# Constraining multi-Higgs flavour models

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## Abstract

To study a flavour model with a non-minimal Higgs sector one must first define the symmetries of the fields; then identify what types of vacua exist and how they may break the symmetries; and finally determine whether the remnant symmetries are compatible with the experimental data. Here we address all these issues in the context of flavour models with any number of Higgs doublets. We stress the importance of analysing the Higgs vacuum expectation values that are pseudo-invariant under the generators of all subgroups. It is shown that the only way of obtaining a physical CKM mixing matrix and, simultaneously, non-degenerate and non-zero quark masses is requiring the vacuum expectation values of the Higgs fields to break completely the full flavour group, except possibly for some symmetry belonging to baryon number. The application of this technique to some illustrative examples, such as the flavour groups \(\Delta (27)\), \(A_4\) and \(S_3\), is also presented.

### Keywords

Scalar Potential Higgs Doublet Spontaneous Symmetry Breaking Higgs Field Quark Sector## 1 Introduction

The origin of the flavour structure of the fermion masses and mixing in the standard model (SM) remains one of the unsolved puzzles in particle physics. Several approaches to this problem have been put forward, most of them based on the use of discrete [1, 2, 3, 4] or continuous [5] flavour symmetries. In particular, mainly motivated by the measurement of several neutrino oscillation parameters, the use of discrete symmetries has recently become more popular (for recent reviews, see e.g. Refs. [6, 7, 8, 9]).

In flavour model building, one commonly chooses a flavour symmetry group \(K\) and then studies how fermion masses and mixing are constrained by this symmetry. Because of the SM gauge symmetry there is, of course, an additional global hypercharge symmetry \(U(1)_Y\) in the Lagrangian. Thus, in such studies it is important to distinguish the chosen flavour group \(K\) from the full flavour group, denoted henceforth by \(G=K\times U(1)_Y\). After spontaneous symmetry breaking (SSB) the vacuum expectation values (vevs) of the Higgs fields break completely, in many instances, \(K\) and \(G\), leaving no residual flavour symmetry in the model. But, in some rather interesting cases, the vevs could fully break \(K\) but not \(G\), and thus some residual symmetry remains. Clearly, any accidental symmetry of the Lagrangian, such as \(U(1)_B\), corresponding to baryon number, is not relevant for the analysis of the flavour sector neither any residual symmetry of \(G\) which is contained in it.

In the literature, there are studies focusing on symmetry groups exclusively applied to the Higgs sector, either with two [10, 11, 12] or more than two doublets [13, 14, 15], as well as studies focusing either on flavour symmetry groups acting in the fermion sector with assumed vev alignments [6, 7, 8, 9, 16, 17] or on the residual symmetries remaining on the fermion mass matrices [18, 19, 20, 21, 22]. Our aim is to provide a contribution towards the connection of all these subjects. In particular, we shall study in a systematic fashion the impact of a chosen flavour group on the Higgs potential and its vevs, on the Yukawa coupling matrices and thus on the residual physical properties of the mass matrices and the Cabibbo–Kobayashi–Maskawa (CKM) quark mixing matrix.

The introduction of additional scalars is a common feature to all flavour models, which brings with it a large number of possible model implementations for a given flavour group. Studying all these implementations, even for two or three Higgs doublet models, is not an easy task, specially for large flavour groups. We shall show that most of the unphysical scenarios can be identified just by analysing how scalars transform under the flavour group and how they break it. Through this analysis we shall prove that under very general conditions some choices of flavour groups are excluded for a given scalar content. Although we shall restrict our discussion to the quark sector of the theory, the results presented here can easily be extended to the lepton sector.

The paper is organised as follows. In Sect. 2, after briefly reviewing our notation and the experimental input, we analyse the invariance of the Yukawa coupling matrices under a full flavour group. Then we proceed to the formulation of a theorem which improves that in Ref. [18], by including also the constraints arising from the right-handed quark sector. A summary of the possible CKM mixing patterns and quark spectra is presented, according to the existence or not of a residual symmetry in the whole quark sector. Most importantly, by introducing the notion of pseudo-invariance of the vev under all subgroups of the flavour group, we address for the first time the problem of knowing whether such a residual symmetry is possible or not. This issue has not been addressed before in the literature, and is explained in detail in Sect. 3, where we illustrate the technique by applying it to the flavour groups \(\Delta (27)\), \(A_4\) and \(S_3\), commonly found in the literature to explain fermion masses and mixing. Finally, our concluding remarks are given in Sect. 4.

