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

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 $a\to 0$ 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.

arXiv comment37 pages

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• Exact WKB in all sectors II: Potentials with non-degenerate saddles