A complex-angle rotation and geometric complementarity in fermion mixing

TL;DR

This paper introduces a geometric complementarity condition between quark and lepton mixing angles, linking their complex angles in a way that unifies flavor mixing and CP phase dependence, with implications for future measurements.

Contribution

It proposes a novel geometric complementarity framework in complex space that relates quark and lepton mixing angles and CP phases, avoiding previous exceptions.

Findings

The CP phase depends non-trivially on mixing angles.

Precise measurements can constrain the CP phase further.

The scheme maintains lepton basis independence from quark basis.

Abstract

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.