Higgs potential in S_3 invariant model for quark/lepton mass and mixing

We analyzed the S_3 invariant Higgs potential with S_3 singlet and doublet Higgs. We obtained a relation (|v_1|/|v_2|)^2=-sin2phi_2/sin2phi_1 from this S_3 invariant Higgs potential, where v_1, v_2 and phi_1, phi_2 are vacuum pectation values and phases of S_3 doublet Higgs, respectively. This relation could be satisfied exactly by the results |v_1|/ |v_2|=0.207, phi_1=-74.9deg and phi_2=0.74deg obtained from the previous our work analyzing the quark/lepton mass and mixing in S_3 invariant Yukawa interaction. Furthermore, the relation v_S ~ v_D=sqrt{|v_1|^2+|v_2|^2}=174GeV is obtained and then the coupling strength of Higgs to top quark g_{H_Stt}=m_t/v_S is altered as by a factor sqrt{2} from the standard value. Introduced the S_3 doublet Higgs, FCNC are produced in tree level. Predicted branching ratios for rare decays mu^- to e^-e^+e^-, K^0_L to mu^+\mu^- etc., induced by the FCNC are sufficiently below the present experimental upper bounds.