Is N = 8 Supergravity Ultraviolet Finite?

TL;DR

The paper investigates whether N=8 supergravity in four dimensions is ultraviolet finite by leveraging the no-triangle hypothesis and the unitarity method to relate gravity amplitudes to N=4 super-Yang-Mills amplitudes via KLT relations. It presents evidence that higher-loop divergences may be canceled similarly to gauge theory, suggesting a finiteness bound in D=4 and first potential divergences at higher loops than previously predicted. To solidify this claim, it advocates constructing complete higher-loop amplitudes (including non-planar contributions) using unitarity and exploring possible underlying dynamical principles, such as dualities or twistor-based structures. If confirmed, these cancellations could reveal deeper symmetries and dramatically impact our understanding of quantum gravity.

Abstract

Conventional wisdom holds that no four-dimensional gravity field theory can be ultraviolet finite. This understanding is based mainly on power counting. Recent studies confirm that one-loop N = 8 supergravity amplitudes satisfy the so-called `no-triangle hypothesis', which states that triangle and bubble integrals cancel from these amplitudes. A consequence of this hypothesis is that for any number of external legs, at one loop N = 8 supergravity and N = 4 super-Yang-Mills have identical superficial degrees of ultraviolet behavior in D dimensions. We describe how the unitarity method allows us to promote these one-loop cancellations to higher loops, suggesting that previous power counts were too conservative. We discuss higher-loop evidence suggesting that N = 8 supergravity has the same degree of divergence as N = 4 super-Yang-Mills theory and is ultraviolet finite in four dimensions. We comment on calculations needed to reinforce this proposal, which are feasible using the unitarity method.

Figures (5)

• Figure 1: Diagram (a) corresponds to a contribution appearing in the iterated two-particle cut of fig. \ref{['CutsFigure']}(a). In ${\cal N}=4$ super-Yang-Mills the iterated two-particle cuts give a numerator factor of $(l_1+k_4)^2$. In ${\cal N}=8$ supergravity it is $[(l_1+k_4)^2]^2$. Diagram (b) contains a non-planar contribution which is not detectable in the iterated two-particle cut of fig. \ref{['CutsFigure']}(a), but is detectable in the cut of fig. \ref{['CutsFigure']}(b).
• Figure 2: An iterated two-particle cut (a) and a three-particle cut (b).
• Figure 3: From the "no-triangle hypothesis" all one-loop subamplitudes appearing in unitarity cuts in ${\cal N}=8$ supergravity have the same degree of divergence as in ${\cal N}=4$ super-Yang-Mills theory. The cut (a) is an $L$-particle cut of an $L$-loop amplitude. Cut (b) makes use of generalized unitarity; if a leg is external to the entire amplitude, it should not be cut.
• Figure 4: An $L$-loop contribution as proposed in ref. BDDPR. In ${\cal N}=4$ super-Yang-Mills theory the numerator factor is $[(l+ k_4)^2]^{(L-2)}$, while in ${\cal N}=8$ supergravity the factor is $[(l+k_4)^2]^{2(L-2)}$.
• Figure 5: An example of a potential ${\cal N}=8$ supergravity contribution where the no-triangle hypothesis does not provide sufficient information to rule out a violation of the bound (\ref{['SuperYangMillsPowerCount']}). The numerator factor proposed in ref. BDDPR is $[(l+k_4)^2]^2$, although when evaluated on the iterated two-particle cuts used in its construction, it is indistinguishable from $(2\, l\cdot k_4)^2$, which is consistent with the no-triangle bound.