Amplitudes, Hopf algebras and the colour-kinematics duality

Abstract

It was recently proposed that the kinematic algebra featuring in the colour-kinematics duality for scattering amplitudes in heavy-mass effective field theory (HEFT) and Yang-Mills theory is a quasi-shuffle Hopf algebra. The associated fusion product determines the structure of the Bern-Carrasco-Johansson (BCJ) numerators, which are manifestly gauge invariant and with poles corresponding to heavy-particle exchange. In this work we explore the deep connections between the quasi-shuffle algebra and general physical properties of the scattering amplitudes. First, after proving the double-copy form for gravitational HEFT amplitudes, we show that the coproducts of the kinematic algebra are in correspondence with factorisations of BCJ numerators on massive poles. We then study an extension of the standard quasi-shuffle Hopf algebra to a non-abelian version describing BCJ numerators with all possible gluon orderings. This is achieved by tensoring the original algebra with a particular Hopf algebra of orderings. In this extended version, a specific choice of the coproduct in the algebra of orderings leads to an antipode in the resulting Hopf algebra that has the interpretation of reversing the gluons' order within each BCJ numerator.

arXiv comment51 pages, minor corrections, version accepted to JHEP

References in corpus (69)

• Heavy Quark Symmetry
• New Relations for Gauge-Theory Amplitudes
• Perturbative Quantum Gravity as a Double Copy of Gauge Theory
• Classical Polylogarithms for Amplitudes and Wilson Loops
• Scattering of Massless Particles: Scalars, Gluons and Gravitons
• New Color Decompositions for Gauge Amplitudes at Tree and Loop Level
• Black holes and the double copy
• A Duality For The S Matrix
• Reparameterisation Invariance Constraints on Heavy Particle Effective Field Theories
• The Momentum Kernel of Gauge and Gravity Theories
• Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
• The Kinematic Algebra From the Self-Dual Sector
• Simplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Amplitudes
• Algebraic Relations Between Harmonic Sums and Associated Quantities
• Pure Gravities via Color-Kinematics Duality for Fundamental Matter
• Color-Kinematics Duality for Pure Yang-Mills and Gravity at One and Two Loops
• Abelian Z-theory: NLSM amplitudes and alpha'-corrections from the open string
• Heavy Black Hole Effective Theory
• Algebras for Amplitudes
• Color-kinematic Duality for Form Factors
• Scattering amplitudes in N=2 Maxwell-Einstein and Yang-Mills/Einstein supergravity
• Type D Spacetimes and the Weyl Double Copy
• Unifying Relations for Scattering Amplitudes
• Symmetry and Action for Flavor-Kinematics Duality
• Non-abelian $Z$-theory: Berends-Giele recursion for the ${\alpha }^{\prime }$-expansion of disk integrals
• New Relations for Three-Dimensional Supersymmetric Scattering Amplitudes
• A Twistorial Foundation for the Classical Double Copy
• Integrand Oxidation and One-Loop Colour-Dual Numerators in N=4 Gauge Theory
• Color/kinematics duality for general abelian orbifolds of N=4 super Yang-Mills theory
• Amplitude Relations in Non-linear Sigma Model
• New BCJ representations for one-loop amplitudes in gauge theories and gravity
• Quasi-shuffle products revisited
• New Relations for Einstein-Yang-Mills Amplitudes
• Semi-abelian Z-theory: NLSM+phi^3 from the open string
• BCJ numerators from reduced Pfaffian
• Construction of an effective Yang-Mills Lagrangian with manifest BCJ duality
• Two-loop supersymmetric QCD and half-maximal supergravity amplitudes
• Expansion of Einstein-Yang-Mills Amplitude
• A new gauge-invariant double copy for heavy-mass effective theory
• Einstein-Yang-Mills from pure Yang-Mills amplitudes
• Double Copy from Homotopy Algebras
• Expanding Einstein-Yang-Mills by Yang-Mills in CHY frame
• BRST-Lagrangian Double Copy of Yang-Mills Theory
• Color-Kinematics Duality in One-Loop Four-Gluon Amplitudes with Matter
• Amplitude relations in heterotic string theory and Einstein-Yang-Mills
• Covariant Color-Kinematics Duality
• On the kinematic algebra for BCJ numerators beyond the MHV sector
• Kinematic Hopf Algebra for BCJ Numerators in Heavy-Mass Effective Field Theory and Yang-Mills Theory
• Kinematic Jacobi Identity is a Residue Theorem: Geometry of Color-Kinematics Duality for Gauge and Gravity Amplitudes
• BCJ Relation of Color Scalar Theory and KLT Relation of Gauge Theory
• Color-kinematics duality and Sudakov form factor at five loops for N=4 supersymmetric Yang-Mills theory
• Pions as Gluons in Higher Dimensions
• The pure BRST Einstein-Hilbert Lagrangian from the double-copy to cubic order
• Constraints and Generalized Gauge Transformations on Tree-Level Gluon and Graviton Amplitudes
• Biadjoint wires
• Simple encoding of higher derivative gauge and gravity counterterms
• Double Field Theory as the Double Copy of Yang-Mills
• Geometry-Kinematics Duality
• Generalizations of the Double-Copy: the KLT Bootstrap
• Composing Effective Prediction at Five Points
• Colour-Kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms
• Relations for Einstein-Yang-Mills amplitudes from the CHY representation
• On the Lagrangian formulation of the double copy to cubic order
• Non-perturbative aspects of the self-dual double copy
• Non-planar form factors of generic local operators via on-shell unitarity and color-kinematics duality
• Color-kinematics duality, double copy and the unitarity method for higher-derivative QCD and quadratic gravity
• BCJ Numerators from Differential Operator of Multidimensional Residue
• Classical gravitational scattering from a gauge-invariant double copy
• The seeds of EFT double copy

Cited by in corpus (13)

• One-loop Gravitational Bremsstrahlung and Waveforms from a Heavy-Mass Effective Field Theory
• Resummed spinning waveforms from five-point amplitudes
• Celestial chiral algebras, colour-kinematics duality and integrability
• Tree-level amplitudes from the pure spinor superstring
• Classical Spin Gravitational Compton Scattering
• Gauge independent kinematic algebra of self-dual Yang-Mills theory
• Classical Double Copy of Spinning Worldline Quantum Field Theory
• Kinematic Hopf algebra for amplitudes and form factors
• Covariant color-kinematics duality, Hopf algebras and permutohedra
• Double copy for tree-level form factors. Part I. Foundations
• Double copy for tree-level form factors. Part II. Generalizations and special topics
• Color-Kinematic Numerators for Fermion Compton Amplitudes
• Algebraic Consistency and Explicit Construction of One-Loop BCJ Numerators of Yang-Mills and Related Theories