Hybrid seesaw neutrino model in SUSY $SU(5)\times \mathbb{A}_{4}$

Motivated by recent results from neutrino experiments, we study the neutrino masses and mixing in the framework of a SUSY $SU(5)\times \mathbb{A}_{4} $ model. The hybrid of Type I and Type II seesaw mechanisms leads to the nonzero value of the reactor angle $\theta_{13}\neq 0$ and to the recently disfavored maximal atmospheric angle $\theta_{23} \neq45^{\circ}$ by the NOvA experiment. The phenomenological consequences of the model are studied for both normal and inverted mass hierarchies. The obtained ranges for the effective Majorana neutrino mass $m_{\beta \beta}$, the electron neutrino mass $m_{\nu_{e}}$, and the $CP$ violating phase $\delta_{CP}$ lie within the current experimental allowed ranges where we find that the normal mass hierarchy is favored over the inverted one.


INTRODUCTION
The neutrino oscillation experiments performed in the past two decades provided many decisive evidences of nonzero neutrino masses and large neutrino mixing [1][2][3][4][5][6]. The atmospheric, solar, and reactor neutrino experiments have provided the measurements of the mass-squared differences ∆m 2 ij as well as the mixing angles θ ij ; the current neutrino oscillation data can be found in the latest global fit analysis [7][8][9]. To understand the origin of these masses-which are very tiny-and mixing, we must go beyond the standard model (SM) that predicts massless neutrinos. Theoretically, the most prominent way to generate such tiny masses for neutrinos is through the famous seesaw mechanism, which requires the introduction of extra heavy fermions (Type I and Type III seesaws) or scalars (Type * Electronic address: E-mail: h-saidi@fsr.ac.ma II seesaw) into the SM [10,11], giving rise to neutrino masses of Majorana type. For the neutrino mixing angles, it was not until 2012 that the reactor angle θ 13 was discovered to be different from zero [3], but unlike the other two mixing angles θ 12 and θ 23 , its value is relatively small. Furthermore, the NOvA experiment has disfavored recently the maximal atmospheric neutrino mixing sin 2 θ 23 = 0.5 [12]; however, whether its value is less or greater than π/4 is yet to be discovered. In the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix that describes these angles, θ 13 always appears in combination with the Dirac phase, and thus, the discovery of its nonzero value has a crucial influence on the Dirac CP violating (CPV) phase δ CP where its measurement is the ultimate objective of the long baseline neutrino oscillation experiments [13]. The recent progress in neutrino physics motivated theoretical as well as experimental physicists to search for new physics beyond the SM. This concerns the preexisting theories and models such as supersymmetric grand unified theories (SUSY GUTs) which unlike the non-SUSY GUTs solve the hierarchy problem and unification of gauge couplings just by introducing supersymmetry; thus, they are adopted as one of the most appealing extensions of the SM [14]. Moreover, an attractive way to outline the observed neutrino mass hierarchies and mixing within SUSY-GUT models is through discrete flavor symmetries. Indeed, several models beyond SM have used different non-Abelian groups and described successfully all the neutrino mixing angles; see Table 3 of Ref. [15] and Ref. [16]. In fact, these non-Abelian discrete groups are widely adopted to describe the large mixing angles in the lepton sector. In particular, these groups lead to a specific form of the neutrino mass matrix which is consistent with tribimaximal mixing (TBM). This special mixing induces θ 13 = 0 and θ 23 = π/4; however, it is now ruled out by the discovery of the nonzero reactor angle as mentioned above. Thus a small deviation from TBM is required to reconcile with the small value of θ 13 as well as a small deviation from the maximal value of the atmospheric angle θ 23 . In this regard, several ways have been proposed to generate a small deviation of these mixing angles. For