Questionable and unquestionable in the perturbation theory of non-Abelian models

Abstract

We show, by explicit computation, that bare lattice perturbation theory in the two-dimensional O(n) nonlinear $\sigma$ models with superinstanton boundary conditions is divergent in the limit of an infinite number of points $|\Lambda|$. This is the analogue of David's statement that renormalized perturbation theory of these models is infrared divergent in the limit where the physical size of the box tends to infinity. We also give arguments which support the validity of the bare perturbative expansion of short-distance quantities obtained by taking the limit $|\Lambda|\to\infty$ term by term in the theory with more conventional boundary conditions such as Dirichlet, periodic, and free.