Constraining Flavour Symmetries At The EW Scale I: The A4 Higgs Potential

We consider an extension of the Standard Model in which the symmetry is enlarged by a global flavour factor A4 and the scalar sector accounts for three copies of the Standard Model Higgs, transforming as a triplet of A4. In this context, we study the most general scalar potential and its minima, performing for each of them a model independent analysis on the related phenomenology. We study the scalar spectrum, the new contributions to the oblique corrections, the decays of the Z and W, the new sources of flavour violation, which all are affected by the introduction of multiple Higgses transforming under A4. We find that this model independent approach discriminates the different minima allowed by the scalar potential.


Introduction
The current data on neutrino oscillations seem to point at one small and two large angles in the neutrino mixing matrix [1][2][3][4][5][6]. The data are consistent with various mixing patterns, where in particular the agreement with the tri-bimaximal [7,8] mixing pattern is striking [9].
The use of non-Abelian discrete flavour symmetries has been proposed in different models (for a review see [9]) to generate both the mentioned lepton mixing patterns and the quark ones. In general, in those models, one introduces so called flavons, scalar fields In this paper, we will consider the discrete flavour symmetry A 4 and we will assume that there are three copies of the Standard Model Higgs field, that transform among each other as a triplet of A 4 [10][11][12][13]. The presence of this extended Higgs sector has an deep impact on the high energy phenomenology: indeed new contributions to the oblique corrections as well as new sources of CP and flavour violation usually appear in this context. We will analyse the constraints coming from these observables for all the vacuum configurations allowed by the scalar potential and will discuss on the viability of each of them.
The structure of the paper is as follows. In section 2, we will introduce the scalar potential invariant under A 4 and under the gauge group of the Standard Model. In section 3 we will introduce the various physical Higgs fields that are present in the model.
In the subsequent two sections we will present the different minima allowed by the potential and discuss the corresponding Higgs spectrum. These minima may or may not violate CP symmetry, depending on the question whether all vacuum expectation values of the Higgses are real or not. In section 4 we will discuss the cases that do not violate the CP symmetry, while in section 5 we discuss CP breaking minima. Section 6 we will discuss bounds on the allowed parameters using respectively unitarity constraints, decays of the Z and W ± bosons and constraints by oblique corrections. We note that all these bounds are rather model independent, meaning that they depend on the flavour symmetry assignment of the relevant Higgs fields, but not on those of the fermions in the theory. Further bounds can be derived from fermion decays and meson oscillations, but these bounds are always model dependent. We will present some of these in an accompanying paper [14].
Finally, in section 7 we present the results of our analysis and in section 8 we conclude.
In the appendix A we report useful formulae for the analysis of the T, S and U parameters.

The A 4 Scalar Potential
We consider the Standard Model gauge group SU (3) c × SU (2) L × U (1) Y with the addition of a global flavour symmetry A 4 [15,16]. We consider three copies Φ a , a = 1, 2, 3, of the conventional SM Higgs field (i.e. a singlet of SU (3) c , doublet of SU (2) L and with hypercharge Y = 1/2) such that the three Higgses are in a triplet of the flavour group A 4 .
Once the flavour structure of the quarks and leptons is specified, each Φ a will couple to the three fermion families according to the group theory rules in a model dependent way.
We will study these couplings in more detail in [14].
Below, we will write down the most general scalar potential for the three Higgses that is invariant under the flavour and gauge symmetries of the model. After the fields occupy one of the minima of the potential, electroweak symmetry gets broken (while electromagnetism is conserved) and we can develop the fields around their vacuum expectation values as Here v a e iωa is the vacuum expectation value of the a th Higgs field. One or two of the v a can be zero, implying that the corresponding Higgs field does not develop a vev. Furthermore, if all vevs are real (so if all ω a are zero) CP is conserved, while if one or more ω a s are nonzero, CP is broken. Note that in general, there is the freedom to put one of the phases to zero by a global rotation.
The most general potential V [Φ a ] can be written as in agreement with the usual notation adopted in the two Higgs Doublet Models (2HDM) (for a review on this topic see [17]). The parameter µ 2 is typically negative in order to have a stable minimum away from the origin. All the other parameters, λ i , are real parameters which must undergo to the condition for a potential bounded from below: this forces λ 1 and the combination λ 1 + λ 3 + λ 4 + λ 5 cos to be positive.
It is interesting to notice that, contrary to other multi Higgs (MH) scenarios, here we can not recover the SM limit, with one light scalar and all the others decoupled and very heavy. The flavour symmetry constrains the potential parameters in such a way that the scalar masses are never independent from each other. This can be easily understood by a parameter counting: the scalar potential in eq. (2) presents 6 independent parameters and the number of the physical quantities is 8, i.e. the electroweak (EW) vev and the seven masses for the massive scalar fields.
We will study the minima of the potential in eq. (2) under electromagnetism conserving vevs as specified in eq. (1) by studying the first derivative system where Φ I is of the fields Re Φ 1 a , Re Φ 0 a , Im Φ 1 a or Im Φ 0 a and by requiring that the Hessian has non negative eigenvalues, or in other words that all the physical masses are positive except those ones corresponding to the Goldstone bosons (GBs) that vanish.
In sections 4 and 5 we will verify that this potential presents a number of solutions.
Some of them are natural in the sense that they do not require ad hoc values of the potential parameters; these are only constrained by requiring the boundness at infinity and the positivity of all the physical scalar masses. The only potential parameter constrained is the bare mass term µ 2 which is related to the physical Electroweak (EW) vev, v 2 w = v 2 1 +v 2 2 +v 2 3 . Others require specific relations between the adimensional scalar potential parameters and may have extra Goldstone bosons.

