Fermion masses and mixings in a $U(1)_X$ model based on the $\Sigma(18)$ discrete symmetry

We have built a renormalizable $U(1)_X$ model with a $\Sigma (18)\times Z_4$ symmetry, whose spontaneous breaking yields the observed SM fermion masses and fermionic mixing parameters. The tiny masses of the light active neutrinos are produced by the type I seesaw mechanism mediated by very heavy right handed Majorana neutrinos. To the best of our knowledge, this model is the first implementation of the $\Sigma (18)$ flavor symmetry in a renormalizable $U(1)_X$ model. Our model allows a successful fit for the SM fermion masses, fermionic mixing angles and CP phases for both quark and lepton sectors. The obtained values for the physical observables of both quark and lepton sectors are in accordance with the experimental data. We obtain an effective neutrino mass parameter of $\langle m_{ee}\rangle=1.51\times 10^{-3}\, \mathrm{eV}$ for normal ordering and $\langle m_{ee}\rangle =4.88\times 10^{-2} \, \mathrm{eV}$ for inverted ordering which are well consistent with the recent experimental limits on neutrinoless double beta decay.


INTRODUCTION
In recent years neutrino oscillation experiments have confirmed that the leptonic mixing angles and neutrino mass squared differences are measured with high precision which require us to extend the standard model (SM) to successfully explain the current pattern of lepton masses and mixing angles. Among the possible extensions of the SM, the versions with an extra U (1) X gauge symmetry  are promising scenarios since the simplest possibility is to introduce three right-handed neutrinos that we need to incorporate the neutrino masses in the SM. In this type of model, many phenomena including neutrino masses [10][11][12][13], dark matter [13][14][15][16][17][18][19], the muon anomalous magnetic moment [20], inflation [21], leptogenesis [22,23], gravitational wave radiation [24] are explained, however, the most minimal versions of U (1) X models do not include a description of SM fermion masses and mixings.
In this work, we propose a U(1) X renormalizable theory based on the Σ(18) flavor symmetry, supplemented by the Z 4 discrete group capable of reproducing the SM fermion masses and mixings at tree-level. We use the Σ(18) discrete group, since it is the simplest non-trivial group of the type Σ(2N 2 ) with N = 3 which is isomorphic to (Z 3 × Z 3 ) Z 2 . The Σ(18) discrete group has 18 elements which are divided into nine conjugacy classes and has nine irreducible representations: the six singlets 1 +0 , 1 +1 , 1 +2 , 1 −0 , 1 −1 , 1 −2 and the three doublets 2 10 , 2 20 and 2 21 . Mathematical properties of the Σ(18) discrete group are discussed in detail in Ref. [93]. However, for convention, we present briefly the tensor products of Σ (18) in Appendix A. The reason for adding the auxiliary symmetry U(1) X was introduced in Ref. [94] in another different multiHiggs model based on the A 4 discrete symmetry where the global U (1) X symmetry is softly broken in the scalar potential in order to prevent the appearance of a Goldstone boson; thus, we do not further discuss on this issue here. Let us note that the Σ(18) symmetry has not been considered before in this type of models and to the best of our knowledge the model proposed in this work is the first implementation of the Σ(18) flavor symmetry in a renormalizable U (1) X model 1 .
The layout of the remainder of the paper is as follows. In Section II we describe our proposed SM extension by adding the U (1) X , Σ (18) and Z 4 symmetries and considering an extended scalar sector and right handed Majorana neutrinos. In Section III we describe the implications of our model in lepton masses and mixings. Section IV deals with quark masses and mixings. The implications of our model in K −K and B −B mixings are discussed in Section V. The consequences of our model in charged lepton flavor violation are analyzed in section VI. We conclude in Section VII. A brief description of the Clebsch Gordan coefficients for the Σ(18) group is presented in Appendix A.

