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

Thermal field theory 405
Figure 16. Ladder diagrams contributing to photon production at leading order. In a quark–gluon
plasma, the external lines correspond to on-shell photons, the internal lines to dressed quark and
gluon propagators. The quarks are hard but collinear with the external photon and exchange soft
gluons such as to maintain collinearity.
In leading-logarithmic approximation, the required non-Abelian Boltzmann equation that
generalizes the non-Abelian Vlasov equation (6.8) has the form [410–415]
∫
[v · D, W(x, v)] a = v · E a (x) + γ d v ′I(v, v ′ )W(x, v ′ ) (8.3)
with I(v, v ′ ) defined in (8.2). Beyond the leading-log approximation [415–417, 394], the
quantity γI(v, v ′ ) needs to be replaced by a collision operator that involves also the HTL
gauge-boson self-energies. The ultrasoft vertex functions obtained formally in analogy to the
expansion (6.9) however still share many of the remarkable properties of the HTLs, namely
gauge-fixing independence and simple Ward identities [415].
At still smaller scales, ω ≪ |k| ≪ γ , it is in fact possible to simplify the above
effective theory of ultrasoft modes and to construct a local effective theory (Bödeker’s effective
theory) [410,411,418] because on distances |k| −1 much larger than the colour coherence length
γ −1 there are no longer propagating modes with definite colour. Bödeker’s effective theory is
a stochastic theory given by the Langevin equation,
σ c E a = (D × B) a + ζ a , 〈ζ ia (x)ζ jb (x ′ )〉=2σ c Tδ ij δ ab δ 4 (x − x ′ ), (8.4)
where ζ is a Gaussian noise term and σ c the colour conductivity given to leading order by
(7.22); beyond leading-log order, this parameter has been determined in [394]. The effective
theory (8.4) is also UV finite and was put to use in the numerical calculation [419] of the rate
of baryon number violation [122] in the hot symmetry-restored phase of electroweak theory,
which at leading order is governed by non-perturbative non-Abelian gauge field fluctuations
with spatial momenta ∼ g 2 T and frequencies ∼ g 4 T ln(1/g).
8.2. Light-like external momenta
As we have seen in the example of NLO corrections to the dispersion law of longitudinal
plasmons in section 7.8.2, the HTL perturbation theory becomes insufficient not only for
ultrasoft momenta |k| g 2 T but also when amplitudes involve harder momenta that are
however nearly light-like, which gives rise to collinear singularities. This problem has surfaced
in particular in the calculation of the production rate of real (non-thermalized) photons in a
quark–gluon plasma from HTL-resummed perturbation theory [420–422], and it turns out that
damping effects on hard internal lines have to be included as the dominant regulator of collinear
singularities [423].
Moreover, for a complete leading-order calculation, it turns out to be necessary to resum
all ladder diagrams built from dressed propagators of the form shown in figure 16. This
reflects the necessity to take into account the physical phenomenon of Landau–Pomeranchuk–
Migdal suppression 8 of photon emission [425]. However, contrary to the conclusion arrived at
in [426], thermal photon production from a QCD plasma was shown in [427] not to be sensitive
8 For a review in the context of energy loss calculations see, e.g. [424]