$Q_{6}$ As The Flavour Symmetry in a Non-minimal SUSY $SU(5)$ Model

We present a non-minimal renormalizable SUSY $SU(5)$~model, with extended Higgs sector and right-handed neutrinos, where the flavour sector exhibits a $Q_{6}$ flavour symmetry. We analysed the simplest version of this model, in which R-parity is conserved and the right-handed neutrino masses in the flavour doublet are considered with and without degeneration. We find the generic form of the mass matrices both in the quark and lepton sectors. We reproduce, according to current data, the mixing in the CKM matrix. In the leptonic sector, in the general case where the right-handed neutrino masses are not degenerate, we find that the values for the solar, atmospheric, and reactor mixing angles are in very good agreement with the experimental data, both for a normal and an inverted hierarchy. In the particular case where the right handed neutrinos masses are degenerate, the model predicts a strong inverted hierarchy spectrum and a sum rule among the neutrino masses. In this case the atmospheric and solar angles are in very good agreement with experimental data, and the reactor one is different from zero, albeit too small ($\theta^{\ell^{th}}_{13} \sim 3.38 $). This value constitues a lower bound for $\theta_{13}$ in the general case. We also find the range of values for the neutrino masses in each case.


Introduction
Understanding the flavour sector of the Standard Model (SM) has been a puzzle for a long time, due to the large differences in the Yukawa couplings of the different fermions. The hierarchy between the fundamental particles, the amount of CP violation and the structure of the CKM matrix remain as open [1]. In spite of these subtle facts, the success of the SM is remarkable.
One of the strategies to deal with the flavour problem in the quark and lepton sectors has been to study it in the framework of textures (zeros) in the mass matrices. The textures have been explored for a long time, as an attempt to eliminate the irrelevant free parameters in the Yukawa sector. As it is well known, the Fritzsch textures [2,3] can accommodate the quarks mixing angles, in terms of the quarks masses. This approach seems to work out correctly because the Cabbibo angle is obtained with great accuracy [2]. However, this framework presents some problems with the top mass and the V cb element of the CKM matrix [4,5], as can be seen in [6,7]. Recently, deviations to the Fritzsch textures have appeared in order to overcome these problems. Moreover, the charged lepton and neutrino sector have been included in this kind of ansatz and consistent results on the P M N S matrix [8,9] have been observed in this generic approximation [7]. As alternative textures to the Fritzsch ones, the Nearest Neighbour Interaction (N N I) [10,11] textures can also reproduce very well the flavour mixing in the quark and lepton sectors; it is well known that Fritzsch textures could be obtained from the N N I ones as a limiting case [10].
Non-Abelian Flavour symmetries have played an important role in model building to obtain, in an elegant way, N N I textures [10,11] in the fermion mass matrices. In particular, the flavour symmetry group Q 6 [12] has been proposed as responsible of the textures in the quarks as well as in the lepton sector [13][14][15][16][17]. This appealing flavour symmetry allows the appearance of the N N I textures in the quark and lepton sectors so that the mixings are in good agreement with the data [13,17]. The rich phenomenology Q 6 provides in the SUSY scenario is remarkable, one of these features is that it prohibits the dangerous terms that mediate fast proton decay rather than invoking the R-parity symmetry [14,15]. The immediate question that arises is how the Q 6 symmetry can look within a GUT framework, in particular, in the SUSY SU (5) model? The main question that we will address here is if the SUSY SU (5) models are compatible with the Q 6 group in order to accommodate masses and mixings for fermions From a theoretical point of view, grand unified ideas [18][19][20][21] are well motivated for fundamental reasons. In particular, the SU (5) model [21] is considered to be one of the best scenarios to unify the electroweak and strong interactions. However, the model itself faces serious phenomenological and theoretical problems, one of them being that active neutrinos are massless [22,23]. The simplicity of the SU (5) can be retained even if it is promoted to be a supersymmetric model. The SUSY SU (5) [24][25][26] version cures most of the above problems, for example, proton decay may be solved by introducing the already known discrete symmetry, R-parity, which prohibits dangerous terms that can mediate its fast decay. Moreover, recent generic studies about the minimal SUSY SU (5) model have made it clear that it may not be ruled out by the current experimental data on the proton decay rates [27,28]. Taking into account the neutrino mass problem, the SUSY SU (5) version (in general supersymmetric) provides elegant mechanisms to generate massive neutrinos via R-parity violation [29,30].
