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A model with fermion and scalar fields charged under a Peccei-Queen~(PQ)symmetry is proposed. The PQ charges are chosen in such a way that they canreproduce mass matrices with five texture zeros, {which can generate} thefermion masses, the CKM matrix, and the PMNS matrix of the Standard Model~(SM).To obtain this result, at least 4~Higgs doublets are needed. As we will see inthe manuscript this is a highly non-trivial result since the texture zeros ofthe mass matrices impose a large number of restrictions. This model shows aroute to understand the different scales of the SM by extending it with amulti-Higgs sector and an additional PQ symmetry. Since the PQ charges are notuniversal, the model presents flavor-changing neutral currents~(FCNC) at thetree level, a feature that constitutes the main source of restrictions on theparameter space. We report the allowed regions by lepton decays and comparethem with those coming from the semileptonic decays $K^{\pm}\longrightarrow \pi\bar{\nu}\nu$. We also show the excluded regions and the projected bounds offuture experiments for the axion-photon coupling as a function of the axionmass and compare it with the parameter space of our model.

A minimal axion model for mass matrices with five texture-zeros