For <axial gauge>,
$$
\frac{\delta G^a(x)}{\delta\alpha^b(y)}
=n^\mu D_\mu^{ab}(x)\delta^{(4)}(x-y),
$$
so the <Faddeev-Popov determinant> is $\det(n^\mu D_\mu)$. Representing it with a <Faddeev-Popov ghost field> pair gives
$$
S_{\rm gh}=\int d^4x\,\bar c^a n^\mu D_\mu^{ab}c^b.
$$
The delta functional sets $\omega=n\mathbin\cdot A$ in its Gaussian weight, and hence
$$
S_{\rm gf}=-\frac1{2\xi}\int d^4x\,(n^\mu A_\mu^a)^2.
$$
Substitution yields
$$
Z[J]=N\int\mathcal DA\,\mathcal D\bar c\,\mathcal Dc\,
e^{,iS[A]+iS_{\rm gh}+iS_{\rm gf}+i\int J^\mu A_\mu}.
$$
In the strict axial-gauge limit $n\mathbin\cdot A=0$, $n\mathbin\cdot D=n\mathbin\cdot\partial$, so the ghost determinant is independent of $A$ and can be absorbed into $N$.

Solved by gpt-5.6-sol high.

Solution

Paper 304 / 4 / b / Solution

{wiki}

An axial gauge imposes $n^\mu A_\mu=0$ for a fixed vector $n^\mu$. Its Faddeev-Popov operator is $n^\mu D_\mu$; on the strict gauge slice its gauge-field-dependent part vanishes, so its ghosts decouple.


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 Incoming links: Axial gauge