Scattering Equations and Global Duality of Residues

TL;DR

This paper leverages algebraic geometry and Bezoutian matrices to efficiently compute scattering amplitudes in the CHY formalism without solving individual residues, improving computational efficiency for multi-particle scattering.

Contribution

It introduces a novel algebraic geometry approach using Bezoutian matrices to calculate scattering amplitudes more efficiently in the CHY formalism.

Findings

Efficient amplitude computation without explicit residue summation.

Bezoutian matrix size grows linearly with particle number.

Algorithm applicable for both analytic and numeric calculations.

Abstract

We examine the polynomial form of the scattering equations by means of computational algebraic geometry. The scattering equations are the backbone of the Cachazo-He-Yuan (CHY) representation of the S-matrix. We explain how the Bezoutian matrix facilitates the calculation of amplitudes in the CHY formalism, without explicitly solving the scattering equations or summing over the individual residues. Since for $n$-particle scattering, the size of the Bezoutian matrix grows only as $(n-3)\times(n-3)$, our algorithm is very efficient for analytic and numeric amplitude computations.