## 2 A no-go theorem for models with \(N\) Higgs doublets

### 2.1 Notation and experimental input

### 2.2 Yukawa coupling invariance under a full flavour group \(G\)

- (i)
we ignore the up-type quarks;

- (ii)
we assume that \(\Phi \) is in an irreducible representation (irrep) of \(G\);

- (iii)
we assume that there is a non-trivial group element, \(g_1\), leaving the vacuum invariant.

**Proposition**

For the down-type Yukawa terms in Eq. (1), each set of Higgs doublets \(\phi _k\), comprising an irreducible representation of \(G\), either couples to quark fields with linearly independent \(\varGamma ^k\), or decouples completely with \(\varGamma ^k=0\).

If assumption (ii) remains valid but we substitute (iii) by the assumption that the element \(g \in U(1)_B\) is trivial (in the newly defined sense), then we conclude that \(M_d\) has no symmetry, i.e., the group \(G\) has been completely broken by the vacuum. In this case, we call the residual symmetry trivial, i.e. \(G_q\subseteq U(1)_B\).

- (1)If \(G_q\subseteq U(1)_B\), there is no non-trivial symmetry left in the quark sector as a whole. More specifically,
- (a)
If \(G_{d,u} \subseteq U(1)_B \), then there is no non-trivial symmetry left in either (up- or down-) quark sector;

- (b)
If \(G_d \not \subseteq U(1)_B\) and \(G_u\subseteq U(1)_B\), then there is some non-trivial symmetry left in \(M_d\), but not in \(M_u\);

- (c)
If \(G_d \subseteq U(1)_B\) and \(G_u \not \subseteq U(1)_B\), then there is some non-trivial symmetry left in \(M_u\), but not in \(M_d\);

- (d)
If \(G_d \not \subseteq U(1)_B\) and \(G_u \not \subseteq U(1)_B\), then there is some non-trivial symmetry left in \(M_d\), some non-trivial symmetry left in \(M_u\), but there is no non-trivial symmetry left in both \(M_d\) and \(M_u\);

- (a)
- (2)
If \(G_q\not \subseteq U(1)_B\), then there is some non-trivial symmetry left in the quark sector as a whole.

### 2.3 Theorem and proof

Let us consider a model with quarks and scalar fields transforming under some set of representations of a full flavour group \(G\). We denote the representation space where \(G\) acts as the *flavour space*, i.e., the horizontal space of replicated multiplets of quark and scalar fields with the same gauge quantum numbers. The following theorem strongly constrains the viable models.

**Theorem**

*(No-Go)* Given a group \(G\) acting on the flavour space, the only way to obtain a non-block-diagonal CKM mixing matrix and, simultaneously, non-degenerate and non-zero quark masses, is that \(\left<\Phi \right>\) breaks completely the group \(G\), except possibly for some symmetry belonging to baryon number.

*Proof*

Suppose that there is a residual symmetry group \(G_q\) which is left invariant by the vev \(\left<\Phi \right>\). Let us denote a generic element of \(G_q\) as \(g_1\), and then Eq. (12) is assumed.^{1} We want to show that either \(g_1=e\), i.e., the trivial element in \(G\), or \(g_1\) acts on the flavour space as a member of the baryon number symmetry.

Finally, we can analyse Eqs. (25), (26) and (29) jointly. If \(\Phi \) is in a faithful representation, then \(g_1=e\) and no residual symmetry is present. In this case, \(e^{i\theta }=1\) in Eq. (25). If \(g_1\ne e\) (\(e^{i\theta }\ne 1\)), then \(\Phi \) is unfaithful and some residual symmetry will be present in the final Lagrangian, without constraining the mixing. This residual symmetry should be specifically represented by Eqs. (25), (26) and (29), which is just part of the baryon number conservation. This completes the proof of the theorem, which generalises that in Ref. [18], by including the constraints on the right-handed quark sector. Without the latter constraints, theories leading to unphysical massless quarks would not be precluded.^{2} \(\square \)

At this point two remarks are in order. First, any SM-like Lagrangian, as the one in Eq. (1), exhibits automatic conservation of the baryon number \(U(1)_B\), independently of additional gauge or flavour symmetries. Such accidental symmetry imposes no constraints on masses and mixing, and remains conserved after electroweak symmetry breaking. The proof above shows that a non-block-diagonal \(V_{CKM}\) is compatible with some residual symmetry inside \(G\) only if the latter is a subgroup belonging to \(U(1)_B\). Second, we should stress that Eq. (29) applies only to the Higgs doublets that couple to quarks. The Higgs doublets that appear solely in the scalar potential are important when analysing possible vacuum alignments, but they are irrelevant for the statement of the theorem.

