Multiscale renormalization group methods for effective potentials with multiple scalar fields

Multiscale 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 and Weinberg method beyond the weak coupling limit. A model containing two interacting real scalar fields is used to illustrate multiscale RG methods and the multiscale 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 a 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 multiscale O(2)-symmetric form of the effective potential. Extensions of these multiscale methods for effective potentials in models containing multiple scalars with O(M) x O(N) symmetry are also discussed.