Magnetic Moments of Leptons, Charged Lepton Flavor Violations and Dark Matter Phenomenology of a Minimal Radiative Dirac Neutrino Mass Model

In a simple extension of the standard model (SM), a pair of vector like lepton doublets ($L_1$ and $L_2$) and a $SU(2)_L$ scalar doublet ($\eta$) have been introduced to help in accommodating the discrepancy in determination of the anomalous magnetic moments of the light leptons, namely, $e$ and $\mu$. Moreover, to make our scenario friendly to a Dirac like neutrino and also for a consistent dark matter phenomenology, we specifically add a singlet scalar ($S$) and a singlet fermion ($\psi$) in the set-up. However, the singlet states also induce a meaningful contribution in other charged lepton processes. A discrete symmetry $\mathcal {Z}_2 \times \mathcal {Z}_2^\prime$ has been imposed under which all the SM particles are even while the new particles may be assumed to have odd charges. In a bottom-up approach, with a minimal particle content, we systematically explore the available parameter space in terms of couplings and masses of the new particles. Here a number of observables associated with the SM leptons have been considered, e.g., masses and mixings of neutrinos, $(g-2)$ anomalies of $e$, $\mu$, charged lepton flavor violating (cLFV) observables and the dark matter (DM) phenomenology of a singlet-doublet dark matter. Neutrinos, promoted as the Dirac type states, acquire mass at one loop level after the discrete $\mathcal{Z}_2^\prime$ symmetry gets softly broken, while the unbroken $\mathcal{Z}_2$ keeps the dark matter stable. The mixing between the singlet $\psi$ and the doublet vector lepton can be constrained to satisfy the electroweak precision observables and the spin independent (SI) direct detection (DD) cross section of the dark matter. In this analysis, potentially important LHC bounds have also been discussed.


I. INTRODUCTION
The standard model (SM) of particle physics has been quite successful in explaining the interactions of elementary particles [1]. The recent discovery of a Higgs boson with a mass of 125 GeV at the Large Hadron Collider [2,3] has been showing good agreements with the SM expectations [4,5]. However, there exists a few experimental and theoretical issues, which cannot be explained in the SM paradigm, thus, hint towards a more complete theory -beyond SM physics (BSM) at the TeV scale. Among these signatures, the precise measurement of the dark matter (DM) abundance and the non-zero values of the neutrino masses and mixings are of particular interests to us. Here, one may broadly recall the issues at hand. (i) Assuming the origin of the dark matter is related to a new kind of particle, the simplest and most compelling candidate has been considered as a weakly interacting massive particle (WIMP). The experiments like PLANCK [6] and WMAP [7] have already provided precise measurements of DM relic density. WIMPs with masses ∼ 1 TeV can lead to the correct relic density through its annihilations to SM particles. Such a mass scale can be probed at the high-energy collider experiments like the LHC and also at the dark matter direct detection experiments. (ii) Non-zero neutrino masses and substantial mixing among the three light neutrino states require specific extensions of the SM. In the simplest case, one may introduce right handed neutrinos ν R and assumes a Dirac mass term m D for the neutrinos. But, then the neutrino Yukawa couplings are assumed to be 10 −11 to generate a neutrino mass ∼ 0.1 eV. However, being a singlet under the SM gauge group, ν R can also accommodate a large Majorana mass parameter M which violates the lepton number by 2 units. Such a mass term leads to an attractive possibility -called "seesaw mechanism" where the light neutrinos ν L obtain an effective small Majorana mass term [8][9][10].
The tinyness of neutrino masses can be explained naturally without requiring a tiny Yukawa coupling.
Though seesaw mechanism is more favoured, experimentally, the searches to probe the Majorana nature of neutrinos through neutrinoless double beta decay experiments have not yet lead to any conclusive evidence.
So the simple idea of considering neutrino as a Dirac particle has been still quite popular.
There have already been many proposals which may incorporate new particles and appropriate mixings, thus, explains the masses for neutrinos and the dark matter abundance in the extensions of the SM.
However, it is more natural to consider that there exists a tie-up between these two important pieces which may lead to a somewhat economical and an attractive extension of the SM to deal with. Driven by the same pursuit, here we will also furnish a connection between these two important issues assuming neutrino as a Dirac particle. Interestingly and more importantly, we will observe that the precision observables like anomalous magnetic moments of µ (a µ = (g−2)µ 2 the experimental constraints related to the charged lepton flavor violations.
The idea of neutrino as a Dirac particle has revived in the recent past when the main theoretical objection of having a very tiny tree level Yukawa coupling has been addressed through the radiative generation of neutrino masses [11][12][13][14][15][16][17][18][19][20][21][22][23][24] (for a review see [25]). The main idea is simple and can be realized through an additional Z 2 × Z 2 symmetry in the SM set up : (1) one may assume a discrete symmetry (here Z 2 ) to forbid a tree-level Dirac neutrino masses. This symmetry would be finally broken softly to generate a tiny neutrino mass through a radiative mechanism.
(2) New fields may be introduced; in the simplest case, an inert scalar doublet (η + η 0 ) T and neutral singlet fermions can be considered (see below) to radiatively induce neutrino masses in the loop. The new fields may transform odd under the another Z 2 symmetry to prohibit their couplings with the other SM fermions, thus, offers an interesting possibility where the lightest state (a new Z 2 odd fermion or a neutral scalar) may become the cold dark matter (CDM) of the universe.
This class of models where neutrinos acquire masses through dark matter in the loop, thereby connects the two important BSM aspects of the particle physics has been dubbed as "scotogenic" model [26]. In the original idea, the neutrino masses have been assumed to be of Majorana type. However, one may employ the same idea to generate the masses for the neutrinos radiatively considering them as the Dirac particle, if a symmetry like global or gauge U (1) symmetry is assumed to prohibit the Majorana mass term in the Lagrangian [14].
Assuming the lepton number as a good symmetry of the Lagrangian at the backdrop of our work we start our discussion with a simple realization 1 . We consider new leptons/scalars at the electroweak (EW) scale in addition to the usual right handed neutrinos ν R : singlet Dirac fermion(s) (N ), two scalars -an inert scalar doublet η and a real singlet scalar S in the particle content of the SM. A perturbative value of the coupling Y N R η ( ∈ e, µ, τ ) may help to realize tiny nature of the neutrino Yukawa couplings radiatively, if the other interaction terms Y RN Sν R and µ η † HS are included in the interacting Lagrangian.
