Fine structure of fermion mixing and quark-mass hierarchies
Abstract
We investigate a semi-empirical flavour framework based on two linked hypotheses: a golden/pentagonal organisation of fermion mixing and the recurrence of its hierarchy maps in the quark-mass spectrum. The two large lepton angles are specified by the fifth-order Chebyshev conditions T5(θ12ℓ)=0 and T5(θ23ℓ)=−1, with angular arguments in radian measure, while the smaller reactor angle follows the approximate bilinear relation θ13ℓ≃θ12ℓθ23ℓ/π. An alternative golden-partition construction yields nearly identical large-angle predictions and an exact circular--hyperbolic duality. The golden ratio φ also enters two distinct geometrical prescriptions for the Cabibbo angle, while the heptadecagonal seed αC=π/17 governs the suppressed mixing hierarchy. The near-equality tanαC≃cos54∘/π further connects the two geometries. The resulting quark hierarchy obeys the quadratic map θ13q=√(θC2+(θ23q)2)−θC≃(θ23q)2/(2θC). Transferring the quadratic and bilinear maps to quark masses yields the crossed relations md≃ mc2/(2mt) and mu≃ mb ms/(mF/φ), where mF=GF−1/2. The complementary triples (t,c,d) and (b,s,u) reproduce both mass targets within 1% for the adopted reference values. Gatto--Sartori--Tonin-type correspondences and Froggatt--Nielsen scaling provide compatible phenomenological interpretations. These results suggest a common golden/geometrical organising structure for flavour mixing and mass hierarchies, whose underlying dynamical origin remains to be established.
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BibTeX
@misc{ciborowski2026fine,
title = {{Fine structure of fermion mixing and quark-mass hierarchies}},
author = {Jacek Ciborowski},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03704},
eprint = {2611.03704},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03704},
}
RIS
TY - UNPB
TI - Fine structure of fermion mixing and quark-mass hierarchies
AU - Jacek Ciborowski
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.03704
UR - https://arxiv.org/abs/2611.03704
AB - We investigate a semi-empirical flavour framework based on two linked hypotheses: a golden/pentagonal organisation of fermion mixing and the recurrence of its hierarchy maps in the quark-mass spectrum. The two large lepton angles are specified by the fifth-order Chebyshev conditions T_5(θ_12^ℓ)=0 and T_5(θ_23^ℓ)=−1, with angular arguments in radian measure, while the smaller reactor angle follows the approximate bilinear relation θ_13^ℓ≃θ_12^ℓθ_23^ℓ/π. An alternative golden-partition construction yields nearly identical large-angle predictions and an exact circular--hyperbolic duality. The golden ratio φ also enters two distinct geometrical prescriptions for the Cabibbo angle, while the heptadecagonal seed α_C=π/17 governs the suppressed mixing hierarchy. The near-equality tanα_C≃cos54^∘/π further connects the two geometries. The resulting quark hierarchy obeys the quadratic map θ_13^q=√(θ_C^2+(θ_23^q)^2)−θ_C≃(θ_23^q)^2/(2θ_C). Transferring the quadratic and bilinear maps to quark masses yields the crossed relations m_d≃ m_c^2/(2m_t) and m_u≃ m_b m_s/(m_F/φ), where m_F=G_F^−1/2. The complementary triples (t,c,d) and (b,s,u) reproduce both mass targets within 1% for the adopted reference values. Gatto--Sartori--Tonin-type correspondences and Froggatt--Nielsen scaling provide compatible phenomenological interpretations. These results suggest a common golden/geometrical organising structure for flavour mixing and mass hierarchies, whose underlying dynamical origin remains to be established.
ER -