Embedding A4 into left-right flavor symmetry: Tribimaximal neutrino mixing and fermion hierarchy

We address two fundamental aspects of flavor physics: the mass hierarchy and the large lepton mixing angles. On one side, left-right flavor symmetry realizes the democratic mass matrix patterns and explains why one family is much heavier than the others. On the other side, discrete flavor symmetry such as A4 leads to the observed tribimaximal mixing for the leptons. We show that, by explicitly breaking the left-right flavor symmetry into the diagonal A4, it is possible to explain both the observed charged fermion mass hierarchies and quark and lepton mixing angles. In particular we predict a heavy 3rd family, the tribimaximal mixing for the leptons, and we suggest a possible origin of the Cabibbo and other mixing angles for the quarks.


I. INTRODUCTION
The recent experimental developments in neutrino physics allowed us to intensify the studies of the flavor structure of the Standard Model (SM) and its extensions. The hardest task was the understanding of the relation between the mass hierarchies and the large lepton mixing angle between the 2nd and 3rd family. In particular the no-go theorem [1] shows that contrary to expectations, a maximal mixing angle θ 23 can never arise in the symmetric limit of whatever flavor symmetry (global or local, continuous or discrete), provided that such a symmetry also explains the hierarchy among the fermion masses and is only broken by small effects, as we expect for a meaningful symmetry. A milestone in these studies has been the discovery that mass hierarchies and mixing angles can be not directly correlated among them in the flavor symmetry breaking [2,3]. In particular, while the mass hierarchies are in general obtained by using continuous flavor symmetries, such as non-Abelian or U (1) flavor symmetryá la Froggatt-Nielsen, the neutrino experimental data indicate that the lepton mixing angles may be explained by discrete flavor symmetries. This complementarity between hierarchy and mixing angles allow us to escape from the hypothesis of the theorem previously outlined [4,5]. The idea is that the flavor symmetry that predicts a large mass for the 3rd family does not make any prediction on the mixing angles. However, once the symmetry is broken into a discrete one, then the mixing angles are naturally generated. Another guideline in flavor physics is given by the unification of the gauge groups. This ingredient forces the field transformations under the flavor symmetry to be related among them, and strongly reduce the degrees of freedom in the model building.
The finite group of even permutations of 4 objects, A 4 , is the smaller non-abelian finite group that contains a triplet irreducible representation. It is the first alternating group that is not isomorphic to any modulo n group, Z n , or to any direct product of permutation groups, S n . It has been used in the last years [6,7,8,9,10,11,12,13,14,15] to build a huge number of models that predict for the lepton sector the tribimaximal mixing matrix [16] with maximal atmospheric angle [17,18], θ 13 = 0 [19] and sin 2 θ sol = 1/3 [20,21,22,23,24] that agree with neutrino oscillation data.
In [5] a non-supersymmetric SO(10) × A 4 grand unified model, which successfully preserves tri-bimaximal leptonic mixing and can accommodate all known fermion masses, has been discussed. In this paper we will show how the embedding of the discrete group A 4 into a left-right symmetry allows us to explain the large hierarchy between the 3rd and the first two families of quarks and charged leptons. At the same time the charged fermion masses of two light families, that of the neutrinos, and the fermion mixing matrices are related to the explicitly breaking of the left-right symmetry into the diagonal A 4 and are generated when A 4 is spontaneously broken. In particular the Cabibbo angle in the quark sector is induced by higher order operators that explicitly break SO(3) L × SO(3) R but preserve the diagonal A 4 . Our final aim would be to introduce a gauge unification group SO(10)-like. Since in SO(10) all the Standard Model (SM) matter fields of one family belong to the same multiplet, namely a 16-plet, as starting point we will consider a model based on the discrete flavor symmetry A 4 in which left-handed and right-handed fermions belong to the same representation of A 4 .
The group A 4 has four irreducible representations, three singlets 1, 1 ′ , 1 ′′ and a triplet 3. Several extensions of the SM are presents in the literature, depending on the A 4 family symmetry realization and the assignments for left-handed and right-handed fermion fields. As we motivated before, we are interested in a realization where both left-handed and right-handed fields have the same A 4 assignment, in such a way to be able to perform an embedding into a gauge grand unified group like SO (10). Therefore in this paper we will consider a model similar to that proposed in [4,5] where both left-handed and right-handed fields are in the triplet representation of A 4 .

