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

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 effect from the tri-bi-maximal mixing. So, the difference of two models is designed only in terms of the flavor symmetry, by changing the charge assignment of righthanded neutrinos. The PMNS matrix in the first model is obtained from both mass matrices, charged leptons giving rise to non-zero $\theta^l_{13}$ and neutrino masses giving rise to tri-bi-maximal mixing. The physical mixing angles are expressed by a simple relation between $\theta^l_{13}$ and tri-bi-maximal angles to fit the recent experimental results. The other model generates PMNS matrix with non-zero $\theta_{13}$, only from the neutrino mass transformation. The 5 dimensional effective theory of Majorana neutrinos obtained in this framework is tested with phenomenological bounds in the parametric spaces $\sin\theta_{23}, \sin\theta_{12}$ and $m_2/m_3$ vs. $\sin\theta_{13}$.