Quark mixing in an $S_3$ symmetric model with two Higgs doublets

TL;DR

This paper presents an $S_3$ symmetric two-Higgs-doublet model that links small first-generation quark masses to a near block-diagonal CKM matrix, predicting Higgs properties and flavor-changing neutral currents.

Contribution

It introduces a novel $S_3$ symmetric 2HDM framework that naturally explains quark mass hierarchies and CKM structure, with precise predictions for $ aneta$ and FCNC behavior.

Findings

Emergence of an SM-like Higgs from the model

Determination of $ aneta$ from quark mass constraints

Predictive FCNC structure

Abstract

We construct a model where the smallness of the masses of first quark generations implies the near block diagonal nature of the CKM matrix and vice-versa. For this set-up, we rely on a 2HDM structure with an $S_3$ symmetry. We show that an SM-like Higgs emerges naturally from such a construction. Moreover, the ratio of two VEVs, $\tan\beta$ can be precisely determined from the requirement of the near masslessness of the up- and down-quarks. The FCNC structure that arises from our model is also very predictive.

Equations

$$a=\left[\begin{array}{cc}-\frac{1}{2}& \frac{\surd 3}{2}\\ -\frac{\surd 3}{2}& -\frac{1}{2}\end{array}\right],b=\left[\begin{array}{cc}\frac{1}{2}& \frac{\surd 3}{2}\\ \frac{\surd 3}{2}& -\frac{1}{2}\end{array}\right].$$

$$2$$

$$1$$

$$2$$

$$V(\Phi )$$

$$V_4(\Phi )$$

$$V_2(\Phi )$$

$${\varphi }_i=\left(\genfrac{}{}{0ex}{}{{\varphi }_i^{+}}{\frac{1}{\surd 2}(v_i+h_i+i{\zeta }_i)}\right),$$

$$v_1=v\cos \beta ,v_2=v\sin \beta ,$$

$${\mu }_1^2$$

$${\mu }_2^2$$

$$\left(\begin{array}{c}{w}^{\pm }\\ H^{\pm }\end{array}\right)=\left(\begin{array}{cc}\cos \beta & \sin \beta \\ -\sin \beta & \cos \beta \end{array}\right)\left(\begin{array}{c}{\varphi }_1^{\pm }\\ {\varphi }_2^{\pm }\end{array}\right).$$

$$M_{H^{\pm }}^2=\frac{2{\mu }_{12}^2}{\sin 2\beta }-2{\lambda }_3{v}^2.$$

$$\left(\begin{array}{c}z\\ A\end{array}\right)=\left(\begin{array}{cc}\cos \beta & \sin \beta \\ -\sin \beta & \cos \beta \end{array}\right)\left(\begin{array}{c}{\zeta }_1\\ {\zeta }_2\end{array}\right)$$

$$M_A^2=\frac{2{\mu }_{12}^2}{\sin 2\beta }-2({\lambda }_2+{\lambda }_3)v^2.$$

$$V_{\mathrm{mass}}^{\mathrm{S}}=\left(\begin{array}{cc}{h}_1& h_2\end{array}\right)\frac{1}{2}{\mathbb{M}}_S^2\left(\begin{array}{c}{h}_1\\ h_2\end{array}\right)$$

$${\mathbb{M}}_S^2=\left(\begin{array}{cc}{\mu }_{12}^2\frac{v_2}{v_1}+2v_1^2({\lambda }_1+{\lambda }_3)& -{\mu }_{12}^2+2v_1{v}_2({\lambda }_1+{\lambda }_3)\\ -{\mu }_{12}^2+2v_1{v}_2({\lambda }_1+{\lambda }_3)& {\mu }_{12}^2\frac{v_1}{v_2}+2v_2^2({\lambda }_1+{\lambda }_3)\end{array}\right).$$

$$\left(\begin{array}{c}h\\ H\end{array}\right)=\left(\begin{array}{cc}\cos \beta & \sin \beta \\ -\sin \beta & \cos \beta \end{array}\right)\left(\begin{array}{c}{h}_1\\ h_2\end{array}\right)$$

