非交换几何中非紧李群的非平凡味结构
Nontrivial Flavor Structure from Noncompact Lie Group in Noncommutative Geometry
摘要：
本文提出了一种在非交换几何中非紧李群SL（3，C）的变换中诱导非平凡味结构的机制。矩阵$L \in$ SL（3，C）与伴随有前子字段的$a_{L,R} (x) \to L_{L,R} a_{L,R} (x)$相关联。为了保持拉格朗日量的厄米性，我们假设当$ψ^{\dagger} ψ$被$\bar ψψ$取代时，采用相同的技巧来构造洛伦兹不变拉格朗日量。结果表明，狄拉克拉格朗日量既有味普适规范相互作用，又有非平凡的汤川相互作用。去除非物理的酉变换，发现Yukawa矩阵是$Y_{ij} = L_{L}^{\dagger} k L_{R} \to Λ_{L}^{} U^{\dagger}_{L} k U_{R} Λ_{R}$。这里，$k$是系数，$U$是3 $\times$ 3酉矩阵，$Λ$是具有$λ_{1}λ_{2}λ_{3} = 1$的特征值矩阵$Λ= {\rm diag}(λ_{1}, λ_{2}, λ_{3})$。如果$L_{L,R}$起源于对称性破缺，那么味道的层次和混合可以分别解释为这个空间中的“洛伦兹提升”和“旋转”。
英文摘要：
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 $ψ^{\dagger} ψ$ is replaced by $\bar ψψ$ 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 $Y_{ij} = L_{L}^{\dagger} k L_{R} \to Λ_{L}^{} U^{\dagger}_{L} k U_{R} Λ_{R}$. Here, $k$ is a coefficient, $U$ is 3 $\times$ 3 unitary matrix and $Λ$ is the eigenvalue matrix $Λ= {\rm diag}(λ_{1}, λ_{2}, λ_{3})$ with $λ_{1}λ_{2}λ_{3} = 1$. 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.
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