Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Algebraic Approach to the Scattering Equations

Rijun Huang, Junjie Rao, Bo Feng, Yang-Hui He

TL;DR

This work introduces a companion-matrix method from computational algebraic geometry to evaluate CHY integrals for tree-level scattering amplitudes without solving the scattering equations explicitly. By recasting the problem as a zero-dimensional ideal and applying Stickelberger’s theorem, the authors replace sums over roots with traces of rational functions of commuting companion matrices, yielding rational amplitudes directly from linear algebra. They demonstrate the approach across $n=4$–$7$ for scalar $\\phi^3$, Yang–Mills, and gravity theories, deriving known results and illuminating amplitude relations such as KLT and BCJ within a purely algebraic framework. While computationally intensive at higher $n$, the method exposes a new, exact, algebraic perspective on CHY integrals and opens avenues for analytic recursion and identity discovery beyond explicit root solving.

Abstract

We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.

An Algebraic Approach to the Scattering Equations

TL;DR

This work introduces a companion-matrix method from computational algebraic geometry to evaluate CHY integrals for tree-level scattering amplitudes without solving the scattering equations explicitly. By recasting the problem as a zero-dimensional ideal and applying Stickelberger’s theorem, the authors replace sums over roots with traces of rational functions of commuting companion matrices, yielding rational amplitudes directly from linear algebra. They demonstrate the approach across – for scalar , Yang–Mills, and gravity theories, deriving known results and illuminating amplitude relations such as KLT and BCJ within a purely algebraic framework. While computationally intensive at higher , the method exposes a new, exact, algebraic perspective on CHY integrals and opens avenues for analytic recursion and identity discovery beyond explicit root solving.

Abstract

We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.

Paper Structure

This paper contains 20 sections, 131 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of Tree-level Scattering Equations
  3. Specific Examples:
  4. The Mathematical Framework
  5. Warmup
  6. Illustrative Examples
  7. Four-Point Amplitudes
  8. Scalar $\phi^3$ Theory
  9. Yang-Mills Theory
  10. Gravity
  11. Five-Point Amplitudes
  12. Scalar $\phi^3$ Theory
  13. Yang-Mills theory
  14. Gravity and $n$-point KLT Relations
  15. Six-Point Amplitudes
  16. ...and 5 more sections

Figures (1)

  • Figure 1: The pentacle diagram representing the CHY-integrand defined by the two-cycles {(1,2,3,4,5),(1,3,5,2,4)}.