Non-Abelian Recoil Geometry and Infrared Holonomies in Heavy-Quark Transitions

TL;DR

This work introduces a geometric description of heavy-quark transitions in which infrared-dressed states are transported along a two-dimensional recoil space, giving rise to a non-Abelian Berry curvature and an emergent $SU(2)$ holonomy. Single-step decays reproduce Abelian Berry phases and the Isgur-Wise universal behavior, while sequential decays probe genuinely non-Abelian geometry, correlating different channels through two universal modes $\Xi_{\pm}$. The framework naturally explains the $j_\ell=\tfrac{1}{2}$ versus $j_\ell=\tfrac{3}{2}$ puzzle and large isospin mixing, and it provides a unified interpretation of near-threshold exotic states such as the $X(3872)$ as adiabatic eigenstates determined by infrared holonomies. It also clarifies the role of $1/m_Q$ corrections as controlled departures from adiabatic transport, suggesting a broad, geometrical organizing principle for heavy-quark dynamics and near-threshold spectroscopy.

Abstract

We propose a geometric formulation of heavy-quark transitions in which infrared-dressed states are adiabatically transported in the multidimensional recoil space and acquire Berry holonomies. Within this framework, single-step decays are governed by an abelian geometric phase and reproduce the standard Isgur-Wise behaviour, while sequential decays probe genuinely non-Abelian holonomies associated with a two-dimensional recoil space. The resulting geometric structure correlates different decay channels and provides a unified interpretation of mixing effects and quasi-degenerate states in heavy-quark phenomenology. This approach suggests that several long-standing puzzles arise as geometric consequences of infrared dressing rather than as accidental features of the microscopic dynamics.