Brandhuber Andreas, Chen Gang, Johansson Henrik, Travaglini Gabriele, Wen Congkao
Centre for Theoretical Physics, Department of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom.
Department of Physics and Astronomy, Uppsala University, Box 516, 75120 Uppsala, Sweden.
Phys Rev Lett. 2022 Mar 25;128(12):121601. doi: 10.1103/PhysRevLett.128.121601.
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.
我们给出了一个封闭公式，用于描述具有两个大质量粒子和任意数量胶子的重质量有效场论中(D)维树图水平散射振幅的所有伯恩 - 卡拉斯科 - 约翰松（BCJ）分子。通过双拷贝得到的相应引力振幅直接进入黑洞散射和引力波发射的计算。我们的构建基于为分子找到一个运动学代数，我们将其与一个拟洗牌霍普夫代数相关联。由此得到的BCJ分子具有紧凑的形式和有趣的特征：规范不变性是明显的，对于无质量交换尊重局域性，并且它们包含对应于大质量交换的极点。计算(n - 2)个胶子的BCJ分子中的项数得到富比尼数(F_{n - 3})，反映了潜在的拟洗牌霍普夫代数结构。最后，通过考虑适当的因式分解极限，大质量粒子解耦，从而我们得到了(D)维纯杨 - 米尔斯理论的运动学代数和所有树图水平的BCJ分子。