The Physical Higgs Fields
The symmetry breaking of the Higgs fields of equation eq. (1) leads to a large number of charged and neutral Higgs bosons as well as the known Goldstone bosons of the Standard Model.
In the most general case, where CP is not conserved, the neutral real and imaginary components of eq. (1) mix to five CP non-definite states and a GB: Here a = 1, 2, 3 and α = 1 − 5, while α = 6 represents the GB π 0 . In matrixform this Clearly eq. (5) holds also in the CP conserved case: in that case the 6 by 6 scalar mass matrix reduces to a block diagonal matrix with two 3 by 3 mass matrices leading to three CP even states and 2 CP odd states and the GB π 0 .
The three charged scalars mix into two new charged massive states and a charged GB.
where π + is the Goldstone boson eaten by the gauge bosons W + . In general, the S is a complex unitary matrix. In the special case where CP is conserved, its entries are real (and it is thus an orthogonal matrix).

CP Conserved Solutions
In this section, we will study minima of the potential in eq. (2) in which only Re φ 0 a develops a vev, i.e. the CP symmetry is conserved. In this case we expect having 3 neutral scalar CP-even states, 2 CP-odd states and 2 charged scalars as well as a real and a complex GBs originating from respectively the CP-odd states and the charged states.
In the CP-conserved case all the ω a vanish and the first derivative system in eq. (3) where the first three derivatives refer to the real components Φ 0 a and the second ones to the imaginary parts. In the most general case, when neither nor λ 5 is zero, the last three equations allow two different solutions 2) v 1 = 0 and v 2 = v 3 = 0 (and permutations of the indices); in this case v 1 = v w .
Both these solutions are solutions of the first three equations as well, provided that      µ 2 = −(3λ 1 + λ 3 + λ 4 + λ 5 cos )v 2 w /3 for the first case for the second case.
In this cases λ 5 can be chosen positive, as a sign can be absorbed in a redefinition of .
Next, we consider the case where sin is 0. This implies = 0 or π. We may however absorb the minus sign corresponding to the second case in a redefinition of λ 5 that is now allowed to span over both positive and negative values.
Assuming v 1 = 0, we may solve the first equation in eq. (8) with respect to µ 2 . Then by substituting µ 2 in the other two equations we get Next to the two solutions present in the general case, this system has two further possible solutions 3) v 3 = 0, v 2 = v 1 = v w / √ 2 and permutations. This requires 4) (λ 3 + λ 4 + λ 5 ) = 0. This condition implies that in the real neutral direction there is a enlarged-O(3) accidental symmetry that is spontaneously broken by the vacuum configuration, thus we xpect extra GBs. Indeed in this case v 1 , v 2 and v 3 are only restricted to satisfy v 2 1 + v 2 2 + v 2 3 = v 2 w and the parameter µ 2 is given by Finally, the case λ 5 = 0 allows special cases of the solutions 1) to 4), but does not give rise to new solutions. For this reason, we will discuss only the general cases and the case = 0 in the remainder of this section and comment what happens for λ 5 = 0.
