Quark masses and CKM hierarchies from $$S_4'$$ S 4 ′ modular flavor symmetry

Abstract

Abstract We propose models to explain the hierarchies of the quark masses and mixing by utilizing the $$S_4^\prime $$ S 4 ′ modular flavor symmetry. The hierarchy is realized by the modulus $$\tau $$ τ stabilized at $$\textrm{Im}\,\tau \gg 1$$ Im τ ≫ 1 , where the residual $$\mathbb {Z}_4^T$$ Z 4 T symmetry is approximately unbroken and the Froggatt–Nielsen mechanism works. It is found that the quark hierarchies are realized only in a few cases of quark representations. We study two models with assigning the modular weights, so that the observed quark hierarchies are explained in the cases of both small and large ratios of the top to bottom Yukawa couplings. We also argue that $$\mathcal {O}\left( {0.1}\right) $$ O 0.1 hierarchies of the $$\mathcal {O}\left( {1}\right) $$ O 1 coefficients and the spontaneous CP violation can be realized by imposing another $$S_3$$ S 3 modular symmetry.