Higgs potential in a minimalS3invariant extension of the standard model

Minimal $S_3$ invariant Higgs potential with real soft $S_3$ breaking masses is investigated. It is required that without having a problem with triviality, all physical Higgs bosons, except one neutral one, become heavy $\gtrsim 10\mathrm{TeV}$ in order to sufficiently suppress flavor-changing neutral currents. There exist three nonequivalent soft mass terms that can be characterized according to their discrete symmetries, and the one that breaks $S_3$ completely. The $S_2^{\prime }$ invariant vacuum expectation values (VEVs) of the Higgs fields are the most economic VEVs in the sense that the freedom of VEVs can be completely absorbed into the Yukawa couplings so that it is possible to derive, without referring to the details of the VEVs, the most general form for the fermion mass matrices in minimal $S_3$ extension of the standard model. We find that except for the completely broken case of the soft terms, the $S_2^{\prime }$ invariant VEVs are unique VEVs that satisfy the requirement of heavy Higgs bosons. It is found that they also correspond to a local minimum in the completely broken case.