Borel Summation of the Derivative Expansion and Effective Actions

TL;DR

This paper demonstrates that the derivative expansion of the QED effective action is a divergent but Borel summable series in certain magnetic backgrounds, and explores non-perturbative effects in electric fields through Borel analysis.

Contribution

It provides an explicit demonstration of Borel summability of the derivative expansion in specific backgrounds and links divergence properties to non-perturbative phenomena like pair production.

Findings

Derivative expansion is divergent but Borel summable for magnetic backgrounds.

Borel dispersion relations reveal non-perturbative imaginary parts related to pair production.

Resummations of Borel approximations modify pair production rate exponents.

Abstract

We give an explicit demonstration that the derivative expansion of the QED effective action is a divergent but Borel summable asymptotic series, for a particular inhomogeneous background magnetic field. A duality transformation B\to iE gives a non-Borel-summable perturbative series for a time dependent background electric field, and Borel dispersion relations yield the non-perturbative imaginary part of the effective action, which determines the pair production probability. Resummations of leading Borel approximations exponentiate to give perturbative corrections to the exponents in the non-perturbative pair production rates. Comparison with a WKB analysis suggests that these divergence properties are general features of derivative expansions and effective actions.