N=8 Supergravity 4-point Amplitudes

TL;DR

The authors compute and cross-check bosonic 4-point amplitudes in $N=8$ supergravity using a generating-function approach, covering tree and 1-loop results for vectors and scalars and extracting explicit 3-loop UV-divergence candidates. They introduce a supersymmetric complex deformation to study large-m momentum behavior and show 1-loop finite amplitudes decay at large $z$ while the 3-loop counterterm does not, illuminating constraints on potential higher-loop divergences. They establish that the perturbative amplitudes preserve the continuous $E_{(7,7)}$ symmetry at both tree and 1-loop levels and derive a low-energy theorem from this symmetry for complex momenta, with implications for higher-point 1-loop amplitudes. Overall, the results support the observed 3-loop finiteness of $N=8$ supergravity and reveal how $E_{(7,7)}$ symmetry constrains UV structure and soft limits, guiding thoughts on possible all-loop finiteness.

Abstract

We present the explicit expressions in N=8 supergravity for the bosonic 4-particle tree and 1-loop amplitudes including vectors and scalars. We also present the candidate 4-point UV divergences in a form of helicity amplitudes, corresponding to 3-loop manifestly N=8 supersymmetric and Lorentz covariant counterterm. This may shed some light on the 3-loop finiteness of N=8 SG and on a conjectured higher loop finiteness. We perform a supersymmetric deformation to complex momentum of the 4-point generating function including higher-loop counterterms and the 1-loop UV finite amplitudes. Using the explicit form of the scalar part of the 3-loop counterterm and of the 1-loop UV finite scalar 4-point amplitudes we find that they both have an unbroken E7 symmetry. We derive from E7 symmetry the low-energy theorem for the 1-loop n-point amplitudes.