In the basis chosen, the vacuum alignment (v, v, v) preserves the Z 3 subgroup of A 4 . It is convenient to perform a basis transformation into the Z 3 eigenstate basis, 1, 1 ∼ ω, 1 ∼ ω 2 according to When A 4 is broken to Z 3 in the Z 3 eigenstate basis, ϕ ∼ 1 behaves like the standard Higgs doublets: its neutral real component develops a vacuum expectation values ϕ 0R = v w and all its other components correspond to the GBs eaten by the corresponding gauge bosons. The physical real scalar gets a mass given by The neutral components of the other two doublets ϕ and ϕ mix into two complex neutral states and their masses are given by The charged components of ϕ , ϕ do not mix, their masses being 4.2 = 0: The Alignment (v, 0, 0) In the chosen A 4 basis, the vacuum alignments (v, 0, 0) preserves the Z 2 subgroup of A 4 .
As we did with the vacuum alignment that conserved the Z 3 subgroup, in this case it is useful to rewrite the scalar potential by performing the following Z 2 conserving basis Φ 1 is even under Z 2 and behaves like the standard Higgs doublet, while Φ 2 and Φ 3 are odd. For what concerns the neutral states, the 6 × 6 mass matrix is diagonal in this basis and with some degenerated entries: using a notation similar to the 2DHM, we have where the last state corresponds to the GB. The charged scalar mass matrix is also diagonal with where the last state corresponds to the GB. The degeneracy in the mass matrices are imposed by the residual Z 2 symmetry. Contrary to the previous case the neutral scalar mass eigenstates are real and not complex.
This vacuum alignment does not preserve any subgroup of A 4 and it holds From the minimum equations we have that The scalar and pseudoscalar mass eigenvalues are given by For the charged sector we have For λ 5 = 0 the alignment (v, v, 0) has the correct number of GBs, while for λ 5 = 0 we have an extra massless pesudoscalar. However in both cases, λ 5 = 0 or λ 5 = 0, the conditions m 2 h 1 > 0 and m 2 h 3 > 0 can not be simultaneously satisfied. This alignment is therefore a saddle point of the A 4 scalar potential we are studying.
This vacuum alignment, as the previous one, does not preserve any subgroup of A 4 . A part from the condition = 0, we recall that in this case there is the further constraint λ 3 + λ 4 + λ 5 = 0 and λ 5 may assume both positive and negative values since we have reabsorbed in the λ 5 sign the case = π.
Let us define v 2 The mass matrix for the neutral scalar states presents two null eigenvaluesas we expected since the condition λ 3 + λ 4 + λ 5 = 0 enlarges the potential symmetry-and a massive one At the same time the mass matrix for the CP-odd states has one null eigenvalue-the GB π 0 and two degenerate eigenvalues of mass . Notice that for the special case λ 5 = 0 we have the constraint λ 3 = −λ 4 that implies two extra massless pseudoscalars. Finally for the charged scalars we have The total amount of GBs is 5 (7) for the case λ 5 = 0 (λ 5 = 0), so we have 2 (4) extra unwanted GBs: this situation is really problematic. We note that the introduction of terms in the potential that softly break A 4 can ameliorate the situation with the Goldstone bosons. We will analyse soft A 4 breaking terms in more detail in [14]

CP Non-Conserved Solutions
In this subsection, we consider vacua that exhibit spontaneous CP violation. This occurs if the vev of at least one of the Higgses is inherently complex. A global rotation can always absorb one of the three phases of the vevs.
We note that that the two natural vacua of the previous section (v, v, v) and (v, 0, 0) do not have CP violating analogues as they have only one phase that can be reabsorbed to make all vevs real.

