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Tri-Bimaximal Neutrino Mixing and the $T_{13}$ Flavor Symmetry

We present a supersymmetric $T_{13}$ model for the tri-bimaximal neutrino mixing, and the complete flavor group is $T_{13}\times Z_{4}\times Z_2$. At leading order, the residual symmetry of the charged lepton sector is $Z_3$, and the $T_{13}$ symmetry is broken completely in the neutrino sector. The charged lepton mass hierarchies are determined by the spontaneous breaking of the flavor symmetry, both the type I see-saw mechanism and the Weinberg operator contribute to generating the light neutrino masses. Tri-bimaximal mixing is exact at leading order while subleading contributions introduce corrections of order $λ^2_c$ to the three lepton mixing angles. The vacuum alignment and subleading corrections are studied in detail, a moderate hierarchy of order $λ_c$ between the vacuum expectation values of the flavon fields in the charged lepton and neutrino sectors can be accommodated.