The Renormalization Group and the Effective Action

TL;DR

This paper demonstrates how the renormalization group method can systematically sum leading and next-to-leading logarithmic contributions to the effective action and potential, using RG functions and anomaly considerations.

Contribution

It introduces a method to incorporate RG functions at different loop levels to improve the calculation of the effective action and potential, fixing log-independent parts via anomalies and stationary conditions.

Findings

Summation of leading-log contributions using one-loop RG functions.

Inclusion of next-to-leading-log contributions with two-loop RG functions.

Determination of log-independent terms through anomalies and stationary conditions.

Abstract

The renormalization group is used to sum the leading-log (LL) contributions to the effective action for a large constant external gauge field in terms of the one-loop renormalization group (RG) function beta, the next-to-leading-log (NLL) contributions in terms of the two-loop RG function etc. The log independent pieces are not determined by the RG equation, but can be fixed by the anomaly in the trace of the energy-momentum tensor. Similar considerations can be applied to the effective potential V for a scalar field phi; here the log independent pieces are fixed by the condition V'(phi=v)=0.