E7(7) constraints on counterterms in N=8 supergravity

TL;DR

The paper probes the fate of $E_{7(7)}$ symmetry in ${ m N}=8$ supergravity by examining single-soft scalar limits of higher-order counterterms. Using string-theory–based matrix elements and explicit supersymmetric amplitudes, it shows that the 5- and 6-loop candidates $D^4R^4$ and $D^6R^4$ violate $E_{7(7)}$, implying no invariant counterterms below seven loops; at seven loops, an infinite tower of candidates arises, but only specially combined terms (e.g., a tuned $D^8R^4$) can be SSL-compatible up to a finite point, with nontrivial SSL constraints persisting at higher points. The authors develop representation-theoretic and Gröbner-basis methods to classify and construct these operators, confirm consistency with automorphism constraints via Laplace equations on moduli, and map the SSL structure to the multiplicities of ${f 70}$ representations, highlighting how $E_{7(7)}$ imposes strong, nontrivial restrictions that complicate the existence of invariant counterterms at and beyond seven loops. Overall, the work provides a concrete framework linking SSL behavior, operator spectra, and automorphism constraints to assess finiteness and symmetry at high loop orders in ${ m N}=8$ supergravity.

Abstract

We prove by explicit computation that 6-point matrix elements of D^4R^4 and D^6R^4 in N=8 supergravity have non-vanishing single-soft scalar limits, and therefore these operators violate the continuous E7(7) symmetry. The soft limits precisely match automorphism constraints. Together with previous results for R^4, this provides a direct proof that no E7(7)-invariant candidate counterterm exists below 7-loop order. At 7-loops, we characterize the infinite tower of independent supersymmetric operators D^4R^6, R^8, phi^2 R^8,... with n>4 fields and prove that they all violate E7(7) symmetry. This means that the 4-graviton amplitude determines whether or not the theory is finite at 7-loop order. We show that the corresponding candidate counterterm D^8R^4 has a non-linear supersymmetrization such that its single- and double-soft scalar limits are compatible with E7(7) up to and including 6-points. At loop orders 7, 8, 9 we provide an exhaustive account of all independent candidate counterterms with up to 16, 14, 12 fields, respectively, together with their potential single-soft scalar limits.