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Scattering Equations and Global Duality of Residues

Mads Sogaard, Yang Zhang

TL;DR

The paper develops a computational algebraic geometry approach to CHY scattering amplitudes by leveraging the Bezoutian matrix to compute the total sum of multivariate residues without solving the polynomial scattering equations. It shows that global duality in the quotient ring $R/I$ yields compact, efficient expressions for tree-level amplitudes, even when the integrand is rational, by replacing denominator factors with polynomial inverses via Hilbert’s Nullstellensatz and Gröbner-basis techniques. The authors demonstrate the method on four-, five-, eight-, and ten-point amplitudes in $\varphi^3$ theory and Yang–Mills, achieving results that match standard expectations and enabling high-multiplicity computations that would be intractable by residue-sum evaluation. The work offers a scalable, algebraic alternative to solving $(n-3)!$ residues and points to future extensions to loop-level CHY formulations and broader classes of theories.

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.

Scattering Equations and Global Duality of Residues

TL;DR

The paper develops a computational algebraic geometry approach to CHY scattering amplitudes by leveraging the Bezoutian matrix to compute the total sum of multivariate residues without solving the polynomial scattering equations. It shows that global duality in the quotient ring yields compact, efficient expressions for tree-level amplitudes, even when the integrand is rational, by replacing denominator factors with polynomial inverses via Hilbert’s Nullstellensatz and Gröbner-basis techniques. The authors demonstrate the method on four-, five-, eight-, and ten-point amplitudes in theory and Yang–Mills, achieving results that match standard expectations and enabling high-multiplicity computations that would be intractable by residue-sum evaluation. The work offers a scalable, algebraic alternative to solving residues and points to future extensions to loop-level CHY formulations and broader classes of theories.

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 -particle scattering, the size of the Bezoutian matrix grows only as , our algorithm is very efficient for analytic and numeric amplitude computations.

Paper Structure

This paper contains 13 sections, 1 theorem, 66 equations, 2 algorithms.

Table of Contents

  1. Introduction and Review
  2. Examples of CHY Representations
  3. The Polynomial Form of the Scattering Equations
  4. Multivariate Residues and the Bezoutian Matrix
  5. Local Residues
  6. Global Residues and the Bezoutian Matrix
  7. Our Proposal for CHY Integrals
  8. Examples
  9. Four-Point Amplitudes
  10. Five-Point Amplitudes
  11. Eight-Point Amplitudes
  12. Ten-point Amplitudes
  13. Conclusion

Key Result

Theorem 1

$\langle\bullet,\bullet\rangle$ is a nondegenerate inner product in $R/I$.

Theorems & Definitions (3)

  • Theorem 1: Global Duality
  • Example 1: Global Residues and the Bezoutian Matrix
  • Example 2: Polynomial Inverses and Global Residues