Casimir energy of a dilute dielectric ball in the mode summation method

TL;DR

This paper derives the Casimir energy of a dilute dielectric ball using mode summation, employing Bessel function addition theorem and subtraction techniques to clarify previous ambiguities.

Contribution

It presents a simple, closed-form calculation of the Casimir energy for a dilute dielectric ball in the $( ext{epsilon}_1- ext{epsilon}_2)^2$ approximation, clarifying the role of contact terms.

Findings

Derived a closed-form expression for Casimir energy.

Clarified the role of contact terms in the calculation.

Removed linear in $( ext{epsilon}_1- ext{epsilon}_2)$ contributions.

Abstract

In the $(\epsilon_1-\epsilon_2)^2$--approximation the Casimir energy of a dilute dielectric ball is derived using a simple and clear method of the mode summation. The addition theorem for the Bessel functions enables one to present in a closed form the sum over the angular momentum before the integration over the imaginary frequencies. The linear in $(\epsilon_1-\epsilon_2)$ contribution into the vacuum energy is removed by an appropriate subtraction. The role of the contact terms used in other approaches to this problem is elucidated.