Exploring the Quark Flavour Puzzle within the 3 Higgs Model

TL;DR

This paper extends the Standard Model with two extra Higgs doublets and uses discrete flavor symmetries to naturally explain the quark mass hierarchy and mixing, predicting potentially observable flavor-changing neutral currents at the LHC.

Contribution

It introduces a novel flavor symmetry framework based on $Z_7$ that naturally aligns quark masses and mixings within a three-Higgs-doublet model.

Findings

Identifies specific $Z_7$ charge configurations that produce realistic quark masses and CKM matrix.

Shows some solutions predict light flavor-violating Higgs accessible at LHC.

Demonstrates that accidental global symmetries are necessarily anomalous in this setup.

Abstract

We extend the standard model with two extra Higgs doublets. Making use of a symmetry principle, we present flavour symmetries based on cycle groups $Z_N$ that oblige each Higgs doublet to contribute to the mass of only one generation, thus accommodating the quark mass hierarchy. We systematically search for all charge configurations that naturally lead to the alignment in flavour space of the quark sectors, resulting in a CKM matrix near to the identity, determined by the quark mass hierarchy, and with the correct overall phenomenological features. The minimal realisation is by the group $Z_7$. We show that only a limited number of solutions exist, and that any accidental global symmetry that may occur together with the discrete symmetry is necessarily anomalous. A phenomenological study of each class of solutions concerning predictions to the flavour changing neutral current (FCNC) phenomena is also performed: for some solutions, it is possible to obtain realistic quark masses and mixing, while the flavour violating neutral Higgs are sufficient lighter to be accessible at LHC.

Tables

Table 1. Table 1 : The table shows the viable configurations for the right-handed down-quark d R i subscript subscript 𝑑 𝑅 𝑖 {d_{R}}_{i} and their corresponding Γ 1 subscript Γ 1 \Gamma_{1} , Γ 2 subscript Γ 2 \Gamma_{2} , Γ 3 subscript Γ 3 \Gamma_{3} and M d subscript 𝑀 𝑑 M_{d} matrices. It is understood that, for each pattern and coupling, the parameters expressed here by the same symbol, are in fact different, but denoting he same order of magnitude, (or possibly smaller). E.g. in pattern I, coupling Γ 1 subscript Γ 1 \Gamma_{1} , the three δ 𝛿 \delta , δ 𝛿 \delta , δ 𝛿 \delta , stand for δ 1 subscript 𝛿 1 \delta_{1} , δ 2 subscript 𝛿 2 \delta_{2} , δ 3 subscript 𝛿 3 \delta_{3} . The same applies to the ε 𝜀 \varepsilon ’s and c 𝑐 c ’s. For patterns IV, VII, and X, which will be excluded, one of the couplings in Γ 1 subscript Γ 1 \Gamma_{1} turns out to be much larger.

Pattern	$𝒬\left(d_R^{}{}_i{}^{}\right)$	${\Gamma }_1$	${\Gamma }_2$	${\Gamma }_3$	$M_d$
I	$(0,0,0)$
II	$(0,0,1)$
III	$(0,0,-3)$
IV	$(0,0,-2)$
V	$(0,1,0)$
VI	$(0,-3,0)$
VII	$(0,-2,0)$
VIII	$(1,0,0)$
IX	$(-3,0,0)$
X	$(-2,0,0)$

Table 2. Table 2 : A numerical example of a Yukawa coupling configuration for each pattern that gives the correct hierarchy among the quark masses and mixing.

Pattern	$v_1{Y}_1$	$v_2{Y}_2$	$v_3{Y}_3$	$M_d$
I
II
III
V
VI
VIII
IX

Table 3. Table 3 : For all allowed patterns, we find that the matrices N 1 d − D d subscript superscript 𝑁 𝑑 1 subscript 𝐷 𝑑 N^{d}_{1}-D_{d} and N 2 d subscript superscript 𝑁 𝑑 2 N^{d}_{2} are proportional to the following paterns, where λ 𝜆 \lambda is the Cabibbo angle.

