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

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 $\xi \neq 0$.