Unique determination of the effective potential in terms of renormalization group functions

Abstract

The perturbative effective potential $V$ in the massless $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ 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 $\frac{{d}^{4}V}{d{\ensuremath{\phi}}^{4}}{|}_{\ensuremath{\phi}=\ensuremath{\mu}}=\ensuremath{\lambda}$ is used, where $\ensuremath{\mu}$ 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, next-to-leading-logarithm, 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 renormalization group functions from the MS scheme to the Coleman-Weinberg scheme requires the use of multiscale renormalization group methods.