Fermion mass hierarchy and g-2 anomalies in an extended 3HDM Model

We propose an extension of the three-Higgs-doublet model (3HDM), where the Standard Model (SM) particle content is enlarged by the inclusion of two inert $SU_{2L}$ scalar doublets, three inert and two active electrically neutral gauge singlet scalars, charged vector like fermions and Majorana neutrinos. These additional particles are introduced to generate the SM fermion mass hierarchy from a sequential loop suppression mechanism. In our model the top and exotic fermion masses appear at tree level, whereas the remaining fermions get their masses radiatively. Specifically, bottom, charm, tau and muon masses appear at 1-loop; the masses for the light up, down and strange quarks as well as for the electron at 2-loop and masses for the light active neutrinos at 3-loop. Our model successfully accounts for SM fermion masses and mixings and accommodates the observed Dark Matter relic density, the electron and muon anomalous magnetic moments, as well the constraints arising from charged Lepton Flavor Violating (LFV) processes. The proposed model predicts charged LFV decays within the reach of forthcoming experiments.

with neutrino mass generated at 4-loop level (n = 4). However, this model has a low cutoff scale, since it includes non-renormalizable Yukawa terms, needed to implement the radiative mechanisms of the SM fermion mass generation (1)- (4). From the view-point of model building, it is much more preferable to have a renormalizable setup with a moderate amount of particle content and predicting a phenomenology beyond the SM within the reach of future experimental sensitivities. With this in mind, we propose here a renormalizable model implementing the sequential loop-suppression mechanism (1)-(3) with the light active neutrino masses appearing at three loop level (n = 3). This model has a much more economical field content compared to the similar renormalizable models proposed in Refs. [50,79]. For instance, whereas the scalar sector of the model of Ref. [79] has 2 SU 2L scalar doublets, 7 complex electrically neutral gauge singlet scalars and 5 electrically charged singlet scalar fields, thus amounting to 32 scalar degrees of freedom, the model proposed here has three SU 2L scalar doublets, 3 complex and 2 real electrically neutral singlet scalars, which corresponds to 20 scalar degrees of freedom. Furthermore, the scalar sector of the model of Ref. [50] has three SU 3L scalar triplets, three complex electrically neutral singlet scalars and four electrically charged singlet scalar fields, thus amounting to 32 scalar degrees of freedom, which is much larger than the number of scalar degrees of freedom of our current model. Moreover, our model can also successfully accommodate the electron and muon anomalous magnetic moments, the observed Dark Matter relic density, as well the constraints arising from charged Lepton Flavor Violating (LFV) processes. Let us emphasize the difference of our proposed model with respect to recent publications based on radiative mass and hierarchy generation: In Ref. [156] there is no mechanism to generate the masses of the quarks of the first generation. In addition, the model described in [156] does not provide an explanation for the SM lepton mass hierarchy. In Ref. [157], both the first and second generation SM charged fermion masses are produced at one loop, whereas here we generate the lightest SM charged fermion masses at two loop level. Moreover, in [157] the neutrinos remain massless. The paper is organized as follows. In section II we outline the proposed model. In section III we analyze the stability and describe the electroweak symmetry breaking of the scalar potential of the model. The scalar mass spectrum of the model is discussed in section IV. The implications of our model with respect to the SM fermion-mass hierarchy is discussed in section V. In section VI charged LFV decays as well as the constraints on the charged scalar masses are considered. The implications of our model for the muon and electron anomalous magnetic moments are discussed in section VII. The prospects with respect to Dark Matter are analyzed in section VIII. Our conclusions are given in section IX.

II. THE MODEL
Before providing a complete model setup, let us explain the motivations behind introducing extra scalars, fermions and symmetries needed for a consistent implementation of the sequential loop suppression mechanism for generating the SM fermion hierarchies. Our strategy is to ban certain operators, by imposing appropriate symmetries to ensure loop suppression, necessary to reproduce the observable hierarchy of the SM fermion masses. In our model the top quark mass arises at tree level from a renormalizable Yukawa operator, with an order one Yukawa coupling, i.e. q iL φu 3R , i = 1, 2, 3 .
We denote the left-handed quarks by q iL and the right-handed up and down quarks by u iR and d iR , respectively, with i = 1, 2, 3 the family index. The SM like Higgs boson doublet is denoted by φ.
