### 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 ^{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.

Original language | English (US) |
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Article number | 081 |

Journal | Journal of High Energy Physics |

Volume | 2012 |

Issue number | 4 |

DOIs | |

Publication status | Published - May 2 2012 |

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### All Science Journal Classification (ASJC) codes

- Nuclear and High Energy Physics

### Cite this

*Journal of High Energy Physics*,

*2012*(4), [081]. https://doi.org/10.1007/JHEP04(2012)081