Two-Higgs-Doublet Models with a Flavored ${\mathbb{Z}}_2$

Two Higgs-doublet models usually consider an ad-hoc ${\mathbb{Z}}_2$ discrete symmetry to avoid flavor changing neutral currents. We consider a new class of two Higgs-doublet models where ${\mathbb{Z}}_2$ is enlarged to the symmetry group $\mathcal{F}⋊{\mathbb{Z}}_2$, i.e. an inner semi-direct product of a discrete symmetry group $\mathcal{F}$ and ${\mathbb{Z}}_2$. In such a scenario the symmetry constrains the Yukawa interactions but goes unnoticed by the scalar sector. In the most minimal scenario, $$\mathbb{Z}$_3 \rtimes $\mathbb{Z}$_2 = D_3$, flavor changing neutral currents mediated by scalars are absent at tree and one-loop level, while at the same time predictions to quark and lepton mixing are obtained, namely a trivial CKM matrix and a PMNS matrix (upon introduction of three heavy right-handed neutrinos) containing maximal atmospheric mixing. Small extensions allow to fully reproduce mixing parameters, including cobimaximal mixing in the lepton sector (maximal atmospheric mixing and a maximal $CP$ phase).