But still, the simplest way to give mass to the left-handed neutrinos in the minimal SU (5) or SUSY scenario is to consider three right-handed neutrinos which are singlets under the gauge group and invoke the type I see-saw mechanism [31][32][33][34][35].
There are interesting SUSY SU (5) models [36,37], that can very well reproduce the CKM and P M N S matrices in agreement with the experimental results, but where the simplicity in the matter content has been left aside. Most of these models include a large number of flavons which are required to accommodate correctly the mixings. A different approach consists of extending the Higgs sector of the models. By itself, the SUSY SU (5) matter content provides the tools to accommodate masses and mixings via the extension of the Higgs scalar sector. This philosophy has been worked out with success in non-supersymmetric [38][39][40] and supersymmetric [13][14][15][16][17] scenarios where the concept of flavour has been extended to the Higgs sector.
Therefore, we propose here a renormalizable SUSY SU (5) model where the Q 6 group plays an important role in the flavour sector. The need to extend the scalar sector is evident in order to accommodate masses and mixings for quarks and leptons without breaking explicitly the flavour symmetry, so that three families of H u and H d 5-plets and two gauge singlet scalars are introduced. The latter scalars provide mass to the right-handed neutrinos and the type I see-saw mechanism is invoked to get small masses for the active neutrinos. At low energies, consistent results are obtained for the CKM , and in the leptonic sector, the atmospheric θ This value corresponds to a lower bound for θ 13 for the more general model where the righthanded neutrinos are not mass degenerate, similar to the case of S 3 non-supersymmetric models, where relaxing the degeneracy condition in the right-handed neutrino masses gives the right value for θ 13 [40].
The paper is organized as follows: In section 2, we build the extended SUSY SU (5) model which obeys the Q 6 flavour symmetry, in addition, we explain the required matter content to get a unified scenario and we do stress the strong assumptions that we will make in the model. The textures in the quark mass matrices, and therefore their consequences in the lepton sector, are analysed in section 3. Also, we describe how to diagonalize the mass matrices for each sector in order to obtain the mixing matrices. In section 4, we present and discuss the results about the mixing angles for each sector. Finally, we give conclusions on this preliminary analysis of the model.
We will consider the SUSY SU (5) model where three right-handed neutrinos have been included in the matter content. In addition, the scalar sector has to be necessarily extended as we will show. The assigned matter content under Q 6 flavour symmetry and U (1) charges are displayed on Table 1. Let us comment on our notation and the matter content: φ andφ are singlet scalars under the gauge group, the former gives mass to the right-handed neutrinos and the latter is introduced in order to cancel anomalies in the U (1) Abelian group. On the other hand, we do need to include H 45 and H4 5 scalar representations so as to fix the incorrect relation M d = M T e , although there is another way to achieve that see [27]. Regarding the fermion sector, N i denotes the right-handed neutrino, which is a singlet under the SU (5) gauge group; F i and T j stand for the 5-plets and the 10 antisymmetric-plet, respectively. Here, a, b, c are SU (5) indices, and i, j are family indices. More explicitly, Here, H u and H d are the coloured triplet scalars that mediate proton decay. For the time being, it will be assumed that these are heavy enough to keep proton lifetime bounded and under control. In addition, we have to point out that this subtle issue will be left aside, since we are only interested in studying the masses and mixings when implementing Q 6 as a flavour symmetry in this model. Also, we do not dwell on the details of full gauge symmetry breaking from SUSY SU (5) ⊗ U (1) ⊗ Q 6 to the MSSM group. Such issues are part of a wider study of this particular model, that is still in progress. However, there are some ideas involving this line of thought, see for example [41][42][43]. On the other hand, H d and H u are identified as the weak doublets of the MSSM group. In consequence, we employ the usual pattern to break the SUSY SU (5) model via the vacuum expectation values (vev's) for every scalar representation.
For simplicity's sake, we have deliberately neglected the scalar triplets, H u = 0 and H d = 0, in the H u and H d 5-plets, assuming they are extremely heavy.
