Universal correlation between CP phase $\delta$ and the non-unitarity $\alpha$ parameter phases

The non-unitarity of the neutrino mixing matrix is a problem related with a more fundamental question about the origin of the neutrino mass. After a brief discussion on the questions ``why do we need model-independent framework for unitarity test?'', we show interesting properties present in the oscillation probabilities of neutrinos propagating in matter with non-unitarity. That is, (1) partial unitarity and (2) universal phase correlation. % To illuminate these points, we formulate a perturbative framework with the two expansion parameters $\epsilon \equiv \Delta m^2_{21} / \Delta m^2_{31}$ and $\alpha$ matrix elements. The complex triangular $\alpha$ matrix is introduced through the definition of $3 \times 3$ non-unitary flavor mixing matrix $N$ as % \begin{eqnarray} N &=& \left( \bf{1} - \alpha \right) U = \left\{ \bf{1} - \left[ \begin{array}{ccc} \alpha_{ee} & 0 & 0 \\ \alpha_{\mu e} & \alpha_{\mu \mu} & 0 \\ \alpha_{\tau e} & \alpha_{\tau \mu} & \alpha_{\tau \tau} \\ \end{array} \right] \right\} U \label{alpha-matrix-def} \end{eqnarray} % where $U$ is the unitary MNS mixing matrix, and hence $\alpha$ characterizes the size and the flavor dependence of unitarity violation (UV)\footnote{ %%%%%%%%%%%%% Despite that in the physics literature UV usually means ``ultraviolet'', we use UV in this manuscript as an abbreviation for ``unitarity violation'' or ``unitarity violating''.}~~caused by new physics (NP) at low- or high-scales. ~~~~The point (1) above essentially means that despite non-unitary mixing, neutrino evolution must be unitary because the three active neutrinos span a complete state space of neutral leptons. The {\em phase correlation} mentioned in the point (2) refers an intriguing property that the complex $\alpha$ parameters and $\nu$Standard Model CP phase $\delta$ always come into the oscillation probabilities in a correlated way, $e^{- i \delta } \alpha_{\mu e}$, $\alpha_{\tau e}$, and $e^{i \delta} \alpha_{\tau \mu}$, universally in all the oscillation channels. The physical meaning of this result is briefly discussed. }