  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.
