Some new results on an old controversy: Is perturbation theory the correct asymptotic expansion in non-Abelian models?

Abstract

Several years ago it was found that perturbation theory for two-dimensional $O(N)$ models depends on boundary conditions even after the infinite volume limit has been taken termwise, provided $N>2$. There ensued a discussion whether the boundary conditions introduced to show this phenomenon were somehow anomalous and there was a class of ``reasonable'' boundary conditions not suffering from this ambiguity. Here we present the results of some computations that may be interpreted as giving some support for the correctness of perturbation theory with conventional boundary conditions; however the fundamental underlying question of the correctness of perturbation theory in these models and, in particular, the perturbative $\ensuremath{\beta}$ function remain challenging problems of mathematical physics.