One-Loop Effective Action on the Four-Ball

TL;DR

This paper uses $$-function regularization to compute the 1-loop effective action for scalar and Maxwell theories with boundaries, analyzing mode contributions under magnetic boundary conditions.

Contribution

It compares recent techniques for $$-function regularization and explicitly derives mode contributions to the effective action in boundary conditions, revealing no cancellations among modes.

Findings

All perturbative modes contribute to $'(0)$ without cancellation.

The analysis is performed for magnetic boundary conditions in Maxwell theory.

Complete mode contributions are explicitly derived and compared.

Abstract

This paper applies $\zeta$-function regularization to evaluate the 1-loop effective action for scalar field theories and Euclidean Maxwell theory in the presence of boundaries. After a comparison of two techniques developed in the recent literature, vacuum Maxwell theory is studied and the contribution of all perturbative modes to $\zeta'(0)$ is derived: transverse, longitudinal and normal modes of the electromagnetic potential, jointly with ghost modes. The analysis is performed on imposing magnetic boundary conditions, when the Faddeev-Popov Euclidean action contains the particular gauge-averaging term which leads to a complete decoupling of all perturbative modes. It is shown that there is no cancellation of the contributions to $\zeta'(0)$ resulting from longitudinal, normal and ghost modes.