Multi-Higgs-doublet models and singular alignment

We consider a 4-Higgs-doublet model in which each Higgs doublet gives mass to one of the fermion sets {m_t}, {m_b, m_τ, m_c}, {m_μ, m_s}, and {m_d, m_u, m_e}. The sets have the feature that within each of them the masses are similar. Our model explains the mass hierarchies of the sets by hierarchies of the vacuum expectation values of the Higgs doublets associated to them. All Yukawa couplings are therefore of order one. Neutrino masses are generated by a type-I seesaw mechanism with PeV-scale singlet neutrinos. To avoid the appearance of tree-level flavour changing neutral currents, we assume that all Yukawa matrices are singularly aligned in flavour space. We mean by this that the Yukawa matrices are given as linear combinations of the rank 1 matrices that appear in the singular value decomposition of the mass matrix. In general, singular alignment allows to avoid flavour changing neutral currents in models with multiple Higgs doublets.