Notice that no information of the scalar potential and vevs has been used in this proof. That is, the theorem constrains residual symmetries and can be applied to virtually any model. Nevertheless, our interest in this article lies on NHDM, and, as we will show in Sect. 3, the explicit transformation properties of the Higgs scalars, their vevs, and their relation to the subgroup chains are crucial, in particular, for the application of the theorem to specific symmetries of complete models of scalars and fermions.

### 2.4 Some illustrative examples

- (i)
\(Q_L\) and \(\Phi \) faithful: \(\ker D(Q_L) =\ker D(\Phi )=\{e\}\) A direct application of the theorem implies that if \(\left<\Phi \right>\) does not break \(G\) completely, then there is a residual symmetry in the fermion sector leading to a block-diagonal mixing.

- (ii)
\(Q_L\) unfaithful and \(\Phi \) faithful: \(\ker D(Q_L)\ne \{e\}\) and \(\ker D(\Phi )=\{e\}\) The full symmetry in the left sector is smaller than the one in the scalar sector and, in principle, we could have the proper subgroup \(\ker D(Q_L)\) of \(G\) (or smaller) unbroken by \(\left<\Phi \right>\). However, Eq. (29) should hold and \(\left<\Phi \right>\) should break \(G\) completely. As an example, let us take \(G=A_4\) with \(Q_L\sim (\varvec{1},\varvec{1}',\varvec{1}'')\) and \(\Phi \sim d_R\sim u_R\sim \varvec{3}\). If \(\left<\Phi \right>\sim (1,0,0)\), the \(\mathbb {Z}_2\) subgroup generated by \(g_1=\mathrm {diag}\,(1,-1,-1)\) is conserved and is contained in \(\ker D(Q_L)\). The representation \(\mathcal {G}_{Lg_1}={1}\mathrm{l}_3\) satisfies Eq. (25), but Eqs. (26) and (29) are not valid as \(\mathcal {G}_{g_1}=\mathcal {G}_{Rg_1}^d=\mathcal {G}_{Rg_1}^u=\mathrm {diag}\,(1,-1,-1)\). Therefore, we end up with a non-trivial residual symmetry, but the invertibility of \(M_u\) and \(M_d\) assumed in Eq. (26) is lost.

- (iii)
\(Q_L\) faithful and \(\Phi \) unfaithful: \(\ker D(Q_L)=\{e\}\) and \(\ker D(\Phi ) \ne \{e\}\) This case is automatically excluded unless \(\ker D(\Phi )\) acts like baryon number on quarks. The full symmetry of the potential is \(G/\ker D(\Phi )\) and \(\left<\Phi \right>\) can never break \(\ker D(\Phi )\). As an example, let us take \(G=A_4\) with \((\phi _1,\phi _2,\phi _3)\sim (\varvec{1},\varvec{1}',\varvec{1}'')\) and \(Q_L\sim d_R\sim u_R\sim \varvec{3}\). If all \(\phi _k\) get non-zero vevs, they only break \(A_4\) to \(\mathbb {Z}_2\times \mathbb {Z}_2\), which corresponds to the kernel of the representation of \(\Phi \), and is generated by \(\mathrm {diag}\,(1,-1,-1)\) and \(\mathrm {diag}\,(-1,-1,1)\). This subgroup remains in the \(Q_L\) quark sector and the CKM matrix would be trivial.

- (iv)
\(Q_L\) and \(\Phi \) unfaithful: \(\ker D(Q_L) \ne \{e\}\) and \(\ker D(\Phi )\) \(\ne \{e\}\) This case can be discarded given the Yukawa structure in Eq. (1) and the assumption that at least one of the representations for \(Q_L,\Phi ,d_R,u_R\) is a faithful irrep or contains a faithful irrep (i.e. \(G\), and not a smaller group, is the full flavour symmetry); see also Appendix C.