example, the deviation from TBM in flavor symmetry-based models can arise from (i) the diagonalization of the charged lepton mass matrix [17], (ii) perturbing the vacuum expectation value (VEV) alignment [18], (iii) the Yukawa sector [19], or (iv) the Majorana sector [16,20]. These deviations are generally realized by introducing next-to-leading-order effective operators while the leading contribution is produced by one of the seesaw mechanisms. On the other hand, it was claimed in Ref. [21] that the required deviations from the TBM matrix can be interpreted as the interplay of two different seesaw mechanisms making what is known as hybrid neutrino masses. This hybrid has been used by many authors to account for the nonzero reactor angle θ 13 = 0 in the framework of the SM and GUTs; see, for example, Ref. [22].
In this paper we propose a neutrino model in the framework of a supersymmetric SU (5) GUT extended by three right-handed neutrinos N i and a 15-dimensional Higgs H 15 transforming respectively as a triplet and a nontrivial singlet under A 4 flavor symmetry. The theoretical predictions of our proposal concerning the mixing angles and masses are compatible with the latest neutrino experimental data. The main line of our proposal is as follows: First, we consider SUSY SU(5)× A 4 theory and generate the neutrino mass matrix by the hybrid seesaw mechanism. In this hybrid, the dominant mass contribution comes from Type I seesaw, leading to the TBM [23]. A small perturbation responsible for nonzero reactor angle θ 13 and nonmaximal atmospheric angle θ 23 is realized by the 15-dimensional SU(5) Higgs that contains an SU(2) L Higgs triplet ∆ d via Type II seesaw mechanism. Then, we perform a numerical study, where we use the experimental allowed ranges of the mixing angles and the mass-squared differences, to examine the octant degeneracy of θ 23 for both normal and inverted mass hierarchies. Next, we use the current neutrino oscillation data as well as the cosmological limit on the sum of neutrino masses to study the phenomenological consequences of our proposal for both normal and inverted mass hierarchies. We find that the allowed ranges of the effective Majorana neutrino mass m ee , the sum of neutrino masses 3 i=1 |m i |, the effective electron neutrino mass m β , and the Dirac CPV phase δ CP are within the current experimental data.
To perform this study, we use known results on SUSY SU (5) as well as properties of the alternating group A 4 . This flavor symmetry is generally admitted as the most natural and economical discrete group that captures the family symmetry as motivated in the literature [24].
The discrete A 4 possesses two generators S, T and four irreducible representations that can be labeled by their characters as 1 (1,1) , 1 (1,ω) , 1 (1,ω 2 ) , and 3 (−1,0) . These four representations, which are related to the A 4 group order by the formula 1 2 (1,1) + 1 2 (1,ω) + 1 2 (1,ω 2 ) + 3 2 (−1,0) = 12, are also used to host the matter and Higgs content of the SUSY SU(5)× A 4 proposal. For general properties on A 4 group representations and their characters, see [25,26]. This paper is organized as follows. In Sec. 2, we present the superfield content for the neutrino sector in SUSY SU(5)× A 4 . Then, we study the Dirac and Majorana neutrino mass matrices as well as the deviations of θ 13 and θ 23 from their TBM values. In Sec. 3, we study the phenomenological implications of the proposal and provide the predictions regarding the effective Majorana mass m ee , the effective mass m β , the sum 3 i=1 |m i |, and the CPV phase δ CP . In Sec. 4, we give our conclusion. In order to make the paper more self-contained we add Appendix A on the charged sector where we show that a U(1) flavor symmetry is needed to control the couplings of the model. We also add in the same appendix a brief discussion on the well-known dangerous four-and five-dimensional operators leading to the rapid proton decay and show how they are suppressed in our model due to the flavor symmetry.

