On the ground state energy for a penetrable sphere and for a dielectric ball

TL;DR

This paper investigates the ultraviolet divergences in the ground state energy of penetrable spheres and dielectric balls, revealing that certain configurations with non-zero heat kernel coefficient a_2 require additional regularization for finiteness.

Contribution

It demonstrates that for penetrable spheres and dielectric balls, the usual subtraction methods are insufficient due to non-zero heat kernel coefficient a_2, and shows that singular configurations can sometimes be better behaved.

Findings

Non-zero a_2 coefficient for penetrable sphere and dielectric ball.

Standard subtraction does not always regularize the energy.

Singular configurations like the dilute dielectric ball can be well-defined.

Abstract

We analyse the ultraviolet divergencies in the ground state energy for a penetrable sphere and a dielectric ball. We argue that for massless fields subtraction of the ``empty space'' or the ``unbounded medium'' contribution is not enough to make the ground state energy finite whenever the heat kernel coefficient $a_2$ is not zero. It turns out that $a_2\ne 0$ for a penetrable sphere, a general dielectric background and the dielectric ball. To our surprise, for more singular configurations, as in the presence of sharp boundaries, the heat kernel coefficients behave to some extend better than in the corresponding smooth cases, making, for instance, the dilute dielectric ball a well defined problem.