The ground state energy of a massive scalar field in the background of a semi-transparent spherical shell

TL;DR

This paper computes the zero point energy of a massive scalar field around a semi-transparent spherical shell using zeta regularization, deriving analytical expressions, renormalizing, and providing numerical results for various potential strengths.

Contribution

It introduces a method to calculate and renormalize the ground state energy of a scalar field in a spherical shell background using zeta functional regularization and scattering theory.

Findings

Numerical ground state energy results for different potential strengths.

Analytical expressions involving the Jost function.

Renormalization scheme ensuring energy vanishes for large mass.

Abstract

We calculate the zero point energy of a massive scalar field in the background of an infinitely thin spherical shell given by a potential of the delta function type. We use zeta functional regularization and express the regularized ground state energy in terms of the Jost function of the related scattering problem. Then we find the corresponding heat kernel coefficients and perform the renormalization, imposing the normalization condition that the ground state energy vanishes when the mass of the quantum field becomes large. Finally the ground state energy is calculated numerically. Corresponding plots are given for different values of the strength of the background potential, for both attractive and repulsive potentials.