Borel summability of perturbative series in 4d N=2 and 5d N=1 theories

TL;DR

This paper proves that perturbative series in 4d N=2 and 5d N=1 supersymmetric gauge theories are Borel summable in various sectors, enabling exact results through Borel resummation, and supports a semiclassical interpretation of infrared renormalons.

Contribution

It demonstrates Borel summability of perturbative series in multiple instanton sectors for these theories, providing a rigorous foundation for exact calculations from perturbation theory.

Findings

Perturbative series are Borel summable in zero instanton sector.

Perturbative series are Borel summable in arbitrary instanton sectors.

Exact results can be obtained by summing Borel resummations across sectors.

Abstract

We study weak coupling perturbative series in 4d N=2 and 5d N=1 supersymmetric gauge theories with Lagrangians. We prove that the perturbative series of these theories in zero instanton sector are Borel summable for various observables. Our result for 4d $\mathcal{N}=2$ case supports an expectation from a recent proposal on a semiclassical realization of infrared renormalons in QCD-like theories, where the semiclassical solution does not exist in N=2 theories and the perturbative series are unambiguous, namely Borel summable. We also prove that the perturbative series in arbitrary number of instanton sector are Borel summable for a wide class of theories. It turns out that exact results can be obtained by summing over the Borel resummations in each number of instanton sector.