  The perturbative effective potential V in the massless $\lambda\phi^4$ model
with a global O(N) symmetry is uniquely determined to all orders by the
renormalization group functions alone when the Coleman-Weinberg renormalization
condition $\frac{d^4V}{d\phi^4}|_{\phi = \mu} = \lambda$ is used, where $\mu$
represents the renormalization scale. Systematic methods are developed to
express the n-loop effective potential in the Coleman-Weinberg scheme in terms
of the known n-loop minimal subtraction (MS) renormalization-group functions.
Moreover, it also proves possible to sum the leading- and
subsequent-to-leading-logarithm contributions to V. An essential element of
this analysis is a conversion of the renormalization group functions in the
Coleman-Weinberg scheme to the renormalization group functions in the MS
scheme. As an example, the explicit five-loop effective potential is obtained
from the known five-loop MS renormalization group functions and we explicitly
sum the leading logarithm (LL), next-to-leading (NLL), and further
subleading-logarithm contributions to V. Extensions of these results to
massless scalar QED are also presented. Because massless scalar QED has two
couplings, conversion of the RG functions from the MS scheme to the CW scheme
requires the use of multi-scale renormalization group methods.
