Nontrivial Flavor Structure from Noncompact Lie Group in Noncommutative Geometry

TL;DR

This paper introduces a mechanism where noncompact Lie group transformations in noncommutative geometry generate complex flavor structures, leading to new insights into flavor hierarchy and mixing in particle physics.

Contribution

It proposes a novel approach linking noncompact Lie group SL(3,C) transformations to flavor structures within noncommutative geometry, providing a geometric interpretation of flavor hierarchy and mixing.

Findings

Yukawa matrices derived from noncompact Lie group transformations

Flavor hierarchy interpreted as Lorentz boosts

Flavor mixing interpreted as rotations in the transformation space

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 $\psi^{\dagger} \psi$ 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.