A 3-3-1 model with right-handed neutrinos based on the $\Delta\left(27\right)$ family symmetry

We present the first multiscalar singlet extension of the 3-3-1 model with right-handed neutrinos, based on the $\Delta \left( 27\right) $ family symmetry, supplemented by the $Z_{4}\otimes Z_{8}\otimes Z_{14}$ flavor group, consistent with current low energy fermion flavor data. In the model under consideration, the light active neutrino masses are generated from a double seesaw mechanism and the observed pattern of charged fermion masses and quark mixing angles is caused by the breaking of the $\Delta \left( 27\right) \otimes Z_{4}\otimes Z_{8}\otimes Z_{14}$ discrete group at very high energy. Our model has only 14 effective free parameters, which are fitted to reproduce the experimental values of the 18 physical observables in the quark and lepton sectors. The obtained physical observables for the quark sector agree with their experimental values, whereas those ones for the lepton sector also do, only for the inverted neutrino mass hierarchy. The normal neutrino mass hierarchy scenario of the model is disfavored by the neutrino oscillation experimental data. We find an effective Majorana neutrino mass parameter of neutrinoless double beta decay of $m_{\beta \beta }=$ 22 meV, a leptonic Dirac CP violating phase of $34^{\circ }$ and a Jarlskog invariant of about $10^{-2}$ for the inverted neutrino mass spectrum.

be understood in the framework of models with SU (3) C ⊗ SU (3) L ⊗ U (1) X gauge symmetry, called 3-3-1 models for short, where U (1) X is a nonuniversal family symmetry that distinguishes the third fermion family from the first and second ones [25,59,60,72,73,103,105,. These models have several interesting features. First, the existence of three generations of fermions is a consequence of the chiral anomaly cancellation and the asymptotic freedom in QCD. Second, the large mass hierarchy between the heaviest quark family and the two lighter ones can be understood from the fact that the former has a different U (1) X charge from the latter. Third, these models include a natural Peccei-Quinn symmetry, which addresses the strong-CP problem [162][163][164][165]. Finally, versions with heavy sterile neutrinos include cold dark matter candidates as weakly interacting massive particles (WIMPs) [166]. Besides that, the 3-3-1 models can explain the 750 GeV diphoton excess recently reported by ATLAS and CMS [167][168][169][170] as well as the 2 TeV diboson excess found by ATLAS [171].
In the 3-3-1 models, the electroweak gauge symmetry is broken in two steps as follows. First the SU (3) L ⊗ U (1) X symmetry is broken down to the SM electroweak group SU (2) L ⊗ U (1) Y by one heavy SU (3) L triplet field acquiring a Vacuum Expectation Value (VEV) at high energy scale v χ , thus giving masses to non SM fermions and gauge bosons. Second, the usual EWSB mechanism is triggered by the remaining lighter triplets, with VEVs at the electroweak scale υ ρ and v η , thus providing SM fermions and gauge bosons with masses [25].
In Ref. [130] we have proposed a 3-3-1 model with ∆ (27) flavor symmetry supplemented by the U(1) L new lepton global symmetry that enforces to have different scalar fields in the Yukawa interactions for charged lepton, neutrino and quark sectors, thus allowing us to treat these sectors independently. The scalar sector of that model includes 10 SU (3) L scalar triplets and three SU (3) L scalar antisextets. The SU(3) C ⊗ SU(3) L ⊗ U(1) X ⊗ U(1) L ⊗ ∆ (27) assignments of the fermion sector of our previous model, require that these 10 SU (3) L scalar triplets be distributed as follows, 4 for the quark sector, 3 for the charged lepton sector and 3 for the neutrino sector. Furthermore the 3 SU (3) L scalar antisextets are needed to implement a type I seesaw mechanism. In that model, light active neutrino masses are generated from type-I and type-II seesaw mechanisms, mediated by three heavy right handed Majorana neutrinos and three SU (3) L scalar antisextets, respectively. Since the Yukawa terms in that model are renormalizable, to explain the charged fermion mass pattern, one needs to impose a strong hierarchy among the charged fermion Yukawa couplings of the model.
