The $\Delta(27)$ flavor 3-3-1 model with neutral leptons

We build the first 3-3-1 model based on the $\Delta (27)$ discrete group symmetry, consistent with fermion masses and mixings. In the model under consideration, the neutrino masses are generated from a combination of type-I and type-II seesaw mechanisms mediated by three heavy right-handed Majorana neutrinos and three $SU(3)_{L}$ scalar antisextets, respectively. Furthermore, from the consistency of the leptonic mixing angles with their experimental values, we obtain a non-vanishing leptonic Dirac CP violating phase of $-\frac{\pi }{2}$. Our model features an effective Majorana neutrino mass parameter of neutrinoless double beta decay, with values $m_{\beta \beta }=$ 10 and 18 meV for the normal and the inverted neutrino mass hierarchies, respectively.


INTRODUCTION
The discovery of the 126 GeV Higgs boson at the Large Hadron Collider (LHC) [1,2], has filled the vacancy of the Higgs boson needed for the completion of the Standard Model (SM) at the Fermi scale and has provided a confirmation for the mass generation mechanism of the weak gauge bosons. Despite LHC experiments indicate that the decay modes of the new scalar state are very close to the SM expectation, there is still room for new extra scalar states. The search of these new scalar states will shed light on the underlying theory behind Electroweak Symmetry Breaking (EWSB) and are the priority of the LHC experiments. Furthermore, despite its great experimental success, the SM has several unaddressed issues, such as, for example, the observed charged fermion mass and quark mixing pattern, the tiny neutrino masses and the sizeable leptonic mixing angles, which contrast with the small quark mixing angles. The global fits of the available data from the Daya Bay [3], T2K [4], MINOS [5], Double CHOOZ [6] and RENO [7] neutrino oscillation experiments, provide constraints on the neutrino mass squared splittings and mixing parameters [8]. It is well kwown that the charged fermion mass hierarchy spans over a range of five orders of magnitude in the quark sector and a much wider range, which includes extra six orders of magnitude, corresponding to the number of orders of magnitude between the neutrino mass scale and the electron mass. The charged fermion masses can be accommodated in the SM, at the price of having an unnatural tuning among its different Yukawa couplings. Furthermore, experiments with solar, atmospheric and reactor neutrinos [3][4][5][6][7]9] provide clear indications of neutrino oscillations, originated by nonvanishing neutrino masses. All these unexplained issues suggest that new physics have to be invoked to address the fermion puzzle of the SM.
The unexplained flavour puzzle of the SM motivates to consider extensions of the SM that explain the fermion mass and mixing pattern. From the phenomenological point of view, one can assume Yukawa textures  to explain some features of the fermion mass hierarchy. Discrete flavor groups provide a very promising approach to address the flavour puzzle, and been extensively used in several models to explain the prevailing pattern of fermion masses and mixings (see Refs. [35][36][37][38] for recent reviews on flavor symmetries).
Besides that, another unaswered issue in particle physics is the existence of three families of fermions at low energies. The origin of the family structure of the fermions can be addressed in family dependent models where a symmetry distinguish fermions of different families. This issue can be explained by the models based on the SU(3) C ⊗ SU(3) L ⊗ U(1) X gauge symmetry, also called 3-3-1 models, which include a family non-universal U(1) X symmetry [25,58,59,72,73,102,104,. These models have several phenomenological advantages. Firstly, the three family structure in the fermion sector can be explained in the 3-3-1 models from the chiral anomaly cancellation and asymptotic freedom in QCD [157][158][159]. Secondly, the fact that the third family is treated under a different representation, can explain the large mass difference between the heaviest quark family and t he two lighter ones. Finally, these models contain a natural Peccei-Quinn symmetry, necessary to solve the strong-CP problem [152]. Furthermore, the 331 models with sterile neutrinos have weakly interacting massive fermionic dark matter candidates [153].
In the 3-3-1 models, 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 that gets a Vacuum Expectation Value (VEV) at high energy scale v χ , thus giving masses to non SM fermions and gauge bosons, while the Electroweak Symmetry Breaking is triggered by the remaining lighter triplets as well as by SU(3) L antisextets in some version of the model, with VEVs at the electroweak scale υ ρ and υ η , thus providing masses for SM fermions and gauge bosons [25].