The Alignment
In this case the third doublet is inert and therefore we are left only with two doublets that develop a complex vev and after the redefinition, there is only one phase ω 1 . Taking the generic solution (v 1 e iω 1 , v 2 , 0) the minimum equations are given by The last equation can be solved by = −2ω 1 or = −2ω 1 + π. Like in section 4, we can absorb the second case by a redefinition of λ 5 . The other three equations reduce to that are simultaneously solved for v 1 = v 2 = v w / √ 2 and The neutral and charged 6 × 6 mass matrices are quite simple and it is possible having analytical expression for the mass eigenvalues. For the neutral sector we have and for the charged one we have We see that the mass of the fourth neutral boson selects negative values for λ 5 , i.e. the second solution = −2ω 1 + π. It is interesting to see that in the CP conserved limits

The Alignment
In this case all the doublets develop a vev v i = 0, so we may have two physical CP violating phases. We have the freedom to take ω 3 = 0. In this case the first derivatives system is given by The last equation is solved for ω 2 = −ω 1 and v 2 = v 1 = v. Defining v 3 = rv and v 2 1 + v 2 2 + v 2 3 = v 2 w the previous system reduces to the three equations We can solve the third equation in eq. (31) in terms of µ 2 and then the second equation in terms of λ 5 , giving Then the first equation in eq. (31) has two possible solutions, for λ 4 and respectively To test the validity of the solution so far sketched it is necessary to check to be in a true minimum of the potential and not to have extra GBs a part from three corresponding to the GBs eaten by the gauge bosons. However the relations given in eq. (32) and eq. (33) do not allow to get analytical solutions for the scalar masses in case ii). For this reason we will consider only three special limits in this case : r ∼ 0, r ∼ 1 and r very large. We think that these limit situations could be the most interesting ones in the model building realizations. Indeed models present in literature [11,12] fall in the third case, r very large.

Case i)
In this case the constraints λ 4 = −λ 3 puts λ 5 to zero and enlarge substantially the symmetries of the potential: we have an accidental O(3) in the neutral real direction and two accidental U (1)s due to λ 5 = 0. For this reason the neutral spectrum has 5 massless particles, the GB π 0 and 4 other GBs, and only one massive state The charged scalars are The massive states are degenerate as in the CP conserving minima studied in sec. (4.3) for the case λ 5 = 0.

Case ii)
As it is not possible to find analytical solutions, here we will study three special limits of case ii.
• r ∼ 0 In this case we will neglect terms of order r 2 . From eq. (33) we have that for r ∼ 0 thus from eq. (32) we have Under these approximations the 6 x 6 neutral scalar mass matrix gives one null mass state, m 2 π 0 = 0, corresponding to the GB and the following five eigenvalues at leading order, given by where f [λ i ] stays for a linear combination of the adimensional λ parameters of the potential. The previous neutral spectrum present a lightest state that may be too light to be phenomenologically acceptable. Assuming that the λ's potential parameters run in the 'natural' range 0.1 ÷ 10 or, somewhat optimistically, 10 −2 ÷ 10 2 . For what concerns r we are in the limit of r 2 ∼ 0, so as reference value we may take r 2 ∼ 10 −3 ÷ 10 −2 . By combining these two ranges we find upper bounds Since f [λ i ] ∼ 100 may be obtained only for very peculiar combinations of the potential parameters, the previous estimates indicate that for relative tiny value of r the spectrum may present very light neutral states.
On the contrary, in the charged sector we have the two GBs eaten by the corresponding gauge bosons, m 2 C 3 = 0, and two complex massive states with masses • r ∼ 1 In this limit we may write r ∼ 1 + δ and make an expansion in terms of δ neglecting terms of order δ 2 . Thus we have and then Under these approximations the 6 x 6 neutral scalar mass matrix gives the usual null mass state, m 2 π 0 , corresponding to the GB and the following five eigenvalues where again f [λ i ] stays for a linear combination of the λ's potential parameters. A analysis similar to the one for the case with r ∼ 0 shows that the neutral spectrum may present very light states.
In the charged sector we have the GBs eaten by the gauge bosons and two degenerate massive state • r 1 In this case we may perform an expansion in term of 1/r and neglect terms of order 1/r 2 . From eq. (33) we have that and then eq. (32) reduces to Under these approximations we find a massless neutral scalar state, m 2 π 0 = 0, and the other 5 neutral masses are given at leading order by where once more f [λ i ] stays for a linear combination of the λ's potential parameters. The charged scalar mass matrix is diagonal up to terms of order O(1/r 2 ) with two massive degenerate states and the correct number of GBs.
If we consider now eq. (47) we see that as for r ∼ 0 and r ∼ 1 the expressions for m 2 h 1,2 say that we may have two very light neutral scalars. Taking as reference values for r the range 50 ÷ 200 we find giving m 2 1,2 ≤ 50 2 GeV 2 for r ∼ 50 , m 2 1,2 ≤ 10 2 GeV 2 for r ∼ 200 , where 50(10) GeV may be obtained only for very peculiar combination of the potential parameters. In other words we expect that also in the majority of the cases for r in the range 50 − 200 we will have m 2 1,2 very light. In conclusion, taking into account the SM context and the potential given in eq. (2), the solution (e iω 1 , e −iω 1 , r)v w / √ 2 + r 2 with r small, close to 1 or large give rise to very light states. Of course this does not mean that these states will be light for any value of r but it is a quite strong hint that it is possible that this could be what indeed happens.
As mentioned before, the addition of soft A 4 breaking terms to the potential may help in the cases of Goldstone bosons or very light bosons. We will discuss these terms in more detail in [14].

