Properties of scattering forms and their relation to associahedra

  • Leonardo de la Cruz
  • Alexander Kniss
  • Stefan WeinzierlEmail author
Open Access
Regular Article - Theoretical Physics


We show that the half-integrands in the CHY representation of tree amplitudes give rise to the definition of differential forms — the scattering forms — on the moduli space of a Riemann sphere with n marked points. These differential forms have some remarkable properties. We show that all singularities are on the divisor \( {\overline{\mathrm{\mathcal{M}}}}_{0,n}\backslash {\mathrm{\mathcal{M}}}_{0,n} \). Each singularity is logarithmic and the residue factorises into two differential forms of lower points. In order for this to work, we provide a threefold generalisation of the CHY polarisation factor (also known as reduced Pfaffian) towards off-shell momenta, unphysical polarisations and away from the solutions of the scattering equations. We discuss explicitly the cases of bi-adjoint scalar amplitudes, Yang-Mills amplitudes and gravity amplitudes.


Scattering Amplitudes Differential and Algebraic Geometry Perturbative QCD 


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) 2018

Authors and Affiliations

  • Leonardo de la Cruz
    • 1
  • Alexander Kniss
    • 2
  • Stefan Weinzierl
    • 2
    Email author
  1. 1.Higgs Centre for Theoretical Physics, School of Physics and AstronomyThe University of EdinburghEdinburghU.K.
  2. 2.PRISMA Cluster of Excellence, Institut für PhysikJohannes Gutenberg-Universität MainzMainzGermany

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