All-order prescription for facet regions in massless wide-angle scattering

We take a step toward answering a long-standing question in the asymptotic expansion of Feynman integrals: how to systematically determine the regions in the Expansion-by-Regions technique for multiscale processes? Focusing on generic massless wide-angle scattering, we provide an all-order momentum-space prescription for facet regions, which generally dominate – and in most cases exhaust – the contributions in a given asymptotic expansion. This extends the Euclidean-space picture, where regions correspond to specific subgraphs, to the complexities of Minkowski space. Our results are derived from a novel analytical approach combining graph theory and convex geometry; as a key byproduct, we uncover for the first time the algebraic structure underlying momentum modes (collinear, soft, and their hierarchies).