Amplitudes and Ultraviolet Behavior of N=8 Supergravity

TL;DR

We address the perturbative ultraviolet behavior of maximally supersymmetric gravity by computing the four-graviton amplitude in $N=8$ SUGRA using KLT relations to express gravity amplitudes in terms of $N=4$ sYM data and generalized unitarity cuts. The main approach reconstructs loop integrands from gauge-theory inputs and then squares them via KLT to obtain gravity results, enabling explicit two-, three-, and four-loop results. The key findings show that the critical dimensions for potential divergences, $D_c(L)=4+\frac{6}{L}$ for $L>1$, match between $N=8$ SUGRA and $N=4$ sYM through four loops, with four-loop amplitudes finite in $D<\tfrac{11}{2}$. This provides strong direct evidence for perturbative finiteness of $N=8$ supergravity up to four loops and clarifies which higher-derivative counterterms could appear at later loops, notably highlighting the role of a possible seven-loop $D^8R^4$ counterterm.

Abstract

In this contribution we describe computational tools that permit the evaluation of multi-loop scattering amplitudes in N=8 supergravity, in terms of amplitudes in N=4 super-Yang-Mills theory. We also discuss the remarkable ultraviolet behavior of N=8 supergravity, which follows from these amplitudes, and is as good as that of N=4 super-Yang-Mills theory through at least four loops.

Figures (8)

• Figure 1: Unitarity relations for the four-point amplitude at one and two loops. The number of holes in a blob indicates the number of loops in the corresponding amplitude.
• Figure 2: An example of multi-loop generalized unitarity. The one-loop five-point amplitude, appearing on the right side of the ordinary cut, is further cut into products of trees, in three inequivalent ways.
• Figure 3: A generalized cut with real momenta generates several maximal cuts; the latter contain only three-point tree amplitudes.
• Figure 4: Evaluation of a generalized cut in ${{\cal N}=8}$ supergravity at three loops, in terms of planar and non-planar cuts in ${{\cal N}=4}$ sYM.
• Figure 5: The two-loop amplitude in ${{\cal N}=4}$ sYM. The blob on the right represents the color-ordered tree amplitude $A_4^{\rm tree}$. In the brackets, black lines are kinematic $1/p^2$ propagators, with scalar ($\phi^3$) vertices. Green lines are color $\delta^{ab}$ propagators, with structure constant ($f^{abc}$) vertices.