Removing the gauge parameter dependence of the effective potential by a field redefinition

TL;DR

This work addresses the gauge-parameter dependence of the electroweak effective potential and shows that a field-redefinition based on a homogeneous first-order PDE eliminates ξ-dependence. By introducing $Φ(φ,ξ,ξ_0)$ via the method of characteristics, the authors prove $V(φ,ξ) = V(Φ(φ,ξ,ξ_0), ξ_0)$ in the SU(2)×U(1) Standard Model for the renormalization-group improved potential at leading and next-leading logarithmic order. They derive explicit LL and NLL solutions under two RG frameworks and highlight the essential roles of the Higgs anomalous dimension $γ_φ$ and the $Δλ$ term in encoding gauge dependence. The results provide a gauge-parameter-free basis for vacuum stability analyses beyond the Landau gauge and offer analytic insight into how instability scales can be mapped between gauges.

Abstract

The gauge parameter dependence of the effective potential is determined by partial differential equations involving also the Higgs boson field expectation value. Solving these equations by the method of characteristics leads to complete elimination of the gauge parameter dependence of the effective potential. The construction is carried out in the case of the standard model of electroweak unification for the renormalization group improved effective potential up to the next-leading logarithmic order.