$$m_H^2=\frac{2{\mu }_{12}^2}{\sin 2\beta },m_h^2=2({\lambda }_1+{\lambda }_3)v^2.$$

$${\varphi }_1$$

$${\varphi }_2$$

$${\mathcal{L}}_Y^{(u)}$$

$${\mathcal{M}}_q=\frac{v}{\surd 2}\left(\begin{array}{ccc}{B}_q\sin \beta & B_q\cos \beta & A_q\cos \beta \\ B_q\cos \beta & -B_q\sin \beta & A_q\sin \beta \\ C_q\cos \beta & C_q\sin \beta & 0\end{array}\right),$$

$${\mathcal{M}}_q{\mathcal{M}}_q^{†}=\frac{1}{2}{v}^2\left(\begin{array}{ccc}{a}_q^2{\cos }^2\beta +b_q^2& \frac{1}{2}{a}_q^2\sin 2\beta & B_q{C}_q^{\ast }\sin 2\beta \\ \frac{1}{2}{a}_q^2\sin 2\beta & a_q^2{\sin }^2\beta +b_q^2& B_q{C}_q^{\ast }\cos 2\beta \\ B_q^{\ast }{C}_q\sin 2\beta & B_q^{\ast }{C}_q\cos 2\beta & c_q^2\end{array}\right),$$

$$x=\frac{2m^2}{v^2}$$

$$x^3-(a^2+2b^2+c^2)x^2+(a^2+b^2)(b^2+c^2)x-a^2{b}^2{c}^2{\sin }^{2}3\beta =0,$$

$$\mathcal{U}=\left(\begin{array}{ccc}0& 0& 1\\ \sin \beta & -\cos \beta & 0\\ \cos \beta & \sin \beta & 0\end{array}\right).$$

$$M^2=\mathcal{U}\mathcal{M}{\mathcal{M}}^{†}{\mathcal{U}}^{†}=\frac{1}{2}{v}^2\left(\begin{array}{ccc}{c}^2& -bc\cos 3\beta & bc\sin 3\beta \\ -bc\cos 3\beta & b^2& 0\\ bc\sin 3\beta & 0& a^2+b^2\end{array}\right),$$

$$0,\frac{1}{2}{v}^2(b^2+c^2),\frac{1}{2}{v}^2(a^2+b^2).$$

$$\sin 3\beta =0.$$

$$M^2=\mathcal{U}\mathcal{M}{\mathcal{M}}^{†}{\mathcal{U}}^{†}=\frac{1}{2}{v}^2\left(\begin{array}{ccc}{c}^2& bc& 0\\ bc& b^2& 0\\ 0& 0& a^2+b^2\end{array}\right).$$

$$a^2\gg b^2,c^2$$

$$U_u={\mathcal{O}}_u\mathcal{U},U_d={\mathcal{O}}_d\mathcal{U},$$

Full text

The Standard Model (SM) does not provide any connection between quark masses and mixings: they are independent parameters to be fixed by the experimental observations. One attractive way to obtain insight into these parameters is to impose some additional symmetry under which the generations of quarks transform in a non-trivial way. There have been many attempts where, by imposing a discrete symmetry on different generations of SM fermions, some relations between the masses and mixings have been obtained (see [1, 2] for review).

In this article, we present an attempt to relate two features of quark masses and mixings. The first of these two is the fact that the first generation quarks are very light compared to the other ones whereas the second concerns the near block-diagonal structure of the quark mixing matrix, or the Cabibbo-Kobayashi-Maskawa (CKM) matrix. This second feature comes out very clearly in the Wolfenstein parametrization [3] of the CKM matrix, where each element is written in a power series of a small parameter $\lambda$. If we keep only terms up to the linear order in $\lambda$, the CKM matrix is indeed block-diagonal. We propose a connection between these two features by invoking an $S_{3}$ symmetry.

