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 3×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.
All Science Journal Classification (ASJC) codes
- Nuclear and High Energy Physics