  Multi-scale renormalization group (RG) methods are reviewed and applied to
the analysis of the effective potential for radiative symmetry breaking with
multiple scalar fields, allowing an extension of the Gildener & Weinberg (GW)
method beyond the weak coupling limit. A model containing two interacting real
scalar fields is used to illustrate multi-scale RG methods and the multi-scale
RG functions of this model are calculated to one-loop order for the $\beta$
function and two-loop order for the anomalous mass dimension. The introduction
of an extra renormalization scale allows the mapping of the effective potential
in this model onto an RG-equivalent form with an O(2) symmetric structure along
a particular trajectory in the multiple renormalization-scale space, leading to
a simplified form of the effective potential. It is demonstrated that the
physical content of the effective potential in the original model, referenced
to a single conventional renormalization scale, can be extracted from a
particular RG-trajectory that connects to this multi-scale O(2)-symmetric form
of the effective potential. Extensions of these multi-scale methods for
effective potentials in models containing multiple scalars with $O(M)\times
O(N)$ symmetry are also discussed.
