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

Thermal field theory 397
[353, 223, 354, 355] and which also seems to provide a useful non-perturbative description of
Debye screening as they turn out to be gauge-independent in accordance with the arguments
of section 5.1.1.
7.4. Magnetostatic screening
The next-to-leading order correction to the other structure function, ii (0,k), can be derived
in a calculation analogous to the one leading to (7.8) and is found to be given by [352]
{ (α +1)
δ ii (0,k)= g 2 2 +10 k
N ˆm D T
+ 1 [
2 − k2 +4ˆm 2 D
arctan
k ]}
. (7.12)
16 ˆm D 4π ˆm D k 2 ˆm D
Apparently, the magnetic permeability defined by 1/µ = 1 − 1 2 ii/k 2 is a gauge-dependent
quantity beyond leading order. The gauge-dependent terms vanish only at the location of the
pole of the transverse gluon propagator, which is at k = 0. There the correction term vanishes
completely, which means that there is no magnetic mass squared of the order g 3 T 2 . The
magnetic mass must therefore be ≪ g 3/2 T at weak coupling.
For small k ≪ ˆm D , (7.12) has a linear behaviour with a gauge-dependent but positive
definite coefficient. The transverse propagator therefore has the form 1/(k 2 −ck) with c ∼ g 2 T .
This corresponds to a pole at space-like momentum with k ∼ g 2 T [364]. However, in this
regime the inherently non-perturbative contributions ∼ g 4 T 2 to µν become relevant and are
expected to remove this pathology by the generation of a magnetic mass m m ∼ g 2 T . This is
completely analogous to the IR problem of the perturbative expansion of the pressure identified
in [150,151], but while the latter sets in at four-loop order, in the transverse gauge-boson selfenergy
with external momenta g 2 T this problem starts already at two-loop order.
In a HTL-resummed two-loop calculation in the general covariant gauge, [365] has verified
that such a magnetic mass receives contributions exclusively from the ultrasoft momentum
regime k ∼ g 2 T , where the relevant effective theory is three-dimensional Yang–Mills theory,
which is confining and inherently non-perturbative. There exists in fact an elegant attempt
towards an analytic non-perturbative study of this theory through its Schrödinger equation using
results from two-dimensional conformal field theory [366–368]. This leads to the estimate
m m = g 2 NT/2π, which is reasonably close to the results for magnetostatic propagators
obtained in the lattice calculations of [353, 223] quoted earlier.
The other approaches that have tried to obtain an analytic estimate for the magnetic mass
mostly involve a reorganization of perturbation theory by assuming a more or less complicated
mass term for the magnetostatic gluons, setting up a self-consistent gap equation and solving
it. However, the various possibilities lead to rather contradictory results [369–374]. In fact,
lattice calculations indicate that the phenomenon of magnetic screening is not well described
by a simple pole mass [223].
In the Abelian case, one may expect that magnetostatic fields are completely unscreened,
and it can indeed be proved rigorously to all orders of perturbation theory [215, 222] that
ii (0,k → 0) = O(k 2 ). In massless scalar electrodynamics, an unresummed one-loop
calculation actually gives ii (0,k → 0) =
1 8 e2 kT , suggesting that the magnetostatic
propagator has the form 1/[k 2 + e 2 kT/8], which would imply power-law screening. However,
a resummation of the thermal mass of the scalar particles removes this unphysical behaviour
[375, 222].
7.5. Dynamical screening and damping at high temperatures
A logarithmic sensitivity to the non-perturbative physics of the (chromo-)magnetostatic sector
has in fact been encountered early on also in the calculation of the damping rate for a heavy