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).

💡 Research Summary

**
The paper tackles a long‑standing problem in the asymptotic expansion of multi‑loop Feynman integrals: how to systematically identify all contributing regions in the Expansion‑by‑Regions (EbR) method for generic mass‑less wide‑angle scattering. While in Euclidean space the region structure is well understood in terms of “asymptotically irreducible subgraphs”, the Minkowski‑space case is far more intricate because of the richer infrared (IR) structure. Existing geometric approaches associate a (N + 1)‑dimensional Lee‑Pomeransky polytope Δ(G) to a graph G and identify regions with its co‑dimension‑1 faces (lower facets). However, this method only captures “facet regions” that lie on the polytope boundary; hidden regions inside the polytope remain elusive and have so far been treated only case‑by‑case.

The authors present an all‑order, momentum‑space prescription that determines the complete set of facet regions for any mass‑less wide‑angle process, at any loop order and for any hierarchy of external virtualities. Their approach combines convex geometry with graph theory. The key steps are:

1. Virtuality Expansion Framework – They introduce a unified expansion parameter λ that simultaneously describes hard, collinear, and soft scalings. External momenta are classified into three sets (collinear p_i, hard q_j, soft l_k) with generic power‑law scalings in λ. The expansion in λ is called the “virtuality expansion”.

2. Momentum Modes and Lattice Structure – Basic modes are Hard (H), Collinear_i (C_i) and Soft (S). Any mode appearing in the expansion can be written as S^m C^n_i, where m and n are non‑negative integers (n may be ∞ for exactly light‑like momenta). The scaling of a mode X = S^m C^n_i is \