Gauge invariance of the scalar-vector mass ratio in the Coleman-Weinberg model

The scalar-meson to vector-meson mass ratio, due to spontaneous symmetry breakdown, is calculated in the two-loop approximation of massless scalar quantum electrodynamics. Although the effective potential is gauge-dependent, this mass ratio is found to be gauge-independent. This strongly supports the interpretation given by Coleman and Weinberg that the radiative corrections drive the spontaneous symmetry breakdown in this theory. The mass ratio is found to be $\frac{{{m}_{S}}^{2}}{{{m}_{V}}^{2}}=\frac{\frac{3}{2}{e}^{2}}{4{\ensuremath{\pi}}^{2}}\ensuremath{-}\frac{61}{48}{(\frac{{e}^{2}}{4{\ensuremath{\pi}}^{2}})}^{2}$, where ${e}^{2}$ is the physical coupling constant.