The Perturbative Method Fails in Non-Abelian Models

TL;DR

This paper demonstrates that perturbation theory fails in non-Abelian 2D nonlinear sigma models and gauge theories, due to boundary condition dependence and the emergence of super-instantons, highlighting limitations of perturbative approaches.

Contribution

It reveals the failure of perturbation theory in non-Abelian models and introduces super-instantons as a key factor in this breakdown.

Findings

Perturbation theory depends on boundary conditions in non-Abelian models.

Super-instantons dominate the low-temperature regime in these models.

Perturbative answers differ from true nonperturbative results in non-Abelian cases.

Abstract

It is shown that perturbation theory in $2D$ nonlinear $\sigma$-models as well gauge theories in dimension $D\geq 2$ produces answers that depend on boundary conditions even after the infinite volume limit has been taken. This unphysical phenomenon occurs only in the non-Abelian versions of those models, starting at $O(1/\beta^2)$. It is not present in the true (nonperturbatively defined) models and represents a failure of the perturbative method. It is related to a hitherto unnoticed type of low-lying excitation, dubbed super-instanton, that dominates the low-temperature (= weak coupling) regime of these models.