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

388 U Kraemmer and A Rebhan
a high density. HDL resummation for dynamic quantities has been considered first in [311]
with applications to energy loss rates in astrophysical systems through axion-like particle, and
for energy loss of heavy quarks at both large T and µ in [312].
HDL resummation is also at the basis of the results of [143] quoted in section 4.4 on non-
Fermi-liquid behaviour of the specific heat at low temperatures and high chemical potentials,
equation (4.17). In this case the non-analytic terms in g 2 involve a logarithm of g as well as
cubic roots of g 2 , which are the result of the only weak dynamical screening of near-static
magnetic modes whose screening lengths involve cubic roots of m 2 Dω (see (5.19)).
6.3.1. HTL-screened perturbation theory (HTLPT). In [106–109] a modification of the above
scheme has been suggested, where the mass parameters ˆm D and ˆM within L HTL are considered
as independent of the coupling g and used for a variational improvement of the perturbative
series. This is a generalization of screened perturbation theory [100, 103]. It differs from
standard HTL/HDL resummation also in that (6.10) is used for both hard and soft momenta,
which results in additional UV divergences that need to be subtracted at any finite order of the
expansion.
This method has been used to calculate the thermodynamic potential to two-loop order,
where it does improve the apparent convergence of the perturbative results, but the result
deviates significantly from lattice results even at the highest temperatures that are available for
the latter.
A problem of this approach, at least in quantities that are dominated by hard excitations
like the pressure, seems to be that the HTL effective action is not a good approximation at
hard momenta. This is also signalled by the fact that HTL propagators do not satisfy the
relativistic KMS condition at large momenta [17]. While this is taken care of eventually by the
counterterms in (6.10), at any finite order of the expansion there are uncancelled unphysical
hard contributions. In [164] it has been shown that a (simpler) implementation of a variational
perturbation theory in dimensional reduction that uses just the Debye mass term (which is the
static limit of the HTL effective action) avoids this problem and indeed leads to results that
are closer to the lattice result as well as the higher-order calculations in dimensional reduction
when improved as discussed in section 4.2. It therefore appears that HTLPT needs to be
amended such that a different treatment of hard and soft modes is secured.
6.3.2. HTL resummed thermodynamics through -derivable approximations. While HTLPT
when applied to the thermodynamical potential does not work satisfactorily (though it may be
of more use in quantities that depend more dominantly on soft rather than hard scales), it turns
out that a generalization of the self-consistent expression for the entropy (3.18) allows for a
resummation of HTL propagators without the problems of HTLPT and with remarkably good
numerical results when compared with lattice data [96, 98, 99] 4 .
In gauge theories including fermions, the self-consistent two-loop expression for the
entropy (3.18) reads
∫
d 4 k ∂n(ω)
S =−tr
(2π) 4 ∂T
[Im log D−1 (ω, k) − Im (ω, k) Re D(ω, k)]
∫
d 4 k ∂f (ω)
− 2tr
(2π) 4 ∂T [Im log S−1 (ω, k) − Im(ω,k)Re S(ω,k)] (6.11)
4 A similar approach but formulated directly in terms of the (modified) thermodynamic potential, , has been set up
in [313].