Fermion Mass and Mixing in a Low-Scale Seesaw Model based on the S4 Flavor Symmetry

We construct a low-scale seesaw model to generate the masses of active neutrinos based on $S_4$ flavor symmetry supplemented by the $Z_2 \times Z_3 \times Z_4 \times Z_{14}\times U(1)_L$ group, capable of reproducing the low energy Standard model (SM) fermion flavor data. The masses of the SM fermions and the fermionic mixings parameters are generated from a Froggatt-Nielsen mechanism after the spontaneous breaking of the $S_4\times Z_2 \times Z_3 \times Z_4 \times Z_{14}\times U(1)_L$ group. The obtained values for the physical observables of the quark and lepton sectors are in good agreement with the most recent experimental data. The leptonic Dirac CP violating phase $\de _{CP}$ is predicted to be $259.579^\circ$ and the predictions for the absolute neutrino masses in the model can also saturate the recent constraints.


I. INTRODUCTION
Despite its great success, the SM still has serious drawbacks such as the lack of mechanisms that explain the smallness of neutrino masses, the large hierarchy of charged fermion masses, the fermionic mixing angles, the leptonic CP violation, etc. Another puzzle of the SM is that it does not explain why there are three generations of fermions. This puzzle can be addressed in the 3-3-1 models [1]. Hence, the neutrino masses and lepton mixings can be regarded as one of the most important evidence of physics beyond the SM. Among the possible extensions of the SM, discrete symmetries associated with the SM extensions are an useful tool to explain the observed pattern of SM fermion masses and mixing angles.
One of the most simplest possibilities to understand small non-zero neutrino masses is probably the seesaw mechanism, including type I, II, III and/or their combinations which has been briefly reviewed in Ref. [5]. However, in these scenarios, the scale of the masses of the right-handed neutrinos should be very high that cannot be reached in the near future. In the inverse-and linear seesaw mechanism  the small neutrino masses arise as a result of new physics at TeV scale which may be probed at the LHC experiments. In such low-scale models, both renormalizable and non-renormalizable interactions are included, which can explain the fermion masses and mixings. In the basis (ν, N, S), the neutrino mass matrix can be presented in the form of a 3 × 3 block matrix where each element is a submatrix. Depending on the position of the zero elements in the mass matrix, active neutrinos can receive masses through inverse or/and linear seesaw mechanisms that all impose some elements of the mass matrix to be zero or very small and none of them are forbidden by the SM symmetry, however, such terms can be avoided by introducing additional flavor symmetries.
In this paper we propose the possibility of predicting fermion masses and mixing angles in the framework of the low-scale seesaw mechanism with S 4 flavor symmetry. S 4 is the permutation group of four objects, which is also the symmetry group of a cube. It has 24 elements divided into 5 conjugacy classes, with 1, 1 ′ , 2, 3, and 3 ′ as its 5 irreducible representations. We will work in the basis in which 3, 3 ′ are real representations whereas 2 is complex. For the Clebsch-Gordan coefficients of S 4 group one can see, for instance, in the Ref. [29].
The content of this paper goes as follows. In Sec. II we present the necessary elements of the linear seesaw model under the S 4 symmetry as well as introduce the necessary Higgs fields responsible for fermion masses and mixings. Section III deals with quark masses and mixings and Section IV is devoted to lepton masses and mixings. We conclude in Section V.

II. THE MODEL
We consider a three Higgs doublet model with several gauge singlet scalars, where the SM gauge symmetry is supplemented by the group is summarized in Tables I and III where the numbered subscripts on fields in order define components of their S 4 multiplet representations as well as the quantum numbers corresponding to other groups of the model. We use the S 4 discrete group since it is the smallest non Abelian discrete group having irreducible triplet and doublet representations.
The discrete group S 4 is crucial to get a predictive fermion sector consistent with the low energy fermion flavor data. Extra symmetries Z 2 , Z 3 , Z 4 and Z 14 are additional introduced in order to get the desired structure of the fermion mass matrices that will be discussed in detail in Sec.IV.

III. QUARK MASSES AND MIXINGS
The quarks content and the corresponding scalar fields of the model, under the , is given in Table. I. The quark Yukawa terms invari-  ant under the symmetries of the model under consideration take the form: where a (u) are O(1) dimensionless couplings. The values of the O(1) dimensionless couplings given above allows to successfully reproduce the experimental values of the quark mass spectrum, CKM parameters and Jarlskog invariant. As indicated by Table II, our model is consistent with the low energy quark flavor data. Note that we use the M Z -scale experimental values of the quark masses given by Ref. [30] (which are similar to those in [31]    quark sector parameters have to be varied in range around the 3% and 4% of their best fit values, respectively, in order to obtain all SM quark masses inside their 3σ experimentally allowed range.