In this paper we propose a 3-3-1 model based on the SU is discussed in Sec II A. In Sec. II B we focus on the discussion of the neutrino sector as well as in lepton masses and mixing and give our corresponding results. In Sec. III, we discuss the implications of our model in the quark sector. Conclusions are given in Sec.
IV. In the appendices we present several technical details: Appendices A and B give a detailed description of the ∆(27) group and the matrices of the 3 representation of ∆(27), respectively. The Appendix C provides the breaking patterns of ∆(27) by triplets. We prefer to use the notation 3 * for a SU(3) anti-triplet and3 for a ∆ (27) anti-triplet, i. e., all ∆ (27) representations appear with a bar underneath, and the anti-triplets appear also with a bar on top.

II. THE MODEL
The symmetry group of the model under consideration is where the electroweak factor SU(3) L ⊗ U(1) X is extended from those of the SM, and the strong interaction sector is retained. Lets us note that the gauge symmetry of the 331 model is supplemented by the U(1) L global and ∆ (27) symmetries. Each lepton family includes a new neutral fermion (N R ) with vanishing lepton number L(N R ) = 0 arranged under the SU(3) L symmetry as a triplet (ν L , l L , N c R ) and a singlet l R . The residual electric charge operator Q is therefore related to the generators of the gauge symmetry by [84] where T a (a = 1, 2, ..., 8) are SU(3) L charges with TrT a T b = 1 2 δ ab and X is the U(1) X charge. This means that the model under consideration does not contain exotic electric charges in the fundamental fermion, scalar and adjoint gauge boson representations. Since particles with different lepton number are put in SU(3) L triplets, it is better to work with a new conserved charge L commuting with the gauge symmetry and related to the ordinary lepton number by diagonal matrices [84,160] The lepton charge arranged in this way, i.e. L(N R ) = 0, is in order to prevent unwanted interactions due to U(1) L symmetry and breaking due to the lepton parity to obtain the consistent lepton and quark spectra. By this embedding, exotic quarks U, D as well as new non-Hermitian gauge bosons X 0 , Y ± possess lepton charges as of the ordinary leptons: The fermion content and the scalar fields of the model are summarized in Tab. I.
To obtain a realistic lepton spectrum, we suppose that in charged lepton sector ∆ (27) is broken down to {Identity}, i.e, it is completely broken. This can be achieved with the VEV Under this alignment, the mass Lagrangian for the charged leptons reads where As will shown in section II B, in the case v and the exact tri-bimaximal mixing form will obtained. For a detailed study of this problem, the reader can see Ref. [102].
As we know, the realistic lepton mixing form is a small deviation from tri-bimaximal form [9] . This can be achieved with a small difference between v 2 , v 3 and v 1 . Therefore we can separate v 2 , v 3 into two parts, the first is equal to v 1 ≡ v, the second is responsible for the deviation, and the matrix M l in (5) becomes The matrix M l in Eq. (8) can be diagonalized by two steps as follows.
Firstly, we denote where with The matrix that diagonalize M ′ l in (9) takes the form: To get the results in Eq. (12) we have used the following relations which are obtained from the unitary condition of U L .
The left-and right-handed mixing matrices in charged lepton sector are given by: In the case ε 3 = 0 it folows that ε * 2 = ε 2 = ε * 3 = 0, U L = 1 and the lepton mixing U ′ L in Eq. (13) reduces to Tri-bimaximal form (U HP S ) [161] which is ruled out by the recent data [9].
We note that the mass hierarchy of the charged leptons are well separated by only one Higgs triplet φ of ∆ (27), and this is one of the good features of the ∆(27) group.

B. Neutrino masses and mixings
The neutrino masses arise from the coupling ofψ Furthermore, we assume the following VEV patterns for the ∆ (27) scalar triplets σ and ρ: i.e, ∆ (27) is broken into Z 3 groups which consisting of the elements {e, aa ′ , (aa ′ ) 2 } and {e, a ′ , a ′2 } by σ and ρ, respectively.
