The validity of perturbation theory for the O( N) non-linear sigma models

Abstract

Recently it has been claimed that ordinary perturbation theory (OPT) gives incorrect weak coupling expansions for lattice O( N) non-linear sigma models in the infinite volume limit, and in particular that the two-dimensional non-abelian models are not asymptotically free, contrary to previous findings. Here it is argued that the problem occurs only for one-dimensional infinite lattices, and that in general, OPT gives correct expansions if physical quantities are first computed on a finite lattice, and the infinite volume limit is taken at the end. In one dimension the expansion is sensitive to boundary conditions because of the severe infrared behavior, but this is not expected to happen in higher dimensions. It is concluded that spin configurations which are far from the perturbative vacuum have too small a measure in the path integral to invalidate OPT, even though they are energetically allowed for non-zero values of the coupling.