Feynman Path Integral Formulation

3.2 Perturbatively Non-renormalizable Theories: The Sigma Model 73This then gives the desired result for the beta function of the nonlinear sigma model,to lowest order in g, as in Eqs. (3.13) and (3.14).In exactly d = 2 one needs to cutoff the k integral, and one finds instead 〈τ 2 〉 =(N − 2) g2π logΛ + O(g2 ). So in two dimensions the effective coupling is1= 1 g eff g − N − 2 logΛ + O(g) (d = 2) , (3.20)2πwhich implies the asymptotic freedom result for d = 2 and N > 2, namely(see Fig. 3.2).β(g) =−(N − 2) g22π + O(g3 ) (d = 2) . (3.21)β (g)Fig. 3.2 The ε-expansion resultfor the Callan-Symanzikβ-function in the non-linearσ-model for d > 2. The nontrivialultraviolet fixed pointis, to lowest order in ε, locatedat g c = 2π(d − 2)/(N − 2).gcgThe one-loop running of g as a function of a sliding momentum scale μ = k andε > 0 can be obtained by integrating Eq. (3.14). One findsg(k 2 )=g c1 ± a 0 (m 2 /k 2 , (3.22))(d−2)/2 with a 0 a positive constant and m a mass scale; the combination a 0 m d−2 is just theintegration constant for the differential equation, which we prefer to split here ina momentum scale and a dimensionless coefficient for reasons that will becomeclear later. The choice of + or − sign is determined from whether one is to the left(+), or to right (-) of g c , in which case g(k 2 ) decreases or, respectively, increases asone flows away from the ultraviolet fixed point. The renormalization group invariantmass scale ∼ m arises here as an arbitrary integration constant of the renormalizationgroup equations, and cannot be determined from perturbative arguments alone. Itshould also be clear that multiplying both sides of Eq. (3.22) by the ultraviolet cutofffactor Λ 2−d to get back the original dimensionful coupling multiplying the actionS(φ) in Eq. (3.1) does not change any of the conclusions.It is important to point out that the result of Eq. (3.22) is quite different from thenaive expectation based on straight perturbation theory in d > 2 dimensions (wherethe theory is not perturbatively renormalizable)