Pattern	$\left(N_1^d-D_d\right)\sim$	$N_2^d\sim$
I
II
III
IV
V
VI
VII
VIII
IX
X

Equations

$$-{\mathcal{L}}_{\text{Y}}=({\Omega }_a{)}_{ij}{\overline{Q}}_{Li}{\tilde{\varphi }}_a{u}_{R_j}+({\Gamma }_a{)}_{ij}{\overline{Q}}_{Li}{\varphi }_a{d}_{R_j}+h.c.,$$

$$M_u$$

$$M_d$$

$$⟨{\varphi }_1⟩=\frac{v_1}{\sqrt{2}},⟨{\varphi }_2⟩=\frac{v_2{e}^{i{\alpha }_2}}{\sqrt{2}},⟨{\varphi }_3⟩=\frac{v_3{e}^{i{\alpha }_3}}{\sqrt{2}},$$

$$v_1\sim v_2\sim v_3.$$

$$v_1\left|({\Omega }_1{)}_{ij}\right|$$

$$v_1\left|({\Gamma }_1{)}_{ij}\right|$$

$$\left|({\Omega }_1{)}_{ij}\right|$$

$$\left|({\Gamma }_1{)}_{ij}\right|$$

$${\Omega }_1=\left(\begin{array}{ccc}\mathsf{x}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),{\Omega }_2=\left(\begin{array}{ccc}0& 0& 0\\ 0& \mathsf{x}& 0\\ 0& 0& 0\end{array}\right),{\Omega }_3=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& \mathsf{x}\end{array}\right).$$

$${\Gamma }_1=\left(\begin{array}{ccc}\mathsf{x}& \mathsf{x}& \mathsf{x}\\ \mathsf{x}& \mathsf{x}& \mathsf{x}\\ \mathsf{x}& \mathsf{x}& \mathsf{x}\end{array}\right),{\Gamma }_2=\left(\begin{array}{ccc}0& 0& 0\\ \mathsf{x}& \mathsf{x}& \mathsf{x}\\ \mathsf{x}& \mathsf{x}& \mathsf{x}\end{array}\right),{\Gamma }_3=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ \mathsf{x}& \mathsf{x}& \mathsf{x}\end{array}\right).$$

$$\chi \to {\chi }^{\prime }=e^{i\mathcal{Q}(\chi )\frac{2\pi }{N}}\chi ,$$

$$\mathcal{Q}({Q_L}_i)$$

$$\mathcal{Q}({u_R}_i)$$

$$\mathcal{Q}({Q_L}_i)=\mathcal{Q}({\varphi }_i)$$

$$\mathcal{Q}({{\overline{Q}}_L}_i{\varphi }_1{d_R}_j)=\left(\begin{array}{ccc}{d}_1& d_2& d_3\\ d_1-1& d_2-1& d_3-1\\ d_1+2& d_2+2& d_3+2\end{array}\right),$$

$$\mathcal{Q}({{\overline{Q}}_L}_i{\varphi }_2{d_R}_j)=\left(\begin{array}{ccc}{d}_1+1& d_2+1& d_3+1\\ d_1& d_2& d_3\\ d_1+3& d_2+3& d_3+3\end{array}\right),$$

$$\mathcal{Q}({{\overline{Q}}_L}_i{\varphi }_3{d_R}_j)=\left(\begin{array}{ccc}{d}_1-2& d_2-2& d_3-2\\ d_1-3& d_2-3& d_3-3\\ d_1& d_2& d_3\end{array}\right),$$

$$A_{3}\equiv\frac{1}{2}\sum_{i=1}^{3}\biggl{(}2X({Q_{L}}_{i})-X({u_{R}}_{i})-X({d_{R}}_{i})\biggr{)}=0,$$

$$A_{2}\equiv\frac{1}{2}\sum_{i=1}^{3}\biggl{(}3X({Q_{L}}_{i})+X({\ell_{L}}_{i})\biggr{)}=0,$$

$$A_{1}\equiv\frac{1}{6}\sum_{i=1}^{3}\biggl{(}X({Q_{L}}_{i})+3X({\ell_{L}}_{i})-8X({u_{R}}_{i})-2X({d_{R}}_{i})-6X({e_{R}}_{i})\biggr{)}=0,$$

$$V(\varphi )=\sum _i\left[-{\mu }_i^2{\varphi }_i^{†}{\varphi }_i+{\lambda }_i({\varphi }_i^{†}{\varphi }_i{)}^2\right]+\sum _{i<j}\left[+C_i({\varphi }_i^{†}{\varphi }_i)({\varphi }_j^{†}{\varphi }_j)+{\bar{C}}_i{\left|{\varphi }_i^{†}{\varphi }_j\right|}^2\right],$$

$$V{}^{†}{M}_{d}W=diag(m_d,m_s,m_b)⟹M_d=Vdiag(m_d,m_s,m_b)W^{†}$$

$$\lambda \equiv \frac{|V_{us}|}{\sqrt{|V_{us}{|}^2+|V_{ud}{|}^2}},A\equiv \frac{1}{\lambda }\left|\frac{V_{cb}}{V_{us}}\right|,$$

