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 [N. Manko\ifmmode \check{c}\else \v{c}\fi{} Bor\ifmmode \check{s}\else \v{s}\fi{}tnik, Phys. Lett. B 292, 25 (1992).][N. Manko\ifmmode \check{c}\else \v{c}\fi{}-Bor\ifmmode \check{s}\else \v{s}\fi{}tnik, J. Math. Phys. (N.Y.) 34, 3731 (1993).][N. Manko\ifmmode \check{c}\else \v{c}\fi{} Bor\ifmmode \check{s}\else \v{s}\fi{}tnik, J. Math. Phys. (N.Y.) 36, 1593 (1995).][N. S. Manko\ifmmode \check{c}\else \v{c}\fi{} Bor\ifmmode \check{s}\else \v{s}\fi{}tnik, Mod. Phys. Lett. A 10, 587 (1995).][N. S. Manko\ifmmode \check{c}\else \v{c}\fi{} Bor\ifmmode \check{s}\else \v{s}\fi{}tnik and S. Fajfer, Nuovo Cimento Soc. Ital. Fis. B 112, 1637 (1997).][A. Bor\ifmmode \check{s}\else \v{s}\fi{}tnik and N. S. Manko\ifmmode \check{c}\else \v{c}\fi{} Bor\ifmmode \check{s}\else \v{s}\fi{}tnik, in Proceedings to the International Workshop on ``What Comes Beyond the Standard Model, Bled, Slovenia, 1998, edited by N. Manko\ifmmode \check{c}\else \v{c}\fi{} Bor\ifmmode \check{s}\else \v{s}\fi{}tnik, H. B. Nielsen, and C. Froggatt (DMFA, Zalo\ifmmode \check{z}\else \v{z}\fi{}ni\ifmmode \check{s}\else \v{s}\fi{}tvo, 1999), p. 52.][N. S. Manko\ifmmode \check{c}\else \v{c}\fi{} Bor\ifmmode \check{s}\else \v{s}\fi{}tnik and H. B. Nielsen, Phys. Rev. 62, 04010 (2000).][N. S. Manko\ifmmode \check{c}\else \v{c}\fi{} Bor\ifmmode \check{s}\else \v{s}\fi{}tnik, Int. J. Theor. Phys. 40, 315 (2001), and references therein.][A. Bor\ifmmode \check{s}\else \v{s}\fi{}tnik and N. S. Manko\ifmmode \check{c}\else \v{c}\fi{} Bor\ifmmode \check{s}\else \v{s}\fi{}tnik, in Proceedings to the International Workshop on ``What Comes Beyond the Standard Model'', Bled 2000, 2001, 2002 Volume 2, edited by N. Manko\ifmmode \check{c}\else \v{c}\fi{} Bor\ifmmode \check{s}\else \v{s}\fi{}tnik, H. B. Nielsen, C. Froggatt, and D. Lukman (DMFA, Zalo\ifmmode \check{z}\else \v{z}\fi{}ni\ifmmode \check{s}\else \v{s}\fi{}tvo, 2002), p. 27 and the paper (unpublished).][A. Bor\ifmmode \check{s}\else \v{s}\fi{}tnik and N. S. Manko\ifmmode \check{c}\else \v{c}\fi{} Bor\ifmmode \check{s}\else \v{s}\fi{}tnik, in Proceedings to the Euroconference on Symmetries Beyond the Standard Model, Portoro\ifmmode \check{z}\else \v{z}\fi{}, 2003 edited by N. Manko\ifmmode \check{c}\else \v{c}\fi{} Bor\ifmmode \check{s}\else \v{s}\fi{}tnik, H. B. Nielsen, C. Froggatt, and D. Lukman (DMFA, Zalo\ifmmode \check{z}\else \v{z}\fi{}ni\ifmmode \check{s}\else \v{s}\fi{}tvo, 2003), pp. 27--51.]. 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 Poincar\'e group $({p}^{a},{S}^{ab})$ and the second kind of the Clifford algebra objects (${\stackrel{\texttildelow{}}{S}}^{ab}$). 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, which are subgroups of the group $SO(1,13)$: the right handed (with respect to $SO(1,3)$) weak chargeless quarks and leptons and the left handed weak charged quarks and leptons (with the right handed neutrino included). 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. A kind of the Clifford algebra objects generates families of quarks and leptons and contributes to diagonal and off-diagonal Yukawa couplings. The approach predicts an even number of families, treating leptons and quarks equivalently (we do not study a possible appearance of Majorana fermions yet). 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 like in the standard model). 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 Ref. [M. Breskvar, D. Lukman, and N. S. Manko\ifmmode \check{c}\else \v{c}\fi{} Bor\ifmmode \check{s}\else \v{s}\fi{}tnik, hep-ph/0606159.].