Neutrino masses and mixing from hierarchy and symmetry

We construct a model that allows us to determine the three neutrino masses and the mass matrix directly from the experimental mass squared differences Δatm and Δsol, anticipating rational hierarchy (μm1/m2=m2/m3),μ≈1, and S3-S2 symmetry for the mixing matrix. We find that both the mass ratios and mixing angles are dominated by a parameter Λ. For the mixing angles, Λ=1/6≈0.41, is a Clebsch-Gordan coefficient. For the masses, the mass ratios depend on the experimental Δatm and Δsol and with most recent data, remarkably, we also obtain m1/m2=m2/m3=0.41=Λ. This possibly coincidental equality gives a simple mass matrix in the sin(θ13)=0 limit. We find that with Δsol=8.2×10−5 eV2, m1=1.5×10−3 eV, m2=9.2×10−3 eV and m3=5.3(5.5)×10−2 eV for Δatm=2.73(2.95)×10−3 eV2. We obtain the mass matrix M and evaluate it's elements numerically for the presently 'best fit' solution in the allowed range for sin(θ13). We find that all matrix elements are smaller than 0.03 eV. The only candidates for double texture zeroes are Mee and Meτ or Meμ (with θ13→−θ13). The maximum effective mass for neutrinoless ββ decay is |mββ|max≈8×10−3 eV.