Resurgence, a problem of missing exponential corrections in asymptotic expansions

TL;DR

This paper introduces a modified Borel summation method using a generalized Borel transform and directional Laplace transform to better capture exponential corrections and resolve ambiguities in asymptotic expansions of path integrals.

Contribution

It proposes a new approach to Borel summation that accounts for exponential corrections and formalizes resurgence as a connection between discontinuities and asymptotic coefficients.

Findings

New generalized Borel transform improves summation accuracy.

Resurgence is formalized as a link between discontinuities and asymptotic coefficients.

Method reduces ambiguities in Borel summation.

Abstract

It is well known that perturbative expansions of path integrals are divergent. These expansions are to be understood as asymptotic expansions, which encode the limiting behaviour of the path integral for positive small coupling. Conventionally, the method of Borel summation assigns a finite answer to the divergent expansion. Still, the Borel sum might not encode the full information of a function, because it misses exponentially small corrections. In the present work, we consider a slight variation of the conventional Borel summation, in which a generalised Borel transform (an inverse Laplace transform) is followed by a directional Laplace transform. These new tools will allow us to give perhaps better answers to typical problems in Borel summation: missing exponential corrections and ambiguities in the Borel summation. In addition, we will define resurgence as a connection between the discontinuity of a function and the coefficients of its asymptotic expansion. From this definition, we will be able to reduce resurgence to the problem of missing exponential corrections in asymptotic expansions and understand, within a unified framework, different approaches to resurgence found in the literature.