Viable low-scale model with Universal and Inverse Seesaw Mechanisms

We formulate a viable low scale seesaw model, where the masses for the Standard Model (SM) charged fermions lighter than the top quark emerge from a Universal Seesaw mechanism mediated by charged vector-like fermions. The small light active neutrino masses are produced from an Inverse Seesaw mechanism mediated by right-handed Majorana neutrinos. Our model is based on the $A_{4} $ family symmetry, supplemented by cyclic symmetries, whose spontaneous breaking produces the observed pattern of SM fermion masses and mixings. The model can accommodate the anomalous magnetic dipole moment of the muon and predicts strongly suppressed $\mu \rightarrow e\gamma $ and $\tau \rightarrow \mu \gamma $ decay rates, but allows a $\tau \rightarrow e\gamma $ decay within the reach of the forthcoming experiments.


I. INTRODUCTION
The Standard Model (SM) has offered us a theoretical framework with great experimental success. In spite of this, the observed values in quark mixing angles together with the pattern in the charged fermion masses find no explanation. Moreover, the observation of neutrino oscillations have augmented this puzzle as the theory must also be extended to incorporate neutrino masses along with the observed leptonic mixing parameters. The pattern in all the fermion masses may be described by three main aspects: (i) Only one mass at the electro-weak (EW) scale while all others well below it, (ii) Neutrino masses are much smaller than the electron mass, (iii) The charged fermion masses satisfy a hierarchical structure, Several attempts have been made to theoretically describe each of these aspects either individually [1][2][3][4] or various simultaneously, see for example [5][6][7][8][9][10][11][12][13]. In the following, to produce Eq. (1) and (3) we opt to work within a low-scale realization of a universal seesaw model whereas for Eq. (2) we consider an inverse seesaw mechanism.
In universal seesaw models [6,14], the smallness of fermion masses except for the top quark, Eq. (1), can be easily explained by promoting parity symmetry (L↔R) to a fundamental symmetry at high energies, larger than the Fermi scale. These models are based on the SU (2) L × SU (2) R × SU (3) c × U (1) B−L gauge symmetry, where B and L stand for the baryon and lepton number, respectively. On the other hand, the matter content is enlarged by introducing vector-like fermions (singlets under the left and right isospin symmetries) whereas the scalar sector gets minimally enlarged by mirroring the SM Higgs, H ∼ ( 2, 1, 1, −1), to the right sector, H R ∼ (1, 2, 1, −1), transforming as a right doublet. The conventional bidoublet in L-R symmetric theories is here missing. As a consequence neutrinos have no masses and Yukawa interactions are now made with both scalars whereas the singlet fermions acquire their own mass terms. Typically, after both scalars have acquired their vacuum expectation values (VEV), small fermion masses arise as an admixture of both VEVs and the heavy mass of the singlet fermions, m f ∼ v EW (v R /M x ), where v R M x , while the top quark mass has no vector-like fermion companion, and thus its mass is simply given by the standard formula, m t ∼ v EW . For last, the hierarchy given in Eq. (3) may be understood by considering exotic fermion masses with an inverse hierarchy, M x1 M x2 M x3 . In the following, we mimic the main shared features among this class of models and discuss a low-scale scenario.
The smallness of neutrino masses may have a different origin than that of the charged fermions. Already their superlightness seems to point out to this possibility. Hence, here we consider that two different mechanisms are responsible for the observed patterns in the fermion masses. We choose to study the mass nature of neutrinos via an inverse seesaw [4,[15][16][17]. This mechanism leads to an effective mass parameter given by m ν ∼ ( m D M E ) 2 µ where m D is the typical scale of a Dirac mass, M E the heavy scale of the isosinglet leptons which conserve lepton number, and µ the mass scale of the gauge singlet neutrinos responsible for breaking lepton number. It follows that for small µ then m ν becomes small; which is opposite to the standard seesaw, where the smallness of neutrino masses is due to the largeness of the right handed neutrino masses. The advantage of using an inverse seesaw is that lepton flavor violation (LFV) rates do not depend on the small magnitude of the lepton number violating scale, µ, while they vanish in standard seesaw scenarios.
In this work we propose a low scale seesaw model with extended scalar and fermion sectors, consistent with the current pattern of SM fermion masses and mixings. In our model, the masses of the SM charged fermions lighter than the top quark are generated from a universal seesaw mechanism mediated by charged exotic vector-like fermions. The small light active neutrinos masses arise from an inverse seesaw mechanism mediated by three sterile neutrinos. In our model we use the A 4 family symmetry, which is supplemented by other auxiliary symmetries, thus allowing to have a viable description of the current SM fermion mass spectrum and mixing parameters. We have chosen the A 4 family symmetry since it is the smallest order discrete group with one three-dimensional and three distinct one-dimensional irreducible representations, where the three families of fermions can be accommodated rather naturally. This group was used for the first time in Ref. [18] and subsequently used in [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] to provide a viable and predictive description of the SM fermion mass spectrum and mixing parameters.
The outline for the rest of this paper is as follows. In Section II we introduce the model, followed by discussions on the quark and lepton masses and mixing in Sections III and IV, respectively. We devote Section V to study some phenomenological aspects of our model. Finally, in Section VI we conclude.  real scalar singlets (flavons), Quark sector: where k = 1, 2, 3 and we have classified the flavons into two different groups depending in which sector they are relevant.
To control the arbitrariness in the Yukawa interactions we introduce A 4 as our flavor symmetry. A 4 has been found to be phenomenologically interesting and successful in the neutrino sector [21,36,37]. Appendix A has a brief description of the group and multiplication rules. On the other hand, we have found to be insufficient this symmetry to fully develop a predictive theory. For this purpose, we employ the Abelian symmetry, Z 2 × Z 5 with an additional Z 2 and Z 4 × Z 8 , in the quark and lepton sector, respectively. The charge assignments are given in Tables I, II, and  III. The full Yukawa terms can be written as down-type quarks, charged leptons, +y (l) 5 and neutrinos, being the dimensionless couplings in Eqs. (8), (9), (10) and (11) O(1) parameters.
We denote by χ i = v χ i (i = 1, 2), and assume the following VEV patterns for the A 4 triplet SM singlet scalars Φ 1,2,3 , ξ, ζ and S, which are natural solutions of the scalar potential minimization equations for a large region of the parameter space as shown in Refs. [22,[38][39][40][41][42]. As the hierarchy among charged fermion masses and quark mixing angles mass emerges from the spontaneous breaking of the we set the VEVs of the SM singlet scalar fields σ, ξ i , ζ i (i = 1, 2, 3) with respect to the Wolfenstein parameter λ = 0.225 and the model cutoff Λ, as follows

