Sphere-plate Casimir interaction in (D+1)-dimensional spacetime

In this paper, we derive the formula for the Casimir interaction energy between a sphere and a plate in $(D+1)$-dimensional Minkowski spacetime. It is assumed that the scalar field satisfies the Dirichlet or Neumann boundary conditions on the sphere and the plate. As in the $D=3$ case, the formula is of TGTG type. One of our main contributions is deriving the translation matrices which express the change of bases between plane waves and spherical waves for general $D$. Using orthogonality of Gegenbauer polynomials, it turns out that the final TGTG formula for the Casimir interaction energy can be simplified to one that is similar to the $D=3$ case. To illustrate the application of the formula, both large separation and small separation asymptotic behaviors of the Casimir interaction energy are computed. The large separation leading term is proportional to $L^{-D+1}$ if the sphere is imposed with Dirichlet boundary condition, and to $L^{-D-1}$ if the sphere is imposed with Neumann boundary condition, where $L$ is distance from the center of the sphere to the plane. For the small separation asymptotic behavior, it is shown that the leading term is equal to the one obtained using proximity force approximation. The next-to-leading order term is also computed using perturbation method. It is shown that when the space dimension $D$ is larger than 5, the next-to-leading order has sign opposite to the leading order term. Moreover, the ratio of the next-to-leading order term to the leading order term is linear in $D$, indicating a larger correction at higher dimensions.