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 \( \mathbb{C}{\mathbb{P}^{{2}}} \), we interpret the 1-loop MHV integrand as the volume of a polytope in \( \mathbb{C}{\mathbb{P}^{{3}}} \) × \( \mathbb{C}{\mathbb{P}^{{3}}} \), which can be thought of as the space obtained by taking the geometric dual of the Wilson loop in each \( \mathbb{C}{\mathbb{P}^{{3}}} \) 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 \( \mathbb{C}{\mathbb{P}^{{4}}} \). 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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ArXiv ePrint: 1012.6030
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Arkani-Hamed, N., Bourjaily, J., Cachazo, F. et al. A note on polytopes for scattering amplitudes. J. High Energ. Phys. 2012, 81 (2012). https://doi.org/10.1007/JHEP04(2012)081
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DOI: https://doi.org/10.1007/JHEP04(2012)081