Scalar-dependent gauge fixing

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 $p$.