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

Thermal field theory 379
5.1.1. Gauge independence of singularities. In [191, 192] it has been shown that under
variations of the gauge-fixing parameters (in our case α, ˜β, ˜γ ) one has ‘gauge dependence
identities’ (generalized Nielsen identities [206–209]) that are of the form
δ −1
A
(k) = −1
A [−Aµ ν (k)δXν ,µ(k)] ≡ −1
A
(k)δY (k), (5.13)
δ −1
B
(k) = −1
B
[
]
−ñµ
ñ + ˜γ ˜β − α C /|⃗k|
k µ 2ñ 2 ν δX
˜β 2 k 2 ,µ ν
− α ≡ −1 B
(k)δZ(k), (5.14)
D
where δX,µ ν has a diagrammatic expansion that is one-particle-irreducible (1PI) except for at
most one Faddeev–Popov ghost line. No such relation exists for C or D .
Now if δY and δZ are regular on the two ‘mass-shells’ defined by −1
A
= 0 and −1
B = 0,
the relations (5.13) and (5.14) imply that the locations of these particular singularities of
the gluon propagator are gauge-fixing independent, for if −1
A
= 0 = −1
B
, then also
−1
A
+ δ−1 A
= 0 = −1 B
+ δ−1 B .
In the case of B , singularities in δZ(k) include a kinematical pole 1/k 2 hidden in
the ñs and the manifestly gauge-dependent D (cf (5.12)). Excluding these obvious gauge
artefacts, everything depends on whether the possible singularities of δX,µ ν could coincide
with the expectedly physical dispersion laws −1
A
= 0 and −1
B
= 0. Because δXν ,µ is oneparticle
reducible with respect to Faddeev–Popov ghosts, the singularities of the latter have to
be excluded, too. However, these are generically different from those that define the spatially
transverse and longitudinal gauge-boson quasi-particles. Indeed, in leading-order thermal
perturbation theory the Faddeev–Popov ghost self-energy and the physical self-energies receive
contributions carrying different powers of temperature or chemical potential and therefore have
independent and generically different dispersion laws.
However, δX,µ ν may develop singularities also from 1PI subdiagrams, namely when one
line of such a diagram is of the same type as the external one and the remaining ones are
massless. This can give rise to IR or mass-shell singularities and seemingly constitute an
obstruction to the gauge-independence proof [210, 211]. But such singularities will be absent
as soon as an overall IR cutoff is introduced, for example by restricting everything to a finite
volume first. In every finite volume, this obstruction to the gauge-independence proof is
then avoided, and −1
A
= 0 and −1
B
= 0 define gauge-independent dispersion laws if the
infinite-volume limit is taken last of all [212].
5.1.2. Gauge-field quasi-particles at leading order. It turns out that the leading-order
contributions to µν for small frequencies and momenta k 0 , |k| ≪T are entirely transverse
( C = D = 0) and gauge-parameter-independent. They are generated by one-loop diagrams
with hard loop momentum, and for this reason they are termed HTLs [213]. Their form is the
same in Abelian [214,215] and non-Abelian gauge theories [216,217] and also in the presence
of chemical potentials µ f for fermions such that max(T , µ f ) ≫ k 0 , |k|. This universal result
reads
ˆ A = 1 2 ( ˆ µ µ − ˆ B ), (5.15)
ˆ B =− k2
k 2 ˆ 00 (5.16)