Having presented the assigned matter fields under Q 6 flavour symmetry, we will now introduce the superpotential, gauge invariant under the Q 6 discrete group. All necessary details for building the flavour symmetry singlets are given in the appendix, where a brief review of this dihedral group is offered. The trilinear terms in the superpotential are given by From this superpotential, there is a missing term which is invariant under SUSY SU (5) ⊗ Q 6 such as y ij N c iφ N c j , however, it is prohibited by the U (1) group. It is important to remember that SUSY must be broken via soft breaking terms, so these soft breaking terms should be included in a complete study of the full scalar potential, but we do not include them in this preliminary analysis. The scalar superpotential is From the superpotential given in Eq. (3), one must obtain the MSSM effective superpotential that contains the Yukawa mass term after spontaneous symmetry breaking via the vev's of the H u and H d weak doublets scalar superfields. We will work in the following basis In general, the up, down, and charged lepton mass matrices are given by (see [44]) In this particular model, from Eqs. (3) and (7) we have that the up, down and charged lepton mass matrices have respectively the following structures: whereȳ u ≡ (y u 2 +y u 3 )/2. As can be seen, the M u mass matrix turns out almost symmetric due to the flavour structure. At the same time, we were able to correct the wrong relationship between the down quarks and the charged leptons; this was achieved including the H4 5 scalar representation. The Dirac and right-handed mass matrices are given by Therefore, after the type I see-saw mechanism, the neutrino mass term is given by where the M ν = M T D M −1 R M D effective neutrino mass matrix has the following structure: where x = M −1 R 1 and y = M −1 R 2 . Interesting phenomenology can be extracted from M ν because the free parameters might be reduced substantially as a result of imposing conditions on the vev's that take the down quark and charged lepton mass matrices to the correct N N I form as in ref. [13][14][15][16][17].

Masses and mixings in the N N I scenario
There are two ways to obtain the N N I textures in the down quark and charged lepton sector: The first scenario consists in taking the condition h 0u 2 = 0 = h 0d 2 on the vev's, but although the N N I textures are present in the quark and lepton sector one can realize that the M ν effective neutrino mass matrix does not lead to a good agreement with the experimental data, since there is a massless neutrino, as can be checked in straightforward way. Thus, this case is ruled out. In the next subsection we will describe the most viable scheme.

Quark and lepton masses
First, if we assume that h 0u 2 = h 0u 1 ≡ h 0u and h 0d 2 = h 0d 1 ≡ h 0d , we obtain the following mass matrices: Here, the upper (lower) sign corresponds to the M d (M ) mass matrix. In addition, the coefficients can read of Eq. (8). The M (d, ) and M u mass matrices contain implicitly the N N I and F ritzsch textures respectively which appear explicitly as follows: the above mass matrices are diagonalized by unitary matrices, U f (R,L) . According to Eq.(6) one obtains . For simplicity, we have normalized the above expressions so thatm We should point out that the √ 2 factor will be absorbed in theB andC dimensionless free parameters for the quark and lepton sectors, respectively. In addition, notice that for the m u mass matrix given in Eq. (13),C u =B u , according to Eq. (12). We will now show how the degeneracy in the vacuum expectation values for two scalar fields modifies the effective neutrino mass matrix which is now given as where we have defined A ν = √ xy n 1 h 0u 3 , B ν = √ yy n 3 h u and C ν = √ xy n 2 h u . Let us add a comment on the degeneracy on the vacuum expectation values. As we have remarked, the conditions h 0u 2 = h 0u 1 ≡ h 0u and h 0d 2 = h 0d 1 ≡ h 0d have been assumed so far, it is not clear yet that these relations will arise in a natural way upon minimization of the scalar potential. We expect that it will be the case and the study of the full scalar potential may be done along the lines as those given in [13,16].

Quark and lepton mixings
We will describe briefly how to diagonalize the mass matrices, m (d, ) and m u , respectively. Let us first start with the down quark and charged lepton mass matrices which have the N N I textures, we will not enter in great detail since these kind of matrices have been well studied in [10,11]. The above mentioned description is applied to the m u mass matrix where the Fritzsch texture [2,3,7] is present. For a pedagogical method to diagonalize these mass matrices see [38].
