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

Thermal field theory 385
6. HTL effective action and resummation
As we have seen in several examples now, complete weak-coupling expansions in thermal
field theory tend to require a reorganization of the standard loop expansion. Already the
perturbative expansion of the thermal mass (3.8) of a scalar quasi-particle in section 3.3 has
shown that ordinary perturbation theory fails to determine higher-order corrections but runs
into IR problems. A resummation of the leading-order (HTL) mass is necessary (and sufficient
in this case) to reorganize the perturbation series [64]. A thermal mass ∝ gT introduces an
additional (soft for g ≪ 1) mass scale, and whenever loop calculations receive important
contributions from this scale, it is clearly mandatory to use propagators dressed by these
masses.
In the general case, it is however equally important to include vertex corrections. In gauge
theories, this is only natural as Ward identities tie up vertex functions with self-energies. But
regardless of gauge symmetry considerations, if there are contributions to N-point one-loop
vertex functions that are proportional to T 2 like the HTL self-energies, they are as important
as bare vertices when the external momentum scale is ∼ gT . One then has
Ɣ HTL
,N ∼ gN T 2 k 2−N ∼ g N−2 k 4−N ∼ ∂N L cl
∂A N ∣
∣∣∣k∼gT
, (6.1)
for bosonic fields A.
A one-loop vertex function whose leading contribution (for soft external momenta) is
proportional to a power of temperature greater than 1 is called HTL, as this is again dominated
by a hard loop momentum ∼T . In fact, already in spinor QED there are infinitely many HTL
vertex functions, namely those involving two external fermion lines and an arbitrary number
of gauge bosons. In QCD, there are in addition HTLs with an arbitrary number of external
gluons. These have been first identified in [279, 213, 280] and used to set up a resummation
programme for amplitudes involving soft external momenta [213, 281].
6.1. HTL effective action
In the case of scalar φ 4 -theory, the only HTL is a mass term corresponding to a local
HTL effective Lagrangian L HTL
scalar
=− 1 2 ˆm2 th φ2 , with ˆm 2 th
the leading-order term from (3.8).
Remarkably, the infinitely many HTL diagrams of gauge theories have a comparatively simple
and manifestly gauge-invariant integral representation [282, 283, 195, 284]:
L HTL = L HTL
f
+ L HTL
g
∫ ∫
= ˆM 2 dv
4π ¯ψγ µ v µ
iv · D(A) ψ + ˆm2 D
2 tr dv
4π F µα v α v β
(v · D adj (A)) F µβ, (6.2)
2
where v = (1, v) is a light-like four-vector, i.e. with v 2 = 1, and its spatial components are
averaged over by ∫ d v ···. Here, v is the remnant of the hard plasma constituents’ momenta
p µ ∼ Tv µ , namely their light-like four-velocity, and the overall scale T has combined with
the coupling constant to form the scale of the thermal masses, ˆM, ˆm D ∼ gT .
The covariant derivatives in the denominators of (6.2) are responsible for the fact that
there are infinitely many HTLs. Because in QED one has D adj (A) → ∂, the only HTL with
exclusively photons as external lines is the photon self-energy polarization tensor; the other
HTL diagrams of QED have two external fermion lines and an arbitrary number of photon
insertions.