$$\overline{\rho }+i\overline{\eta }\equiv -\frac{V_{ud}^{}{V}_{ub}^{\ast }}{V_{cd}^{}{V}_{cb}^{\ast }}$$

$$\begin{array}[]{l}\sqrt{\frac{m_{d}}{m_{s}}}=\sqrt{{\frac{k_{d}}{k_{s}}}}\ \lambda\\ \\ \frac{m_{s}}{m_{b}}=k_{s}\ \lambda^{2}\end{array}\quad\Longrightarrow\quad\begin{array}[]{l}m_{d}=k_{d}\ \lambda^{4}m_{b}\\ \\ m_{s}=k_{s}\ \lambda^{2}m_{b}\end{array}$$

$$v_{3}=\frac{1}{n_{3}}\left(\begin{array}[]{r}\frac{m_{d}}{m_{b}}V_{11}\\ \frac{m_{s}}{m_{b}}V_{12}\\ V_{13}\end{array}\right)\times\left(\begin{array}[]{r}\frac{m_{d}}{m_{b}}V_{31}\\ \frac{m_{s}}{m_{b}}V_{32}\\ V_{33}\end{array}\right)$$

$$v_{1}=\frac{1}{n_{1}}\left(\begin{array}[]{r}\frac{m_{d}}{m_{b}}V_{31}\\ \frac{m_{s}}{m_{b}}V_{32}\\ V_{33}\end{array}\right)\times v_{3}^{\ast}$$

$$M_{d}=m_{b}\ \left(\begin{array}[]{ccc}-k_{d}\ \lambda^{3}&\left(\overline{\rho}-i\,\overline{\eta}\right)\ A\ \lambda^{3}&0\\ -k_{d}\ \lambda^{2}&A\ \lambda^{2}&-k_{s}\ \lambda^{3}\\ 0&1&0\end{array}\right)$$

$$m_u$$

$$m_c$$

$$\lambda$$

$$\overline{\rho }$$

$$-{\mathcal{L}}_{\text{Neutral Yukawa}}=$$

$$+\frac{1}{v^{\prime }}\overline{u_L}{N}_1^u\left(R_1-i{I}_1\right)u_R+\frac{1}{v^{\prime \prime }}\overline{d_L}{N}_2^d\left(R_2+i{I}_2\right)d_R$$

$$+\frac{1}{v^{\prime \prime }}\overline{u_L}{N}_2^u\left(R_2-i{I}_2\right)u_R+h.c.$$

$$N_1^d=\frac{1}{\sqrt{2}}{V}^{†}\left(v_2{\Gamma }_1-v_1{e}^{i{\alpha }_2}{\Gamma }_2\right)W,$$

Full text

I Introduction

The discovery of the Higgs boson in 2012 at the LHC has attested the success of the standard model (SM) in describing the observed fermions and their interactions. However, there exist many theoretical issues or open questions that have no satisfactory answer. In particular, the observed flavour pattern lacks of a definitive explanation, i.e., the quark Yukawa coupling matrices $Y_{u}$ and $Y_{d}$, which in the SM reproduce the six quark masses, three mixings angles and a complex phase to account for CP violation phenomena, are general complex matrices, not constrained by any gauge symmetry.

Experimentally the flavour puzzle is very intricate. First, there is the quark mass hierarchy in both sectors. Secondly, the mixings in the SM, encoded in the Cabibbo-Kobayashi-Maskawa (CKM) unitary matrix, turns out to be close to the identity matrix. If one takes also the lepton sector into account, the hierarchy there is even more puzzling Emmanuel-Costa:2015tca . On the other hand, in the SM there is in general no connection between the quark masses hierarchy and the CKM mixing pattern. In fact, if one considers the Extreme Chiral Limit, where the quark masses of the first two generations are set to zero, the mixing does not necessarily vanish Botella:2016krk , and one concludes that the CKM matrix $V$ being close to the identity matrix has nothing to do with the fact that the quark masses are hierarchical. Indeed, in order to have $V\approx\mathbf{1}$, one must have a definite alignment of the quark mass matrices in the flavour space, and to explain this alignment, a flavour symmetry or some other mechanism is required Botella:2016krk .

