arXiv:1202.3423v2 [hep-th] 28 Jul 2012 CERN–PH–TH/2012/041 UCLA/12/TEP/101 Absence of Three-Loop Four-Point Divergences in N =4 Supergravity Zvi Bern a,b , Scott Davies a , Tristan Dennen a , and Yu-tin Huang a,c a Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095-1547, USA b Theory Division, Physics Department, CERN, CH–1211 Geneva 23, Switzerland c School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA We compute the coefficient of the potential three-loop divergence in pure N = 4 supergravity and show that it vanishes, contrary to expectations from symmetry arguments. The recently uncovered duality between color and kinematics is used to greatly streamline the calculation. We comment on all-loop cancellations hinting at further surprises awaiting discovery at higher loops. PACS numbers: 04.65.+e, 11.15.Bt, 11.30.Pb, 11.55.Bq Recent years have seen a resurgence of interest in the possibility that certain supergravity theories may be ul- traviolet finite. This question had been carefully studied in the late 70’s and early 80’s in the hope of using su- pergravity to construct fundamental theories of gravity. The conclusion of these early studies was that nonrenor- malizable ultraviolet divergences would almost certainly appear at a sufficiently large number of quantum loops, though this remains unproven. Although supersymme- try tends to tame the ultraviolet divergences, it does not appear to be sufficient to overcome the increasingly poor ultraviolet behavior of gravity theories stemming from the two-derivative coupling. The consensus opinion from that era was that all pure supergravity theories would likely diverge at three loops (see e.g. ref. [1]), though with assumptions, certain divergences are perhaps de- layed a few extra loop orders [2]. More recently, direct calculations of divergences in su- pergravity theories have been carried out [3–6], shed- ding new light on this issue. From these studies we now know that through four loops maximally supersymmet- ric N = 8 supergravity is finite in space-time dimen- sions, D< 6/L + 4 for L =2, 3, 4 loops. These cal- culations also tell us that the bound is saturated. In D = 4, E 7(7) duality symmetry [7] has recently been used to imply ultraviolet finiteness below seven loops [8], also explaining the observed lack of divergences. In a parallel development, string theory and a first quantized formalism use supersymmetry considerations to arrive at similar conclusions [9]. The latter approach leads to D- dimensional results consistent with the explicit calcula- tions through four loops, but predicts a worse behavior starting at L = 5. At seven loops, the potential four- graviton counterterm of N = 8 supergravity [10] appears to be consistent with all known symmetries [8, 11]. (See ref. [12] for a more optimistic opinion.) More generally, 1/N -BPS operators serve as potential counterterms for N =4, 5, 6, 8 supergravity at L =3, 4, 5, 7 loops, respec- tively, suggesting that in D = 4 ultraviolet divergences will occur at these loop orders in these theories [11]. It therefore might seem safe to conclude that N = 4 super- FIG. 1: A sample cut at three loops displaying cancellations in N = 4 supergravity special to four dimensions. gravity [13] in particular will diverge at three loops. On the other hand, studies of scattering amplitudes suggest that additional ultraviolet cancellations will be found beyond these. We know that even pure Einstein gravity at one loop exhibits remarkable cancellations as the number of external legs increases [14]. Through uni- tarity, such cancellations feed into nontrivial ultraviolet cancellations at all loop orders [15]. In addition, the proposed double-copy structure of gravity loop ampli- tudes [16] suggests that gravity amplitudes are more con- strained than symmetry considerations suggest. In this Letter we show that the ultraviolet properties of N =4 supergravity are indeed better than had been anticipated. To motivate the possibility of hidden cancellations in N = 4 supergravity, consider the unitarity cut displayed in fig. 1 isolating a one-loop subamplitude in a three- loop amplitude. As noted in refs. [14, 17], at one loop a five-point diagram in an N = 4 supergravity amplitude effectively can have up to five powers of loop momenta in the numerator, similar to the power counting of pure Yang-Mills theory. There are also three additional pow- ers of numerator loop momentum coming from the tree amplitude on the right-hand side of the cut, giving a total of at least eight powers of numerator loop momen- tum. Taking into account three loop integrals and ten propagators suggests that this amplitude should diverge at least logarithmically in D = 4. (The power count- ing analysis of this cut performed in ref. [17] assumed that additional powers of numerator loop momenta com- ing from the tree amplitude in the cut can be ignored, contrary to our analysis.) However, this type of power counting is too na¨ ıve and does not account for the special property that no one-