Infrared Behaviour of Systems With Goldstone Bosons

Abstract

We develop various complementary concepts and techniques for handling quantum fluctuations of Goldstone bosons.We emphasise that one of the consequences of the masslessness of Goldstone bosons is that the longitudinal fluctuations also have a diverging susceptibility characterised by an anomalous dimension $(d-2)$ in space-time dimensions $2<d<4$.In $d=4$ these fluctuations diverge logarithmically in the infrared region.We show the generality of this phenomenon by providing three arguments based on i). Renormalization group flows, ii). Ward identities, and iii). Schwinger-Dyson equations.We obtain an explicit form for the generating functional of one-particle irreducible vertices of the O(N) (non)--linear $σ$--models in the leading 1/N approximation.We show that this incorporates all infrared behaviour correctly both in linear and non-linear $σ$-- models. Our techniques provide an alternative to chiral perturbation theory.Some consequences are discussed briefly.