The T7 flavor symmetry in 3-3-1 model with neutral leptons

We construct a 3-3-1 model based on non-Abelian discrete symmetry $T_7$ responsible for the fermion masses. Neutrinos get masses from only anti-sextets which are in triplets $\underline{3}$ and $\underline{3}^*$ under $T_7$. The flavor mixing patterns and mass splitting are obtained without perturbation. The tribimaximal form obtained with the breaking $T_7 \rightarrow Z_3$ in charged lepton sector and both $T_7 \rightarrow Z_3$ and $Z_3 \rightarrow \{\mathrm{Identity}\}$ must be taken place in neutrino sector but only apart in breakings $Z_3 \rightarrow \{\mathrm{Identity}\}$ (without contribution of $\si'$), and the upper bound on neutrino mass $\sum_{i=1}^3m_i$ at the level is presented. The Dirac CP violation phase $\delta$ is predicted to either $\frac{\pi}{2}$ or $\frac{3\pi}{2}$ which is maximal CP violation. From the Dirac CP violation phase we obtain the relation between Euler's angles which is consistent with the experimental in PDG 2012. On the other hand, the realistic lepton mixing can be obtained if both the direction for breakings $T_7 \rightarrow Z_3$ and $Z_3 \rightarrow \{\mathrm{Identity}\}$ are taken place in neutrino sectors. The CKM matrix is the identity matrix at the tree-level.

can be considered as a good approximation for the recent neutrino experimental data.
A fundamental relation holds among some of the generators of the group [150,151]: where Q indicates the electric charge, T 3 and T 8 are two of the SU (3) generators and X is the generator of U (1) X . β is a key parameter that defines a specific variant of the model. The model thus provides a partial explanation of the family number, as also required by flavor symmetries such as T 7 for 3-dimensional representations. In addition, due to the anomaly cancelation one family of quarks has to transform under SU(3) L differently from the two others. T 7 can meet this requirement with three inequivalent representations 1, 1 ′ , 1 ′′ . Note that T 7 has not been considered before in the kind of the 3-3-1 model.
There are two typical variants of the 3-3-1 models as far as lepton sectors are concerned. In the minimal version, three SU(3) L lepton triplets are (ν L , l L , l c R ), where l R are ordinary right-handed charged-leptons [152][153][154][155][156] . In the second version, the third components of lepton triplets are the right-handed neutrinos, (ν L , l L , ν c R ) [157][158][159][160][161][162]. To have a model with the realistic neutrino mixing matrix, we should consider another variant of the form (ν L , l L , N c R ) where N R are three new fermion singlets under standard model symmetry with vanishing lepton-numbers [169][170][171][172] .
CP violation plays a crucial role in our understanding of the observed baryon asymmetry of the Universe [173]. In the SM, CP symmetry is violated due to a complex phase in the CKM matrix [13,14]. However, since the extent of CP violation in the SM is not enough for achieving the observed BAU, we need new source of CP violation for successful BAU. On the other hand, CP violations in the lepton sector are imperative if the BAU could be realized through leptogenesis. So, any hint or observation of the leptonic CP violation can strengthen our belief in leptogenesis [173].
The violation of the CP symmetry is a crucial ingredient of any dynamical mechanism which intends to explain both low energy CP violation and the baryon asymmetry. Renormalizable gauge theories are based on the spontaneous symmetry breaking mechanism, and it is natural to have the spontaneous CP violation as an integral part of that mechanism. Determining all possible sources of CP violation is a fundamental challenge for high energy physics. In theoretical and economical viewpoints, the spontaneous CP breaking necessary to generate the baryon asymmetry and leptonic CP violation at low energies brings us to a common source which comes from the phase of the scalar field responsible for the spontaneous CP breaking at a high energy scale [173].
In this paper, we investigate another choice with T 7 , the smallest group with two non-equivalent 3-dimensional irreducible representations, contains two triplet irreducible representations and three singlets which play a crucial role in consistently reproducing fermion masses and mixing. As we will see, T 7 model has some new features since fewer Higgs multiplets are needed in order to allow the fermions to gain masses and to break symmetries and the physics we will see is different from the former. The CP violation is the first time considered under SU (3) L × U (1) X model based on T 7 flavor symmetry in which the T 7 symmetry avoids the mass degeneracy of lepton masses. The light neutrino masses can be generated at tree level, and the vacuum alignment problem which arises in the presence of two T 7 -triplet scalar fields 3, 3 * can naturally explain the measured value of θ 13 and thereby the hierarchy of neutrino masses. The seesaw mechanism can explain the smallness of the measured neutrino masses and the maximal Dirac CP violation.
The rest of this work is organized as follows. In Sec. 2 and 3 we present the necessary elements of the 3-3-1 model with the T 7 symmetry as well as introducing necessary Higgs fields responsible for the charged lepton masses. In Sec. 4, we discuss on quark sector. Sec. 5 is devoted for the neutrino mass and mixing. We summarize our results and make conclusions in the section 7. Appendix A presents a brief of the T 7 theory. Appendix B provides the lepton number (L) and lepton parity (P l ) of particles in the model.

