  The effective potential $V$ is considered in massless $\lambda\phi^4_4$
theory. The expansion of $V$ in powers of the coupling $\lambda$ and of the
logarithm of the background field $\phi$ is reorganized in two ways; first as a
series in $\lambda$ alone, then as a series in $\ln\phi$ alone. By applying the
renormalization group (RG) equation to $V$, these expansions can be summed.
Using the condition $V^\prime(v) = 0$ (where $v$ is the vacuum expectation
value of $\phi$) in conjunction with the expansion of $V$ in powers of
$\ln\phi$ fixes $V$ provided $v\neq 0$. In this case, the dependence of $V$ on
$\phi$ drops out and $V$ is not analytic in $\lambda$. Massless scalar
electrodynamics is considered using the same approach.
