# BCJ numerators from reduced Pfaffian

## TL;DR

This paper introduces graphic rules based on spanning trees to directly evaluate BCJ numerators for Yang-Mills and NLSM from the reduced Pfaffian expansion in CHY integrands, simplifying calculations across dimensions.

## Contribution

It provides a straightforward graphical method derived from Laplace expansion to compute BCJ numerators for YM and NLSM, with explicit formulas and reduced complexity.

## Key findings

- For YM, each BCJ numerator has exactly (n-1)! terms.
- For NLSM, the number of nonzero numerators is at most (n-2)! - (n-3)!.
- The method simplifies the calculation of BCJ numerators across arbitrary dimensions.

## 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.