Finiteness of SO(3) and SO(4) supergravity and nonrenormalizability of the scalar multiplet coupling in explicit one-loop examples

Photon-photon scattering with all possible colors in SO(3) supergravity and with equal colors in SO(4) supergravity is shown to be one-loop finite by explicit evaluation. Scalar-scalar scattering in (2,3/2,1,1/2) supergravity diverges at this level. Our previous conjecture that only $\mathrm{SO}\mathrm{}\left(n\right)\mathrm{}$ models are renormalizable thus is confirmed in these examples.