II. THE MODEL
The electroweak gauge group of the SM is supplemented by a Σ(18) × Z 4 discrete symmetry and a global symmetry U(1) X where ψ iL , l iR (i = 1, 2, 3) and ϕ, ϕ carry X = 1 while all other fields have X = 0. In addition to the SM model particle content, three right-handed neutrinos (ν 1R , ν αR ), one SU (2) L doublet φ with X = 0 are assigned as 2 10 , two SU (2) L doublets ϕ, ϕ with X = 1, respectively, put in 1 −0 and 2 20 under Σ(18) and two SU (2) L singlets χ, ρ with X = 0 respectively put in 2 10 and 1 +1 under Σ(18) are introduced. The particle content of the model are summarized in Tables I and II. 1 In this model, fermion masses and mixing angles are generated from renormalizable Yukawa interactions. Non-Abelian discrete groups S3, T , Q4, D4, Q6 contain one-and two-dimensional representations, however, their singlet/doublet components are combined in different ways. Furthermore, Σ(18) contains three two dimensional representations 210, 220, 221 where 2 * 10 = 220 and 2 * 20 = 210 while 221 is a real representation together with its tensor products presented in Appendix A make Σ(18) group has some advantages compared to the other discrete groups and our proposed model is completely different from previous works.
The charged lepton masses can arise from the couplings ofψ (1,α)L l (1,α)R to scalars and the neutrino masses are generated by the couplings ofψ (1,α)L ν (1,α)R andν c (1,α)R ν (1,α)R to scalars while quarks masses can arise from couplings ofQ (β,3)L u (β,3)R andQ (β,3)L d (β,3)R to scalars. Under G symmetry these couplings are summarized in Table III. In order to generate all SM fermion masses, we introduce seven scalars as shown in Table II where H, φ and φ give the charged-lepton and quarks masses, whereas ϕ, ϕ are responsible for generating the Dirac mass terms and χ, ρ yield the Majorana mass terms. The Yukawa interactions for leptons and quarks invariant under all the symmetries of the model are 2 : It is important to note that the U (1) X and Σ(18) symmetries forbid some Yukawa interactions thus giving rise to the desired textures for the lepton and quark sectors as shown in Eqs. (32), (34) and (75) and this is an interesting feature of these symmetries. For instance, for the known scalars in Table II, in the charged lepton sector, the following interactions (ψ 1L l 1R )φ, (ψ 1L l αR )φ, (ψ αL l 1R )φ, (ψ 1L l αR )H, (ψ αL l 1R )H are forbidden by the Σ(18) symmetry; in the neutrino sector, the following interactions In order to generate the observed pattern of SM fermion masses and mixing angles, from the potential minimization condition, we consider the following VEV configuration for the scalar fields: In order to proof that the scalar fields with the VEV alignments as chosen in Eq. (3) is obtained from the minimization condition of V total in Appendix B, let us put which leads to and the minimization condition of V total become ∂V total ∂v j = 0, Furthermore, for simplicity and without loss of generality, we consider the following benchmark point of the Yukawa couplings: λ Hφ λ Hϕ λ Hφϕ The expressions of the scalar potential minimum equations in Eq. (7) thus reduce to the expressions in Appendix C in which the system of Eqs. (C1)-(C8) always have the solution where β Φ (Φ = H, φ, φ , ϕ, ϕ , χ, ρ) and β Hφ are defined in Appendix D.
We will show that, with λ Φ and λ Hφ in Eqs.
In models with more than one SU (2) L Higgs doublet, as in the present model, the Flavor Changing Neutral Current (FCNC) processes exist however they can be suppressed by adding discrete symmetries which have been presented in Refs. [90,[95][96][97][98][99][100][101]. In addition, the large amount of parametric freedom allows to find a suitable region of parameter space where these FCNC can be suppressed. A numerical analysis of the FCNC, along with other phenomenological aspects in a multiHiggs doublet model with the D 4 discrete symmetry is presented in [59]. The implications of our model in the FCNC interactions are discussed in section 4.

