Renormalization for a Scalar Field in an External Scalar Potential

TL;DR

This paper applies renormalization techniques to a scalar field interacting with an external potential, identifying divergences and demonstrating their absorption into potential coupling constants, advancing understanding of boundary divergences.

Contribution

It extends renormalization methods to scalar fields in external potentials, showing how divergences are absorbed into potential parameters and exploring residual dependencies.

Findings

Divergences are isolated into covariant functionals of the potential.

Divergences can be absorbed into coupling constants of the potential.

Residual logarithmic dependence on the potential remains, akin to renormalization group effects.

Abstract

The Pauli--Villars regularization procedure confirms and sharpens the conclusions reached previously by covariant point splitting. The divergences in the stress tensor of a quantized scalar field interacting with a static scalar potential are isolated into a three-parameter local, covariant functional of the background potential. These divergences can be naturally absorbed into coupling constants of the potential, regarded as a dynamical object in its own right; here this is demonstrated in detail for two different models of the field-potential coupling. here is a residual dependence on the logarithm of the potential, reminiscent of the renormalization group in fully interacting quantum field theories; these terms are finite but numerically dependent on an arbitrary mass or length parameter, which is purely a matter of convention. This work is one step in a program to elucidate boundary divergences by replacing a sharp boundary by a steeply rising smooth potential.