Resurgence and $1/N$ Expansion in Integrable Field Theories

TL;DR

This paper investigates the relationship between resurgence theory and the $1/N$ expansion in integrable field theories with renormalons, revealing cases where resurgent trans-series fully decode the $1/N$ corrections and others where non-perturbative effects are not captured.

Contribution

It provides explicit solutions for the large $N$ free energy in integrable models, demonstrating the interplay between resurgence and $1/N$ expansions, including new analytical results and insights into non-perturbative corrections.

Findings

Resurgent trans-series fully decode $1/N$ corrections in some models.

Explicit median resummation solution for the principal chiral field.

Non-perturbative corrections in the Gross-Neveu model are not predicted by resurgence.

Abstract

In theories with renormalons the perturbative series is factorially divergent even after restricting to a given order in $1/N$, making the $1/N$ expansion a natural testing ground for the theory of resurgence. We study in detail the interplay between resurgent properties and the $1/N$ expansion in various integrable field theories with renormalons. We focus on the free energy in the presence of a chemical potential coupled to a conserved charge, which can be computed exactly with the thermodynamic Bethe ansatz (TBA). In some examples, like the first $1/N$ correction to the free energy in the non-linear sigma model, the terms in the $1/N$ expansion can be fully decoded in terms of a resurgent trans-series in the coupling constant. In the principal chiral field we find a new, explicit solution for the large $N$ free energy which can be written as the median resummation of a trans-series with infinitely many, analytically computable IR renormalon corrections. However, in other examples, like the Gross-Neveu model, each term in the $1/N$ expansion includes non-perturbative corrections which can not be predicted by a resurgent analysis of the corresponding perturbative series. We also study the properties of the series in $1/N$. In the Gross-Neveu model, where this is convergent, we analytically continue the series beyond its radius of convergence and show how the continuation matches with known dualities with sine-Gordon theories.