Feynman Path Integral Formulation

3.3 Non-linear Sigma Model in the Large-N Limit 81which 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 (Wilsonand Fisher, 1972). Note that in the large N limit the constant of proportionality inEq. (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 abovetwo dimensions is that all particles are massless in perturbation theory, yet they allbecome massive in the strong coupling phase T > T c , with masses proportional tothe non-perturbative scale m.Again one can perform a renormalization group analysis as was done in the previoussection in the context of the 2 + ε expansion. To this end one defines dimensionlesscoupling 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( 1m(g) ≃ c d ·Λ − 1 ) 1/(d−2), (3.60)g c gwith the numerical coefficient given by c d =[ 1 2 (d − 2)π|csc( ) 1dπ2|] d−2 .Onewelcomefeature of this large-N result is the fact that it provides an explicit value forthe coefficient in Eq. (3.29), namely( ) 1/(d−2) gcc d =, (3.61)a 0and 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 tobe scale- and cutoff-independent as in Eq. (3.26)Λ d m[Λ,g(Λ)] = 0 , (3.62)dΛor, 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 resultβ(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, inagreement with the 2 + ε result of Eq. (3.14). Furthermore Eq. (3.64) gives themomentum dependence of the coupling at fixed cutoff. After integration, one findsfor the momentum (μ) dependence of the coupling at fixed cutoff Λg(μ) 1=g c 1 − c(μ 0 /μ) d−2 ≈ 1 + c(μ 0/μ) d−2 + ... (3.65)