<font>SU(3)</font><sub><font>L</font></sub> ⋊ (ℤ<sub>3</sub> × ℤ<sub>3</sub>) GAUGE SYMMETRY AND TRI-BIMAXIMAL MIXING

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<jats:p> We study an effective gauge theory whose gauge group is a semidirect product G = G<jats:sub> c </jats:sub> ⋊ Γ with G<jats:sub> c </jats:sub> and Γ being a connected Lie group and a finite group, respectively. The semidirect product is defined through a projective homomorphism (i.e. homomorphism up to the center of G<jats:sub> c </jats:sub>) from Γ into G<jats:sub> c </jats:sub>. To be specific, we take SU (3)<jats:sub> L </jats:sub> as G<jats:sub> c </jats:sub> and ℤ<jats:sub>3</jats:sub> × ℤ<jats:sub>3</jats:sub> as Γ. We notice that the irreducible representations of the gauge group G necessarily contain three G<jats:sub> c </jats:sub>-multiplets in spite of the Abelian nature of Γ = ℤ<jats:sub>3</jats:sub> × ℤ<jats:sub>3</jats:sub>. This triplication phenomenon is due to the semidirect product structure of G. We suggest that the appearance of three families is attributable to this triplication. We give a toy model on the lepton mixing and show that under a particular vacuum alignment the tri-bimaximal mixing matrix is reproduced. </jats:p>

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