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

Kang Zhou

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arXiv · 2604.13810

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

Kang Zhou

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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.

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Kang Zhou. 2026-04-15. Towards New Hidden Zero and $2$-Split of Loop-Level Feynman Integrands in ${\rm Tr}(\phi^3)$ Model. https://arxiv.org/abs/2604.13810

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