Gravity loop integrands from the ultraviolet

TL;DR

This work investigates the ultraviolet (UV) structure of gravity loop integrands, revealing surprising improvements in four dimensions that persist up to seven loops. By analyzing multi-particle unitarity cuts and implementing specialized shifts, the authors show that the large-momentum behavior of gravity integrands is constrained in a way that can uniquely fix several N=8 supergravity amplitudes using solely homogeneous, vanishing-cut conditions. In D=4, Gram-determinant cancellations drive additional suppression at infinity, hinting at a deeper geometric structure and potential recursion relations that extend tree-level BCFW-like ideas to loop integrands. The results also differentiate four-dimensional gravity from general D, suggesting special features of four-dimensional amplitudes and opening the path toward a gravity analogue of positive geometry and the Amplituhedron. Overall, the paper advances our understanding of gravity amplitudes at high energy, offering a framework to reconstruct loop integrands from UV properties and proposing new tree-level recursions tied to infinity behavior.

Abstract

We demonstrate that loop integrands of (super-)gravity scattering amplitudes possess surprising properties in the ultraviolet (UV) region. In particular, we study the scaling of multi-particle unitarity cuts for asymptotically large momenta and expose an improved UV behavior of four-dimensional cuts through seven loops as compared to standard expectations. For N=8 supergravity, we show that the improved large momentum scaling combined with the behavior of the integrand under BCFW deformations of external kinematics uniquely fixes the loop integrands in a number of non-trivial cases. In the integrand construction, all scaling conditions are homogeneous. Therefore, the only required information about the amplitude is its vanishing at particular points in momentum space. This homogeneous construction gives indirect evidence for a new geometric picture for graviton amplitudes similar to the one found for planar N=4 super Yang-Mills theory. We also show how the behavior at infinity is related to the scaling of tree-level amplitudes under certain multi-line chiral shifts which can be used to construct new recursion relations.

Figures (4)

• Figure 1: Ambiguity in labeling loop momenta in a given contribution to the integrand.
• Figure 2: UV scaling of $\mathcal{N}=8$ SUGRA, (planar) $\mathcal{N}=4$ SYM, pure GR, and pure (planar)YM multi-particle unitarity cuts under four-dimensional deformations with results up to seven loops. The Scaling axis labels the leading $t$ behavior of the cuts as $t\rightarrow\infty$. The thin lines denote the scaling in D-dimensions, where the continuous part has been checked explicitly and the dashed part is conjectured. There is an overall improvement of one power in the large $t$ limit of gravity cuts with respect to $D$-dimensions; the same is not true for Yang-Mills.
• Figure 3: The integral topologies appearing in our ansatz for the two-loop four-point $\mathcal{N}=8$ SUGRA amplitude.
• Figure 4: Cut topologies considered in the UV construction of the three-loop four-point amplitude in $\mathcal{N}=8$ SUGRA.