Scattering equations: from projective spaces to tropical grassmannians

  • Freddy Cachazo
  • Nick Early
  • Alfredo GuevaraEmail author
  • Sebastian Mizera
Open Access
Regular Article - Theoretical Physics


We introduce a natural generalization of the scattering equations, which connect the space of Mandelstam invariants to that of points on ℂℙ1, to higher-dimensional projective spaces ℂℙk − 1. The standard, k = 2 Mandelstam invariants, sab, are generalized to completely symmetric tensors \( {\mathrm{s}}_{a_1{a}_2\dots {a}_k} \) subject to a ‘massless’ condition \( {\mathrm{s}}_{a_1{a}_2\dots {a}_{k-2}bb}=0 \) and to ‘momentum conservation’. The scattering equations are obtained by constructing a potential function and computing its critical points. We mainly concentrate on the k = 3 case: study solutions and define the generalization of biadjoint scalar amplitudes. We compute all ‘biadjoint amplitudes’ for (k, n) = (3, 6) and find a direct connection to the tropical Grassmannian. This leads to the notion of k = 3 Feynman diagrams. We also find a concrete realization of the new kinematic spaces, which coincides with the spinor-helicity formalism for k = 2, and provides analytic solutions analogous to the MHV ones.


Scattering Amplitudes Differential and Algebraic Geometry 


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) 2019

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  2. 2.Massachusetts Institute of TechnologyCambridgeU.S.A.
  3. 3.Department of Physics & AstronomyUniversity of WaterlooWaterlooCanada
  4. 4.CECs Valdivia & Departamento de FísicaUniversidad de ConcepciónConcepciónChile

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