$SU(3)_{\rm L} \rtimes (\mathbb{Z}_3 \times \mathbb{Z}_3)$ gauge symmetry and Tri-bimaximal mixing

TL;DR

This paper explores a novel gauge symmetry combining $SU(3)_L$ with a $bZ_3 imes bZ_3$ finite group, demonstrating how it can naturally produce the tri-bimaximal lepton mixing pattern through a specific vacuum alignment.

Contribution

It introduces a new gauge symmetry framework using a semidirect product of a Lie group and a finite group, showing its potential to explain lepton mixing patterns.

Findings

The gauge symmetry $SU(3)_L times (bZ_3 imes bZ_3)$ admits three-dimensional projective representations.

A toy model demonstrates the reproduction of the tri-bimaximal mixing matrix.

The model suggests a novel approach to flavor symmetry in particle physics.

Abstract

We study an effective gauge theory whose gauge group is a semidirect product $G = G_c \rtimes \mathit{\Gamma}$ with $G_c$ and $\mathit{\Gamma}$ being a connected Lie group and a finite group, respectively. The semidirect product is defined through a projective homomorphism $\gamma$ (i.e., homomorphism up to the center of $G_c$) from $\mathit{\Gamma}$ into $G_c$. The (linear) representation of $G$ is made from $\gamma$ and a projective representation of $\mathit{\Gamma}$ over $\mathbb{C}$. To be specific, we take $SU(3)_L$ as $G_c$ and $\mathbb{Z}_3 \times \mathbb{Z}_3$ as $\mathit{\Gamma}$. It is noticed that the irreducible projective representations of $\mathit{\Gamma}$ are three-dimensional in spite of its Abelian nature. We give a toy model on the lepton mixing which illustrates the peculiar feature of such gauge symmetry. It is shown that under a particular vacuum alignment the tri-bimaximal mixing matrix is reproduced.