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Polynomial reduction and evaluation of tree- and loop-level CHY amplitudes

Michael Zlotnikov

TL;DR

This work presents a universal degree-reduction procedure that converts any gauge-fixed CHY integrand into a standard form, spanned by ladder-type monomials with maximal multivariate degree $\frac{(n-3)(n-4)}{2}$, enabling a purely residue-based evaluation via the global residue theorem. By proving that rational CHY integrands are equivalent to standard-form polynomials on the scattering-equation support (via Hilbert’s Nullstellensatz and constructive linear-algebra steps), the authors shift the computational challenge from solving scattering equations to polynomial reduction and residue collection. The resulting prescription evaluates amplitudes by collecting only simple residues at infinity, with explicit tree- and one-loop examples demonstrating analytic results and agreement with conventional methods. The approach is universal for CHY integrands in arbitrary dimension and theories, and it opens avenues for automation, structural insight into amplitude polynomials, and potential closed-form expressions focusing on the highest-degree ladder basis. Overall, the paper provides a robust algebraic framework that recasts CHY amplitude evaluation as a tractable, residue-based computation anchored in standard-form polynomials.

Abstract

We develop a polynomial reduction procedure that transforms any gauge fixed CHY amplitude integrand for $n$ scattering particles into a $σ$-moduli multivariate polynomial of what we call the $\textit{standard form}$. We show that a standard form polynomial must have a specific $\textit{ladder type}$ monomial structure, which has finite size at any $n$, with highest multivariate degree given by $(n-3)(n-4)/2$. This set of monomials spans a complete basis for polynomials with rational coefficients in kinematic data on the support of scattering equations. Subsequently, at tree and one-loop level, we employ the global residue theorem to derive a prescription that evaluates any CHY amplitude by means of collecting simple residues at infinity only. The prescription is then applied explicitly to some tree and one-loop amplitude examples.

Polynomial reduction and evaluation of tree- and loop-level CHY amplitudes

TL;DR

This work presents a universal degree-reduction procedure that converts any gauge-fixed CHY integrand into a standard form, spanned by ladder-type monomials with maximal multivariate degree , enabling a purely residue-based evaluation via the global residue theorem. By proving that rational CHY integrands are equivalent to standard-form polynomials on the scattering-equation support (via Hilbert’s Nullstellensatz and constructive linear-algebra steps), the authors shift the computational challenge from solving scattering equations to polynomial reduction and residue collection. The resulting prescription evaluates amplitudes by collecting only simple residues at infinity, with explicit tree- and one-loop examples demonstrating analytic results and agreement with conventional methods. The approach is universal for CHY integrands in arbitrary dimension and theories, and it opens avenues for automation, structural insight into amplitude polynomials, and potential closed-form expressions focusing on the highest-degree ladder basis. Overall, the paper provides a robust algebraic framework that recasts CHY amplitude evaluation as a tractable, residue-based computation anchored in standard-form polynomials.

Abstract

We develop a polynomial reduction procedure that transforms any gauge fixed CHY amplitude integrand for scattering particles into a -moduli multivariate polynomial of what we call the . We show that a standard form polynomial must have a specific monomial structure, which has finite size at any , with highest multivariate degree given by . This set of monomials spans a complete basis for polynomials with rational coefficients in kinematic data on the support of scattering equations. Subsequently, at tree and one-loop level, we employ the global residue theorem to derive a prescription that evaluates any CHY amplitude by means of collecting simple residues at infinity only. The prescription is then applied explicitly to some tree and one-loop amplitude examples.

Paper Structure

This paper contains 33 sections, 82 equations, 1 table.

Table of Contents

  1. Introduction
  2. Note added:
  3. CHY formulation of tree level scattering amplitudes
  4. Warm up: five point tree level scalar amplitude
  5. Amplitude structure and evaluation prescription
  6. Degree reduction of polynomials to a standard form
  7. Conventions:
  8. Definition 1:
  9. Definition 2:
  10. Theorem 1:
  11. Proof:
  12. Step 1: Reduction of monomials to $0\leq p_{j}-p_{j-1}\leq 1$ for all $j\in\{1,2,...,m_{max}\}$
  13. Step 2: Reduction of monomials to $m_{max}\leq n-4$
  14. Conclusion
  15. Polynomial reduction of rational expressions
  16. ...and 18 more sections