Feynman Path Integral Formulation

3.2 Perturbatively Non-renormalizable Theories: The Sigma Model 77
theories, including gravity, there is no local order parameter, so this quantity has no
obvious generalization there.
In general the ε-expansion is only expected to be asymptotic. This is already seen
from the expansion for ν which has recently been computed to four loops (Hikami
and Brezin, 1978; Bernreuther and Wegner, 1986; Kleinert, 2000)
ν −1 = ε +
ε2
N − 2 + ε 3
2(N − 2)
− [30 − 14N + N 2 +(54 − 18N)ζ (3)]
ε 4
× + ... (3.39)
4(N − 2) 3
which needs to be summed by Borel-Padé methods to obtain reliable results in three
dimensions. For example, for N = 3 one finds in three dimensions ν ≈ 0.799, which
can be compared to the 4 − ε result for the λφ 4 theory to five loops ν ≃ 0.705, to
the seven-loop perturbative expansion for the λφ 4 theory directly in 3d which gives
ν ≃ 0.707, with the high temperature series result ν ≃ 0.717 and the Monte Carlo
estimates ν ≃ 0.718, as compiled for example in a recent comprehensive review
(Guida and Zinn-Justin, 1998).
There exist standard methods to deal with asymptotic series such as the one in
Eq. (3.39). To this purpose one considers a general series
and defines its Borel transform as
f (g)=
F(b)=
∞
∑
n=0
∞
∑
n=0
f n g n , (3.40)
f n
n! bn . (3.41)
One can attempt to sum the series for F(b) using Padé methods and conformal transformations.
The original function f (g) is then recovered by performing an integral
over the Borel transform variable b
where the familiar formula
f (g)= 1 g
∫ ∞
0
∫ ∞
0
dbe −b/g F(b) , (3.42)
dzz n e −z/g = n!g n+1 , (3.43)
has been used. Bounds on the coefficients f n suggest that in most cases F(z) is
analytic in a circle of radius a around the origin, and that the integral will converge
for |z| small enough, within a sector |argz| < α/2 with typically α ≥ π (Le Guillou
and Zinn-Justin, 1980).