Fermion mass hierarchy in an extended left-right symmetric model

We present a Left-Right symmetric model that provides an explanation for the mass hierarchy of the charged fermions within the framework of the Standard Model. This explanation is achieved through the utilization of both tree-level and radiative seesaw mechanisms. In this model, the tiny masses of the light active neutrinos are generated via a three-loop radiative inverse seesaw mechanism, with Dirac and Majorana submatrices arising at one-loop level. To the best of our knowledge, this is the first example of the inverse seesaw mechanism being implemented with both submatrices generated at one-loop level. The model contains a global U (1) X symmetry which, after its spontaneous breaking, allows for the stabilization of the Dark Matter (DM) candidates. We show that the electroweak precision observables, the electron and muon anomalous magnetic moments as well as the Charged Lepton Flavor Violating decays, µ → eγ , are consistent with the current experimental limits. In addition, we analyze the implications of the model for the 95 GeV diphoton excess recently reported by the CMS collaboration and demonstrate that such anomaly could be easily accommodated. Finally, we discuss qualitative aspects of DM in the considered model


I. INTRODUCTION
Despite the remarkable accomplishments of the Standard Model (SM) in describing the fundamental interactions, it fails to address several profound inquiries that remain unanswered.In the context of the present work, we plan to address specific questions such as the SM flavor structure (i.e. the observed pattern of SM fermion masses and mixing), the origin of Dark Matter (DM) and the parity violation in weak interactions, whose answers lie certainly beyond the SM.Even though the energy scale of New Physics is still a mystery, current experimental searches keep restricting the possibilities for new phenomena at experimentally accessible energy scales and pushing potential New Physics signatures towards high energies.For this reason, it is interesting to look into well motivated SM extensions, whose signatures dwells at energies much higher than the electroweak scale, such as for instance the Left-Right (LR) symmetric models.These are particularly interesting since they provide a robust explanation of parity violation in weak interactions, as a low-energy effect of the spontaneously broken at high scales LR-symmetry.
In this study, we present a LR-symmetric extension of the SM, which offers a comprehensive explanation for the fermion mass hierarchy.To start, the masses of the third generation of SM charged fermions as well as the charm quark mass are generated by means of a seesaw-like mechanism.This implies the existence of a mixing between the SM charged fermions and the heavy charged vector-like (with respect to the gauge symmetry) fermions 1 .In addition, the masses of the light (up, down and strange) quarks, as well as those of electron and muon, arise from a radiative seesaw mechanism at one-loop level.The complete picture of this setup considers the tiny masses of the light active neutrinos generated via the inverse seesaw mechanism at the three-loop level, with the Dirac as well as the Majorana mass submatrices induced at the one-loop level.So, this is an alternative to other radiative models where the Majorana mass submatrices arise at the loop level [2][3][4][5][6][7][8][9][10][11][12][13] while the Dirac mass submatrix is generated at tree level.To the best of our knowledge, the scenario proposed in this work is the first example of the inverse seesaw mechanism with both Dirac and Majorana submatrices generated at one-loop level.Other variants of the LR symmetric model have been recently considered in Refs.[7,[13][14][15][16][17][18].Let us note that the model studied here introduces the minimal ingredients for a LR-symmetric theory to account for the fermion mass hierarchy, which means that the proposed set-up turns out to be minimalistic in comparison with other existing proposals, such as the one presented in [13].
For example, the scalar sector of our model has two SU (2) L scalar doublets (8 degrees of freedom), two SU (2) R scalar doublets (8 degrees of freedom) and three electrically neutral gauge-singlet complex scalar fields (6 degrees of freedom), thus, amounting to a total of 16 physical scalar degrees of freedom (after subtracting the number of Goldstone bosons).The charged exotic fermion sector of the model has 7 charged exotic vector-like fermions.Thus, in total the model has 23 physical degrees of freedom (compared to 32 degrees of freedom of the model of Ref. [13]).
It is worth noticing that the charged exotic vector-like leptons, which mediate the seesaw mechanisms yielding the SM charged lepton masses, contribute to the muon and electron anomalous magnetic moments.Therefore, a scenario of this type offers a natural link between the fermion mass generation mechanism and the g − 2 anomalies.Another feature of the model is the rich phenomenology at colliders related to the scalar and gauge boson sectors.
The paper is organized as follows.In sec.II we introduce the model theoretical setup, describing its field content and symmetries.The scalar potential and scalar particles of the model are studied in sec.III.In sec.IV we explore the quark, charged lepton and neutrino mass spectra and mixing, as well as study the role of heavy exotic fermions.Phenomenological implications of the model are examined in secs.V-VIII.Finally, in sec.IX we briefly discuss possible DM particle candidates provided by the model.

