First analytic correction beyond PFA for the electromagnetic field in sphere-plane geometry

TL;DR

This paper derives the first analytic correction beyond the proximity force approximation for the electromagnetic Casimir energy in a sphere-plane setup at small separations, including logarithmic terms.

Contribution

It provides the first explicit analytic correction to the PFA for electromagnetic fields in sphere-plane geometry, revealing logarithmic contributions at next-to-leading order.

Findings

Derived the next-to-leading order correction in small separation expansion.

Identified logarithmic terms in the correction not present in scalar cases.

Compared analytic results with numerical and experimental data.

Abstract

We consider the vacuum energy for a configuration of a sphere in front of a plane, both obeying conductor boundary condition, at small separation. For the separation becoming small we derive the first next-to-leading order of the asymptotic expansion in the separation-to-radius ratio $\ep$. This correction is of order $\ep$. In opposite to the scalar cases it contains also contributions proportional to logarithms in first and second order, $\ep \ln \ep$ and $\ep (\ln \ep)^2$. We compare this result with the available findings of numerical and experimental approaches.