Fermion content
The gauge symmetry is based on SU(3) C ⊗ SU(3) L ⊗ U(1) X , where the electroweak factor SU(3) L ⊗ U(1) X is extended from those of the Standard Model (SM), and the strong interaction sector is retained. 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 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 the particles in the lepton triplet have different lepton number (1 and 0), so the lepton number in the model does not commute with the gauge symmetry unlike the SM. Therefore, 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 [169][170][171][172]174] 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: Under the [SU(3) L , U(1) X , U(1) L , T 7 ] symmetries as proposed, the fermions of the model transform as follows 3) where the subscript numbers on field indicate to respective families which also in order define components of their T 7 multiplets. U and D 1,2 are exotic quarks carrying lepton numbers L(U ) = −1 and L(D 1,2 ) = 1, known as leptoquarks. In the following, we consider possibilities of generating the masses for the fermions. The scalar multiplets needed for the purpose are also introduced.

Charged lepton masses
The charged lepton masses arise from the couplings ofψ L l 1R ,ψ L l 2R andψ L l 3R to scalars, whereψ L l iL (i = 1, 2, 3) transforms as 3 * under SU(3) L and 3 * under T 7 . To generate masses for charged leptons, we need a SU (3) L Higgs triplets that lying in 3 under T 7 and transforms as 3 under SU(3) L , Following the potential minimization conditions, we have the followings alignments: (1) The first alignment: φ 1 = φ 2 = φ 3 then T 7 is broken into Z 3 consisting of the elements {e, b, b 2 }.
(2) The second alignment: (3) The third alignment: (4) The fourth alignment: (5) The fifth alignment: In this work, we argue that only the first alignment of VEV in charged -lepton sector is taken place, i.e, T 7 → Z 3 , and this can be achieved by the Higgs triplet φ with the VEV alignment φ = ( φ 1 , φ 1 , φ 1 ) under T 7 , where The Yukawa interactions are The mass Lagrangian for the charged leptons is then given by

Couplings
Higgs multiplets The mass Lagrangian for the charged leptons reads This matrix can be diagonalized as, where As will see in section 5, in this case, the exact tribimaximal mixing form is obtained, by choosing the right vev's in the neutrino sector.
The experimental values for masses of the charged leptons at the weak scale are given as [8] :

Quark masses
To generate masses for quarks with a minimal Higgs content, we additionally introduce the following Higgs triplets: The Higgs content and Yukawa couplings in the quark sector are summarized in Table 1.
The Yukawa interactions are We suppose that T 7 is broken into Z 3 like the case of the charged lepton sector, i,e, the VEVs of η and χ are given as η = ( η 1 , η 1 , η 1 ) with

4)
and The mass Lagrangian for quarks is given by The exotic quarks get masses The mass matrices for ordinary up-quarks and down-quarks are, respectively, obtained as follows: The structure of the up-and down-quark mass matrices in (4.8) is similar to those in [33], i.e, in the model under consideration there is no CP violation in the quark mixing matrix. The mass matrices M u , M d in (4.8) are diagonalized as follows The unitary matrices, which couple the left-handed up-and down-quarks to those in the mass bases are unit matrices. Therefore we get the Cabibbo-Kobayashi-Maskawa (CKM) matrix Note that the property in (4.11) is common for some models based on the discrete symmetry groups [169,170]. The up and down quark masses are The current mass values for the quarks are given by [8]: It is obvious that if |u| ∼ |v|, the Yukawa coupling hierarchies are |h u and Higgs scalar multiplets are slightly heavier than those of down-quarks (h d 2 ≪ h d 3 ), respectively.

