An algebraic approach to BCJ numerators

Abstract

One important discovery in recent years is that the total amplitude of gauge theory can be written as BCJ form where kinematic numerators satisfy Jacobi identity. Although the existence of such kinematic numerators is no doubt, the simple and explicit construction is still an important problem. As a small step, in this note we provide an algebraic approach to construct these kinematic numerators. Under our Feynman-diagram-like construction, the Jacobi identity is manifestly satisfied. The corresponding color ordered amplitudes satisfy off-shell KK-relation and off-shell BCJ relation similar to the color ordered scalar theory. Using our construction, the dual DDM form is also established.

30 pages, 7 figures

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• Perturbative spacetimes from Yang-Mills theory
• The double copy: Bremsstrahlung and accelerating black holes
• Inelastic Black Hole Scattering from Charged Scalar Amplitudes
• Color-Kinematics Duality for Pure Yang-Mills and Gravity at One and Two Loops
• The Kerr-Schild double copy in curved spacetime
• Colour-Kinematics Duality for One-Loop Rational Amplitudes
• Scattering amplitudes in N=2 Maxwell-Einstein and Yang-Mills/Einstein supergravity
• Scattering Amplitudes
• The Kinematic Algebras from the Scattering Equations
• Integrand Oxidation and One-Loop Colour-Dual Numerators in N=4 Gauge Theory
• Amplitude Relations in Non-linear Sigma Model
• Berends-Giele recursion for double-color-ordered amplitudes
• String-inspired BCJ numerators for one-loop MHV amplitudes
• Construction of an effective Yang-Mills Lagrangian with manifest BCJ duality
• Expansion of Einstein-Yang-Mills Amplitude
• A new gauge-invariant double copy for heavy-mass effective theory
• Gluons and gravitons at one loop from ambitwistor strings
• On the kinematic algebra for BCJ numerators beyond the MHV sector
• Covariant Color-Kinematics Duality
• Off-Shell Color-Kinematics Duality for Chern-Simons
• New heavenly double copies
• Kinematic Lie Algebras From Twistor Spaces
• On Three-Algebra and Bi-Fundamental Matter Amplitudes and Integrability of Supergravity
• Off-shell Currents and Color-Kinematics Duality
• Dual-color decompositions at one-loop level in Yang-Mills theory
• A Vertex Operator Algebra Construction of the Colour-Kinematics Dual numerator
• BCJ Identities and $d$-Dimensional Generalized Unitarity
• Kinematic Numerators and a Double-Copy Formula for N = 4 Super-Yang-Mills Residues
• Note on symmetric BCJ numerator
• The Construction of Dual-trace Factor in Yang-Mills Theory
• Note on Construction of Dual-trace Factor in Yang-Mills Theory
• BCJ relations from a new symmetry of gauge-theory amplitudes
• Color kinematic symmetric (BCJ) numerators in a light-like gauge
• Homotopy double copy and the Kawai-Lewellen-Tye relations for the non-abelian and tensor Navier-Stokes equations
• Gauge invariance induced relations and the equivalence between distinct approaches to NLSM amplitudes
• BCJ Numerators from Differential Operator of Multidimensional Residue
• Next-to-MHV Yang-Mills kinematic algebra
• Off-shell BCJ Relation in Nonlinear Sigma Model
• Note on off-shell relations in nonlinear sigma model
• Boundary Behaviors for General Off-shell Amplitudes in Yang-Mills Theory
• A Tale Of Two Amplitudes In High Energy Physics
• Recursion in multiplet bases for tree-level MHV gluon amplitudes