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

364 U Kraemmer and A Rebhan
on ¯µ is such that it cancels the one implicit in g 2 = g 2 ( ¯µ),
dg
dln( ¯µ) = 3
4π 2 g3 , (3.7)
which is exact for N →∞.
More complicated effects of zero-temperature renormalization on reorganized thermal
perturbation theories have been discussed in a scalar φ6 3 model in [67]. Another solvable toy
model is given by the large-N f limit of QCD or QED, which has been worked out in a thermal
field theory context in [68–71]. These theories also require the introduction of a cutoff to
avoid the Landau singularity and triviality. In contrast to the O(N →∞) φ 4 -theory, they
involve complicated momentum-dependent dispersion laws as well as damping effects. While
the large-N φ6 3 -theory has unphysical instabilities above a certain temperature (aside from the
Landau singularity), large-N f QED and QCD is well-behaved and very useful as a benchmark
for approximations to real QED and QCD.
3.3. Perturbation series
The integral appearing in (3.6) can be evaluated, e.g., using Mellin transformation techniques
[72] to obtain a series expansion of m th whose first few terms read [62]
m 2 (
) (
)
th
T = 2 g2 − 3g3
π
+ 3g4
¯µ
3 − γ − ln + 27g5
¯µ
2γ +2ln
2π 2 4πT 8π 3 4πT − 1 +O(g 6 ). (3.8)
This result shows that ordinary perturbation theory is unable to go beyond the one-loop
result (3.4), for ordinary perturbation theory is an expansion in powers of g 2 . Equation (3.8)
however involves odd powers of g. Indeed, because of the masslessness of the scalar theory,
ordinary perturbation theory encounters IR divergences starting at two-loop order, which
are exacerbated by the Bose distribution function behaving as n(q 0 ) ∼ T/q 0 for q 0 → 0.
For example, the second diagram in figure 3 involves two massless propagators with equal
momentum. The inserted tadpole does not vanish at zero momentum, but is given by the
constant term (3.4). At zero temperature, such an insertion (if non-zero), would make this
diagram logarithmically IR divergent; at finite temperature it is instead linearly IR divergent.
The higher-loop diagrams shown in figure 3 are even more IR divergent. But, as we have seen,
the full propagator contains the thermal mass, so all these divergences are spurious—they just
signal the need for using a resummed, i.e. massive propagator.
A systematic method to perform the required resummation of ordinary perturbation theory
is to add a thermal mass term − 1 2 m2 th φ2 to the Lagrangian (3.1) and to subtract it as a counterterm
which is treated as a one-loop-order quantity. The corresponding calculation at N = 1,
where (3.6) is only part of the full result, has been performed up to and including order g 5
in [64, 73].
3.4. Dimensional reduction
An important technical concept for studying static quantities such as thermodynamic potentials
and (static) screening masses is that of ‘dimensional reduction’ [74–80], which in the case of
scalar field theories has been worked out in [81]. In this approach one separates hard (k ∼ T )
from soft (k gT ) modes and integrates out the former. Since in the Matsubara formalism
all non-static modes are necessarily hard, this yields a three-dimensional effective theory
containing zero-modes only, and whose parameters (masses and coupling constants) are to be
determined by perturbative matching to the full theory. To lowest-order in scalar φ 4 -theory,
this yields (now for N = 1)
L 3 = 1 2 (∇φ 0) 2 + 1 2 m2 3 φ2 0 + g2 3 φ4 0 + ···, (3.9)