II. MASS OF THE 3RD FAMILY FROM THE LEFT-RIGHT FLAVOR SYMMETRY
The study of models based on the flavor symmetry U (3) L × U (3) R [25] or its subgroups both continuos [26] or discrete [27,28,29] has a long history. Usually, by imposing a discrete symmetry like S 3L × S 3R , the charged fermion mass matrix obtained is the so-called democratic mass matrix [30] given by This matrix has only one eigenvalues different from zero, m f , and can be assumed to be the mass of the 3rd family. The unitary matrix that diagonalizes the symmetric matrix M 0f has one angle and the three phases undeterminated. One possible parametrization is given by The unknow angle and phases are fixed only after breaking the democratic structure of M 0f with a small perturbation δM f , i.e.
The effect of δM f is to give a small mass to the first and second family and to fix the mixing angles. Another feature of the models based on a symmetry that gives democratic charged fermion mass matrices is that up and down quarks are diagonalized by almost identical matrices and therefore the CKM can be fitted to be close to the identity. Some attempts of including the neutrinos in this kind of models are quite successful and can fit with good agreement the data [31]. Nevertheless, models that have a democratic mass matrix for the charged fermions and at the same time predict the tribimaximal mixing matrix for the leptons are still missing.
In the following we will build a "supersymmetry inspired" model, in the sense that the scalars and the SM matter fields we introduce belong to supermultiplets and the Lagrangian arises by a superpotential. In the supersymmetric model proposed in [32] the correct alignment of the vevs in the lepton sector, that gives the tribimaximal mixing matrix, has been successfully obtained. However it is difficult to obtain the same result in a context that is non supersymmetric. By the way, in order to focus on the origin of the mass hierarchies and mixing angle and to make lighter the reading we will report only the Yukawa Lagrangian involving the SM-like fields.
Let's now extend the flavor symmetry and let's think to A 4 as a discrete subgroup of the continuous global group To implement the idea of explaining both the hierarchy and the mixing angles by starting with the same flavor symmetry, we use two kinds of symmetry breaking: the explicit one and the spontaneous one. We impose that the fermion weak doublets L, Q transform with respect to SO(3) L × SO(3) R as ∼ (3, 1) while the right-handed fermions E, U, D as ∼ (1, 3). As explained in [26], by assuming that a discrete S 3L × S 3R is left survived in the spontaneous breaking of SO(3) L × SO(3) R , the charged fermion mass matrices must have the democratic structure. To write down an invariant term, we introduce a weak scalar singlet Φ ij ∼ (3, 3) bi-triplet with respect to SO(3) L × SO(3) R . The charged fermion masses arise from the following Lagrangian

MSSM fields
Fields of the explicit breaking into A4 where H u , H d are the scalar components of usual weak doublets of the MSSM. The constants h l , h u , and h d are of order one while Λ is a cut-off. The α, β and i, j are weak and flavor indeces respectively, and ǫ αβ is the antisymmetric tensor. In ref. [26] has been shown that the minimization of a potential, invariant with respect to and the resulting charged fermion masses are the democratic mass matrix. The masses of the first and second families and that of the neutrinos arise once we include, in the Yukawa Lagrangian of eq. (4), terms that explicitly break the continuous SO(3) L ×SO(3) R but that preserve the discrete diagonal subgroup A 4 . For example, we can assume the presence of an hidden scalar sector that breaks spontaneously the continuous The explicit breaking terms will be of the form where the fields in eq. (6) transform according to table I and Λ is the cut-off scale of the model. The labels α, β are again weak indeces, σ n are the Pauli matrices and ∆ n ,∆ n are weak triplets, as reported in table I. We have introduced an additional Z 5 symmetry that affects only the scalar sector and avoids the presence of unwanted Yukawa couplings as done for instance in [8,14]. We remember that, if a = (a 1 , a 2 , a 3 ) and b = (b 1 , b 2 , b 3 ) are two A 4 triplets, Under the hypothesis that the breaking of SO(3) L × SO(3) R into A 4 happens in a hidden scalar sector and then it is transmitted to the fermions through the integration of the heavy fields, it is quite natural to assume that the explicit breaking terms in eq. (6), to be added to the Lagrangian of eq.  (3) is broken into A 4 as explained in [33].
