Scalar-dependent gauge fixing

Scalar-dependent gauge fixing chooses a gauge condition containing both a gauge field and a scalar field. After spontaneous symmetry breaking, expanding a scalar kinetic term often produces mixing between the vector field and a scalar phase fluctuation. A suitable gauge condition can cancel that mixing in the quadratic action.

For example, a condition schematically $G=\partial \cdot A-\xi m\chi$ leads to a gauge fixing term $-G^2/(2\xi )$, with the sign chosen consistently with the original mixing. The Faddeev-Popov operator follows by varying the full $G$, including its scalar dependence. Thus ghosts can acquire scalar-dependent interactions even for an Abelian theory when the gauge condition is nonlinear. Cancelling the quadratic mixing does not imply that all vector–scalar interactions disappear, or that physical predictions can depend on the gauge-fixing parameter.

For the polar parametrization of the Abelian Higgs field with $D=\partial +ieA$, adding $-H^2/(2\xi )$ cancels quadratic vector-phase mixing. The cross terms reduce, modulo a total derivative, to $-2e(1+h/f)\pi A^{\mu }{\partial }_{\mu }h$; interactions remain. The Faddeev-Popov operator is proportional to $\square +\xi e^2(f+h{)}^2$, so the Faddeev-Popov ghosts couple to $h$. The vector propagator falls as $k^{-2}$ at fixed finite $\xi$, but the polar kinetic and nonlinear gauge fixing vertices include terms of mass dimension above four. Manifest polynomial power counting instead uses Cartesian scalar variables with conventional R-xi gauge.

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.

The Faddeev-Popov ghost field couples to both the gauge field and the adjoint scalar field. In a convention with $D_{\mu }={\partial }_{\mu }+[A_{\mu },\cdot ]$, the ghost action after integration by parts contains $f^{abc}({\partial }_{\mu }{\bar{c}}^a)A_{\mu }^b{c}^c$ and $f^{abc}(n\cdot \partial {\bar{c}}^a){\varphi }^b{c}^c/\sqrt{2}$.

For gauge fixing with $F^a=\partial \cdot A^a+(n\cdot \partial {\varphi }^a)/\sqrt{2}$, integrating out the Nakanishi-Lautrup field produces gauge-scalar mixing. The Schur complement of the free gauge-field kernel cancels the extra $(n\cdot p{)}^2/2$ in the scalar kernel, leaving $⟨{\varphi }^a(p){\varphi }^b(-p)⟩={\delta }^{ab}/p^2$. This is independent of the gauge vector $n$, but still depends on the full momentum $p$.