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.
Categories: hep-ph