III. LEPTON MASSES AND MIXINGS
From the lepton Yukawa terms given by Eq. (1) and the tensor product of Σ (18) in Appendix A, we can rewrite the Yukawa interactions in the lepton sector: 3 Here, we have used the notations: With the help of Eq.(3), we get the mass terms for leptons as follows: which can be written in the matrix form: where and Let us define a Hermitian matrix m l as follows with Comparing the result in Eq. (39) with the experimental values of the charged lepton masses given in Ref. [102], m e 0.51999 MeV, m µ 105.65837 MeV, m τ = 1776.86 MeV, we obtain: In the case α 2 = β 1 = β 2 and |v 1 | ∼ |v 2 |, we get: As we will see below, since the charged lepton mixing matrix U L is non trivial in our model, it can contribute to the final leptonic mixing matrix, defined by U = U + L U ν where U L refers to the left-handed charged-lepton mixing matrix and U ν is the neutrino mixing matrix.
Regarding the neutrino sector, from Eq. (34), the light active neutrino mass matrix arises from the type-I seesaw mechanism as follows: which has three exact eigenvalues where and the corresponding mixing matrix is: where P = diag(1, 1, i) and K, K ∓ , N ∓ are defined where and From the explicit expressions of m 2,3 , K, K ∓ and N ∓ in Eqs. (45), (46), (48) and (49), the following relations hold: with The effective neutrino mass matrix M ν in Eq. (44) is diagonalized as where m 2,3 and K, K ∓ , N ∓ are respectively given in Eqs. (45) and (48).
The final leptonic mixing matrix then reads: In the three neutrino oscillation picture, the lepton mixing matrix can be parametrized as 4 [102] U whereby, θ 12 , θ 23 , θ 13 can be defined via the elements of the leptonic mixing matrix: As it is well known, the neutrino mass spectrum is currently unknown and it can be NO or IO depending on the sign of ∆m 2 32 [102] which will be presented in the next section.

A. Normal spectrum
In NO, the Jarlskog invariant J CP which determines the magnitude of CP violation in neutrino oscillations [102], determined from Eqs. (50) and (54), takes the form Comparing Eq. (57) with its corresponding expression in the standard parametrization of the neutrino mixing matrix given in Ref. [102], J CP = s 13 c 2 13 s 12 c 12 s 23 c 23 sin δ, we get: cos θ l sin θ l sin α s 13 c 2 13 s 12 c 12 s 23 c 23 . 4 Here, δ is the Dirac CP violating phase and cij = cos θij, sij = sin θij with θ12, θ23 and θ13 being the solar, atmospheric and reactor angle, respectively. P contains two Majorana phases (α21, α31) which play no role in neutrino oscillations, P = diag 1, e i α 21 2 , e i α 31

2
, and thus will be ignored.
which is unitary and consistent with the constraint on the absolute values of the entries of the lepton mixing matrix given in Ref. [117]. Now, by using the recent best-fit values for the squared-neutrino mass differences [102], ∆m 2 21 = 7.53 × 10 −5 eV 2 , ∆m 2 32 = 2.453 × 10 −3 eV 2 for the NO, we get a solution κ 1 = 2.95 × 10 −2 , κ 2 = 2.08 × 10 −2 , The absolute neutrino mass, defined as the sum of the mass of the three neutrino mass eigenstates, is found to be 3 i=1 m N ν i = 5.90 × 10 −2 eV. At present, the sum of the three neutrino masses has not been precisely determined, however, the result obtained from our model is well consistent with the strongest bound from cosmology, m ν < 0.078 eV [103].  (51), we get the following relations:

B. Inverted spectrum
Similar to the NO, from Eqs. (54) and (56) for IO, we get a solution: and the Jarlskog invariant J CP determined from (54) and sin δ take the form: cos θ l sin θ l sin α s 13 c 2 13 s 12 c 12 s 23 c 23 .
With the help of Eq. (50), it is easy to show that J N CP = J I CP and sin δ N = sin δ I thus the relations in Eqs. (57) and (58) are satisfied for both normal and inverted orderings and the differences start from Eqs. (60) and (66).
In the case the CP violating phase takes the best-fit values [102], δ = 1.36π we find sin θ l = −0.537 (θ l = 327.5 • ) and the other model parameters are explicitly given in Tab. V.
which is unitary and consistent with the constraint on the absolute values of the entries of the lepton mixing matrix given in Ref. [117]. Now, by using the recent best-fit values for the squared-neutrino mass differences [102], ∆m 2 21 = 7.53 × 10 −5 eV 2 , ∆m 2 32 = −2.546 × 10 −3 eV 2 for IO, we get a solution Using our best fit results given above, we find that the sum of three light neutrino masses is given by

C. Effective neutrino mass parameters
The effective neutrino mass parameters governing the beta decay and neutrinoless double beta ek m k , where U ek (k = 1, 2, 3) are the leptonic mixing matrix elements and m k correspond to the masses of three light neutrinos.
Using the model parameters obtained in subsections III A and III B, we find the following numerical values for the above mentioned mass parameters: The resulting effective neutrino mass parameters in Eqs. (72) and (73) [110,111] m ee < 0.11 ÷ 0.5 eV. Hence, our obtained effective neutrino mass parameter is beyond the reach of the present and forthcoming 0νββ-decay experiments.

IV. QUARK MASSES AND MIXINGS
In this section, we show that our model is able to successfully reproduce the observed pattern of SM quark masses and mixing parameters. Indeed, from the quark Yukawa terms given by Eq.
(2) and the tensor product of Σ (18) in Appendix A, we can rewrite the Yukawa interactions in the quark sector in the form: With the VEV alignments of H and φ as chosen in Eq. (3), the mass Lagrangian of quarks reads where the up-and down-quark mass matrices are with Now we turn our attention to the experimental values of the SM quark masses and CKM parameters [112,113]: We look for the eigenvalue problem solutions reproducing the experimental values of the quark masses and the CKM parameters given by Eq. (77), finding the following solution: This show that our model is consistent with and successfully accommodate the experimental values of the physical observables of the quark sector: the six quark masses, the quark mixing angles and the CP violating phase in the quark sector.
In this section we discuss the implications of our model in the FCNC interactions in the down type quark sector. The FCNC Yukawa interactions in the down type quark sector give rise to meson oscillations. Here we focus on the K −K mixing, whose corresponding ∆M K parameter arises from the following effective Hamiltonian: In our analysis of the K −K mixing we follow the approach of [114,115]. As in Ref. [114,115], the K −K mixing in our model mainly arise from the tree level exchange of neutral CP even and CP odd scalars, thus giving rise to the following operators: where the corresponding Wilson coefficient are given by: with Here N = 11 is the number of CP even scalars of our model, whereas N − 4 = 7 is the number of CP odd scalars. Let us note that our model is an extended 8HDM where the scalar sector is enlarged by the inclusion of 3 real gauge singlet scalars.
On the other hand, the K −K mass splitting has the form: Then, it follows that: Using the following parameters [114,115]: We get the following constraint arising from K −K mixing: Given the large amount parametric freedom in both fermion and scalar sectors of our model, such constraint can be fulfilled. To show explicitly that the above given constraint resulting from K −K mixing is successfully fullfilled and given the large amount of parameters in our model, we consider a simplified benchmark scenario where: Here we identified H 0 1 with 126 GeV SM like Higgs boson. We plot in Figure 4 the allowed parameter space in the m H − m A plane consistent with the constraint arising from K −K mixing in the aforementioned simplified benchmark scenario of our model. Here, for the sake of simplicity we have set y = 2 × 10 −5 . Figure  ratio is given by [116]: To simplify our analysis we choose the benchmark scenario described in section 4. We display in Figure  The tensor products between doublets of Σ(18) are given by [93]: where x i , y i (i = 1, 2) are the components of two different representations.