Article

# A complex-angle rotation and geometric complementarity in fermion mixing

08/2008; DOI:10.3938/jkps.53.1228
Source: arXiv

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.

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### Keywords

angles

common axis

complementarity constraint

complex angle

CP phase

CP phase $\delta$

geometric

geometric complementarity condition

mass eigenvectors

non-trivial dependence

precise measurements

quark basis

quark-lepton unification

quarks

real angles