On UV Finiteness of the Four Loop N=8 Supergravity

TL;DR

Addresses the UV finiteness of the 4-loop, 4-point amplitude in $N=8$ supergravity and the possibility of a 5-point divergence. Using linearized superspace with the dimensionless superfield $W_{ijkl}$, the author tests all potential SUSY completions of a $R^5$ term, including various $W^5$ configurations and derivative insertions. The main result is that no $N=8$ SUSY invariant corresponding to $R^5$ exists, so the 4-loop amplitudes remain UV finite and all higher-point amplitudes are also finite. This finding aligns with harmonic superspace analyses and explicit computations, and motivates future work using light-cone superspace and the $E_{7(7)}$ symmetry to pursue an all-loop finiteness proof.

Abstract

The 4-loop 4-point amplitude in N=8 d=4 supergravity is UV finite due to supersymmetry. Even better UV behavior of the 4-loop 4-point amplitude, analogous to that of N=4 SYM theory, has been recently established by computation in [1]. All n-point 4-loop amplitudes with n greater than 5 are finite on dimensional grounds. However, the situation with the 5-point amplitudes remained unclear. In this paper we will show that the 5-point 4-loop amplitude must be finite due to supersymmetry, despite the fact that $R^5$ has a supersymmetric generalization for N=1, N=2 and N=4 SUSY. This means that all 4-loop amplitudes in N = 8 supergravity are UV finite. We also discuss the current expectations for higher loops.