PHYSICAL REVIEW D 98, 086021 (2018) Editors' Suggestion Ultraviolet properties of N = 8 supergravity at five loops Zvi Bern,1 John Joseph Carrasco,2 Wei-Ming Chen,1 Alex Edison,1 Henrik Johansson,3,4 Julio Parra-Martinez,1 Radu Roiban,5 and Mao Zeng1 1Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy, University of California at Los Angeles, Los Angeles, California 90095, USA 2Institute of Theoretical Physics, Universit´e Paris-Saclay, CEA-Saclay and CNRS, F-91191 Gif-sur-Yvette cedex, France 3Department of Physics and Astronomy, Uppsala University, 75108 Uppsala, Sweden 4Nordita, Stockholm University and KTH Royal Institute of Technology, Roslagstullsbacken 23, 10691 Stockholm, Sweden 5Institute for Gravitation and the Cosmos, Pennsylvania State University, University Park, Pennsylvania 16802, USA (Received 1 May 2018; published 19 October 2018) We use the recently developed generalized double-copy construction to obtain an improved repre- sentation of the five-loop four-point integrand of N ¼ 8 supergravity whose leading ultraviolet behavior we analyze using state-of-the-art loop-integral expansion and reduction methods. We find that the five-loop 8 4 critical dimension where ultraviolet divergences first occur is Dc ¼ 24=5, corresponding to a D R counterterm. This ultraviolet behavior stands in contrast to the cases of four-dimensional N ¼ 4 supergravity at three loops and N ¼ 5 supergravity at four loops whose improved ultraviolet behavior demonstrates enhanced cancellations beyond implications from standard symmetry considerations. We express this Dc ¼ 24=5 divergence in terms of two relatively simple positive-definite integrals with vanishing external momenta, excluding any additional ultraviolet cancellations at this loop order. We note nontrivial relations between the integrals describing this leading ultraviolet behavior and integrals describing lower-loop behavior. This observation suggests not only a path towards greatly simplifying future calculations at higher loops, but may even allow us to directly investigate ultraviolet behavior in terms of simplified integrals, avoiding the construction of complete integrands. DOI: 10.1103/PhysRevD.98.086021 I. INTRODUCTION pointlike gravity theories. For example, all-loop-order unitarity cuts exhibit remarkable infrared and ultraviolet Since the discovery of supergravity theories [1],a cancellations [2] whose consequences remain to be fully complete understanding of their ultraviolet properties has explored. Indeed, we know of examples in N ¼ 4 [3] and remained elusive. Despite tremendous progress over the N ¼ 5 [4] supergravity theories that display “enhanced years, many properties of gravitational perturbation theory cancellations” [5–9], where quantum corrections exclude remain unknown. Power-counting arguments, driven by the counterterms thought to be consistent with all known dimensionality of Newton’s constant, suggest that all symmetries. In addition, there are indications that anoma- pointlike theories of gravity should develop an ultraviolet lies in known symmetries of supergravity theories play a divergence at a sufficiently high-loop order. However, if a role in the appearance of ultraviolet divergences [10,11]. pointlike theory were ultraviolet finite, it would imply the existence of an undiscovered symmetry or structure that Restoration of these symmetries in S-matrix elements by should likely have a fundamental impact on our under- finite local counterterms may lead to the cancellation of standing of quantum gravity. Explicit calculations in recent known divergences. In this paper, we take a step forward by years have revealed the existence of hidden properties, not presenting a detailed analysis of the ultraviolet behavior of the five-loop four-point scattering amplitude in the max- readily apparent in Lagrangian formulations. One might N 8 1 wonder whether these tame the ultraviolet behavior of imally supersymmetric theory, ¼ supergravity [12], Published by the American Physical Society under the terms of 1Strictly speaking the maximally supersymmetric theory the Creative Commons Attribution 4.0 International license. is only recognized as N ¼ 8 supergravity in four dimensions. Further distribution of this work must maintain attribution to While we concern ourselves with mainly higher dimensions, in the author(s) and the published article’s title, journal citation, this paper we take the liberty to apply the four-dimensional and DOI. Funded by SCOAP3. nomenclature. 