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

TL;DR

This paper proves that certain candidate counterterms in N=8 supergravity violate E7(7) symmetry, providing evidence that the theory is finite below 7 loops and analyzing the symmetry properties of potential counterterms at higher loops.

Contribution

It explicitly computes soft scalar limits of operators in N=8 supergravity to show they violate E7(7) symmetry, ruling out certain counterterms and constraining finiteness.

Findings

6-point matrix elements violate E7(7) symmetry

No E7(7)-invariant counterterms below 7 loops

Candidate counterterms at 7 loops and beyond violate E7(7)

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.