Feynman Path Integral Formulation

82 3 Gravity in 2 + ε Dimensions
β (g)
gc
g
Fig. 3.3 The β-function for
the non-linear σ-model in the
large-N limit for d > 2.
with cμ0 d−2 the integration constant. The sign of c then depends on whether one is
on the right (c > 0) or on the left (c < 0) of the ultraviolet fixed point at g c .
One notices therefore again that the general shape of β(g) is of the type shown
in Fig. 3.3, with g c a stable non-trivial UV fixed point, and g = 0 and g = ∞ two
stable (trivial) IR fixed points. Once more, at the critical point g c the β-function
vanishes and the theory becomes scale invariant. Furthermore one can check that
again ν = −1/β ′ (g c ) where ν is the exponent in Eq. (3.59). As before one can
re-write the physical mass m for 2 < d < 4as
( ∫ g
ξ −1 dg ′ )
(g)=m(g) ∝ Λ exp −
β(g ′ , (3.66)
)
as was done previously in Eq. (3.34).
Another general lesson one learns is that Eq. (3.62),
[
Λ ∂
∂Λ + β(g) ∂ ]
m[Λ,g(Λ)] = 0 , (3.67)
∂g
can be used to provide a non-perturbative definition for the β-function β(g). If one
sets m = ΛF(g), with F(g) a dimensionless function of g, then one has the simple
result
β(g)=− F(g)
F ′ (g) . (3.68)
Thus the knowledge of the dependence of the mass gap m on the bare coupling g
fixes the shape of the β function, at least in the vicinity of the fixed point. It should
be clear then that the definition of the β-function per se, and therefore the scale
dependence of g(μ) which follows from it [as determined from the solution of the
differential equation μ ∂g
∂μ
= β(g(μ))]isnot necessarily tied to perturbation theory.
When N is large but finite, one can develop a systematic 1/N expansion in order
to evaluate the corrections to the picture presented above (Zinn-Justin, 2002).
Corrections to the exponents are known up to order 1/N 2 , but the expressions are
rather complicated for arbitrary d and will not be reproduced here. In general it appears
that the 1/N expansion is only asymptotic, and somewhat slowly convergent
for useful values of N in three dimensions.