SU (5) GUT WITH A 4 FLAVOR SYMMETRY
In this section, we first describe the superfield content of our SU(5) × A 4 GUT proposal.
Then, we use a hybrid seesaw mechanism to study the deviation of the θ 13 and θ 23 angles in this proposal. After that, we study the mass-squared differences as functions of the space parameters of the model and the θ 23 and θ 13 mixing angles. Here we focus our attention on the neutrino sector in SUSY SU(5) GUT promoted by an A 4 flavor symmetry. Thus, we give only the superfield content needed to generate the mass terms for the neutrinos. In our construction of SUSY SU(5) × A 4 GUT, we proceed as follows: (i) First, we extend the fermion sector of SU(5) GUT by adding three right-handed neutrinos N i which are SU(5) gauge singlets and sit together in the A 4 triplet 3 −1,0 . These models that describe successfully all the mixing angles. Some of these models that used at least three flavon superfields in the neutrino sector are given in Ref. [27].
(ii) Second, we extend the Higgs sector of SUSY SU(5) GUT by adding a 15-dimensional Higgs 15 ∆ d ≡ H 15 which contains a Y = 2 SU(2) L Higgs triplet ∆ d . This leads to a Majorana mass matrix M II ν via the Type II seesaw mechanism as exhibited by the Yukawa coupling5 m ⊗ 15 ∆ d ⊗5 m . When added to m I ν , the matrix M II ν will play the role of a perturbation inducing a deviation from the TBM values. Notice that H 15 has been first used in non-SUSY SU (5) without flavor symmetry to achieve the gauge coupling unification and the generation of tiny neutrino masses [28]. Notice also that the deviation from TBM by Type II seesaw mechanism with discrete flavor A 4 has also been considered in SM to reconcile with the experimental value of θ 13 [29]. In our SUSY SU(5) × A 4 proposal which extends this approach to supersymmetric GUT models building, we took into account the latest experimental results on neutrino masses and mixing, and we successfully produced the nonzero value of θ 13 as well as the nonmaximal value of θ 23 .
So the superfield content of our proposal is as follows: (a) matter containing three gener-  Table I.
where the decompositions of 5 H d and 15 ∆u are understood.
2.2. Deviation of θ 13 and θ 23 in SU (5) × A 4 hybrid seesaw We start with the leading approximation where the neutrino mass matrix is generated through Type I seesaw mechanism and is consistent with TBM predicting the mixing angles: sin 2 θ 12 = 1 3 , sin 2 θ 23 = 1 2 , and sin 2 θ 13 = 0. Then, we make use of the 15-dimensional SU(5) Higgs 15 ∆ d that contains an SU (2) L Higgs triplet ∆ d leading to Majorana mass term via Type II seesaw mechanism. Hence, the total neutrino mass matrix combines both Type I and Type II seesaws, allowing a reconciliation with the experimental values of the mixing angles θ 13 and θ 23 .

TBM from Type I seesaw mechanism
The Type I seesaw formula incorporates both Dirac and Majorana mass matrices where the Dirac mass matrix m D is obtained from the superpotential term involving the couplings among the superfields N i , F i , and H 5 while the Majorana mass matrix M R is obtained from the superpotential involving the coupling of right-handed neutrinos N i with themselves.
As we mentioned before, both F i and N i live in the A 4 triplet 3 −1,0 while the Higgs H 5 is assigned to the trivial singlet. The leading order superpotential for neutrino Yukawa couplings respecting gauge and A 4 symmetries is given by where λ 1 is a Yukawa coupling constant. Using the tensor product of A 4 irreducible representations in the Altarelli-Feruglio basis [25,30], the superpotential (2.2) reads When the Higgs doublet develops its VEV as the usual H u = υ u , we get the Dirac mass matrix of neutrinos as As for the Majorana mass matrix, the superpotential respecting gauge and flavor symmetries of our model are given by where we have added the second term involving the flavon Φ to satisfy the TBM texture and to generate appropriate masses for the neutrinos. This term-which is at the renormalizable level-will contribute to all the entries in the Majorana mass matrix. By using the multiplication rules of A 4 , the superpotential W R develops into and by taking the VEV of the flavons Φ as Φ 1 = Φ 2 = Φ 3 = υ Φ , we find the Majorana neutrino mass matrix M R given by The light neutrino mass matrix is obtained using Type I seesaw mechanism formula m I ν = −m D M −1 R m T D with the Dirac mass matrix as in Eq. (2.4), and we find where we have adopted the following parametrization Moreover, the values of the parameters a and b are related as a = 1 − 2b; this property will be used in our numerical study. The matrix m I ν respects the well-known µ − τ reflection symmetry [31], and the condition among the elements m I (2.10)

