Determination of the mixing between active neutrinos and sterile neutrino through the quark-lepton complementarity and self-complementarity

It is suggested that there is an underlying symmetry which relates the quark and lepton sectors. Namely, among the mixing matrix elements of Cabibbo-Kobayashi-Maskawa for quarks and Pontecorvo-Maki-Nakawaga-Sakata for leptons there exist complementarity relations at a high energy scale (such as the seesaw or even the grand unification theory scales). We assume that the relations would remain during the matrix elements running down to the electroweak scale. Observable breaking of the rational relation is attributed to the existence of sterile neutrinos that mix with the active neutrino to result in the observable Pontecorvo-Maki-Nakawaga-Sakata matrix. We show that involvement of a sterile in the ($3+1$) model induces that $|U_{e4}{|}^2=0.040$, $|U_{\mu 4}{|}^2=0.009$, and ${\sin }^{2}2\alpha =0.067$. We also find a new self-complementarity ${\vartheta }_{12}+{\vartheta }_{23}+{\vartheta }_{13}+\alpha \approx 90°$. The numbers are generally consistent with those obtained by fitting recent measurements, especially in this scenario, to the existence of a sterile neutrino that does not upset the LEP data; i.e., the number of neutrino types is very close to 3.