Among many attempts made in the literature to address the flavour puzzle, extensions of the SM with new Higgs doublet are particularly motivating. This is due to fact that the number of Higgs doublets is not constrained by the SM symmetry. Moreover, the addition of scalar doublets gives rise to new Yukawa interactions and as a result it provides a richer framework in approaching the theory of flavour. On the other hand, any new extension of the Higgs sector must be very much constrained, since it naturally leads to flavour changing neutral currents. At tree level, in the SM, all the flavour changing transitions are mediated through charged weak currents and the flavour mixing is controlled by the CKM matrix Cabibbo:1963yz ; Kobayashi:1973fv . If new Higgs doublets are added, one expects large FCNC effects already present at tree level. Such effects have not been experimentally observed and they constrain severely any model with extra Higgs doublets, unless a flavour symmetry suppresses or avoids large FCNC Branco:2011iw .

Minimal flavour violating models Joshipura:1990pi ; Antaramian:1992ya ; Hall:1993ca ; Mantilla2017 ; Buras:2001 ; Dambrosio:2002 are examples of a multiHiggs extension where FCNC are present at tree-level but their contributions to FCNC phenomena involve only off-diagonal elements of the CKM matrix or their products. The first consistent models of this kind were proposed by Branco, Grimus and Lavoura (BGL) Branco:1996bq , and consisted of the SM with two Higgs doublets together with the requirement of an additional discrete symmetry. BGL models are compatible with lower neutral Higgs masses and FCNC’s occur at tree level, with the new interactions entirely determined in terms of the CKM matrix elements.

The goal of this paper is to generalize the previous BGL models and to, systematically, search for patterns where a discrete flavour symmetry naturally leads to the alignment of the flavour space of both the quark sectors. Although the quark mass hierarchy does not arise from the symmetry, the effect of both is such that the CKM matrix is near to the identity and has the correct overall phenomenological features, determined by the quark mass hierarchy, Branco:2011aa . To do this we extend the SM with two extra Higgs doublets to a total of three Higgs $\phi_{a}$. The choice for discrete symmetries is to avoid the presence of Goldstone bosons that appear in the context of any global continuous symmetry, when the spontaneous electroweak symmetry breaking occurs. For the sake of simplicity, we restrict our search to the family group $Z_{N}$, and demand that the resulting up-quark mass matrix $M_{u}$ is diagonal. This is to say that, due to the expected strong up-quark mass hierarchy, we only consider those cases where the contribution of the up-quark mass matrix to quark mixing is negligible.

If one assumes that all Higgs doublets acquire vacuum expectation values with the same order of magnitude, then each Higgs doublet must couple to the fermions with different strengths. Possibly one could obtain similar results assuming that the vacuum expectation values (VEVs) of the Higgs have a definite hierarchy instead of the couplings, but this is not considered here. Combining this assumption with the symmetry, we obtain the correct ordered hierarchical pattern if the coupling with $\phi_{3}$ gives the strength of the third generation, the coupling with $\phi_{2}$ gives the strength of the second generation and the coupling with $\phi_{1}$ gives the strength of the first generation. Therefore, from our point of view, the three Higgs doublets are necessary to ensure that there exists three different coupling strengths, one for each generation, to guarantee simultaneously an hierarchical mass spectrum and a CKM matrix that has the correct overall phenomenological features e.g. $\left|V_{cb}\right|^{2}+\left|V_{ub}\right|^{2}=O(m_{s}/m_{b})^{2}$,  and denoted here by $V\approx\mathbf{1.}$

Indeed, our approach is within the BGL models, and such that the FCNC flavour structure is entirely determined by CKM. Through the symmetry, the suppression of the most dangerous FCNC’s, by combinations of the CKM matrix elements and light quark masses, is entirely natural.

The paper is organised as follows. In the next section, we present our model and classify the patterns allowed by the discrete symmetry in combination with our assumptions. In Sec. III, we give a brief numerical analysis of the phenomenological output of our solutions. In Sec. IV, we examine the suppression of scalar mediated FCNC in our framework for each pattern. Finally, in Sec. V, we present our conclusions.

II The Model

We extend the Higgs sector of the SM with two extra new scalar doublets, yielding a total of three scalar doublets, as $\phi_{1}$, $\phi_{2}$, $\phi_{3}$. As it was mentioned in the introduction, the main idea for having three Higgs doublets is to implement a discrete flavour symmetry, that leads to the alignment of the flavour space of the quark sectors. The quark mass hierarchy does not arise from the symmetry, but together with the symmetry the effect of both is such that the CKM matrix is near to the identity and has the correct overall phenomenological features, determined by the quark mass hierarchy.