Here the last term µ can be regarded as the soft symmetry breaking parameter. As in the case of a "scotogenic" model, with proper charge assignments under Z 2 × Z 2 symmetry, Dirac masses for the SM neutrinos, proportional to the soft breaking scale µ , would be generated radiatively through a N − η − S loop. Similarly, observable abundance of the dark matter N would follow naturally. However, this simple model fall short to account for the BSM contributions in the measurement of the anomalous magnetic moment of muon a µ [33], though can help to acclimatize the measurement of a e . Primarily, the non-SM contribution, controlled by the N − η ± loop, comes out to be negative while the discrepancy in the muon anomalous magnetic moment ∆a µ requires a positive boost, thus, disfavours this simple set-up (for a generic discussion on the new physics contributions to a µ , see [34][35][36]).
We next consider the vector like (VL) leptons in place of singlet Dirac like state N in the SM setup, without changing the basic structure of the model. For a color singlet VL, left and right handed components transform similarly under the SM gauge symmetries, and one may observe that ∆a µ can be accommodated through the mixings with the SM leptons [37][38][39][40][41]. However, addressing a e along with a µ invites a further modification. We, thus introduce a pair of SU (2) vector like leptons L 1 ≡ (L 0 1 L − 1 ) T , L 2 ≡ (L 0 2 L − 2 ) T with same hypercharge (but charged differently under Z 2 × Z 2 symmetry) which can be found to be suitable when coupled to new states; e.g., an inert Higgs doublet η, a real singlet scalar S and a SM singlet fermion ψ in the present context. As in the previous case, S acts to realize the soft breaking of Z 2 symmetry; thus to generate Dirac masses for the neutrinos while ψ has its role to realize the proper dark matter abundance. In fact, L 1 and ψ can enjoy the same transformation properties under the Z 2 × Z 2 symmetry; thus the neutral L 0 1 and ψ can mix to provide with a suitable candidate for dark matter (χ 0 ) and to accommodate (g − 2) e anomaly through neutral fermions and charged scalars running in the loop. The charged components of the new leptons help to explain the other anomaly in (g − 2) µ .
Naturally, neutrino mass as well as cLFV processes receive contributions from the diagrams that involve both of the VL leptons in the loops. In [40], authors find that a vector like lepton doublet in presence of a right handed neutrino and inert Higgs doublet may indeed be helpful in explaining (g − 2) µ deviation while the tiny Majorana masses for the neutrinos can also be generated in a "scotogenic" model. Here we will try to find if the both anomalous (g − 2) µ and (g − 2) e can be accommodated with the said particle contents while neutrinos acquire Dirac masses through dark matter χ 0 in the loop.
In dark matter phenomenology, singlet-doublet DM χ 0 comprised of L 0 1 and singlet ψ, could just be able to produce the correct relic abundance [42][43][44][45][46][47][48][49][50]. Admitting only VL doublet lepton L 0 1 , one finds a large DM-nucleon elastic cross-section through Z mediated processes, thus has essentially been ruled out by the experiments such as XENON1T [51] or LUX [52]. As a natural deviation, one finds that a singlet-doublet fermion dark matter, through its SM singlet component may escape the stringent direct detection bounds.
For practical purposes, the dark matter particle has to be essentially dominated by the singlet component, while only a very small doublet part can be allowed. For the same reason, we purposefully introduce ψ in the particle content.
We organise our paper as follows. In sec II, we explain the details of our model including the new particles and their charges under the complete gauge group which would be considered. After electroweak symmetry breaking (EWSB), our model predicts additional neutral and charged leptons. Consequently, relevant interactions of the new particles with the SM particles can be realized. Theoretical and experimental bounds on their couplings and masses have been summarized in III. These include (i) anomalous magnetic moments and different charged lepton flavor violating decays of the SM leptons, (ii) vacuum stability of the tree level scalar potential, (iii) Electroweak precision observables (EWPO) and (iv) collider physics constraints. In the results sections, we present radiative generation of the neutrino masses and mixing angles in sec IV. As discussed, one of the motivations is to show that our model can accommodate anomalous magnetic moments of the lighter charged leptons. We depict the parameter space of our model in sec V, where discrepancies in a µ/e can simultaneously be satisfied. Subsequently, we probe our model parameters with different charged lepton flavor violating (cLFV) observables, namely α → β γ, α → 3 β and flavor violating decays of Z boson. DM phenomenology including the relic density and the direct detection of a singlet-doublet fermionic DM have been covered in sec VI. Finally, we conclude this work in sec VII.

II. THE MODEL: RELEVANT LAGRANGIAN AND SCALAR POTENTIAL AT THE TREE LEVEL
As stated, the proposed model is a simple extension of the Standard Model where we augment two scalars, namely a real singlet (S) and a SU (2) L doublet η ≡ (η + η 0 ) T , two vector like lepton doublets T , a singlet fermion ψ and the usual SM singlet right handed neutrinos ν R . All the new states are charged under an additional Z 2 × Z 2 symmetry (see Table I).
The allowed interactions of the new fields and the SM fields can be read from the following Lagrangian : where L new , the new physics Lagrangian is given by, Here, D µ is the SU (2) L × U (1) Y covariant derivative and V (η, H, S) is the scalar potential. We define field Φ as iτ 2 Φ * . We are following the convention Q EM = T 3 + Y . For clarity, we refrain from explicit showing of SU (2) contractions. Except for the right handed neutrinos, single generation of all the other new states would suffice for our purpose (see Table I  Finally, we may express the scalar potential V (η, H, S) in Eq. (2) which adheres the proposed symmetry as follows: There can be a few additional terms like which are allowed by gauge and Lorentz invariance, but due to the imposed Z 2 × Z 2 symmetry these terms transform non-trivially and hence are forbidden (see e.g., last four terms in Table I(b)). This in turn ensures that the new scalars S, η do not acquire any induced VEV.
