Æther coupling effects on casimir energy for self-interacting scalar field within extra dimension

This paper presents comprehensive calculations for thermal and first-order radiative corrections to the Casimir energy in systems involving self-interacting massive and massless scalar fields coupled with æther in a fifth compact dimension. The method used to compute the radiative correction to the Casimir energy differs from conventional approaches by applying a unique renormalization scheme that is consistent with specific boundary conditions or backgrounds. Despite this divergence from conventional methodologies, our results demonstrate consistency within established physical limits. Furthermore, employing a toy model, we calculated the total Casimir energy density in the bulk, taking into account both thermal and radiative corrections. We also provide a thorough characterization of the total Casimir energy density in the compact dimension, detailing its magnitude and sign using graphical representations and quantitative data.