Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin-Barnes Representation

TL;DR

This paper introduces a non-perturbative method combining Mellin-Barnes representation and Borel resummation to derive hyperasymptotic expansions, significantly improving the accuracy of perturbative calculations in zero-dimensional φ^4 theory.

Contribution

It develops an iterative hyperasymptotic procedure using inverse factorial expansions, linking perturbative and non-perturbative results through resurgence, applicable across complex coupling phases.

Findings

Hyperasymptotic expansions outperform standard perturbation theory.

Optimal truncation schemes exponentially suppress remainders.

Method is validated numerically for improved accuracy.

Abstract

Using a method mixing Mellin-Barnes representation and Borel resummation we show how to obtain hyperasymptotic expansions from the (divergent) formal power series which follow from the perturbative evaluation of arbitrary "$N$-point" functions for the simple case of zero-dimensional $\phi^4$ field theory. This hyperasymptotic improvement appears from an iterative procedure, based on inverse factorial expansions, and gives birth to interwoven non-perturbative partial sums whose coefficients are related to the perturbative ones by an interesting resurgence phenomenon. It is a non-perturbative improvement in the sense that, for some optimal truncations of the partial sums, the remainder at a given hyperasymptotic level is exponentially suppressed compared to the remainder at the preceding hyperasymptotic level. The Mellin-Barnes representation allows our results to be automatically valid for a wide range of the phase of the complex coupling constant, including Stokes lines. A numerical analysis is performed to emphasize the improved accuracy that this method allows to reach compared to the usual perturbative approach, and the importance of hyperasymptotic optimal truncation schemes.