A bstractIn this paper we define, independent of theories, two discriminant matrices involving a solution to the scattering equations in four dimensions, the ranks of which are used to divide the solution set into a disjoint union of subsets. We further demonstrate, entirely within the Cachazo-He-Yuan formalism, that each subset of solutions gives nonzero contribution to tree-level NkMHV gauge/gravity amplitudes only for a specific value of k. Thus the solutions can be characterized by the rank of their discriminant matrices, which in turn determines the value of k of the NkMHV amplitudes a solution can support. As another application of the technique developed, we show analytically that in Einstein-Yang-Mills theory, if all gluons have the same helicity, the tree-level single-trace amplitudes must vanish.

中文翻译：

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表征支持树级 N k MHV 规范/重力振幅的散射方程的解

摘要在本文中，我们独立于理论定义了两个判别矩阵，它们涉及四维散射方程的解，其秩用于将解集划分为不相交的子集联合。我们进一步证明，完全在 Cachazo-He-Yuan 形式主义中，每个解的子集仅对于特定的 k 值对树级 NkMHV 规范/重力振幅给出非零贡献。因此，解可以通过它们的判别矩阵的秩来表征，这反过来决定了解可以支持的 NkMHV 幅度的 k 值。作为该技术的另一个应用，我们分析表明，在爱因斯坦-杨-米尔斯理论中，如果所有胶子具有相同的螺旋度，则树级单迹振幅必须消失。