Feynman Path Integral Formulation

3.3 Non-linear Sigma Model in the Large-N Limit 81
which gives for the correlation length exponent the non-gaussian value ν = 1/(d −
2), with the gaussian value ν = 1/2 being recovered as expected at d = 4 (Wilson
and Fisher, 1972). Note that in the large N limit the constant of proportionality in
Eq. (3.59) is completely determined by the explicit expression for Ω d (m).
Perhaps one of the most striking aspects of the non-linear sigma model above
two dimensions is that all particles are massless in perturbation theory, yet they all
become massive in the strong coupling phase T > T c , with masses proportional to
the non-perturbative scale m.
Again one can perform a renormalization group analysis as was done in the previous
section in the context of the 2 + ε expansion. To this end one defines dimensionless
coupling constants g = Λ d−2 T and g c = Λ d−2 T c as was done in Eq. (3.1).
Then the non-perturbative result of Eq. (3.59) becomes
( 1
m(g) ≃ c d ·Λ − 1 ) 1/(d−2)
, (3.60)
g c g
with the numerical coefficient given by c d =[ 1 2 (d − 2)π|csc( ) 1
dπ
2
|] d−2 .Onewelcome
feature of this large-N result is the fact that it provides an explicit value for
the coefficient in Eq. (3.29), namely
( ) 1/(d−2) gc
c d =
, (3.61)
a 0
and thereby for the numerical factor a 0 appearing in Eqs. (3.29) and (3.22).
Again the physical, dimensionful mass m in Eqs. (3.59) or (3.60) is required to
be scale- and cutoff-independent as in Eq. (3.26)
Λ d m[Λ,g(Λ)] = 0 , (3.62)
dΛ
or, more explicitly, using the expression for m in Eq. (3.60),
[
Λ ∂
∂Λ + β(g) ∂ ]
Λ ( 1 − 1 ∂g g c g )1/(d−2) = 0 , (3.63)
which implies for the O(N) β-function in the large N limit the simple result
β(g)=(d − 2)g(1 − g/g c ) . (3.64)
The latter is valid again in the vicinity of the fixed point at g c , due to the assumption,
used in Eq. (3.59), of m ≪ Λ. Note that it vanishes in d = 2, and for g = 0, in
agreement with the 2 + ε result of Eq. (3.14). Furthermore Eq. (3.64) gives the
momentum dependence of the coupling at fixed cutoff. After integration, one finds
for the momentum (μ) dependence of the coupling at fixed cutoff Λ
g(μ) 1
=
g c 1 − c(μ 0 /μ) d−2 ≈ 1 + c(μ 0/μ) d−2 + ... (3.65)