Many works on flavour model building using $S_{3}$ symmetry have been done in the past [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. In these constructions one usually employs, for the scalar sector, a three Higgs doublet structure which goes well with the aesthetic idea of having three replicas of Higgs doublets in conformity with three generations of fermions [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. Even more complicated scalar structures are not uncommon [42, 43, 44, 45, 46, 47, 48]. However, in this paper, we rely on a two Higgs-doublet model (2HDM) scalar structure [49] which is much more economical in terms of independent parameters. Although the idea of a 2HDM with $S_{3}$ symmetry has been conceived lately [24], some distinct implications have not been emphasised earlier. For example, we will show that an $S_{3}$ symmetric 2HDM potential naturally delivers an SM-like Higgs boson which can be identified with the scalar resonance observed at the LHC with signal strengths in close agreement with the SM predictions [50]. We will also demonstrate how, in our scenario, the requirement of near-masslessness for the first generation of quarks dictates a particular value of $\tan\beta$, which will simultaneously render the CKM matrix block-diagonal. For intuitive understanding of the model Lagrangian and the conclusions that follow from it, a brief overview of the $S_{3}$ symmetry is in order.

The discrete symmetry group $S_{3}$ has three irreducible representations: $\mathbf{1}$, $\mathbf{1}^{\prime}$ and $\mathbf{2}$. We pick a basis such that the generators in the ${\bf 2}$ representation are given by

[TABLE]

Note that $a$ is of order 3, whereas $b$ is of order 2. The rest of the elements can be obtained by taking products of powers of these two elements. In this basis the quark fields transform under $S_{3}$ in the following way:

[TABLE]

where the $Q_{A}$’s ($A=1,2,3$) are the usual left-handed $\rm SU(2)$ quark doublets, whereas the $u_{AR}$’s and $d_{AR}$’s are the right-handed up-type and down-type quark fields respectively, which are singlets of the $\rm SU(2)$ part of the gauge symmetry. Note that the square brackets, in Eqs. (1) and (2) as well as in the subsequent text, denote the doublet representation of $S_{3}$, and has nothing to do with the representation of the enclosed fields under $\rm SU(2)$. Similarly, in the Higgs sector, there are two $\rm SU(2)$ doublets $\phi_{i}$ ($i=1,2)$, and their transformation under the $S_{3}$ symmetry is as follows:

[TABLE]

We write the potential of the theory as follows:

[TABLE]

which is not $S_{3}$-symmetric unless the co-efficients satisfy some special conditions. If these conditions are not met, $V_{2}(\Phi)$ contains terms which softly break the $S_{3}$ symmetry, and we allow for such terms. We will consider various scenarios with the quadratic terms in a short while.

The parameters in the quartic part of the potential must be real because of hermiticity of the Lagrangian. In the quadratic part $V_{2}(\Phi)$, the parameters $\mu_{1}^{2}$ and $\mu_{2}^{2}$ are also real. The parameter $\mu_{12}^{2}$ can be complex, but its phase can be absorbed by redefining either $\phi_{1}$ or $\phi_{2}$. Thus, all parameters in $V(\Phi)$ can be taken to be real without any loss of generality. It has been argued [32] that in this case the vacuum expectation values (VEVs) can also be taken to be real. Denoting the VEV of $\phi_{i}$ by $v_{i}$ we write the doublets after symmetry breaking in the form

[TABLE]

and use the standard notation

[TABLE]

where the $W$ and $Z$-boson masses are proportional to $v\approx 246$ GeV. Assuming both $v_{1}$ and $v_{2}$ to be non-zero, the minimisation condition of the potential $V(\Phi)$ can be written as

[TABLE]

Let us discuss the physical scalar spectrum of the model. We begin with the charged boson sector. One combination of $\phi_{1}^{\pm}$ and $\phi_{2}^{\pm}$, to be denoted by $w^{\pm}$, will constitute an unphysical field that does not appear in the physical spectrum. The orthogonal combination, $H^{\pm}$, will be a physical charged scalar. The two combinations will be given by