To generate the bottom, charm, tau and muon masses at one loop level, it is necessary to forbid the operators: at tree level and to allow the following operators, crucial to close the one loop level diagram of the upper left panel of Figure 1: , This requires to add an unbroken Z symmetries is crucial for the implementation of the radiative seesaw mechanism that produces one-loop level masses for the bottom, charm, tau and muon without invoking soft-breaking mass terms. Notice that the fermionic sector is enlarged by electroweak charged exotic fermions F r , where r = 1 for quarks and r = 2 for charged leptons, and that the Yukawa operators as well as the trilinear scalar operator shown in Eq. (7) correspond to the three vertices of the one loop level diagram in the upper left panel of Figure 1. Considering the simplest possibility, where such charged exotic fermions F r are SU 2L singlets, the scalar sector has to be extended by the inclusion of an extra SU 2L scalar doublet η and an electrically neutral electroweak gauge-singlet scalars σ and χ. The scalar fields η and σ are both charged under the preserved Z (2) 2 symmetry, whereas the scalar χ is neutral under this symmetry. The singlet scalar field χ is needed to provide masses to the charged exotic fermions F r . This scalar field χ is assumed to be charged under the spontaneously broken Z (1) 2 symmetry. Furthermore, the Yukawa term (y F ) r F rL χF rR , which involves the electroweak charged exotic fermions, must also be included as well, in order to close the one loop level diagram of Figure 1. Besides that, small masses for the light SM charged fermions, i.e., the up, down and strange quarks as well as the electron, are generated at two loop level. This implies to forbid the following operators that would give rise to tree and one-loop-level masses for these particles: Here fiL = uiL, diL, eiL (i = 1, 2, 3), fR = u2R, d3R, l2R, l3R, fR = u1R, d1R, d2R, l1R. The electroweak singlet charged exotic fermions, see (13), are denoted by FrR, FrL, FsR and FsL, where r = 1 for quarks, r = 2 for charged leptons, s = 1 for up type quarks and charged leptons, and s = 2 for down type quarks and neutrinos. Furthermore, in the neutrino loop diagram we have p, k ∈ {1, 2}.
However the following operators are required to provide two loop level masses for the light SM charged fermions: 1 for up-type quarks and charged leptons, 2 for down-type quarks.
Such operators are crucial to close the two-loop-level diagram of the upper right panel of Figure 1. For this to happen, the fermion sector is extended as well, by adding the electroweak charged exotic fermions F s , F s where s = 1 for up-type quarks and charged exotic leptons and s = 2 for down-type quarks. The simplest choice is to assign these charged exotic fermions F s , F s to SU 2L singlets. Then, in order to build the Yukawa interactions that determine three of the four vertices of the two loop level diagram of Figure 1, we also need to add an extra SU 2L scalar doublet ϕ and another electrically neutral electroweak gauge-singlet scalars ρ, ξ and ζ. The scalar fields ϕ, ρ and ξ are assumed to have complex charges under an additional spontaneously broken Z 4 symmetry, whereas the scalar ζ has a real charge under this Z 4 symmetry. We further assume that the Z 4 symmetry is spontaneously broken down to a preserved Z 2 symmetry, which implies that the scalar fields ρ and ξ do not acquire vacuum expectation values whereas the scalar ζ does. Furthermore, in order to close the aforementioned two loop diagram, one has to include the mass term m F s F sL F sR and the Yukawa interaction y F s¯ F sL ζ F sR involving the electroweakly charged exotic fermions. Notice that the Yukawa operators, as well as the quartic scalar operator shown in Eq. (9), correspond to the four vertices of the two loop level diagram of the upper right panel of Figure 1.
In what regards the neutrino sector, we require that the light active neutrino masses only appear at three-loop level.
