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BCJ numerators from reduced Pfaffian

Yi-Jian Du, Fei Teng

TL;DR

The authors develop a constructive framework to obtain BCJ numerators in DDM form directly from CHY tree-level integrands by expanding the reduced Pfaffian with Laplace recursion. They introduce graphic, spanning-tree rules that translate the expansion into explicit YM and NLSM numerators, yielding $(n-1)!$-term YM numerators per order and at most $(n-2)!-(n-3)!$ nonzero NLSM numerators. The YM results are cross-validated against existing constructions, while NLSM numerators are obtained via dimensional reduction and a recursive, phi^3-augmented approach, clarifying connections to bi-adjoint amplitudes. The work unifies CHY, color–kinematic duality, and explicit numerator construction, offering a practical toolkit for analytic amplitude calculations and potential extensions to loop-level and string-theoretic contexts.

Abstract

By expanding the reduced Pfaffian in the tree level Cachazo-He-Yuan (CHY) integrands for Yang-Mills (YM) and nonlinear sigma model (NLSM), we can get the Bern-Carrasco-Johansson (BCJ) numerators in Del Duca-Dixon-Maltoni (DDM) form for arbitrary number of particles in any spacetime dimensions. In this work, we give a set of very straightforward graphic rules based on spanning trees for a direct evaluation of the BCJ numerators for YM and NLSM. Such rules can be derived from the Laplace expansion of the corresponding reduced Pfaffian. For YM, the each one of the $(n-2)!$ DDM form BCJ numerators contains exactly $(n-1)!$ terms, corresponding to the increasing trees with respect to the color order. For NLSM, the number of nonzero numerators is at most $(n-2)!-(n-3)!$, less than those of several previous constructions.

BCJ numerators from reduced Pfaffian

TL;DR

The authors develop a constructive framework to obtain BCJ numerators in DDM form directly from CHY tree-level integrands by expanding the reduced Pfaffian with Laplace recursion. They introduce graphic, spanning-tree rules that translate the expansion into explicit YM and NLSM numerators, yielding -term YM numerators per order and at most nonzero NLSM numerators. The YM results are cross-validated against existing constructions, while NLSM numerators are obtained via dimensional reduction and a recursive, phi^3-augmented approach, clarifying connections to bi-adjoint amplitudes. The work unifies CHY, color–kinematic duality, and explicit numerator construction, offering a practical toolkit for analytic amplitude calculations and potential extensions to loop-level and string-theoretic contexts.

Abstract

By expanding the reduced Pfaffian in the tree level Cachazo-He-Yuan (CHY) integrands for Yang-Mills (YM) and nonlinear sigma model (NLSM), we can get the Bern-Carrasco-Johansson (BCJ) numerators in Del Duca-Dixon-Maltoni (DDM) form for arbitrary number of particles in any spacetime dimensions. In this work, we give a set of very straightforward graphic rules based on spanning trees for a direct evaluation of the BCJ numerators for YM and NLSM. Such rules can be derived from the Laplace expansion of the corresponding reduced Pfaffian. For YM, the each one of the DDM form BCJ numerators contains exactly terms, corresponding to the increasing trees with respect to the color order. For NLSM, the number of nonzero numerators is at most , less than those of several previous constructions.

Paper Structure

This paper contains 27 sections, 79 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Color-kinematic duality and double copy construction
  3. CHY integrands and their relations
  4. Definitions
  5. Expansion of reduced Pfaffian
  6. Graphic rules for BCJ numerators: YM
  7. Step 1: Constructing the trees that contribute to the order $\pmb\beta$.
  8. Step 2: Separation of gluons and gravitons.
  9. Step 3: Evaluation of graphs.
  10. Four-point numerators
  11. Five-point numerators
  12. Graphic rules for BCJ numerators: NLSM
  13. Dimensional reduction
  14. Recursive expansion
  15. Graphic rules
  16. ...and 12 more sections

Figures (3)

  • Figure 1: The spanning trees corresponding to $n^{\text{YM}}(1,\{23\},4)$ and $n^{\text{YM}}(1,\{32\},4)$. The evaluation is performed under the reference order $\pmb\rho=\{32\}$.
  • Figure 2: The spanning trees corresponding to $n(1,\{234\},5)$ for YM. All gluon vertices are highlighted in red.
  • Figure 3: The $5$-point spanning trees that contribute to $n^{\text{NLSM}}(1,\{2345\},6)$.