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A note on polytopes for scattering amplitudes

  • N. Arkani-Hamed
  • J. Bourjaily
  • F. Cachazo
  • A. Hodges
  • J. Trnka
Article

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.

Keywords

Scattering Amplitudes Duality in Gauge Field Theories 

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Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • N. Arkani-Hamed
    • 1
  • J. Bourjaily
    • 1
    • 2
  • F. Cachazo
    • 3
  • A. Hodges
    • 4
  • J. Trnka
    • 1
    • 2
  1. 1.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.
  2. 2.Department of PhysicsPrinceton UniersityPrincetonU.S.A.
  3. 3.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  4. 4.Wadham CollegeUniversity of OxfordOxfordU.K.

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