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One-parameter Neutrino Mass Matrix and Symmetry Realization

We investigate the Majorana neutrino mass matrix $M_ν$ with one parameter in the context of two texture zeros and its symmetry realization by non-Abelian discrete symmetry. From numerical calculation, we confirm that the textures $(M_ν)_{11,12}=0$ and $(M_ν)_{11,13}=0$ are consistent with the current experimental constraints, and show the correlations between non-zero elements of $M_ν$. The ratios of non-zero elements of $M_ν$ are constrain in small regions, and we find simple examples of $M_ν$ with one real mass parameter. We also discuss symmetry realization of the mass matrix by the type-II seesaw mechanism based on the binary icosahedral symmetry $A_5'$ .