Three dimensional Casimir piston for massive scalar fields

TL;DR

This paper analyzes the Casimir force on a three-dimensional rectangular piston due to a massive scalar field, deriving explicit finite expressions and examining how the force varies with distance, boundary conditions, and mass.

Contribution

It provides explicit formulas for the Casimir force in a 3D piston setup with massive scalar fields under various boundary conditions, showing the force's behavior and mass effects.

Findings

Casimir force is always attractive under studied conditions.

Force magnitude behaves like 1/a^4 as a approaches zero.

Mass reduces the force magnitude at large distances.

Abstract

We consider Casimir force acting on a three dimensional rectangular piston due to a massive scalar field subject to periodic, Dirichlet and Neumann boundary conditions. Exponential cut-off method is used to derive the Casimir energy in the interior region and the exterior region separated by the piston. It is shown that the divergent term of the Casimir force acting on the piston due to the interior region cancels with that due to the exterior region, thus render a finite well-defined Casimir force acting on the piston. Explicit expressions for the total Casimir force acting on the piston is derived, which show that the Casimir force is always attractive for all the different boundary conditions considered. As a function of a -- the distance from the piston to the opposite wall, it is found that the magnitude of the Casimir force behaves like $1/a^4$ when $a\to 0^+$ and decays exponentially when $a\to \infty$. Moreover, the magnitude of the Casimir force is always a decreasing function of a. On the other hand, passing from massless to massive, we find that the effect of the mass is insignificant when a is small, but the magnitude of the force is decreased for large a in the massive case.