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

We prove by explicit computation that 6-point matrix elements of (DR4)-R-4 and (DR4)-R-6 in N = 8 supergravity have non-vanishing single-soft scalar limits, and therefore these operators violate the continuous E-7(7) symmetry. The soft limits precisely match automorphism constraints. Together with previous results for R-4, this provides a direct proof that no E-7(7)-invariant candidate counterterm exists below 7-loop order. At 7-loops, we characterize the infinite tower of independent supersymmetric operators (DR6)-R-4, R-8, phi R-2(8),... with n > 4 fields and prove that they all violate E-7(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 (DR4)-R-8 has a non-linear supersymmetrization such that its single-and double-soft scalar limits are compatible with E-7(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. (C) 2010 Elsevier B.V. All rights reserved.