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

Thermal field theory 365
where φ 0 = √ T ∫ β
0 dτφ(τ, x) and, to lowest-order, m2 3 = g2 T 2 , g3 2 = g2 T .
Calculating now also soft one-loop corrections, one obtains in dimensional regularization
∫
δm 2 = 12g3
2 d 3 k 1
(2π) 3 k 2 + m 2 =− 3
3
π g2 3 m 3 =− 3 π g3 T 2 , (3.10)
which is exactly the first correction given in (3.8). (In cutoff regularization there would be
an extra term ∝ g3 2, cancelling a contribution ∝ g2 T to m 2 3
.) Using this method, the
perturbative expansion of m th for N = 1 has been worked out to order g 5 in [73].
3.5. Apparent convergence
The large-N limit provides an instructive opportunity to study the convergence properties
of a resummed perturbation theory with the exact result obtained by simply solving (3.6)
numerically. This has been carried out in great detail in [62] with the result that only for g
sufficiently smaller than 1 there is quick convergence, which deteriorates if the renormalization
scale ¯µ is very different from T . An optimal value turned out to be ¯µ ≈ 2πT, the scale of
the bosonic Matsubara frequencies, which has been argued previously in [81] to be a natural
choice.
However, the convergence of the resummed thermal perturbation series seems to be
surprisingly poor, given that the exact result following from (3.6) is a rather unspectacular
function. Naturally, a perturbative result is a truncated polynomial and thus bound to diverge
more and more rapidly at large coupling as the order is increased. This may be the case even
when the physical effects described by the lowest-order terms are still predominant.
In [82] it has been shown that the alternative so-called non-linear δ-expansion scheme [83]
yields approximations to m 2 th
that converge almost uniformly in g, but this scheme has not yet
found applications in more complicated (gauge) field theories.
A simpler alternative is provided by Padé approximants [84], that is rewriting a given
perturbative result as a perturbatively equivalent rational function in g by replacing
F n (g) = c 0 + c 1 g 1 + ···+ c n g n → F [p,n−p] (g) =
c 0 + a 1 g 1 + ···+ a p g p
. (3.11)
1+a p+1 g 1 + ···+ a n gn−p For example, truncating (3.8) above order g 3 gives an approximation F 3 to m 2 th
that stops
growing as a function of g at g = 2π/9 ≈ 0.7 and goes back to zero and then to negative
values for g π/3 ≈ 1 (line labelled ‘(3)’ in figure 4). The true (large-N) thermal mass
from (3.6) however is a monotonically growing function (line labelled ‘exact’ in figure 4).
The result F 3 in fact ceases to be an improvement over the leading-order result F 2 = g 2 T 2
roughly where it stops growing. On the other hand, the simplest possibility for a perturbatively
equivalent Padé approximant, F [2,1] , is a monotonic function in g,
m 2 th
T = g 2
2 1+3g/π +O(g4 ) (3.12)
(long-dashed line in figure 4), and it gives a substantial improvement for g 1.
Higher-order Padé approximants converge rather well in the simple scalar model [62]
(except when they happen to have a pole of the denominator at positive coupling), and they
have been proposed for improving also the unsatisfying convergence of perturbative results for
the thermodynamical potential in finite-temperature QCD [85, 86, 62], which will be discussed
in more detail in section 4.2. There they do increase the apparent convergence of the resummed
perturbation series for the first few orders, but at higher orders they look less convincing. In fact,
at these higher orders the perturbation series also involves ln(g)-contributions, which make