Deviation using Type II seesaw mechanism
Now we turn to study the deviation from TBM, which consists of inducing a small perturbation in the neutrino mass matrix. This deviation is motivated by the fact that the current experimental data on solar and atmospheric mixing angles are inadequate with the TBM values. The current 3σ ranges of the three mixing angles obtained from the global analysis in Ref. [9] are given by for a normal (inverted) mass hierarchy. As mentioned above, the perturbation is carried out through Type II seesaw, which implies the introduction of a scalar SU (2) L triplet ∆ d belonging to the 15-dimensional representation H 15 of the SU(5) gauge group. The

SU(5) × A 4 -invariant superpotential induces the Yukawa coupling involving ∆ d as
neutrino mass matrix reads as follows: where we factored this matrix by m 0 to form a dimensionless deviation parameter ε as well as to ease the hybridization between the seesaw mechanisms. Even though the tiny mass of neutrinos is encoded in the VEV of the Higgs triplet-which is expressed as the ratio of the Higgs doublets VEVs and the Higgs triplet mass [11]-in ordinary seesaw Type II models, in the present paper we will discuss its contribution only through the deviation parameter ε as we will see later when we perform a numerical study concerning the oscillation parameters.
In addition, it is well known that the phenomenological constraint from the ρ parameter that measures the ratio between the neutral and charged currents [32] restricts the VEVs of the Higgs multiplets higher than dimension two [33]. As in our model the calculation of the ρ parameter requires taking into consideration at least three kinds of Higgs superfieldsnamely an SU(2) triplet that belongs to 15 ∆ d with hypercharge Y = 2, an SU(2) triplet that belongs to 15 ∆u with Y = −2, and an SU(2) triplet that belongs to 24 H with Y = 0-we leave detailed investigations to future work. Now, we turn to the total neutrino mass matrix generated by the hybrid seesaw mechanism that consists of combining the contribution of Type II seesaw in Eq. (2.13) and the one arisen from the Type I seesaw in Eq.
where a, b, and c are as given in Eq. (2.9). The neutrino mass matrix is diagonalized by a transformation such as m diag ν =Ũ T m νŨ where the system of eigenvectors and eigenvalues can be developed as power series of ε; we find up to order O(ε 2 ), the matrixŨ given in terms of its eigenvectors as and eigenvalues Consequently, the mixing angles θ 13 and θ 23 become while the solar angle maintains its TBM (maximal) value sin θ 12 = 1/ √ 3. We have now a nonvanishing reactor angle θ 13 and a small shift from the TBM value for the atmospheric angle θ 23 .

Mass-squared differences and mixing angles
Concerning neutrino masses, the current neutrino oscillation experiments are only sensitive to mass-squared differences where we distinguish between two mass hierarchies: normal mass hierarchy (NH) where m 1 < m 2 < m 3 and inverted mass hierarchy (IH) where Their 3σ experimental ranges are given by [9] 0.0000703 ≤ ∆m 2 21 ≤ 0.0000809, (0.002399)0.002407 ≤ ∆m 2 3l ≤ 0.002643(0.002635) (2.18) with l = 1 (l = 2) for NH (IH). In our proposal, by using the masses in Eq. (2.16), the solar ∆m 2 21 and atmospheric ∆m 2 3l mass-squared differences up to first order in ε are expressed as By using the mixing angles in Eq. (2.17), we show in the left panel (right panel) of Fig. 1 the correlation among the parameters sin θ 23 , ε, and sin θ 13 for the NH case (IH case). The experimental inputs of the mass squared differences ∆m 2 31 (∆m 2 32 ) as well as their expressions given in Eq. (2.19) are taken into account. Before we discuss the ranges of the oscillation the parameter c is allowed to vary freely. Moreover, as the parameter of deviation ε has to be small, we have taken its range to be around O( 1 10 ). We have also fixed m 0 in the range 0, 1 10 since it is well known that the mass of the right-handed neutrinos-proportional to m R -lies at a scale beyond the reach of present experiments, and it is usually taken at the GUT scale in grand unified theories.
As a follow-up to the above discussion, it is clear that the intervals of the parameters a, b, and c-expressed as a function of α = (λ 2 υ Φ /3m R )-are fixed according to Eq. (2.9).
However, in order to find their restricted ranges compatible with the oscillation experiments, we plot in Fig. 2  tively. These new ranges will be used as inputs to perform a numerical study concerning the phenomenology of neutrino in the next section.

