On the Relations between Fermion Masses and Isospin Couplings in the Microscopic Model

Quark and lepton masses and mixings are considered in the framework of the microscopic model. The most general ansatz for the interactions among tetrons leads to a Hamiltonian $H_T$ 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. Compact expressions for the eigenfunctions of $H_T$ are given. The almost exact isospin conservation of the system dictates the form of the lepton states and makes them independent of all the couplings in $H_T$. As a consequence, a parameter-free analytic expression for the PMNS matrix is derived which fits numerically all the measured matrix components. The formula includes a prediction of the leptonic Jarlskog invariant $J_{PMNS}=-0.0106$. An outlook is given on the treatment of the CKM matrix.