Neutrino mass and mixing
The neutrino masses arise from the couplings ofψ c L ψ L to scalars, whereψ c L ψ L transforms as 3 * ⊕ 6 under SU(3) L and 3 ⊕ 3 * ⊕ 3 * under T 7 . It is worth mentioning that, with the T 7 group, 3 × 3 × 3 has two invariants and 3 × 3 × 3 * has one invariant. For the known scalar triplets (φ, χ, η), there is no available interaction because of the L-symmetry. We will therefore propose new SU(3) L anti-sextets instead coupling toψ c L ψ L responsible for the neutrino masses. To obtain a realistic neutrino spectrum, the antisextets transform as follows 2 Following the potential minimization conditions, we have the followings alignments: (1) The first alignment: (2) The second alignment: (3) The third alignment: (4) The fourth alignment: (5) The fifth alignment: To obtain a realistic neutrino spectrum, in this work we argue that both the breakings T 7 → Z 3 and T 7 → {identity} (Instead of Z 3 → {identity}) must be taken place in neutrino sector. However, the VEVs of σ does only one of these tasks. The T 7 → Z 3 can be achieved by a SU (3) L anti-sextet σ given in (5.1) with the VEVs is set as σ = ( σ 1 , σ 1 , σ 1 ) under T 7 , where To achieve the second direction of the breakings T 7 → {Identity} (equivalently to Z 3 → {Identity}), we additionally introduce another SU (3) L anti-sextet Higgs scalar which is either put in 3 or 3 * under T 7 . This is equivalent to breaking the subgroup Z 3 of the first direction into {Identity}, and it can be achieved within each case below.

with the VEVs chosen by
Note that σ ′ differs from σ only in the VEVs alignment. Combining both cases, after calculation, we obtain the Yukawa interactions: The mass Lagrangian for the neutrinos is given by The neutrino mass Lagrangian can be written in matrix form as follows where 9) and the mass matrices are then obtained by Three observed neutrinos gain masses via a combination of type I and type II seesaw mechanisms derived from (5.8) and (5.10) as We can diagonalize the mass matrix (5.12) as follows 14) and the corresponding neutrino mixing matrix: Combining (3.8) and (5.15), we get the lepton mixing matrix: It is worth noting that in our model, K given in (5.16) is an arbitrary number. Hence in general the lepton mixing matrix given in (5.17) is different to U HP S in (1.1), but similar to the original version of trimaximal mixing considered in [175] which is based on the ∆(27) group. In the case where T 7 is broken in Identity (Instead of Z 3 → Identity) only by s, i.e, without contribution of σ ′ (or λ ′ σ = v ′ σ = Λ ′ σ = 0), the lepton mixing matrix (5.17) being equal to U HP S as given in (1.1). This is a good features of T 7 with tensor product 3 ⊗ 3 given in (A.2).
Since cos δ = 0 so that sin δ must be equal to ±1, it is then δ = π 2 or δ = 3π 2 . Thus, our model predicts the maximal Dirac CP violating phase which is the same as in Refs. [175,176], and this is one of the most striking prediction of the model under consideration.
Up to now the precise evaluation of θ 23 is still an open problem while θ 12 and θ 13 are now very constrained [8]. From (5.23), our model can provide constraints on θ 23 from θ 12 and θ 13 which satisfy [8] as follows.
(i) In the case δ = π 2 , from (5.23) we have the relation among three Euler's angles as follows:    Until now values of neutrino masses (or the absolute neutrino masses) as well as the mass ordering of neutrinos is unknown. The tritium experiment [178,179] provides an upper bound on the absolute value of neutrino mass A more stringent bound was found from the analysis of the latest cosmological data [180] m i ≤ 0.6 eV, The neutrino mass spectrum can be the normal mass hierarchy (|m 1 | ≃ |m 2 | < |m 3 |), the inverted hierarchy (|m 3 | < |m 1 | ≃ |m 2 |) or nearly degenerate (|m 1 | ≃ |m 2 | ≃ |m 3 |). The mass ordering of neutrino 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 neutrino mass spectrum can be provided. Combining (5.14) and the two experimental constraints on squared mass differences of neutrinos as shown in (1.2) and the values of K in (5.25) or in (5.29), we have the solutions as shown bellows.   [182][183][184][185][186][187],   We also notice that in the normal spectrum, |m 1 | ≈ |m 2 | < |m 3 |, so m 1 given in (C.1) is the lightest neutrino mass. Hence, it is denoted as m 1 ≡ m light . In Figs. 8a and 8b, we have plotted the value |m ee |, |m β | and |m light | as functions of m 2 with m 2 ∈ (0.0087, 0.05) eV and m 2 ∈ (−0.05, −0.0087) eV, respectively.
Note that Λ σ , Λ s , Λ σ ′ are needed to the same order and not to be so large that can naturally be taken at TeV scale as the VEV v χ of χ. This is because v σ , v s and v σ ′ carry lepton number, simultaneously breaking the lepton parity which is naturally constrained  to be much smaller than the electroweak scale [165,166,[169][170][171]. This is also behind a theoretical fact that v χ , Λ σ are scales for the gauge symmetry breaking in the first stage from SU(3) L ⊗U(1) X → SU(2) L ⊗U(1) Y in the original form of 3-3-1 models [165,166,174]. They will provide masses for the new gauge bosons: Z ′ , X and Y . Also, the exotic quarks gain masses from v χ while the neutral fermions masses arise from Λ σ , Λ s , Λ σ ′ . The second stage of the gauge symmetry breaking from SU(2) L ⊗ U(1) Y → U(1) Q is achieved by the electroweak scale VEVs such as u, v responsible for ordinary particle masses. In combination with those of type II seesaw as determined, in our model, the following limit is often taken into account [165,166,[169][170][171]: Our model contains a lot of SU(3) L scalar triplets that may modify the precision electroweak parameter such as S, T, U [193] and ρ parameters. The most serious one can result from the tree-level contributions to the ρ parameter. To see this let us approximate W, Z mass and ρ parameter 4 : and where v 2 W ≃ (3u 2 + 3v 2 ) = (246 GeV) 2 is naturally given according to (6.1) with u ∼ v ∼ 100 GeV. Since λ s = 6v 2 σ + 2v 2 s + 2v 2 σ ′ is in eV scale responsible for the observed neutrino masses, the ρ in (6.3) is absolutely close to the unity and in agreement with the data [8].
The mixings between the charged gauge bosons W − Y and the neutral ones Z ′ − W 4 are in the same order since they are proportional to vσ Λσ , and in the limit λ s , λ σ , v s , v σ → 0 these mixing angles tend to zero. In addition, from (6.1) and (6.2), it follows that M 2 W is much smaller than M 2 Y .