III. QUARK MASSES AND MIXINGS
Due to the symmetry assignments, the top quark does not mix with the exotic vector-like quarks and thus its mass is simply given as in the SM via its Yukawa interaction with the Higgs doublet, m t ∼ y t v EW . On the other hand, the vector-like quarks do mix with all the others SM quarks. Then, from Eqs. (8) and (9), the quark mass matrices take the form, where As the masses of the vector-like quarks are much larger than the employed VEVs, M T , M B {v EW , v χ , v i }, the implementation of the Universal Seesaw yields the following 3 × 3 low-scale quark mass matrices,  where the up-type quark masses are given by, and we have defined the parameters, with y f k denoting the product of two different Yukawa couplings which can be merged into a single one as both are O(1) parameters. From Eq. (19) and the known hierarchy in the up-quark masses we can estimate the ratio among the heavy masses and VEVs to be, which if we assume a single heavy scale, In total we have 5 complex parameters, that is, 10 free parameters to fit 7 observables. We restrict the number of free parameters to 7. For this purpose, we only keep free 2 of the 5 phases, and limit ourselves to the particular case where, Hence, the set of independent parameters becomes, We then perform a numerical fit to the set of parameters. The experimental input parameters are the three down-type quark masses, the magnitude of the three independent mixing matrix elements, and the Jarlskog invariant. The masses are taken at the M z scale with a symmetrised 1σ error taken to be the larger one. The employed input parameters are summarized in Table IV. To measure the quality of the fit we use the function, Its minimization leads to the best-fit values, implying the observed down-type quark masses and mixing shown in Table IV. Moreover, we find that the two smallest mixing angles are correlated among them and also with the Jarlskog invariant, see Fig. 1. For last, notice that our model prefers small values of the Jarlskog invariant compared to the latest fit from the PDG [43]. From the best fit values, Eq. (24), we can estimate the required ratio among the heavy masses and VEVs in order to reproduce the observed mild hierarchy among the fitted parameters, in such a way that all Yukawa couplings may still remain as O(1) parameters. All these ratios can be rewritten in terms of the heaviest mass, Models with vector-like fermions are being tested at the LHC. The ATLAS collaboration has reported several analyses, in particular [45][46][47]. At the moment, mass exclusion limits for exotic isosinglet quarks give M B > 1. 22 TeV and M T > 1.31 TeV as found in Ref. [47]. These lower bounds were set by only assuming that the exotic quarks can decay on SM particles. That is, the vector-like quarks would first be produced at collider experiments via pair-production, a process dominated by the strong interactions, gg →BB (T T ). Then, each exotic quark would decay to: T → W b, Zt, Ht or B → W t, Zb, Hb, where it has only been considered the third fermion family. Now, in our case, in the interaction basis, our model has no initial mixing between the top-quark and the vector-and top-like fermions, but only between the up and charm quarks with the exotic partners. On the other hand, in the down-quark sector, all the standard quarks mix with the exotic partners. Therefore, in the mass basis, we may only expect from the previous decay modes that only those from the exotic bottom-like quarks will survive. Observation of an excess of events in any of the final states related to the vector-like B quarks with respect to the SM background, e.g. when Z → + − the final state has one dilepton and at least two b-jets, would be a signal supporting this model at the LHC. A detailed study of the collider phenomenology of our model is beyond the scope of this paper and is left for future studies.