II. MODEL SETUP
We start this section by explaining the reasoning that justifies the inclusion of extra scalars, fermions and symmetries needed for implementing an interplay of tree-level and radiative seesaw mechanisms, enabling us to explain the SM charged fermion mass hierarchies, as well as the three-loop level inverse seesaw mechanism, with one loop induced Dirac and Majorana submatrices, in order to generate tiny masses of the light active neutrinos.In our theoretical construction, detailed below, the basis ν L , ν C R , N C R , has the following structure: where ν iL (i = 1, 2, 3) correspond to the active neutrinos, whereas ν iR and N iR (i = 1, 2) are the sterile neutrinos.Furthermore, the entries of the full neutrino mass matrix of Eq. ( 1) should obey the hierarchy µ ij << m ij << M ij (i = 1, 2, 3, n, p = 1, 2), where the submatrices µ and m νD are radiatively generated at one loop level, whereas the submatrix M arises at tree level.Here we consider a theory based on the SU (3 symmetry, which is supplemented by the inclusion of the global U (1) X symmetry.The U (1) X global symmetry is assumed to be spontaneously broken down to a preserved matter parity symmetry, which is crucial for ensuring the radiative nature of the inverse seesaw mechanism that produces the tiny active neutrino masses, as well as the seesawlike mechanism that generates the masses of the light (up, down and strange) quarks, as well as those of electron and muon.To generate the Majorana submatrix µ at one loop level, we require the following operators: where L Rj (j = 1, 2, 3) are χ R are SU (2) R lepton and scalar doublets, respectively.Besides that, N Ri and Ω Rn (n = 1, 2) are right handed Majorana neutrinos, whereas φ and ρ are electrically neutral gauge singlet scalars.
Notice that the second and third operators in Eq. ( 2) generate a one-loop level Majorana neutrino mass submatrix µ, associated with the breaking of lepton number, while the first operator in this equation generates the mass scale of the heavy sterile neutrinos which mediate the inverse seesaw mechanism.
On the other hand, a radiative generation of the Dirac neutrino mass submatrix m νD at one-loop level requires the inclusion of the following operators: which are also crucial to generate one loop level muon and electron masses.Here L Li (ϕ L ) and L Ri (ϕ R ) are SU (2) L and SU (2) R lepton (scalar) doublets, E n (n = 1, 2) are heavy vector-like charged leptons and σ is a electrically neutral scalar singlet.
It is worth mentioning that the operators presented in Eqs. ( 2) and (3) allow for a successful implementation of the three-loop level inverse seesaw mechanism that produces the tiny neutrino masses.This is due to the fact that the light active neutrino mass matrix arising from the inverse seesaw mechanism will have a quadratic and linear dependence with the one loop induced Dirac m νD and Majorana µ submatrices, respectively.Given that our proposed model does not consider a bidoublet scalar in the particle spectrum, the invariance of the Yukawa interactions with respect to the symmetries of the model forbids charged fermions Yukawa terms involving a SM fermion-anti fermion pair.Consequently, heavy charged vector-like (with respect to the gauge symmetry) fermions have to be included in the particle spectrum in order to trigger a seesaw-like mechanism that generates the masses of the SM charged fermions.
The symmetries of the model imply that the masses of the third generation of SM charged fermions as well as the charm quark mass will be generated through the mixing with heavy vector-like (with respect to the gauge symmetries) fermions, arising from the following operators: where L and SU (2) R lepton (quark) doublets, respectively, and χ L and χ R are SU (2) L and SU (2) R scalar doublets, respectively.Furthermore, T n (n = 1, 2), B and E are heavy vector-like up, Here, i = 1, 2, 3 and n = 1, 2.
down type quarks and charged leptons, respectively.The heavy quarks T 1 , T 2 , B as well as the heavy lepton E get their masses from mass terms and Yukawa interactions involving the gauge-singlet scalar field σ.We assume that the singlet scalar field σ is charged under a global U (1) X symmetry introduced in the model.
Furthermore, in order to radiatively generate the masses of the light up, down and strange quarks, as well as the electron and muon masses, the following operators are required: where ϕ L and ϕ R are SU (2) L and SU (2) R scalar doublets, respectively.These scalar doublets will be assumed to have odd U (1) X charge.The U (1) X global symmetry is assumed to be spontaneously broken down to a preserved matter parity symmetry, which will imply that the scalar doublets ϕ L and ϕ R will not acquire vacuum expectation values (VEVs).Consequently, the radiative generation of masses for the light (up, down and strange) quarks, as well as those of electron and muon can be achieved at the one-loop level.The implementation of such a radiative seesaw mechanism also requires the inclusion of the heavy vector-like up-type quarks T ′ , down-type quarks B ′ n and charged leptons E ′ n (n = 1, 2) in the fermion spectrum of the model under consideration.The model under consideration is based on the gauge symmetry SU (3) C ×SU (2) L ×SU (2) R ×U (1) B−L supplemented by the global U (1) X symmetry, where the full symmetry G exhibits the following breaking scheme: The spontaneous breaking of the global U (1) X symmetry is assumed to occur together with the LR symmetry breaking at the same energy scale.We further assume that this scale is about v R ∼ O (10) TeV.With the scalar field assignments given in Table II, the global U (1) X symmetry is spontaneously broken down to a preserved Z 2 discrete symmetry.The latter allows a successful implementation of a radiative seesaw mechanism at one-loop level, that generates masses of the light (up, down and strange) quarks, as well as those of electron and muon.Furthermore, thanks to the preserved Z 2 symmetry, the masses of the light active neutrinos arise from a three-loop inverse seesaw mechanism, while the masses of the third family of SM charged fermions as well the charm quark mass are generated by means of a seesaw mechanism responsible for their tree-level mixing with heavy charged vector-like fermions.The fermion assignments under the SU (3 X group are displayed in Table I. Notice that the fermion sector of the original LR symmetric model has been enlarged by introducing only gauge group singlet Majorana neutrinos N Ri (i = 1, 2, 3) and Ω Rn (n = 1, 2) as well as vector-like exotic fermions: three exotic up-type quarks T n (n = 1, 2), T ′ , one exotic down-type quark B, three charged leptons E, E ′ n .Since gauge singlet and vector-like fermions do not contribute to the gauge group chiral anomaly, our model, following the original LR-symmetric model, is also anomaly free.
