On UV Finiteness of the Four Loop N=8 Supergravity

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.