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.


Introduction
One very compelling extension of the standard model (SM) is the left-right symmetric (LRS) model [1,2] based on the electroweak gauge group SU(2) L × SU(2) R × U(1) B−L . Unlike the SM, the left-chiral and right-chiral fermions are treated similarly in these models.
In the minimal LRS model, the left-chiral and right-chiral quarks are assigned the following representations under the gauge group: where the index i runs from 1 to 3 to accommodate three generations. Allowing the Yukawa couplings for the quarks requires the presence of a scalar bidoublet Φ : (2, 2, 0) .
It has two neutral components and therefore two possible vacuum expectation values (VEVs). The quarks obtain their masses after symmetry breaking through the VEVs of this Φ field. The mass matrix has many free parameters. There are nine Yukawa couplings involving Φ that relate three generations of left-chiral quarks with three generations of right-chiral quarks. Besides, since the representation of Φ under the gauge group is real, there are nine more couplings where Φ is replaced by its complex conjugate, suitably sandwiched by the antisymmetric Pauli matrix so that its transformation property is exactly the same as that of Φ. Because of this large number of parameters, the quark mass matrices do not have much of a predictive power.
In this article, we impose an S 3 symmetry between the generations and show that the number of Yukawa couplings is drastically reduced to the extent that predictions are possible. Such a symmetry has been explored extensively in the context of the SM gauge group . However, to our knowledge, this discrete symmetry in the context of the left-right symmetric gauge group [40,41] has not been explored very much. We will show that the enhanced gauge symmetry, along with the discrete symmetry, leads to relations between the quark masses and mixings.

The model with a horizontal S 3 symmetry
We extend the minimal LRS model with an extra S 3 symmetry that acts between different generations of fermions. This S 3 symmetry has three different irreducible representation, 1 , 1 and 2 , where the numbers signify the dimension of the representation matrices. The group has one order-2 and one order-3 generators. In the 2 representation, we take them to be For this choice of basis, we assign the following representations to the fermions following the same rule for left and right chiral quarks. In order to obtain an acceptable mass pattern, we now need scalars to be in the 2 representation of S 3 . This means that we need to add an extra bidoublet over and above what was shown in Eq. (2) [42]. Calling the two scalar multiplets Φ 1 and Φ 2 , we assign them the representation under the S 3 symmetry. Keeping in mind the fact that for any term in the Lagrangian where there is a Φ, there is another term containing Φ, we can write down the most general Yukawa couplings involving quarks as: After symmetry breaking, both Φ 1 and Φ 2 develop VEVs: The resulting mass matrices for the quarks are of the form and F and G are matrices which have exactly the same form, except that they involve the Yukawa couplings with tilde marks. We will assume that all Yukawa couplings are real, and so are the VEVs. Our task is now to perform the diagonalization of the mass matrices given in Eq. (9) and show that, under some reasonable assumptions, the diagonalization can be performed and the quark mixing matrix can be obtained in a form that is consistent with the present data.
Of course the matrices shown in Eq. (9) cannot be diagonalized in general with the help of unitary transformations. One needs bi-unitary transformations, which induces different transformations on the left-chiral and right-chiral quarks. For the sake of the CKM matrix, we need only the mixing of the left-chiral fermions. The relevant mixing matrices can be obtained by considering the diagonalization of M q M † q , where the index q takes two values, u and d, to distinguish the up-sector quarks from the down-sector quarks. Let us write where D 2 q is a diagonal matrix whose diagonal elements are the mass-squared values of the quarks of type q (i.e., u or d). Then the CKM matrix will be given by We therefore need to find the diagonalizing matrices U u and U d . For this, we need to proceed in steps, making some assumptions which we now describe.

