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

Thermal field theory 387
equations of motion are given by
∫
[D µ ,F µν ] a = j νa = ˆm 2 dv
D
4π vν W a (x, v), (6.7)
[v · D, W(x, v)] a = v · E a (x), (6.8)
where E is the chromo-electric field strength and W a (x, x) describes the fluctuations of the
phase space density of hard gluons. At the expense of introducing W a (x, x) as a new soft
degree of freedom, this formulation permits a completely local description of the physics of
soft gauge-boson modes [300, 301].
The HTL diagrams can be constructed from solving (6.7) in terms of jµ a [A] and formally
expanding
jµ a ab
=−ˆ µν Aν b + 1 abc ˆƔ
2 µνρ Aν b Aρ c + ···. (6.9)
The kinetic-theory approach also allows one to consider strong deviations from
equilibrium. In [302–304] the analogue of the HTL self-energies with anisotropic momentum
distributions has been considered, which turns out to lead to space-like singularities in the
gauge-boson propagator. The latter are related to plasma instabilities that could play an
important role in thermalization issues [305, 306].
6.3. HTL /HDL resummation
Since for soft momenta ∼gT HTL self-energies and vertices are equally important as the
tree-level self-energies and vertices, the former may not be treated perturbatively but should
rather be combined with the latter to form effective self-energies and vertices. This can be
done formally by replacing
L cl → L cl + L HTL − δ × L HTL , (6.10)
where δ is a parameter that is sent to 1 in the end, after the last term has been treated as a
‘thermal counterterm’ by assuming that δ counts as a one-loop quantity.
Because L HTL has been derived under the assumption of soft external momenta, this
prescription is in fact only to be followed for soft propagators and vertices [213]. Propagators
and vertices involving hard momenta (if present) do not require this resummation, and in
fact for obtaining a systematic expansion in the coupling g, one has to expand out all HTL
insertions on hard internal lines. In practice, a separation between hard and soft scales may be
implemented by introducing an intermediate scale with gT ≪ ≪ T , assuming g ≪ 1,
for example ∝ √ gT (see, e.g. [307, 308]).
The resulting systematic expansions in g typically involve single powers in g and
logarithms of g, in contrast to conventional perturbation theory, which would involve only
g 2 as an expansion parameter. This is because increasing the loop order by one involves
a factor g 2 T that in an ultrarelativistic situation and soft external momenta gT is made
dimensionless by a thermal mass m ∼ gT , so that the effective expansion parameter becomes
g 2 T/m ∼ g.
The same resummation scheme arises in the presence of a chemical potential. In the
effective action, the chemical potential enters only in the mass parameters ˆM 2 and ˆm 2 D according
to (5.28) and (5.18). For T ≈ 0 but large µ f , the HTLs have also been nicknamed ‘ HDLs’ [309].
At T ≈ 0, resummation of the HDL effective action is necessary for soft momenta gµ,but
because of the absence of Bose enhancement, this does not give rise to single powers of g in
the perturbations series but only to logarithms of g in addition to powers of g 2 , as found long
ago in the ring resummation scheme of Gell-Mann and Brueckner [310] for an electron gas at