The neutrino Yukawa interactions invariant under the symmetries of the model are given by 4 : 4 The following terms are invariant under the symmetries of the model: (ψ but they are all vanish , i.e., they have no contribution to the neutrino mass matrices M L,D,R .
Then, it follows that the neutrino mass terms are We can rewrite (19) in the matrix form where with The effective neutrino mass matrix, in the framework of type I and type II seesaw mechanisms, is given by 5 where In the case without the ρ contribution (v ρ = 0) we have c D = 0 and M eff in (23) becomes The mass matrix in Eq. (24) gives the degenerate mass of neutrinos and the corresponding leptonic mixing matrix yields the tri-bimaximal mixing form U + L U ν = U HP S , which is ruled out by the recent neutrino experimental data. However, the ρ contribution will improve this. Indeed, the mass matrix (23) is diagonalized as follows and the corresponding neutrino mixing matrix: where Combining (13) and (26), the lepton mixing matrix takes the form: where with We see that all the elements of the matrix U Lep in Eq. (29) depend only on two parameters ε 3 ans K. From experimental constraints on the elements of the lepton mixing matrix given in Refs. [162][163][164], we can find out the regions of K and ε 3 that satisfy experimental data on lepton mixing matrix. Indeed, in the case α i = β i = 1/ . By using the experimental constraint values of u 11 given in [162][163][164], 0.801 ≤ |u 11 | ≤ 0.845 we get 1.1 ≤ |K| ≤ 1.5 which is depicted in Fig. 1. To get the specific value of ε 3 , a specific value of K would be chosen with an experimental value of u 11 . In the case K = √ 2 ≃ 1.4142, combining with the constraint values on the element u 11 of lepton mixing matrix [162][163][164], u 11 = 0.805, we obtain a solution 6 : Then, it follows that the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) leptonic mixing matrix takes the form: Using Eq. (27) and K = √ 2, we obtain In the standard Particle Data Group(PDG) parametrization, the lepton mixing matrix can be parametrized as where P = diag(1, e iα , e iβ ), and c ij = cos θ ij , s ij = sin θ ij with θ 12 , θ 23 or which are all very consistent with the recent data on neutrino mixing angles. Furthermore, comparing the lepton mixing matrix given in Eq. (31) with the standard parametrization in Eq.(34), one obtains vanishing Majorana phases, i.e., α = 0, β = 0 as well as nonvanishing leptonic Dirac CP violating phase δ = − π 2 and Jarskog invariant close to −3.2 × 10 −2 . It is worth mentioning that having leptonic mixing parameters consistent with their experimental values, require that the parameter K to be equal or very close to √ 2. The other parameters that determine the leptonic mixing angles are Re (ε 3 ) and Im (ε 3 ), i.e, which are of the order of 10 −4 . Besides that we have numerically checked the leptonic mixing parameters have a low sensitivity with Re (ε 3 ) and Im (ε 3 ) but are highly sensitive under small variations around K = √ 2, for example having K = 0.9 √ 2 ≃ 1.27 leads to sin 2 θ 13 = 0.009, which is outside the 3σ experimentally allowed range. In the region of parameter space consistent with the experimental values of the leptonic mixing parameters, we have numerically checked that the leptonic Dirac CP violating phase is equal to − π 2 . Other phases different than − π 2 are obtained for values of the K parameters outside the vicinity of K = √ 2, that leads to a reactor mixing angle θ 13 unacceptably small.
At present, the absolute neutrino masses as well as the mass ordering of neutrinos is unknown. The result in [165] shows that while the upper bound on the sum of light active neutrino masses is given by [166] 3 The neutrino mass spectrum can be described by the normal mass hierarchy (|m 1 | ≃ |m 2 | < |m 3 |), the inverted hierarchy (|m 3 | < |m 1 | ≃ |m 2 |) or the nearly degenerate (|m 1 | ≃ |m 2 | ≃ |m 3 |) ordering. The neutrino mass ordering depends on the sign of ∆m 2 23 , which is currently unknown. In the case of 3-neutrino mixing, in the model under consideration, the two possible signs of ∆m 2 23 correspond to two types of allowed neutrino mass spectra.

