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

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

2009-12-25

arxiv.org/abs/0912.4957v1

Mod.Phys.Lett.A25:2981-2989,2010

Physics

High Energy Physics

High Energy Physics - Phenomenology

10 pages

Scientific paper

10.1142/S0217732310034092

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.

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