Going back to the expressionM f = u † f R m f u f L , we are interested in obtaining the u f L left-handed matrices that appear in the CKM matrix, and for this we must build the bilineal form: From this relation, we can factorize the CP phases that comes [45], such that, we should keep in mind that the |Ã f |, |B f |, |C f | and the |D f | free parameters are real and dimensionless, and that for the up mass matrix, |B u | = |C u |.
Having factorized out the phases associated with CP violation in the bilineal form, we where the three eigenvectors have the following form (17) Here, N f 1 (i = 1, 2) and N f 3 stand for the normalization factors whose definition must be read directly from the above expression. On the other hand, three free parameters can be fixed in terms of the physical masses and |D f | ≡ y f remains as the only free parameters [10,11]. Explicitly, these are given by where Here, there are two free parameters, y f , for f = d, . These parameters should be tuned in order to get reliable mixing matrices as we will see later. So far, we have found the U (d, )L left-handed matrices that diagonalize the M (d, ) mass matrices which have the N N I textures.
Let us now focus on m u . Going back to Eq. (15), we must remember that |B u | = |C u |, then one can determine the three free parameters in terms of the physical masses. Explicitly, we obtain Following the same procedure, the O uL orthogonal matrix that diagonalizes (m † u m u ) is fixed in terms of above parameters. Thus, using the expression given in Eq. (17), we get Therefore, the full left-handed unitary matrices are given, in general, by U f L = U π/4 u f L where u f L = Q f O f L . Then, the CKM matrix may be completely determined and given by In this way, the CKM matrix can be obtained analytically or numerically, however, we are now just interested in getting a numerical expression for it which will be done in the next section.
In the leptonic sector, the P M N S matrix is defined by V P M N S = U † L U νL K where U L = U π/4 Q O L and the U νL neutrino contribution will be obtained next. We should point out that we will neglect the K Majorana CP phases, which are currently unobservable.
Before diagonalizing the effective neutrino mass matrix, let us show the set of neutrino observables which is considered along the analytic and numerical analysis [46,47]. This is given below ∆m 2 [10 −5 eV 2 ] = m 2 ν 2 − m 2 ν 1 = 7.59 +0.20 Here, the data appearing in parentheses stand for the inverted case. Leaving aside the experimental results, we consider the M ν mass matrix given in Eq. (14). Due to the form of M ν we can rotate the left-handed neutrino field as follows: ν L = U ννL , where U ν = u π/4 u ν so that one gets: The m ν block matrix can be easily diagonalized. First, let us factorize the CP phases of m ν [45]. So that, m ν = P νmν P ν , where P ν andm ν are given in Eq. (23). We can associate immediately |A ν | 2 = m ν 3 . Thus, we just have to diagonalize the left upper block of m ν A necessary condition to factorize the phases in the above way is that the A 2 ν and B 2 ν phases must be aligned, although they may be different in magnitude. On the other hand, we appropriately choose u ν = P † ν O ν . Here, O ν is a real orthogonal matrix that diagonalizes them ν matrix. Using the left upper block of m ν , we fix the |B ν | 2 and |C ν | 2 free parameters through the following equations Solving for the rest of the free parameters we find that where R ν ≡ (m ν 2 + m ν 1 − m ν 3 ) 2 − 4m ν 2 m ν 1 . As we can observe, there are two solutions for |B ν | 2 and |C ν | 2 , respectively. However, following a straightforward analysis, it is clear that one solution is discarded by demanding that two free parameters (for the normal and for the inverted hierarchy) should be real and positive definite since they come from a real symmetric matrix. As a result of this, we realize that the normal spectrum is ruled out. For the inverted case (m ν 2 > m ν 1 > m ν 3 ), |B ν | 2 − and |C ν | 2 + turn out being real and positive, if and only if, the m ν 3 lightest neutrino mass is very small. Actually, from the definition of R ν we obtain the following sum rule where the equality in the above expression means an upper bound for the m ν 3 lightest mass.
