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

Thermal field theory 363
Figure 3. One-loop correction to the self-energy in scalar φ 4 -theory, and some-higher-loop
diagrams with increasing degree of IR divergence when the propagators are massless.
smaller but still exponentially large cutoff, the n = 4 theory seems perfectly acceptable. For
our purposes we shall just have to restrict ourselves to temperature scales smaller than (3.3)
when considering the finite-temperature effects in this scalar theory.
3.2. Thermal masses
To one-loop order, the scalar self-energy diagram is the simple tadpole shown in the first
diagram of figure 3, which is quadratically divergent in cutoff regularization, but strictly zero in
dimensional regularization. The Bose distribution function occurring at non-zero temperatures
provides a cutoff at the scale of the temperature which gives
= (m (1)
th )2 = 4!g 2 ∫
d 3 q
(2π) 3 θ(q0 )δ(q 02 − q 2 )
{
n(q 0 ) + 1 2
}
= g 2 T 2 (3.4)
and so the initially massless scalar fields acquire a temperature-dependent mass. As we shall
see, in more complicated theories the thermal self-energy will generally be a complicated
function of frequencies and momenta, but the appearance of a thermal mass scale ∼gT is
generic.
It should be noted, however, that thermal masses are qualitatively different from ordinary
Lorentz-invariant mass terms. In particular they do not contribute to the trace of the energy–
momentum tensor as m 2 T 2 , as an ordinary zero-temperature mass would do [63]. So while
the dispersion law of excitations is changed by the thermal medium, the theory itself retains
its massless nature.
At higher orders in perturbation theory, the thermal contributions to the scalar self-energy
become non-trivial functions of frequency and momentum, which is complex-valued, implying
a finite but parametrically small width of thermal (quasi-)particles [64, 65], so that the latter
concept makes sense perturbatively.
In the large-N limit the self-energy of the scalar field remains a momentum-independent
real-mass term also beyond one-loop order and is given by the Dyson equation,
= 4!g 2 0 µ4−n ∫
d n q
(2π) n−1 {n(q 0 ) + 1 2
}
θ(q 0 )δ(q 2 − ). (3.5)
(Note that in (3.4) we had simply replaced g0 2 by g2 , disregarding the difference as being of
higher than one-loop order.) The appearance of a thermal mass introduces quadratic ultraviolet
(UV) divergences in , which are however of exactly the form required by the renormalization
of the coupling according to (3.2). Including the latter, one finds a closed equation for the
thermal mass of the form
{∫
m 2 th = d n ( )}
q
24g2 (2π) n−1 n(q0 )θ(q 0 )δ(q 2 − m 2 th ) + 1
32π 2 m2 th ln m2 th
¯µ − 1 2
=: 24g 2 {I T (m th ) + I f 0 (m th, ¯µ)}. (3.6)
The last term in the braces in (3.6), which has been neglected in [58, 66, 10], is responsible
for a non-trivial interplay between thermal and vacuum contributions. Its explicit dependence