It is interesting to find an alternative and better explanation for the SM fermion mass and mixing hierarchy, by formulating a 3-3-1 model with less scalar content than our previous model of Ref. [130]. To this end, we propose an alternative and improved version of the 3-3-1 model based on the SU(3) C ⊗ SU(3) L ⊗ U(1) X ⊗ ∆ (27) ⊗ Z 4 ⊗ Z 8 ⊗ Z 14 symmetry that successfully describes the observed fermion mass and mixing pattern and is consistent with the current low energy fermion flavor data. The particular role of each discrete group factor is explained in detail in Sec. II. The scalar sector of our model includes 3 SU (3) L scalar triplets and 22 SU (3) L scalar singlets, assigned into triplet and singlet irreducible representations of the ∆(27) discrete group. This scalar sector of our current ∆(27) flavor 3-3-1 model is more minimal than that one of our previous model of Ref. [130] and does not include SU (3) L scalar antisextets. Furthermore, our current model does not include the U(1) L new lepton global symmetry presented in our previous ∆(27) flavor 3-3-1 model. Unlike our previous ∆(27) flavor 3-3-1 model of Ref. [130], in our current 3-3-1 model, the charged fermion mass and quark mixing pattern can successfully be accounted for, by having all Yukawa couplings of order unity and arises from the breaking of the ∆ (27) ⊗ Z 4 ⊗ Z 8 ⊗ Z 14 discrete group at very high energy, triggered by SU (3) L scalar singlets acquiring vacuum expectation values much larger than the TeV scale.
In the following we summarize the most important differences of our current ∆(27) flavor 3-3-1 model with our previous 3-3-1 model also based on the ∆(27) family symmetry. First of all, the scalar sector of our current 3-3-1 model has 3 SU (3) L scalar triplets plus 22 very heavy SU (3) L scalar singlets. On the other hand, our previous ∆(27) flavor 3-3-1 model has a scalar sector composed of 10 SU (3) L scalar triplets and three SU (3) L scalar antisextets. Second, the charged fermion mass and quark mixing pattern can successfully be accounted for in our current 3-3-1 model with ∆ (27) family symmetry by having the Yukawa couplings of order unity, whereas in our previous ∆(27) flavor 3-3-1 model, a strong hierarchy of the Yukawa couplings is needed to accommodate the current pattern of charged fermion masses and the CKM quark mixing matrix is predicted to be equal to the identity matrix. Third, in our current 3-3-1 model with ∆ (27) family symmetry the light active neutrino masses arise from a double seesaw mechanism whereas in our previous ∆(27) flavor 3-3-1 model, type I and type II seesaw mechanisms generate the masses for the light active neutrinos. Finally, our current 3-3-1 model with ∆(27) family symmetry, does not include the U(1) L new lepton global symmetry presented in our previous ∆(27) flavor 3-3-1 model, but has instead a Z 4 ⊗ Z 8 ⊗ Z 14 discrete symmetry, whose breaking at very high energy gives rise to the observed pattern of charged fermion masses and quark mixing angles.
It is noteworthy that our previous ∆(27) flavor 3-3-1 model model corresponds to an extension of the original 3-3-1 model with right handed Majorana neutrinos (which includes 3 SU (3) L scalar triplets in its scalar spectrum), where 7 extra SU (3) L scalar triplets and 3 SU (3) L scalar antisextets are added to build the charged fermion and neutrino Yukawa terms needed to give masses to SM charged fermions and light active neutrinos. On the other hand, in our current ∆(27) flavor 3-3-1 model, preserves the content of particles of the 3-3-1 model with right handed Majorana neutrinos, but we add additional very heavy SU (3) L singlet scalar fields with quantum numbers that allow to build Yukawa terms invariant under the local and discrete groups. Consequently our current model corresponds to the first multiscalar singlet extension of the original 3-3-1 model with right-handed neutrinos, based on the ∆(27) family symmetry. As these singlet scalars fields are assumed to be much heavier than the 3 SU (3) L scalar triplets, our model at low energies reduces to the 3-3-1 model with right-handed neutrinos.
In this paper we propose the first implementation of the ∆(27) flavor symmetry in a multiscalar singlet extension of the original 3-3-1 model with right-handed neutrinos. In our model, light active neutrino masses arise from a double seesaw mechanism mediated by three heavy right-handed Majorana neutrinos. This paper is organized as follows. In Sect. II we outline the proposed model. In Sect. III we discuss the implications of our model in masses and mixings in the lepton sector. In Sect. IV we present a discussion of quark masses and mixings, followed by a numerical analysis. Finally we conclude in Sect. V. Appendix A provides a description of the ∆ (27) discrete group. Appendix B includes a discussion of the scalar potential for two ∆(27) scalar triplets and its minimization equations.