Moreover the exotic fermions of our model are assigned to singlet representations of the SU (2) L × SU (2) R group.The above mentioned exotic fermion content is the minimal one required to generate tree-level masses via a seesaw mechanism for the third generation of SM charged fermions and for the charm quark, one-loop level masses for the light (up, down and strange) quarks and for the electron and muon, as well as Dirac neutrino submatrix at one-loop level and the one-loop Majorana submatrix µ of the inverse seesaw mechanism.The exotic lepton spectrum of the model allows to generate the light active neutrino masses via an inverse seesaw mechanism at three-loop level.
The scalar boson assignments under the SU (3 The fermion and scalar boson charge assignments under the SU (3 X symmetry are shown in Tables I and II, respectively. Notice that the scalars χ L and χ R have been introduced to implement the seesaw mechanism that produce tree-level masses for the third generation of SM charged fermions.Furthermore, the SU (2) R scalar doublet χ R is required to trigger the spontaneous breaking of the SU (2) R × U (1) B−L × U (1) X symmetry, whereas the SU (2) L scalar doublet χ L spontaneously breaks the SM electroweak gauge symmetry.Besides, the scalar fields σ and ρ spontaneously break the U (1) X global symmetry down to a preserved Z 2 matter parity symmetry defined as The gauge singlets σ and ρ are included to generate tree-level masses for the charged exotic vector-like fermions and heavy neutral leptons Ω Rn (n = 1, 2), respectively.In addition, a successful implementation of the loop-level radiative seesaw mechanism that produces the masses for the first and second family of SM charged fermions, requires the inclusion of the SU (2) R and SU (2) L inert scalar doublets ϕ R and ϕ L , respectively.Moreover, in order to generate the µ submatrix of the inverse seesaw at one-loop level, the inert gauge singlet scalar field φ is also required in the scalar spectrum.The residual M P matter parity symmetry implies that the scalar fields having odd U (1) X charges do not acquire VEVs, enabling an appropriate implementation of the radiative seesaw mechanisms.
The VEVs of the scalars χ L and χ R read: The scalar fields having odd U (1) X charges do not acquire VEVs, due to the preserved matter parity symmetry defined as (−1) X+2s .
With the above particle content, the following relevant Yukawa terms arise:   Finally, to close this section we provide a discussion about collider signatures of exotic fermions of our model.From the Yukawa interactions it follows that the charged exotic fermions have mixing mass terms with the SM charged fermions, which allows the former to decay into any of the scalars of the model and SM charged fermions.These heavy charged exotic fermions can be produced in association with the charged fermions and can be pair produced at the LHC via gluon fusion (for the exotic vector-like quarks only) and also through the Drell-Yan mechanism.Consequently, observing an excess of events in the multijet and multilepton final states can be a signal in support of this model.Regarding the sterile neutrino sector, it is worth mentioning that the sterile neutrinos can be produced at the LHC in association with a SM charged lepton, via quark-antiquark annihilation mediated by a W ′ gauge boson.8 From the experimental point of view, vector-like quarks are among the most searched for New Physics states at the LHC.Being colored particles, their production rates in gluon-gluon fusion is expected to be rather high, pushing the lower bounds on the the mass scale beyond a TeV scale, typically landing between 1.4 and 2.0 TeV depending on the underlined assumptions.The majority of existing searches assume dominant couplings to the third-generation of SM quarks (see e.g.Refs.[19][20][21][22][23][24]).There are a few searches focusing on couplings to light SM quarks, such as Ref. [25] for a pair-production channel, with subsequent decays into light jets via neutral and charged currents, and in Ref. [26] for a single-production channel, with the same decay modes.In the case of charged-current decays, for instance, a down-type vector-like quark can decay into a W -boson and an up-type SM quark, induced by a possible mixing with down-type SM quarks.
In a more recent analysis of Ref. [27], Machine learning techniques have been effectively utilized to obtain robust exclusion bounds and compute the statistical significance for potential discoveries at future upgrades of the Large Hadron Collider (LHC).These techniques primarily rely on analyzing the pair-production of vector-like quarks and their subsequent charge-current decays.In a particular promising channel, when one of the W ± 's decays leptonically into a charged lepton l = e, µ and the corresponding neutrino, while the other one decays hadronically into light jets, it has been shown that the down-type vector-like quarks can be excluded both at the run-III and the HL phase of the LHC for masses of up to 800 GeV.
Typical production channels of exotic vector-like leptons concern the vector-boson fusion topologies, as well as via γ/Z decays in a Drell-Yan type reaction -for pair production -and via charged-channel W ± decays -for single production.The latter process can be accompanied either by missing energy (in case of associated production of SM neutrino in the final state) or a charged lepton plus W (in case of association production of the neutral doublet partner of a doublet-type exotic lepton).The potential for possible discoveries of exotic vector-like leptons at both the LHC and future electron and muon colliders has been recently explored in Refs.[28,29] utilising sophisticated deep learning methods.It has been demonstrated that weak-doublet vector-like leptons can be probed at the LHC up to a TeV mass scale, while the weak-singlet vector-like leptons can be barely probed beyond several hundreds of GeV.In the considered model, only weak-singlet exotic leptons are present, for which the existing exclusion limits are weaker than for the doublet ones.This motivates future searches at leptonic colliders, which can exclude larger parameter-space regions, probing larger mass scales beyond those accessible at the high-luminosity LHC upgrade.

