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

394 U Kraemmer and A Rebhan
Originally, the Debye mass (squared) has been defined as the IR limit 00 (ω = 0,k → 0),
which indeed is correct at the HTL level (cf (5.17)).
In QED, this definition has the advantage of being directly related to the electric
susceptibility, i.e. the second derivative of the thermodynamic pressure with respect to the
chemical potential, µ, so that the higher-order terms available for the latter also determine
those of QED
00
(ω = 0,k → 0) through [215, 8]
∣ ( √ ···)
00 (0,k → 0)| µ=0 = e 2 ∂2 P ∣∣∣µ=0
= e2 T 2
1 − 3e2 3e
3
∂µ 2 3 8π + 2 4π + . (7.3)
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This result is gauge-independent because in QED all of µν is.
In the case of QCD, there is no such relation. Moreover, δm 2 D / ˆm2 D ∼ g rather than g3
because of gluonic self-interactions and Bose enhancement. The calculation of this quantity
should be much easier than the dynamic ones considered earlier because in the static limit the
HTL effective action collapses to just the local, bilinear HTL Debye mass term,
HTL static
L −→ − 1 2 ˆm2 D trA2 0 . (7.4)
This is also gauge-invariant because A 0 behaves like an adjoint scalar under time-independent
gauge transformations. Resummed perturbation theory for static quantities thus reduces to a
resummation of the HTL Debye mass in the electrostatic propagator [310, 114, 8].
However, in QCD this simple (‘ring’) resummation leads to the gauge-dependent
result [345]
√
00 (0, 0)
ˆm 2 = 1+α N 6
g, (7.5)
D
4π 2N + N f
where α is the gauge parameter of general covariant gauge (which coincides with general
Coulomb gauge in the static limit).
This result was initially interpreted as meaning either that the non-Abelian Debye mass
could not be obtained in resummed perturbation theory [346] or that one should use a
physical gauge instead [196, 8]. In particular, the temporal axial gauge was put forward
because in this gauge there is, as in QED, a linear relationship between electric field strength
correlators and the gauge propagator. However, because static ring resummation clashes with
the temporal gauge, inconclusive and contradicting results were obtained by different authors
[347, 348, 196]. A consistent calculation in fact requires vertex resummations [349, 350],
but this does not resolve the gauge-dependence issue because the non-Abelian field strength
correlator is gauge-variant [351].
On the other hand, in view of the gauge-dependence identities discussed in section 5.1.1,
the gauge-dependence of (7.5) is no longer surprising. Gauge-independence can only be
expected ‘on-shell’, which here means ω = 0butk 2 →−ˆm 2 D .
Indeed, the exponential fall-off of the electrostatic propagator is determined by the position
of the singularity of B (0,k), and not simply by its IR limit. This implies in particular that
one should use a different definition of the Debye mass already in QED, despite the gaugeindependence
of (7.3), namely [352]
m 2 D = 00(0,k)| k 2 →−m . (7.6)
2 D
For QED (with massless electrons), the Debye mass is thus not given by (7.3) but rather
as [352]
m 2 D = 00(0,k → 0) +[ 00 (0,k)| k 2 =−m 2 − 00(0,k → 0)]
D
√
= e2 T 2 (1 − 3e2 3e
3
[
3 8π + e2
+ ···− ln ˜µ
2 4π 3 6π 2 πT + γ E − 4 ] )
+ ··· , (7.7)
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