Proposition: Gauge-fixing normalization

The auxiliary equation gives

Eliminating $b$ in the action produces exactly $\frac{1}{4}\int (\mathsf{C}r{)}_{\mu }(\mathsf{C}r{)}^{\mu }$. Thus the reduced metric operator is ${\mathsf{P}}_{\mathrm{gr}}$, and the ghost term in this convention is $-\int {\bar{c}}_{\mu }\square c^{\mu }$, without an extra normalization factor.

The independent $b$ is required for off-shell nilpotence. After its elimination, ${s_0}^2\bar{c}=\frac{1}{2}\square c$ is only an on-shell identity.

Proof. Complete the square:

Equation (eq:operators) gives the reduced operator. The minus sign of the ghost term follows from the odd parity of $\bar{c}$ in the variation of (eq:Psi0). Finally, $s_0(\mathsf{C}r)=\mathsf{C}Kc=\square c$.