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 $T_5({\theta }_{12}^{\ell })=0$ and $T_5({\theta }_{23}^{\ell })=-1$, with angular arguments in radian measure, while the smaller reactor angle follows the approximate bilinear relation ${\theta }_{13}^{\ell }\simeq {\theta }_{12}^{\ell }{\theta }_{23}^{\ell }/\pi$. An alternative golden-partition construction yields nearly identical large-angle predictions and an exact circular--hyperbolic duality. The golden ratio $\phi$ also enters two distinct geometrical prescriptions for the Cabibbo angle, while the heptadecagonal seed ${\alpha }_C=\pi /17$ governs the suppressed mixing hierarchy. The near-equality $\tan {\alpha }_C\simeq \cos 54^{\circ }/\pi$ further connects the two geometries. The resulting quark hierarchy obeys the quadratic map ${\theta }_{13}^q=\sqrt{{\theta }_C^2+({\theta }_{23}^q{)}^2}-{\theta }_C\simeq ({\theta }_{23}^q{)}^2/(2{\theta }_C)$. Transferring the quadratic and bilinear maps to quark masses yields the crossed relations $m_d\simeq m_c^2/(2m_t)$ and $m_u\simeq m_b{m}_s/(m_F/\phi )$, 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.

arXiv comment25 pages