Related Experiment Video
Updated: Sep 27, 2025

Image-based Lagrangian Particle Tracking in Bed-load Experiments
Published on: July 20, 2017
Kinematic Hopf Algebra for Bern-Carrasco-Johansson Numerators in Heavy-Mass Effective Field Theory and Yang-Mills
Andreas Brandhuber1, Gang Chen1, Henrik Johansson2,3
1Centre for Theoretical Physics, Department of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom.
Abstract:
We present a closed formula for all Bern-Carrasco-Johansson (BCJ) numerators describing D-dimensional tree-level scattering amplitudes in a heavy-mass effective field theory with two massive particles and an arbitrary number of gluons. The corresponding gravitational amplitudes obtained via the double copy directly enter the computation of black-hole scattering and gravitational-wave emission. Our construction is based on finding a kinematic algebra for the numerators, which we relate to a quasishuffle Hopf algebra. The BCJ numerators thus obtained have a compact form and intriguing features: gauge invariance is manifest, locality is respected for massless exchange, and they contain poles corresponding to massive exchange. Counting the number of terms in a BCJ numerator for n-2 gluons gives the Fubini numbers F_{n-3}, reflecting the underlying quasishuffle Hopf algebra structure. Finally, by considering an appropriate factorization limit, the massive particles decouple, and we thus obtain a kinematic algebra and all tree-level BCJ numerators for D-dimensional pure Yang-Mills theory.
Related Concept Videos
Spin–Spin Coupling Constant: Overview
Qualitatively, any spin plus-half nucleus polarizes the spins of its electrons to the minus-half state. Consequently, the paired electron in the hydrogen–carbon bond must...
Conservation of Angular Momentum
Moment-of-Momentum Equation
Conservation of Angular Momentum: Application
Euler's Equations of Motion
Dimensionless Groups in Fluid Mechanics

