Gravity on-shell diagrams

Journal of High Energy Physics, Nov 2016

We study on-shell diagrams for gravity theories with any number of super-symmetries and find a compact Grassmannian formula in terms of edge variables of the graphs. Unlike in gauge theory where the analogous form involves only d log-factors, in gravity there is a non-trivial numerator as well as higher degree poles in the edge variables. Based on the structure of the Grassmannian formula for \( \mathcal{N}=8 \) supergravity we conjecture that gravity loop amplitudes also possess similar properties. In particular, we find that there are only logarithmic singularities on cuts with finite loop momentum and that poles at infinity are present, in complete agreement with the conjecture presented in [1].

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Gravity on-shell diagrams

Received: June Gravity on-shell diagrams Open Access 0 1 4 c The Authors. 0 1 4 0 Department of Physics, University of California 1 Pasadena , CA 91125 , U.S.A 2 Center for Quantum Mathematics and Physics (QMAP) 3 Walter Burke Institute for Theoretical Physics, California Institute of Technology 4 nite loop momentum and that poles We study on-shell diagrams for gravity theories with any number of supernd a compact Grassmannian formula in terms of edge variables of the graphs. Unlike in gauge theory where the analogous form involves only d log-factors, in gravity there is a non-trivial numerator as well as higher degree poles in the edge variables. Based on the structure of the Grassmannian formula for N = 8 supergravity we conjecture that gravity loop amplitudes also possess similar properties. In particular, we there are only logarithmic singularities on cuts with at in nity are present, in complete agreement with the conjecture presented in [1]. Scattering Amplitudes; Spacetime Singularities 1 Introduction 2 3 2.1 On-shell diagrams Grassmannian formulation Non-planar on-shell diagrams First look: MHV leading singularities Three point amplitudes with spin s Grassmannian formula Properties of gravity on-shell diagrams Calculating on-shell diagrams More examples Structure of singularities Background material on Grassmannian formulation of on-shell diagrams From on-shell diagrams to scattering amplitudes Non-planar N = 4 sYM amplitudes Gravity from Yang-Mills Collinear behavior Introduction The study of scattering amplitudes was revolutionized in the last two decades by the advent of modern on-shell techniques [2{8], making accessible calculations of new amplitudes with large numbers of loops and legs. The ability to calculate higher loop amplitudes is exciting both from the practical point of view of a collider physicists as well as from the formal side. Studying the structure of this theoretical \data" led to an enormous advance in our Yang-Mills (sYM), due to its relative simplicity at loop level compared to QCD for example. Taking the large N limit of the gauge group, the planar theory is even simpler and spawned most of the newly discovered structures, including dual conformal symmetry [9{ 11], Yangian symmetry [12], integrability [13, 14], a dual interpretation of amplitudes in terms of Wilson loops [15{20], the expansion of amplitudes in special kinematic limits at nite coupling using OPE methods [21{23], the hexagon-function bootstrap [24{26] heavily using symbols and cluster polylogarithmics [27{30], as well as a variety of other structures. More recently, scattering amplitudes were expressed in terms of on-shell diagrams and the positive Grassmannian [31{37] (see related work in refs. [38{41]). The Grassmannian formulation of on-shell diagram is of geometric avor but expanding the amplitudes in terms of those objects still required recursion relations and unitarity. In the following, ArkaniHamed and one of the authors achieved a completely geometric description of scattering amplitudes as \volumes" in the amplituhedron [42] (see also refs. [43{49]). Interestingly, the novel formulations of on-shell diagrams and scattering amplitudes make surprising connections to active areas of mathematics, ranging from algebraic geometry to combinatorics (see e.g. refs. [50{55]). extension beyond the planar limit. If the geometric picture is indeed a more general feature eld theory we should see hints in theories other than the simplest toy example. In collaboration with Bern et al. we initiated this line of thought by nding evidence for an extension of dual conformal invariance, the formulation in terms of on-shell diagrams, as theory [1, 56]. In this paper, we focus on the dual description of gravity on-shell diagrams in terms of the Grassmannian. On-shell diagrams are interesting objects by themselves. On one hand, they have direct physical relevance as cuts of loop amplitudes and serve as important reference data in the generalized unitarity method [4{8]. Furthermore, they are building blocks for tree amplitudes via the BCFW recursion relation [2, 3]. On the other hand, they are completely well-de ned functions and one might wonder about their analytic properties. Taking the importance of the Grassmannian description of on-shell diagrams we initiate the exploration of the Grassmannian formulation for gravity. in the dual formulation, led us to explore the d log structure of amplitudes [1, 56], our new gravity formula in eq. (3.24) shows novel features that inspire us to test analogous properties on amplitudes directly. In particular, our Grassmannian formula involves nontrivial numerator factors that make manifest the vanishing of the gravity on-shell forms when the legs of any three-point amplitude inside a diagram become collinear. We demonstrate on 1-loop and 2-loop examples that loop amplitudes possess the same behavior on collinear cuts. Our analysis indicates that this is a highly non- (...truncated)


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Enrico Herrmann, Jaroslav Trnka. Gravity on-shell diagrams, Journal of High Energy Physics, 2016, pp. 136, Volume 2016, Issue 11, DOI: 10.1007/JHEP11(2016)136