Duality Constraints on Counterterms in $\mathcal{N}=5,6$ Supergravities

The UV finiteness found in calculations of the 4-point amplitude in $\mathcal N=5$supergravityatlooporder$L=3, 4$ has not been explained, which motivates our study of the relevant superspace invariants and on-shell superamplitudes for both $\mathcal{N}=5$ and $\mathcal{N}=6$. 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 $\mathcal{N}=5$, 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 $L=4$ $\mathcal{N}=5$ UV finiteness. For $\mathcal{N}=6$ supergravity there are no 6-point invariants at either $L= 3$ or 4, so the SSL argument predicts UV finiteness.