Natural Quasi-Alignment with two Higgs Doublets and RGE Stability

In the context of two Higgs doublet models, we study the conditions required in order to have stable quasi-alignment in flavour space. We show that stability under the RGE imposes strong constraints on the flavour structure of the Yukawa couplings associated to each one of the Higgs doublets. In particular, we find a novel solution, where all Yukawa couplings are proportional to the so-called democratic matrix. This solution is rather unique, since it is the only stable solution which is a good starting point for reproducing the observed pattern of quark masses and mixing. We also show that this stable solution can be obtained by imposing on the Lagrangian a $Z_3 \times Z^\prime_3$ flavour symmetry. Quark masses of the lighter quark generations are generated through the breaking of this discrete symmetry, and, at this stage, scalar mediated FCNC arise, but are naturally suppressed by the smallness of light quark masses. In this way, we relate Higgs alignment to the hierarchy of quark masses through a discrete family symmetry.