A Model of Neutrino Masses and Mixing from Hierarchy and Symmetry

Abstract

We construct a model that allows us to determine the three neutrino masses directly from the experimental mass squared differences, $\Delta_{\rm atm}$ and $\Delta_{\rm sol}$, together with the assumption that $\Lambda = \surd (1/6) = \surd (m_2/m_3)$. The parameter, $\Lambda$, basically a Clebsch-Gordan coefficient with the value of about 0.4 stems from the group $S_3$, and is NOT an expansion parameter, in contrast with the Wolfenstein parameter, $0.22 <\lambda < 0.25$ needed to explain quark masses. For a variety of initial values of $\Delta_{\rm atm}$, we find that the lowest mass, $m_1$, varies from $1.4 - 3.6 10^{-3} eV$, the next lowest mass, $m_2$ varies only slightly from $8.4 - 9.0 10^{-3} eV$, and the heaviest mass, $m_3$, ranges from $5.0 - 5.4 10^{-2} eV$. The elements of the mixing matrix, $U$, and of the mass matrix, $M$, are examined with particular emphasis on the role of small angle $\theta_{13}$. The phase, $\delta$, of the mixing matrix $U$ has a serious effect in the mass matrix only for the matrix elements $M_{e\mu}$ and $M_{e\tau}$, because these are the only ones for which the real part vanishes in the allowed range for $\theta_{13}$. Their dependence on $s_{13}$ for various values of $\delta$ is given explicitly. We study the elements of the mass matrix, $M$, for our solution 1, that of the perfect rational hierarchy, for the case $\delta = 0$, and find that all of them are smaller than $0.03 eV$. The only candidates for double texture zero models are $M_{ee}$ and $M_{e\mu} = M_{\mu e}$.