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

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.