Let us start by considering the most general quark Yukawa coupling Lagrangian invariant in our setup

[TABLE]

with the Higgs labeling $a=1,2,3$ and $i,j$ are just the usual flavour indexes identifying the generations of fermions. In the above Lagrangian, one has three Yukawa coupling matrices $\Omega_{1}$, $\Omega_{2}$, $\Omega_{3}$ for the up-quark sector and three Yukawa coupling matrices $\Gamma_{1}$, $\Gamma_{2}$, $\Gamma_{3}$ for the down sector, corresponding to each of the Higgs doublets $\phi_{1}$, $\phi_{2}$, $\phi_{3}$. Assuming that only the neutral components of the three Higgs doublets acquire vacuum expectation value (VEV), the quark mass $M_{u}$ and $M_{d}$ are then easily generated as

[TABLE]

[TABLE]

with $v_{1}$, $v_{2}$ and $v_{3}$ being the VEV moduli and $\alpha_{2}$, $\alpha_{3}$ just complex phases. We have chosen the VEV of $\phi_{1}$ to be real and positive, since this is always possible through a proper gauge transformation. As stated, we assume that the moduli of VEVs $v_{i}$ are of the same order of magnitude, i.e.,

[TABLE]

Each of the $\phi_{a}$ couples to the quarks with a coupling $(\Omega_{a})_{ij},(\Gamma_{a})_{ij}$ which we take be of the same order of magnitude, unless some element vanishes by imposition of the flavour symmmetry. In this sence, each $\phi_{a}$ and $(\Omega_{a},\Gamma_{a})$ will generate it’s own respective generation: i.e., our model is such that by imposition of the flavour symmmetry, $\phi_{3}$, $\Omega_{3}$, $\Gamma_{3}$ will generate $m_{t}$ respectivelly $m_{b}$, that $\phi_{2}$, $\Omega_{2}$, $\Gamma_{2}$ will generate $m_{c}$ respectivelly $m_{s}$, and that $\phi_{1}$, $\Omega_{1}$, $\Gamma_{1}$ will generate $m_{u}$ respectivelly $m_{d}$. Generically, we have

[TABLE]

[TABLE]

Next, we focus on the required textures for the Yukawa coupling matrices $\Omega_{a}$ and $\Gamma_{a}$ that naturally lead to an hierarchical mass quark spectrum and at the same time to a realistic CKM mixing matrix. These textures must be reproduced by our choice of the flavour symmetry. As referred in the introduction, we search for quark mass patterns where the mass matrix $M_{u}$ is diagonal. Therefore, one derives from Eqs. (2a), (6a) the following textures for $\Omega_{a}$

[TABLE]

The entry $\mathsf{x}$ means a non zero element. In this case, the up-quark masses are given by $m_{u}=v_{1}\left|(\Omega_{1})_{11}\right|$, $m_{c}=v_{2}\left|(\Omega_{2})_{22}\right|$ and $m_{t}=v_{3}\left|(\Omega_{3})_{33}\right|$.

Generically, the down-quark Yukawa coupling matrices must have the following indicative textures

[TABLE]

We distinguish rows with bold $\boldsymbol{\mathsf{x}}$ in order to indicate that it is mandatory that at least one of matrix elements within that row must be nonvanishing. Rows denoted with $\mathsf{x}$ may be set to zero, without modifying the mass matrix hierarchy. These textures ensure that not only is the mass spectrum hierarchy respected but it also leads to the alignment of the flavor space of both the quark sectors Branco:2011aa and to a CKM matrix $V\approx\mathbf{1}$. For instance, if one would not have a vanishing, or comparatively very small, $(1,3)$ entry in the $\Gamma_{2}$, this would not necessarily spoil the scale of $m_{s}$, but it would dramatically change the predictions for the CKM mixing matrix.

In order to force the Yukawa coupling matrices $\Omega_{a}$ and $\Gamma_{a}$ to have the indicative forms outlined in Eqs. (7) and (8), we introduce a global flavour symmetry. Since any global continuous symmetry leads to the presence of massless Goldstone bosons after the spontaneous electroweak breaking, one should instead consider a discrete symmetry. Among many possible discrete symmetry constructions, we restrict our searches to the case of cycle groups $Z_{N}$. Thus, we demand that any quark or boson multiplet $\chi$ transforms according to $Z_{N}$ as

[TABLE]

where $\mathcal{Q}(\chi)\in\{0,1,\dots,N\}$ is the $Z_{N}$-charge attributed for the multiplet $\chi$.