We shall comment on case (1d) of Sect. 2.2, which is the most predictive and hence one of the most common approaches in model building, since each quark sector has its own non-trivial residual symmetry while the full flavour group \(G\) is completely broken. This will lead, in general, to strong constraints to the mixing angles and the CP phase. For instance, in Ref. [23], the authors studied which would be the best residual symmetries in order to accommodate the experimental CKM mixing matrix. This was done in the context of the von Dyck flavour groups [21, 22]. It is shown that the flavour groups \(D_N\) and \(\Delta (6N^2)\), with \(N\) an integer multiple of 7, are capable of determining the mixing angle \(\theta _{12}\) close to its experimental value, if we fix \(\theta _{23}=\theta _{13}=0\). Non-zero values for all three mixing angles require, on the other hand, infinite von Dyck groups.

Possible CKM mixing patterns and quark spectra depending on the presence of global residual symmetries, modulo \(U(1)_B\). The Yes/No distinguishes if one/two quarks of the same type are massless

Residual symmetry | Non-zero mixing angles | CP violation | Quark spectrum |
---|---|---|---|

Yes | \({\le }1\) | No | All masses \(\ne 0\) |

Yes | \({\le }3\) | Yes/No | Some masses \(= 0\) |

No | \({\le }3\) | Yes | All masses \(\ne 0\) |

## 3 Application of the theorem to flavour models

The no-go theorem previously demonstrated strongly constrains viable flavour models based on a full flavour symmetry \(G\) of the Lagrangian. However, it is crucial for its application that one can determine the symmetry properties of the vevs. In this section, we shall perform this task by introducing the notion of pseudo-invariance of the vevs under subgroups of the original group. We illustrate the technique by applying it to some examples of flavour groups commonly found in the literature to explain fermion masses and mixing.

Before proceeding with the examples, it is worth recalling the distinction between what is meant by “full flavour group” and by simply “flavour group”. As said before, in model building, we may add symmetries that act only on the flavour space of the model. Those symmetries are usually taken to be subgroups of \(SU(N)\) (in particular \(SU(3)\), when we are dealing with three fermion generations). We refer to these symmetries as flavour symmetries \(K\). However, the full flavour symmetry group \(G\) is, in general, larger since it contains the global hypercharge transformation, i.e. \(G=K\times U(1)_Y\).^{3} The accidental baryon symmetry \(U(1)_B\) is not included in the full flavour symmetry \(G\) because the Higgs fields do not transform under it, but it might happen that \(G\) intersects \(U(1)_B\). We further assume that there is no additional accidental symmetry in the Yukawa Lagrangian (1), apart from \(G\) and \(U(1)_B\) (see Refs. [24, 25] for methods for detecting them).

We remark that Eq. (31) (and consequently Eq. (30)) is merely an eigenvalue equation. Therefore, from a simple group-theoretical method, and without analysing the scalar potential, we can extract a set of vacuum alignments that will be automatically excluded by the theorem, namely those corresponding to the eigenvectors of \(\mathcal {G}_{\tilde{g}_1}\). Notice, however, that the vev alignments obtained through this procedure may not be global minima of the scalar potential. Yet, this is a straightforward group-theoretical check in the spirit familiar to model building. One could instead follow a geometrical method [14] to find first the global minima of the scalar potential. In the latter case, if all the minima preserve some subgroup of the initial symmetry, the theorem applies directly without the need of solving the eigenvalue equation (31). The drawback is that the geometrical approach is not universal, and it is not guaranteed that it could easily be applied to more complicated Higgs sectors, for example to some high symmetry groups in 4HDM.

### 3.1 The flavour group \(\Delta (27)\) in 3HDM

^{4}Therefore, for the flavour group \(\Delta (27)\) all the global minima of the potential are excluded by the theorem. Said otherwise, one cannot construct a viable model based on \(\Delta (27)\) where \(\Phi \) is in a faithful (triplet) irrep.