III. THE SCALAR POTENTIAL
The scalar potential of the model under consideration takes the form: In order to simplify our analysis of the scalar sector, instead of considering all the terms in the scalar potential, it is enough to consider up to the λ 15 term, since the rest of the terms would be important only in the high energy regime.Furthermore, we neglect the terms of the scalar potential involving a mixing between the singlet scalar fields and the SU (2) L,R doublet scalars.For the purpose of focusing on the low-energy phenomenology, we consider the following low-energy scalar potential: The minimization of this scalar potential gives rise to the effective mass terms: The squared mass matrix for the CP-even neutral fields, in the basis of Re The mixing between the χ L and χ R fields can be approximated by v L v R and this value is suppressed in the case where the VEV of v L is the SM Higgs VEV of 246 GeV and that of v R is equal to 10 TeV, This shows that the CP-even neutral component of the SU (2) L scalar doublet corresponds to the 126 GeV SM like Higgs boson, which in our model has couplings very close to the SM expectation, which is consistent with the experimental data.On the other hand, the mixing between Re ϕ 0 L and Re ϕ 0 R depends on the free parameters λ 7 , µ 2 3 and µ 2 4 , so it can be sizeable.Furthermore, this mixing gives rise to the heavy CP-even scalars H 1 and H 2 defined as follows: The squared mass matrix for the CP-odd neutral fields in the basis of Im Then, in the CP-odd neutral scalar sector there are four massless neutral fields, from which two of them, namely, Im χ 0 L and χ 0 R , correspond to the Goldstone bosons associated with the longitudinal components of the Z and Z ′ gauge bosons.The remaining massless CP-odd neutral scalars, i.e., Im σ and Im ρ, correspond to Majorons, massless scalars arising from the spontaneous breaking of the U (1) X global symmetry.These Majorons are harmless since they are gauge singlets, annd can acquire non-vanishing masses by including the soft-breaking mass terms µ (2) sb ρ 2 + h.c in the scalar potential.As in the CP-even scalar sector, the mixing between Im ϕ 0 L and Im ϕ R depends on the free parameters λ 7 , µ 2  3 and µ 2 4 .Furthermore, there are three massive scalars in the CP-odd scalar sector, i.e, Im φ as well as A 1 and A 2 , which are defined as follows: .
The squared mass matrix for the electrically charged scalar fields in the basis of χ ± L , χ ± R , ϕ ± L , ϕ ± R takes the form: Then the electrically charged scalar mass spectrum is composed of the massless scalar eigenstates χ ± L and χ ± R , as well as the massive scalars H ± 1 and H ± 2 .The electrically charged massless scalars χ ± L and χ ± R are the Goldstone bosons associated with the longitudinal components of the W ± and W ′± gauge bosons, respectively.Furthermore, the massive scalars H ± 1 and H ± 2 are defined as follows:

IV. FERMION MASS MATRICES
From the Yukawa interactions, we find that the mass matrices for charged fermions in the basis ) are respectively given: x (E) 2 x Furthermore, due to the preserved matter parity symmetry arising from the spontaneous breaking of the U (1) X global symmetry, the exotic up-type quarks T ′ and B ′ n (n = 1, 2) do not mix with the remaining up-type quark fields.For the same reason, the charged exotic leptons E ′ n (n = 1, 2), do not mix with the remaining charged leptons.As seen from Eqs. ( 21), ( 22) and ( 23), the exotic heavy vector-like fermions T n (n = 1, 2), B and E mix with the SM fermions.The masses of these vector-like fermions are much larger than the electroweak symmetry breaking scale, since the gauge singlet scalars η and ρ are assumed to acquire VEVs around the scale of breaking of the LR symmetry, taken to be much larger than the Fermi scale.Consequently, the charged exotic vector-like fermions T n (n = 1, 2), B and E induce a seesaw mechanism that gives rise to tree level masses to the third family of SM charged fermions, as well as to the charm quark.The remaining charged exotic vector-like fermions T ′ , B ′ n and E ′ n (n = 1, 2) mediate a one-loop level radiative seesaw mechanism that generates the masses of the light up, down and strange quarks as well as of the electron and muon masses.Thus, the SM charged fermion mass matrices take the form: Where ∆ U , ∆ D and ∆ E are the one-loop contributions to the SM charged fermion mass matrices which are given by: where f (m 1 , m 2 ) is given by: For θ ≪ θ H , θ A , the SM charged fermion mass matrices can be parametrized as follows: where The SM charged fermion mass matrices can be rewritten, in the scenario θ ≪ θ H , θ A , as follows: In what follows we show that our model can successfully reproduce the following experimental values of the quark masses [30], the CKM parameters [31] and the charged lepton masses [31]: m e (M eV ) = 0.4883266 ± 0.0000017, m µ (M eV ) = 102.87267± 0.00021, m τ (M eV ) = 1747.43± 0.12, where J is the Jarlskog parameter.By solving the eigenvalue problem for the SM charged fermion mass matrices, we find a solution for the parameters that reproduces the values in Eq. (38).It is given by The above values successfully reproduce the SM charged fermion masses and CKM parameters.The Yukawa couplings of the charged fermion sector feature a moderate hierachy, since their order of magnitude is located in the range 10 −2 , 1 .On the other hand, the electrically charged fermionic seesaw mediators have O (1) TeV and O (10) TeV mases.Despite the aforementioned moderate hierarchy in the Yukawa couplings, this situation is much better than in the SM, where a hierarchy of about 5 orders of magnitude is present in the charged fermion sector.Regarding the charged lepton sector, in our numerical analysis we consider charged exotic leptons with masses of a few TeVs, which allows us to successfully reproduce the muon and electron (g − 2) anomalies.Furthermore, it is worth mentioning that the effective Yukawa couplings are proportional to a product of two other dimensionless couplings, so a moderate hierarchy in those couplings can yield a quadratically larger hierarchy in the effective couplings, thus allowing to explain the SM charged fermion mass and quark mixing pattern.Small quark mixing angles are attributed to the hierarchy in the rows of the SM quark mass matrices, whose entries are proportional to the product of two dimensionless Yukawa couplings, as shown in Eqs. ( 35), ( 36) and (37).Having all charged fermion Yukawa couplings of the same order of magnitude would require the implementation of the sequential loop suppression mechanism [32] where the first family of SM charged fermions will get masses at two loop level.This will imply a non minimal extension of the scalar and fermion sectors of the model and such implementation is beyond the scope of this work and will be done elsewhere.
On the other hand, it is worth mentioning that the Yukawa interactions of the 126 GeV SM like Higgs boson (which is identified with the CP even neutral part of the SU (2) L scalar doublet χ L ) with SM fermion-antifermion pairs are dynamically generated below the left-right symmetry breaking scale, after integrating out the charged vector like seesaw mediators.This can be seen from the Feynman diagrams of Figure 1, where the external scalar lines are associated with the χ L and χ R scalar fields, whereas the external fermionic lines are mostly composed of the SM charged fermions (after rotating to the physical basis).Given that there is only one non vevless SU (2) L scalar doublet χ L in the scalar sector of the model and there is no bidoublet scalar, our model is free from tree level flavor changing neutral currents and the couplings of the 126 GeV SM like Higgs boson with SM particles are very close to the SM expectation, thus implying the alignment limit is naturally fulfilled in our model.
Concerning the neutrino sector, we find that the neutrino Yukawa interactions give rise to the following neutrino mass terms: where the neutrino mass matrix reads: and the submatrices are given by: The µ block is generated at one-loop level due to the exchange of Ω rR (r = 1, 2) and φ in the internal lines, as shown in Figure 2.
The light active masses arise from an inverse seesaw mechanism, and the physical neutrino mass matrices are: where M ν corresponds to the mass matrix for light active neutrinos ν a (a = 1, 2, 3), whereas M (1) ν and M (2) ν are the mass matrices for sterile neutrinos (N − a , N + a ), which are superpositions of mostly ν aR and N aR , as In the limit µ → 0, which corresponds to unbroken lepton number, the light active neutrinos become massless.The smallness of the µ parameter is responsible for a small mass splitting between the three pairs of sterile neutrinos, thus implying that the sterile neutrinos form pseudo-Dirac pairs.Notice that the physical neutrino spectrum is composed of three light active neutrinos and six exotic neutrinos.These exotic neutrinos are pseudo-Dirac, with masses ∼ ± 1  2 M + M T and a small splitting µ.

