. Location of poles in A and B of the HTL gauge propagator. The right part with
k 2 0 corresponds to propagating normal modes, the left part to (dynamical) screening.
with
ˆ µ µ = ˆm 2 D ,
ˆ 00 = ˆm 2 D
)
(1 − k0
2|k| ln k0 + |k|
, (5.17)
k 0 −|k|
⎧⎪
e 2 T 2
⎨ + e2 µ 2 e
for QED,
ˆm 2 D = 3 π 2
⎪ ⎩
g 2 NT 2 ∑
(5.18)
f
+
g2 µ 2 f
for SU(N) with N
3 2π 2
f flavours.
The spectral representation of the resulting propagators A,B involves simple poles and
continuous parts for k 2 < 0, which are given in detail in appendix A.1.
In figure 9, the location of the poles of the corresponding propagators A and B are
displayed in quadratic scales. The light-cone is marked by a dashed line. A simple mass
hyperboloid would be given by a line parallel to the latter. Evidently, the dispersion laws of
the HTL quasi-particle excitations are more complicated—they involve momentum-dependent
masses: in the long-wavelength limit |k| →0, there is a common lowest (plasma) frequency
ˆω pl = ˆm D / √ 3 for propagating normal modes.
For larger frequencies and momenta ω,|k| ≫ ˆm D it turns out that mode A approaches
asymptotically a mass hyperboloid with mass m ∞ = ˆm D / √ 2. So in this momentum region
the physical spatially transverse polarizations of the gauge bosons acquire indeed a constant
thermal mass. The additional mode B, whose dispersion curve approaches the light-cone
exponentially, is found to have a spectral strength (the residue of the corresponding pole) that
decays exponentially as k/ ˆm D →∞[218], showing its exclusively collective nature.
For real |k| but ω 2 < k 2 , ˆ µν has an imaginary part ∼ ˆm 2 D from the logarithm in (5.17),
which prevents the appearance of poles in this region. This imaginary part corresponds to
the possibility of Landau damping, which is the transfer of energy from soft fields to hard
plasma constituents moving in phase with the field [219, 6] and is an important part of the
spectral density of HTL propagators. At higher, subleading orders of perturbation theory,
it is, however, not protected against gauge-dependences in non-Abelian gauge theories, in
contrast to the location of the singularities that determine the dispersion laws of quasi-particles.
However, at asymptotically large times, Landau damping is (generically) dominated by the