When φ takes vev as φ = v φ (1, 1, 1) we have for the charged leptons with γ l i = δ l i v φ /Λ and the two δ i arise by the two different contractions of A 4 . Similar expressions are obtained for the quarks. The effect of the explicit breaking terms in the mass matrices is translated in a perturbation of the democratic mass matrix of eq. (5), that is with the obvious correspondences m f corresponding to the U of eq. (2) with θ = π/4, α = 2π/3, β = 5π/6 and γ = π/2. M f of eq. (8) gives an heavy 3rd family mass m f 3 and small 1st and 2nd family masses satisfying (10)

B. Neutrino sector
The Yukawa interactions for the neutrinos are the following where the scalars ξ ′ and φ ′ are singlets of the weak SU (2) L and transform with respect to A 4 as 1 ′ and 3 respectively. The scalars ∆ and∆ are singlets of A 4 and triplets of the weak SU (2) L . When the triplet field φ ′ takes vev in the A 4 direction φ ∼ (0, 0, 1) -notice that this alignment is different from the one used in many models as for example in [5,8] -, the resulting neutrino mass matrix is given by The charged leptons are diagonalized by L →Ũ ω L, so we obtain a tribimaximal mixing for the lepton sector, that is and the the neutrino masses result to have the same expressions of [8].  In eq. (6) we have reported the leading A 4 invariant terms that arise after the explicitly breaking of SO(3) L ×SO(3) R . We include now the higher order operators suppressed by powers of the cut-off scale Λ. The first terms at order 1/Λ 2 that change the structure of the charged fermion mass matrices above are With the inclusion of these contributions the charged fermion mass matrices of eq. (8) become The g f i , i = 1, 2, arise from the possible different contractions of 3-plet of A 4 to give a singlet 1 ′′ and the factor 3 is introduced to simplify the subsequent formulas. In the basis rotated byŨ ω of eq. (9), namelyM f ≡Ũ † ω M f ef fŨ ω , the charged fermion mass matrices are now approximatively given bỹ Let's assume that the ǫ f i are arbitrary parameters of O(λ 5 ) , where λ is the Cabibbo angle. The crucial point is that this assumption has the consequences that the higher order operators give negligible effects in the down and charged lepton sectors, since for the down and charged leptons we have (r d,l 1 , r d,l 2 , r d,l 3 ) ∼ (λ 4 , λ 2 , 1) andM d,l may be considered diagonals. On the contrary for the up quarks we have that (r u 1 , r u 2 , r u 3 ) ∼ (λ 7 , λ 4 , 1) and therefore the off-diagonal entries (1,2) and (2,1) cannot be neglected: the matrix M u is diagonalized by a rotation in the 12 plane with sin θ 12 ≈ λ. This rotation produces the Cabibbo angle in the CKM. In fact while M d is still diagonalized by U ω , we have that M u is diagonalized by are unitary matrix, rotations in the 12 plane, and therefore the CKM mixing matrix is given by The charm and top quark masses are almost unaffected by the corrections and still are given byṽ u r u 2 andṽ u r u 3 respectively. The up quark mass is obtained by tuning the ǫ u i and is given more or less by In [13] the full CKM was obtained by breaking the Z 2 symmetry that survives when a triplet of A 4 takes vev in the direction (1, 0, 0). In our model we suggest that the origin of the Cabibbo angle is instead in the A 4 invariant subleading corrections to the Yukawa interactions. The breaking of the residual Z 2 symmetry allows instead to generate the complete CKM. The main difference between our model and some previous models, where the subleading corrections in the charged lepton matrix are too small to generate a Cabibbo angle in order to keep the lepton mixing angles inside the bounds given by the experimental data, is related to the different assignment and the U (1) flavor symmetry one introduces in order to explain the mass hierarchies. For example, in [32] the left-handed fields belonged to a triplet of A 4 , while at the right-handed fields was given the assignment 1, 1 ′′ , 1 ′ and they have U (1) charges (2q, q, 0) where q is a real number.
