Unexpected Cancellations in Gravity Theories

TL;DR

This work shows that pure Einstein gravity exhibits significant one-loop cancellations that reduce the expected powers of loop momentum in bubble and triangle integrals, hinting at improved ultraviolet behavior beyond naive power counting. Using unitarity and spinor-helicity techniques, the authors demonstrate universal bounds (∼ four powers for bubbles and ∼ six for triangles) and corroborate these results with explicit six-point examples and all-n analyses up to ten points. They connect these cancellations to tree-level large-z scaling and factorization, arguing that the observed structure is partly generic to gravity and partially enhanced by supersymmetry in ${ m N} ext{=}8$ supergravity, consistent with the no-triangle hypothesis. The findings suggest a broader, potentially finite ultraviolet behavior for gravity theories based on the Einstein–Hilbert action, with important implications for higher-loop computations and effective actions. The work lays a framework for future higher-loop tests and for translating these cancellations into concrete statements about UV properties and EFT structures.

Abstract

Recent computations of scattering amplitudes show that N=8 supergravity is surprisingly well behaved in the ultraviolet and may even be ultraviolet finite in perturbation theory. The novel cancellations necessary for ultraviolet finiteness first appear at one loop in the guise of the "no-triangle hypothesis". We study one-loop amplitudes in pure Einstein gravity and point out the existence of cancellations similar to those found previously in N=8 supergravity. These cancellations go beyond those found in the one-loop effective action. Using unitarity, this suggests that generic theories of quantum gravity based on the Einstein-Hilbert action may be better behaved in the ultraviolet at higher loops than suggested by naive power counting, though without additional (supersymmetric) cancellations they diverge. We comment on future studies that should be performed to support this proposal.

Figures (6)

• Figure 1: A gravity Feynman diagram. Individual diagrams display bad behavior as $z \rightarrow \infty$.
• Figure 2: The different types of box, bubble and triangle integrals are characterized by the momenta $K_i$ at each corner. The $K_i$ are sums of momenta of external particles. Non-trivial cancellations of numerator loop momentum occur within a given integral type.
• Figure 3: The (a) quadruple, (b) triple and (c) ordinary double cut. (The minus sign in the definition of $K_2$ in the triple cut follows the conventions of ref. Forde.) We take $l_0 \equiv l$.
• Figure 4: Example bubble and triangle integrals appearing in the six-point amplitude.
• Figure 5: An example of a Feynman diagram giving the worst behaved contribution to the bubble integral in fig. \ref{['SixBubbleTriangleFigure']}(a). Under a Passarino-Veltman reduction a hexagon integral in gravity gives bubble integrals with up to eight powers of loop momentum in the numerator. The dashed line represents the channel used to evaluate the contribution of this diagram to the bubble integral in fig. \ref{['SixBubbleTriangleFigure']}(a) via the unitarity cuts.