Renormalized Effective Actions in Radially Symmetric Backgrounds I: Partial Wave Cutoff Method

TL;DR

This paper introduces a novel renormalization technique combining the Schwinger proper-time method and DeWitt expansion to compute finite one-loop effective actions in radially symmetric backgrounds, improving efficiency over previous methods.

Contribution

It develops an elegant, unified approach for extracting large partial-wave contributions in effective action calculations, applicable to complex gauge field backgrounds.

Findings

Efficient evaluation of radial determinants using Gel'fand-Yaglom method.

A new renormalization procedure combining Schwinger proper-time and DeWitt expansion.

Applicability to SU(2) Yang-Mills backgrounds and instanton configurations.

Abstract

The computation of the one-loop effective action in a radially symmetric background can be reduced to a sum over partial-wave contributions, each of which is the logarithm of an appropriate one-dimensional radial determinant. While these individual radial determinants can be evaluated simply and efficiently using the Gel'fand-Yaglom method, the sum over all partial-wave contributions diverges. A renormalization procedure is needed to unambiguously define the finite renormalized effective action. Here we use a combination of the Schwinger proper-time method, and a resummed uniform DeWitt expansion. This provides a more elegant technique for extracting the large partial-wave contribution, compared to the higher order radial WKB approach which had been used in previous work. We illustrate the general method with a complete analysis of the scalar one-loop effective action in a class of radially separable SU(2) Yang-Mills background fields. We also show that this method can be applied to the case where the background gauge fields have asymptotic limits appropriate to uniform field strengths, such as for example in the Minkowski solution, which describes an instanton immersed in a constant background. Detailed numerical results will be presented in a sequel.