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

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 $(d\mathrm{/}R{)}^{3\mathrm{/}2}\log (d\mathrm{/}R)$, where $d$ is the separation and $R$ the sphere radius.