= Solution
Let $G_1,\ldots,G_r$ be the separated gauge-invariant insertions. Their <BRST symmetry> variations vanish: the <gauge field strength> and its <gauge covariant derivatives> transform by adjoint commutators, and invariant color contractions remove these commutators. Thus $sG_i=0$.
Make the infinitesimal change of integration variables $\Phi\mapsto\Phi+\epsilon s\Phi$ in the normalized expectation of $\mathcal O(x)\prod_iG_i$. The action is invariant up to its boundary term and, by assumption, the measure has no Jacobian anomaly. Therefore
$$
0=\left\langle s\left(\mathcal O(x)\prod_iG_i\right)\right\rangle
=\left\langle(s\mathcal O(x))\prod_iG_i\right\rangle.
$$
Since $s\mathcal O$ is generated by the <BRST charge>, this is the <graded BRST Ward identity>
$$
\boxed{\left\langle[Q_{\mathrm{BRST}},\mathcal O(x)\}\prod_iG_i\right\rangle=0.}
$$
Here $[Q,\mathcal O\}=Q\mathcal O-(-1)^{|\mathcal O|}\mathcal OQ$ is the <graded commutator>. For even $\mathcal O$ it is the ordinary commutator printed in the question; for odd $\mathcal O$ it is an anticommutator. Separation of the other insertions from $x$ avoids the additional coincident-point contact terms. An operator need not itself be <BRST-closed> for this identity to hold.
\b[A BRST-exact insertion has zero correlation with physical, gauge-invariant insertions.] This is the decoupling of <BRST-exact insertions in physical correlation functions>. In the associated <BRST cohomology>, physical information is represented by closed states or operators modulo exact ones, schematically $\ker Q/\operatorname{im}Q$, using the nilpotence of the <BRST charge> in the anomaly-free theory. Gauge-fixing fields thereby do not supply additional physical observables.
For example, the gauge parameter dependence is exact:
$$
\frac{\partial\mathcal L}{\partial\xi}=-\frac12h^ah^a=s\left(-\frac12b^ah^a\right).
$$
Differentiating a normalized physical correlation function with respect to $\xi$ inserts this expression; the <BRST Ward identity> makes that derivative zero under the same invariant-measure and boundary assumptions. This illustrates <gauge-fixing parameter independence from BRST symmetry>.
Back to article page
