Is maximal $\mathcal{N}=8$, $D=4$ supergravity ultraviolet finite to all orders of perturbation theory?

Whether four-dimensional maximal ($\mathcal{N}=8$) supergravity is perturbatively ultraviolet finite to all loop orders is a famous open question in quantum field theory and quantum gravity, posed in this form by Bern, Dixon and Roiban and restated in Freedman's fortieth-anniversary-of-supergravity lecture. Explicit amplitude computations establish finiteness through five loops in $D=4$, and $E_{7(7)}$ duality arguments exclude candidate counterterms below seven loops; but the first possible divergence is predicted at seven loops, no computation reaches that order, and no all-orders finiteness proof or divergence is known. The problem is to settle the all-orders ultraviolet behavior of the $\mathcal{N}=8$ supergravity S-matrix: either prove finiteness by identifying an organizing principle beyond the known symmetries, or determine the loop order and explicit form of the first divergence in four dimensions.

Background

Supergravity is the supersymmetric extension of general relativity, obtained by pairing the graviton with a spin-3/2 gravitino. The maximal four-dimensional theory, $\mathcal{N}=8$ supergravity (the unique local theory with 32 real supersymmetry charges), arises as the low-energy limit of type-II string theory and contains the graviton, 8 gravitini, 28 vector fields, 56 spin-1/2 fermions, and 70 scalars. When quantized perturbatively, pure Einstein gravity is non-renormalizable: Goroff and Sagnotti showed that a local counterterm cubic in the Weyl tensor, $C_{\mu\nu}{}^{\rho\sigma}C_{\rho\sigma}{}^{\tau\omega}C_{\tau\omega}{}^{\mu\nu}$, diverges at two loops, and no four-dimensional gravity theory was conventionally expected to be ultraviolet (UV) finite.

In perturbation theory an $L$-loop UV divergence of the S-matrix is encoded in a local counterterm built from roughly $L+1$ powers of the curvature: $R^4$ would first appear at three loops, $D^4R^4$ at five loops, $D^6R^4$ at six, and $D^8R^4$ at seven (here $D^kR^4$ denotes four Riemann tensors plus $k$ extra derivatives). Supersymmetry forbids the two-loop counterterm, and the maximal theory has a further on-shell continuous duality symmetry, $E_{7(7)}$, acting nonlinearly on the 70 scalars and mixing the 28 vectors with their magnetic duals. Because single-soft scalar emission probes this symmetry, $E_{7(7)}$ constrains which counterterms can appear.

Modern computations use color-kinematics duality (a property of gauge-theory amplitudes whose kinematic numerators obey Jacobi-like identities) and the double copy (gravity amplitudes obtained by replacing gauge-theory color factors by a second copy of the kinematic numerators), which made multi-loop $\mathcal{N}=8$ supergravity amplitudes computable. The complete four-loop four-graviton amplitude is UV finite in $D=4$ and even $D=5$ (Bern, Carrasco, Dixon, Johansson, Roiban, 2009). Soft-scalar analyses proved that the candidate $R^4$, $D^4R^4$ and $D^6R^4$ counterterms violate $E_{7(7)}$, so no $E_{7(7)}$-invariant counterterm exists below seven loops (Beisert, Elvang, Freedman, Kiermaier, Morales, Stieberger, 2010). The five-loop computation (Bern et al., 2018) found the first divergence at the critical dimension $D_c=24/5$, corresponding to $D^8R^4$, leaving the theory finite in $D=4$ at five loops. The observed pattern $D_{\text{crit}}=2+14/L$ conjecturally predicts the first four-dimensional divergence at seven loops ($L=7$ gives $D_{\text{crit}}=4$), but no counterterm is known to be compatible with the full nonlinear supersymmetry and nonlinear $E_{7(7)}$, and enhanced cancellations beyond the standard symmetries have repeatedly surprised expectations. A 2024 review (Nicolai) summarizes the state of the art: finiteness through five loops is established, the seven-loop computation is out of reach, and an all-orders proof cannot be envisaged with order-by-order techniques. Freedman's anniversary lecture states the issue precisely: if the theory is finite to all orders, a new principle beyond the obvious symmetries must be found.

Problem Statement

Determine the ultraviolet behavior of maximal $\mathcal{N}=8$, $D=4$ supergravity to all orders of perturbation theory. Specifically, settle whether the perturbative S-matrix of the theory is ultraviolet finite in four dimensions at every loop order $L\ge3$ (finiteness at one and two loop orders being long established), by one of two routes: (a) an all-orders argument — an organizing principle, symmetry, or non-renormalization theorem beyond the manifest supersymmetry and linearized $E_{7(7)}$ duality — proving that no ultraviolet-divergent counterterm compatible with the full nonlinear supersymmetry and nonlinear $E_{7(7)}$ invariance of the theory contributes at any loop order; or (b) an explicit determination of the first divergence, establishing the loop order at which a four-dimensional counterterm first contributes with nonvanishing coefficient and its local operator structure and coefficient — in particular resolving whether the $D^8R^4$ counterterm contributes at seven loops as the conjectured critical-dimension pattern $D_{\text{crit}}=2+14/L$ suggests, or whether enhanced cancellations or nonlinear $E_{7(7)}$ constraints exclude it. This is the open question as posed in Freedman's lecture (arXiv:1702.02203): 'divergences cancel through 6-loop order, but cannot be excluded beyond this level. If the theory is finite to all orders, a new principle that goes beyond the obvious symmetries must be found.'

References

1. Z. Bern, L. J. Dixon, R. Roiban, 'Is N = 8 Supergravity Ultraviolet Finite?', Phys. Lett. B 644 (2007) 265, arXiv:hep-th/0611086, doi:10.1016/j.physletb.2006.11.030, https://arxiv.org/abs/hep-th/0611086
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