Deviation from Tri-Bimaximal Neutrino mixing from the Charged Lepton Sector

The tri-bimaximal mixing matrix is most likely the leading order term in the description of leptonic mixing. We derive deviations from tri-bimaximal mixing(TBM) due to corrections from the charged lepton sector. We assume a decoupled 2-3 symmetry, which is a $Z_2$ symmetry, in the charged lepton sector and another $Z_2$ symmetry in the neutrino sector. The TBM is obtained in the flavor symmetric limit. We consider deviations from the TBM form arising from the breaking of the 2-3 symmetry in the charged lepton sector but do not consider deviations from the breaking of the flavor symmetry in the neutrino sector. In particular we find the size of $\sin{\theta_{13}}$ to be related to the deviation $\sin{\theta_{12}}$ from $\sqrt{1/3}$ while the deviation of $\sin{\theta_{23}}$ from its tri-bimaximal value is $ \sim {m_{\mu} \over m_{\tau}}$.