IV. LEPTON MASSES AND MIXINGS
Using Eq. (11) we get the following mass matrix for charged leptons: where the different submatrices are given by: Here we have adopted a simplifying benchmark scenario with the following particular assumptions about the charged lepton sector model parameters and VEVs of some of the gauge singlet scalars: Thus, the universal seesaw mechanism gives rise to the following SM charged lepton mass matrix: where the charged lepton masses are: ). Let us note that the charged lepton masses are linked with the scale of electroweak symmetry breaking through their power dependence on the Wolfenstein parameter λ = 0.225, with O(1) coefficients.
The neutrino Yukawa terms of Eq. (11) give origin to the following neutrino mass terms: where the neutrino mass matrix reads: and the submatrices read: The light active masses arise from an inverse seesaw mechanism and the resulting physical neutrino mass matrices take the form: where M ν corresponds to the active neutrino mass matrix whereas M (1) ν and M (2) ν are the sterile mass matrices. Thus, the light active neutrino mass matrix is given by: The full neutrino mass matrix given by Eq. (33) can be diagonalized by the following rotation matrix [48]: where Notice that the physical neutrino spectrum is composed of three light active neutrinos and six exotic neutrinos. The exotic neutrinos are pseudo-Dirac, with masses ∼ ± 1 2 M 2 + M T 2 and a small splitting µ. Furthermore, R ν , R On the other hand, using Eq. (40) we find that the neutrino fields ν L = (ν 1L , ν 2L , ν 3L ) T , ν C R = ν C 1R , ν C 2R and N C R = N C 1R , N C 2R are related with the physical neutrino fields by the following relations: where Ψ jL and Ψ    Table V. Fig. 2 shows the correlation between the solar mixing parameter sin 2 θ 12 and the leptonic CP violating phase. To obtain this figure the lepton sector model parameters were randomly generated in a range of values where the neutrino mass squared splittings and leptonic mixing parameters are inside the 3σ experimentally allowed range. As seen from Fig. 2, our model predicts a solar mixing parameter sin 2 θ 12 and leptonic Dirac CP violating phase in the ranges 0.27 sin 2 θ 12 0.38 and 140 • δ 260 • , respectively.
Another relevant observable that can be determined in this model, is the effective Majorana neutrino mass parameter of neutrinoless double beta decay, which provides information on the Majorana nature of neutrinos. The effective Majorana neutrino mass parameter takes the form: where U ej and m ν k are the the PMNS leptonic mixing matrix elements and the neutrino Majorana masses, respectively. The neutrinoless double beta (0νββ) decay amplitude is proportional to m ee . In Figure 3 we display the correlation between the effective Majorana neutrino mass parameter m ee and the leptonic Dirac CP violating phase δ CP . As indicated by Figure 3, our model predicts an effective Majorana neutrino mass parameter in the range 0.020 eV m ee 0.040 eV, thus implying that the values for the effective Majorana neutrino mass parameter predicted by our model are within the reach of the next-generation bolometric CUORE experiment [52], as well as the next-to-next-generation ton-scale 0νββ-decay experiments [53][54][55][56].

