Running coupling and non-perturbative corrections for O$(N)$ free energy and for disk capacitor

TL;DR

This paper analyzes the $O(N)$ nonlinear sigma models' energy density at finite density using Wiener-Hopf, incorporating all perturbative and non-perturbative effects, and explores resurgence theory's applicability to these solutions.

Contribution

It introduces a running coupling to represent asymptotic series as pure power series and investigates non-perturbative effects and resurgence theory in the $O(3)$ model and capacitor analogy.

Findings

Wiener-Hopf method effectively captures all contributions.

Resurgence theory fails for the $O(3)$ case due to instanton effects.

Resurgence with median resummation correctly predicts non-perturbative terms in the capacitor problem.

Abstract

We reconsider the complete solution of the linear TBA equation describing the energy density of finite density states in the $O(N)$ nonlinear sigma models by the Wiener-Hopf method. We keep all perturbative and non-perturbative contributions and introduce a running coupling in terms of which all asymptotic series appearing in the problem can be represented as pure power series without logs. We work out the first non-perturbative contribution in the $O(3)$ case and show that (presumably because of the instanton corrections) resurgence theory fails in this example. Using the relation of the $O(3)$ problem to the coaxial disks capacitor problem we work out the leading non-perturbative terms for the latter and show that (at least to this order) resurgence theory, in particular the median resummation prescription, gives the correct answer. We demonstrate this by comparing the Wiener-Hopf results to the high precision numerical solution of the original integral equation.