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A Combinatoric Shortcut to Evaluate CHY-forms

Gang Chen, Yeuk-Kwan E. Cheung, Tianheng Wang, Feng Xu

TL;DR

The paper develops a combinatorial, tableau-based method to determine the coefficients of a differential operator $\mathbb{D}$ that computes multidimensional residues in generalized CHY forms. By recasting CHY integrals into prepared forms and exploiting local duality and intersection-number constraints, the authors obtain analytic coefficient solutions and apply them to the one-loop $n$-gon and the one-loop five-point SYM amplitudes, reproducing results obtained via Q-Cuts. The approach turns the evaluation of CHY integrals into a tractable combinatorial problem and demonstrates exact agreement with known forward-limit channel analyses. The framework is theory-agnostic and promises efficient CHY evaluations at higher points and loops, with potential extensions to two-loop structures.

Abstract

In \cite{Chen:2016fgi} we proposed a differential operator for the evaluation of the multi-dimensional residues on isolated (zero-dimensional) poles.In this paper we discuss some new insight on evaluating the (generalized) Cachazo-He-Yuan (CHY) forms of the scattering amplitudes using this differential operator. We introduce a tableau representation for the coefficients appearing in the proposed differential operator. Combining the tableaux with the polynomial forms of the scattering equations, the evaluation of the generalized CHY form becomes a simple combinatoric problem. It is thus possible to obtain the coefficients arising in the differential operator in a straightforward way. We present the procedure for a complete solution of the $n$-gon amplitudes at one-loop level in a generalized CHY form. We also apply our method to fully evaluate the one-loop five-point amplitude in the maximally supersymmetric Yang-Mills theory; the final result is identical to the one obtained by Q-Cut.

A Combinatoric Shortcut to Evaluate CHY-forms

TL;DR

The paper develops a combinatorial, tableau-based method to determine the coefficients of a differential operator that computes multidimensional residues in generalized CHY forms. By recasting CHY integrals into prepared forms and exploiting local duality and intersection-number constraints, the authors obtain analytic coefficient solutions and apply them to the one-loop -gon and the one-loop five-point SYM amplitudes, reproducing results obtained via Q-Cuts. The approach turns the evaluation of CHY integrals into a tractable combinatorial problem and demonstrates exact agreement with known forward-limit channel analyses. The framework is theory-agnostic and promises efficient CHY evaluations at higher points and loops, with potential extensions to two-loop structures.

Abstract

In \cite{Chen:2016fgi} we proposed a differential operator for the evaluation of the multi-dimensional residues on isolated (zero-dimensional) poles.In this paper we discuss some new insight on evaluating the (generalized) Cachazo-He-Yuan (CHY) forms of the scattering amplitudes using this differential operator. We introduce a tableau representation for the coefficients appearing in the proposed differential operator. Combining the tableaux with the polynomial forms of the scattering equations, the evaluation of the generalized CHY form becomes a simple combinatoric problem. It is thus possible to obtain the coefficients arising in the differential operator in a straightforward way. We present the procedure for a complete solution of the -gon amplitudes at one-loop level in a generalized CHY form. We also apply our method to fully evaluate the one-loop five-point amplitude in the maximally supersymmetric Yang-Mills theory; the final result is identical to the one obtained by Q-Cut.

Paper Structure

This paper contains 16 sections, 78 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Prescription for determining differential operators
  3. Toy model: one-loop four-point SYM integrand
  4. Preliminary: notations & tableaux
  5. Coefficient generating algorithm
  6. Trivial coefficients
  7. Non-trivial elementary coefficients
  8. Summary of the coefficient generating method
  9. One-loop five-point SYM amplitude
  10. Simplify the one-loop generalized CHY integrand
  11. Transformations to the prepared CHY integrand
  12. Evaluating the prepared form of the five point integrand
  13. Analysis for the spurious poles
  14. One-loop n-gon amplitude: A direct evaluation
  15. The analysis for the spurious pole
  16. ...and 1 more sections

Figures (10)

  • Figure 1: Tableaux for the four-point scattering equations. The red tiles in the tableaux are allowed to perform a permutation among the rows. The meaning of such permutations are to be discussed shortly..
  • Figure 2: Tableaux indicating the permutations of the red tiles among the rows.
  • Figure 3: Box Diagram.
  • Figure 4: Example of a tile move.
  • Figure 5: The tableaux $a_{\{0,0,1,2,\cdots,n-3,n-2\}},$
  • ...and 5 more figures