Examination of unitarity condition (positive definiteness of expression for transition probabilities) at three neutrino oscillations in vacuum

Abstract

This work has shown that at strict fulfilment of condition $\Delta m^2_{1 3} = \Delta m^2_{1 2} + \Delta m^2_{2 3}$ the expression for probability of $\nu_e \to \nu_e$ transitions $P_{\nu_e \to \nu_e}(t)$ is positively defined at every values of $\theta$ and $\beta$ while at any arbitrarily small deviation from this condition it becomes negative. In order to make this expression for probability transitions positively defined, it is necessary to put a limitation on angle mixing $\beta$ at fixed value of $\theta= 32.45^o$ (i.e. the value for $\beta$ must be $\beta \le 15^o \div 17^o$).