[TABLE]

The mass of the physical charged scalar can be easily calculated:

[TABLE]

In the pseudoscalar sector, there is one combination, $z$, which becomes unphysical after symmetry breaking, and there is one physical pseudoscalar field $A$. They are given by

[TABLE]

with

[TABLE]

The mass matrix for the scalar part can be written as,

[TABLE]

with

[TABLE]

The diagonalisation of ${\mathbb{M}}_{S}^{2}$ will lead to two neutral physical scalars $H$ and $h$,

[TABLE]

with

[TABLE]

At this point one should note that in the case of 2HDMs, the combination $H^{0}=(v_{1}h_{1}+v_{2}h_{2})/v$ has SM-like couplings at the tree level. But in general $H^{0}$ is not a physical eigenstate. In the limit where $H^{0}$ is aligned with one of the physical $CP$-even scalars, is known as the alignment limit for 2HDMs. Eq. (14) shows that this is indeed the case in the present model, viz., that the eigenstate $h$ is the same as $H^{0}$. Thus the alignment limit emerges naturally [51] in our scenario. Hence by identifying $h$ with the 125 GeV scalar observed at the LHC, our model becomes consistent, by design, with the LHC Higgs data [50].

Looking at the spectrum, we can identify the following different scenarios in regard to Eq. (4c).

If $\mu_{1}^{2}=\mu_{2}^{2}$ and $\mu_{12}^{2}=0$, $V_{2}(\Phi)$ is completely $S_{3}$-symmetric. In fact, the potential is invariant under a much bigger symmetry: an $\rm SO(2)$ symmetry under which

[TABLE]

Thus, after gauge symmetry breaking when the $\phi_{i}$’s develop vacuum expectation values (VEVs), we will have a massless scalar, a Goldstone boson as seen clearly from Eq. (15). This is not the scenario that we advocate. 2. 2.

If $\mu_{1}^{2}\neq\mu_{2}^{2}$ and $\mu_{12}^{2}=0$, the potential is not $S_{3}$ symmetric, but Eq. (15) shows that we will still have a massless boson. Thus, this is not our desired scenario either. 3. 3.

If $\mu_{1}^{2}=\mu_{2}^{2}$ and $\mu_{12}^{2}\neq 0$, there exists no massless scalar, but Eq. (7) shows that we will now have $\tan\beta=1$ or $v_{1}=v_{2}$ because the potential has an exchange symmetry $\phi_{1}\leftrightarrow\phi_{2}$. As we discuss later, this scenario will be detrimental to our aim. 4. 4.

If $\mu_{1}^{2}\neq\mu_{2}^{2}$ and $\mu_{12}^{2}\neq 0$, there is no massless scalar and also $\tan\beta$ can be arbitrary. This is the scenario that will be useful for us, implying that the soft-breaking terms are absolutely necessary.

We now present the most general Yukawa couplings involving the $u_{R}$ quarks that is consistent with the gauge and $S_{3}$ symmetries. The $S_{3}$ symmetry cuts down on the number of Yukawa couplings drastically, and we obtain only the following couplings involving right-chiral $u$-type quarks:

[TABLE]

We have used the standard abbreviation $\widetilde{\phi}_{i}=i\sigma_{2}\phi_{i}^{*}$. The Yukawa couplings of the $d_{R}$ quarks can be obtained by replacing $u_{AR}$ by $d_{AR}$, $\{A,B,C\}_{u}$ by $\{A,B,C\}_{d}$, and $\widetilde{\phi}_{i}$ by $\phi_{i}$ in Eq. (17). Although the Yukawa couplings, in general, may be complex, we will discuss later that all but one phase can be absorbed in the field redefinitions.

