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)