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

Abstract

Current experimental data indicate that two unitarity triangles of the CKM quark flavor mixing matrix V are almost the right triangles with α\approx 90^\circ. We find that α= 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 ϕis nearly equal to α(i.e., ϕ- α\approx 1.1^\circ). Both ϕand αare 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 α= 90^\circ at M_Z from ϕ= 90^\circ at M_X. We conjecture that there might also exist a maximal CP-violating phase ϕ\approx 90^\circ in the MNS lepton flavor mixing matrix U. The approximate quark-lepton complementarity relations, which hold in the standard parametrizations of V and U, can also hold in our particular parametrizations of V and U simply due to the smallness of |V_{ub}| and |V_{e3}|.