U(3)-Flavor Nonet Scalar as an Origin of the Flavor Mass Spectra

According to an idea that the quark and lepton mass spectra originate in a VEV structure of a U(3)-flavor nonet scalar \Phi, the mass spectra of the down-quarks and charged leptons are investigated. The U(3) flavor symmetry is spontaneously and completely broken by non-zero and non-degenerated VEVs of $\Phi$, without passing any subgroup of U(3). The ratios (m_e+m_\mu+m_\tau)/(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau})^2 and \sqrt{m_e m_\mu m_\tau}/(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau})^3 are investigated based on a toy model.