Nontrivial Flavor Structure from Noncompact Lie Group in Noncommutative Geometry

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 $\psi^{\dagger} \psi$ 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 $Y_{ij} = L_{L}^{\dagger} k L_{R} \to \Lambda_{L}^{} U^{\dagger}_{L} k U_{R} \Lambda_{R}$. Here, $k$ is a coefficient, $U$ is 3 $\times$ 3 unitary matrix and $\Lambda$ is the eigenvalue matrix $\Lambda = {\rm diag}(\lambda_{1}, \lambda_{2}, \lambda_{3})$ 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.