To this end, right-handed Majorana-neutrinos have to be added in the fermionic spectrum. In addition, one should prevent the appearance of tree, one and two-loop level masses for the light active neutrinos. Generating light active neutrino masses at three-loop level, as in the Feynman diagram of the bottom left panel of Figure 1, requires the presence of the operators and forbidding: where ν sR , Ω sR and Ψ sR (s = 1, 2) are gauge singlet right-handed Majorana neutrinos. By an appropriate choice of charges (shown below) under the aforementioned Z 2 × Z 4 symmetry, the three-loop level radiative seesaw mechanism for light active neutrinos can be implemented. With the aim of implementing the sequential loop suppression mechanism that generates the pattern of SM fermion masses, we consider an extension of the inert 3HDM, where the SM gauge symmetry is supplemented by a Z 2 × Z 4 discrete symmetry is that it is the smallest cyclic symmetry that allows us to realize the loop-suppression scenario (1)-(4) with n = 3 in a renormalizable 3HDM setup without invoking soft symmetry breaking. The scalar sector of the model consists of three SU 2L scalar doublets, i.e., φ, η, ϕ and five scalar singlets σ, ρ, ξ, χ and ζ, with the Z We assume that the Z and Z 4 symmetries are spontaneously broken. We further assume that the Z 4 symmetry is spontaneously broken down to a preserved Z 2 symmetry. These assumptions imply that the scalar fields η, ϕ, σ, ρ, ξ, charged under the Z (2) 2 symmetry and (or) having complex Z 4 charges, do not acquire vacuum expectation values. This conditions are inevitable in the present setup for implementing the scenario (1)-(4). Let us note that the SU 2L scalar doublet φ is the only scalar field that acquires a non-vanishing vacuum expectation value (VEV) equal to about 246 GeV and thus corresponds to the SM Higgs doublet. A justification of the extension of the scalar sector of the model is provided in the following. The SU 2L inert scalar doublet η as well as the inert SM gauge singlet scalar σ are introduced to generate the one-loop level masses for the bottom, charm quarks, tau and muon leptons. The scalar singlets χ and ζ are introduced to provide masses to the charged exotic fermions. Moreover, the implementation of the two-loop level radiative seesaw mechanisms, generating the up, down, strange quark masses as well as the electron mass, requires to introduce an extra SU 2L inert scalar doublet, namely ϕ and inert SM gauge singlet scalars, i.e., ρ and ξ. The particles ϕ and ρ are also crucial to give three-loop level masses for the light active neutrinos. The three loop level neutrino mass diagram is closed thanks to the gauge singlet scalars χ and ζ.
The fermion sector of the SM is extended by the SU 2L singlet exotic quarks T ,T , T , B,B, B and singlet leptons E,Ẽ, E , ν s (s = 1, 2), Ω, Ψ with electric charges 2 × Z 4 assignments of the fermion sector are: The quark, lepton and scalar assignments under Tables I, II and III, respectively.
φ η ϕ σ ρ ξ χ ζ SU3c 1 1 1 1 1 1 1 1 Now, let us justify the exotic fermion content of our model. The gauge-singlet neutral leptons ν s , Ω s , Ψ s (s = 1, 2) are introduced to generate the three-loop level masses for two light active neutrinos. Let us note that the neutrino oscillation experimental data requires to have at least two light massive active neutrinos [158]. Furthermore, note that the SU 2L singlet exotic quarks T ,T , T , B,B, B and singlet leptons E s (s = 1, 2),Ẽ, E introduced in our model, correspond to the minimal amount of charged exotic fermion content needed to yield one-loop level masses for the bottom, charm quarks, tau and muon leptons, as well as two-loop level masses for the light up, down, strange quarks and the electron, without including soft-breaking mass terms. With the specified particle content, we have the following quark, charged lepton and neutrino Yukawa terms invariant under the Z −L After electroweak gauge-symmetry breaking, the above-given Yukawa interactions yield the SM fermion masses via sequential loop suppression. Furthermore, the non SM-like scalars (excepting the scalar singlets χ and ζ) are not allowed to acquire VEVs for the following reasons: Firstly, in this way we avoid to generate tree-level masses for the SM fermions lighter than the top quark. Secondly, we open the possibility to have stable scalar Dark Matter candidates. Eventually we also avoid to encounter tree level Flavor Changing Neutral Currents (FCNCs).

