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

TL;DR

This work addresses how to systematically determine all regions contributing to the asymptotic expansion of massless wide-angle scattering in the Expansion-by-Regions framework. It introduces an all‑order momentum-space prescription based on convex geometry and graph theory, where facet regions are tied to minimum spanning (2‑)trees and momentum modes form a lattice under join/meet operations. The main results are two subgraph requirements—the connectivity theorems and an infrared-compatibility condition—that are necessary and sufficient for facet regions, together with a weight-solving algorithm that guarantees unique edge-weight assignments. The authors validate the framework with multi-loop examples (three-loop 2→2 and six-loop 1→3 decay plus soft emission), show agreement with computational region counts, and discuss extensions to massive cases and potential applications to factorization and SCET. Overall, the paper provides a rigorous, all-order, momentum-space method to enumerate dominant region contributions in a broad multiscale setting.

Abstract

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).

Figures (50)

• Figure 1: The one-loop form factor with all the edges massless. Even for expanding this relatively simple graph near the same infrared limit ($p_1^2,p_2^2\ll 1$ and $q_1^2\sim 1$), the region structure depends strongly on the precise form of the expansion.
• Figure 2: Three examples where the loop momenta are all soft. The expanded integrals corresponding to (a) and (b) are scaleless, and should not be taken into account in EbR. In contrast, (c) leads to a scaleful integral and should be considered as one region.
• Figure 3: General wide-angle scattering graph $\mathcal{G}$, with external momenta $\{p_i^\mu\}_{i=1,\dots,K}$, $\{q_j^\mu\}_{i=1,\dots,L}$, and $\{l_k\}_{k=1,\dots,M}$ subject to eq. (\ref{['eq:virtuality_expansion']}).
• Figure 4: Two examples illustrating some special cases of mode subgraphs, where the external momenta and internal vertices are explicitly specified. (a): the $H$ subgraph $\Gamma_H$ consists of the vertex $v_0$ only. (b): $\Gamma_H$ consists of the vertex $v_0$ only; additionally, the $S^2$ subgraph $\Gamma_{S^2}$ consists of two connected components.
• Figure 5: The fundamental pattern for facet regions in the virtuality expansion of massless wide-angle scattering. The graph comprises a connected hard subgraph $\mathcal{H}$, jet subgraphs $\mathcal{J}_1,\dots,\mathcal{J}_K$, and a soft subgraph $\mathcal{S}$. Off-shell external momenta $q_1,\dots,q_L$ attach exclusively to $\mathcal{H}$. Each on-shell external momentum $p_i$ attaches to its jet $\mathcal{J}_i$ (if nonempty) or directly to $\mathcal{H}$ (otherwise). Doubled lines represent any number of edges.

Theorems & Definitions (50)

• Theorem 3.1
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Corollary 3.3.1
• proof
• Corollary 3.3.2
• proof
• Theorem 3.4