A Note on Polytopes for Scattering Amplitudes

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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 CP², we interpret the 1-loop MHV integrand as the volume of a polytope in CP³x CP³, which can be thought of as the space obtained by taking the geometric dual of the Wilson loop in each CP³ 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 CP⁴. 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.