# Radiative Correction to the Casimir Energy for Massive Scalar Field on a Spherical Surface

## TL;DR

This paper calculates the first-order radiative correction to the Casimir energy for a massive scalar field on a spherical surface, introducing a position-dependent renormalization and a novel regularization method called Box Subtraction Scheme.

## Contribution

It presents a systematic perturbation approach with position-dependent counter-terms and introduces the Box Subtraction Scheme for divergence regularization in Casimir energy calculations.

## Key findings

- The correction depends on boundary conditions and background space.
- The Box Subtraction Scheme effectively removes divergences without ambiguity.
- Results are consistent for both massive and massless scalar fields.

## Abstract

In this paper, the first order radiative correction to the Casimir energy for a massive scalar field in the $\phi^4$ theory on a spherical surface with $S^2$ topology was calculated. In common methods for calculating the radiative correction to the Casimir energy, the counter-terms related to free theory are used. However, in this study, by using a systematic perturbation expansion, the obtained counter-terms in renormalization program were automatically position-dependent. We maintained that this dependency was permitted, reflecting the effects of the boundary conditions imposed or background space in the problem. Additionally, along with the renormalization program, a supplementary regularization technique that we named Box Subtraction Scheme (BSS) was performed. This scheme presents a useful method for the regularization of divergences, providing a situation that the infinities would be removed spontaneously without any ambiguity. Analysis of the necessary limits of the obtained results for the Casimir energy of the massive and massless scalar field confirmed the appropriate and reasonable consistency of the answers.