As usual µ 2 H can take negative values. As stated, to generate the mass terms for the SM neutrinos, Z 2 symmetry can be broken explicitly by introducing a soft breaking term at the scalar potential, Since µ breaks the Z 2 , it may be argued to be very small, thus may be helpful in fitting neutrino masses.
Similarly,L 1 L 2 can also accommodate a soft beraking term. The mass term can also be generated at the two loops (∝ µ ) which we assume to be small for further consideration. If the VL states would be considered to transform identically under Z 2 , then we will have a restricted class of the Yukawa terms and consequently accommodating the anomalous magnetic momemts of µ and e simultaneously cannot be realized in this proposed model with the given particle content. However, we may consider a global U (1) symmetry (the charge assignments could read as L 1 , S, Ψ, η=1 while L 2 =-1 with all the SM particles including ν R assume zero charges), then our model and its phenomenology would be completely unchanged.
Infact, it will make the dark matter stable thus Z 2 can be assumed to be replaced.
Before discussing the phenomenology, let us briefly outline the role of different discrete symmetries in the present analysis. We assume Z 2 to be an exact symmetry which always ensures that (i) a tree level  Table I along with their transformations under the proposed symmetry group. Here √ and × refer to the occasions when a particular interaction term turns out to be even or odd under a symmetry operation respectively. Possible completion of the model at the GUT scale : Here we discuss a possibility to embed our low energy model to a larger gauge group e.g., SO (10). Specific gauge breaking chains may include, e.g., left-right (LR) symmetric phase at the intermediate scale [53][54][55][56][57], with M GU T denoting the breaking scale of SO(10) gauge group which is subsequently broken to the SM at M LR < M GU T . There are a few reasons for considering the LR models: (i) the particle content contains automatically the right-handed neutrino, (ii) a TeV scale LR symmetric intermediate phase may be obtained within a class of renormalizable SO(10) GUTs with a perfect gauge coupling unification [58].
Here one has to account for a few copies of one or two types of extra fields; e.g., additioanl triplet and/or doublet scalars under SU (2) R . However, for different possibitites, we refer the reader to Ref. [58]. Of course, the new scalars can effect the low energy phenomenology e.g., (g − 2) µ through a gauge invariant interaction at the LR scale . The matter content of the model along with their possible transformations at   each intermediate stage is given in table II. Here Q, Q c , L and L c (we follow the notation in [56]) are the quark and lepton families with the addition of (three) right-handed neutrino(s) ν R . The SM Higgs and the inert doublet can be included as bidoublets under SU (2) L × SU (2) R . More than a single bidoublet is required for a correct Yukawa Lagrangian at the low scale [58].  Similarly, Y 6(1i) Sψ L ν Ri can be cast as SψL c which, under, SO(10) goes as 1 × 16 × 16. Though the particle contents can easily be accommodated under a unified gauge group, one has to admit a minor change, e.g., ψ in Eq. (2) should refer to the neutral component of SU (2) R doublet in Table II. Alternatively, one may also consider the symmetry breaking chain as SO (10) which was earlier considered in Ref. [40,59].
Mixings and couplings of the VL states with bosons and fermions: As can be seen from Eq. (2) that lepton phenomenology is primarily governed by the new Yukawa couplings Y i (i = 1...6). Apparently, the first four couplings are more important for the phenomenology in the lepton sector, while Y 5 primarily controls the DM physics. The Yukawa interactions involving the singlet states may contribute to neutrino masses and also the dark matter relic abundance. For a generic study, we keep all the couplings with Y i (i = 1...6) in the flavor space.
Let us first start our discussion with the interactions mediated by Y 5 in Eq. (2). The Yukawa interaction, Y 5L1H ψ generates a mass matrix, in the basis of (ψ, L 0 1 ). We can rotate this to the mass basis with the help of (2 × 2) orthogonal matrix, The two mass eigenstates can be defined as, with the masses are given by, The mixing angle is defined as, If we assume a small mixing angle i.e., θ << 1 then χ 1 is dominantly doublet-like with a small admixture of singlet ψ, while χ 0 is mostly singlet-like. Since the direct detection experiments require DM to be mostly singlet dominated, we can propose χ 0 as the DM candidate with the condition that M χ 0 < M χ 1 , which is further ensured by the choice M ψ < M L 1 . The Yukawa coupling Y 5 , now being a dependent parameter, can be expressed in terms of M χ 1 , M χ 0 and θ through the following relation, At this point we can recast the Yukawa terms in Eq. (2) in this new basis of (χ 0 , χ 1 ) as: All the Yukawa couplings appearing above need to satisfy a generic condition |Y | ≤ 4π so to remain perturbative at the TeV scale. Similarly, the terms appearing in the covariant derivative can be collected to write down the couplings with the gauge bosons.
Note that, all the other terms in Eq.
(2) will not be affected by this basis change.

III. BOUNDS RELATED TO DIFFERENT EXPERIMENTS AND THEORIES
Here we review different bounds related to experimental searches and theories. We will use the limits in delineating the parameter space consistent with the anomalous magnetic moments of leptons, charged lepton flavor violations and the dark matter abundance.

A. Anomalous magnetic moment and different LFV decays
Bounds on anomalous magnetic moment: From the first precision measurement of the magnetic dipole moment of the muon a µ at BNL (Brookhaven National Laboratory), the persistent discrepancy in its determination compared to its SM prediction has been undoubtedly one of the most promising hints towards a new physics signal at the TeV scale. The discrepancy can be expressed through its experimental measurements (≡ a exp µ ) and the SM prediction (≡ a SM µ ). The difference in the two values can be seen to be driven by the BSM contributions (≡ ∆a µ ). For the last many years, the experimental data produced a roughly 3.7σ deviation from the standard model (SM) value [60][61][62][63]. For a better understanding of the known physics, it was imperative to resolve the tension related to the hadronic vacuum polarization (HVP) of a SM µ [64][65][66][67][68][69][70][71] (see also [63] and references therein). The tension lies in the fact that a recent lattice-QCD [64] estimation of the HVP may bring the SM prediction of a µ into agreement with experiments which seems to be in contradiction with e + e − → hadrons cross section data and global fits to electroweak precision observables [69,71]. The Fermilab-based Muon g-2 experiment has just reported a new result [72,73] which, if combined with the BNL result reads 4.2σ deviation from the SM value 2 . ∆a µ = (25.1 ± 5.9) × 10 −10 .
