Exact Casimir interaction of perfectly conducting three-spheres in four euclidean dimensions

TL;DR

This paper derives an exact formula for the electromagnetic Casimir interaction between two perfectly conducting three-spheres in four-dimensional Euclidean space, confirming known approximations and revealing a non-analytic correction at next-to-next-to-leading order.

Contribution

It provides the first exact analytical expression for the Casimir interaction of three-spheres in four dimensions, including validation of approximation methods and discovery of a non-analytic correction.

Findings

Exact Casimir interaction formula derived

Short distance expansion matches PFA at leading order

Non-analytic correction appears at next-to-next-to-leading order

Abstract

Exploiting conformal symmetry, we derive a simple exact formula for the classical electromagnetic Casimir interaction of two perfectly conducting three-spheres, including the sphere-plate geometry as a special case, in four euclidean dimensions. We verify that the short distance expansion of the Casimir energy agrees to leading order with the Proximity Force Approximation (PFA), while the next-to-leading-order is in agreement with a recently proposed derivative expansion of the Casimir energy. At the next-to-next-to-leading order we find a non-analytic correction to PFA, which for a sphere-plate system is of the order of $(d/R)^{3/2} \log(d/R)$, where $d$ is the separation and $R$ the sphere radius.