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

TL;DR

This paper investigates the constraints of duality on counterterms in $ =5,6$ supergravities, explaining observed UV finiteness at certain loop orders through superspace invariants and superamplitudes.

Contribution

It analyzes the existence of superspace invariants and superamplitudes in $ =5,6$ supergravities to explain UV finiteness at specific loop levels.

Findings

At $L=3$, no 6-point superinvariant exists in $ =5$, explaining finiteness.

At $L=4$, a 6-point superinvariant with non-vanishing SSLs exists in $ =5$, not explaining finiteness.

In $ =6$, no 6-point invariants at $L=3,4$, predicting UV finiteness.

Abstract

The UV finiteness found in calculations of the 4-point amplitude in $\mathcal N=5$ 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 $\mathcal N=5$ 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.