Thus, as stated earlier, from Eq. (16) it is clearly visible that one needs a positive BSM contribution to satisfy the experimental constraint on ∆a µ . In the context of a e , the experimental value has been updated in 2018 [90] from a precision measurement of the fine-structure constant [91] that relies on the caesium recoil measurements. This measurement also shows a possible disagreement between the experimental observation and theory prediction, though with a less significance ∼ 3σ.
More importantly, here the measured value is lower than the corresponding SM prediction. Following the improved estimates, specially in the evaluation of a e , attempts have been made to link the both discrepancies with a common new physics origin [92][93][94][95][96][97][98][99][100][101][102][103][104][105][106][107][108][109]. Here we note that a very recent determination of the fine structure constant [110], obtained from the measurement of the recoil velocity on rubidium atoms, result into a positive discrepancy of about 1.6σ. Clearly the discrepancy in the measurement of a e can only be settled in the future. This work focuses on caesium recoil measurements, thus, Eq. (17) in the subsequent sections.
Bounds on charged lepton flavor violating decays: Charged lepton flavor violating processes, specifically α → β γ or α → 3 β through photon penguins may be influenced by the same dipole operators which provides the BSM contributions to a µ/e . Non observations of any cLFV processes so far, can potentially constrain the new physics parameters. Currently, the radiative decay of α → β γ, specifically µ → eγ, is the leading candidate among the cLFV observables to put a stringent constraint on the parameter space. In the future upgrades, the MEG collaboration can reach a sensitivity of about 6 × 10 −14 after 3 years of acquisition time [111]. Similarly, in the near future, µ → 3e can be probed by the Mu3e experiment [112,113] with a branching ratio sensitivity of 10 −16 . A significant improvement is expected compared to the present limit, set by the SINDRUM experiment [114]. An impressive improvement on most of the LFV modes of the rare τ decays can be expected from searches in B factories [115,116].
TABLE III. Current Experimental bounds and future sensitivities for the LFV processes.

B. Condition of Vacuum stability and the masses of scalars
The scalar potential must be bounded from below i.e., it does not acquire negative infinite value in any of the field directions for large field values. This physical requirement puts certain constraints on the scalar couplings. Considering the tree level scalar potential, these conditions are listed below [120].
After the electroweak symmetry breaking only H field gets a VEV, v 246 GeV. Thus, scalar fields can be expressed as : Substituting H and η in Eq. (3) one finds We identify h as our SM like Higgs scalar with mass M h 125 GeV. Again for simplicity we assume that the new scalars are heavier to forbid the constraints coming from the invisible Z and h decays. Similarly the mass splitting between the charged and the neutral components of the doublet η are considered to be (20)), if the couplings λ ηH and λ ηH can be assumed to be very small. In fact, λ ηH is absent under a global or gauge U (1) symmetry.
However, such a mass splitting may play a significant role for its discovery at the LHC (see e.g., [121]).

C. Electroweak precision observables (EWPO)
In the presence of two BSM scalars (η, S), two vector like lepton doublets (L 1 , L 2 ) and a singlet fermion (ψ), our model may introduce corrections to the gauge boson vacuum polarization amplitudes or electroweak precision observables (EWPO). These observables were initially discussed by Peskin and Takeuchi as S, T and U parameters in Ref. [122]. Later Barbieri  The current experimental constraints are [62,123], Inert doublet η may particularly effect T orT parameter through λ ηH and λ ηH [125]. But in the limit, M η I = M η R = M η ± ≡ M η , which we assume in the subsequent analysis, the electroweak parameters seem to be unaffected by the presence of new scalars. Hence the correction is completely due to the effect of vector like fermions (VLF), i.e., in our model ∆(Ŝ,T ) = (Ŝ,T ) V LF . Therefore, the correction inT parameter appearing due to the mixing between L 1 and ψ for q 2 → 0 limit can be expressed as [124], where M L 1 is the mass term for L − 1 , g is the SU (2) L coupling constant, θ is the mixing angle between L 0 1 and ψ as discussed earlier, M W stands for the mass of W boson and is the correction to gauge boson propagators in presence of the new VLF's. Div = 1 + ln(4π) − γ is the usual divergent piece appearing in the dimensional regularisation and µ denotes the renormalization scale.
One can easily see that for m a = m b , Eq. (24) vanishes. Hence Eq. (23) simplifies tô It can be noted that the divergent part of the first term of Eq. (25) is cancelled by the divergences encapsulated in the last two terms. Moreover in the limit, when the mass splitting between M L 1 and M χ 1 vanishes, (i.e., sin θ → 0) one findsT → 0.
In our model, the correction inŜ can be parameterized as, where the ' ' signifies derivative with respect to q 2 . The general expression forΠ (m a , m b , 0) is given as [124,126], For m a = m b the above expression reduces tõ It can be directly verified that the divergent parts along with the scaling factor µ get cancelled when Eq. (27) or Eq. (28) Notably, the change in ∆M is negligible to the variation with M χ 0 for a small mixing angle (sin θ ≤ 0.1) (see Fig. 1(c) and (d)). In other words, the EWPOs are insensitive to the lightest neutral fermion mass M χ 0 as long as the mixing angle is not much high. In the subsequent section, we consider sin θ ≤ 0.01, thus, in this regime, the mass of the charged component of the VL, M L 1 can easily be fixed through M χ 1 while satisfying all the bounds coming from EWPOs.

D. Constraints from the collider observables
For vector like quarks, the LHC pair production cross section is determined from QCD, so model independent bounds can be placed in the parameter space. However, for the vector like leptons, the pairproduction cross section is mediated by the s-channel electroweak vector boson exchanges, thus depends and it has been observed that even a lighter smuon mass is also allowed depending on the value of mχ0 1 (e.g. mμ ∼ 200 GeV is allowed for mχ0 1 ∼ 120 GeV).
