Pushforwards via Scattering Equations with Applications to Positive Geometries

TL;DR

This paper connects the CHY formalism and positive geometries in scattering amplitudes, introducing methods to compute canonical forms in kinematic space without solving scattering equations explicitly.

Contribution

It develops three novel methods to push forward rational differential forms via scattering equations without explicit solutions, using algebraic geometry techniques.

Findings

New methods for pushforward of differential forms without solving scattering equations

Extension of algebraic geometry techniques to rational differential forms

Enhanced understanding of the link between CHY formalism and positive geometries

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.