A bstractThe tree-level BCJ relations reveal a duality between color and kinematic structure, this can be used to reduce the number of independent color-ordered amplitudes. At one-loop level in Yang-Mills theory, we investigate a similar BCJ relation in this paper. The four-point one-loop example in $\mathcal{N}=4$ SYM gives hints about the relations between integrands. The five-point example suggests that the general formula can be proven by the unitary-cut method. We will then prove a ‘general’ BCJ relation for one-loop integrands by D-dimension unitary cut, which can be regarded as a non-trivial generalization of the (fundamental)BCJ relation given by Boels and Isermann in [17, 18].

中文翻译：

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杨米尔斯理论中单环层次的一般BCJ关系

摘要树级 BCJ 关系揭示了颜色和运动结构之间的二元性，这可用于减少独立颜色排序幅度的数量。在 Yang-Mills 理论的单环层次上，我们在本文中研究了类似的 BCJ 关系。$\mathcal{N}=4$ SYM 中的四点单循环示例给出了有关被积函数之间关系的提示。五点例子表明，通用公式可以用幺正切法证明。然后，我们将通过 D 维酉割证明单环被积函数的“一般”BCJ 关系，这可以看作是 Boels 和 Isermann 在 [17, 18] 中给出的（基本）BCJ 关系的非平凡推广.