Box Singularity Conditions in Box Diagrams of Decay Processes

TL;DR

This work analyzes box singularities in box diagrams of three‑body decays, extending the Landau/triangle singularity framework to two box topologies. It derives analytic conditions for the occurrence of box singularities, including both general configurations and degenerate collinear cases, and translates the velocity‑based criteria into pure mass‑condition inequalities. The results yield a practical, two‑tier method: check on‑shell velocity relations in the diagram (Coleman–Norton interpretation) and/or apply the derived mass inequalities on the Dalitz plane to determine the presence of Landau singularities. Together, these findings sharpen the understanding of how nonperturbative, kinematic singularities manifest in box diagrams and their potential impact on invariant mass spectra in hadronic processes.

Abstract

Since 1959, singularities within single-loop diagrams have been studied. They are believed to significantly influence our understanding of experimental observables. In this study, we explore the singularities that arise from box diagrams in the decay process, which can be classified into two distinct categories. A comprehensive analysis of box singularities has been conducted, wherein we have derived and presented specific conditional formulae to ascertain the occurrence of singularities, along with the related physical scenarios.

Figures (13)

• Figure 1: The triangle loop diagram.
• Figure 2: Two types of box diagrams. The notation of particles and momentum is shown in (b).
• Figure 3: The relevant cuts for the two types of box diagrams.
• Figure 4: The diagram of the internal particle trajectories in real space-time when box singularity happens in the case of Fig. \ref{['fg:3loopa']}.
• Figure 5: The directions of each particle in the case of $\theta_1=\theta=\pi$. Black line: the direction is determined; red line: the direction is determined only when singularity occurs.