Ghost coupling in a linear Cartesian Abelian Higgs gauge (${\mathcal{L}}_{\mathrm{gh}}=\bar{c}\square c-\xi g^{2}v(v+h)\bar{c}c$)

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.