II. THE MODEL
The first 3-3-1 model with right handed Majorana neutrinos in the SU (3) L lepton triplet was considered in [134]. However that model cannot describe the observed pattern of fermion masses and mixings, due to the unexplained hierarchy among the large number of Yukawa couplings in the model. Below we consider a multiscalar singlet extension of the SU (3) C ⊗ SU (3) L ⊗ U (1) X (3-3-1) model with right-handed neutrinos, which successfully describes the SM fermion mass and mixing pattern. In our model the full symmetry G experiences the following three-step spontaneous breaking: and the symmetry breaking scales obey the relation We define the electric charge of our 3-3-1 model as a combination of the SU (3) generators and the identity, as follows: with I = diag(1, 1, 1), T 3 = 1 2 diag(1, −1, 0) and T 8 = ( 1 2 √ 3 )diag(1, 1, −2) for triplet. From the requirement of anomaly cancellation, it follows that the fermions of our model are assigned in the following (SU (3) C , SU (3) L , U (1) X ) left-and right-handed representations: , Regarding leptons, we group left handed leptons and right handed Majorana neutrinos in ∆ (27) triplets, whereas right handed charged leptons are assigned as ∆ (27) triplets, as follows: The Z 4 ⊗ Z 8 ⊗ Z 14 assignments for leptons are: The Z 4 ⊗ Z 8 ⊗ Z 14 assignments for quarks are: Here With the above particle content, the following Yukawa terms for the quark and lepton sectors arise: We assume that all of these dimensionless couplings are real, except for y ρτ , taken to be complex. In the following we provide a justification for the aforementioned assumption. As shown in Sect. III, having h ρτ complex is required to yield leptonic mixing angles consistent with the current neutrino oscillation experimental data. Furthermore, as shown in Sect. IV, the quark assignments under the different group factors of our model will give rise to SM quark mass texture where the Cabbibo mixing arise from the down type quark sector, whereas the up type quark sector contributes to the remaining mixing angles. As indicated by the current low energy quark flavor data encoded in the Standard parametrization of the quark mixing matrix, the complex phase responsible for CP violation in the quark sector is associated with the quark mixing angle in the 1-3 plane. Consequently, in order to reproduce the experimental values of quark mixing angles and CP violating phase, y (U) 13 is required to be complex. An explanation of the role of each discrete group factor of our model is provided in the following. The ∆ (27), Z 4 and Z 8 discrete groups are crucial for reducing the number of model parameters, thus increasing the predictivity of our model and giving rise to predictive and viable textures for the fermion sector, consistent with the observed pattern of fermion masses and mixings, as will be shown later in Sects. III and IV. The Z 4 and Z 14 symmetries reduce the number of parameters in the neutrino sector. Besides that, the Z 4 and Z 8 discrete group determine the allowed entries of the SM quark mass matrices. As a result of the Z 4 ⊗ Z 8 charge assignments for the SM quark sector given by Eq. (10), the Cabbibo mixing will arise from the down type quark sector, whereas the up sector will contribute to the remaining mixing angles. Furthermore, thanks to the ∆ (27) discrete symmetry, SM quarks do not mix with the exotic ones. This arises from the fact that the right-handed SM and exotic quarks are assigned as nontrivial and trivial ∆ (27) singlets, respectively. The Z 14 symmetry give rises to the hierarchical structure of the charged fermion mass matrices that yields the