The No-Triangle Hypothesis for N=8 Supergravity

TL;DR

This paper investigates the perturbative structure of four-dimensional ${\cal N}=8$ supergravity through the no-triangle hypothesis, which posits that one-loop amplitudes are expressed solely in terms of scalar box integrals: $M^{\rm 1\hbox{-}loop}_{{\cal N}=8}=\sum_{i\in \mathcal{C}} c_i\, I_4^i$. By combining IR soft-divergence constraints, unitarity (two- and triple-cut) analyses, and factorisation principles, the authors prove the six-point NMHV amplitude can be written as boxes only, and provide strong evidence that seven-point and higher-point amplitudes share this structure. The approach leverages relations to ${\cal N}=4$ SYM (box-dominance there), KLT-type gravity–YM connections, and large-$z$ shift tests to rule out triangles, bubbles, and rational contributions, thereby suggesting a deeper symmetry. If these box-only properties persist at higher points and loops, they would imply a softer UV behaviour for ${\cal N}=8$ supergravity and point toward potential finiteness, possibly connected to twistor-string dualities.

Abstract

We study the perturbative expansion of N=8 supergravity in four dimensions from the viewpoint of the ``no-triangle'' hypothesis, which states that one-loop graviton amplitudes in N=8 supergravity only contain scalar box integral functions. Our computations constitute a direct proof at six-points and support the no-triangle conjecture for seven-point amplitudes and beyond.