The Effective Potential in Non-Conformal Gauge Theories

TL;DR

This paper investigates the effective potential in non-conformal gauge theories, especially scalar electrodynamics with a mass term, using the renormalization group to sum logarithmic corrections and analyze conditions for spontaneous symmetry breaking.

Contribution

It extends the analysis of the effective potential to theories with multiple scales and mass terms, showing how to sum logarithms and examine the potential's flatness or minima.

Findings

Effective potential can be flat or have a non-zero vacuum expectation value.

Summation of logarithmic corrections is feasible with multiple scales.

Vacuum expectation value is either zero or the potential is independent of the scalar.

Abstract

By using the renormalization group (RG) equation it has proved possible to sum logarithmic corrections to quantities that arise due to quantum effects in field theories. In particular, the effective potential V in the Standard Model in the limit that there are no massive parameters in the classical action (the "conformal limit") has been subject to this analysis, as has the effective potential in a scalar theory with a quartic self coupling and in massless scalar electrodynamics. Having multiple coupling constants and/or mass parameters in the initial action complicates this analysis, as then several mass scales arise. We show how to address this problem by considering the effective potential in scalar electrodynamics when the scalar field has a tree level mass term. In addition to summing logarithmic corrections by using the RG equation, we also consider the consequences of the condition V'(v)=0 where v is the vacuum expectation value of the scalar. If V is expanded in powers of the logarithms that arise, then it proves possible to show that either v is zero or that V is independent of the scalar. (That is, either there is no spontaneous symmetry breaking or the vacuum expectation value is not determined by minimizing V as V is "flat".)