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

378 U Kraemmer and A Rebhan
with, in momentum space, A µν being the spatially transverse tensor (2.33), and the others
chosen as [131, 151, 196, 197, 11, 55]
B µν (k) = ñµ ñ ν
≡ η µν − kµ k ν
ñ 2
k 2 − A µν (k), (5.3)
C µν (k) = 1
|k| {ñµ k ν + k µ ñ ν } , (5.4)
D µν (k) = kµ k ν
, (5.5)
k 2
where ñ µ = (η µσ − k µ k σ /k 2 )δσ 0 in the plasma rest frame.
Only A and B are transverse with respect to the four-momentum k, A µν k ν = B µν k ν = 0.
C obeys the weaker relation C µν k µ k ν = 0, and D projects onto k. This particular basis is a
convenient choice because A, B and D are idempotent and mutually orthogonal; C is only
orthogonal to A, but its product with the other tensors has vanishing trace.
In Abelian gauge theory with linear gauge fixing, the Ward identities imply transversality
of the polarization tensor, µν k ν ≡ 0. In non-Abelian gauge theories transversality holds only
in certain gauges such as axial gauges [198] (with only the temporal axial gauge respecting
rotational invariance) and background-covariant gauges [199–201]. Contrary to the experience
at zero temperature, at finite temperature the polarization tensor turns out to be non-transverse
already at one-loop order in general covariant and Coulomb gauges [197, 202], with the
fortuitous exception of the Feynman gauge (at one-loop order).
For a gauge breaking Lagrangian (2.34), BRS invariance only requires that
[203, 11, 204, 205]
f ˜ µ f˜
ν G µν =−α. (5.6)
This entails that [204, 205]
D (k 2 − B ) = 2 C (5.7)
so that generally the polarization tensor contains three independent structure functions.
Equation (5.7) also implies that at one-loop order D ≡ 0 but not beyond in those gauges
where C ≢ 0.
Rotationally invariant gauge-fixing vectors can be written generally as f ˜ µ = ˜β(k)k µ +
˜γ(k)ñ µ with ˜β ≠ 0. This includes covariant gauges ( ˜β = 1, ˜γ = 0), Coulomb gauges
( ˜β =ñ 2 , ˜γ =−k 0 ) and temporal gauges ( ˜β = k 0 /k 2 , ˜γ = 1).
For these, the structure functions in the full propagator:
−G µν = A A µν + B B µν + C C µν + D D µν (5.8)
are determined by
A = [k 2 − A ] −1 , (5.9)
[
B = k 2 − B − 2 ˜β ˜γ |k| C − α 2 C + ˜γ ] −1 2 ñ 2 D
, (5.10)
˜β 2 k 2 − α D
C =−˜β ˜γ |k|−α C
˜β 2 k 2 − α D
B , (5.11)
D = ˜γ 2 ñ 2 + α(k 2 − B )
˜β 2 k 2 − α D
B . (5.12)