An $S_3$ flavored left-right symmetric model of quarks

We construct a model based on the electroweak gauge group $\rm SU(2)_L \times SU(2)_R \times U(1)_{B-L}$ augmented by an $S_3$ symmetry. We assign nontrivial $S_3$ transformation properties to the quarks and consequently we need two scalar bidoublets. Despite the extra bidoublet we have only six Yukawa couplings thanks to the discrete symmetry. Diagonalization of the quark mass matrices shows that at the leading order only the first two generations mix, resulting in a block diagonal CKM matrix, and the first generation quarks are massless. Inclusion of subleading terms produce an acceptable CKM matrix up to corrections of order $\lambda^4$. As for the first generation quark masses, we obtain satisfactory value of $m_u/m_d$. The masses themselves, though being in the same ballpark, turn out to be somewhat smaller than the accepted range.