  By using the renormalization group (RG) equation it has proved possible to
sum logarithmic corrections to quantities that arise due to quantum effects in
field theories. In particular, the effective potential V in the Standard Model
in the limit that there are no massive parameters in the classical action (the
"conformal limit") has been subject to this analysis, as has the effective
potential in a scalar theory with a quartic self coupling and in massless
scalar electrodynamics. Having multiple coupling constants and/or mass
parameters in the initial action complicates this analysis, as then several
mass scales arise. We show how to address this problem by considering the
effective potential in scalar electrodynamics when the scalar field has a tree
level mass term. In addition to summing logarithmic corrections by using the RG
equation, we also consider the consequences of the condition V'(v)=0 where v is
the vacuum expectation value of the scalar. If V is expanded in powers of the
logarithms that arise, then it proves possible to show that either v is zero or
that V is independent of the scalar. (That is, either there is no spontaneous
symmetry breaking or the vacuum expectation value is not determined by
minimizing V as V is "flat".)