### 3.2 The flavour group \(A_4\) in 3HDM

### 3.3 The flavour group \(S_3\) in 3HDM

## 4 Conclusions

In this work we have studied the connection between the breaking of flavour symmetries by the Higgs vevs and the existence of residual symmetries in the quark mass matrices. We have performed two tasks. First, in order to avoid nonphysical quark masses and mixing, we have developed a simple but powerful no-go theorem, highlighting the importance of including the transformation properties of the right-handed fields. Second, we have shown that, in many instances, exploring the eigenspaces of all the subgroups of the original flavour group is sufficient to study all the relevant vevs. The vevs of the scalars coupled to the quarks have to break the full flavour symmetry (or break it into a subgroup acting as baryon number) in order to avoid nonphysical quark masses and mixing.

In this context, the notion of full flavour group turns out to be important. While, in general, one refers to the flavour group as the group added to the SM acting globally on the flavour space, the SM gauge group already contains the global hypercharge transformation that should also be taken into account. The inclusion of this additional flavour transformation builds what we call the full flavour group. Then the problem of finding the vevs left invariant by the full flavour group and, therefore, excluded by the theorem, turns into the problem of finding all pseudo-invariant vevs of the flavour group, discussed here for the first time.

As we have shown through some examples, it is possible to find a set of excluded vev alignments (minima or not of the scalar potential) just by determining the eigenvectors of the Higgs representation for each flavour group element. If the global minima are known, as it is the case of 3HDM with \(A_4\) or \(\Delta (27)\) symmetric potentials, one then needs only to check whether these minima are contained in the set of excluded alignments. For 3HDM with \(A_4\) (or \(S_4\)) and \(\Delta (27)\) symmetric potentials, this is indeed verified and thus these models are excluded. In this case, a phenomenologically viable description of the quark sector requires that (1) the symmetry is explicitly broken by new interaction terms, or (2) higher-order interaction terms in the Higgs potential are present, which could then lead to the complete breaking of the \(A_4\) or \(\Delta (27)\) group upon minimisation of the potentials, or (3) additional non-invariant scalar multiplets are added to the theory. In more realistic but non-minimal models, one or more of these options are necessarily implemented. The same type of analysis can be carried out for other groups and, in particular, for smaller groups which have a more complex scalar potential.

While we only treated quarks in our discussion, the extension of the theorem to the whole fermion sector, including leptons, is straightforward. If neutrinos are Dirac-type particles the analogy is direct. There is the experimental possibility of a massless neutrino, however, as shown in our analysis, all cases with CP violation always imply a massless fermion in each sector. The case where neutrinos are Majorana-type particles is more interesting since there is a larger number of possible implementations. Lepton number is no longer conserved and thus the theorem will be slightly modified.

*Note added:* While our paper was undergoing the review process, Ref. [31] appeared, in which a classification of lepton mixing matrices is performed based on the assumption that the residual symmetries in the charged-lepton and neutrino mass matrices originate from a finite flavour symmetry group.

## Footnotes

- 1.
Here we only consider Higgs doublets that couple to quarks.

- 2.
We give one such example at the end of Sect. 3.2.

- 3.
The center of \(SU(N)\) is \(Z(SU(N))=\mathbb {Z}_N\), which is already included in the \(U(1)\) global transformations of \(G\). To avoid redundancy we may work with the projective group \(PSU(N)=SU(N)/\mathbb {Z}_N\), and define the flavour symmetry as a subgroup of \(PSU(N)\) instead.

- 4.
We do not include minima which differ by an (irrelevant) overall phase. For example, one could have \((\omega , \omega ^2, 1)\), but this minimum equals the phase \(\omega \) multiplied by \((1, \omega , \omega ^2)\), which is already taken into account.

## Notes

### Acknowledgments

J.P.S. is grateful to John Jones for clarifications about the notation in Ref. [27]. We are grateful to Yossi Nir for enlightening discussions regarding Ref. [18]. The work of R.G.F. and J.P.S. was partially supported by FCT—*Fundação para a Ciência e a Tecnologia*, under the projects PEst-OE/FIS/UI0777/2013 and CERN/FP/123580/2011, and by the EU RTN Marie Curie Project PITN-GA-2009-237920. The work of C.C.N. was partially supported by Brazilian CNPq and Fapesp. The work of I.P.I. is supported by the RF President grant for scientific schools NSc-3802.2012.2, and the Program of Department of Physics SC RAS and SB RAS “Studies of Higgs boson and exotic particles at LHC”. The work of H.S. is funded by the European FEDER, Spanish MINECO, under the grant FPA2011-23596, and the Portuguese FCT project PTDC/FIS-NUC/0548/2012.

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