Explicit BCJ numerators of nonlinear sigma model

TL;DR

This work explores color-kinematics duality in the nonlinear sigma model by constructing explicit BCJ numerators through a KLT-inspired approach in two parametrizations (Cayley and pion). It introduces a systematic off-shell extension using Berends-Giele currents and derives a general rule that reduces numerators to pole-free polynomials in Mandelstam variables, with detailed demonstrations up to eight points. The Cayley parametrization yields a compact expression via the momentum kernel, while the pion parametrization provides permutation-symmetric numerators and off-shell BCJ relations, both validated by explicit four-, six-, and eight-point results. The findings advance understanding of the kinematic algebra in NLSM and offer practical algorithms for generating BCJ numerators at arbitrary multiplicities, with discussion of deeper algebraic interpretations and future directions.

Abstract

In this paper, we investigate the color-kinematics duality in nonlinear sigma model (NLSM). We present explicit polynomial expressions for the kinematic numerators (BCJ numerators). The calculation is done separately in two parametrization schemes of the theory using Kawai-Lewellen-Tye relation inspired technique, both lead to polynomial numerators. We summarize the calculation in each case into a set of rules that generates BCJ numerators for all multilplicities. In Cayley parametrization we find the numerator is described by a particularly simple formula solely in terms of momentum kernel.

Figures (2)

• Figure 1: All possible configurations for the six-point numerator.
• Figure 2: All arrow graphs for the six-point numerator.