In four dimensions, a tree-level double-trace Einstein-Yang-Mills (EYM) amplitude with two negative-helicity gluons (the $(g^-,g^-)$-configuration) satisfies a symmetric spanning forest formula, which was derived from the graphic expansion rule. On another hand, in the framework of Cachazo-He-Yuan (CHY) formula, the maximally-helicity-violating (MHV) amplitudes are supported by the MHV solution of scattering equations. The relationship between the symmetric formula for double-trace amplitudes, and the MHV sector of Cachazo-He-Yuan (CHY) formula in four dimensions is still not clear. In this note, we promote a series of transformations of the spanning forests in four dimensions and then show a systematic way for decomposing the MHV sector of the CHY formula of double-trace EYM amplitudes. Along this line, the symmetric formula of double-trace MHV amplitudes is directly obtained by the MHV section of CHY formula. We then prove that EYM amplitude with an arbitrary total number of negative-helicity particles (gravitons and gluons) has to vanish when the number of negative- (or positive-) helicity gluons is less than the number of traces.

中文翻译：

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关于四个维度的多道 EYM 幅度的注释

在四个维度上，具有两个负螺旋度胶子（$(g^-,g^-)$-配置）的树级双迹 Einstein-Yang-Mills (EYM) 幅度满足对称生成森林公式，即源于图形扩展规则。另一方面，在Cachazo-He-Yuan（CHY）公式的框架下，最大螺旋违反（MHV）振幅由散射方程的MHV解支持。双道振幅对称公式与四维卡查佐-和元（CHY）公式的MHV扇区之间的关系尚不清楚。在这篇笔记中，我们在四个维度上促进了一系列跨越森林的转换，然后展示了一种系统的方法来分解双迹 EYM 幅度的 CHY 公式的 MHV 扇区。沿着这条线，双迹MHV幅值的对称公式由CHY公式的MHV部分直接得到。然后，我们证明当负（或正）螺旋胶子的数量小于轨迹数时，具有任意总数的负螺旋粒子（引力子和胶子）的 EYM 振幅必须消失。