A complex-angle rotation and geometric complementarity in fermion mixing

The mixing among flavors in quarks or leptons in terms of a single rotation angle is defined such that three flavor eigenvectors are transformed into three mass eigenvectors by a single rotation about a common axis. We propose that a geometric complementarity condition exists between the complex angle of quarks and that of leptons in $\mathbb{C}^2$ space. The complementarity constraint has its rise in quark-lepton unification and is reduced to the correlation among $\theta_{12}, \theta_{23}, \theta_{13}$ and the CP phase $\delta$. The CP phase turns out to have a non-trivial dependence on all the other angles. We will show that further precise measurements in real angles can narrow down the allowed region of $\delta$. In comparison with other complementarity schemes, this geometric one can avoid the problem of the $\theta_{13}$ exception and can naturally keep the lepton basis being independent of quark basis.