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Two Categories of mu-tau Symmetry

Up Fate of mu-tau Symmetry Two Categories of mu-tau Symmetry CP Violation and mu-tau Synmmetry

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Two Categories of Approximately μ-τ Symmetric Neutrino Mass Textures

  • TOKAI-HEP/TH-0602  with Kenichi Fuki, published in Nucl. Phys. B 783 No.1-2 (2007) 31-56 (Received date: 8 August, 2006).
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Our approximately $\mu$-$\tau$ symmetric neutrino mass textures fall into two different categories, whose behaviors in the $\mu$-$\tau$ symmetric limit are characterized by either $\sin\theta_{13}\rightarrow 0$ (referred to as C1)), or $\sin\theta_{12}\rightarrow 0$ (referred to as C2)), where $\theta_{12}$ and $\theta_{13}$, respectively stand for the $\nu_e$-$\nu_\mu$ and $\nu_e$-$\nu_\tau$ mixing angles.  We present ten phenomenologically viable neutrino mass textures: two for the normal mass hierarchy, three for the inverted mass hierarchy, and five for the quasi degenerate mass pattern.  Tiny $\mu$-$\tau$ symmetry breaking ensures that $\sin^2\theta_{13}\ll 1$ for C1), and $\Delta m^2_\odot/\Delta m^2_{atm} (\equiv R) \ll 1$ for C2), where $\Delta m^2_\odot = m^2_2-m^2_1$ for solar neutrinos, and $\Delta m^2_{atm}=\vert m^2_3-(m^2_1+m^2_2)/2\vert$ for atmospheric neutrinos with $m_{1,2,3}$ being neutrino masses. A correlation among small quantities is provided by $\cos 2\theta_{23}\sim\sin\theta_{13}$ for C1), and by either $\cos 2\theta_{23}\sim R$, or $\cos 2\theta_{23}\sin\theta_{13}\sim R$ for C2), where $\theta_{23}$ is the $\nu_\mu$-$\nu_\tau$ mixing angle.  It is further shown that $\tan 2\theta_{12}\sim\cos 2\theta_{23}/\sin\theta_{13}$ is satisfied for C2).  We find the following properties for each mass ordering: 1) For the normal mass hierarchy, the other smallness of $R$ (or $\sin^2\theta_{13}$) in the case of C1) (or C2)) can be ascribed to an approximate electron number conservation; 2) For the normal and inverted mass hierarchies and for the quasi degenerate mass pattern II (with $m^2_{1,2,3}\gg \Delta m^2_{atm}$) exhibiting $m_1\sim m_2 \sim m_3$, all in the case of C2), either $R\sim\tan 2\theta_{12}\sin^2\theta_{13}$ or $R\sim\tan 2\theta_{12}\sin\theta_{13}$ is satisfied; 3) For the inverted mass hierarchy and the quasi degenerate mass pattern I (with $\vert m_{1,2,3}\vert\sim \sqrt{\Delta m^2_{atm}}$), both in the case of C1) with $m_1\sim -m_2$, the relation $R\sim \sin^2\theta_{13}$ arises from the contribution of ${\mathcal{O}}(\sin^2\theta_{13})$ in the estimation of neutrino masses; and 4) For the quasi degenerate mass pattern II, we observe that $m_{\beta\beta}$ can be as large as 0.5 eV, where $m_{\beta\beta}$ is the effective neutrino mass in $(\beta\beta)_{0\nu}$-decay, and that larger values of $m_{\beta\beta}$ favor larger values of $\sin \theta_{13}$ in the case of C1) for $m_1\sim m_2 \sim m_3$ and smaller values of $\sin \theta_{13}$ in the cases of C1) for $m_1\sim m_2 \sim -m_3$ and C2) for $m_1\sim m_2 \sim \pm m_3$.

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