observed charged fermion mass and quark mixing pattern. Let us note that the five dimensional Yukawa operators 1 Λ L L ρS 10,0 e R , 1 Λ L L ρS 11,0 µ R and 1 Λ L L ρS 12,0 τ R are invariant under the ∆ (27) family symmetry but do not respect the Z 14 symmetry, as the right-handed charged leptons transform nontrivially under the Z 14 cyclic group. Let us note that the Z 14 symmetry is the smallest lowest cyclic symmetry, from which charged lepton Yukawa term of dimension 12 can be built, by inserting σ 7 Λ 7 on the 1 Λ L L ρS 10,0 e R operator. It is noteworthy that the small value of the electron mass can naturally arise from the aforementioned charged lepton Yukawa term of dimension 12. Furthermore, since the breaking of the ∆ (27) ⊗ Z 4 ⊗ Z 8 ⊗ Z 14 discrete group gives rise to the charged fermion mass and quark mixing pattern, we set the VEVs of the SU (3) L singlet scalar fields φ, ξ n (n = 1, 2), τ j , S j (j = 1, 2, 3) and σ, with respect to the Wolfenstein parameter λ = 0.225 and the model cutoff Λ, as follows: Let us note that the SU (3) L singlet scalar fields φ, ξ n (n = 1, 2), τ j , S j , Ω j , Θ j (j = 1, 2, 3) and σ having the VEVs of the same order of magnitude are the ones that appear in the SM charged fermion Yukawa terms, thus playing an important role in generating the SM charged fermion masses and quark mixing angles. As we will explain in the following, we are going to implement a double seesaw mechanism for the generation of the light active neutrino masses. To implement a double seesaw mechanism, we need very heavy right handed Majorana neutrinos, which implies that the SU (3) L singlet scalars should acquire very large vacuum expectation values. In addition, in order to simplify our analysis of the scalar potential for the ∆(27) scalar triplets, we need that the ∆(27) scalar triplets Ξ and Φ contributing to the right handed Majorana neutrino masses should acquire much lower VEVs than the ∆(27) scalar triplet S that gives rise to the charged lepton masses. That hierarchy in their VEVs will allow to neglect the mixings between these fields as follows from the method of recursive expansion of Ref. [172] and to treat their scalar potentials independently. Because of these reasons, we assume that the VEVs of SU (3) L singlet scalar fields Ξ j , Φ j (j = 1, 2, 3) satisfy the following hierarchy: Furthermore, implementing a double seesaw mechanism also requires that the ∆(27) scalar triplets Ω and Θ contributing to the Dirac neutrino Yukawa terms, should acquire VEVs much lower than the electroweak symmetry breaking scale v = 246 GeV. Consequently, the scalar fields of our model obey the following hierarchy: v Thus, the SU (3) L scalar singlets presented in the right-handed Majorana neutrino Yukawa interactions, acquire very large vacuum expectation values, which implies that the Majorana neutrinos acquire very large masses, hence allowing to implement a double seesaw mechanism to generate the light active neutrino masses. Consequently, the neutrino spectrum is composed of very light active neutrinos as well as heavy and very heavy sterile neutrinos. In summary, for the reasons mentioned above and considering a very high model cutoff Λ ≫ v χ , we set the vacuum expectation values (VEVs) of the SU (3) L scalar singlets at a very high energy, much larger than v χ ≈ O(1) TeV, with the exception of the VEVs of Ω j and Θ j (j = 1, 2, 3), taken to be much smaller than the electroweak symmetry breaking scale v = 246 GeV. It is noteworthy the SU at the scale Λ int , by the vacuum expectation values of the SU (3) L singlet scalar fields φ, ξ n (n = 1, 2), τ j , S j , (j = 1, 2, 3) and σ.
