Calculation of renormalized fermion effective actions in radially symmetric non-Abelian backgrounds

TL;DR

This paper extends a method to calculate fermion one-loop effective actions in radially symmetric non-Abelian backgrounds, providing semi-analytical and numerical results for massless and massive cases, and analyzing instanton interactions.

Contribution

It introduces an extended partial wave cutoff method for 4-D fermion effective actions in non-Abelian backgrounds, including semi-analytical evaluation in the massless limit.

Findings

Explicit calculation of fermion effective action in certain non-Abelian backgrounds

Validation of large mass expansion against numerical results

Analysis of fermion mass dependence on instanton-antiinstanton interactions

Abstract

Our recent method to calculate renormalized functional determinants, the partial wave cutoff method, is extended for the evaluation of 4-D fermion one-loop effective action with arbitrary mass in certain types of radially symmetric, non-Abelian, background gauge fields (including instanton-like and instanton-antiinstanton-like configurations). A detailed study on functional determinants for matrix-valued radial differential operators is presented, explicating both our analytic treatment on the high partial wave contribution and the application of the generalized Gel'fand-Yaglom formula to determine the low partial wave contribution. In general, some numerical work is needed for the low partial wave part. In the massless limit, however, the factorizable nature of our partial-wave radial differential operators can be exploited to evaluate semi-analytically even the low partial wave part, and we thus have the full fermion effective action calculated explicitly in a class of non-Abelian background gauge fields. With nonzero mass, we also perform necessary numerical analysis as regards the low partial wave contribution to produce numerically exact results for the massive effective action. Comparing these against the results of the large mass expansion, the validity range of the large mass expansion is addressed. Also studied is the fermion mass dependence of the effective instanton-antiinstanton interaction.