We have chosen the up-quark mass matrix $M_{u}$ to be diagonal. This restricts the flavour symmetry $Z_{N}$. We have found that, in order to ensure that all Higgs doublet charges are different, and to have appropriate charges for fields ${Q_{L}}_{i}$ and ${u_{R}}_{i}$, we must have $N\geq 7$. We simplify our analysis by fixing $N=7$ and choose:

[TABLE]

[TABLE]

It turns out that these choices do not restrict the results, i.e. the possible textures that one can have for the $\Gamma_{i}$ matrices. Other choices would only imply that we reshuffle the charges of the multiplets.

With the purpose of enumerating the different possible textures for the $\Gamma_{i}$ matrices implementable in $Z_{7}$, we write down the charges of the trilinears $\mathcal{Q}({\overline{Q}_{L}}_{i}\phi_{a}{d_{R}}_{j})$ corresponding to each $\phi_{a}$ as

[TABLE]

We summarise in Table 1 all the allowed textures for the $\Gamma_{a}$ matrices and the resulting $M_{d}$ mass matrix texture, excluding all cases which are irrelevant, e.g. matrices that have too much texture zeros and are singular, or matrices that do not accommodate CP violation. It must be stressed, that these are the textures obtained by the different charge configurations that one can possibly choose. However, if one assumes a definite charge configuration, then the entire texture, $M_{d}$ and $M_{u}$ and the respective phenomenology are fixed. As stated, the list of textures in Table 1 remains unchanged even if one chooses any other set than in Eqs. (10), (11). As stated, that all patterns presented here are of the Minimal Flavour Violation (MFV) type Joshipura:1990pi ; Antaramian:1992ya ; Hall:1993ca ; Mantilla2017 ; Buras:2001 ; Dambrosio:2002 .

Pattern I in the table was already considered in Ref. Botella:2009pq in the context of $Z_{8}$. We discard Patterns IV, VII and X, because contrary to our starting point, at least one of three non-zero couplings with $\phi_{1}$ will turn out be of the same order as the larger coupling with $\phi_{2}$ in order to meet the phenomenological requirements of the CKM matrix.

Notice also, that the structure of other $M_{d}$’s cannot be trivially obtained, e.g. from Pattern I, by a transformation of the right-handed down quark fields.

Our symmetry model may be extended to the charged leptons and neutrinos, e.g. in the context of type one see-saw. Choosing for the lepton doublets ${L}_{i}$ the charges $\mathcal{Q}({L}_{i})=(0,-1,2)$, opposite to the Higgs doublets in Eq. (11), and e.g. for the charges $\mathcal{Q}({e_{R}}_{i})=(0,-2,4)$ of the right-handed fields ${e_{R}}_{i}$, we force the charged  lepton mass matrix to be diagonal. Then for the right-handed neutrinos ${\nu_{R}}_{i}$, choosing $\mathcal{Q}({\nu_{R}}_{i})=(0,0,0)$, we obtain for the neutrino Dirac mass matrix a pattern similar to pattern I. Of course, for this case, the heavy right-handed neutrino Majorana mass matrix is totally arbitrary. In other cases, i.e. for other patterns and charges, in particular for the right-handed neutrinos, we could introduce scalar singlets with suitable charges, which would then lead to certain heavy right-handed neutrino Majorana mass matrices.

Next, we address an important issue of the model, namely, whether accidental $U(1)$ symmetries may appear in the Yukawa sector or in the potential. One may wonder whether a continuous accidental $U(1)$ symmetry could arise, once the $Z_{7}$ is imposed at the Lagrangian level in Eq. (1). This is indeed the case, i.e., for all realizations of $Z_{7}$, one has the appearance of a global $U(1)_{X}$. However, any consistent global $U(1)_{X}$ must obey to the anomaly-free conditions of global symmetries Babu:1989ex , which read for the anomalies $SU(3)^{2}\times U(1)_{X}$, $SU(2)^{2}\times U(1)_{X}$ and $U(1)_{Y}^{2}\times U(1)_{X}$ as

[TABLE]

We also comment on the scalar potential of our model. The most general scalar potential with three scalars invariant under $Z_{7}$ reads as

[TABLE]

where the constants $\mu_{i}^{2}$, $\lambda_{i}$, $C_{i}$ and $\bar{C}_{i}$ are taken real for $i,j=1,2,3$. Analysing the potential above, one sees that it gives rise to the accidental global continuous symmetry $\phi_{i}\rightarrow e^{i\alpha_{i}}\phi_{i}$, for arbitrary $\alpha_{i}$, which upon spontaneous symmetry breaking leads to a massless neutral scalar, at tree level. Introducing soft-breaking terms like $m_{ij}^{2}\phi_{i}^{\dagger}\,\phi_{j}\,+\text{H.c.}$ can erase the problem. Another possibility without spoiling the $Z_{7}$ symmetry is to add new scalar singlets, so that the coefficients $m_{ij}^{2}$ are effectively obtained once the scalar singlets acquire VEVs.

