Properties of the vacuum expectation values in $R_{\xi }$ and general gauge

TL;DR

This paper studies the behavior of vacuum expectation values in different gauges of the Abelian Higgs model, revealing gauge-dependent renormalization group properties and IR divergence patterns.

Contribution

It provides a detailed analysis of the VEV's renormalization group evolution and IR divergences in the $R_{\xi}$ and general gauges, highlighting differences from previous gauge choices.

Findings

Gamma function of VEV differs from scalar field in $R_{\xi}$ gauge.

Gamma function of VEV matches scalar field in general gauge.

IR divergences in two-loop VEVs occur only in Landau gauge ($\xi=0$).

Abstract

We have investigated the renormalization group evolution and the perturbative expansion of the vacuum expectation value(VEV) of the Abelian Higgs model both in the $R_{\xi }$ and the general gauge which extrapolate between the Fermi and the $\overline{R_{\xi }\text{ }}$ gauge. In $R_{\xi }$ gauge, the gamma function of the VEV was different from that of the scalar field. On the contrary, the gamma function of the VEV was same as that of the scalar field in general gauge. Both in $R_{\xi }\text{ }$ and $\overline{% R_{\xi }\text{ }}$ gauge, the two-loop VEVs have an IR divergences in Landau gauge($\xi =0$) and these IR divergence do not occur when $\xi \neq 0$.