Large and small terms
In order to perform the diagonalization, we will first make some assumptions about the relative magnitudes of different parameters. The first thing we assume is that the primed VEVs are much smaller compared to the unprimed ones: The opposite assumption κ 1 , κ 2 κ 1 , κ 2 will serve as well, and amounts to fixing a convention. Such an assumption can naturally suppress the mixing between the gauge bosons in the left and right sectors. The terms in Eq. (9) proportional to the unprimed VEVs will therefore be considered dominant, and the other terms, proportional to the primed VEVs, will be considered as perturbations. In this section, we consider diagonalization of the quark mass matrices in the limit κ 1 = κ 2 = 0, i.e., in the zeroth order of smallness.
There are only two VEVs at this level of approximation, κ 1 and κ 2 . Since the other VEVs have been assumed to be negligible, we can write where v = 174 GeV is the breaking scale of the SM. We also define Henceforth, instead of using κ 1 and κ 2 directly, we will use the parameters v and β. Thus, in this zeroth order approximation, the mass matrices of the quarks are of the form where, for the ease of notation, we have renamed the Yukawa couplings by a subscript q according to which mass matrix they contribute to: The kind of mass matrix shown in Eq. (16) was obtained in our earlier work [39] in the context of SU(2) L × U(1) Y model. In order to perform a diagonalization of the mass matrices at this level of approximation, we note that Since the first generation quark masses are very small, we assume that they are zero at this level, and arise entirely from smaller corrections to the mass matrices. Then the determinant must vanish at this level. Without arbitrarily making some of the Yukawa couplings vanish, this can be achieved, in both up and down sectors, if we have This value can be nontrivially obtained by setting We assume that this is indeed the value of β that comes out of the minimization of the Higgs potential at this level of approximation, i.e., on assuming κ 1 = κ 2 = 0. Then, taking one finds At this stage, then, the CKM matrix is given by This shows that at the zeroth order, we have only the Cabibbo angle that mixes the first two generations of quarks, whereas the third generation is unmixed. This state of affairs is certainly consistent with the fact that the Cabibbo angle in the largest angle in the CKM matrix, and all others are much smaller. In Sec. 4, we will see how the small angles can arise from the small corrections that we have left out so far. Before that, we want to summarize the information that we have already obtained about the masses and consequently about the Yukawa couplings. From Eq. (23), we see that at the zeroth level of approximation, Although these masses will receive some corrections which will be introduced later, such modifications are expected to be small, and therefore we can use Eq. (25) as a very good approximation to the actual masses. Knowledge of the hierarchy of quark masses then tells us that so that the third generation is much heavier than the second, and further to ensure that the top mass is much bigger than the bottom mass. Using Eq. (26) and the definition of Eq. (22), we can write the Yukawa couplings in the form where m 3q and m 2q denote the masses of the third and second generation quarks in the sector marked by q, i.e., At this point, perhaps it is worth reemphasizing the main conclusion of this section. Here we have considered an approximate reality where m u = m d = 0 and V i3 = V 3i = 0 (i = 1, 2) as well. The vanishing of V 3i (i = 1, 2) will follow automatically from the vanishing of V i3 (i = 1, 2) due to the unitarity of the CKM matrix. We are thus left with four zeros, viz., m u = 0, m d = 0 and V i3 = 0 (i = 1, 2), which are disconnected in the SM, i.e., they are four different accidents in the framework of the SM. But, in our model, one needs only one accident, given by Eq. (19), to achieve all these zeros, i.e., the four zeros are connected. Therefore, concerning the small parameters in the quark Yukawa sector, our construction provides a sense of aesthetic connection that is absent in the SM. Moreover, this approximate reality with κ a = 0 forbids W L -W R mixing. In the next section, we will see that turning on small values of κ a leads to small CKM elements as well as the first generation quark masses. Therefore, in our scenario, these small masses and mixings in the quark sector owe their origin to the same parameters which govern the smallness of the W L -W R mixing.

