Connection Between $\nu_e, \nu_\mu, \nu_\tau$ and $\nu_1, \nu_2, \nu_3$ Neutrino States and Time Dependence of Neutrino Wave Functions and Transition Probabilities at Three Neutrino Oscillations in Vacuum

Abstract

For description of the $d, s, b$ quark mixings the Cabibbo-Kobayashi-Maskawa matrices are used but they do not contain the time dependence. In this work the analogous matrix is obtained for the case of three neutrino ($\nu_{e}, \nu_{\mu }, \nu_{\tau}$) mixings (oscillations) in vacuum in the general case, when CP violation is absent. In contrast to the quark case this matrix contains the time dependence. The matrix for probability of neutrino transitions (oscillations) in vacuum is also obtained. Naturally, it contains the time dependence. The matrix which does not contain the time dependence is obtained by using time $t$ averaging of this matrix. Elements of this matrix can be used to describe neutrino decays.