The Five-Loop Four-Point Integrand of N=8 Supergravity as a Generalized Double Copy

TL;DR

<3-5 sentence high-level summary> This work develops and applies a generalized double-copy framework to construct the five-loop, four-point integrand of $N=8$ supergravity from the corresponding $N=4$ super-Yang–Mills data. It combines BCJ discrepancy-function corrections with a maximal-cut unitarity strategy to systematically add contact terms to the naive double copy, enabling gravity predictions from generic gauge-theory representations. The authors provide explicit formulas for level-2 and higher-level contact terms, verify consistency with a broad set of generalized unitarity cuts, and demonstrate ultraviolet finiteness in $D=22/5$ through vacuum-integral analysis and Baikov-based integration techniques. The results offer a practical path toward high-loop gravity amplitudes and illuminate the structure of ultraviolet cancellations in perturbative quantum gravity, with potential extensions to other double-copy theories and higher dimensions.

Abstract

We use the recently developed generalized double-copy procedure to construct an integrand for the five-loop four-point amplitude of N=8 supergravity. This construction starts from a naive double copy of the previously computed corresponding amplitude of N=4 super-Yang-Mills theory. This is then systematically modified by adding contact terms generated in the context of the method of maximal unitarity cuts. For the simpler generalized cuts, whose corresponding contact terms tend to be the most complicated, we derive a set of formulas relating the contact contributions to the violations of the dual Jacobi identities in the relevant gauge-theory amplitudes. For more complex generalized unitarity cuts, which tend to have simpler contact terms associated with them, we use the method of maximal cuts more directly. The five-loop four-point integrand is a crucial ingredient towards future studies of ultraviolet properties of N=8 supergravity at five loops and beyond. We also present a nontrivial check of the consistency of the integrand, based on modern approaches for integrating over the loop momenta in the ultraviolet region.

Figures (11)

• Figure 1: The three diagrams with only cubic vertices contributing to a four-point tree amplitude.
• Figure 2: A half-ladder tree graph, used to define the color factor in Eq. (\ref{['PartialRepresentation']}).
• Figure 3: Sample maximal and next-to-maximal cuts that are determined by the naive double copy. The exposed lines connecting the blobs are on shell. The labels refer to those used in the Mathematica attachment AttachedFile.
• Figure 4: Sample N$^{k}$MCs for a five-loop four-point amplitude. The exposed lines connecting the blobs are on shell. The labels refer to those used in the Mathematica attachment AttachedFile.
• Figure 5: Examples of parent diagrams used in the naive double copy. These are diagrams with only cubic vertices and 16 propagators carrying loop momentum. In our construction of the five-loop four-point amplitude of ${{\cal N}=8}$ supergravity there are a total of 410 such nonvanishing diagrams. The labeling $(0\! : j)$ indicates that it is a level 0 diagram with no collapsed propagators and $j$ is the diagram number, following the labels in the Mathematica attachment.