Radiative Correction to the Dirichlet Casimir Energy for $λφ^{4}$ Theory in Two Spatial Dimensions

In this paper, we calculate the next to the leading order Casimir energy for real massive and massless scalar fields within $λφ^{4}$ theory, confined between two parallel plates with the Dirichlet boundary condition in two spatial dimensions. Our results are finite in both cases, in sharp contrast to the infinite result reported previously for the massless case. In this paper we use a renormalization procedure introduced earlier, which naturally incorporates the boundary conditions. As a result our radiative correction term is different from the previously calculated value. We further use a regularization procedure which help us to obtain the finite results without resorting to any analytic continuation techniques.