$S_3$ flavored left-right symmetric model of quarks

We construct a model based on the electroweak gauge group $\mathrm{SU}(2{)}_{\mathrm{L}}\times \mathrm{SU}(2{)}_{\mathrm{R}}\times \mathrm{U}(1{)}_{\mathrm{B}\text{−}\mathrm{L}}$ 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 produces an acceptable CKM matrix up to corrections of $\mathcal{O}({\lambda }^4)$. As for the first generation quark masses, we obtain a satisfactory value for $m_u/m_d$. The masses themselves, though in the same ballpark, turn out to be somewhat smaller than the accepted range.

Figure 1

Shaded areas in yellow, orange, green and gray represent allowed regions from $|V_{ub}|$, $|V_{cb}|$, $|V_{ts}|$ and $|V_{td}|$ respectively. The area shaded in cyan represents the common solution region.