Quark Mixing Model with Three Higgs Doublets and $S_3$ Modular Symmetry at $τ=i$

Abstract: We study the quark sector of a model with three Higgs electroweak doublets, complemented with a modular $S_3$ symmetry. We show that a proper assignment of the quark and Higgs fields in $S_3$ and their modular weights allows to write a mass matrix with a phenomenologically viable texture. Moreover, by considering the conditions on the Higgs VEVs derived from the minimization of the scalar potential together with evaluation of the Yukawa couplings at the fixed point $τ=i$, leads to a significant reduction of the free parameters. We present two setups, one with three effective parameters, which is overly constrained, and one with four effective parameters, where the $V_{CKM}$ matrix can be fitted precisely.