A bstractAll tree-level amplitudes in Einstein-Yang-Mills (EYM) theory and gravity (GR) can be expanded in terms of color ordered Yang-Mills (YM) ones whose coefficients are polynomial functions of Lorentz inner products and are constructed by a graphic rule. Once the gauge invariance condition of any graviton is imposed, the expansion of a tree level EYM or gravity amplitude induces a nontrivial identity between color ordered YM amplitudes. Being different from traditional Kleiss-Kuijf (KK) and Bern-Carrasco-Johansson (BCJ) relations, the gauge invariance induced identity involves polarizations in the coefficients. In this paper, we investigate the relationship between the gauge invariance induced identity and traditional BCJ relations. By proposing a refined graphic rule, we prove that all the gauge invariance induced identities for single trace tree-level EYM amplitudes can be precisely expanded in terms of traditional BCJ relations, without referring any property of polarizations. When further considering the transversality of polarizations and momentum conservation, we prove that the gauge invariance induced identity for tree-level GR (or pure YM) amplitudes can also be expanded in terms of traditional BCJ relations for YM (or bi-scalar) amplitudes. As a byproduct, a graph-based BCJ relation is proposed and proved.

中文翻译：

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一种测量不变性诱导恒等式的图形方法

摘要爱因斯坦-杨-米尔斯 (EYM) 理论和重力 (GR) 中的所有树级振幅都可以根据颜色有序杨-米尔斯 (YM) 展开，其系数是洛伦兹内积的多项式函数，并由图形规则。一旦施加了任何引力子的规范不变性条件，树级别 EYM 或重力振幅的扩展会在颜色有序的 YM 振幅之间引起非平凡的同一性。与传统的 Kleiss-Kuijf (KK) 和 Bern-Carrasco-Johansson (BCJ) 关系不同，规范不变性诱导恒等式涉及系数的极化。在本文中，我们研究了规范不变性诱导恒等式与传统 BCJ 关系之间的关系。通过提出一个精致的图形规则，我们证明了单道树级 EYM 振幅的所有规范不变性诱导恒等式都可以根据传统的 BCJ 关系精确扩展，而无需参考任何极化属性。当进一步考虑极化的横向性和动量守恒时，我们证明了树级 GR（或纯 YM）振幅的规范不变性诱导恒等式也可以根据 YM（或双标量）振幅的传统 BCJ 关系扩展。作为副产品，提出并证明了基于图的 BCJ 关系。我们证明了树级 GR（或纯 YM）幅度的规范不变性诱导恒等式也可以根据 YM（或双标量）幅度的传统 BCJ 关系进行扩展。作为副产品，提出并证明了基于图的 BCJ 关系。我们证明了树级 GR（或纯 YM）幅度的规范不变性诱导恒等式也可以根据 YM（或双标量）幅度的传统 BCJ 关系进行扩展。作为副产品，提出并证明了基于图的 BCJ 关系。