Feynman Path Integral Formulation

3.2 Perturbatively Non-renormalizable Theories: The Sigma Model 71(a) (b) (c)Fig. 3.1 One-loop diagrams giving rise to coupling and field renormalizations in the non-linearσ-model. Group theory indices a flow along the thick lines, dashed lines should be contracted to apoint.experience rapid small fluctuations perpendicular to its average slow motion on theunit sphere, and that only these fluctuations contribute to leading order.In two dimensions the quantum correction (the second term on the r.h.s.) increasesthe value of the effective coupling at low momenta (large distances), unlessN = 2 in which case the correction vanishes. In fact the quantum correction can beshown to vanish to all orders in this case; the vanishing of the β-function in twodimensions for the O(2) model is true only in perturbation theory, for sufficientlystrong coupling a phase transition appears, driven by the unbinding of vortex pairs(Kosterlitz and Thouless, 1973). For N > 2asg(μ) flows toward increasingly strongcoupling it eventually leaves the regime where perturbation theory can be consideredreliable. But for bare g ≈ 0 the quantum correction is negligible and the theoryis scale invariant around the origin: the only fixed point of the renormalizationgroup, at least in lowest order perturbation theory, is at g = 0. For fixed cutoff Λ,thetheory is weakly coupled at short distances but strongly coupled at large distances.The results in two dimensions for N > 2 are qualitatively very similar to asymptoticfreedom in four-dimensional SU(N) Yang-Mills theories.Above two dimensions, d − 2 = ε > 0 and one can redo the same type of perturbativecalculation to determine the coupling renormalization. The relevant diagramsare shown in Fig. 3.1. One finds for the effective coupling g e , i.e. the coupling whichincludes the leading radiative correction (using dimensional regularization, which ismore convenient than an explicit ultraviolet cutoff Λ for performing actual perturbativecalculations),1g e= Λ εg[1 − 1 ]N − 2ε 2π g + O(g2 ). (3.13)The requirement that the dimensionful effective coupling g e be defined independentlyof the scale Λ is expressed as ΛdΛ d g e = 0, and gives for the Callan-Symanzikβ-function (Callan, 1970; Symanzik, 1970) for g