III. STABILITY AND ELECTROWEAK SYMMETRY BREAKING OF THE HIGGS POTENTIAL
The renormalizable Higgs potential, invariant under the symmetries of the model, has the form: The scalar fields can be written as From the condition to have a vanishing gradient of the potential with the neutral component of the doublet φ getting a VEV v/ √ 2 as well as the scalar singlets χ, ζ acquiring VEVs v χ and v ζ , respectively, whereas all other VEVs are vanishing, we find the constraints on the potential parameters: From the symmetry of the potential (18) we find that for positive quartic parameters λ 1 , λ 2 , λ 3 , as well as κ 1 , κ 2 , κ 3 , κ 4 , κ 5 , the potential is bounded from below, that is, it is stable. However, we also have to ensure that it provides the experimentally acceptable electroweak symmetry breaking of SU (2) L × U (1) Y → U (1) em and gives the correct VEV of about v ≈ 246 GeV. Let us emphasize that in general it is not sufficient to check that the potential has a vanishing gradient, leading to the condition (21). In particular, the corresponding local stationary point can correspond to a saddle point or maximum, and not a minimum. Moreover, there can be deeper stationary points. A systematic approach to find the global minimum for any 3HDM has been presented in [159]. The case of two Higgs-boson doublets accompanied by an arbitrary number of Higgs-boson singlets has been also studied [160]. For the potential considered here we have, in addition to the three Higgs-boson doublet fields φ, η, ϕ, also three complex singlet fields, σ, ρ and ξ and two real scalars χ and ζ. We adopt the formalism of three doublets presented in [159] to the case of additional Higgs-singlet fields that we have here. The essential step is to introduce bilinears for the Higgs-boson doublets [161,162] and decompose the complex Higgs singlets into its real and imaginary parts. First, all gauge-invariant scalar products of the three doublet fields φ, η, ϕ are arranged in a matrix, This matrix can be expressed in a basis of matrices λ α (α = 0, 1, . . . , 8), where λ 0 = 2 3 1 3 is the conveniently scaled identity matrix and λ a (a = 1, . . . , 8) are the Gell-Mann matrices. In this basis we can write The real coefficients, called bilinears K α , are obtained from We can invert this relation and express the gauge-invariant scalar products of the doublets, which appear in the potential, in terms of the bilinears: Further, we decompose the complex singlets into its real and imaginary parts, With the replacements (27) and (28), the potential can be written in terms of the bilinears as well as the real and imaginary parts of the singlets, V (K 0 , . . . , K 8 , σ R , σ I , ρ R , ρ I , ξ R , ξ I , χ, ζ). All gauge degrees of freedom are systematically avoided and all fields and parameters are real in this form. We now look for all stationary points of the potential, in order to find the global minimum, or in the degenerate case, the global minima. For a stable potential, the global minimum is given by the deepest stationary point. We now classify the stationary points with respect to the rank of the matrix K. Any stationary point with rank 2 of K corresponds to a fully broken electroweak symmetry, rank 0 to an unbroken electroweak symmetry, and rank 1 to a physically acceptable breaking of SU (2) L × U (1) Y → U (1) em . The rank conditions result in different sets of polynomial equations. Explicitly, the set of equations corresponding to the rank 2 are, Here u denotes a Lagrange multiplier. For the solutions with rank 0 of K, we set all bilinears to zero and look for the stationary points of the corresponding potential, that is, solutions of the set of equations, ∇ σ R ,σ I ,ρ R ,ρ I ,ξ R ,ξ I ,χ,ζ V (K 0 = 0, . . . , K 8 = 0, σ R , σ I , ρ R , ρ I , ξ R , ξ I , χ, ζ) = 0.
With respect to rank 1 solutions of K, we can parametrize the matrix K in terms of the three-component complex getting for the bilinears The potential can now be written as V (K 0 , w † , w, σ R , σ I , ρ R , ρ I , ξ R , ξ I , χ, ζ) and the corresponding set of polynomial equations reads where u again denotes a Lagrange mulitplier. We solve the three sets of equations (29), (30), (33) and the solution with the lowest potential value is (are) the global minimum (minima). A solution is only physically acceptable if it originates from the set (33), corresponding to the observed electroweak symmetry-breaking. In addition, we have to check that the vacuum gives the observed VEV v. Numerically, we accept solutions which provide a vacuum-expectation value in the range 245 GeV < v < 247 GeV. The sets of equations can be solved via homotopy continuation; see for instance [163]. The homotopy continuation algorithms can be found implemented in the open-source software package PHCpack [164]. We have numerically checked that there is large parameter space available fulfilling the stationarity and stability conditions of the potential. Also the potential can provide sufficiently heavy scalars, apart from the SM-like Higgs boson, in accordance with the experimental constraints.