In the framework that we considered, we shall place M L 1 (M χ 1 ) at 800 GeV, but the other VL L 2 has to be set at a lower value (e.g.∼ 200 GeV) in order to satisfy (g − 2) µ constraints. Here we may note a few observations which would be detailed in the next sections. First of all, we will find that, the potentially important contribution in the evaluation of ∆a µ would be driven by the interaction involving coupling Y 3µ and in the perturbative unitarity regime (will be discussed in Sec. V) Y 3µ can only take ∼ O(1) values. We will further observe that all other Yukawa couplings of L 2 would be orders of magnitude suppressed either from the neutrino masses and mixings or from (g − 2) e and cLFV observables. Thus, the dominant decay of L 2 can be considered as L 2 → µS followed by S → χ 0 ν (M L 1 (M χ 1 ) > M L 2 > M S > M χ 0 would be followed throughout this analysis). So, naturally, P P → (L ± 2 L ∓ 2 ) → 2µ + E / T through Z boson exchange can be considered as the most useful constraint for the present analysis. Here we may borrow the limits from Ref. [131] as direct production of sleptons or VL states would have same cross-section. Thus, based upon our previous discussion, we would consider M L 2 = 190 GeV and mχ0 1 = 120 GeV respectively for the calculation of different observables in the leptonic sector.
In our model, η couples to leptons, so can only be produced through electroweak gauge bosons at the LHC. Also, recall that η does not acquire any VEV, thus do not take part in electroweak symmetry breaking. In a model specific study, one would expect dilepton +E / T [121,132] through charged η pair production, or mono-lepton + E / T through charged and neutral η productions via Z boson or W boson exchanges. An observable signal may be expected during high luminosity run of LHC through multilepton searches for M η ≤ 250 GeV [121]. Here, assuming all the charged and neutral components of η are of similar masses, we consider M η > 100 GeV which is closely based on the exclusions at LEP [133]. However, our result does not depend much on M η .
Radiative mass generation for the neutrinos that adheres lepton number conservation. In the second diagram, , Y 6(1i) )cosθ have been used.
. As discussed, here neutrinos are massless at the tree level due to the imposed Z 2 × Z 2 symmetry while they may receive appropriate radiative corrections through the symmetry breaking term in Eq. (4). Thus one may develop a Dirac mass term for the SM neutrinos at one loop order after the Higgs field acquires a VEV. Additionally, the neutrino loops contain a stable particle χ 0 that could be treated as the cold DM of the universe [see Fig. 2(b)]. This intrinsically sets up a bridge between the phenomenology of light neutrinos and the other sectors like dark matter. The (3 × 3) neutrino mass matrix can be read as: Similarly, and = µ H is the symmetry breaking term, with a mass dimension of 2. For each element in f ∈ (L 2 , χ 1 , χ 0 ), the vertices y and z take (3 × 1) and (1 × 3) elements respectively which can be read as where the PMNS matrix can be parameterized as [1]:  phases (α 21(31) = 0) and the 3σ uncertainties, the magnitudes of the neutrino mass matrix elements in units of eV for NH and IH can be estimated as: Here, following Eqs. (35) and (31) we may note a few observations related to the neutrino masses and mixings. In fact Eq. (31) can be cast as Eq. (32) to delineate the domain for Y 2(1i) and Y 6(1i) that may produce correct values for |M νij | in Eq. (35) in the NH scenario. For simplicity, we recast the parameter as Y 2(1i) = Y 2i and Y 6(1i) = Y 6i (see also the discussion in Sec.V). We also fix {Y 3 , Y 4 } at the given values (see Table V) which would be allowed by (g − 2) and cLFV constraints. We would further detail it in Sec. V. The lower and upper limits in Table V would refer to the minimum and maximum value of the |M ν ij | in Eq. (35). In the lepton phenomenology, apart from tuning the µ and e anomaly, new scalars η, S, charged fermions L ± 2 , L ± 1 and neutral leptons χ 1 and χ 0 may lead to observable signatures to lepton flavor violating processes such as α → β γ, or α → 3 β through the Yukawa couplings Y 1 , Y 3 and Y 4 that tie the SM leptons to BSM particles. The free parameters can be listed as: The can be given as ( α = β = ): where, the superscripts 'n' and 'c' correspond to the neutral and the charged lepton contributions in Fig. 3(a) and Fig. 3(b) respectively. The three individual contributions of Eq. (37) can be expressed as, The Form factors are defined in Appendix B. It is instructive to identify the positive and negative contributions of ∆a ( ∈ e, µ) in Eq. (38)- (40).
In the above, sin 2 θ → 0 has been taken for illustration. Additionally, we consider that all the couplings are real and positive. In Eq. (41), the first two terms arise from the diagram with a charged fermion and a neutral scalar in the loop. The third term involves a neutral fermion and a charged scalar in the loop.
Here the DM state χ 0 may provide with a positive contribution in ∆a µ , owing to the mixing between L 0 1 and ψ. The negative parts in ∆a (see Eq. (42)) involves only a neutral fermion and a charged scalar in the loop which is shown in Fig. 3(a). Thus, considering the opposite signs of ∆a µ and ∆a e in mind, one can easily expect that ∆a µ should have a major contribution from Eq. , where, We begin our discussion with µ-specific couplings However, at the same time it becomes unfriendly to obtain a correct ∆a e (since the same bracketed term in Eq. (46) potentially contributes to e magnetic moment). For a practical choice, we set Y 4µ = 0 as we will see that ∆a Y 4e Y 1e e term would have to be properly tuned to fit ∆a e . In other words, M χ 0 /M χ 1 will be chosen to have a negative contribution from ∆a Y 4e Y 1e e to have a consistent ∆a e .
