On the origin of families of fermions and their mass matrices

Abstract

We are proposing a new way of describing families of quarks and leptons, using the approach unifying all the internal degrees of freedom, proposed by one of us. Spinors, living in d(=1+13)-dimensional space, carry in this approach only the spin and interact with only the gravity through vielbeins and two kinds of the spin connection fields - the gauge fields of the Poincare group and the second kind of the Clifford algebra objects. All the quarks and the leptons of one family appear in one Weyl representation of a chosen handedness of the Lorentz group, if analyzed with respect to the Standard model gauge groups: the right handed (with respect to SO(1,3)) weak chargeless quarks and leptons and the left handed weak charged quark and leptons. A part of the starting Lagrange density of a Weyl spinor in d=1+13 transforms right handed quarks and leptons into left handed quarks and leptons manifesting as the Yukawa couplings of the Standard model. The second kind of Clifford algebra objects generates families and contributes to diagonal and off diagonal Yukawa couplings. The approach predicts an even number of families, treating leptons and quarks equivalently. In this paper we investigate within this approach the appearance of the Yukawa couplings within one family of quarks and leptons as well as among the families (without assuming any Higgs fields). We present the mass matrices for four families and investigate whether our way of generating families might explain the origin of families of quarks and leptons as well as their observed properties - the masses and the mixing matrices. Numerical results are presented in the paper following this one.