Duality Constraints on Counterterms in N=5,6 Supergravities

The UV finiteness found in calculations of the 4-point amplitude in N = 5 supergravity at loop order L = 3, 4 has not been explained, which motivates our study of the relevant superspace invariants and on-shell superamplitudes for both N = 5 and N = 6. The local 4-point superinvariants for L = 3,4 are expected to have nonlinear completions whose 6-point amplitudes have non-vanishing SSL's (soft scalar limits), violating the behavior required of Goldstone bosons. For N = 5, we find at L = 3 that local 6-point superinvariant and superamplitudes, which might cancel these SSL's, do not exist. This rules out the candidate 4-point counterterm and thus gives a plausible explanation of the observed L = 3 finiteness. However, at L = 4 we construct a local 6-point superinvariant with non-vanishing SSL's, so the SSL argument does not explain the observed L = 4 N = 5 UV finiteness. For N = 6 supergravity there are no 6-point invariants at either L = 3 or 4, so the SSL argument predicts UV finiteness.