Recently, formulas for the mixing matrices of quarks and leptons have been put forward in [1]. My contribution here describes the relevant foundational and technical aspects which have led to those results.
The work has been carried out in the framework of the microscopic model. The most general ansatz for the interactions among tetrons leads to a Hamiltonian H involving Dzyaloshinskii-Moriya (DM), Heisenberg and torsional isospin forces. Diagonalization of the Hamiltonian provides for 24 eigenvalues which are identified as the quark and lepton masses. While the masses of the third and second family arise from DM and Heisenberg type of isospin interactions, light family masses are related to torsional interactions among tetrons. Neutrino masses turn out to be special in that they are given in terms of tiny isospin non-conserving DM, Heisenberg and torsional couplings.
The approach not only leads to masses, but also allows to calculate the quark and lepton eigenstates, an issue, which is important for the determination of the CKM and PMNS mixing matrices. The almost exact isospin conservation of the system dictates the form of the lepton states and makes them independent of all the couplings in H. Much in contrast, there is a strong dependence of the quark states on the coupling strengths, and a promising hierarchy between the quark family mixings shows up.