Towards New Hidden Zero and $2$-Split of Loop-Level Feynman Integrands in ${\rm Tr}(\phi^3)$ Model

TL;DR

This paper generalizes the concepts of hidden zeros and 2-split structures from tree-level to loop-level Feynman integrands in the ${ m Tr}({ ext{phi}^3)}$ model, revealing simpler kinematic conditions and a recursive structure.

Contribution

It introduces a novel factorization mechanism for loop-level integrands, extending the hidden zeros and 2-split concepts with a simple, recursive formula.

Findings

Loop-level hidden zeros differ from previous literature.

The 2-split condition at loop-level is derived from the zero condition.

The L-loop integrand can be expressed as a sum over L+1 terms with 2-split structure.

Abstract

We extend the hidden zeros and $2$-split of tree-level ${\rm Tr}(\phi^3)$ amplitudes to loop-level Feynman integrands, apart from some physically irrelevant scaleless integrals. Our method is based on a certain factorization mechanism that occurs in Feynman diagrams when summing over shuffle permutations. The loop-level hidden zeros and $2$-split identified in this work differ from those in the literature. In our result, the kinematic conditions for loop-level hidden zeros and $2$-split are remarkably simple. Their connection is as tight as at tree-level, with the same procedure for obtaining the $2$-split condition from the zero condition. The resulting $2$-split formula at loop-level represents a generalization of that at tree-level: the $L$-loop integrand is expressed as a sum over $L+1$ terms, each of which exhibits a $2$-split structure.