The critical ultraviolet behaviour of N=8 supergravity amplitudes

TL;DR

This work analyzes the critical ultraviolet behaviour of the four-graviton amplitude in $N=8$ supergravity to all orders, using the Bern–Carrasco–Johansson double-copy construction and insights from $N=4$ super-Yang–Mills. It shows that starting at four loops the UV structure is controlled by the factorization of the $\partial^8\,\mathcal{R}^4$ operator, implying a logarithmic divergence at seven loops in $D=4$ with a counterterm $\partial^8\,\mathcal{R}^4$. The analysis hinges on representing $N=8$ amplitudes as squares of $N=4$ SYM numerators (with 1PI and 1PR contributions and inverse derivative factors) and on the KLT relation $\mathcal{R}^4=\mathcal{F}^4\tilde{\mathcal{F}}^4$, ensuring consistent UV counting via the double-copy. Overall, the results argue that the seven-loop divergence in $D=4$ is a robust consequence of the $\partial^8\,\mathcal{R}^4$ structure within the proposed parameterization.

Abstract

We analyze the critical ultraviolet behaviour of the four-graviton amplitude in N=8 supergravity to all order in perturbation. We use the Bern-Carrasco-Johansson diagrammatic expansion for N=8 supergravity multiloop amplitudes, where numerator factors are squares of the Lorentz factor of N=4 super-Yang-Mills amplitudes, and the analysis of the critical ultraviolet behaviour of the multiloop four-gluon amplitudes in the single- and double-trace sectors. We argue this implies that the superficial ultraviolet behaviour of the four-graviton N=8 amplitudes from four-loop order is determined by the factorization the k^8 R^4 operator. This leads to a seven-loop logarithmic divergence in the four-graviton amplitude in four dimensions.