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

Thermal field theory 369
m eff ( ¯φ) ∼ λT , and the loop expansion parameter of the perturbation series, which is λT /m eff ,
ceases to be small. Hence, perturbation theory cannot decide whether there is a first-order
phase transition as implied by a second degenerate minimum in the potential or not. In fact,
universality arguments show that the phase transition must be second-order for a φ 4 -theory
with Z 2 symmetry [61].
The situation is different in gauge theories. Again, resummed perturbation theory signals a
first-order phase transition, but when the Higgs mass is small compared with the massive vectorboson
mass, it is sufficiently strongly first-order so that the perturbation series does not break
down where the degenerate minima appear. The (resummed) effective potential has been calculated
perturbatively to one-loop [116–118] and two-loop order [114,119,120] (see also [121]).
Of particular interest is the question whether the electroweak sector of the Standard Model (or
extensions thereof) admits a first-order phase transition, which would be of great interest for
baryogenesis in the Early Universe [122]. Non-perturbative studies indeed confirmed a firstorder
transition for sufficiently small Higgs masses [123,124], but found an endpoint [125–127]
at Standard Model Higgs mass m H 80 GeV, so that a first-order phase transition in the Standard
Model is excluded by the current experimental bounds on the Higgs mass (though not in
possible (e.g. supersymmetric) extensions of the Standard Model [128, 129]).
4. QCD thermodynamics
The (resummed) perturbative evaluation of the thermodynamic potential of QCD at high
temperature has been pushed in recent years up to the order g 6 ln g [130–137]. At higher orders
in g this is much facilitated by the possibility of employing effective field theory methods,
which in this case lead to a dimensional reduction to a three-dimensional Yang–Mills plus
adjoint Higgs theory [75]. Completion of these results at order g 6 is in fact impossible without
inherently non-perturbative input, but further progress has been made most recently by the
extension to non-zero quark chemical potential within dimensional reduction [138].
At zero temperatures and high chemical potentials the pressure is known to order g 4 [139–
142, 138], while the low-temperature expansion of the pressure leads to the phenomenon of
non-Fermi-liquid behaviour of entropy and specific heat [143].
4.1. Dimensional reduction
Dimensional reduction in hot QCD leads to an effective three-dimensional Lagrangian [75–77],
L E = 1 2 trF ij 2 +tr[D i,A 0 ] 2 + m 2 E trA2 0 + 1 2 λ E(trA 2 0 )2 + ···, (4.1)
where the parameters are determined perturbatively by matching [136, 144]. In lowest-order 2
and at zero chemical potential, one has a dimensionful coupling, gE 2 = g2 T , and [77]
(
m 2 E = 1+ N )
f
g 2 T 2 , λ E = 9 − N f
6
12π 2 g4 T, (4.2)
though λ E starts to contribute to the pressure only at order g 6 . At this order, however, the selfinteractions
of the massless magnetostatic gluons start to contribute, and these contributions
are inherently non-perturbative because the three-dimensional theory for the zero modes A i (⃗x)
is a confining theory [149–151].
The thermal pressure of the four-dimensional theory can be decomposed into contributions
from the hard modes ∼T , calculable by standard perturbation theory, and soft contributions
2 Some higher-dimension terms in the effective theory (4.1) have been determined in [78, 145, 146], and, including
the electroweak sector, in [147, 148].