Soft theorems of tree-level Tr(ϕ3), YM and NLSM amplitudes from 2-splits

A bstract In this paper, we extend the method proposed in [1] for deriving soft theorems of amplitudes, which relies exclusively on factorization properties including conventional factorizations on physical poles, as well as newly discovered 2-splits on special loci in kinematic space. Using the extended approach, we fully reproduce the leading and sub-leading single-soft theorems for tree-level Tr( ϕ 3 ) and Yang-Mills (YM) amplitudes, along with the leading and sub-leading double-soft theorems for tree-level amplitudes of non-linear sigma model (NLSM). Furthermore, we establish universal representations of higher-order single-soft theorems for tree-level Tr( ϕ 3 ) and YM amplitudes in reduced lower-dimensional kinematic spaces. All obtained soft factors maintain consistency with momentum conservation; that is, while each explicit expression of the resulting soft behavior may changes under re-parameterization via momentum conservation, the physical content remains equivalent. Additionally, we find two interesting by-products: first, the single-soft theorems of YM amplitudes and the double-soft theorems of NLSM, at leading and sub-leading orders, are related by a simple kinematic replacement. This replacement also transmutes gauge invariance to Adler zero. Second, we obtain universal sub-leading soft behaviors for the resulting pure YM and NLSM currents in the corresponding 2-splits.