来自世界面的运动学分子：来自标记树的立方树
Kinematic numerators from the worldsheet: cubic trees from labelled trees
AI总结：
本文研究利用世界面公式计算任意维度多理论树级散射振幅问题，通过Cayley函数提出标记树系数线性组合的运动学分子通用公式，构建了可计算多理论树振幅的高效Mathematica算法。
摘要：
在本短文中，我们重新探讨了利用世界面公式显式计算任意维度下各类理论中的树级散射振幅的问题。已知一旦世界面上的任意半被积函数约化为对数函数，世界面公式就能生成树振幅的立方树展开，其运动学分子自动满足雅可比恒等式。我们回顾了一类被称为“Cayley函数”的自然世界面函数，它们与标记树一一对应，且已知的半被积函数可在其上自然展开，展开系数是运动学分子的极为紧致的构造基元。我们提出了一个通用公式，将所有立方树的运动学分子表示为标记树系数的线性组合，该组合按构造满足雅可比恒等式，且将用主分子表示的常见组合作为特例包含在内。我们的结果提供了一种高效算法，并已在一个Mathematica程序包中实现，可用于计算包括非线性σ模型、特殊Galileon、Yang-Mills-标量、Einstein-Yang-Mills和Dirac-Born-Infeld在内的理论中的所有树振幅。
英文摘要：
In this note we revisit the problem of explicitly computing tree-level scattering amplitudes in various theories in any dimension from worldsheet formulas. The latter are known to produce cubic-tree expansion of tree amplitudes with kinematic numerators automatically satisfying Jacobi-identities, once any half-integrand on the worldsheet is reduced to logarithmic functions. We review a natural class of worldsheet functions called "Cayley functions", which are in one-to-one correspondence with labelled trees, and natural expansions of known half-integrands onto them with coefficients that are particularly compact building blocks of kinematic numerators. We present a general formula expressing kinematic numerators of all cubic trees as linear combinations of coefficients of labelled trees, which satisfy Jacobi identities by construction and include the usual combinations in terms of master numerators as a special case. Our results provide an efficient algorithm, which is implemented in a Mathematica package, for computing all tree amplitudes in theories including non-linear sigma model, special Galileon, Yang-Mills-scalar, Einstein-Yang-Mills and Dirac-Born-Infeld.
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