Abstract
We present a closed formula for all Bern-Carrasco-Johansson (BCJ) numerators describing -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 gluons gives the Fubini numbers , 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 -dimensional pure Yang-Mills theory.
- Received 8 December 2021
- Accepted 1 March 2022
DOI:https://doi.org/10.1103/PhysRevLett.128.121601
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Published by the American Physical Society