Abstract
In this paper, we present new expressions for n-point NMHV tree-level gravity amplitudes. We introduce a method of factorization diagrams which is a simple graphical representation of R-invariants in Yang-Mills theory. We define the gravity analogues which we call mathcal{G} -invariants, and expand the NMHV gravity amplitudes in terms of these objects. We provide explicit formulas of NMHV gravity amplitudes up to eight points in terms of mathcal{G} -invariants, and give the general definition for any number of points. We discuss the connection to BCFW representation, special behavior under large momentum shift, the role of momentum twistors and the intricate web of spurious poles cancelation. Because of the close connection between R-invariants and the (tree-level) Amplituhedron for Yang-Mills amplitudes, we speculate that the new expansion for gravity amplitudes should correspond to the triangulation of the putative Gravituhedron geometry.
Highlights
Recursion relations [6] and the positive Grassmannian [7,8,9,10,11,12]
The important open question is the existence of the dual Amplituhedron geometry [33,34,35]; in this picture the scattering amplitude should be given by a volume rather than a differential form
We focus on four-dimensional on-shell graviton scattering amplitudes, and initiate the search for the positive geometry which we tentatively call Gravituhedron
Summary
The central object of our interest is the n-pt tree-level Nk−2MHV graviton amplitude with the following helicity configuration, Mn(1−2− . . . k−(k+1)+ . . . n+). N), and the sum is over permutation of n−2 labels making Sn−2 manifest This formula served as a motivation for the explicit solution of BCFW recursion relations for any NkMHV amplitude which takes similar form [120]. Where AMn HV is a the gluon Parke-Taylor factor, Rn;ij is a R-invariant we review and GNn;MijHV are certain kinematical factors with both numerator and denominator terms This formula looks reminiscent of the KLT relations and contains squares of YangMills building blocks. As with all BCFW expressions in (2.23), (2.24), (2.25) the choice of {a, b} shift makes the permutational symmetries of the amplitude in external labels non-manifest. In Yang-Mills the individual BCFW terms manifest the O(1/z) behavior of the amplitude for any good shift This is closely related to the manifest dual conformal symmetry. We leave the attempts to use our formula to discover putative Gravituhedron geometry for future work
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