 Complex quadratic Abelian gauge condition

The formal variation of this condition is $\delta F=({\partial }^2+2iA\cdot \partial )\alpha$. Thus its Faddeev-Popov ghost field operator can be chosen as ${\mathcal{O}}_A=-{\partial }^2-2iA\cdot \partial$. With $D_{\mu }={\partial }_{\mu }+i{A}_{\mu }$, one has $D^2={\partial }^2+2iA\cdot \partial +iF[A]$, so ${\mathcal{O}}_A=-D^2$ on the formal gauge slice. The reality obstruction for a complex Euclidean gauge condition prevents interpreting this slice as an ordinary real Euclidean gauge fixing without an additional complex-contour prescription.