Abstract
We define recursion relations for N = 8 supergravity amplitudes using a generalization of the on-shell diagrams developed for planar N = 4 super-Yang-Mills. Although the recursion relations generically give rise to non-planar on-shell diagrams, we show that at tree-level the recursion can be chosen to yield only planar diagrams, the same diagrams occurring in the planar N = 4 theory. This implies non-trivial identities for non-planar diagrams as well as interesting relations between the N = 4 and N = 8 theories. We show that the on-shell diagrams of N = 8 supergravity obey equivalence relations analogous to those of N = 4 super-Yang-Mills, and we develop a systematic algorithm for reading off Grassmannian integral formulae directly from the on-shell diagrams. We also show that the 1-loop 4-point amplitude of N = 8 supergravity can be obtained from on-shell diagrams.
Highlights
On-shell diagrams and their correspondence to positive cells of the Grassmannian suggest a geometric interpretation of scattering amplitudes as the volume of a new object known as the Amplituhedron [21,22,23]
We show that the on-shell diagrams of N = 8 supergravity obey equivalence relations analogous to those of N = 4 super-Yang-Mills, and we develop a systematic algorithm for reading off Grassmannian integral formulae directly from the on-shell diagrams
We show that the 1-loop 4-point amplitude of N = 8 SUGRA can be obtained from on-shell diagrams, which suggests the possiblity of formulating loop-level BCFW recursion in this theory as well
Summary
As mentioned in the introduction, the difficulties of Feynman diagrams can be overcome by using BCFW recursion relations to express higher-point on-shell amplitudes in terms of lower-point on-shell amplitudes. The building blocks for on-shell diagrams are 3-point MHV and anti-MHV amplitudes, which encode the scattering of three gluons or gravitons with helicity {− − +} and {+ + −}, respectively. More general on-shell diagrams are constructed by connecting 3-point vertices and integrating over the on-shell supermomenta associated with the internal edges between two vertices: dμ =. In order to construct on-shell diagrams corresponding to higher-point amplitudes, one uses the BCFW bridge:. Parameterizing the momentum through the internal edge by αλ1λn, one finds that λ1λˆ1 = λ1 λ1 − αλn λnλn = (λn + αλ1) λn This diagram corresponds to BCFW shifting legs (1, n). On-shell diagrams for higher-point tree-level scattering amplitudes can be constructed by connecting on-shell diagrams for lower-point amplitudes with a BCFW bridge and summing over all permutations of the unshifted legs, as depicted in figure 1
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