Neumann boundary conditions inhibiting the SSB in the Coleman-Weinberg mechanism

In this work we show that homogeneous Neumann boundary conditions inhibit the Coleman-Weinberg mechanism for spontaneous symmetry breaking in the scalar electrodynamics if the length of the finite region is small enough ($a = e^{2}M^{-1}{\phi}$,where$M{\phi}$ is the mass of the scalar field generated by the Coleman-Weinberg mechanism)