Pushforwards via scattering equations with applications to positive geometries

Tomasz Łukowski, Robert Moerman, Jonah Stalknecht

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Abstract

In this paper we explore and expand the connection between two modern descriptions of scattering amplitudes, the CHY formalism and the framework of positive geometries, facilitated by the scattering equations. For theories in the CHY family whose S-matrix is captured by some positive geometry in the kinematic space, the corresponding canonical form can be obtained as the pushforward via the scattering equations of the canonical form of a positive geometry defined in the CHY moduli space. In order to compute these canonical forms in kinematic spaces, we study the general problem of pushing forward arbitrary rational differential forms via the scattering equations. We develop three methods which achieve this without ever needing to explicitly solve any scattering equations. Our results use techniques from computational algebraic geometry, including companion matrices and the global duality of residues, and they extend the application of similar results for rational functions to rational differential forms.
Original languageEnglish
Article number3
Number of pages41
JournalJournal of High Energy Physics (JHEP)
Volume2022
Issue number10
Early online date3 Oct 2022
DOIs
Publication statusE-pub ahead of print - 3 Oct 2022

Keywords

  • Regular Article - Theoretical Physics
  • Scattering Amplitudes
  • Differential and Algebraic Geometry

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