Thus assuming Y 4µ = 0, one finds ∆a µ = ∆a µ . A prominent cancellation between the two terms in ∆a Y 1µ µ can always be observed irrespective of the value of Y 1µ , and, thus, one finds ∆a µ ∆a Y 3µ µ . Thus, naturally, we may choose Y 1µ at any value within its perturbative limit while satisfying the experimental bounds on ∆a µ . We will see that a smaller Y 1µ (which will be chosen in the subsequent analysis) would be highly desired to satisfy µ → eγ constraint. is fixed at zero. Clearly, the doublet scalar does not have any influence to the result. As said earlier, only L 2 − S loop can manage to attune ∆a µ , and thus, one requires somewhat larger values for Y 3µ . This can be further verified through Fig. 4(a). Note that, here mass of the singlet M S needs to be smaller to make Y 3µ within the perturbative bound, and this can only be realized if our model considers light dark matter (since M χ 0 < M S needs to be satisfied). However, a heavier χ 0 can also accommodate ∆a µ without having any difficulties. Recall that setting Y 4µ = 0 will automatically make vanishing contributions from Eqs. (45) and (46), which include M χ 0 . Thus, because of the choice of our parameters, χ 0 can affect ∆a µ only through Eq. (44), which can only lead to insignificant contribution. A further confirmation can be made through to be consistent over the entire χ 0 range. We note here that, in Fig. 4(a) and Fig. 4(b), we refrain from considering LHC bounds based on with two leptons and missing transverse energy (see Sec.III D) on the parameter space. This helps us to study the dependence of different parameters on the ∆a µ numerically and to choose a valid parameter space which is consistent with the LHC searches. For instance, a light L 2 accompanied with a light scalar S may easily accommodate ∆a µ with a perturbative value of Y 3µ ∼ 2. We have checked that Y 3µ remains perturbative upto TeV scale even when one includes dominant radiative In our next precision calculation, we will now see the role of different parameters in obtaining a correct value for ∆a e . Note that, here, for practical purposes, one finds ∆a e ∆a are ∝ m 2 e , thus, are much suppressed and can be neglected for the parameter space, we are interested in. Additionally, we choose Y 3e = 0 to forbid the positive part in Eq. (43). So we may re-express ∆a e as follows: As before, in the numerical analysis, we fixed M S = 130 GeV, M η = 300 and 1200 GeV, M L 2 = 190 GeV and M χ 0 = 120 GeV. Fig. 5(a) depicts the variation of Y 1e as a function of M χ 1 , when Y 4e is fixed at 0.2. And similarly for the Fig. 5(b), where Y 4e appears as the variable and Y 1e is fixed at 0.2. In both of these plots red and green dots represent the scenarios corresponding to M η = 300 GeV and 1200 GeV  and hence a correct value of ∆a e would be difficult to obtain.
As a final remark, it is now evident that the presence of the two VL states L 1 and L 2 are necessary to accommodate the both ∆a µ and ∆a e . The second doublet L 2 may provide the sole contribution to muon magnetic moment, while the other one can be used to tune the magnitude and sign of the e magnetic moment. Moreover, we will find that, satisfying different cLFV processes may become much easier in this scenario.

B. cLFV constraints
In this model framework, in computing the cLFV observables we closely follow Refs. [135,136]. Oneloop effective vertices, relevant for the different two and three body processes α → β γ or α → 3 β are generated through the interactions among BSM fermions (χ a , L ± a ), scalars η and S and the SM leptons.
We start with the form factors for α → β γ, where the relevant diagrams have been depicted in Fig. 3.
The details of the calculation are presented in Appendix C. Here we recast the form factors A Finally, the coefficients in the above can be clubbed to get the total contributions.
The decay width is given by [135,136] where α em is the electromagnetic fine structure constant and τ α is the lifetime of α .

α → 3 β
Here we calculate the decay width for the processes where a heavier SM lepton decays into three lighter leptons of the same flavor, i.e., − α → − β − β + β . We present the relevant γ-penguin, Z-penguin and Box diagrams contributions to get the complete decay width and hence the branching ratio for α → 3 β processes. The details of the calculation can be found in Appendix C.
The index a reads 0, 1 for neutral and 1, 2 for charged fermions and s 1 = η 0 , s 2 = S for the charged lepton loops. The corresponding leg-corrections (not shown) are also taken into account.
• Photon penguin contribution: As shown in Fig. 6, the monopole contributions can be recast in terms of our model parameters, The dipole contributions can be read from Eq. (48) and Eq. (49).
• Z penguin contribution: Dominant Feynman diagrams are shown in Fig. 7. We have calculated the coefficients as follows: The expressions for the form factors are given below [137][138][139]: As before, F L,R = F • Box diagram contributions: Leading contributions are shown in Fig. 8. The dominant B-factors can be calculated as, The generic functional forms for these D 0 andD 0 are again available at Appendix B. Finally, there may be Higgs penguin diagrams as well, but the Higgs couplings to the SM leptons are much suppressed (∼ O(≤ 10 −2 )) compared to that of γ and Z, and hence we can ignore them 3 .

Numerical Results
Here, we will particularly identify the allowed regions of parameter space associated with free parameters and masses as introduced in Eq. (36), in regard to different cLFV decays. Some of the free parameters, as already tuned by ∆a i (i ∈ e, µ), collider or the electroweak precision searches would be set within their allowed domains. In Figs. 9 and 10, the variation of branching ratios for the different cLFV processes with respect to the relevant couplings have been shown for M η = 300 GeV and 1200 GeV respectively.
We have followed a particular color code for all these plots, i.e., the red signifies Br( α → β γ) while blue stands for Br( α → 3 β ). The horizontal lines specify the present experimental bounds [see Table III] on the   Br( α → β γ) and Br( α → 3 β ) have been marked with the black and magenta horizontal lines respectively.
So, at this point, we are left with only four flavor specific free parameters, i.e., Y 1µ , Y 1τ , Y 3τ and Y 4τ . Our aim would be to constrain these free couplings using the present and future limits of the cLFV branching ratios for α → β γ and α → 3 β processes (where α, β = e, µ, τ ). Thus, we have varied the free couplings randomly, and calculated the corresponding values for Br( α → β γ) and Br( α → 3 β ). Focusing on a particular flavor at a time, in the following, we present the possible 2-body and 3-body decays.
• Br(µ → eγ) and Br(µ → 3e) : The first rows of the Figs. 9 and 10 depict the variation of µ → e branching fractions. Here the relevant couplings can be read as Y (1,3,4)i ∼ (i = e, µ). However, only Y 1µ can be regarded as the free parameter since all the other couplings have already been fixed by the precision measurements of µ and e anomalous magnetic moments. As can be evident from the plot, for Y 1µ ≤ 10 −4 both the Br(µ → eγ) and Br(µ → 3e) can be made satisfied. This explains our choice for Y 1µ in the earlier (g − 2) µ analysis. Thus, to have a simultaneous validation of the (g − 2) µ and cLFV constraints (i.e. Br(µ → eγ) and Br(µ → 3e)) one certainly needs a much smaller value of Y 1µ (∼ 10 −4 ).
