New asymptotic techniques for the partial wave cut-off method for calculating the QED one loop effective action

TL;DR

This paper introduces two new asymptotic techniques to approximate solutions of differential equations in the partial-wave-cutoff method, enabling more efficient calculation of the QED one-loop effective action for symmetric backgrounds.

Contribution

The paper develops and tests two complementary asymptotic methods that provide approximate analytical solutions, improving the implementation of the partial-wave-cutoff method in QED calculations.

Findings

Asymptotic methods yield accurate effective action calculations.

Methods are effective across various background fields and mass regimes.

Analysis of the massless limit reveals divergence structures.

Abstract

The Gel'fand-Yaglom theorem has been used to calculate the one-loop effective action in quantum field theory by means of the "partial-wave-cutoff method". This method works well for a wide class of background fields and is essentially exact. However, its implementation has been semi-analytical so far since it involves solving a non-linear ordinary differential equation for which solutions are in general unknown. Within the context of quantum electrodynamics (QED) and $O(2)\times O(3)$ symmetric backgrounds, we present two complementary asymptotic methods that provide approximate analytical solutions to this equation. We test these approximations for different background field configurations and mass regimes and demonstrate that the effective action can indeed be calculated with good accuracy using these asymptotic expressions. To further probe these methods, we analyze the massless limit of the effective action and obtain its divergence structure with respect to the radial suppression parameter of the background field, comparing our findings with previously reported results.