On the kinematic algebra for BCJ numerators beyond the MHV sector

TL;DR

The paper develops a dimension-agnostic framework for the kinematic algebra behind BCJ numerators beyond the MHV sector by introducing tensor currents and fusion products. It derives a closed all-multiplicity NMHV BCJ numerator in the bi-scalar sector, decomposed into polarization-power-one and power-two parts, with explicit, compact formulas and a detailed account of the associated generalized gauge freedom. The construction clarifies how tensor currents encode gauge freedom and crossing symmetry, and it connects these structures to BCJ relations via a diagrammatic tensor-monomial/topology language. The work sets the stage for extending the algebra to full NMHV amplitudes and higher polarizations, and it points to potential Lagrangian or off-shell formulations to realize the kinematic Lie algebra more broadly.

Abstract

The duality between color and kinematics present in scattering amplitudes of Yang-Mills theory strongly suggest the existence of a hidden kinematic Lie algebra that controls the gauge theory. While associated BCJ numerators are known on closed forms to any multiplicity at tree level, the kinematic algebra has only been partially explored for the simplest of four-dimensional amplitudes: up to the MHV sector. In this paper we introduce a framework that allows us to characterize the algebra beyond the MHV sector. This allows us to both constrain some of the ambiguities of the kinematic algebra, and better control the generalized gauge freedom that is associated with the BCJ numerators. Specifically, in this paper, we work in dimension-agnostic notation and determine the kinematic algebra valid up to certain ${\cal O}\big((\varepsilon_i \cdot \varepsilon_j)^2\big)$ terms that in four dimensions compute the next-to-MHV sector involving two scalars. The kinematic algebra in this sector is simple, given that we introduce tensor currents that generalize standard Yang-Mills vector currents. These tensor currents controls the generalized gauge freedom, allowing us to generate multiple different versions of BCJ numerators from the same kinematic algebra. The framework should generalize to other sectors in Yang-Mills theory.

Figures (4)

• Figure 1: Fusion product in the canonical ordering. The algebra simplifies by demanding that $\hat{\varepsilon}_1$ correspond to a scalar, since the internal currents must carry this polarization.
• Figure 2: The graph for general connected topology. The white blobs denote generic trees composed of red edges and solid dots.
• Figure 3: The graph for general disconnected topology.
• Figure 4: Typical monomials of disconnected topologies at seven points.