2470-0010=2018=98(8)=086021(36) 086021-1 Published by the American Physical Society ZVI BERN et al. PHYS. REV. D 98, 086021 (2018) and observe properties that should help us determine its reach. Fortunately the results of our current analysis, four-dimensional ultraviolet behavior at even higher loops. when combined with earlier work at lower loops Its many symmetries suggest that, among the pointlike [5,6,11,19,20,24], reveal highly nontrivial constraints on theories of gravity, the maximally supersymmetric theory the subloops of integrals describing the leading ultraviolet has the softest ultraviolet behavior. These symmetry prop- behavior through five loops. These patterns suggest not erties also make it technically easier to explore and only new efficient techniques to directly determine the understand its structure. Over the years there have been ultraviolet behavior at ever higher loops, but potentially many studies and predictions for the ultraviolet behavior of undiscovered principles governing the ultraviolet consis- N ¼ 8 supergravity [13,14]. The current consensus, based tency. In this work we will describe these observed on standard symmetry considerations, is that N ¼ 8 super- constraints, leaving their detailed study for the future. gravity in four dimensions is ultraviolet finite up to at least The results of this paper are the culmination of many seven loops [15–17]. Through four loops, direct compu- advances in understanding and computing gauge and tation using modern scattering amplitude methods prove gravity scattering amplitudes at high-loop orders. The that the critical dimension of N ¼ 8 supergravity where unitarity method [25,26] has been central to this progress divergences first occur is [18–20] because of the way that it allows on-shell simplifications to be exploited in the construction of new higher-loop 6 amplitudes. We use its incarnation in the maximal-cut D ¼ þ 4; ð2 ≤ L ≤ 4Þð1:1Þ c L (MC) organization [26] to systematically build complete integrands [27,28]. where L is the number of loops. This matches the formula The unitarity method combines naturally with double- [18,21] for N ¼ 4 super-Yang-Mills theory [22] which is copy ideas, including the field-theoretic version of the known to be an ultraviolet finite theory in D ¼ 4 [23].At string-theory Kawai, Lewellen and Tye (KLT) relations one loop the critical dimension, for both N ¼ 4 super- between gauge and gravity tree amplitudes [29] and the N 8 8 Yang-Mills theory and ¼ supergravity [13],isDc ¼ . related Bern, Carrasco and Johansson (BCJ) color- We define the theories in dimensions D>4 via dimen- kinematics duality and double-copy construction [30,31]. sional reduction of N ¼ 1 supergravity in D ¼ 11 and The double-copy relationship reduces the problem of con- N ¼ 1 super-Yang-Mills theory in D ¼ 10 [13]. structing gravity integrands to that of calculating much In this paper we address the longstanding question of simpler gauge-theory ones. For our calculation, a generali- whether Eq. (1.1) holds for N ¼ 8 supergravity at five zation [27] of the double-copy procedure has proven loops. Symmetry arguments [16] suggest D8R4 as a valid invaluable [28]. counterterm and that the critical dimension for the five-loop The analysis in Ref. [28] finds the first representation divergence should be Dc ¼ 24=5 instead of that suggested of an integrand for the five-loop four-point amplitude of by Eq. (1.1), Dc ¼ 26=5. (See also Refs. [15,17].) Such N ¼ 8 supergravity. The high power counting of that arguments, however, cannot ascertain whether quantum representation obstructs the necessary integral reductions corrections actually generate an allowed divergence. needed to extract its ultraviolet behavior. Here we use Indeed, explicit three-loop calculations in N ¼ 4 super- similar generalized double-copy methods [27] to construct gravity and four-loop calculations in N ¼ 5 supergravity an improved integrand that enormously simplifies the reveal that while counterterms are allowed by all known integration. The key is starting with an improved gauge- symmetry considerations, their coefficients vanish [5,6]. theory integrand, which we build by constraining a mani- These enhanced cancellations are nontrivial and only fest-power-counting ansatz via the method of maximal cuts. manifest upon applying Lorentz invariance and a repar- The needed unitarity cuts are easily obtained from the ametrization invariance to the loop integrals [8]. This gauge-theory integrand of Ref. [32]. implies that the only definitive way to settle the five-loop The earlier representation of the supergravity integrand, question is to directly calculate the coefficient of the given in Ref. [28], is superficially (though not actually) 8 4 potential D R counterterm in D ¼ 24=5, as we do here. quartically divergent in the dimension of interest. The new This counterterm is of interest because it is the one that representation shifts these apparent quartic divergences to would contribute at seven loops if N ¼ 8 supergravity