Analytic resurgence in the O(4) model

TL;DR

This paper analytically investigates the large order behavior and resurgence properties of the perturbative series for the ground state energy in the integrable two-dimensional O(4) sigma model, confirming previous numerical findings.

Contribution

It provides an analytical study of the resurgence phenomena in the O(4) model's perturbative expansion, including the behavior of Borel singularities and alien derivatives.

Findings

Confirmed the leading resurgence behavior analytically.

Identified the structure of Borel plane singularities.

Demonstrated resurgence in a simplified toy model.

Abstract

We study the perturbative expansion of the ground state energy in the presence of an external field coupled to a conserved charge in the integrable two-dimensional O$(4)$ nonlinear sigma model. By solving Volin's algebraic equations for the perturbative coefficients we study the large order asymptotic behaviour of the perturbative series analytically. We confirm the previously numerically found leading behaviour and study the nearest singularities of the Borel transformed series and the associated alien derivatives. We find a 'resurgence' behaviour: the leading alien derivatives can be expressed in terms of the original perturbative series. A simplified 'toy' model is also considered: here the perturbative series can be found in a closed form and the resurgence properties are very similar to that found in the real problem.