Bounds From The Higgs Phenomenology
In this section we analyse the phenomenology corresponding to the different vacua discussed above: unitarity, Z and W ± decays and oblique parameters. In this way we manage to constrain the parameter space and, in some cases, to rule out the studied vacuum configuration.

Unitarity
In this section we account for the tree level unitarity constraints coming from the additional scalars present in the theory. We examine the partial wave unitarity for the neutral twoparticle amplitudes for s M 2 W , M 2 Z . We can use the equivalence theorem, so that we can compute the amplitudes using only the scalar potential described in eq. (2). In the regime of large energies, the only relevant contributions are the quartic couplings in the scalar potential [18][19][20][21] and then we can write the J = 0 partial wave amplitude a 0 in terms of the tree level amplitude T as where F represents a function of the λ i couplings. Using for simplicity the notation we can write the 30 neutral two-particle channels as follows: Once written down the full scattering matrix a 0 , we find a block diagonal structure. The first 12 × 12 block concerns the channels while the other three 6 × 6 blocks are related to the channels once we specify the labels (a, b) as (1, 2), (1, 3) and (2,3). Notice that up this point the analysis is completely general and is valid for all the vacua presented. We specify the vacuum configuration, expressing the quartic couplings λ i in terms of the masses of the scalars. Afterwards, putting the constraint that the largest eigenvalues of the scattering matrix a 0 is in modulus less than 1, we find upper bounds on the scalar masses which we use in our numerical analysis.

Z And W ± Decays
From an experimental point of view gauge bosons decays into scalar particles are detected by looking at fermionic channels, such as for example Z → hA → 4f in the 2HDM, or fermions or all have been precisely been calculated and measured, we may focus on the decays Z, W ± → all. Doing this we overestimate the allowed regions in the parameter space, but we have a first and model independent cut arising by the gauge bosons decay.
Once we will pass to a model dependent analysis the region may only be restricted, not enlarged. Furthermore, defining the contribution from new physics as ∆Γ, since we expect the error we commit being quite small.
From LEP data we have with ∆Γ Z ∼ 0.0023 GeV and ∆Γ W ± ∼ 0.042 GeV [22]. Therefore we may calculate the for the different multi Higgs (MH) vacuum configuration studied and select the points that Here we have indicated the generic Z → h i h j referring to our notation introduced in section 2. Clearly when CP is conserved the h i have defined CP and only couplings to CP odd states are allowed. Of course this is not true for the configuration when CP is spontaneously broken.
In the vacuum analysis we did we have seen that in few situations we have extra massless or very light particles. For those cases the gauge bosons decays put strong bounds. For what concerns the Z decays we have where g is the SU (2) gauge coupling, c W the cosine of the Weinberg angle θ W and the parameter k Z is given by with U defined in eq. (6).
Similarly for the W ± decays we have where, in analogy to the Z decay, the parameter k W is given by with S defined in eq. (7).

Large Mass Higgs Decay
Electroweak data analysis considering the data from LEP2 [23] with β ≤ 1. In our case for example β is given by with f a = v a /v w and ω a the corresponding CP phase. Taking into account that h 1 is less produced then the SM Higgs and that its Γ M H W W (m h 1 ) is reduced with respect to the SM one, we can roughly constrain the upper bound for masses m h 1 ≥ 194 GeV.

