The Effectve Potential in Massless Theories

The effective potential V in a massless self-coupled scalar theory and massless scalar electrodynamics is considered. Both the MS and Coleman-Weinberg renormalization schemes are examined. The renormalization scheme dependence of V is determined. Upon summing all of the logarithmic contributions to V, it is shown that the implicit and explicit dependence on the renormalization scale {\mu} cancels. In addition, if there is spontaneous symmetry breaking, then the dependence on the background field {\Phi} cancels, leaving V flat but with non-perturbative contributions. The quartic scalar coupling in the Coleman-Weinberg renormalization scheme consequently vanishes.