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

Abstract

We study an effective gauge theory whose gauge group is a semidirect product $G = G_c \rtimes \mathitΓ$ with $G_c$ and $\mathitΓ$ being a connected Lie group and a finite group, respectively. The semidirect product is defined through a projective homomorphism $γ$ (i.e., homomorphism up to the center of $G_c$) from $\mathitΓ$ into $G_c$. The (linear) representation of $G$ is made from $γ$ and a projective representation of $\mathitΓ$ over $\mathbb{C}$. To be specific, we take $SU(3)_L$ as $G_c$ and $\mathbb{Z}_3 \times \mathbb{Z}_3$ as $\mathitΓ$. It is noticed that the irreducible projective representations of $\mathitΓ$ 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.