Ghost coupling in the Abelian Higgs model (${\mathcal{L}}_{\mathrm{gh}}=-\bar{c}[\square +\xi e^2(f+h{)}^2]c$)

With the specified scalar-dependent gauge fixing, the Faddeev-Popov determinant depends on the radial Higgs mode even though the gauge group is Abelian. Besides a ghost mass squared $\xi e^2{f}^2$, the Lagrangian density contains $-2\xi e^{2}fh\bar{c}c-\xi e^2{h}^2\bar{c}c$. These Faddeev-Popov ghosts do not decouple for generic $\xi$; their interactions vanish in the $\xi \to 0$ Landau gauge limit. Free ghosts in pure Abelian Lorenz gauge rely on a field-independent determinant, rather than merely on commutativity of the gauge group.