[Paper Review] Explicit BCJ numerators of nonlinear sigma model

Yi-Jian Du, C. D. Fu|arXiv (Cornell University)|Jun 19, 2016

Algebraic structures and combinatorial models56 references17 citations

TL;DR

This paper presents explicit polynomial expressions for BCJ numerators in the nonlinear sigma model (NLSM) using two parametrization schemes—Cayley and another KLT-inspired approach—demonstrating that kinematic numerators can be systematically constructed as polynomials in momentum invariants. The key result is a simple formula in Cayley parametrization solely in terms of the momentum kernel, providing a direct algebraic realization of color-kinematics duality in NLSM.

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.

Motivation & Objective

• To construct explicit polynomial BCJ numerators for scattering amplitudes in the nonlinear sigma model (NLSM).
• To demonstrate that color-kinematics duality in NLSM can be realized through systematic, polynomially structured kinematic numerators.
• To provide a set of constructive rules for generating BCJ numerators at arbitrary multiplicity in two distinct parametrization schemes.
• To uncover a simple, momentum kernel-based formula for numerators in Cayley parametrization, suggesting an underlying algebraic structure.

Proposed method

• Employing a KLT-relation-inspired technique to derive kinematic numerators in two parametrization schemes: Cayley and an alternative scheme.
• Deriving explicit polynomial expressions for numerators using momentum invariants and the momentum kernel.
• Applying a set of recursive rules to generate BCJ numerators for all multiplicities, ensuring anti-symmetry and Jacobi identities.
• Using the Cayley parametrization to reveal a particularly simple formula for numerators expressed solely in terms of the momentum kernel.
• Verifying consistency with known BCJ amplitude relations and the duality structure in NLSM.
• Handling non-cubic vertices and contact terms systematically through the parametrization and polynomial construction.

Experimental results

Research questions

• RQ1 Can explicit polynomial BCJ numerators be constructed for the nonlinear sigma model beyond known amplitude relations?
• RQ2 What is the algebraic structure underlying the kinematic numerators in NLSM, particularly in Cayley parametrization?
• RQ3 How do generalized gauge freedom and non-uniqueness of numerators affect the construction of consistent BCJ numerators?
• RQ4 Can the momentum kernel alone fully determine the BCJ numerators in a simple, closed-form expression?
• RQ5 How do the two parametrization schemes (Cayley and KLT-inspired) compare in terms of simplicity and generality of the resulting numerator expressions?

Key findings

• The paper derives explicit polynomial BCJ numerators for NLSM amplitudes in both Cayley and KLT-inspired parametrizations, confirming the existence of such numerators beyond abstract duality.
• In Cayley parametrization, the BCJ numerator is given by a remarkably simple formula expressed solely in terms of the momentum kernel, significantly simplifying the structure.
• The authors provide a set of constructive rules that generate BCJ numerators for all multiplicities, enabling systematic computation of amplitudes.
• The numerators satisfy the required anti-symmetry and Jacobi identities, confirming the validity of the color-kinematics duality in the NLSM.
• The construction handles non-cubic vertices and contact terms naturally, showing that the duality persists even in non-cubic theories.
• The results support the broader conjecture that color-kinematics duality is a universal feature of gauge and gravity theories, even in models like NLSM not built from cubic vertices.