After symmetry breaking, the mass matrices that arise in the quark sector have the following form:

[TABLE]

where the subscripted index $q$ can take the value $u$ for the up-type quarks, and $d$ for the down-type quarks. It is well-known that these matrices can be diagonalized through bi-unitary transformations, e.g., one can find two unitary matrices $U_{u}$ and $V_{u}$, for the up-sector, such that $U_{u}{\cal M}_{u}V_{u}^{\dagger}$ is diagonal. The CKM matrix is then given by $U_{u}U_{d}^{\dagger}$.

The matrices $U_{u}$ and $U_{d}$ are the unitary matrices which diagonalize, through similarity transformations, the hermitian matrices ${\cal M}_{u}{\cal M}_{u}^{\dagger}$ and ${\cal M}_{d}{\cal M}_{d}^{\dagger}$ respectively. From Eq. (18), we obtain

[TABLE]

where $a_{q}=|A_{q}|$ etc. Clearly, the three eigenvalues of ${\cal M}_{u}{\cal M}_{u}^{\dagger}$ would be the mass squared of the three up sector quarks, namely $m_{u}^{2},m_{c}^{2}$ and $m_{t}^{2}$, and the three eigenvalues of ${\cal M}_{d}{\cal M}_{d}^{\dagger}$ would be $m_{d}^{2}$, $m_{s}^{2}$ and $m_{b}^{2}$. The eigenvalues can be obtained by solving the characteristic equation of the general matrix in Eq. (19). Introducing the shorthand notation

[TABLE]

for any fermion with mass $m$, this characteristic equation has the following form:

[TABLE]

with subscripts $u$ or $d$ attached to the Yukawa couplings, as the case may be. Note that this equation is free from the phase of $BC^{*}$, which is the only phase that is present in Eq. (19).

Looking at the Lagrangian of Eq. (17) and the corresponding Lagrangian involving $d_{iR}$, we see why only one phase is present in Eq. (19). Any phase of $A_{u}$ and $A_{d}$ can be absorbed by redefining the fields $u_{3R}$ and $d_{3R}$. After this, both $B_{u}$ and $B_{d}$ can be made real by redefining the fields $u_{1R},u_{2R}$ and $d_{1R},d_{2R}$. Finally, either $C_{u}$ or $C_{d}$ can be made real by choosing the phase of $Q_{3}$, but one of them remains complex. Alternatively, one can make both $C_{u}$ or $C_{d}$ real first, by redefining the right-chiral quark fields, and then either $B_{u}$ or $B_{d}$ can be made real by choosing the phases of $Q_{1}$ and $Q_{2}$. Either way, one of the $C_{q}$’s or one of the $B_{q}$’s can be complex in the most general case. In what follows, we will assume that all Yukawa couplings are real, and use the lower-case symbols for them.

Before entering into a discussion of the eigenvalues obtained as solutions of Eq. (21), let us have some idea of the form of the diagonalizing matrix. As a first step, we can diagonalize only the terms in Eq. (19) that are proportional to $a_{q}^{2}$. This is done, e.g., by a matrix

[TABLE]

Note that this matrix does not depend on the Yukawa couplings, and is therefore the same for the up-type and down-type mass matrices. Applying a similarity transformation with this matrix on ${\cal MM}^{\dagger}$, we obtain

[TABLE]

with subscripts $u$ and $d$ attached for quarks of positive and negative charges respectively.

In the preamble of the article, we said that we want to relate the almost-masslessness of first generation quarks with the almost-block-diagonal form of the CKM matrix. We now narrow down the scenario in which we can have one zero eigenvalue in both up-type and down-type quark sector, as well as a block-diagonal CKM matrix.

First we note that if one solution of Eq. (21) is zero, then the $x$-independent term should vanish in that equation. In this case, the eigenvalues of ${\cal MM}^{\dagger}$ are given by

[TABLE]

For the diagonalizing matrix, we now consider two different cases.

