  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.