V. OBLIQUE T , S AND U PARAMETERS
The extra scalars affect the oblique corrections of the SM, whose values are measured in high precision experiments.Consequently, they act as a further constraint on the validity of any New Physics model.The oblique corrections are parametrized in terms of the three well-known quantities T , S and U .In this section, we calculate one-loop contributions to the oblique parameters T , S and U , defined as [33-35] where Π 11 (0), Π 33 (0), and Π 30 q 2 are the vacuum polarization amplitudes with {W 1 µ , W 1 µ }, {W 3 µ , W 3 µ } and {W 3 µ , B µ } external gauge bosons, respectively, and q is their momentum.We note that in the definitions of the T , S and U parameters, the New Physics is assumed to be heavy compared to M W and M Z .
The contributions arising from New Physics to the T , S and U parameters are: where we introduced the functions [36]

VI. MUON AND ELECTRON ANOMALOUS MAGNETIC MOMENTS
In this section, we will analyze the implications of our model in the muon and electron anomalous magnetic moments.The contributions to the muon and electron anomalous magnetic moments take the form: Re ( Furthermore, the loop function I (e,µ) S(P ) (m E , m) has the form [39][40][41][42][43]: and the dimensionless parameters β 1k , β 2k , γ k1 , γ k2 , κ 1 , κ 2 , ϑ 1 , ϑ 2 are given by: where V lL and V lR are the rotation matrices that diagonalize M E according to the relation:  Considering that the muon and electron anomalous magnetic moments are constrained to be in the ranges [44,45], (∆a µ ) exp = (2.51 ± 0.59) × 10 −9 (∆a e ) exp = (4.8± 3.0) × 10 −13 , (66) we plot in Figure 4 the correlation between the electron and muon anomalous magnetic moments, and in Figure 5 we show the correlations of the muon and electron anomalous magnetic moments with the charged exotic lepton mass m E .In our numerical analysis we have fixed m H 0 1 = 1.4 TeV, m H 0 2 = 1.6 TeV, m H 0 3 = 14 TeV and we have varied the masses of the charged exotic leptons in the ranges 1 TeV ⩽ m E ⩽100 TeV and 1 TeV ⩽ m E ′ k ⩽10 TeV (k = 1.2), and we have used the extended Casas-Ibarra parametrization [46,47] for the SM charged lepton mass matrix given in Eq. (31), to guarantee that the obtained points of the model parameter space consistent with the muon and electron anomalous magnetic moments are also in excellent agreement with the experimental values of the SM charged lepton masses.It is worth mentioning that the parameter space considered in our analysis is consistent with the measured values of the SM charged fermion masses and fermionic mixing parameters.As indicated by Figures 4 and 5, our model can successfully accommodate the experimental values of the muon and electron anomalous magnetic moments.

