We present an identity satisfied by the kinematic factors of diagrams describing the tree amplitudes of massless gauge theories. This identity is a kinematic analog of the Jacobi identity for color factors. Using this we find new relations between color-ordered partial amplitudes. We discuss applications to multiloop calculations via the unitarity method. In particular, we illustrate the relations between different contributions to a two-loop four-point QCD amplitude. We also use this identity to reorganize gravity tree amplitudes diagram by diagram, offering new insight into the structure of the KLT relations between gauge and gravity tree amplitudes. This can be used to obtain novel relations similar to the KLT ones. We expect this to be helpful in higher-loop studies of the ultraviolet properties of gravity theories.

We present a technique which utilizes unitarity and collinear limits to construct ansätze for one-loop amplitudes in gauge theory. As an example, we obtain the one-loop contribution to amplitudes for n gluon scattering in N = 4 supersymmetric Yang-Mills theory with the helicity configuration of the Parke-Taylor tree amplitudes. We prove that our N = 4 ansatz is correct using general properties of the relevant one-loop n-point integrals. We also give the "splitting amplitudes" which govern the collinear behavior of one-loop helicity amplitudes in gauge theories.

We compute the leading-color (planar) three-loop four-point amplitude of N = 4 supersymmetric Yang-Mills theory in 4 − 2ǫ dimensions, as a Laurent expansion about ǫ = 0 including the finite terms. The amplitude was constructed previously via the unitarity method, in terms of two Feynman loop integrals, one of which has been evaluated already. Here we use the Mellin-Barnes integration technique to evaluate the Laurent expansion of the second integral. Strikingly, the amplitude is expressible, through the finite terms, in terms of the corresponding one-and two-loop amplitudes, which provides strong evidence for a previous conjecture that higher-loop planar N = 4 amplitudes have an iterative structure. The infrared singularities of the amplitude agree with the predictions of Sterman and Tejeda-Yeomans based on resummation. Based on the four-point result and the exponentiation of infrared singularities, we give an exponentiated ansatz for the maximally helicity-violating n-point amplitudes to all loop orders. The 1/ǫ 2 pole in the four-point amplitude determines the soft, or cusp, anomalous dimension at three loops in N = 4 supersymmetric YangMills theory. The result confirms a prediction by Kotikov, Lipatov, Onishchenko and Velizhanin, which utilizes the leading-twist anomalous dimensions in QCD computed by Moch, Vermaseren and Vogt. Following similar logic, we are able to predict a term in the three-loop quark and gluon form factors in QCD.2

In a previous paper we observed that (classical) tree-level gauge theory amplitudes can be rearranged to display a duality between color and kinematics. Once this is imposed, gravity amplitudes are obtained using two copies of gauge-theory diagram numerators. Here we conjecture that this duality persists to all quantum loop orders and can thus be used to obtain multiloop gravity amplitudes easily from gauge-theory ones. As a nontrivial test, we show that the three-loop four-point amplitude of N = 4 super-Yang-Mills theory can be arranged into a form satisfying the duality, and by taking double copies of the diagram numerators we obtain the corresponding amplitude of N = 8 supergravity. We also remark on a non-supersymmetric two-loop test based on pure Yang-Mills theory resulting in gravity coupled to an anti-symmetric tensor and dilaton.PACS numbers: 04.65.+e, 11.15. Bt, 11.25.Db, 12.60.Jv Although gauge and gravity theories have rather different physical behaviors we know that they are intimately linked. The celebrated AdS/CFT correspondence [1] is the most striking such example, linking maximally supersymmetric gauge theory to supergravity in AdS space. We also know that at weak coupling the tree-level (classical) scattering amplitudes of gauge and gravity theories are deeply intertwined because of the Kawai, Lewellen and Tye (KLT) relations [2].Recent years have seen a renaissance in the study of scattering amplitudes driven in part by the resurgence of collider physics with the recent start up of the Large Hadron Collider at CERN and by the realization that scattering amplitudes have far simpler and richer structures than visible from Feynman diagrams. Striking examples are the discoveries of twistor-space [3] and Grassmannian structures [4] in four dimensions for N = 4 super-Yang-Mills (sYM) theory, as well as interpolations between weak and strong coupling [5][6][7]. In another development we noted [8] that at tree level we could impose a duality between color and kinematics for gauge theories, without altering the amplitudes. This has important consequences in clarifying the tree-level relation between gravity and gauge theory. As we shall argue this duality also greatly clarifies the multiloop structure of (super)gravity theories.The key tool for our studies of loop amplitudes has been the unitarity method [9]. An important refinement which simplifies multiloop studies is the method of maximal cuts [10,11], which relies on generalized unitarity [12]. Here we will make use of these tools to present an all-loop extension of recently discovered tree-level relations. As we shall explain, this allows us to immediately write down multiloop gravity amplitudes directly from gauge-theory multiloop amplitudes once they have been organized to respect the duality between kinematics and color.To understand the relationship between tree-level gravity and gauge theory amplitudes, consider a gauge-theory amplitude where all particles are in the adjoint color representation. By exercising the trivial ability to absorb...

We identify a large class of one-loop amplitudes for massless particles that can be constructed via unitarity from tree amplitudes, without any ambiguities. One-loop amplitudes for massless supersymmetric gauge theories fall into this class; in addition, many non-supersymmetric amplitudes can be rearranged to take advantage of the result. As applications, we construct the one-loop amplitudes for n-gluon scattering in N = 1 supersymmetric theories with the helicity configuration of the Parke-Taylor tree amplitudes, and for six-gluon scattering in N = 4 super-Yang-Mills theory for all helicity configurations.