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

TL;DR

This paper investigates the unitarity condition in three-neutrino oscillations, showing that the transition probability remains positive only under specific mass-squared difference conditions and mixing angle constraints.

Contribution

It demonstrates the precise conditions under which the neutrino transition probability remains positive, highlighting the importance of mass-squared differences and mixing angles.

Findings

Probability is positive only when $ riangle m^2_{13} = riangle m^2_{12} + riangle m^2_{23}$

Deviations from this condition lead to negative probabilities

Mixing angle $eta$ must be limited to $eta extless= 15^ ext{o} extless= 17^ ext{o}$ for positivity

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 $\beta \le 15^o \div 17^o$).