Soft theorems of tree-level ${\rm Tr}(\phi^3)$, YM and NLSM amplitudes from $2$-splits

TL;DR

This paper extends a method based on factorization and $2$-splits to derive soft theorems for tree-level amplitudes in ${ m Tr}(^3)$, YM, and NLSM, revealing universal structures and relations among soft behaviors.

Contribution

It introduces an extended approach using $2$-splits to derive and unify soft theorems across different theories at tree level.

Findings

Reproduces leading and sub-leading soft theorems for ${ m Tr}(^3)$ and YM.

Establishes double-soft theorems for NLSM amplitudes.

Identifies universal representations of higher-order soft theorems in reduced kinematic spaces.

Abstract

In this paper, we extend the method proposed in \cite{Arkani-Hamed:2024fyd} 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 ${\rm Tr}(\phi^3)$ 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 ${\rm Tr}(\phi^3)$ 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 theorems for the resulting pure YM and NLSM currents in the corresponding $2$-splits.