Uncovering Hidden Patterns: Approximate Resurgent Resummation from Truncated Series

TL;DR

This paper develops a method to analyze and resum divergent series solutions of non-linear differential equations using Borel-Padé approximants and resurgent analysis, with applications in quantum field theory.

Contribution

It introduces an approximate resummation technique for divergent series based on resurgent properties and classical analysis, applicable to non-linear differential equations in physics.

Findings

Effective approximation of Borel-Ecalle resummation for divergent series

Insight into the analytic structure of solutions to non-linear ODEs

Potential applications in quantum field theory and related areas

Abstract

We analyze truncated series generated as divergent formal solutions of non-linear ordinary differential equations. Motivating the study is a specific non-linear, first-order differential equation, which is the basis of the resurgent formulation of renormalized perturbation theory in quantum field theory. We use the Borel-Pad\'e approximant and classical analysis to determine the analytic structure of the solution using the first few terms of its asymptotic series. Afterward, we build an approximant, consistent with the resurgent properties of the equation. The procedure gives an approximate expression for the Borel-Ecalle resummation of the solution useful for practical applications. Connections with other physical applications are also discussed.