 Solution

The gauge-invariant functional integral counts every gauge orbit repeatedly. Insert a gauge-orbit identityDividing by the gauge-group volume and averaging $f$ with a Gaussian weight produces the term $-G^2/(2\xi )$. The determinant is represented by a pair of independent anticommuting scalar fields $c,\bar{c}$, giving the prescribed 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 commuting Gaussian integral; field-independent constants and overall determinant signs are absorbed in normalization.

Although the gauge group is Abelian, the determinant here is field dependent because the gauge condition contains the scalar fluctuation $b$. Thus Abelian ghosts need not decouple in a broken-phase $R_{\xi }$ gauge. The fields implement gauge-orbit counting and are not physical external particles.