Neutrino masses from an approximate mixing matrix with $θ_{13}\neq 0$

An approximate neutrino mixing matrix is formutated by using the standard neutrino mixing matrix as a basis and experimental data of neutrino oscillations as inputs. By using the resulted approximate neutrino mixing matrix to proceed the neutrino mass matrix and constraining the resulted neutrino mass matrix with zero texture: $M_ν(1,1)=M_ν(1,3)=M_ν(3,1)=0$, we can have neutrino masses as function of mixing angle $θ_{13}$ with normal hierarchy: $m_{1}<m_{2}<m_{3}$. By taking the central value of mixing angle $θ_{13}=9^{o}$ that gives $ε=0.16$ and using the squared mass difference: $Δm_{32}^{2}$ for normal hierarchy, we then obtained neutrino masses: $m_{1}=0.00847$ eV, $m_{2}=0.01215$ eV, and $m_{3}=0.05062$ eV which can predict the squared mass difference for solar neutrino precisely with the experimental result: $Δm_{21}^{2}=7.59\times 10^{-5}~{\rm eV^{2}}$

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