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This work proposes two models of neutrino masses that predict non-zero$\theta_{13}$ under the non-Abelian discrete flavor symmetry$\mathbb{S}_3\otimes\mathbb{Z}_2$. We advocate that the size of $\theta_{13}$is understood as a group theoretical consequence rather than a perturbed effectfrom the tri-bi-maximal mixing. So, the difference of two models is designedonly in terms of the flavor symmetry, by changing the charge assignment ofrighthanded neutrinos. The PMNS matrix in the first model is obtained from bothmass matrices, charged leptons giving rise to non-zero $\theta^l_{13}$ andneutrino masses giving rise to tri-bi-maximal mixing. The physical mixingangles are expressed by a simple relation between $\theta^l_{13}$ andtri-bi-maximal angles to fit the recent experimental results. The other modelgenerates PMNS matrix with non-zero $\theta_{13}$, only from the neutrino masstransformation. The 5 dimensional effective theory of Majorana neutrinosobtained in this framework is tested with phenomenological bounds in theparametric spaces $\sin\theta_{23}, \sin\theta_{12}$ and $m_2/m_3$ vs.$\sin\theta_{13}$.

Non-vanishing $U_{e3}$ under $S_3$ symmetry