Quark and Lepton Mass Matrix Structures Suggested by the Observed Unitary Triangle Shape

Abstract

Under the hypothesis that the CP violating phase parameter δin the CKM matrix V takes own value so that the radius R(δ) of the circle circumscribed about the unitary triangle takes its minimum value, possible phase conventions of the CKM matrix are investigated. We find that two of the 9 phase conventions can give favorable predictions for the observed shape of the unitary triangle. One of the successful two suggests phenomenologically interesting structures of the quark and lepton mass matrices, which lead to |V_{us}|\simeq \sqrt{m_d/m_s}=0.22, |V_{ub}|\simeq \sqrt{m_u/m_t} =0.0036 and |V_{cb}|\simeq \sqrt{m_c/2m_t}=0.043 for the CKM matrix V, and to \sin^2 2θ_{atm}=1, \tan^2 θ_{solar} \simeq |m_{ν1}/m_{ν2}|$ and |U_{13}|\simeq \sqrt{m_e/2 m_τ} for the lepton mixing matrix U under simple requirements for the textures.