Properties of scattering forms and their relation to associahedra

  • Leonardo de la Cruz
  • Alexander Kniss
  • Stefan Weinzierl
Open Access
Regular Article - Theoretical Physics
  • 22 Downloads

Abstract

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.

Keywords

Scattering Amplitudes Differential and Algebraic Geometry Perturbative QCD 

Notes

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
  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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