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.

51 pages, minor corrections, version accepted to JHEP

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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