V. PHENOMENOLOGY
As previously stated, the physical sterile neutrino spectrum containts six almost degenerate TeV scale neutrinos, which mix the active ones, with mixing angles of the order of . In return, these new couplings will induce one-loop level phenomena through which we may impose constraints to our model.
In this section we will discuss the implications of our model in the lepton flavor violating decays and in the anomalous magnetic dipole moment of the muon.

A. Charged LFV decays
The heavy sterile neutrinos together with the W gauge boson induce the one loop level decay l i → l j γ, whose corresponding branching ratio reads [15,60,61]: where s W = sin(θ W ), and Thus, the charged lepton flavor violating processes µ → eγ, τ → µγ and τ → eγ have the following branching ratios: where Γ τ = 2.27 × 10 −12 GeV is the tau decay width. On the other hand, the upper experimental bound of the charged lepton flavor violating process τ → eγ is given by: In Figure 4 we display the allowed parameter space in the m N − f plane consistent with the constraints arising from charged lepton flavor violating decays. As seen from Figure 4, the obtained values for the branching ratio of τ → eγ decay are located in the range 5 × 10 −10 Br (τ → eγ) 3 × 10 −9 , for sterile neutrino masses m N lower than about 1 TeV. Consequently, our model is compatible with the charged lepton flavor violating decay constraints provided that the sterile neutrinos are lighter than about 1 TeV.
The current discrepancy between the experimental and predicted value is still inconclusive and amounts to 3.5 standard deviations [43], where the errors at 1 σ are from experiment and theory, respectively. In the following, we consider the average error between the theoretical and experimental one.
Contributions to ∆a µ arising from scenarios like this one where the active neutrinos mix with heavy-right handed neutrinos have been already computed. The relevant expression is given by [61,62], and f = m N f mµ and κ µ = mµ M W . In our particular case, the vector and axial-vector couplings to the W bosons are identical and thus the expression is reduced to, Notice that only two couplings will contribute, R v 1µ and R v 3µ . Both of them can be approximated to R v times order one parameters, {x, y, r} ∼ O(1). Figure 5 exemplifies the available parameter space in the m N − f plane which accommodates ∆a µ at 3 σ. Variations of the set of parameters in the range 0.3 {x, y} 1 and −4.5 r 2.9 are shown in the gray background whereas for the particular case in which all order one parameters are taken equal to one are depicted by the colored bands.

VI. CONCLUSIONS
We have proposed a viable low scale seesaw model based on the A 4 family symmetry and other auxiliary cyclic symmetries, where the SM particle spectrum is enlarged by the inclusion of several charged vector-like fermions, right-handed Majorana neutrinos and scalar singlets, consistent with the low energy SM fermion flavor data. The masses for the SM charged fermions lighter than the top quark emerge from a Universal Seesaw mechanism mediated by charged vector-like fermions, whereas the small light active neutrino masses are generated from an Inverse Seesaw mechanism. The smallness of the µ parameter of the inverse seesaw is attributed to a right-handed neutrino nonrenormalizable mass term, generated after the spontaneous breaking of the discrete symmetries of the model. The spontaneous breaking of these discrete symmetries takes place at large energies and gives rise to the observed SM fermion mass spectrum and mixing parameters. We have studied the implications of our model in the lepton flavor violating decays and in the anomalous magnetic dipole moment of the muon. We have found that the µ → eγ and τ → µγ are strongly suppressed in our model, whereas the τ → eγ decay can attain values within the reach of the current sensitivity of the forthcoming charged lepton flavor violation experiments. Furthermore, the obtained values for the branching ratio for the τ → eγ are lower than its current experimental bound for sterile neutrino masses lower than about 1 TeV. Finally, we have found that our model successfully accommodates the experimental value of the anomalous magnetic dipole moment of the muon.