VII. THE 95 GeV DIPHOTON EXCESS
In this section we discuss the implication of our model for a possible interpretation of the 95 GeV diphoton excess recently observed by the CMS collaboration.In what follows we discuss the possibility that the excess of events in the diphoton final state at the invariant mass of about 95 GeV be due to the real component σ R of the scalar singlet σ of the model under consideration, whose mass is assumed to be equal to 95 GeV.The EW scalar singlet σ R is mainly produced via a gluon fusion mechanism, involving the heavy exotic T 2 , T ′ and B ′ n (n = 1, 2) quarks running in the internal lines of the triangular loop.The diphoton decay of the scalar singlet σ R is mediated by triangular loops involving the virtual exchange of vector-like quarks, charged vector-like leptons E and E ′ and electrically charged scalars.Consequently, the cross section for the production of the diphoton scalar resonance at the LHC takes the form: where m σ R ≃ 95GeV is the resonance mass, Γ σ R its total decay width, f g (x) is the gluon distribution function, and √ s = 13 TeV is the LHC center of mass energy.
The 95 GeV diphoton excess can be interpreted as a scalar resonance with a signal strength given by [48,49]: where σ SM corresponds to the total cross section for a hypothetical SM Higgs boson at the same mass.
The corresponding decay widths of the resonance into photon and gluon pairs are respectively given by: where K gg ∼ 1.5 is a QCD loop enhancement factor that accounts for the higher order QCD corrections, ) and the loop functions F 1/2 (ζ) and F 0 (ζ) are given by: where For the sake of simplicity we consider a benchmark scenario characterized by the absence of mixings between the 95 GeV resonance σ R and the remaining scalar fields.In addition, we assume that the 95 GeV resonance σ R is lighter than the remaining scalar fields.Consequently, under the aforementioned assumptions, the 95 GeV resonance σ R does not feature tree-level decays into scalar pairs and thus it predominantly decays into photon and gluon pairs.It is worth mentioning that the 95 GeV resonance σ R can also decay into SM fermion-antifermion pairs (except for the top-antitop quark pair), however, those decays are strongly suppressed by the fourth power of the very small mixing angles between the SM charged fermions lighter than the top quark and the heavy vector-like fermions.Here, we consider the benchmark scenario specified in Eq. (39).The parameters relevant for this part of our analysis are which is consistent with the measured values of the SM charged fermion masses and fermionic mixing parameters, as well as with the constraints arising from the g − 2 muon and electron anomalies, and the experimental measurements of the oblique T , S and U parameters, for charged scalar mass m H ± in the range 1.3 TeV ⩽ m H ± ⩽ 1.5 TeV.
Figure 6 displays the diphoton signal strength for a hypothetical 95 GeV scalar resonance as a function of the charged scalar mass m H ± .The magenta and orange horizontal lines correspond to the upper and lower experimental limits within the 1σ range, respectively.For the computation of the total cross section, the MSTW next-to-leading-order (NLO) gluon distribution functions [50], evaluated at the factorisation scale µ = m σ R , have been used.As seen from Fig. 6, our model successfully accommodates the 95GeV diphoton excess.

VIII. CHARGED LEPTON FLAVOR VIOLATION
In this section we will discuss the implications of the model for charged lepton flavor violation.As mentioned in the previous section, the sterile neutrino spectrum of the model is composed of six nearly degenerate heavy neutrinos.These sterile neutrinos, as well as the light active neutrinos together with the SM W gauge and heavy W ′ gauge bosons, induce the l i → l j γ decay at one loop level, whose branching ratio is given by [51][52][53]: where (76) where the µ − − e − conversion ratio is defined [42] as follows: Consequently, for our model we expect that the resulting rates for the LFV transitions µ → 3e, µAl → eAl and µTi → eT will be of the order of 10 −15 , i.e, two orders of magnitude lower than the obtained rates for the µ → eγ decay, thus implying that in our model the corresponding values are below the current experimental bounds of about 10 −12 for these lepton flavor violating transitions.

