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The Gel'fand-Yaglom theorem has been used to calculate the one-loop effectiveaction 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 essentiallyexact. However, its implementation has been semi-analytical so far since itinvolves solving a non-linear ordinary differential equation for whichsolutions are in general unknown. Within the context of quantum electrodynamics(QED) and $O(2)\times O(3)$ symmetric backgrounds, we present two complementaryasymptotic methods that provide approximate analytical solutions to thisequation. We test these approximations for different background fieldconfigurations and mass regimes and demonstrate that the effective action canindeed be calculated with good accuracy using these asymptotic expressions. Tofurther probe these methods, we analyze the massless limit of the effectiveaction and obtain its divergence structure with respect to the radialsuppression parameter of the background field, comparing our findings withpreviously reported results.

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