Two Higgs doublet models with an $S_3$ symmetry

We study all implementations of the $S_3$ symmetry in the two Higgs doublet models with quarks, consistent with nonzero quark masses and a Cabibbo-Kobayashi-Maskawa (CKM) matrix, which is not block diagonal. We study the impact of the various soft-breaking terms and vacuum expectation values and find an interesting relation between the mixing angles $\alpha$ and $\beta$. We also show that, in this minimal setting, only two types of assignments are possible: Either all field sectors are in singlets or all field sectors have a doublet.