A bstractIn this paper, we study the relation between the Cachazo-He-Yuan (CHY) formula and the maximal-helicity-violating (MHV) amplitudes of Yang-Mills and gravity in four dimensions. We prove that only one special rational solution of the scattering equations found by Weinzierl supports the MHV amplitudes. Namely, localized at this solution, the integrated CHY formula produces the Parke-Taylor formula for MHV Yang-Mills amplitudes as well as the Hodges formula for MHV gravitational amplitudes, with an arbitrary number of external gluons/gravitons. This is achieved by developing techniques, in a manifestly Möbius covariant formalism, to explicitly compute relevant reduced Pfaffians/determinants. We observe and prove two interesting properties (or identities), which facilitate the computations. We also check that all the other (n − 3)! − 1 solutions to the scattering equations do not support the MHV amplitudes, and prove analytically that this is indeed true for the other special rational solution proposed by Weinzierl, that actually supports the anti-MHV amplitudes. Our results reveal a mysterious feature of the CHY formalism that in Yang-Mills and gravity theory, solutions of scattering equations, involving only external momenta, somehow know about the configuration of external polarizations of the scattering amplitudes.

中文翻译：

---

CHY 公式和 MHV 振幅

摘要在本文中，我们从四个维度研究了 Cachazo-He-Yuan (CHY) 公式与 Yang-Mills 和重力的最大螺旋度违反 (MHV) 振幅之间的关系。我们证明，只有 Weizierl 发现的散射方程的一种特殊有理解支持 MHV 振幅。即，局部于该解，积分 CHY 公式生成 MHV Yang-Mills 振幅的 Parke-Taylor 公式以及 MHV 引力振幅的 Hodges 公式，具有任意数量的外部胶子/引力子。这是通过开发技术来实现的，在明显的莫比乌斯协变形式中，明确计算相关的约化 Pfaffians/决定因素。我们观察并证明了两个有趣的属性（或身份），这有助于计算。我们还检查所有其他 (n − 3)! − 1 散射方程的解不支持 MHV 幅度，并通过分析证明这对于 Weizierl 提出的其他特殊有理解确实是正确的，它实际上支持反 MHV 幅度。我们的结果揭示了 CHY 形式主义的一个神秘特征，即在杨米尔斯和引力理论中，仅涉及外部动量的散射方程的解以某种方式知道散射幅度的外部极化的配置。