Non-linear σ models: A perturbative approach to symmetry restoration

We examine the extent to which pertubation theory, about the naive Goldstone vacuum in two dimensions, is able to reveal the true state of affairs - that there are no Goldstone modes and that the symmetry is restored for arbitrarily small coupling constant. We exploit the recent observation, by Elitzur, of the infrared finiteness of globally invariant Green functions to compute them to third non-trivial order in a variety of models. Applying the renormalization group to improve these calculations we argue that one can see cluster decomposition beginning to restore the symmetry. The perturbation theory's ignorance of the global topology means that we cannot follow the decay of the Green function out to infinity because of a "non-perturbative barrier". We comment on the relevance of this to "perturbative confinement" in QCD. We also examine the way in which the global topology influences the phase diagrams when we add the two-dimensional analogue of Higgs fields.