= Ghost-scalar determinant cancellation in a complex quadratic gauge
A massless complex bosonic <charged scalar field> with $D_\mu=\partial_\mu+iA_\mu$ 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.
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