IV. THE HIGGS MASS SPECTRUM
Here we restrict ourselves to real parameters of the potential (18), that is, in particular we consider a CP conserving scalar sector. Then, we find that the spectrum of the physical CP even neutral scalars is composed of the 126 GeV SM-like Higgs boson, i.e h, two heavy CP even scalars H 1 and H 2 as well as the inert scalars transforming nontrivially under the Z (2) 2 symmetry and (or) having complex Z 4 charges, namely ϕ 0 R , ρ R , ξ R , S 1 and S 2 . For the sake of simplicity, we consider the scenario of the decoupling limit which is motivated by the experimental fact that the couplings of the 126 GeV SM-like Higgs boson are very close to the SM expectation. In the decoupling limit φ 0 R corresponds to the 126 GeV SM-like Higgs boson, i.e h. The squared masses of the φ 0 R , ϕ 0 R , ρ R , ξ R scalars are given by: The scalar fields H 1 and H 2 are physical mass eigenstates of the following squared scalar mass matrix written in the χ, ζ basis: This matrix can be diagonalized as follows: Consequently, the physical scalar mass eigenstates states of the matrix M 2 H are given by: Their squared masses are: The scalar fields S 1 and S 2 are physical mass eigenstates of the following squared scalar mass matrix written in the η 0 R , σ R basis: This matrix can be diagonalized as follows: Consequently, the physical scalar mass eigenstates states of the matrix M 2 S are given by: Their squared masses are: Concerning the CP odd scalar sector, we find that it is composed of one massless pseudoscalar state, i.e, φ 0 I , which is identified with the neutral SM Nambu-Goldstone boson G 0 Z eaten up by the longitudinal component of the Z gauge boson, as well as four physical pseudoscalar fields ϕ 0 I , ρ I , ξ I , P 1 and P 2 . The squared masses of the φ 0 I , ϕ 0 I , ρ I scalars are given by: The scalar fields P 1 and P 2 are the physical mass eigenstates of the following squared scalar mass matrix written in the η 0 I , σ I -basis: which can be diagonalized by the transformation: Consequently, the physical scalar mass eigenstates P 1,2 are given by: Their squared masses are: In the charged scalar sector we find two massless Nambu-Goldstone states, φ ± , absorbed by the longitudinal components of W ± gauge bosons, as well as four physical charged scalars, η ± , ϕ ± with the masses: This completes the list of the scalar sector of our model.

V. SM FERMION MASS HIERARCHY
The SM fermion mass matrices are generated in our model according to the diagrams in Fig. 1, with the Yukawa interactions in (14)- (17). We write the mass matrices for the charged fermions in the form Here we have taken into account the loop level at which the columns of these matrices are generated, in particular l ≈ (1/4π) 2 is the loop suppression factor. The powers of this loop factor in (49), (50), explicitly display the following picture that we have in our model: The third column in M U is generated at the tree-level, engendering mass to the top quark. The second and first columns of M U arise at the one and two-loop levels, respectively, and are associated with the charm and up quark masses. The light down and strange quark masses are also generated at two-loop level. On the other hand, the third column of M D arise at one-loop level. As for the SM charged lepton mass matrix M l , its first column, responsible for the electron mass, appears at the two-loop level, whereas its second and third columns, providing masses to the muon and the tau lepton, respectively, are generated at one loop.
As we pointed out before, the objective of the model is to generate the observed hierarchy of the fermion mass spectrum in terms of loop suppression. Therefore, it is crucial that the quark masses and mixings predicted by the model are reproduced with parameters a = 1, 2, 3), Let us check this essential point in detail: We use the experimental values of the quark masses [165], the CKM parameters [166] and the charged lepton masses [166]: As we can see, all the entries (the absolute values) of the above matrices are of order unity with rather mild deviations. This demonstrates that the proposed model is able to explain the existing pattern of the observed quark spectrum. via the sequential loop suppression mechanism. Finally, the small masses of the active neutrinos are generated at the three-loop level, as follows from the last diagram of Figure 1. Thus, for the neutrino mass matrix we can write

VI. CHARGED LEPTON-FLAVOR VIOLATION CONSTRAINTS
In this section we will derive constraints from the non-observation of the charged Lepton Flavor Violating (LFV) process µ → eγ. The dominant contribution to the decay l i → l j γ occurs in our model at one-loop level and, according to the diagram in Figure 2, is mediated by a virtual electrically charged scalar ϕ + , originating from the SU (2) L inert doublet ϕ, and by the right-handed Majorana neutrinos ν sR (s = 1, 2). There is also a contribution arising from the charged exotic leptons E 2 and the electrically neutral scalars S k , P k . However this contribution is sub-leading since it only appears at the two loop level, as shown in Appendix A. Therefore, we can safely neglect it.
