Towards new hidden zero and 2-split of loop-level Feynman yntegrands in Tr(ϕ3) model

A bstract We extend the hidden zeros and 2-split of tree-level Tr( ϕ 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.