Absence of Triangles in Maximal Supergravity Amplitudes

TL;DR

The paper develops unordered one-loop integral reductions for colourless theories like gravity, deriving new identities that cancel two powers of loop momentum per reduction step. Applying these to N=8 supergravity, it provides a direct proof of the no-triangle hypothesis: in four dimensions, one-loop n-graviton amplitudes are expressible solely in terms of scalar box integrals (with dimension-shifted pieces canceling). The results extend to less or no supersymmetry, outlining endpoint scalar integrals and the persistence of constructibility for N≥3, while outlining the presence of bubbles or rational pieces for lower SUSY. Overall, the work links crossing symmetry, worldline/string-based formalisms, and novel reduction identities to explain the simplified UV/IR structure of one-loop gravity amplitudes and suggests avenues for higher-loop generalizations.

Abstract

From general arguments, we show that one-loop n-point amplitudes in colourless theories satisfy a new type of reduction formula. These lead to the existence of cancellations beyond supersymmetry. Using such reduction relations we prove the no-triangle hypothesis in maximal supergravity by showing that in four dimensions the n-point graviton amplitude contain only scalar box integral functions. We also discuss the reduction formulas in the context of gravity amplitudes with less and no supersymmetry.