Massive scalar Casimir interaction beyond proximity force approximation

TL;DR

This paper investigates the Casimir interaction between a sphere and a plate mediated by a massive scalar field, deriving formulas for energies beyond the proximity force approximation and analyzing mass effects.

Contribution

It extends the TGTG formula to massive scalar fields, providing new asymptotic expansions and insights into mass dependence of Casimir interactions.

Findings

Derived TGTG formulas for massive scalar interactions.

Obtained small separation asymptotic expansions including next-to-leading order.

Analyzed the complex small mass asymptotic behavior with odd powers and logarithmic terms.

Abstract

Since massive scalar field plays an important role in theoretical physics, we consider the interaction between a sphere and a plate due to the vacuum fluctuation of a massive scalar field. We consider combinations of Dirichlet and Neumann boundary conditions. There is a simple prescription to obtain the functional formulas for the Casimir interaction energies, known as TGTG formula, for the massive interactions from the massless interactions. From the TGTG formulas, we discuss how to compute the small separation asymptotic expansions of the Casimir interaction energies up to the next-to-leading order terms. Unlike the massless case, the results could not be expressed as simple algebraic expressions, but instead could only be expressed as infinite sums over some integrals. Nonetheless, it is easy to show that one can obtain the massless limits which agree with previously established results. We also show that the leading terms agree with that derive using proximity force approximation. The dependence of the leading order terms and the next-to-leading order terms on the mass of the scalar field is studied both numerically and analytically. In particular, we derive the small mass asymptotic expansions of these terms. Surprisingly, the small mass asymptotic expansions are quite complicated as they contain terms that are of odd powers in mass as well as logarithms of mass terms.