A Unique Determination of the Effective Potential in Terms of Renormalization Group Functions

TL;DR

This paper demonstrates that the effective potential in a massless λϕ^4 model with O(N) symmetry can be uniquely determined from renormalization group functions, and provides methods to express and sum contributions to all orders.

Contribution

It develops systematic methods to relate the n-loop effective potential in the Coleman-Weinberg scheme to known MS scheme RG functions, including explicit five-loop calculations.

Findings

Explicit five-loop effective potential derived from MS RG functions.

Sum of leading and subleading logarithm contributions to the effective potential.

Extension of methods to massless scalar QED with multiple couplings.

Abstract

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.