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

Thermal field theory 381
gauge-independent location of the branch cuts at ω =±|k|, resulting in power-law relaxation
of perturbations [220]. There is also an exponential component of Landau damping due to
a pole at purely imaginary ω and real |k| on the unphysical sheet reached by continuation
through the branch cut between ω =±|k| [221].
√
k 2 = iκ
For real ω< ˆω pl , there are no poles for real |k| but instead for imaginary
corresponding to exponential (dynamical) screening of (time-dependent) external sources.
These poles, displayed on the left part of figure 9, are in fact closely related to the just mentioned
poles at purely imaginary ω and real |k|.
In the static limit ω → 0, only mode B is screened with (Debye) screening length ˆm −1
D .
This corresponds to exponential screening of (chromo-)electrostatic fields. The transverse
mode A on the other hand is only weakly screened when ω ≪ m D with a frequency-dependent
inverse screening length [217]:
( πm
2
) 1/3
κ A ≃ D ω
, ω ≪ m D . (5.19)
4
When ω → 0, mode A describes magnetostatic fields that are found to be completely
unscreened in the HTL approximation. In QED the absence of magnetic screening is intuitively
clear and can be proved rigorously to all orders of perturbation theory [215,222] but not in the
non-Abelian case.
In QCD, the absence of a magnetostatic screening mass causes problems for perturbation
theory as will be discussed further in section 7.4. In fact, lattice simulations of gauge-fixed
propagators in non-Abelian theories do find a screening behaviour in the transverse sector,
though the corresponding singularity is evidently quite different from a simple pole [223].
5.2. Fermions
The fermion self-energy at non-zero temperature or density has one more structure function than
usually. In the ultrarelativistic limit where masses can be neglected, it can be parametrized by
(ω,k) = a(ω,|k|)γ 0 + b(ω, |k|)ˆk · γ. (5.20)
(For a massive fermion, this would also include a mass correction, i.e. = aγ 0 + bˆk · γ + c1.)
This can be rewritten as
γ 0 (ω,k) = + (ω, |k|) + (ˆk) − − (ω, |k|) − (ˆk), (5.21)
where ± ≡ b ± a and the spin matrices
± (ˆk) ≡ 1 ± γ 0 γ · ˆk
(5.22)
2
project onto spinors whose chirality is equal ( + ), or opposite ( − ), to their helicity. Dyson’s
equation, S −1 =−̸k + , then implies
γ 0 S −1 = −1
+ + + −1
− − (5.23)
with −1
± ≡−[ω ∓ (|k| + ± )]. This is trivially inverted to yield the fermion propagator
Sγ 0 = + + + − − , (5.24)
whose singularities are conveniently summarized by the equation
det S −1 (ω, k) = 0, (5.25)
where the determinant refers to spinor indices. The potential gauge dependences of the
singularities of S are described by an identity of the form [192]
δ det S −1 (ω, k) = det S −1 (ω, k) δtrX(ω,k), (5.26)