Renormalizability properties of antisymmetric tensor fields coupled to gravity

The coupling of antisymmetric tensor fields $A_{\mu \nu \rho }$ to gravity is one-loop finite while coupling $A_{\mu \nu }$ to gravity and gravity to $A_{\mu \nu \rho }$ and to a scalar $\phi$ is nonrenormalizable. The counterterms are on-shell zero and equal to those for the coupling of gravity to $\phi$, respectively, as one might expect from the classical result that $A_{\mu \nu \rho }$ vanishes and $A_{\mu \nu }$ is equivalent to $\phi$ on shell.