PHENOMENOLOGICAL IMPLICATIONS
In this section, by using the model parameters that are restricted by the 3σ experimental values of the mixing angles and the mass-squared differences, we show by means of scatter plots for both hierarchies the physical observables m ee and m β related respectively to neutrinoless double beta decay and tritium beta decay experiments, and we also provide scatter plot predictions on the sum of neutrino masses as well as on the Dirac CP violating phase.

Neutrinoless double beta decay
One of the most known neutrino mass related experiments is the neutrinoless double beta decay (0νββ) process, which has not been observed yet. Its discovery would prove that neutrinos are Majorana particles, and it would also prove that the lepton number L is violated. The decay amplitude for the 0νββ process is proportional to the effective Majorana neutrino mass given by [35] where m i are the three neutrino masses and U ei are the elements of the first row of the PMNS matrix [36]. In our proposal, this mixing matrix is given bỹ where α and β are the Majorana CP violating phases andŨ is given in Eq. (2.15). Currently, the most recent bounds of m ee come from the KamLAND-Zen [37] and GERDA [38] experiments; they are respectively given by      [39]. In particular, the obtained ranges can be tested in future experiments like KamLAND-Zen, which plans to reach a sensitivity below 50 meV on |m ee |, and thus, it will start to constrain the inverted mass hierarchy region [40].

Tritium beta decay
The tritium beta decay is the most sensitive direct way to measure the absolute neutrino mass scale ignoring the nature of neutrinos [41]. The limit (at 95% C.L.) from the Troitsk and Mainz experiments of the effective electron neutrino mass are, respectively, given by m β < 2.12 eV and m β < 2.3 eV [42,43], while the current generation of neutrino mass measurement comes from the KATRIN experiment with a sensitivity of m β < 0.2 eV (at 90 % C.L.) [44]. The quantity m β (or m νe ) is defined in terms of the mass eigenvalues m i and mixing matrix elements U ei : In terms of our model parameters, it is expressed as (  for both mass hierarchies. However, the expected future sensitivity from Project 8 [45] is as low as 0.04 eV, which means that only the range corresponding to NH is allowed.

Sum of neutrino masses
Although the absolute mass scale of the neutrinos remains unknown, the sum of the three light neutrino masses 3 i=1 |m i | is constrained by a cosmological upper bound given by the Planck Collaboration's limit 3 i=1 |m i | < 0.17 eV [46]. In our model, the sum of neutrino masses is expressed in terms of the model parameters as (3.6) Using the 3σ ranges of mass-squared differences (2.18) and mixing angles (   for IH. Thus, for both mass hierarchies, the sum of neutrino masses gets more restricted as compared to the Planck limit, and these ranges may be tested in future cosmological observations.

Dirac CP violation
The Dirac CPV phase δ CP is one among the unknown quantities in the physics of neutrino, and its measurement becomes more important when recent experiments reported the nonzero value of the reactor angle θ 13 as they are related in the PMNS matrix. Moreover, estimations on the CPV phase δ CP can be obtained by considering the Jarlskog invariant quantity J CP which is defined as J CP = Im{U µ3 U * e3 U e3 U * µ3 } and by using the PMNS matrix. It is expressed as [47] J CP = cos θ 12 sin θ 12 cos θ 23 sin θ 23 cos θ 2 13 sin θ 13 sin δ CP , where the allowed ranges at 3σ of sin θ 12 , sin θ 23 , and sin θ 13 are given in Eq. (2.11) while the allowed 3σ ranges of CPV phase δ CP are giving by [9] 0 ≤ δ CP ≤ 2π for NH , 0.8π ≤ δ CP ≤ 2.17π for IH.   corresponding δ CP and sin θ 23 ranges extracted from Fig. 6.