E. Effective Majorana neutrino mass parameter
In what follows we proceed to compute the effective Majorana neutrino mass parameter, whose value is proportional to the amplitude of neutrinoless double beta (0νββ) decay. The effective Majorana neutrino mass parameter has the form: where U 2 ej is the squared of the PMNS leptonic mixing matrix elements andand m ν k correspond to the masses of the Majorana neutrinos.
where their G assignments are reported in table I and the VEV pattern of the ∆ (27) triplet η is given as with The quark Yukawa interactions are Then, it follows that the quark mass terms take the form Consequently, the exotic quarks do not mix with the SM quarks. From the quark mass terms given above, it follows that the exotic quark masses are and the SM up-type and down-type quark mass matrices take the form: In the quark sector, we assume that the ∆(27) discrete group is broken down to the Z 3 subgroup, which consists of the elements {1, b, b 2 }. This breaking is triggered by the ∆ (27) scalar triplet η, with the VEV alignment described in Eq. (50). In the case v and the quark mixing matrix V CKM = V d † L V u L = 1, which is acceptable since the quark mixing matrix is very close to the identity matrix [9]. By an appropriate choice of parameters in the SM quark mass matrices given by Eq. (53), we can successfully reproduce the experimental values of quark masses and quark mixing angles. Furthermore it is noteworthy to mention that our model is an extension of the 3-3-1 model considered in [175]. As pointed out in Refs. [175], the flavor constraints can be fullfilled by considering the scale of breaking of the SU(3) L ⊗ U(1) X gauge symmetry much larger than the electroweak symmetry breaking scale v = 246 GeV, which corresponds to the alignment limit of the mass matrix for the CP-even Higgs bosons. Consequently, following [175], we expect that the FCNC effects as well as the constraints arising from K 0 −K 0 , B 0 −B 0 and D 0 −D 0 mixings will be fullfilled in our model, by considering the scale of breaking of the SU(3) L ⊗ U(1) X gauge symmetry much larger than scale of breaking of the electroweak symmetry. In that alignment limit, our model effectively becomes a nine Higgs doublet model, whose scalar sector includes 9 CP even neutral Higges, 8 CP odd neutral Higges and 16 charged Higges. That scalar sector is not predictive as its corresponding scalar potential has many free uncorrelated parameters that can be adjusted to get the required pattern of scalar masses. Therefore, the loop effects of the heavy scalars contributing to certain observables can be suppressed by the appropriate choice of the free parameters in the scalar potential. Fortunately, all these adjustments do not affect the charged fermion and neutrino sector, which is completely controlled by the fermion-Higgs Yukawa couplings. In addition, in models with discrete flavor symmetries, like ours, the deviation of the CKM matrix from the identity can be given by the FCNC effects with the left-handed quarks, but in the alignment limit previously described, such deviations are highly suppressed by the mass of the extra quarks [84].

IV. CONCLUSIONS
We constructed the first SU (3)   The ∆(27) discrete group is a subgroup of SU(3) and is isomorphic to the semi-direct It is also a simple group 7 of the type ∆(3n 2 ) with n = 3. The ∆(27) discrete group has 27 elements divided into 11 conjugacy classes, so it has 11 irreducible representations, including two triplets (3 and its conjugate3) and 9 singlets 1 i (i = 1, 2, ..., 9). Any element of ∆ (27) can be written as a multiplication of three generators, i.e., b, a and a ′ , in the form b k a m a ′ n , satisfying the relations where b is a generator of Z 3 , and a, a ′ belong to Z ′ 3 and Z ′′ 3 , respectively. The character table of ∆(27) is given in Tab. II, where n is the number of elements, h is the order of each element, and ω = e 2 is the cube root of unity, obeying 1 + ω + ω 2 = 0 and ω 3 = 1. The conjugacy classes generated from b, a and a ′ are presented  7 In fact, the simplest group of the type ∆(3n 2 ) is ∆(3) ≡ Z 3 . The next group, ∆ (12), is isomorphic to A 4 .