Resurgence and semiclassical expansion in two-dimensional large-$N$ sigma models

TL;DR

This paper investigates the resurgence structure of the 2D large-$N$ sigma model, revealing how IR cutoff scales influence ambiguities and their cancellation, and compares semiclassical and exact results, with implications for related models.

Contribution

It introduces a detailed analysis of IR cutoff effects on resurgence in 2D sigma models, showing how ambiguities evolve and cancel, and extends the framework to the ${ m C}P^{N-1}$ model with boundary conditions.

Findings

Ambiguities depend on the IR cutoff scale $a$ and are canceled when all contributions are considered.

Some ambiguities vanish as the cutoff $a$ approaches zero, altering the resurgence structure.

The resurgence structure exhibits discontinuous changes with the compactification radius in the ${ m C}P^{N-1}$ model.

Abstract

The resurgence structure of the 2d $O(N)$ sigma model at large $N$ is studied with a focus on an IR momentum cutoff scale $a$ that regularizes IR singularities in the semiclassical expansion. Transseries expressions for condensates and correlators are derived as series of the dynamical scale $\Lambda$ (nonperturbative exponential) and coupling $\lambda_{\mu}$ renormalized at the momentum scale $\mu$. While there is no ambiguity when $a > \Lambda$, we find for $a < \Lambda$ that the nonperturbative sectors have new imaginary ambiguities besides the well-known renormalon ambiguity in the perturbative sector. These ambiguities arise as a result of an analytic continuation of transseries coefficients to small values of the IR cutoff $a$ below the dynamical scale $\Lambda$. We find that the imaginary ambiguities are cancelled each other when we take all of them into account. By comparing the semiclassical expansion with the transseries for the exact large-$N$ result, we find that some ambiguities vanish in the $a \rightarrow 0$ limit and hence the resurgence structure changes when going from the semiclassical expansion to the exact result with no IR cutoff. An application of our approach to the ${\mathbb C}P^{N-1}$ sigma model is also discussed. We find in the compactified model with the ${\mathbb Z}_N$ twisted boundary condition that the resurgence structure changes discontinuously as the compactification radius is varied.