A. MFV at large tan β . If the Yukawa Lagrangian contains more than a single Higgs field, we can still assume that the Yukawa couplings are the only irreducible breaking sources of SU (3) 3 $_{q}$, but we can change their overall normalization. A particularly interesting scenario is the two-Higgs-doublet model where the two Higgs fields, φ$_{U}$ and φ$_{D}$ , are coupled separately to up- and down-type quarks: This Lagrangian is invariant under an extra U(1) symmetry with respect to the one-Higgs La- grangian in Eq. (2.5): a symmetry under which the only charged fields are D$_{R}$ and E$_{R}$ (charge +1) and φ$_{D}$ (charge − 1). This symmetry, denoted U(1)$_{PQ}$, prevents tree-level FCNCs and implies that Y u,d are the only sources of SU (3) 3 q breaking appearing in the Yukawa interaction (similar to the one-Higgs-doublet scenario). Consistently with the MFV hypothesis, we can then assume that Y u,d are the only relevant sources of SU (3) 3 q breaking appearing in all the low-energy effective operators. This is sufficient to ensure that flavor-mixing is still governed by the CKM matrix, and naturally guarantees a good agreement with present data in the Δ F = 2 sector. However, the extra symmetry of the Yukawa interaction allows us to change the overall normalization of Y u,d with interesting phenomenological consequences in specific rare modes. These effects are related only to the large value of bottom Yukawa, and indeed can be found also in other NP frameworks where there is no extended Higgs sector, but the bottom Yukawa coupling is of order one [30]. Assuming the Lagrangian in Eq. (4.5), the normalization of the Yukawa couplings is controlled by the ratio of the vacuum expectation values of the two Higgs fields, or by the parameter For tan β ≫ 1 the smallness of the b quark and τ lepton masses can be attributed to the smallness of 1 / tan β rather than to the corresponding Yukawa couplings. As a result, for tan β ≫ 1 we cannot anymore neglect the down-type Yukawa coupling. Moreover, the U(1)$_{PQ}$ symmetry cannot be exact: it has to be broken at least in the scalar potential in order to avoid the presence of a massless pseudoscalar Higgs. Even if the breaking of U(1)$_{PQ}$ and SU (3) 3 q are decoupled, the presence of U(1)$_{PQ}$ breaking sources can have important implications on the structure of the Yukawa interaction, especially if tan β is large [23, 32–34]. We can indeed consider new dimension- four operators such as tan β = 〈 φ$_{U}$ 〉 / 〈 φ$_{D}$ 〉 . (4.6) ϵ Q$_{L}$Y $^{d}$D$_{R}$ ˜ φ$_{U}$ or ϵ Q$_{L}$Y $^{u}$Y u $^{†}$Y $^{d}$D$_{R}$ ˜ φ$_{U}$ , (4.7) −L 2HDM Yukawa = Y d ij Q$_{Li}$φ$_{D}$D$_{Rj}$ + Y u ij Q$_{Li}$φ$_{U}$ U$_{Rj}$ + Y e ij L$_{Li}$φ$_{D}$E$_{Rj}$ + h . c .. (4.5) 16