Nontrivial Flavor Structure from Noncompact Lie Group in Noncommutative Geometry

Abstract

In this paper, we propose a mechanism which induces nontrivial flavor structure from transformations of a noncompact Lie group SL(3,C) in noncommutative geometry. Matrices $L\in$ SL(3,C) are associated with accompanied by the preon fields as $a_{L,R}(x)\to L_{L,R}{a}_{L,R}(x)$. In order to retain the Hermiticity of the Lagrangian, we assume the same trick when ${\psi }^{†}\psi$ is replaced by $\bar{\psi }\psi$ to construct a Lorentz invariant Lagrangian. As a result, the Dirac Lagrangian has both of flavor-universal gauge interactions and nontrivial Yukawa interactions. Removing the unphysical unitary transformations, Yukawa matrices found to be $Y_{ij}=L_L^{†}k{L}_R\to {\Lambda }_L^{}{U}_L^{†}k{U}_R{\Lambda }_R$. Here, $k$ is a coefficient, $U$ is 3 $\times$ 3 unitary matrix and $\Lambda$ is the eigenvalue matrix $\Lambda =\mathrm{diag}({\lambda }_1,{\lambda }_2,{\lambda }_3)$ with ${\lambda }_1{\lambda }_2{\lambda }_3=1$. If $L_{L,R}$ are originated from a broken symmetry, the hierarchy and mixing of flavor can be interpreted as the "Lorentz boost" and the "rotation" in this space respectively.

Comments: This paper has been withdrawn by the author because the paper has several faults