The expansions of tree-level amplitudes for one theory into amplitudes for another theory, which have been studied in various recent literatures, exhibit hidden connections between different theories that are invisible in traditional Lagrangian formulism of quantum field theory. In this paper, the general expansion of tree EM (Einstein-Maxwell) amplitudes into KK basis of tree YM (Yang-Mills) amplitudes have been derived by applying the method based on differential operators. The obtained coefficients are shared by the expansion of tree $\phi^4$ amplitudes into tree BS (bi-adjoint scalar) amplitudes, the expansion of tree sYMS (special Yang-Mills-scalar) amplitudes into tree BS amplitudes, as well the expansion of tree DBI (Dirac-Born-Infeld) amplitudes into tree special extended DBI amplitudes.

中文翻译：

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爱因斯坦-麦克斯韦和其他理论的树振幅的扩展

将一种理论的树级振幅扩展为另一种理论的振幅，这在最近的各种文献中都有研究，显示出不同理论之间的隐藏联系，这些联系在量子场论的传统拉格朗日公式中是不可见的。在本文中，应用基于微分算子的方法导出了树EM（Einstein-Maxwell）振幅到树YM（Yang-Mills）振幅的KK基的一般扩展。获得的系数由树 $\phi^4$ 振幅扩展到树 BS（双伴随标量）振幅，树 sYMS（特殊杨米尔斯标量）振幅扩展到树 BS 振幅，以及树 DBI (Dirac-Born-Infeld) 振幅扩展为树特殊扩展 DBI 振幅。