{
  "id": "2111.15649",
  "version": "v2",
  "published": "2021-11-30T18:31:51.000Z",
  "updated": "2022-03-03T17:13:21.000Z",
  "title": "Kinematic Hopf Algebra for BCJ Numerators in Heavy-Mass Effective Field Theory and Yang-Mills Theory",
  "authors": [
    "Andreas Brandhuber",
    "Gang Chen",
    "Henrik Johansson",
    "Gabriele Travaglini",
    "Congkao Wen"
  ],
  "comment": "10 pages, 8 figures, Physical Review Letters version",
  "categories": [
    "hep-th",
    "math.CO"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-03-03T17:13:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "heavy-mass effective field theory",
      "bcj numerator",
      "kinematic hopf algebra",
      "quasi-shuffle hopf algebra structure",
      "dimensional pure yang-mills theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}