Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Feynman Rules of Higher-order Poles in CHY Construction

Rijun Huang, Bo Feng, Ming-xing Luo, Chuan-Jie Zhu

TL;DR

This work extends the CHY framework by generalizing integration rules to higher-order poles, enabling direct reading of amplitudes from CHY-integrands with double, triple, duplex-double, and triplex-double poles. The authors develop a deduction strategy that starts from the simplest CHY-integrands and uses symmetry, kinematics, and extensive cross-checks to formulate explicit Feynman rules (I, II, III, IX) that encode quasi-local propagators. They validate these rules across a broad suite of examples, including mixed-pole configurations up to nine points, demonstrating accurate reproduction of analytic CHY results and highlighting the practical utility of pattern-matching computations. The work positions these rules as a bridge to efficient CHY computations beyond simple poles, discusses connections to alternative approaches like the $ abla$- or $\\"Lambda$-formalisms, and outlines future avenues for analytic derivation and handling of new pole structures in more general theories.

Abstract

In this paper, we generalize the integration rules for scattering equations to situations where higher-order poles are present. We describe the strategy to deduce the Feynman rules of higher-order poles from known analytic results of simple CHY-integrands, and propose the Feynman rules for single double pole and triple pole as well as duplex-double pole and triplex-double pole structures. We demonstrate the validation and strength of these rules by ample non-trivial examples.

Feynman Rules of Higher-order Poles in CHY Construction

TL;DR

This work extends the CHY framework by generalizing integration rules to higher-order poles, enabling direct reading of amplitudes from CHY-integrands with double, triple, duplex-double, and triplex-double poles. The authors develop a deduction strategy that starts from the simplest CHY-integrands and uses symmetry, kinematics, and extensive cross-checks to formulate explicit Feynman rules (I, II, III, IX) that encode quasi-local propagators. They validate these rules across a broad suite of examples, including mixed-pole configurations up to nine points, demonstrating accurate reproduction of analytic CHY results and highlighting the practical utility of pattern-matching computations. The work positions these rules as a bridge to efficient CHY computations beyond simple poles, discusses connections to alternative approaches like the - or -formalisms, and outlines future avenues for analytic derivation and handling of new pole structures in more general theories.

Abstract

In this paper, we generalize the integration rules for scattering equations to situations where higher-order poles are present. We describe the strategy to deduce the Feynman rules of higher-order poles from known analytic results of simple CHY-integrands, and propose the Feynman rules for single double pole and triple pole as well as duplex-double pole and triplex-double pole structures. We demonstrate the validation and strength of these rules by ample non-trivial examples.

Paper Structure

This paper contains 13 sections, 122 equations, 35 figures.

Table of Contents

  1. Introduction
  2. Review of integration algorithm
  3. The Feynman rules for higher-order poles
  4. The Feynman rule $\mathcal{R}_{{\scriptsize\hbox{ule}}}^{{\tiny\hbox{I}}}$ for a single double pole
  5. The Feynman rule $\mathcal{R}_{{\scriptsize\hbox{ule}}}^{{\tiny\hbox{II}}}$ for a single triple pole
  6. The Feynman rule $\mathcal{R}_{{\scriptsize\hbox{ule}}}^{{\tiny\hbox{III}}}$ for duplex-double pole
  7. The Feynman rule $\mathcal{R}_{{\scriptsize\hbox{ule}}}^{{\tiny\hbox{IX}}}$ for triplex-double pole
  8. Supporting examples
  9. Examples with only double poles and simple poles
  10. Examples with only triple poles and simple poles
  11. Examples with only duplex-double poles and simple poles
  12. Examples with mixed types of higher-order poles
  13. Discussion and conclusion

Figures (35)

  • Figure 1: The left-most graph is an example of 4-regular graph for certain CHY-integrand. The remaining five diagrams are contributing Feynman diagrams for amplitude of that CHY-integrand.
  • Figure 2: Left: The 4-regular graph of a 4-point CHY-integrand and its corresponding Feynman diagram. The double-line propagator denotes a double pole of this CHY-integrand. Right: Labels of external momenta for the Feynman rule $\mathcal{R}_{{\scriptsize\hbox{ule}}}^{{\tiny\hbox{I}}}$ when legs are massive.
  • Figure 3: The 4-regular graph of a given five-point CHY-integrand with a double pole, and two Feynman diagrams corresponding to this CHY-integrand. To indicate the ordering of two Feynman diagrams, we have drawn two pinched CHY-graphs.
  • Figure 4: Left: The 4-regular graph of a given CHY-integrand and its corresponding Feynman diagram. The triple-line propagator denotes a triple pole of this CHY-integrand. Right: Labels of external momenta for the Feynman rule $\mathcal{R}_{{\scriptsize\hbox{ule}}}^{{\tiny\hbox{II}}}$ when legs are massive.
  • Figure 5: The 4-regular graph of a given five-point CHY-integrand with a triple pole, and two Feynman diagrams corresponding to this CHY-integrand.
  • ...and 30 more figures