Instanton effects vs resurgence in the $O(3)$ sigma model

TL;DR

This paper examines the ground-state energy of the 2D $O(3)$ sigma model, analyzing perturbative and non-perturbative effects, and finds that instantons and renormalons contribute differently to the model's asymptotics.

Contribution

It provides a detailed comparison between instanton effects and resurgence phenomena in the $O(3)$ sigma model, highlighting the distinct origins of exponential contributions.

Findings

Leading non-perturbative contributions match analytical calculations.

Perturbative coefficients' asymptotics are linked to renormalons.

Instanton contributions are confirmed through analytical expansion.

Abstract

We investigate the ground-state energy of the integrable two dimensional $O(3)$ sigma model in a magnetic field. By determining a large number of perturbative coefficients we explore the closest singularities of the corresponding Borel function. We then confront its median resummation to the high precision numerical solution of the exact integral equation and observe that the leading exponentially suppressed contribution is not related to the asymptotics of the perturbative coefficients. By analytically expanding the integral equation we calculate the leading non-perturbative contributions up to fourth order and find complete agreement. These anomalous terms could be attributed to instantons, while the asymptotics of the perturbative coefficients seems to be related to renormalons.