The Ultraviolet Behavior of N=8 Supergravity at Four Loops

TL;DR

The construction of the complete four-loop four-particle amplitude of N=8 supergravity is described, and the amplitude is ultraviolet finite, not only in four dimensions, but in five dimensions as well.

Abstract

We describe the construction of the complete four-loop four-particle amplitude of N=8 supergravity. The amplitude is ultraviolet finite, not only in four dimensions, but in five dimensions as well. The observed extra cancellations provide additional non-trivial evidence that N=8 supergravity in four dimensions may be ultraviolet finite to all orders of perturbation theory.

Figures (4)

• Figure 1: Vacuum graphs from which one can build the contributing four-point graphs by attaching external legs. They are also useful for classifying the UV divergences.
• Figure 2: Four of the 50 distinct graphs corresponding to the integrals composing the result for $M_4^{\rm 4\hbox{-}loop}$.
• Figure 3: Evaluating these 11 cuts, along with 15 two-particle reducible cuts, suffices to uniquely determine the four-loop four-point amplitude. Each blob denotes a tree amplitude.
• Figure 4: Two of the vacuum relations used to analyze the $D=5$ divergence. They are valid in $D=5-2\epsilon$ to order $1/\epsilon$. Dots denote doubled propagators and $l^2_{1,2}=(l_1+l_2)^2$ represents a numerator factor inside the integral.