Positive amplitudes in the amplituhedron

  • Nima Arkani-Hamed
  • Andrew Hodges
  • Jaroslav TrnkaEmail author
Open Access
Regular Article - Theoretical Physics


The all-loop integrand for scattering amplitudes in planar \( \mathcal{N}=4 \) SYM is determined by an “amplitude form” with logarithmic singularities on the boundary of the amplituhedron. In this note we provide strong evidence for a new striking property of the superamplitude, which we conjecture to be true to all loop orders: the amplitude form is positive when evaluated inside the amplituhedron. The statement is sensibly formulated thanks to the natural “bosonization” of the superamplitude associated with the amplituhedron geometry. However this positivity is not manifest in any of the current approaches to scattering amplitudes, and in particular not in the cellulations of the amplituhedron related to on-shell diagrams and the positive grassmannian. The surprising positivity of the form suggests the existence of a “dual amplituhedron” formulation where this feature would be made obvious. We also suggest that the positivity is associated with an extended picture of amplituhedron geometry, with the amplituhedron sitting inside a co-dimension one surface separating “legal” and “illegal” local singularities of the amplitude. We illustrate this in several simple examples, obtaining new expressions for amplitudes not associated with any triangulations, but following in a more invariant manner from a global view of the positive geometry.


Supersymmetric gauge theory Scattering Amplitudes 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Nima Arkani-Hamed
    • 1
  • Andrew Hodges
    • 2
  • Jaroslav Trnka
    • 3
    Email author
  1. 1.School of Natural SciencesInstitute for Advanced StudyPrincetonUnited States
  2. 2.Wadham CollegeUniversity of OxfordOxfordUnited Kingdom
  3. 3.Walter Burke Institute for Theoretical PhysicsCalifornia Institute of TechnologyPasadenaUnited States

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