Case 1: Some Yukawa couplings vanish

Surely, the $x$-independent term in Eq. (21) can vanish if at least one of the Yukawa couplings is zero. Looking at Eq. (23), we see that $a=0$ does not make $M^{2}$ block-diagonal, so we reject this possibility. If either $b$ or $c$ vanishes, the matrix $M^{2}$ becomes completely diagonal. This means that for $b$ or $c=0$, the same matrix $\cal U$ will diagonalize both ${\cal M}_{u}{\cal M}_{u}^{\dagger}$ and ${\cal M}_{d}{\cal M}_{d}^{\dagger}$ making the CKM matrix a unit matrix. Therefore making some Yukawa coupling vanish to obtain one zero mass does not produce the desirable block-diagonal structure of the CKM matrix.

One should recall that making $\mu_{1}^{2}=\mu_{2}^{2}$ in Eq. (4c) had led to $\tan\beta=1$, which in view of Eq. (21) demands that one of the Yukawas must be zero in order to obtain zero mass eigenvalue. For this reason we discard this particular form of $V_{2}(\Phi)$.

Case 2: $\sin 3\beta=0$

However, there is a second and more attractive possibility. From the characteristic equation, Eq. (21), one can see that zero eigenvalue can also be ensured if

[TABLE]

Discarding the trivial solution $\beta=0$, we obtain the solution $\beta=\frac{1}{3}\pi$ which implies that $\tan\beta=\surd 3$ i.e., $v_{2}=\surd 3v_{1}=\surd 3v/2$.444While this VEV alignment is useful for our consequent discussions we note that in case of a three Higgs doublet model with $S_{3}$ symmetry, the minimisation of potential leads to a vev alignment $v_{1}=\surd 3v_{2}$ and such alignment implies a residual $\mathbb{Z}_{2}$ symmetry [38]. In the present case, however, no such implications are possible. Looking at Eq. (23) now, we see that this value of $\beta$ also makes the matrix $M^{2}$ block-diagonal, and one obtains

[TABLE]

Notice that the third generation has been singled out, and therefore $v\sqrt{(a^{2}+b^{2})/2}$ can be readily identified with the mass of the third generation quark. In order that it be much heavier than the quarks in the first two generations, we need

[TABLE]

in both up and down sectors.

Complete diagonalization would require a further similarity transformation affecting the upper $2\times 2$ block. This will involve the values of the Yukawa couplings. Thus, we obtain

[TABLE]

where

[TABLE]

with

[TABLE]

From Eq. (28) the CKM matrix can now be written as,

[TABLE]

Thus the difference $\theta_{u}-\theta_{d}$, which can be identified with the Cabibbo angle, $\theta_{C}$.

In passing, we make a point about the VEV alignment, i.e., the value of $\beta$, dictated by Eq. (25). It reflects our choice of the representation for $S_{3}$. Had we chosen a different representation, the value of $\beta$ would in general be different. But the physical implications should be independent of the representation, and so the block-diagonal form of the CKM matrix would still result.

Having reproduced the leading order effects of the mixing matrix in the Wolfenstein parametrization as a consequence of the masslessness of the first generation quarks, we now explore whether one can do better. So far, the conclusions that we derived came from Eq. (25), which is a statement about the relative magnitude of the VEVs of the two Higgs doublets. Note that this relation is not protected by any symmetry. Suppose we deviate from Eq. (25) by a small amount such that

[TABLE]

Since $\delta$ is expected to be small, we do not expect the heavier quark masses to be altered very much by this change. The sums of eigenvalues etc. will also not change appreciably. The only thing that will change dramatically is the product of all eigenvalues, which should be the $x$-independent term in Eq. (21). Therefore the first generation quark masses will be given, in the notation of Eq. (20), by

[TABLE]

where in the last step we have used the hierarchy mentioned in Eq. (27). Now, using Eq. (30), we can write

[TABLE]

Since $\theta_{u}-\theta_{d}=\theta_{C}$, Eqs. (34a) and (34b) can be solved for $\delta$ and $\theta_{u}$ or $\theta_{d}$. Taking all the uncertainties into account we have found $\delta>0.2$ which is inconsistent with our assumption of small $\delta$ in Eq. (32). Therefore, this minimal framework is not sufficient to reproduce the observed masses of the first generation quarks.