All the µ specific couplings are already fixed: Y 3µ and Y 4µ have been set to their earlier values and Y 1µ = 10 −4 is considered (in accordance with Figs. 9(a) and 10(a)). Thus we have varied the τ specific free parameters Y jτ (j = 1, 4, 3) and calculated the branching ratios. The allowed ranges of these couplings where Br(τ → µγ) and Br(τ → 3µ) are satisfied, can be seen from Figs. 9 (b), (c), (d) and 10 (b), (c), (d) respectively. Clearly, only meaningful constraint can be derived for Y 3τ which reads as Y 3τ ≤ 0.04. The bound can be placed using Br(τ → 3µ) which seems to be much stringent compared to Br(τ → µγ). This is a result of the Z-penguin dominance in that region of the parameter space.
To illustrate it further, we focus on the dominant parts of γ penguin contributions. In case of photon initiated 2-body Br( α → β γ), or 3-body Br( α → 3 β ) decays, dipole terms become more important, and specially the most significant parts read as: The other terms related to dipole or monopole terms are proportional to the products of the other However, generically, considering the couplings for any α, β are of the same size, these terms are few orders of magnitude smaller compared to A • Br(τ → eγ) and Br(τ → 3e) : Third rows of Figs. 9 and 10 show the plots for these two processes.
Here the only free parameters are Y jτ (j = 1, 4, 3), as the electronic couplings are fixed by the (g − 2) e results. Indeed, the τ specific parameters are same as in the τ → µ analysis. The ranges of Y jτ couplings where Br(τ → eγ) and Br(τ → 3e) can be simultaneously satisfied, have been shown in Figs. 9 (e), (f) and (g) and 10 (e), (f) and (g) respectively. We may observe that Z-penguin diagrams become dominant over photon penguins in Figs. 9, 10 (f) since Y 4τ Y 4e can now contributes significantly. From these plots (Figs. 9 (e), (f) and 10 (e), (f)), we are able to constrain the two τ -specific couplings as: Y 1τ ≤ 0.5 and Y 4τ ≤ 0.7. Note that, the variation of BRs with respect to Y 3τ has been appearing as two horizontal lines, implying that the BRs are apparently independent of this coupling. This result is a sole outcome of the choice Y 3e = 0. Since in both Br(τ → eγ) and Br(τ → 3e), the coupling structure appears as Y 3e Y 3τ , putting Y 3e = 0 automatically ensures the invariance of the BRs with respect to Y 3τ .
So finally, collecting all the constraints, i.e., from the anomalous magnetic moment data and non observation of the cLFV processes, we find that all the flavor specific couplings Y 1,3,4 may assume ∼ O(1 − 10 −4 ) values, some of which may be tested in the near future.
C. Z and h observables 1) Invisible decays Z, h → χ 0 χ 0 : In this model, a light DM is natural and the parameter space associated with it can be observed to be consistent with the all low energy data. It is well known that for a light DM, invisible decays of Z and h which lead to Z, h → χ 0 χ 0 can be substantial to constrain the parameter space. The corresponding decay widths are given by, sin 2θ, with M χ 1 fixed at 800 GeV. We also plot the valid regions in sin θ − M χ 0 plane. For depicting our results, we use (i) the observed invisible partial width of Z boson, Γ inv Z = 499 ± 1.5 MeV which is below the SM prediction Γ inv SM = 501.44 ± 0.04 MeV at 1.5σ C.L. [143] and (ii) the experimental bound on invisible h decay reads as Br inv < 0.26 [144]. Note also that, Γ SM h = 4.07 MeV, has been taken [143].
Clearly, a more stringent bound on the model parameters comes from the invisible h decay, compared to that of the Z decay, but for sin θ 0.01 the entire parameter space is allowed.
The new fermions f = χ 0 , χ 1 , L 1 , L 2 and the scalars s = η, S can lead to Z → i j decays. Rare charged lepton flavour violating (cLFV) Z decays also inherit a possible complementarity test with low-energy cLFV searches. The current LHC limits put stringent bounds compared to the old limits obtained by the LEP experiments on the three flavor violating decay modes of Z boson. Similarly, future sensitivity can be estimated from [145] which considers the future e + e − colliders CEPC/FCC-ee [146,147] experiments assuming 3 × 10 12 visible Z decays. The present limits and the future bounds can be read as, b) Br(Z → e ± τ ∓ ) ≤ 5 × 10 −6 [149,150] ; 10 −9 [145] c) Br(Z → µ ± τ ∓ ) ≤ 6.5 × 10 −6 [149,150] ; 10 −9 [145] The branching ratio can be expressed as [151,152], where, sin 2θ W = 2 sin θ W cos θ W , F L and F R are defined via Eqs.  Table VI which will subsequently be helpful to obtain a correct relic density for the DM. Substituting these values in Eq. (65), we get the following branching ratios: The first branching fraction is much supressed due to the choice of the Yukawa couplings. Thus, the chances of observing the LFV decays of Z bosons even in the future are not quite attractive.
3) h → ± ∓ : The radiative corrections to Yukawa couplings of SM leptons (y ) can also be generated through the new neutral fermions χ 0 , χ 1 in the loop (see Fig. 13). The new physics contributions at one loop can be calculated as, where, in terms of our definitions of Yukawa couplings, we define Y 0 α = Y 4 and Y 1 α = Y 1 for ∈ e, µ, τ . Similarly, Y 5 has been recast via Eq. (13) with sin θ = 0.01. The corresponding decay width is [143], where, Y h eff = Y SM +Ỹ h . Now, for the same masses and Yukawa couplings as discussed for the flavor violating Z decays (also see Table VI), Γ(h → ee/µµ) has been found to be practically unchanged to the corresponding SM value. 4) Contribution to W ± ∓ ν vertex: The one-loop correction to W → ν process as shown in Fig. 14 FIG. 14. Representative diagram for one-loop correction to the W → ν vertex. results in, where, C 0 , B 0 are the standard PV integrals. M f 1 and M f 2 correspond to the masses of VL leptons f 1 and f 2 respectively, while m stands for the mass of SM lepton. We are assuming the neutrinos to be massless.
