Abelian symmetries in the two-Higgs-doublet model with fermions

We classify all possible implementations of an Abelian symmetry in the two-Higgs-doublet model with fermions. We identify those symmetries which are consistent with nonvanishing quark masses and a Cabibbo-Kobayashi-Maskawa quark-mixing matrix (CKM), which is not block-diagonal. Our analysis takes us from a plethora of possibilities down to 246 relevant cases, requiring only 34 distinct matrix forms. We show that applying $Z_n$ with $n\ge 4$ to the scalar sector leads to a continuous $U(1)$ symmetry in the whole Lagrangian. Finally, we address the possibilities of spontaneous $CP$ violation and of natural suppression of the flavor-changing neutral currents. We explain why our work is relevant even for non-Abelian symmetries.