Interplay of Infrared Divergences and Gauge-Dependence of the Effective Potential

TL;DR

This work analyzes infrared (IR) divergences and gauge dependence of the perturbative effective potential in gauges with massless Goldstones, focusing on Fermi gauge in the Abelian Higgs model and extending to the Standard Model. It demonstrates that IR divergences from Goldstone loops can be cured by an IR resummation that shifts the Goldstone mass $G$ to $\overline G=G+\Pi_g$, and that this resummation also eliminates residual gauge dependence at radiatively generated minima. A novel issue arises in Fermi gauge: the first derivative $V'$ develops an IR divergence due to a $p^4$ pole from Goldstone–longitudinal gauge mixing, but this does not affect physical observables like the Higgs mass or Nielsen identities; this divergence can be removed by a one-loop field redefinition or regulated with an IR regulator. The results imply that physical predictions remain gauge-invariant and IR-finite even in Fermi gauge, and the methods extend straightforwardly to the Standard Model. Together, these insights provide a consistent, gauge-aware framework for using the effective potential in high-precision SM and beyond-SM analyses.

Abstract

The perturbative effective potential suffers infrared (IR) divergences in gauges with massless Goldstones in their minima (like Landau or Fermi gauges) but the problem can be fixed by a suitable resummation of the Goldstone propagators. When the potential minimum is generated radiatively, gauge-independence of the potential at the minimum also requires resummation and we demonstrate that the resummation that solves the IR problem also cures the gauge-dependence issue, showing this explicitly in the Abelian Higgs model in Fermi gauge. In the process we find an IR divergence (in the location of the minimum) specific to Fermi gauge and not appreciated in recent literature. We show that physical observables can still be computed in this gauge and we further show how to get rid of this divergence by a field redefinition. All these results generalize to the Standard Model case.