III. GRAND UNIFIED GROUP SO(10) × SU (3) As already explained in the introduction our final aim would be the construction of a grand unified SO(10)-like model. Let us assume the group A 4 as flavor symmetry and the "constrain" of assigning right and left-handed fermion fields to the same representations. Since A 4 has four irreducible representations, three singlets 1, 1 ′ and 1 ′′ , and a triplet 3, clearly we have just few possibilities. For example if we assign the three 16-plets to 1, 1 ′ and 1 ′′ we obtain a mass matrix for the charged fermions of the form where α and β are arbitrary parameters, that gives for instance the wrong prediction m c = m t . The situation is better only when the three 16-plets transform as a triplet of A 4 . Indeed, it has been showed in [5] that the assignment of both left-handed and right-handed SM fields to triplets of A 4 , that is therefore compatible with SO(10), can be lead to the charged fermion textures proposed by E.Ma [4] and given by with h f 0 , h f 1 and h f 2 distinct parameters. In [5], in order to obtain a mass matrix of the form of M f in eq. (17) without spoiling the predictions of the neutrino sector, higher order operators were introduced containing simultaneously the SO(10) representations 45 T 3R and 45 Y that took vevs in the isospin and hypercharge directions respectively. A renormalizable SO(10) × A 4 model has been recently studied in [34] where however the A 4 flavor symmetry does not enforce a tribimaximal mixing in the lepton sector.
The group SO(3) L × SO(3) R is not compatible with SO(10) since the 16-plet contains both left-handed and right-handed fields that belong to different representations of SO(3) L × SO(3) R . We have therefore to search for a continuous group larger than SO(3) L ×SO(3) R , with rank bigger than 2+2 = 4, and that has a triplet as fundamental representation. The group SU (3) seems us a good candidate. The scalar field Φ ij ∼ (3, 3) of the model we have just considered will correspond to the6 representation of SU (3) whose vev is compatible with the democratic mass matrix. Without entering into the details of the realization of an SO(10) × SU (3) model [35,36,37,38,39] that we leave for a future work, we want to suggest how its realization could be achieved using non renormalizable operators. We can think that such operators arise by integrating out some heavy extra fermions that are coupled to the matter fields at the renormalizable level, for instance see [40,41,42,43]. The effective SO (10)  where the scalar fields 126 s,t are a singlet 1 ′ and a triplet of A 4 respectively, the 45 ′ T3R , 45 ′ Y are other scalars that transform as 45 of SO(10), singlet and triplet of A 4 respectively. The 10 is a singlet of A 4 . It is not difficult to show that when type-II seesaw is dominant, the first term in δL A4 generates the light neutrino mass matrix of the form of eq. (12). The second term in δL A4 gives a contribution like in eq. (6) and, after the breaking of A 4 into Z 3 , it generates the first and second family masses, see eqs. (8)- (10).

IV. CONCLUSIONS
In this paper we have proposed an embedding of the discrete A 4 flavor symmetry in the larger continuous group SO(3) L × SO(3) R that explains in a natural way the huge hierarchy between the 3rd family charged fermion masses and the others two. This is a consequence of the fact that SO(3) L × SO(3) R breaks spontaneously into S 3L × S 3R and gives a democratic mass matrix that has only one massive eigenstate. If such eigenstate is assumed to be the 3rd family state, we still have an undeterminated 12 angle in the charged lepton sector that is fixed by breaking the democratic mass matrix. The crucial feature of our model is that once we break explicitly SO(3) L × SO(3) R into A 4 we automatically generate first and second family charged fermion masses m 1,2 ≪ m 3 . In order to fit the hierarchy between the masses of the first and second families, we require a tuning. Assuming that the light neutrino Yukawa interactions come from the couplings with an A 4 singlet ξ ∼ 1 ′ and an A 4 triplet φ ′ that are scalar electroweak singlets and that φ ′ acquires vev in the direction (0, 0, 1), we have showed that the lepton mixing matrix is the tribimaximal one. The CKM is given by the identity matrix. Afterward we suggest how to generate the Cabibbo angle in the quark sector through the introduction of higher order corrections. In particular in our model higher order operators give corrections of the same magnitude in each entries of all charged fermion mass matrices. Assuming that the ratio between the correction and m c is of the order of the Cabibbo angle λ, we obtain that a rotation of order λ in the 12 plane appears in the up mass matrix. However the down and charged lepton mass matrices are almost unaffected by corrections. This mismatching gives up to the Cabibbo angle.