# $2$-split from Feynman diagrams and Expansions

## TL;DR

This paper proves the 2-split property of tree-level amplitudes in various field theories using Feynman diagrams and expands these amplitudes into simpler components, revealing universal structures.

## Contribution

It provides a rigorous proof of the 2-split behavior for BAS plus other theories and derives universal expansions of currents into BAS currents.

## Key findings

- Proved 2-split property for BAS⊕X amplitudes using Feynman diagrams.
- Established expansions of X amplitudes into BAS⊕X amplitudes.
- Derived universal expansions of pure X currents into BAS currents.

## Abstract

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$\oplus$X amplitudes with $\mathrm{X}={\mathrm{YM},\mathrm{NLSM},\mathrm{GR}}$. The proof relies crucially on a particular pattern in the Feynmam rules of various vertices. Building on this, we use the expansion of X amplitudes into BAS$\oplus$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.