 Ghost-scalar determinant cancellation in a complex quadratic gauge

A massless complex bosonic charged scalar field with $D_{\mu }={\partial }_{\mu }+i{A}_{\mu }$ contributes $[\det (-D^2){]}^{-1}$ to a Gaussian background-field integral. A Faddeev-Popov ghost field pair contributes $\det {\mathcal{O}}_A$. On the formal complex quadratic Abelian gauge condition, ${\mathcal{O}}_A=-D^2$, so the two determinants cancel and their effective-action contributions sum to zero. Use matching regulators, boundary conditions and removed zero modes. This is a formal cancellation of closed loops; the reality obstruction for a complex Euclidean gauge condition precludes deducing that physical scalar quantum electrodynamics has no quantum effects.