CONCLUSION
In this work, we have constructed a renormalizable hybrid seesaw neutrino model in the framework of SUSY SU(5) GUT extended by a discrete A 4 family symmetry. The dominant TBM pattern is obtained from Type I seesaw mechanism while Type II seesaw is responsible for a small deviation from TBM. Both seesaws are controlled by the action of the A 4 flavor symmetry through its algebraic properties. We found that the predictions of our proposal concerning the mixing angles and masses are consistent with the recent measurements. In particular, we showed that the deviation by Type II seesaw leads to a nonmaximal atmospheric angle θ 23 as reported recently by the NOvA experiment and a nonvanishing reactor angle θ 13 . Thus, we made a full analysis depending on the octant of θ 23 .
We also studied the phenomenological consequences of our proposal where we showed through scatter plots the allowed ranges for the physical observables and model parameters which we have restricted by using the 3σ ranges of the neutrino oscillation parameters for both mass hierarchies. We found also that the sum of neutrino masses and CPV phase are within the allowed experimental regions. Furthermore, we found that the ranges of the physical observables involving the effective Majorana neutrino mass m ββ and the electron neutrino mass m β are preferred in the case of normal mass hierarchy. For the latter, the obtained range of m β in the inverted mass hierarchy case is forbidden by future sensitivity from Project 8.  Table III.Furthermore, in order to achieve the correct mass hierarchy and to get These couplings which destroy the form of neutrino mass matrix (2.14) that led to the desired oscillation parameters must be suppressed. This is possible if we assume that υ Φ ≪ Λ, which is acceptable according to Eqs. (2.7) and (2.9). On the other hand, even if the VEV of the flavon Φ is around the cutoff scale-say Φ ≃ Λ-this would just give terms that are relative to the leading ones:  With the A 4 ×U(1) charge assignments shown in Table IV, the usual renormalizable Yukawa where Y 1 , Y 2 , Y 3 , and Y 45 are the Yukawa mass matrices and Λ represents the cutoff scale of the model. Notice that the coupling T 2 F iH 45 Φ Λ is also allowed by the symmetries of the model, but again its suppression is guaranteed by the condition υ Φ ≪ Λ. Using A 4 tensor products, the superpotential W d,e develops into The masses arise from the breaking of A 4 × U(1) family symmetry as well as the breaking of the electroweak symmetry. Therefore, by taking the flavon triplet VEVs along the directions where r = υ d υ η /Λ, h = υ d υ σ /Λ, and t = υ d υ ρ /Λ. By assuming we diagonalize the mass matrices M d and M e where we find that the masses of down-type quarks and charged leptons are respectively given by where these masses imply the Georgi-Jarlskog relations given in Eq. (A.3). Notice that these mass relations are admissible at the GUT scale at leading order and can be improved assuming the SUSY threshold corrections and appropriate values of tan β = υu υ d ; for more details on the SUSY threshold corrections procedure see Refs. [49,50]. On the other hand, an alternative way to go beyond the b−τ unification in GJ predictions at high scale is through higher dimensional effective operators [49,51]. These operators involve additional Higgses in 24 H or 75 H and a nontrivial SU(5) messenger fields X and X allowing for relations such as m τ = 3 2 m b . All possible relations between down-quark and the charged lepton masses are listed in Table 1 of Ref. [49] and Table 2 of Ref. [51]. One of these GUT scale relations using fermion and scalar messenger fields is studied in the framework of SU(5) × A 4 in Ref. [52].
Regarding the up-type quark sector, besides the top quark mass which is preferred to arise from a renormalizable coupling, the remaining up and charm quark masses are derived from higher dimensional Yukawa couplings involving flavon superfields. Indeed, in our model, two different flavons χ and ϕ couple to the first and second generations, respectively. Thus, the superpotential of the up-type quarks respecting gauge and flavor symmetries takes the form where y u , y c , and y t are the Yukawa coupling constants for up-, charm-, and top-type quarks.
