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Cross-ratio Identities and Higher-order Poles of CHY-integrand

Carlos Cardona, Bo Feng, Humberto Gomez, Rijun Huang

TL;DR

This work introduces cross-ratio identities as a systematic tool to decompose CHY-integrands with higher-order poles into sums of simple-pole terms, enabling straightforward evaluation by simple-pole integration rules. A finite-step decomposition algorithm is proposed, using pole-order indicators to guide iterative application of cross-ratio identities; when needed, the Λ-algorithm is combined with these identities to handle challenging configurations and large datasets. The authors derive and utilize general identities for arbitrary poles, demonstrate the method on illustrative six- and eight-point examples, and develop recurrence relations for Parke–Taylor–squared geometries (PT^2 ⊕ PT^2) that further improve efficiency. The results provide a practical, analytic framework for computing CHY amplitudes in theories like Yang–Mills and gravity, potentially reducing computational complexity in high-multiplicity scattering. The approach offers a complement or alternative to monodromy-based identities, with explicit algorithms and notational conventions that facilitate implementation and SEO-friendly summarization.

Abstract

The evaluation of generic Cachazo-He-Yuan(CHY)-integrands is a big challenge and efficient computational methods are in demand for practical evaluation. In this paper, we propose a systematic decomposition algorithm by using cross-ratio identities, which provides an analytic and easy to implement method for the evaluation of any CHY-integrand. This algorithm aims to decompose a given CHY-integrand containing higher-order poles as a linear combination of CHY-integrands with only simple poles in a finite number of steps, which ultimately can be trivially evaluated by integration rules of simple poles. To make the method even more efficient for CHY-integrands with large number of particles and complicated higher-order pole structures, we combine the $Λ$-algorithm and the cross-ratio identities, and as a by-product it provides us a way to deal with CHY-integrands where the $Λ$-algorithm was not applicable in its original formulation.

Cross-ratio Identities and Higher-order Poles of CHY-integrand

TL;DR

This work introduces cross-ratio identities as a systematic tool to decompose CHY-integrands with higher-order poles into sums of simple-pole terms, enabling straightforward evaluation by simple-pole integration rules. A finite-step decomposition algorithm is proposed, using pole-order indicators to guide iterative application of cross-ratio identities; when needed, the Λ-algorithm is combined with these identities to handle challenging configurations and large datasets. The authors derive and utilize general identities for arbitrary poles, demonstrate the method on illustrative six- and eight-point examples, and develop recurrence relations for Parke–Taylor–squared geometries (PT^2 ⊕ PT^2) that further improve efficiency. The results provide a practical, analytic framework for computing CHY amplitudes in theories like Yang–Mills and gravity, potentially reducing computational complexity in high-multiplicity scattering. The approach offers a complement or alternative to monodromy-based identities, with explicit algorithms and notational conventions that facilitate implementation and SEO-friendly summarization.

Abstract

The evaluation of generic Cachazo-He-Yuan(CHY)-integrands is a big challenge and efficient computational methods are in demand for practical evaluation. In this paper, we propose a systematic decomposition algorithm by using cross-ratio identities, which provides an analytic and easy to implement method for the evaluation of any CHY-integrand. This algorithm aims to decompose a given CHY-integrand containing higher-order poles as a linear combination of CHY-integrands with only simple poles in a finite number of steps, which ultimately can be trivially evaluated by integration rules of simple poles. To make the method even more efficient for CHY-integrands with large number of particles and complicated higher-order pole structures, we combine the -algorithm and the cross-ratio identities, and as a by-product it provides us a way to deal with CHY-integrands where the -algorithm was not applicable in its original formulation.

Paper Structure

This paper contains 14 sections, 108 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The Cross-Ratio Identities
  3. A Systematic Decomposition Algorithm
  4. The algorithm
  5. An illustrative example
  6. The $\Lambda$-Algorithm and the Cross-Ratio Identities
  7. Some notations
  8. A simple example
  9. $({\rm Parke}-{\rm Taylor)^2}\oplus({\rm Parke}-{\rm Taylor)^2}$ geometry and recurrence relations
  10. Conclusion
  11. A Practical Algorithm to Determine the $\pm$ Sign in Integration Rules
  12. The Identities from Monodromy Relation and Cross-Ratio Identities
  13. $({\rm Parke- Taylor})\times ({\rm Parke- Taylor})$ Geometry and Its Recurrence Relation
  14. Recurrence Relation for ${\cal I}^{{\rm PT^2}\oplus {\rm PT^2}}_{3,3}, {\cal I}^{{\rm PT^2}\oplus {\rm PT^2}}_{4,3}, {\cal I}^{{\rm PT^2}\oplus {\rm PT^2}}_{5,3}$ and ${\cal I}^{{\rm PT^2}\oplus {\rm PT^2}}_{6,3}$

Figures (6)

  • Figure 1: The 4-regular graphs of four-point CHY-integrands with simple, double and triple pole respectively.
  • Figure 2: The diagrammatic presentation of how CHY-integrands with different pole structures can be related by cross-ratio identities.
  • Figure 3: The 4-regular graph of a six-point CHY-integrand with three double poles.
  • Figure 4: The 4-regular graph of a six-point CHY-integrand with one triple pole and two double poles.
  • Figure 5: The 4-regular graph of an eight-point CHY-integrand.
  • ...and 1 more figures