Broken $S_3$ Neutrinos

Motivated by recent measurements which strongly support a nonzero reactor mixing angle $\theta_{13}$, we study a deviation from $S_3$ neutrino discrete symmetry by explicitly breaking the neutrino mass matrix with a general retrocirculant matrix. We show that nonzero $\theta_{13}$ and nonzero CP violation parameter $J_{CP}$ arise due to the difference between $y_2$ and $y_3$. We demonstrate that it is possible to obtain the experimentally favored results for neutrino masses and mixing angles from this mass matrix. Furthermore, we estimate the effective masses $m_{\beta}$ and $m_{\beta \beta}$ and total neutrino mass $\sum |m_i|$ predicted by this mass matrix.