Fast Summation of Divergent Series and Resurgent Transseries in Quantum Field Theories from Meijer-G Approximants

TL;DR

This paper introduces a Meijer-G-function-based resummation method that significantly improves the approximation of divergent series and resurgent transseries in quantum field theories over traditional techniques.

Contribution

The paper presents a novel Meijer-G approximant approach that generalizes Borel-Hypergeometric methods for better resummation of divergent series in QFT.

Findings

Outperforms Borel-Padé and Borel-Padé-Écalle methods

Successfully applied to various QFT models

Accurately computes instanton contributions

Abstract

We demonstrate that a Meijer-G-function-based resummation approach can be successfully applied to approximate the Borel sum of divergent series, and thus to approximate the Borel-\'Ecalle summation of resurgent transseries in quantum field theory (QFT). The proposed method is shown to vastly outperform the conventional Borel-Pad\'e and Borel-Pad\'e-\'Ecalle summation methods. The resulting Meijer-G approximants are easily parameterized by means of a hypergeometric ansatz and can be thought of as a generalization to arbitrary order of the Borel-Hypergeometric method [Mera {\it et al.} Phys. Rev. Lett. {\bf 115}, 143001 (2015)]. Here we illustrate the ability of this technique in various examples from QFT, traditionally employed as benchmark models for resummation, such as: 0-dimensional $\phi^4$ theory, $\phi^4$ with degenerate minima, self-interacting QFT in 0-dimensions, and the computation of one- and two-instanton contributions in the quantum-mechanical double-well problem.