Constraints By Oblique Corrections
The consistence of a MH model has to be checked also by means of the oblique corrections.
These corrections can be classified [25][26][27][28][29] by means of three parameters, namely T SU , that maybe written in terms of the physical gauge boson vacuum polarizations as [30] T = 4π where s W , c W are sine and cosine of θ W and e is the electric charge. EW precision measurements severely constrain the possible values of the three parameters T , S and U . In For a Higgs boson mass of m h = 117 GeV (and in brackets the difference assuming instead m h = 300 GeV), the data allow [22] S exp = 0.10 ± 0.10(−0.08) T exp = 0.03 ± 0.11(+0.09) U exp = 0.06 ± 0.10(+0.01) .
The constraints in eq. (67) must be rescaled not only for the different values of the Higgs boson mass but also for a different scalar or fermion field content: for example, if we assume to have a MH scenario this gives a contribution T MH to the T-parameter and we need A detailed analysis on the T SU in a MH model has been presented in [31,32]

Results
We have performed a numerical analysis for all vacuum configurations considered, neglecting the alignment (v, v, 0) since in this case there are tachyonic states. Our aim was to find a region in the parameter space where all the Higgs constraints were satisfied for each configuration considered. We have analysed the points generated through subsequent constraints, from the weaker one to the stronger according to • points Y: true minima -all the squared masses positive-(yellow points in the figures); • points B: unitarity bound (blue points); • points G: Z and W ± decays (green points); • points R: T SU parameters (red points). On the contrary for the CP breaking alignment (ve iω 1 , ve −iω 1 , rv) we have personalized the plots for reasons that will be clear in the following.
Notice that in all the following discussion, we refer as m 1 (m 2 ) to the (next-to-the-) lightest neutral state and as m ch 1 as the lightest charged mass state. Indeed by looking at fig. 1 we see that we may find R (allowed) points for very tiny m 1 masses and up to ∼ 500 GeV when the unitarity bound starts to show its effect. However by looking at the crowded points in fig. 1 it seems that case 2) is slightly preferred with respect to case 1). Finally for the G points -those that satisfy the minimum, unitarity and decays conditions-we have compared the contributions to the oblique parameters T and S to see which of the two is more constraining. It turns out to be T , while we have not reported U because its behavior is very similar to S. level -long,normal,short dashing respectively. The T parameter turns out to be the most constraining one. for values from 0 up to 700 GeV, thus reflecting case 2). Then the points corresponding to case 1) have a sharp cut at m 1 = 194 GeV, that rejects many blue points, i.e. those satisfying the unitarity constrain but not the decays one. We have reported also m 1 versus m 3 to check that indeed, when m 1 → 0, m 3 is bounded by m Z as we expected. Our intuitions are also confirmed by the plot m 1 − m ch 1 . As for the Z 3 preserving case the most constraining oblique parameter is T .

CP Conserved Solutions
In this case we do not have any surviving symmetry which forbid some couplings. However from sec. 4.3 we know that the conditions = 0, λ 3 + λ 4 + λ 5 = 0 give rise to two extra massless CP even particles. Therefore we expect that 1) when the lightest massive state is CP odd, then its mass is bounded by the Z decay through eq. (58); 2) when the lightest massive state is CP even, then its mass could reach smaller values since the Z decay bound would constrain the combination of its mass with the lightest CP odd state mass. The T parameter turns out to be the most constraining one.
Moreover in both cases we expect the mass of the lightest charged scalar bounded by W decay, according to eq. (60), due to its coupling with W and the massless particles.
By fig. 3 we see that it seems that case 2) happens very rarely because the cut at m 1 ∼ m Z is in evidence. As for the Z 3 and Z 2 preserving minima the T parameter is the most constraining one. does not preserve any A 4 subgroup. Moreover since even CP is broken any symmetry cannot help us in sketching the behavior we expect. In general any state, having a CP even and a CP odd component, may couple to Z and to another neutral state. However we expect limit situations in which for example CP is almost conserved and the 2 lightest states have almost the same CP parity. Thus for those cases we do not expect any lower bound on m 1 and m 2 . On the contrary the coupling between the W with the lightest neutral and the lightest charged scalars does not go to zero when CP is almost restored.
Then we expect that the quantity m 2 1 + m 2 ch 1 is bounded by the W decay ( fig. 4). For what concerns the upper bound on the lighetst neutral mass state we do not expect any clear cut because we may not identify a SM-like Higgs. In conclusion, the solutions for the alignment (ve iω 1 , ve −iω 1 , rv) with λ 5 = 0, λ 4 = −λ 3 are not easy to find, but the Higgs phenomenology does not completely rule out this vacuum configuration. We could introduce a weight to estimate how much a solution is stable or fine-tuned but this goes over the purposes of this work. We expect that this situation with 4 extra massless particles could be very problematic when considering the model dependent constraints [14].
In the analytical discussion done in sec. 5.2.2 we have seen that at least in the special limit r ∼ 0 (r ∼ 1 and r >> 1) we expect the presence of one (two) very light particles.
From all the numerical scans we performed we found out that solutions for the vacuum alignment (ve iω 1 , ve −iω 1 , rv) with the constraints of case ii) are very difficult to be found.
Moreover from fig. 6 we see that for any value of r the two lightest states are always very light, thus confirming our rough analytical approximations. Indeed both m 1 and m 2 are lighter then we expected -especially m 2 for r ∼ 0-thus indicating that some cancellations have to occur to give all the masses greater then 0. This supports the difficulty to find solutions, difficulty that cannot to be ascribed to any constrain we imposed, because even The presence of a single R point in fig. 6 is not statistically relevant, but more interesting is the order of magnitude of m 1,2 : even in case ii) we expect that the alignment (ve iω 1 , ve −iω 1 , rv) may present serious problems once we add model dependent constraints [14].

