Resurgence in sine-Gordon quantum mechanics: Exact agreement between multi-instantons and uniform WKB

TL;DR

This paper calculates multi-instanton effects in sine-Gordon quantum mechanics, demonstrating exact agreement with uniform WKB results and confirming the resurgence phenomenon by showing how nonperturbative contributions cancel ambiguities in perturbation series.

Contribution

It extends the Bogomolnyi--Zinn-Justin prescription to multi-instanton configurations and provides a unified framework for exact nonperturbative calculations in sine-Gordon quantum mechanics.

Findings

Multi-instanton amplitudes computed with a new subtraction scheme.

Exact cancellation of imaginary parts confirms resurgence.

Quantitative corrections to neutral bion contributions in ${ m C}P^{N-1}$ model.

Abstract

We compute multi-instanton amplitudes in the sine-Gordon quantum mechanics (periodic cosine potential) by integrating out quasi-moduli parameters corresponding to separations of instantons and anti-instantons. We propose an extension of Bogomolnyi--Zinn-Justin prescription for multi-instanton configurations and an appropriate subtraction scheme. We obtain the multi-instanton contributions to the energy eigenvalue of the lowest band at the zeroth order of the coupling constant. For the configurations with only instantons (anti-instantons), we obtain unambiguous results. For those with both instantons and anti-instantons, we obtain results with imaginary parts, which depend on the path of analytic continuation. We show that the imaginary parts of the multi-instanton amplitudes precisely cancel the imaginary parts of the Borel resummation of the perturbation series, and verify that our results completely agree with those based on the uniform-WKB calculations, thus confirming the resurgence : divergent perturbation series combined with the nonperturbative multi-instanton contributions conspire to give unambiguous results. We also study the neutral bion contributions in the ${\mathbb C}P^{N-1}$ model on ${\mathbb R}^1\times S^{1}$ with a small circumference, taking account of the relative phase moduli between the fractional instanton and anti-instanton. We find that the sign of the interaction potential depends on the relative phase moduli, and that both the real and imaginary parts resulting from quasi-moduli integral of the neutral bion get quantitative corrections compared to the sine-Gordon quantum mechanics.