Then we find for the branching ratio corresponding to the diagram in Fig. 2 the following expression [167][168][169][170] Br Here and m ϕ ± are the masses of the charged scalar components of the SU (2) L inert doublet ϕ, whereas m ν sR (s = 1, 2) correspond to the masses of the right-handed Majorana neutrinos ν sR . To simplify our analµ e γ ϕ ± ϕ ∓ ν kR ysis we choose a benchmark scenario where the right-handed Majorana neutrinos ν sR are all degenerate with respect to a common mass m N . In our numerical analysis we vary these masses in the following ranges 1 TeV m ϕ ± 30 TeV and 10 MeV m N 100 MeV. We also vary the dimensionless couplings in the window 0.1 x (ν) is 1 (i = 1, 2, 3 and s = 1, 2). Let us note that we scanned only over the MeV scale masses for the right-handed Majorana neutrinos ν sR , since these masses are generated at the two-loop level, as seen from the two-loop sub-diagram of the third Feynman diagram of Fig. 1. This is the same loop level at which masses of light and strange quarks, lying in the MeV region, are generated. The results of our analysis are displayed in Figures 3 and 4. In Figure 3 we plot the allowed parameter space in the m ϕ ± − x (ν) js plane consistent with the existing µ → eγ experimental constraints. This plot is obtained by randomly generating the parameters m N , m ϕ ± , x (ν) is and x (ν) js in a range of values where the µ → eγ branching ratio is below its upper experimental limit of 4.2 × 10 −13 [170]. As can be seen from Figure 3, this condition is satisfied for the charged scalar masses m ϕ ± larger than about 3.5 TeV. We also find that in the same region of parameter space, our model predicts branching ratios for the τ → µγ and τ → eγ decays up to 10 −10 , which is below their corresponding upper experimental bounds of 4.4 × 10 −9 and 3.3 × 10 −9 , respectively. Consequently, the model is compatible with the current charged lepton-flavor-violating decay constraints. The branching ratio for the µ → eγ decay as a function of the charged scalar mass m ϕ ± is shown in Fig. 4 for different values of the x (ν) js couplings. This Figure shows that the branching ratio for the µ → eγ decay decreases as the charged scalar masses m ϕ ± acquire larger values. The horizontal line corresponds to the experimental upper bound of 4.2 × 10 −13 [170] for the branching ratio of the µ → eγ decay. Here we set m N = 50 MeV. We have checked that the branching ratio for the µ → eγ decay has a very low sensitivity to the mass m N of the right-handed Majorana neutrinos ν sR (s = 1, 2). Given that future experiments, such as Mu2e and COMET [171], are expected to measure or at least constrain leptonflavor conversion in nuclei with much better precision than the radiative lepton LFV decays, we proceed to derive constraints imposed on the model parameter space by µ − e conversion in nuclei. The µ − − e − conversion ratio is defined [170] as: Br(μ→eγ) max Using an Effective Lagrangian approach for describing LFV processes, as done in [172], and considering the low momentum limit, where the off-shell contributions from photon exchange are negligible with respect to the contributions arising from real photon emission, the dipole operators shown in Ref. [172] dominate the conversion rate, thus, yielding the following relations [170,172]: Notice that the aforementioned relations are valid for the case of photon dominance in the µ − − e − conversion, which applies to our model due to the absence of tree-level flavor changing neutral scalar interactions. Therefore, experimental upper bounds on the conversion rates (56) will translate in our model to upper limits on Br(µ → eγ).
The sensitivity of the CERN Neutrino Factory, which will use a Titanium target [173], is expected at the level of ∼ 10 −18 . The expected sensitivities of the next generation experiments such as Mu2e and COMET [171], with an Aluminum target, are expected to be about ∼ 10 −17 . Thus, according to Eqs. (57), the future limits will result in about three order of magnitude improvement in Br(µ → eγ).
In Figure 5 we show the CR(µTi → eTi) (top plot) and CR(µAl → eAl) (bottom plot), as function of the charged scalar mass m ϕ ± for different values of the dimensionless couplings x Here the value of a exp µ is a combined result of the BNL E821 experiment [182] and the recently announced FNAL Muon g-2 measurement [154] , showing the 4.2σ tension between the SM and experiment. The last positive value for ∆a e corresponds to the recently published new measurement of the fine-structure constant with an accuracy of 81 parts per trillion [181]. In this section we analyze predictions of our model for these observables. The leading contributions to ∆a e,µ arising in the model are shown in Figs. 6, 7.
