One-loop Effective Actions in Shape-invariant Scalar Backgrounds

TL;DR

This paper develops a recursive method to compute the one-loop effective action in scalar backgrounds with shape-invariance, effectively handling continuum modes and applying to domain-wall backgrounds in various dimensions.

Contribution

It introduces a recursive procedure for calculating one-loop effective actions in shape-invariant backgrounds, addressing continuum mode subtleties using the regularized Witten index.

Findings

Provides full expressions for effective actions in (1+1) and (3+1) dimensions.

Applies to domain-wall-type scalar backgrounds.

Handles continuum mode contributions effectively.

Abstract

The field-theoretic one-loop effective action in a static scalar background depending nontrivially on a single spatial coordinate is related, in the proper-time formalism, to the trace of the evolution kernel (or heat kernel) for an appropriate, one dimensional, quantum-mechanical Hamiltonian. We describe a recursive procedure applicable to these traces for shape-invariant Hamiltonians, resolving subtleties from the continuum mode contributions by utilizing the expression for the regularized Witten index. For some cases which include those of domain-wall-type scalar backgrounds, our recursive procedure yields the full expression for the scalar or fermion one-loop effective action in both (1+1) and (3+1)-dimensions.