Having fixed |B ν | 2 − and |C ν | 2 + in terms of the physical neutrino masses, the O ν matrix is well determined by them. Explicitly, O ν is given by Therefore, the M ν neutrino mass matrix is diagonalized by Here, S 23 = U T π/4 u π/4 is the permutation matrix which is an element of the S 3 family group, at the same time, it is an element of the Q 6 family since S 3 is a subgroup of it. Therefore, the P M N S mixing matrix has the following form 28) Comparing this matrix with the standard parametrization given in [48], we find that the reactor, the atmospheric and the solar mixing angles are well determined as follows Let us point out a remarkable coincidence between the above formulas and those showed in [38,39], their functional behaviour seems to be the same, at least. As we already commented briefly, it is not a surprise since the Q 6 family group is the double covering of the S 3 one, so that in this particular model the S 23 presence in the leptonic sector is not simply a coincidence. Of course, we expect that our results turn out being different to the S 3 case, since the charged lepton and neutrino contributions are different in both models, as we will see next.
4 Numerical analysis for the mixing matrices

CKM mixing matrix
The CKM matrix is defined as with the form of the mass matrices found in the previous sections. So far, there are three free parameters (y d and two CP-violating phases in Q q ) if the quark mass ratios are taken as inputs. As it is well known, the physical masses depend on the scale at which they are measured, in this model the CKM matrix may be obtained numerically with masses at the GUT scale. However, the mass ratios do not change drastically at different energy scales as one can verify directly from [49]. Therefore, we will assume that the form of the mass matrices will remain the same from the GUT scale to the electroweak scale. Of course, in the more detailed analysis that is currently in progress, the effects of the extra Higgs fields in the model and the running of the renormalization group will be taken into account. Thus, for the rest of the analysis we will assume we are already at the electroweak scale.
At low energies, we have used the following values for the quark mass ratios given in [50,51] In order to obtain the numerical values for the three free parameters, we perform a χ 2 analysis on the parameter space, to find their best fit points. It is built as in [50,51] where we have taken the following experimental values for the V CKM elements we used to construct the χ 2 function: |V ex ud | = 0.97427 ± 0.00015, |V ex us | = 0.2253 ± 0.007, |V ex ub | = 0.00351 ± 0.00015, J ex q = (2.96 ± 0.18) × 10 −5 [48]. Notice that using the Jarlskog invariant in the χ 2 function implies unitarity as a constraint. The best values for the free parameters are thus found to be at 70 % C. L with χ 2 = 0.0515 as the minimal value. These correspond to the following values for the V CKM elements

P M N S mixing matrix
As we observe from Eq. (29), the reactor and atmospheric angles turn out to be independent of the neutrino masses. These observables only depend explicitly on the y e free parameter, the charged lepton masses and the η 2e Dirac phase; the latter may be ignored since we are Figure 1: Allowed region for the three free parameters in the quark sector at 70% (green) and 90% (orange) confidence level.
just interested in the absolute values of the two mixing angles. On the other hand, the solar mixing angle depends on the y e free parameter, theη 3e phase, as well as the charged lepton and neutrino masses.
In order to show that in this model we can describe the masses and mixing of flavour of the leptons we will also make a χ 2 fit using the theoretical expressions for the atmospheric and reactor angles given in Eq. (29) and compare them with the current experimental data for these mixing angles. Thus, for the atmospheric mixing angle we consider the following experimental value sin 2 θ 23 = 0.52 ± 0.06. On other hand, within this theoretical framework we considered the particular case when the first and second right-handed neutrinos are mass degenerate. So we can only determine a lower bound for the value of the reactor angle [40]. Therefore, to perform the χ 2 fit we considered the values for the reactor angle reported by the MINOS experiment; sin 2 2θ 13 = 0.076 ± 0.068. We considered also the following charged lepton masses values: m e = 0.51099 MeV, m µ = 105.6583 MeV and m τ = 1776.82 MeV [52]. As a result of this χ 2 analysis, we obtained that for the best fit with χ 2 = 0.85 as the minimum value, the free parameter y e at 1σ has the following range of values: y e = 0.8478 +0,0045 −0.0046 . Moreover, the best theoretical values for the atmospheric and reactor angles that come from this analysis, at 1σ, are: As can be observed, the atmospheric angle value is in good agreement with the experimental values, however, the obtained reactor mixing value is smaller than the central value obtained in the global fits. In fact, this theoretical value of the reactor angle is more than 3σ away from the values reported in ref. [53,54]. But it is worth noting that our model predicts that the reactor angle must be different from zero and that is large in comparison to the one obtained from the tribimaximal mixing matrix. Moreover, as already stated above, it is a lower bound for the value obtained in a more general model where the right-handed neutrino masses are not degenerate. In figure 2, we show the dependence of the atmospheric and reactor angles on the parameter y e .
