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Abstract
Along-standing challenge in hadron spectroscopy is the lack of a unified principle connecting the algebraic relations describing static quantum number splittings (flavor, spin), such as the Gell-Mann–Okubo(GMO) formula, with the Regge trajectories that characterize dynamical excitations (angular momentum, radial). Moreover, the simple fractional coefficients (e.g., 1/3, 1/2) appearing in numerous empirical formulas lack a deep explanation rooted in quantum chromodynamics (QCD). This work systematically establishes and validates the theory of Hierarchical Fractional Quantization (HFQ) and its dynamical extension, starting from the symmetry structure of QCD. The core thesis is that the observable properties of hadrons arise from a sequence of symmetry-breaking hierarchies with well-separated energy scales. The contribution of each hierarchy factorizes into a characteristic energy scale determined by QCD dynamics and a rational coefficient determined a priori and uniquely by the representation theory of Lie groups (Clebsch–Gordan coefficients). First, via symmetry-breaking chains and the Wigner–Eckart theorem, we deductively derive from first principles of QCD the static HFQ mass formula Mi = M0+ Nk=1Ek·R(k)i ,thereby unifying the Gell-Mann–Okubo relation, equal-spacing rule, Coleman–Glashow formula, and others as natural special cases of different breaking hierarchies. Subsequently, by introducing the physical hypotheses that flavor breaking acts as an additive shift and that color-magnetic interaction strength decays with excitation, we fuse static HFQ with linear Regge trajectories, deducing for the first time the Hierarchical Regge Trajectory (HRT) formula: M2 = κ(2n + L) + kE(M2)kfk(n,L)R(k)i , achieving a dynamical unification of static quantum number splittings and angular momentum/radial excitations.
Supplementary materials
Title
SupplementaryMaterial: HierarchicalFractionalQuantizationFramework
Description
This supplementary document provides the complete mathematical foundation, extended physical applications, computational implementation, and systematic verification of the Hierarchical Fractional Quantization (HFQ) framework. The material is organized as follows:
• Appendix A: HFQ in Heavy-Ion Collisions and Finite-Temperature QCD– Application to quark-gluon plasma and collective flow phenomena.
• Appendix B: Classification and Predictions for Exotic Hadrons– Systematic treatment of tetraquarks, pentaquarks, and hybrid states within HFQ.
• Appendix C: Beyond Standard Model Extensions– Application to GUTs, supersymmetry,extra dimensions, and composite Higgs models.
• Appendix D: Quantum Information and Entanglement Perspective– Analysis of baryon wavefunction entanglement and quantum chaos.
• Appendix E: Modular Symmetry Analysis of Hadronic Regge Trajectories– Systematic study of Regge trajectory generating functions and their approximate modular covariance.
• Appendix F: Mathematical Structure, Algorithmic Implementation, and Uncertainty Quantification– Complete computational framework with error analysis.
All appendices form a self-contained, consistent extension of the main manuscript.
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