Radiative Correction of the Casimir Energy for the Scalar Field with the Mixed Boundary Condition in 3 + 1 Dimensions

In the present study, the zeroth- and first-order radiative correction of the Casimir energy for massive and massless scalar fields, confined with mixed boundary conditions (Dirichlet- Neumann) between two parallel plates in $\phi^4$ theory, were computed. Two issues in performing calculations in this work are essential: first, to renormalize the bare parameters of the problem, a systematic method was used, which allows all influences from the boundary conditions to be imported in all elements of the renormalization program. This idea yields our counterterms appearing in the renormalization program to be position-dependent. Using the box subtraction scheme as a regularization technique is the other noteworthy point in the calculation. In this scheme, by subtracting the vacuum energies of two similar configurations from each other, regularizing divergent expressions and their removal process were significantly facilitated. All the obtained answers for the Casimir energy with the mixed boundary condition were consistent with well-known physical grounds. We also compared the Casimir energy for the massive scalar field confined with four types of boundary conditions (Dirichlet, Neumann, a mix of them and Periodic) in 3+1 dimensions with each other, and the sign and magnitude of their values were discussed.