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Kinematic Hopf Algebra for BCJ Numerators in Heavy-Mass Effective Field Theory and Yang-Mills Theory

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arxiv 2111.15649 v2 pith:JYPY25FV submitted 2021-11-30 hep-th math.CO

Andreas Brandhuber , Gang Chen , Henrik Johansson , Gabriele Travaglini , Congkao Wen This is my paper · ORCID
classification hep-thmath.CO
keywords algebranumeratorstheoryhopfkinematicmassiveamplitudescorresponding
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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 quasi-shuffle 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 $\mathsf{F}_{n-3}$, reflecting the underlying quasi-shuffle Hopf algebra structure. Finally, by considering an appropriate factorisation 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.

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Cited by 4 Pith papers

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    The heavy-mass effective field theory kinematic numerators are derived as the field theory limit of nested commutators of string vertex operators, reproducing and extending earlier fusion-rule results.

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