R-xi gauge ($R_{\xi }$ gauge)

For the Abelian Higgs mechanism with scalar phase $\chi$ and vector mass $m$, take ${\mathcal{L}}_{\mathrm{gf}}=-(\partial \cdot A+\xi m\chi {)}^2/(2\xi )$ when the scalar kinetic mixing is $-m{A}_{\mu }{\partial }^{\mu }\chi$. The cross term cancels this mixing after integration by parts. The resulting vector propagator isAt fixed finite $\xi$ it decays as $1/k^2$. Would-be Goldstone bosons and the required Faddeev-Popov ghosts remain in intermediate calculations; BRST symmetry identifies the physical states and ensures cancellation of unphysical modes. The limit $\xi \to \infty$ gives the unitary gauge propagator and is not uniform in ultraviolet momentum.

For $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$, transformations $\delta A_{\mu }=-{\partial }_{\mu }\theta$, $\delta \varphi =-ig\theta \varphi$, and Cartesian shift $\varphi =(v+h+ib)/\sqrt{2}$, the gauge function $G=\partial \cdot A-\xi gvb$ varies by $[-\square +\xi g^{2}v(v+h)]\theta$. Exponentiating the Faddeev-Popov determinant with the convention $-\bar{c}(\delta G/\delta \theta )c$ gives ghost mass squared $\xi g^2{v}^2$ and one Higgs-ghost vertex $-i\xi g^{2}v$. A scalar-dependent Abelian gauge therefore has interacting ghosts; a pure linear Lorenz gauge has a field-independent determinant.