Four-graviton scattering to three loops in ${\mathcal N}=8$ supergravity

TL;DR

This work delivers the first analytic three-loop four-graviton scattering amplitude in ${\mathcal N}=8$ supergravity, obtained from a known integrand via Laporta-reduction to master integrals and expressed in terms of harmonic polylogarithms with uniform weight $2L$ (here $L=3$, weight $6$). The result is fully permutation-symmetric and written as a sum over permutations of a seed function $f^{(3)}(x)$, with a simple final-entry differential structure. The authors rigorously treat analytic continuation across physical regions, verify the infrared structure and exponentiation, and extract soft/Regge-limit behavior that matches existing predictions. These findings reinforce the perspective that gravity amplitudes exhibit remarkable simplicity and maximal transcendental weight, with implications for higher-point amplitudes and potential hidden symmetries. The work provides a robust platform for exploring the structure of scattering amplitudes in ${\mathcal N}=8$ supergravity and for future investigations into the BCJ/double-copy framework and integrability hints.

Abstract

We compute the three-loop scattering amplitude of four gravitons in ${\mathcal N}=8$ supergravity. Our results are analytic formulae for a Laurent expansion of the amplitude in the regulator of dimensional regularisation. The coefficients of this series are closed formulae in terms of well-established harmonic poly-logarithms. Our results display a remarkable degree of simplicity and represent an important stepping stone in the exploration of the structure of scattering amplitudes. In particular, we observe that to this loop order the four graviton amplitude is given by uniform weight $2L$ functions, where $L$ is the loop order.