Universal Interpretation of Hidden Zero and $2$-Split of Tree-Level Amplitudes Using Feynman Diagrams, Part $\mathbf{I}$: ${\rm Tr}(\phi^3)$, NLSM and YM

TL;DR

This paper introduces a universal diagrammatic interpretation for hidden zeros and 2-splits in tree-level amplitudes across ${ m Tr}()3$, NLSM, and YM theories, based on a new factorization mechanism called SFASL.

Contribution

It extends the SFASL mechanism from ${ m Tr}()3$ to NLSM and YM, unifying the interpretation of zeros and splits in various theories.

Findings

The interpretation links hidden zeros to on-shell conditions $k_j^2=0$.

2-splits are explained as diagram separations along two lines.

The mechanism unifies amplitude structures across different theories.

Abstract

In this paper, we propose a universal diagrammatic interpretation of hidden zeros and $2$-splits of tree-level amplitudes. Originally developed for ${\rm Tr}(\phi^3)$ amplitudes in our previous work, this interpretation is now extended to tree-level amplitudes in Nonlinear sigma model (NLSM) and Yang-Mills (YM) theories. The interpretation is based on a certain factorization behavior of Feynman diagrams under specific kinematic constraints, which we term shuffle factorization along a specific line (SFASL). This mechanism allows us to separate Feynman diagrams along specific lines after summing over shuffle permutations. When applied to NLSM and YM amplitudes, we perform proper extensions of the SFASL used in the ${\rm Tr}(\phi^3)$ case. Through the SFASL, the interpretation for the hidden zeros and $2$-splits of tree amplitudes of ${\rm Tr}(\phi^3)$, NLSM, and YM can be unified as: the hidden zeros are ascribed to the on-shell condition $k_j^2=0$ of a massless particle, while the $2$-splits are caused by separating each Feynman diagram along two lines, akin to unzipping two zippers.