2-split from Feynman diagrams and expansions

A bstract In this paper, we investigate the 2-split behavior of tree-level amplitudes of bi-adjoint scalar (BAS), Yang-Mills (YM), non-linear sigma model (NLSM), and general relativity (GR) theories under certain kinematic conditions. Our approach begins with a proof, based on the Feynman diagram method, of the 2-split property for tree-level BAS ⊕ X amplitudes with X = YM , NLSM , GR. The proof relies crucially on a particular pattern in the Feynman rules of various vertices. Building on this, we use the expansion of X amplitudes into BAS ⊕ X amplitudes to establish the 2-split behavior. As a byproduct, we derive universal expansions of the resulting pure X currents into BAS currents, which closely parallel the corresponding on-shell amplitude expansions.