Some new results on an old controversy: is perturbation theory the correct asymptotic expansion in nonabelian models?

TL;DR

This paper investigates whether perturbation theory provides the correct asymptotic expansion in nonabelian models, addressing boundary condition dependence and its implications for the validity of perturbative methods.

Contribution

It presents computational results supporting the correctness of perturbation theory with conventional boundary conditions in nonabelian models.

Findings

Perturbation theory may be valid with standard boundary conditions.

Boundary condition dependence persists even in the infinite volume limit.

The fundamental question of perturbation theory's correctness remains open.

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 $\beta$ function remain challenging problems of mathematical physics.