  In this article, we present the zero and first-order radiative correction to
the Dirichlet Casimir energy for massive and massless scalar field confined in
a rectangle. This calculation procedure was conducted in two spatial dimensions
and for the case of the first-order correction term is new. The renormalization
program that we have used in this work, allows all influences from the dominant
boundary conditions (e.g. the Dirichlet boundary condition) be automatically
reflected in the counterterms. This permission usually makes the counterterms
position-dependent. Along with the renormalization program, a supplementary
regularization technique was performed in this work. In this regularization
technique, that we have named Box Subtraction Scheme (BSS), two similar
configurations were introduced and the zero point energies of these two
configurations were subtracted from each other using appropriate limits. This
regularization procedure makes the usage of any analytic continuation
techniques unnecessary. In the present work, first, we briefly present
calculation of the leading order Casimir energy for the massive scalar field in
a rectangle via BSS. Next, the first order correction to the Casimir energy is
calculated by applying the mentioned renormalization and regularization
procedures. Finally, all the necessary limits of obtained answers for both
massive and massless cases are discussed.
