Bootstrapping solutions of scattering equations

TL;DR

This paper introduces a numerical algebraic geometry-based method to efficiently solve scattering equations by tracking solutions through homotopies in kinematic space, enabling solutions to be bootstrapped from known cases.

Contribution

It presents a novel homotopy-based approach for solving scattering equations, leveraging soft limits to bootstrap solutions from the simplest case.

Findings

Solutions can be tracked via homotopy in kinematic space

All solutions can be bootstrapped from the four-particle case

The method improves efficiency in solving scattering equations

Abstract

The scattering equations are a set of algebraic equations connecting the kinematic space of massless particles and the moduli space of Riemann spheres with marked points. We present an efficient method for solving the scattering equations based on the numerical algebraic geometry. The cornerstone of our method is the concept of the physical homotopy between different points in the kinematic space, which naturally induces a homotopy of the scattering equations. As a result, the solutions of the scattering equations with different points in the kinematic space can be tracked from each other. Finally, with the help of soft limits, all solutions can be bootstrapped from the known solution for the four-particle scattering.