IX. DARK MATTER CANDIDATE
As we described in section II, in this model there is a residual matter parity symmetry (8), surviving the spontaneous breaking of the global U (1) X group.This symmetry is responsible for stabilizing the dark matter particle candidate in the our model.In this case, the fields that compose the dark sector have no electric charge and are odd under the matter parity symmetry.Following these criteria, one can identify the neutral components of the VEV-less scalars ϕ L , ϕ R , and φ, and the Majorana fermions N Ri (i = 1, 2, 3) as part of the dark sector.All these fields are involved in the fermion mass generation at the loop-level, see Figs. 1 and 2. Notice that the Majorana fermions N Ri mix the with active neutrinos, which will cause their decay.For this reason, the N Ri are discarded as dark matter candidates.Therefore, the dark matter candidate in this model has a scalar nature and would be the lightest neutral component of the scalars (ϕ 0 L , ϕ 0 R , φ).Here, (ϕ 0 L , ϕ 0 R ) generate the corrections to the quark and charged lepton mass matrices, Fig. 1, and φ generates the µ term in Eq. ( 41) needed to get the neutrino masses, see Fig. 2.This model has enough freedom to consider different simplified situations in the parameter space.One of these limits is to assume that the CP-even (odd) component of the singlet scalar φ is the DM candidate.One can see from Eqs. ( 14) and ( 17) that this field does not mix with any other dark scalar.In this case, the mass hierarchy between dark scalars, m φ < m ϕ 0 R , m ϕ 0 L , is satisfied.Notice that in such a benchmark the dark matter relic abundance will be dominated by the DM annihilation into scalars, similar to Refs.[56,57].The DM annihilation channel (s-channel with Higgs exchange) into SM particles is expected to appear by effective couplings that mix SM and heavy fermions.The DM annihilation processes into heavy fermions might be possible, if allowed by kinematics.
In what follows we analyze the implications of the model for the dark matter relic density as well as in the direct detection for dark matter.In order to simplify our analysis we consider a benchmark scenario where the scalar DM candidate φ I annihilates into a pair of SM particles as well as into a pair of the heavy CP even state H arising from the SU (2) R scalar doublet χ R .In this benchmark scenario we take all dark sector scalar couplings excepting λ 24 and λ 25 to be small.Then, in the previously described simplified benmark scenario, the scalar dark matter candidate mainly annihilates into W W , ZZ, tt, bb and hh, via a Higgs portal scalar interaction λ 24 (φ † φ)(χ † L χ L ), as well into HH thanks to the quartic scalar interaction λ 25 (φ † φ)(χ † R χ R ).The corresponding annihilation cross sections are given by [58]: 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; H ≃ Re χ 0 R , λ 24 and λ 25 are the quartic scalar coupling corresponding to the interactions λ 24 (φ † φ)(χ † L χ L ) and λ 25 (φ † φ)(χ † R χ R ), respectively.It is known that the Dark Matter relic abundance of the present Universe can be determined as follows [31,59]: 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 [59] A = T 32π 4 with K 1 and K 2 being the modified Bessel functions of the second kind of order 1 and 2, respectively [59].For the relic density calculation, we take T = m φ I /20 as in Ref. [59], which corresponds to a typical freeze-out temperature.We require that the obtained values for the dark matter relic abundance to be consistent with the experimentally allowed range measured by the Planck collaboration [60]: Ω DM h 2 = 0.1200 ± 0.0012 (81) Besides that, direct detection constraints should also be taken into account.They are obtained from the comparison of the spin-independent cross section for the scattering of a dark matter off a nucleon, to the most recent upper bounds on σ SI arising from the Xenon1T experiment [61].As shown in Ref. [61], these upper bounds are of the order of 10 −44 cm 2 for dark matter masses in the range 1.5 TeV ⩽ m DM ⩽ 2 TeV, with m DM = m φ I .Here m N is the nucleon mass and f ≃ 1 3 corresponds to the form factor [62,63].Furthermore, we display in Figure 8 the correlation of the spin independent cross section with the dark matter mass.Here we have set m H = 1 TeV and we have varied the quartic scalar couplings λ 24 and λ 25 in the range 0 − 4π.As indicated in Figure 8 our model is consistent with the constraints arising from dark matter direct detection and can successfully reproduce the experimental values of the dark matter relic density.

X. CONCLUSIONS
We have proposed a renormalizable extended Left-Right symmetric theory with an additional global U (1) X symmetry, capable of explaining and accommodating the observed SM fermion mass hierarchy and the tiny values of the light active neutrino masses.As the same time, the model has been demonstrated to be consistent with the current amount of dark matter relic density observed in the Universe, the muon anomalous magnetic moment, the oblique T , S and U parameters, as well as the 95 GeV diphoton excess.In the proposed model, the global U (1) X symmetry is spontaneously broken down to a matter parity symmetry, thus allowing the existence of stable scalar dark matter candidates in the particle spectrum of our model and ensuring the radiative nature of the seesaw mechanisms that generate the masses of the light up, down and strange quarks, as well as those ones of the electron, muon and active neutrinos.
As an important feature of our model, the third generation of SM charged fermions and the charm quark obtain their masses via a tree-level seesaw mechanism triggered by their mixing with heavy charged vector-like fermions.The masses of the light up, down and strange quarks, as well as the electron and muon masses, are generated via a one-loop radiative seesaw mechanism mediated by heavy charged vector-like fermions and non-SM scalars.The tiny active neutrino masses are produced through a three-loop inverse seesaw mechanism, with the Dirac and Majorana submatrices generated at one-loop level.In the low-energy limit, the model features one naturally light 126 GeV SM-like boson, strongly decoupled from the other heavy scalars, as well as the absence of tree-level FCNC processes, rendering the model safe against the existing flavor physics bounds.

T ) 2 ≃
(a) Correlation between the oblique S and T parameters.(b) Correlation between the oblique S and U parameters.(c) Correlation between the oblique U and T parameters.

Figure 4 :
Figure 4: Correlation between the electron and muon anomalous magnetic moments.

Figure 5 :
Figure 5: Correlation of the muon and electron anomalous magnetic moments with the charged exotic lepton mass mE.

Figure 6 :
Figure 6: Diphoton signal strength for a hypothetical 95 GeV scalar resonance as a function of the charged scalar mass m H ± .The magenta and orange horizontal lines correspond to the upper and lower experimental limits within the 1σ range, respectively.

Figure 7 :
Figure 7: Correlation between the branching ratio for the µ → eγ decay and the mass mN of the sterile neutrinos.

Figure 8 :
Figure 8: Correlation of the spin independent cross section σSI with the dark matter mass mDM .

Table II :
Scalar boson charge assignments under the SU (3