III Numerical analysis

In this section, we give the phenomenological predictions obtained by the patterns listed in Table 1. Note that, although these patterns arrize directly from the chosen discrete charge configuration of the quark fields, one may further preform a residual flavour transformation of the right-handed down quark fields, resulting in an extra zero entry in $M_{d}$. Taking this into account, all the parameters in each pattern may be uniquely expressed in terms of down quark masses and the CKM matrix elements $V_{ij}$. This follows directly from the diagonalization equation of $M_{d}:$

[TABLE]

with $V$ being the CKM mixing matrix, since $M_{u}$ is diagonal. Because of the zero entries in $M_{d}$, it is easy to extract the right-handed diagonalization matrix $W$, completely in terms of the down quark masses and the $V_{ij}$. Thus, all parameters, modulo the residual transformation of the right-handed down quark fields, are fixed, i.e., all parameters in each pattern may be uniquely expressed in terms of down quark masses and the CKM matrix elements $V_{ij}$, including the right-handed diagonalization matrix $W$ of $M_{d}$. More precisely, all matrix elements of $V$ are written in terms of Wolfenstein real parameters $\lambda$, $A$, $\overline{\rho}$ and $\overline{\eta}$, defined in terms of rephasing invariant quantities as

[TABLE]

[TABLE]

with phenomenologically, $k_{d}$ and $k_{s}$ being factores of order one. Writing $W^{\dagger}$ in Eq. (15) as $W^{\dagger}=(v_{1},v_{2},v_{3})$, with the $v_{i}$ vectors formed by the $i$-th column of $W^{\dagger}$, we find e.g. for pattern II,

[TABLE]

where $n_{3}$ is the norm of the vector obtained from the external product of the two vectors. Taking into account the extra freedom of transformation of the right-handed fields, we may choose $M_{31}^{d}=0$, corresponding to $c_{1}=0$ in Table 1, and we conclude that

[TABLE]

Obviously, then $v_{2}=\frac{1}{n_{2}}v_{1}^{\ast}\times v_{3}^{\ast}$. This process is replicated for all patterns. Thus, $V$ and $W$, are entirely expressed in terms of Wolfenstein parameters and $k_{d}$ and $k_{s}$ of Eq. (17). These two matrices will be later used to compute the patterns of the FCNC’s in Table 3. Indeed, in this way, we find e.g. for pattern II,  in leading order order,

[TABLE]

which corresponds to the expected power series where the couplings in $\Gamma_{1}$ to the first Higgs $\phi_{1}$ are comparatively smaller than then couplings in $\Gamma_{2}$, and these smaller to the couplings in $\Gamma_{3}$. Similar results are obtained for all patterns in Table 1, except for patterns IV, VII and X, where e.g. for pattern IV, we find that the coupling in $(\Gamma_{1})_{33}$ is proportional to$\ \lambda$, which is too large and contradicts our initial assumption that all couplings in $\Gamma_{1}$ to the first Higgs $\phi_{1}$ must be smaller than the couplings in $\Gamma_{2}$ to the second Higgs $\phi_{2}$. Therefore, we exclude Patterns IV, VII and X.

We give in Table 2 a numerical example of a Yukawa coupling configuration for each pattern. We use the following quark running masses at the electroweak scale $M_{Z}$:

[TABLE]

[TABLE]

IV Predictions of flavour changing neutral currents

In the SM, flavour changing neutral currents (FCNC) are forbidden at tree level, both in the gauge and the Higgs sectors. However, by extending the SM field content, one obtains Higgs Flavour Violating Neutral Couplings Branco:2011iw . In terms of the quark mass eigenstates, the Yukawa couplings to the Higgs neutral fields are:

[TABLE]

where the $N_{i}^{u,d}$ are the matrices which give the strength and the flavour structure of the FCNC,

[TABLE]

[TABLE]

In order to satisfy experimental constraints arising from $K^{0}-\overline{K^{0}}$, $B^{0}-\overline{B^{0}}$ and $D^{0}-\overline{D^{0}}$, the off-diagonal elements of the Yukawa interactions $N_{1}^{d}$ and $N_{2}^{d}$ must be highly suppressed Botella:2014ska AndreasCrivellin2013 . For each of our 10 solutions in Table 1, we summarize in Table 3 all FCNC patterns, for each solution, and for $v_{1}=v_{2}=v_{3}$ and $\alpha_{2}=\alpha_{3}=0$. These patterns are of the BGL type, since in Eq. (25) all matrices can be expressed in terms of the CKM mixing matrix elements and the down quark masses. As explained, to obtain these patterns, we express the CKM matrix $V$ and the matrix $W$ in terms of Wolfenstein parameters.