Now, for completeness, we comment on the flavour changing neutral currents (FCNC) in our model. To set up the notations we first lay out the Yukawa Lagrangian for 2HDM in the following form:

[TABLE]

where we have kept the notation for the field the same as in Eq. (17) but put them in boldface font to remind ourselves that the generation indices have been suppressed. Unlike Eq. (17), here we also take into account the Yukawa Lagrangian for the down sector too. Here $\Delta_{1,2}$ and $\Gamma_{1,2}$ represent the Yukawa matrices in the up and down sectors respectively. By comparing Eqs. (35) and (17) we can write,

[TABLE]

and the $\Gamma_{k}$’s can be obtained by replacing the subscript $u$ by the subscript $d$ in the matrices. Now, the Yukawa Lagrangian in terms of physical fields can be written as

[TABLE]

where $D_{u}$ and $D_{d}$ are the diagonal mass matrices in the up and down sectors respectively. Note that the SM-like scalar, $h$, does not have any FCNC couplings. This is a direct consequence of the natural alignment that we have talked about earlier. The matrices $N_{u}$ and $N_{d}$, in Eq. (37), carry the information of FCNC in the up and down sectors respectively and are given by,

[TABLE]

Note that the expressions for $V_{u}$ and $V_{d}$ can be obtained from diagonalizing $\mathcal{M^{\dagger}M}$ for both up and down sectors. The matrices $\mathcal{M^{\dagger}M}$ can be obtained from $\mathcal{MM^{\dagger}}$ by making the interchange $a\leftrightarrow c$ in the Yukawa couplings. Because of this interchange, the matrix $V$ is different from $U$ in two respects. First, the matrix corresponding to $\cal U$ should have the last two rows interchanged so that the eigenvalues can occur in the same order. Second, the angle $\theta_{q}$ should be replaced by $\theta^{\prime}_{q}$, which will be given by

[TABLE]

In view of the hierarchy mentioned in Eq. (27), we can use these to write

[TABLE]

neglecting higher order terms in $b_{q}/a_{q}$. Replacing the Yukawa couplings by the mass eigenvalues and the angles $\theta_{q}$, we obtain

[TABLE]

neglecting corrections of order $m_{c}/m_{t}$. A similar expression for $N_{d}$ can be obtained from Eq. (41) by replacing $\theta_{u},m_{c},m_{t}$ with $\theta_{d},m_{s},m_{b}$ respectively. Thus the FCNCs are uniquely determined by $\theta_{u}$ or $\theta_{d}$. One should keep in mind that this represents the FCNC couplings at the leading order, i.e., when the CKM matrix is block-diagonal. In a more complete framework where the CKM matrix can be reproduced exactly these FCNC matrices are expected to get small corrections.

A trivial but viable solution to the FCNC problem would be to make all the scalars except $h$ sufficiently heavy. Moreover, the bounds from the electroweak $T$-parameter can also be evaded if the non-standard scalars, $H,A$ and $H^{\pm}$ are nearly degenerate [52, 53].

In summary, we connect two apparently disjoint experimental observations namely, the tiny masses of first generation of quarks and the near block-diagonal structure of the CKM matrix in a simple set-up of 2HDM with an $S_{3}$ symmetry. We attribute these two features of the quark sector to Nature’s choice of a particular value of $\tan\beta$. An added bonus of our model is the existence of a light scalar, which can be identified with the 125 GeV Higgs observed at the LHC, in view of a naturally emerging alignment limit. Admittedly, the exact CKM matrix and correct non-zero masses for the first generation of quarks could not be reproduced in this minimalistic scenario. Perhaps our set-up can be taken as a constituent towards a more elaborate framework which can address the full quark structure.

Acknowledgements :

DD thanks Arcadi Santamaria and Anirban Kundu for useful discussions. The work of UKD is supported by Department of Science and Technology, Government of India under the fellowship reference number PDF/2016/001087 (SERB National Post-Doctoral fellowship).