Conclusions
Flavour models based on non-Abelian discrete symmetries under which the SM scalar doublet (and its replicants) transforms non trivially are quite appealing for many reasons.
First of all there are no new physics scales, since the flavour and the EW symmetries are simultaneously broken. Furthermore this kind of models are typically more minimal with respect to the ones in which the flavour scale is higher than the EW one: in particular the Due to the restricted number of parameters and the abundance of sensitive observables in these models, there are many constraints to analyze: the most stringent ones arise by FCNC and LFV processes [14] but even Higgs phenomenology put several constraints on this class of models.
In this paper we focussed on the A 4 discrete group, but this analysis can be safely generalized for any non-Abelian discrete symmetry. We consider three copies of the SM Higgs fields, that transform as a triplet of A 4 . This setting has already been chosen in several papers [10][11][12][13] due to the simple vacuum alignment mechanism.
We have considered all the possible vacuum configurations allowed by the A 4 × SM scalar potential. These configurations can either conserve or violate CP. For all of them we have considered only model independent constraints, related to the Higgs-gauge boson Lagrangian, and postponing the model dependent analysis to an accompanying paper [14].
The first model independent constraint comes from the partial wave unitarity for the neutral two-particle amplitudes, which puts upper bounds on the scalar masses. Then we have explained how the light scalar mass region can be constrained considering the gauge boson decays. Moreover we have seen how to put an upper bound on the lightest neutral state mass considering the Higgs decay channel h → W + W − . Finally the most stringent bounds arise by the oblique parameters T SU .
We have shown that the Higgs-gauge boson model independent analysis can be used to study the parameter space of the difference vacuum configurations. Among the possible solutions which minimize the scalar potential, only one is ruled out due to the presence of tachyonic states. Furthermore, some other configurations may be obtained only by tuning the potential parameters, giving rise to scalar spectrums characterized by very light or even massless particles. Finally, for the remaining ones, we find that they may share common features and this increases the difficulty in discriminating among them. Nevertheless, the model independent approach restricts in a non trivial way the parameter space. In conclusion, we underline that more constraining results can be found considering specific realizations which adopt the different vacuum configurations: we present this analysis in [14].

Note Added In Proof
While completing this paper we received ref. [33], where a study of the scalar potential with three copies of the SM Higgs doublet transforming as a triplet of A 4 is performed. This analysis is restricted only to few vacuum configurations allowed by the scalar potential and a complete phenomenological study is missing.

Aknowledgments
The work of RdAT and FB is part of the research program of the Dutch Foundation for

Appendix A: Analytical Formulae for T SU Parameters
In this Appendix we provide a sort of translator from the papers [26,29] to our notations and furnish the formulae we have used when different from their.
Reminding their notation we are in the case in which n d = 3 and n n , n c = 0 so we do not have the matrices T and R. Then we have (A.1) Moreover they put the GBs as first mass eigenstates while we put them as the last ones and contrary to them we use the standard definition for the photon.
We have rewritten they expression for