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

Natural Quasi-Alignment with two Higgs Doublets and RGE Stability