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 sector 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.

在四个维度上，具有两个负螺旋性胶子的树级双迹 Einstein-Yang-Mills (EYM) 振幅（( g ^- , g ^-)-configuration) 满足对称生成森林公式，该公式由图形扩展规则导出。另一方面，在 Cachazo-He-Yuan (CHY) 公式的框架下，最大螺旋度违反 (MHV) 振幅由散射方程的 MHV 解支持。双迹振幅的对称公式与四维Cachazo-He-Yuan (CHY) 公式的MHV 扇区之间的关系仍不清楚。在这篇文章中，我们在四个维度上推进了一系列生成森林的变换，然后展示了一种系统的方法来分解双迹 EYM 振幅的 CHY 公式的 MHV 部分。沿着这个思路，双迹MHV幅值的对称公式直接由CHY公式的MHV扇区得到。