For simplicity we set θ S = θ P = θ and y  (45)). Furthermore, we work on a simplified benchmark scenario with a diagonal SM charged lepton mass matrix, where the charged exotic leptonsẼ, E ; E 1 and E 2 , only contribute to the electron, muon and tau masses, respectively. Then, the contribution to the muon anomalous magnetic moment takes the form where the loop integral F (x) is defined in Eq. (55) and was previously computed in Ref. [167], whereas I S(P ) (m E , m) has the form [170,[183][184][185][186] In our numerical analysis we consider a benchmark scenario with θ = π 4 , m ν 1R = m ν Cr(μAl→eAl) max js couplings (j = 1, 2, 3, n = 1, 2). The black horizontal line in each plot corresponds to the expected sensitivities of the next generation of experiments that will use titanium [173] and aluminum [171] as targets, respectively.
Here we have set mN = 50 MeV.
in the ranges 6 TeV M E1 8 TeV. Note that these masses for the right-handed Majorana neutrinos ν sR (s = 1, 2) and for the electrically charged scalar ϕ + are consistent with the constraints arising from the charged lepton flavor processes µ → eγ, τ → µγ and τ → eγ, as shown in the previous section. Considering that the muon anomalous magnetic moment is constrained to be in the range shown in (58), we plot in Fig. 8 the muon anomalous magnetic moment as a function of the charged exotic lepton mass M E1 . Figure 8 shows that the muon anomalous magnetic moment decreases when the charged exotic lepton mass is increased. The anomalous magnetic moment of the electron ∆a e can be computed in an analogous way as ∆a µ . The difference is that the neutral (pseudo-)scalars and exotic charged leptons contribution to the ∆a e appears at two-loop level, as shown in Appendix A, and is therefore sub-leading. Thus, ∆a e is dominated by the effective vertex diagram in Fig 7, involving the electrically charged scalar ϕ + , which couples to the right-handed Majorana neutrinos ν sR (s = 1, 2). From this diagram we find in an approximate form [167] ∆a e ≈ − y 2 1 m 2 e 16π 2 m 2 Consequently, our model predicts negative values for this observable in accordance with [180]. However, in order to reproduce either [180] or [181], the experimental values shown in (59), we need that the mass of the electrically charged scalar ϕ ± lies in the interval 100 GeV m ϕ ± 150 GeV. These values are incompatible with the µ → eγ constraints analyzed in Section VI. The latter require m ϕ ± ≥ 3.5 TeV, which will yield in this case a bit too small value for the electron anomalous magnetic moment, which nonetheless are consistent with the above mentioned 2σ experimentally allowed range.

VIII. DARK MATTER RELIC DENSITY
In this section we provide a discussion of our model in view of Dark Matter (DM). We do not intend to provide a sophisticated analysis of the DM constraints, which is beyond the scope of the present paper. Note that due to the preserved Z (2) 2 discrete symmetry and to the residual Z 2 symmetry (arising from the spontaneous breaking of the Z 4 subgroup), our model has several stable scalar DM candidates. As follows from the scalar assignments to the Z 2 ×Z 4 symmetry, given by Eq. (12), we can assign this role to any of the following scalar particles: ϕ 0 R , ρ R , ξ R , S 1 , S 2 , ϕ 0 I , ρ I , ξ I , P 1 or P 2 . Furthermore, our model has a fermionic dark matter candidate, which can be the lightest among the two right-handed Majorana neutrinos ν sR (s = 1, 2), since in our model they are the only right-handed Majorana 6000 6500 7000 7500 8000 2. × 10 -9 2.5 × 10 -9 3. × 10 -9 3.5 × 10 -9

M E 1 [GeV]
Δa μ Figure 8: Muon anomalous magnetic moment as a function of the charged exotic lepton mass ME 2 .
neutrinos whose masses appear at two-loop level. Based on Eqs. (34) and (43) of Section III, we take ρ I as our scalar Dark Matter candidate. To guarantee the stability of ρ I , we assume that this field is lighter than the charged exotic fermions, and in this way its decay modes into exotic and SM charged fermions are kinematically forbidden. The relic density of the Dark Matter in the present Universe is estimated as follows (c.f. Ref. [166,187]) where σv is the thermally averaged annihilation cross section, A is the total annihilation rate per unit volume at temperature T and n eq is the equilibrium value of the particle density, which are given in [187] with K 1 and K 2 being the modified Bessel functions of the second kind of order 1 and 2, respectively [187]. For the relic density calculation, we take T = m ρ I /20 as in Ref. [187], which corresponds to a typical freeze-out temperature.