Including the smaller terms
We now try to see the effects of non-zero values of κ 1 and κ 2 . The extra contributions that appear in the mass matrices will be denoted by M , i.e., These contributions will come from two sources. First, there are terms proportional to κ 1 and κ 2 in Eq. (9). Second, the minimization of the Higgs potential will now not give Eq. (20), but rather sin 3β = 3δ , with some small value of δ. All correction terms in the mass matrices will have one factor of some Yukawa coupling. Motivated by the hierarchy among the Yukawa couplings noted in Sec. 3, we will keep only the terms proportional to A u as the dominant corrections to M (0) q defined in Eq. (16), with the understanding that the contribution from other terms are proportional to much smaller Yukawa couplings, and are negligible at the level of accuracy that we seek for. Keeping these in mind, we can write the dominant corrections are as follows: In order to set up a uniform notation for both up and down sectors, let us introduce some shorthands through the relations so that u = δ , χ u = −π/6 , (34a) We now need to examine the mass matrices including these corrections, and the diagonalization procedure.
The first thing that we notice is that, after the inclusion of M , the determinant of the mass matrix is no more zero and is given by This quantity should be equal to the product of the three mass eigenvalues. Therefore, the mass of the first generation quark will be given by where we have substituted B q and C q using Eq. (28) and defined In a less cluttered but lengthier way, we can break up Eq. (36) as We now look at the diagonalization of the matrices M q M † q . Referring back to Eq. (11) and its zeroth level analog, Eq. (23), we propose to incorporate the correction to the diagonalizing matrix by writing where X q is supposed to inflict small corrections on U (0) q . We now parametrize X q by writing ignoring higher order terms in α q and γ q . We have checked that including a rotation in the 12 sector as well contributes only at a subleading order. Therefore, for our purposes, Eq. (40) constitutes a reasonable approximation for the correction to U q . If we now evaluate the left side of Eq. (11), using Eq. (39) for U q and M q as the sum of the expression of Eq. (16) with β = π/3 and the corrections from Eq. (32), we should obtain a diagonal matrix, to the accuracy employed in defining the small parameters. From this condition, one should be able to determine the relevant parameters of X q .
First, we check the diagonal elements. The lower two diagonal elements will pick up small corrections to the formulas of the second and third generation quarks given in Eq. (25), and are unimportant for our purpose. The first diagonal element should give the mass squared of the first generation quark. Evaluation of Eq. (11) gives But the mass value has already been found in Eq. (36) from the consideration of the determinant. Putting in the value from there and neglecting terms which provide corrections of order m 2 2q /m 2 3q , we can determine α q as: Quite nicely, the 13 element of the left side of Eq. (11) also vanishes at the leading order under the same condition, confirming the consistency of the approximation. Further, the vanishing of the 23 element at the leading order gives the expression for γ q as  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.

The CKM matrix
Using Eq. (12) in conjunction with Eq. (39), we can now write the CKM matrix as where V CKM has been defined already in Eq. (24) and the Cabibbo angle, θ C , is defined as In writing Eq. (44) we have also used the definition of X q given in Eq.
While comparing with the experimental values, we should keep in mind that the inherent uncertainty of O λ 4 in Eq. (44) is much larger than the experimental uncertainties in Eq. (46). Therefore, for the ij-th element of the CKM matrix, we take where the central values are taken from Eq. (46). We assume that sin θ C ≡ −λ ≈ −0.225 has been measured quite accurately and use Eq. (45) to express θ u in terms of θ d . Our goal is to see, using Eq. (47), whether there exists a common region in the sin θ d vs. | d − u | plane, which is allowed by |V ub |, |V cb |, |V ts | and |V td | simultaneously. We display our result in Fig. 1 where we see that there is indeed some common solution region. Note that there are two different allowed regions from |V ub |, which correspond to different signs for sin θ u . With | d − u | and sin θ d nearly fixed from Fig. 1, now we have only one parameter, namely u (or equivalently d ) to play around. Thus, using Eq. (38), we still need to reproduce two light quark masses, m u and m d , with only one parameter remaining at our disposal. As a matter of fact, the cyan region on the left in Fig. 1, is disfavored because it gives too small values for the down-quark mass. Keeping this in mind, we choose the following values which correspond to a benchmark point somewhere in the cyan region on the right in Fig. 1. Using these values we find We see that these values of the CKM elements are acceptable within an error bar of O λ 4 . As commented earlier, we also obtain the light quark masses from this exercise. Taking The values of m u as well as the ratio m u /m d are within tolerable ranges, but the absolute value of m d comes out to be a bit too low. Having only one free remaining parameter prevents us from obtaining a good fit for both the up and down quark masses. We have checked that, as long as the down quark mass is concerned, the values given in Eq. (49) reflects the best case scenario.

Summary
We have considered a model where the left-right symmetric gauge group SU(2) L × SU(2) R × U(1) B−L is augmented by an S 3 symmetry. The discrete symmetry drastically reduces the number of Yukawa couplings in the model. In fact, there are only six Yukawa couplings. Because of the small number of parameters, we can relate many aspects of quark masses and mixings satisfactorily in our model. We have demonstrated that the smallness of the first generation quark masses is related to the smallness of the 13 and 23 elements of the CKM matrix as well as to the smallness of the W L -W R mixing. We have also shown that, under some reasonable assumptions about the relative magnitudes of the VEVs, the CKM matrix can be reproduced within an accuracy of O λ 4 . The only sore point seems to be the light quark masses that we obtain from the model, which, although being in the same ballpark, turn out to be a bit smaller than expected. Yet the model deserves careful attention since we believe that the successes of the model with the CKM matrix outweigh the dissatisfaction with the light quark masses.