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arXiv · 1012.6030

A Note on Polytopes for Scattering Amplitudes

Nima Arkani-Hamed·Jacob L. Bourjaily·Freddy Cachazo·Andrew Hodges·Jaroslav Trnka
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Abstract

In this note we continue the exploration of the polytope picture for scattering amplitudes, where amplitudes are associated with the volumes of polytopes in generalized momentum-twistor spaces. After a quick warm-up example illustrating the essential ideas with the elementary geometry of polygons in CP^2, we interpret the 1-loop MHV integrand as the volume of a polytope in CP^3x CP^3, which can be thought of as the space obtained by taking the geometric dual of the Wilson loop in each CP^3 of the product. We then review the polytope picture for the NMHV tree amplitude and give it a more direct and intrinsic definition as the geometric dual of a canonical "square" of the Wilson-Loop polygon, living in a certain extension of momentum-twistor space into CP^4. In both cases, one natural class of triangulations of the polytope produces the BCFW/CSW representations of the amplitudes; another class of triangulations leads to a striking new form, which is both remarkably simple as well as manifestly cyclic and local.

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Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Andrew Hodges, Jaroslav Trnka. 2010-12-29. A Note on Polytopes for Scattering Amplitudes. https://doi.org/10.1007/jhep04(2012)081

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