= Solution
The gauge-invariant <Path integral>[functional integral] counts every <gauge orbit> repeatedly. Insert a gauge-orbit identity
$$
1=\Delta_{\mathrm{FP}}[A,\phi]\int\mathcal D\theta\,\delta(G[A^\theta,\phi^\theta]-f),\qquad
\Delta_{\mathrm{FP}}=\det\left(\frac{\delta G(x)}{\delta\theta(y)}\right).
$$
Dividing by the <Gauge group>[gauge-group] volume and averaging $f$ with a <Gaussian integral>[Gaussian] weight produces the term $-G^2/(2\xi)$. The <determinant> is represented by a pair of independent <Faddeev-Popov ghost fields>[anticommuting scalar fields] $c,\bar c$, giving the prescribed <Faddeev-Popov ghost>[ghost] density $-\bar c\mathcal M c$, where $\mathcal M=\delta G/\delta\theta$. The <Grassmann integral> gives a <determinant> rather than the inverse <determinant> of a <Commutator>[commuting] <Gaussian integral>; field-independent constants and overall <determinant> signs are absorbed in <normalization>.
Although the <gauge group> is <Abelian group>[Abelian], the <determinant> here is field dependent because the <Gauge fixing>[gauge condition] contains the <Scalar field>[scalar] fluctuation $b$. Thus <Abelian group>[Abelian] <Faddeev-Popov ghosts>[ghosts] need not decouple in a broken-phase $R_\xi$ gauge. The fields implement gauge-orbit counting and are not physical external <particles>.
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