In the following we comment on the possible VEV patterns for the ∆(27) scalar triplets S, Ξ, Φ, Ω, and Θ. Since the VEVs of the ∆(27) scalar triplets satisfy the following hierarchy: the mixing angles of S and Ξ with Φ, Ω, and v S are very small since they are suppressed by the ratios of their VEVs, which is a consequence of the method of recursive expansion proposed in Ref. [172]. Thus, the scalar potential for the ∆(27) scalar triplet S can be treated independently from the scalar potentials for the two sets of ∆(27) scalar triplets Ξ, Φ, and Ω and Θ. Furthermore, because of the reason mentioned above, one can treat the scalar potential for Ξ, Φ independently from the one that involves Ω and Θ. As shown in detail in Appendix B, the following VEV patterns for the ∆(27) scalar triplets are consistent with the scalar potential minimization equations for a large region of parameter space:

III. LEPTON MASSES AND MIXINGS
From the lepton Yukawa terms given by Eq. (13), we find that the mass matrix for charged leptons takes the form: α and β being the complex phases of h ρτ , respectively, and the charged lepton masses given by: Regarding the neutrino sector, we see that the neutrino mass terms take the form: where the ∆ (27) family symmetry constrains the neutrino mass matrix to be of the form: As the SU (3) L scalar singlets presented in the right-handed Majorana neutrino Yukawa interactions, acquire very large vacuum expectation values, the Majorana neutrinos are very heavy, thus giving rise to a double seesaw mechanism that generates small masses for the active neutrinos. The neutrino mass matrix is diagonalized by a rotation matrix, which is approximately given by [152]: with and the neutrino mass matrices for the physical states take the form: where M are the heavy and very heavy sterile neutrino mass matrices, respectively. Thus, the double seesaw mechanism produces a neutrino spectrum composed of light active neutrinos, heavy and very heavy sterile neutrinos. Furthermore, let us note that the neutrino mass matrices M are diagonalized by the rotation matrices R ν , U R and U χ , respectively [152]. Using Eq. (24), we find that the light active neutrino mass matrix takes the form: Then we find that, for the normal (NH) and inverted (IH) neutrino mass hierarchies, the light active neutrino mass matrix is diagonalized by a rotation matrix R ν , according to: Using the rotation matrices in the charged lepton sector V L , given by Eq. (18), and in the neutrino sector R ν , given by Eqs. (28) and (29) for normal (NH) and inverted (IH) neutrino mass hierarchies, respectively, we find that the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) leptonic mixing matrix takes the form: Let us note that, according to Eqs. (18), (28) and (29), the lepton sector of our model is described by 8 effective free parameters that are fitted to reproduce the experimental values of the 8 physical observables in the lepton sector, i.e., the three charged lepton masses, the two neutrino mass squared splittings and the three leptonic mixing angles. Despite this parametric freedom, we found that the normal hierarchy scenario of our model leads to a large value of the reactor mixing angle, not consistent with the experimental data on neutrino oscillations. On the contrary, for the case of inverted hierarchy, as we will see in the following, our obtained physical parameters in the lepton sector are in excellent agreement with the experimental data. We fit the parameters A, B, C, α and β to reproduce the experimental values of the neutrino mass squared splittings and three leptonic mixing angles. By varying the  Table I: Experimental ranges of neutrino squared mass differences and leptonic mixing angles, from Ref. [8], for the case of inverted neutrino mass spectrum.
parameters A, B, C, α and β, we find the following best fit result:  Table I we see that the leptonic mixing parameters sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 and the neutrino mass squared splittings are in excellent agreement with the experimental data. We found a leptonic Dirac CP violating phase close to 34 • and a Jarlskog invariant of about 10 −2 . Now we compute the effective Majorana neutrino mass parameter, which is proportional to the neutrinoless double beta (0νββ) decay amplitude. The effective Majorana neutrino mass parameter is given by: being U 2 ej the PMNS mixing matrix elements and m ν k the Majorana neutrino masses. From Eqs. (31), (32) and (33), we obtain the following value for the effective Majorana neutrino mass parameter, for the case of an inverted mass hierarchy: Then we get a value for the Majorana neutrino mass parameter within the declared reach of the next-generation bolometric CUORE experiment [173] or, more realistically, of the next-to-next-generation ton-scale 0νββ-decay experiments. It is worth mentioning that the upper limit of the Majorana neutrino mass parameter is m ββ ≤ 160 meV, which corresponds to T 0νββ 1/2 ( 136 Xe) ≥ 1.6 × 10 25 yr at 90% C.L, as follows from the EXO-200 experiment [174]. It is expected an improvement of this bound within a not too far future. The GERDA "phase-II"experiment [175,176] is expected to reach T 0νββ 1/2 ( 76 Ge) ≥ 2 × 10 26 yr, corresponding to m ββ ≤ 100 meV. A bolometric CUORE experiment, using 130 T e [173], is currently under construction and has an estimated sensitivity close to T 0νββ 1/2 ( 130 Te) ∼ 10 26 yr, which corresponds to m ββ ≤ 50 meV. Besides that, there are proposals for ton-scale next-to-next generation 0νββ experiments with 136 Xe [177,178] and 76 Ge [175,179] which claim sensitivities over T 0νββ 1/2 ∼ 10 27 yr, corresponding to m ββ ∼ 12 − 30 meV. For a recent review, see for example Ref. [180]. Consequently, our model predicts T 0νββ 1/2 , which is at the level of the sensitivities of the next generation or next-to-next generation 0νββ experiments.

