Neutrino oscillations in the scheme of mass mixings and problem of smallness of angle mixing $\theta_{1 3}$

Abstract

In the framework of the mass mixing scheme we have considered mixings and oscillations of $\nu_e, \nu_\mu, \nu_\tau$ neutrinos and obtained expressions for angle mixings and lengths of oscillations in dependence on components of the nondiagonal mass matrix. Then analysis of these obtained results was done by using modern experimental data on neutrino oscillations. It has been shown that in this approach the lengths of neutrino oscillations $L_{2 3}$ and $L_{1 3}$ are not compulsory to be equal. It means that the angle mixing $\theta_{1 3}$ can be not very small, i.e., $L_{1 3}$ can be larger than $L_{2 3}$. In the conventional approach $L_{1 3} \approx L_{2 3}$ ($L_{1 2} \gg L_{2 3}$) and angle mixing of $\theta_{1 3}$ is very small. Angle mixings $\theta_{2 3}, \theta_{1 2}$ are big. Then there ia a problem: why is mixing angle $\theta_{1 3}$ so small? A natural solution of the problem is to suppose that $(m_2^2 - m_1^2) \neq (m_3^2 - m_1^2) - (m_3^2 - m_2^2)$, then $L_{1 3} > L_{2 3}$. It will be realized if there are 4 neutrino oscillations instead of 3 neutrino oscillations. Then the value of $\theta_{1 3}$ is necessary to search at distances more than $L_{2 3}$.