As usual, the up-type quark masses arise from the breaking of the flavor and electroweak symmetries. Thus, when the flavons ϕ and χ and the Higgs H u develop their VEVs as we obtain a diagonal mass matrix of the up-type quarks given by with the mass eigenvalues as The large mass of the top quark is obtained at tree level, while the mass hierarchy among the first two generations of up-type quarks can be obtained by assuming a hierarchy between the VEVs of the flavons χ and ϕ.
As for the mixing in the quark sector, it is defined as |U Q | = U † up U d where U d is the matrix that diagonalizes the mass matrix of down quarks M d while U up is the one that diagonalizes the mass matrix of up quarks M up . Since this latter is diagonal (A.12), U up is just the identity matrix, and thus, the total mixing matrix is the one that diagonalizes the mass matrix of the down quarks M d (A.8); we find Notice that the zero entries in the mixing matrix (A.14) can be seen to be a first approximation to the mixing matrix V CKM of the quark sector [35]. From this matrix, it is clear that the charged lepton mixing angles θ l 13 and θ l 23 are both equal to zero; thus in our model the mixing from the charged lepton sector does not affect the mixing angles of the neutrino sector given in Eq. (2.17). Notice by the way that the total mixing in the lepton sector U PMNS = U † eŨ is proportional toŨ with a small shift of the mixing angle θ l 12 . We end this appendix by giving comments concerning the well-known four-and fivedimensional operators that contribute to fast proton decay in supersymmetric SU(5) GUT models. In this respect, the dangerous proton decay terms arise from the dimension four λ ijk 10 i m5 j m5 k m and dimension five λ ij λ kl 10 i m 10 j m 10 k m5 l m operators. These operators are dangerous in the sense that they lead to proton decay rates far larger than the experimental limits. As regards to the former operators, they contribute to the proton decay through the term violating baryon number (U c 1 D c 1 D c k ) combined with the term (Q i L j D c k ) that violates the lepton number with family indices as i, j = 1, 2 and k = 2, 3. In fact, these operators which are renormalizable can be avoided by imposing the usual R parity as in the case of the MSSM [53]. However, in our SU(5) × A 4 × U(1) proposal, these four-dimensional operators that are given by m Ω with Ω = ϕ, χ, ρ, σ, η as the various flavon superfields used throughout the different sectors studied in this work. It is easy to check from Tables III and IV that these couplings are also not allowed as they are not invariant under the U(1) symmetry.
Regarding the five-dimensional couplings λ ijkl 10 i m 10 j m 10 k m5 l m , they are mediated by the heavy color triplet Higgsino and it is well known that their dressing diagrams 2 to form sixdimensional operators are the most disturbing operators that lead to fast proton decay in SUSY SU(5) models [55,56]. These operators that are derived from the renormalizable up and down Yukawa couplings λT T H 5 and λ ′ T F H5 are absent in our model since they behave, respectively, as nontrivial singlets and triplet under the A 4 flavor symmetry. However, the last couplings-which are required to generate masses for the charged fermions-are allowed through their interactions with the flavon superfields as given in the Yukawa couplings (A. 10) and (A.6). Thus, our model contains higher order operators of the kind 1 M T T T T F Ω Λ n where M T is the mass of the colored Higgs triplet and n = 1, 2; for n = 1 we have Ω = η, and for n = 2 the relevant combinations are Ω 2 = σχ, ρχ, ηϕ. Hence, the suppression of these operators compared to the usual five-dimensional couplings T T T F is now enhanced by the factors Ω Λ n coming from the flavon superfields required by A 4 invariance, thus