The heat kernel coefficients for the dielectric cylinder

TL;DR

This paper computes the heat kernel coefficients for electromagnetic fields around a dielectric cylinder, revealing that the vacuum energy vanishes at dilute order but not beyond, confirming previous findings through a new approach.

Contribution

It provides the first calculation of heat kernel coefficients for a dielectric cylinder, demonstrating the behavior of vacuum energy at different orders.

Findings

The coefficient $a_{2}$ is zero at order $( ext{permittivity}-1)^2$

Vacuum energy vanishes at dilute order

Vanishing of vacuum energy is method-independent

Abstract

We calculate the \hkks for the \elm field in the background of a dielectric cylinder with non equal speeds of light inside and outside. The coefficient $a_{2}$ whose vanishing makes the vacuum energy of a massless field unique, turns out to be zero in dilute order, i.e., in order $(\ep-1)^{2}$, and nonzero beyond. As a consequence, the vanishing of the vacuum energy in the presence of a dielectric cylinder found by Casimir-Polder summation must take place irrespectively of the methods by which it might be calculated.