IV. LEPTON MASSES AND MIXINGS
The lepton fields and the corresponding scalars in lepton sectors, under the , is given in Table III. The lepton Yukawa terms invariant Table III:

assignments for leptons and scalars.
under the symmetries of the model are: In the case where S 4 is spontaneously broken down to {identity} by the VEVs alignment we get the lepton flavor changing interactions as follows From (6), it follows that, in the model under consideration, the usual Yukawa couplings are associated with the factor v Λ and the lepton flavor changing decays consist of the contribution of three Feynman diagrams as in Fig. 3.
The current experimental data on lepton flavor changing decays read [32]: Br(µ − → e − γ) < 4.2 × 10 −13 , Br(τ − → e − γ) < 3.3 × 10 −8 and Br(τ − → µ − γ) < 4.4 × 10 −8 . The partial decay width is given by [36,37] where the above form factors C L and C R are determined from the process amplitude [36,37] For the case m i ≫ m j , we get where G F = g 2 /(4 √ 2m 2 W ). In the model under consideration, one has [37,38] where M H is the mass scale of the heavy scalars (which provide the dominant contributions to the LFV decays) running in the internal lines of the loop. For further details on the form factors D L,R , the reader is referred to Refs. [36][37][38][39].
Combining (9) and (10) Let us turn into lepton mass issue. From (4), the lepton mass terms read where the mass matrix for charged leptons is given by: This matrix can be diagonalized as, where where ω = e i2π/3 is the cube root of unity.
The best fit values for the masses of charged-leptons are given in Ref. [2]: m e ≃ 0.51099 MeV, m µ ≃ 105.65837 MeV, m τ ≃ 1776.86 MeV. Then, we find the relations We also assume that in the neutrino sector, the S 4 discrete group is spontaneously broken down to the Klein four group K by the VEV alignment ϕ = (0, v ϕ , 0) of ϕ and the VEVs of ξ, ρ as ξ = v ξ , ρ = v ρ . In this case, the neutrino mass matrices become Let us note that the matrices given by Eqs.
In this work, we introduce the Z 2 × Z 3 × Z 4 × Z 14 × U(1) L symmetry 1 , which in addition to the S 4 symmetry to prevent some Yukawa interactions thus giving rise to the predictive textures for the neutrino sector shown in Eqs. (16) - (21). For instance, since the product of two S 4 triplets contains a S 4 triplet, the coupling ψ L N R can transform under 1, 1, 1, 1, 0), which implies that in order to generate the mass matrix m νN , one needs one S 4 singlet transforming as (1, -1, 1,1,1,0 In the basis (ν , N, S), the full neutrino mass matrix predicted by our model takes the form: The light active neutrino masses are obtained by diagonalizing the matrix given by Eq. (22) and this is done by introducing the following matrices The effective neutrino mass matrix M eff in Eq. (22) can be rewritten in the form: which is similar to the one resulting from a type-I seesaw mechanism. Then, the light active neutrino mass matrix takes the form: Replacing Eqs. (16) - (21) in Eq. (24) yields the following mass matrix for light active neutrinos: where with a 1,2 , b 1,2 , c 1,2 , d 1,2 and g 1,2,3,4 defined in Eqs. (16) - (21). The mass matrix m ν for light active neutrinos is diagonalized by the rotation matrix U ν , and the light active neutrino masses m 1,2,3 are given by By combining Eqs. (14) and (28) we find that the leptonic mixing matrix takes the form: We see that all the elements of the matrix U lep in Eq. (30) depend only on two parameters α and β. From experimental constraints on the elements of the lepton mixing matrix given in Ref. [33], we can find out the regions of α and β to establish experimental constraints for lepton mixing matrix. In the standard Particle Data Group (PDG) parametrization, the leptonic mixing matrix can be parameterized in three Euler's angles as follows: i.e, s 13 , t 12 and t 23 in Eqs. (31) and (33) depend only on one parameter β. Eqs. (31) - (33) yields: The data in Particle Data Group 2018 [2] shows that s 13 ∈ (0.145258, 0.15) rad so t 23 ∈  values of θ 23 and θ 12 given in Ref. [2]. On the other hand, with this best value of θ 13 , we get β = −0.363663 rad (∼ 339.163 • ) and Dirac CP violation phase δ CP = 259.579 • which is a viable value of the CP violating Dirac phase [2]. The leptonic mixing matrix in Eq. (36) takes the explicit form which is an unitary matrix.
The expression (36) shows that α is free parameter so we can choose the VEV alignment φ in the charged-lepton sector as φ = v(1, 1, e iβ ), i.e, α may get the value α = 0. In this case, the leptonic mixing matrix becomes: i.e, the ranges of the magnitude of the elements of the three-flavour leptonic mixing matrix is consistent with those of given in Ref. [33]. At present, the values of neutrino masses (or the absolute neutrino masses) as well as the mass ordering of neutrinos are still unknown.
The result in Ref. [34] shows that m i ≤ 0.6 eV (i = 1, 2, 3) while the upper bound on the sum of light active neutrino masses is given by [35], The experimental neutrino oscillation data given in Eq. (1) are compatible with two possible signs of ∆m 2 23 which is currently unknown and correspond to two types of neutrino mass spectra.
In this model, C ≡ m 2 ∈ (0.051, 0.065) eV is a good region of C that can reach the inverted neutrino mass hierarchy which is depicted in Fig. 8   CP violating phase δ CP is predicted to be 259.579 • which is consistent with the most recent neutrino oscillation experimental data [2]. The predictions for the absolute neutrino masses in the model can also saturate the recent constraints.