Having determined the allowed values for the y e free parameter, then, the solar mixing angle just depends on the neutrino masses and one Dirac phase. We can reduce further the free parameters, noticing that the neutrino masses can be determined using the sum rule given in Eq. (26). This is written in terms of the observables ∆m 2 and ∆m 2 AT M as From the above expression and using the experimental results of ∆m 2 and ∆m 2 AT M , we can get an upper bound for the m ν 3 lightest neutrino mass. Therefore, the allowed values for the lightest neutrino mass are: 0 ≤ m ν 3 ≤ 4 × 10 −6 eV. As a result, the m ν 2 and m ν 1 neutrino masses are easily calculated in the following way As we already commented, our model predicts an inverted ordering among the neutrino masses and the above values clarify explicitly our statement. Since we have calculated the neutrino masses, these cease to be considered as free parameters in the solar mixing angle expression given in Eq. (29). Furthermore, we obtain easily the solar mixing value that is allowed by the fixed free parameters, which are y e and the neutrino masses. Actually, with the particular value for theη 3e = π Dirac phase, and using the y e value at 90% at C.L, we obtain for the solar mixing angle θ th 12 = 36.62 ± 4.06 (36) where we have used m ν 2 = 0.05080 eV, m ν 1 = 0.04987 eV and m ν 3 = 3.9 × 10 −6 eV. As we observe the solar mixing angle value in Eq. (36) is in good agreement with the experimental data. In figure 3 we show explicitly the dependence of the solar angle on the y e parameter and the m ν 3 lightest neutrino mass, respectively. It is clear from these results that the solar angle has a strong dependence on the m ν 3 mass, and that the sum rule for the neutrino masses does play an important role to determine the above mixing angle in good agreement with the experimental data.

Outlook and Remarks
We have studied a non-minimal supersymmetric SU (5) model where the Q 6 flavour symmetry plays an important role in accommodating the masses and mixings for quarks an leptons. For the former sector, the CKM mixing matrix has been obtained with great accuracy and it is consistent with the experimental results. For leptons, the flavour symmetry implies an appealing sum rule for the neutrino masses, which leads to an inverted hierarchy and is crucial to determine the solar mixing angle. As a main result, we have that the atmospheric θ −0.02 , is more than 3σ away from the central value from the global fits. It is worth pointing out that the model predicts a non-zero value, unlike the tri-bimaximal case, and this value would constitute the lower bound for θ 13 in the more general model, where the right-handed neutrinos in the Q 6 doublet are not mass degenerate. It is expected than in a more general study of the model it will be possible to obtain a value for the reactor mixing angle within one or two σ of the central value, as is the case in the non-supersymmetric S 3 models [40].
Although in this preliminary analysis we have found the form of the mass and mixing matrices for quarks and leptons and we have shown that they lead to realistic values, we have left aside subtle issues as the full analysis of the scalar superpotential, the details of the proton decay and all the phenomenology that the model provides by itself. These topics will be taken into account in a complete study of the model, as we already commented. In general, the model seems to work out very well and it may be considered as a realistic one.
A Q 6 flavour symmetry The Q 6 group has twelve elements which are contained in six conjugacy classes, therefore, it contains six irreducible representations. We will use the notation given in [15], there are various notations and extensive studies for this group, see for example [12,55]. The Q 6 family symmetry has 2 two-dimensional irreducible representations denoted by 2 1 and 2 2 , 4 onedimensional ones which are denoted by 1 +,0 , 1 +,2 , 1 −,1 and 1 −,3 . As it is well known, 2 1 is a pseudo real and 2 2 is a real representation. In addition, for 1 ±,n we have that n = 0, 1, 2, 3 is the factor exp (inπ/2) that appears in the matrix given by B. The ± stands for the change of sign under the transformation given by the A matrix. So that the first two one-dimensional representations are real and the two latter ones are complex conjugate to each other.