Instantons, renormalons and the theta angle in integrable sigma models

TL;DR

This paper investigates the impact of the theta angle on non-perturbative effects in integrable sigma models, revealing how instanton and renormalon corrections are affected and providing a formula for topological susceptibility.

Contribution

It provides explicit non-perturbative corrections for integrable sigma models with a theta angle, including effects on instantons and renormalons, and derives a formula for topological susceptibility.

Findings

Instanton corrections change sign with theta angle.

Renormalon corrections remain unchanged or change sign depending on the context.

Derived an explicit formula for topological susceptibility in the O(3) sigma model.

Abstract

Some sigma models which admit a theta angle are integrable at both $\vartheta=0$ and $\vartheta=\pi$. This includes the well-known $O(3)$ sigma model and two families of coset sigma models studied by Fendley. We consider the ground state energy of these models in the presence of a magnetic field, which can be computed with the Bethe ansatz. We obtain explicit results for its non-perturbative corrections and we study the effect of the theta angle on them. We show that imaginary, exponentially small corrections due to renormalons remain unchanged, while instanton corrections change sign, as expected. We find in addition corrections due to renormalons which also change sign as we turn on the theta angle. Based on these results we present an explicit non-perturbative formula for the topological susceptibility of the $O(3)$ sigma model in the presence of a magnetic field, in the weak coupling limit.