Conclusions
In this paper, we have constructed the T 7 model based on SU(3) C ⊗ SU(3) L ⊗ U(1) X gauge symmetry responsible for fermion masses and mixing. Neutrinos get masses from only anti-sextets which are in triplets 3 and 3 * under T 7 . The flavor mixing patterns and mass splitting are obtained without perturbation. The number of Higgs multiplets needed in order to allow the fermions to gain masses are less than those of S 3 , S 4 and D 4 [170][171][172]. The tribimaximal form obtained with the breaking T 7 → Z 3 in charged lepton sector and both T 7 → Z 3 and Z 3 → {Identity} must be taken place in neutrino sector but 4 We have used the notation sW = sin θW , cW = cos θW , tW = tan θW , and the continuation of the gauge coupling constant g of the SU(3)L at the spontaneous symmetry breaking point [172,190,191] only apart in breakings Z 3 → {Identity} (without contribution of σ ′ ), and the upper bound on neutrino mass 3 i=1 m i at the level is presented. From the Dirac CP violation phase we obtain the relation between Euler's angles which is consistent with the experimental in PDG 2012. On the other hand, the realistic lepton mixing can be obtained if both the direction for breakings T 7 → Z 3 and Z 3 → {Identity} are taken place in neutrino sectors. The CKM matrix is the identity matrix at the tree-level. The Dirac CP violation phase δ is predicted to either π 2 or 3π 2 which is maximal CP violation.

A T 7 group and Clebsch-Gordan coefficients
The tetrahedral group A 4 has 12 elements and four equivalence classes with three inequivalent one-dimensional representations and one three-dimensional one, which is the smallest group with only a real 3 representation. The Frobenius group T 7 has 21 elements and five equivalence classes with three inequivalent one-dimensional representations and two three-dimensional once, which is the smallest group with a pair of complex 3 and 3 * representations. It is generated by where ρ = exp(2πi/7), so that a 7 = 1, b 3 = 1, and ab = ba 4 . The character table of T 7 (with ξ = −1/2 + i √ 7/2) is given in table 2. Let us put 3(1, 2, 3) which means some 3 multiplet such as x = (x 1 , x 2 , x 3 ) ∼ 3 or y = (y 1 , y 2 , y 3 ) ∼ 3 and so on, and similarly for the other representations. Moreover, the numbered multiplets such as (..., ij, ...) mean (..., x i y j , ...) where x i and y j are the multiplet components of different representations x and y, respectively. In the following the components of representations in l.h.s will be omitted and should be understood, but they always exist in order in the components of decompositions in r.h.s. All the group multiplication rules of T 7 as given below.

B The numbers
In the following we will explicitly point out the lepton number (L) and lepton parity (P l ) of the model particles (notice that the family indices are suppressed):