Right Unitarity Triangles, Stable CP-violating Phases and Approximate Quark-Lepton Complementarity

Understand

Current experimental data indicate that two unitarity triangles of the CKM quark flavor mixing matrix V are almost the right triangles with \alpha \approx 90^\circ.

• We find that \alpha = 90^\circ actually points to {\rm Re}(V_{tb}V_{ud}V^*_{td}V^*_{ub}) = 0.
• Considering a very suggestive parametrization of V, we show that its CP-violating phase \phi is nearly equal to \alpha (i.e., \phi - \alpha \approx 1.1^\circ).
• Both \phi and \alpha are stable against the renormalizaton-group evolution from the electroweak scale M_Z to a superhigh energy scale M_X or vice versa, and thus it is impossible to obtain \alpha = 90^\circ at M_Z from \phi = 90^\circ at M_X.