The scalar DM candidate ρ I annihilates mainly into W W , ZZ, tt, bb and hh, via a Higgs portal scalar interaction φ † φ ρ I ρ I , where φ is the SM Higgs doublet. The corresponding annihilation cross sections are given by: [188]: where √ s is the centre-of-mass energy, N c = 3 is the color factor, m h = 125.7 GeV and Γ h = 4.1 MeV are the SM Higgs boson h mass and its total decay width, respectively; α 3 is the quartic scalar coupling corresponding to the interaction α 3 φ † φ ρ † ρ .  Figure 9 shows that the relic density is an increasing function of the mass m ϕ and a decreasing function of the quartic scalar coupling α 3 . Consequently, an increase in the mass m ρ I of the scalar field ρ I will require a larger quartic scalar coupling α 3 , in order to account for the measured value of the Dark Matter relic density, as indicated in Fig. 10.
It is worth mentioning that the Dark Matter relic density constraint yields a linear correlation between the quartic scalar coupling α 3 and the mass m ρ I of the scalar Dark Matter candidate ρ I , as shown in Fig. 10. We have numerically checked that in order to reproduce the observed value, Ωh 2 = 0.1198 ± 0.0026 [189], of the relic density, the mass m ρ I of the scalar field ρ I has to be in the range 400 GeV m ρ I 800 GeV, for a quartic scalar coupling α 3 in the range 1 α 3 1.5. In what concerns prospects for the direct DM detection, the scalar DM candidate would scatter off a nuclear target in a detector via Higgs boson exchange in the t-channel, giving rise to a constraint on the coupling of the φ † φ ρ I ρ I interaction.

IX. CONCLUSIONS
We have constructed an extension of the 3HDM based on the Z 2 × Z 4 symmetry, where the SM particle content is enlarged by two inert SU 2L scalar doublets, three inert and two active electrically neutral gauge singlet scalars, charged vector like fermions and Majorana neutrinos. These fields are introduced in order to generate the SM fermion mass hierarchy from a sequential loop suppression mechanism: tree-level top quark mass; 1-loop bottom, charm, tau and muon masses; 2-loop masses for the light up, down and strange quarks as well as for the electron; and 3-loop masses for the light active neutrinos. In our model, the Z (2) 2 symmetry is preserved, whereas the Z (1) 2 symmetry is completely broken and the Z 4 symmetry is broken down to a conserved Z 2 symmetry, thus allowing the stability of the Dark Matter as well as a successful implementation of the aforementioned sequential loop suppression mechanism, without the inclusion of soft symmetry breaking terms. For studying the electroweak symmetry breaking in our model we applied the bilinear formalism of the 3HDM. We demonstrated that our model successfully accommodates the current fermion mass spectrum and fermionic mixing parameters, the electron and muon anomalous magnetic moments, as well as the constraints arising from charged lepton flavor violating processes. We have also shown that in our model the branching ratios of the decays µ → eγ, τ → µγ and τ → eγ can reach values of the order of 10 −13 , which is within the reach of the future experimental sensitivity, thus making our model testable by the forthcoming experiments. Finally, we have examined the scalar DM particle candidate of the model and have shown that the prediction is compatible with the observed DM relic density abundance for scalar masses in the range 400 GeV m ρ I 800 GeV. Ωh 2 Figure 9: Relic density Ωh 2 , as a function of the mass mρ I of the ρ I scalar field, for several values of the quartic scalar coupling α3. The curves from top to bottom correspond to α3 = 1, 1.2, 1.5, respectively. The horizontal line shows the observed value Ωh 2 = 0.1198 [189] for the relic density.
Let us show that the contribution to l i → l j γ decay of the charged exotic leptons E 2 and the electrically neutral scalars S k , P k vanishe at one loop. Their one-loop contribution is given by the first two diagrams in Fig. 6, with one µ replaced by e.
In the mass eigenstate basis l i the corresponding contribution to the branching fraction is given by: (m µ δ a2 δ b2 + m τ δ a3 δ b3 ) (1 − δ ab ) F 1loop = 0, what was to be shown. Here we have taken into account that the SM charged lepton mass matrix has the form: