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

Abstract

We study an effective gauge theory whose gauge group is a semidirect product $G=G_c⋊\Gamma$ with $G_c$ and $\Gamma$ 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 $\Gamma$ into $G_c$. The (linear) representation of $G$ is made from $\gamma$ and a projective representation of $\Gamma$ over $\mathbb{C}$. To be specific, we take $SU(3{)}_L$ as $G_c$ and ${\mathbb{Z}}_3\times {\mathbb{Z}}_3$ as $\Gamma$. It is noticed that the irreducible projective representations of $\Gamma$ 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.

Comments: 10 pages