The tree level Higgs mediated $\Delta S=2$ amplitude must be suppressed. This may allways be achieved if one chooses the masses of the flavour violating neutral Higgs scalars sufficiently heavy. However, from the experimental point of view, it would be interesting to have these masses as low as possible. Therefore, we also estimate the lower bound of these masses, by considering the contribution to $B^{0}-\overline{B^{0}}$ mixing. We choose this mixing, since for our patterns, the $(3,1)$ entry of the matrix $N_{1}^{d}$ is the less suppressed in certain cases and would require very heavy flavour violating neutral Higgses. The relevant quantity is the off-diagonal matrix element $M_{12}$, which connects the B meson with the corresponding antimeson. This matrix element, $M_{12}^{NP}$, receives contributions Botella:2014ska both from a SM box diagram and a tree-level diagram involving the FCNC:

[TABLE]

where the New Physics (NP) short distance tree level contribution to the meson-antimeson contribution is:

[TABLE]

with $v^{2}=v_{1}^{2}+v_{2}^{2}+v_{2}^{2}$ and

[TABLE]

In order to obtain a conservative measure, we have tentatively expanded the original expression in Botella:2014ska and, for the three Higgs case, included all neutral Higgs mass eigenstates.

Adopting as input values the PDG experimental determinations of $f_{B}$, $m_{B}$ and $\Delta\,m_{B}$ and considering a common VEV for all Higgs doublets, we impose the inequality $M_{12}^{NP}<\Delta m_{B}$. The following plots show an estimate of the lower bound for the flavour-violating Higgs masses for two different patterns. We plot two masses chosen from the set $\left(m_{1}^{R},m_{2}^{R},m_{1}^{I},m_{2}^{I}\right)$, while the other two are varied over a wide range. In Fig. 1, we illustrate these lower bounds for Pattern III, which are restricted by the $(3,1)$ entry of $N_{1}^{d}$ matrix and suppressed by a factor of $\lambda$. For Pattern VIII, in Fig. 2 we find the flavour violating neutral Higgs to be much lighter and possibly accessible at LHC.

V Conclusions

We have presented a model based on the SM with 3 Higgs and an additional flavour discrete symmetry. We have shown that there exist flavour discrete symmetry configurations which lead to the alignment of the quark sectors. By allowing each scalar field to couple to each quark generation with a distinctive scale, one obtains the quark mass hierarchy, and although this hierarchy does not arise from the symmetry, the effect of both is such that the CKM matrix is near to the identity and has the correct overall phenomenological features. In this context, we have obtained 7 solutions fulfilling these requirements, with the additional constraint of the up quark mass matrix being diagonal and real.

We have also verified if accidental $U(1)$ symmetries may appear in the Yukawa sector or in the potential, particularly the case where a continuous accidental $U(1)$ symmetry could arise, once the $Z_{7}$ is imposed at the Lagrangian level. This was indeed the case, however we shown that the anomaly-free conditions of global symmetries are violated. Thus, the global $U(1)_{X}$ symmetry is anomalous and therefore only the discrete symmetry $Z_{7}$ persists.

As in this model new Higgs doublets are added, one expects large FCNC effects, already present at tree level. However, such effects have not been experimentally observed. We show that, for certain of our specific implementations of the flavour symmetry, it is possible to suppress the FCNC effects and to ensure that the flavour violating neutral Higgs are light enough to be detectable at LHC. Indeed, in this respect, our model is a generalization of the BGL models for 3HDM, since the FCNC flavour structure is entirely determined by CKM.

Acknowledgements.

This work is partially supported by Fundação para a Ciência e a Tecnologia (FCT, Portugal) through the projects CERN/FP/123580/2011, PTDC/FIS-NUC/0548/2012, EXPL/FIS-NUC/0460/2013, and CFTP-FCT Unit 777 (PEst-OE/FIS/UI0777/2013) which are partially funded through POCTI (FEDER), COMPETE, QREN and EU. The work of D.E.C. is also supported by Associação do Instituto Superior Técnico para a Investigação e Desenvolvimento (IST-ID). N.R.A is supported by European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 674896. N.R.A is grateful to CFTP for the hospitality during his stay in Lisbon.