We analyzed the $S_3$ invariant Higgs potential with $S_3$ singlet and doublet Higgs. We obtained a relation $(|v_1|/|v_2|{)}^2=-\sin 2{\varphi }_2/\sin 2{\varphi }_1$ from this $S_3$ invariant Higgs potential, where $v_1$, $v_2$ and ${\varphi }_1$, ${\varphi }_2$ are vacuum expectation values and phases of $S_3$ doublet Higgs, respectively. This relation could be satisfied exactly by the results $|v_1|/|v_2|=0.207$, ${\varphi }_1=-74.9°$, and ${\varphi }_2=0.74°$ obtained from our previous work analyzing the quark/lepton mass and mixing in the $S_3$ invariant Yukawa interaction. Furthermore, the relation $v_S\sim v_D=\sqrt{|v_1{|}^2+|v_2{|}^2}=174\mathrm{GeV}$ is obtained, and then the coupling strength of the Higgs to the top quark $g_{H_{S}tt}=m_t/v_S$ is altered by a factor $\sqrt{2}$ from the standard value. Introducing the $S_3$ doublet Higgs, flavor changing neutral currents are produced at tree level. The predicted branching ratios for rare decays ${\mu }^{-}\to e^{-}{e}^{+}{e}^{-}$, $K_L^0\to {\mu }^{+}{\mu }^{-}$, etc., induced by the flavor changing neutral currents, are sufficiently below the present experimental upper bounds.

• Received 8 March 2012

DOI:https://doi.org/10.1103/PhysRevD.85.105013