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What does the graviton-matter Feynman scattering diagram look like?
Yes, this diagram will contribute to physical processes. Just a comment: please remember that feynman diagrams are just perturbative calculation tools, this particular one will depend on the arbitrary gauge choise, so it self does not represent the full physical process.
Divergence in Supergravity
N=8 Sugra is 4 dimensional theory. If you look at possible counter terms that may arise like R^2, R^3 i.e. different contractions of Riemann tensor. Some of them will be not allowed by different symmetries e.g. susy. The statement would be that on dimensional grounds counter terms at 7 loops in D=4 N=8 SUGRA, 5 loops in Maximally Supersymmetric SUGRA in 5 dimensions, will be allowed by the same symmetries. Therefore, if you find that one of them in absent, then there must be some new/unknown symmetry that kills it, which should also kill the other.