Renormalization and vacuum energy for an interacting scalar field in a \delta-function potential

TL;DR

This paper investigates the renormalization of a self-interacting scalar field with a delta-function potential, analyzing vacuum energy and effective potential in complex spacetime, highlighting non-analytic behaviors and the importance of surface interactions.

Contribution

It provides a two-loop renormalization analysis for scalar fields with delta-function potentials and explores the effective potential in non-simply connected spacetime, emphasizing non-analytic effects.

Findings

Counterterms are computed using dimensional regularization.

Effective potential depends on boundary condition phase factors.

Non-analytic behavior invalidates standard weak field perturbation theory.

Abstract

We study a self-interacting scalar field theory in the presence of a \delta-function background potential. The role of surface interactions in obtaining a renormalizable theory is stressed and demonstrated by a two-loop calculation. The necessary counterterms are evaluated by adopting dimensional regularization and the background field method. We also calculate the effective potential for a complex scalar field in a non-simply connected spacetime in the presence of a \delta-function potential. The effective potential is evaluated as a function of an arbitrary phase factor associated with the choice of boundary conditions in the non-simply connected spacetime. We obtain asymptotic expansions of the results for both large and small \delta-function strengths, and stress how the non-analytic nature of the small strength result vitiates any analysis based on standard weak field perturbation theory.