IV. QUARK MASSES AND MIXINGS
From the quark Yukawa terms of Eq. (12), it follows that the SM quark mass matrices take the form: where a k (k = 1, 2, 3), b 1 , c 1 , g 1 , f 1 , f 2 and e 1 are O(1) parameters. Here λ = 0.225 is one of the Wolfenstein parameters and v = 246 GeV the scale of electroweak symmetry breaking. From the SM quark mass textures given above, it follows that the Cabbibo mixing emerges from the down type quark sector, whereas the up type quark sector generates the remaining mixing angles. Besides that, the low energy quark flavor data indicates that the CP violating phase in the quark sector is associated with the quark mixing angle in the 1-3 plane, as follows from the Standard parametrization of the quark mixing matrix. Consequently, in order to get quark mixing angles and a CP violating phase consistent with the experimental data, we assume that all dimensionless parameters given in Eq. (35) are real, except for a 1 , taken to be complex. Furthermore, as follows from the different ∆(27) singlet assignments for the quark fields, the exotic quarks do not mix with the SM quarks. We find that the exotic quark masses are: Since the the breaking of the ∆ (27) ⊗ Z 4 ⊗ Z 8 ⊗ Z 14 discrete group gives rise to the observed pattern of charged fermion masses and quark mixing angles, and in order to simplify the analysis, we set e 1 = f 1 as well as c 1 = a 3 = 1 and g 1 = b 1 , motivated by naturalness arguments and by the relation m c ∼ m b , respectively. Consequently, there are only 6 effective free parameters in the SM quark sector of our model, i.e., |a 1 |, a 2 , b 1 , f 1 , f 2 and the phase γ q . We fit these 6 parameters to reproduce the 10 physical observables of the quark sector, i.e., the six quark masses, the three mixing angles and the CP violating phase. By varying the parameters |a 1 |, a 2 , b 1 , f 1 , f 2 and γ q , we find the quark masses, the three quark mixing angles and the CP violating phase δ reported in Table II, which correspond to the best fit values: In Table II we show the model and experimental values for the physical observables of the quark sector. We use the M Z -scale experimental values of the quark masses given by Ref. [181] (which are similar to those in [182]). The experimental values of the CKM parameters are taken from Ref. [183]. As indicated by Table II, the obtained quark masses, quark mixing angles, and CP violating phase are highly consistent with the experimental low energy quark flavor data. Note that in our previous paper [130], the CKM matrix, at the tree level, is the identity, which should be improved by higher order loop corrections.     [173] or, more realistically, of the next-to-next generation ton-scale 0νββ-decay experiments.
The ∆(27) discrete group is a subgroup of SU (3), has 27 elements divided into 11 conjugacy classes. Then the ∆(27) discrete group contains the following 11 irreducible representations: two triplets, i.e., 3 [0] [1] (which we denote by 3) and its conjugate 3 [0] [2] (which we denote by 3) and 9 singlets, i.e., 1 k,l (k, l = 0, 1, 2), where k and l correspond to the Z 3 and Z ′ 3 charges, respectively [36]. The ∆(27) discrete group, which is a simple group of the type ∆(3n 2 ) with n = 3, is isomorphic to the semi-direct product group (Z ′ 3 × Z ′′ 3 ) ⋊ Z 3 [36]. It is worth mentioning that the simplest group of the type ∆(3n 2 ) is ∆(3) ≡ Z 3 . The next group is ∆ (12), which is isomorphic to A 4 . Consequently the ∆(27) discrete group is the simplest nontrivial group of the type ∆(3n 2 ). Any element of the ∆(27) discrete group can be expressed as b k a m a ′ n , being b, a and a ′ the generators of the Z 3 , Z ′ 3 and Z ′′ 3 cyclic groups, respectively. These 3 ω 2r+sp 0 0 generators fulfill the relations: The characters of the ∆(27) discrete group are shown in Table III. Here n is the number of elements, h is the order of each element, and ω = e 2 is the cube root of unity, which satisfies the relations 1 + ω + ω 2 = 0 and ω 3 = 1. The conjugacy classes of ∆(27) are given by: {a ′2 , a 2 , aa ′ }, h = 3, C The tensor products between ∆(27) triplets are described by the following relations [36]: The multiplication rules between ∆(27) singlets and ∆ (27) triplets are given by [36]: .