Toward diagnosing neutrino non-unitarity through CP phase correlations

We discuss correlations between the $\nu$SM CP phase $\delta$ and the phases that originate from new physics which causes neutrino-sector unitarity violation (UV) at low energies. This study is motivated to provide one of the building pieces for a machinery to diagnose non-unitarity, our ultimate goal. We extend the perturbation theory of neutrino oscillation in matter proposed by Denton {\it et al.}~(DMP) to include the UV effect expressed by the $\alpha$ parametrization. By analyzing the DMP-UV perturbation theory to first order, we are able to draw a completed picture of the $\delta$ - UV phase correlations in the whole kinematical region covered by the terrestrial neutrino experiments. There exist the two regions with the characteristically different patterns of the correlations: (1) the chiral-type $[e^{- i \delta } \alpha_{\mu e}, ~e^{ - i \delta} \alpha_{\tau e}, ~\alpha_{\tau \mu}]$ (PDG convention) correlation in the entire high-energy region $\vert \rho E \vert \gsim 6~(\text{g/cm}^3)$ GeV, and (2) (blobs of the $\alpha$ parameters) - $e^{ \pm i \delta}$ correlation in anywhere else. Some relevant aspects for measurement of the UV parameters, such as the necessity of determining all the $\alpha_{\beta \gamma}$ elements at once, are also pointed out.