Clearly,Ṽ lν will include the desired corrections at one loop to W ± ∓ ν vertex due to presence of the BSM states. However, we find the total contribution to be much suppressed. For having an estimate about the most significant part in it, we consider f 1 = L ± 2 , f 2 = L 0 2 , = µ and s = S. In this case, the general couplings in Eq. (68) can be read as, Y 1 = Y 2 = Y 3µ . We set the masses and couplings in accordance with our previous discussion i.e., M L 2 = 190 GeV, M S = 130 GeV and Y 3µ = 2.3. With these choice of parameters, one can directly get,Ṽ lν ∼ 10 −6 , thus smaller than its tree level values.
As evident from the discussion, in our model, gauge boson-leptonic vertex does not receive any meaningful contribution at all. In fact, both Z ± ∓ and W ± ∓ ν can be considered at their SM values, thus, processes involving leptonic or semileptonic decays of mesons, e.g., K L → µµ, K L → πνν, or B s → µµ, or precisely measured CKM elements can be completely determined by the SM physics.

VI. DARK MATTER PHENOMENOLOGY
This model may offer a singlet-doublet dark matter; phenomenology of such scenarios have been studied in detail [42][43][44][45][46][47][48][49][50]. Here we would simply check that if all the couplings which are already constrained by the different precision and collider bounds, can provide us with an acceptable DM relic density, consistent with SI DM-nucleon elastic cross section bounds. After EWSB, χ 0 -a dominantly singlet-like state, odd under Z 2 × Z 2 symmetry can be considered to be the lightest particle -thus a valid DM candidate while the other neutral state χ 1 carries a strong doublet-like nature for a small mixing angle θ. In general, the singlet-doublet mixing parameter θ is completely controlled by the SI direct detection bounds (much stronger than the EWPO constraints); usually, only a very tiny θ is allowed. We have fixed all other BSM particles (L ± 1 , L ±,0 2 , η, S) at a heavier mass scale, discussed as in our previous exercises. Since a small M χ 0 is preferred from cLFV and ∆a , we may focus on the parameter space with a light DM.
The relic abundance of DM in the universe as obtained from the PLANCK data is Ω DM h 2 = 0.1198 ± 0.0012 [153]. The singlet-like fermionic DM χ 0 , being the lightest odd particle and stable under the imposed Z 2 × Z 2 symmetry, was in thermal equilibrium in the early universe through its interaction with the SM particles. But at a point of time (or temperature: T ≤ T f reeze out ) it gets decoupled from the thermal bath when the interaction rate fell shorter than the expansion rate of the universe. The relic density of the DM can be obtained by solving the Boltzmann equation, given by, where H is the Hubble constant, σ ef f v is the thermal averaged cross section of the DM annihilating to the SM particles and n signifies the number of interacting particles, with the subscript 'eq' designating its equilibrium value. Though, for doing the numerical analysis we have used micrOMEGAs [154,155]. After implementing the model parameters in LanHEP [156], the output files have been used as the input for micrOMEGAs, to solve the Boltzmann equation numerically and for calculating the relic density. Here, the mass parameters have been fixed at the same values as was done in Sec. V, with M η assuming the lower value, i.e. 300 GeV. For the flavor dependent Yukawa couplings, which are restricted by the cLFV and (g − 2) bounds, we choose them at the representative values, shown in Table VI. We also note here that though the choices for Y 4τ or Y 3τ are somewhat different than the values in Fig. 9 and in Fig. 10, we have checked that the cLFV constraints are completely unaffected.
The other meaningful coupling for DM phenomenology is Y 6i (=Y 6(1i) , as in Eq. (14)) )-the interaction between DM, singlet scalar S and the right handed neutrinos ν Ri . The same coupling controls the calculation of neutrino masses [see Sec. IV]. Here we set Y 6 without affecting the neutrino masses and mixings, e.g., Y 6τ = 0.13 is taken.
FIG. 15. The most dominant annihilation channels contributing to the relic density. This is particularly true when the other input parameters are fixed at the values shown in Table VI. In this model, there may be a number of annihilation channels which can contribute to the relic density calculation. The order of dominance of these channels changes with the choice of the other input parameters. Here, Fig. 15 shows the most dominant annihilation channel for the chosen parameter space. We have listed the annihilation channels at M χ 0 = 120 GeV in Table VII.   is given by [157][158][159], where, 'A' and 'Z' represent the mass number and atomic number of the target nucleus respectively, ≈ m N defines the reduced mass, m N being the mass of nucleon (proton or neutron). The second contribution in direct detection comes from the h-mediated diagram and the corresponding SI cross section per nucleon is given as, where the DM-nucleon effective interaction strength can be parameterized as, Where N = n, p and α q = Y 5 sin 2θ T q is the nuclear matrix element as determined in the chiral perturbation theory from the pion-nucleon scattering sigma term, and the gluonic part f (N ) T G is given by, Thus for a fixed M χ 1 , the above equation becomes only a function of M χ 0 (DM mass) and the mixing angle θ. Here we note that, Higgs contribution to SI scattering can be completely evaded if one considers the light-quark Yukawa couplings to assume non-Standard Model (non-SM)-like values [160] 4 . For generating the numerical results we have used the code "micrOMEGAs", as was done for studying the relic density, and analysed the variation of SI scattering cross section as a function of DM mass for sin θ = 0.01. In   Table VI].
Mostly, for the entire parameter space, the σ SI becomes effectively independent of the DM mass, since the Z-mediated scattering process (shown in Fig. 17) appears as the dominant contributor to the total SI cross section over this mass regime. From the observational side we have mainly considered the LUX [52], PandaX-II [162] and XENON 1T [51] limits, which show that the calculated SI cross section, proportional to sin 4 θ, lies much below the present bounds for the entire mass range. However, the future projected limit coming from LZ collaboration [163] may probe only a parts of the parameter space [see Fig. 18]. Further, due to Z-mediation there is a small amount of SD cross section as well, but it is observed to be far below the existing limits. Moreover, note that, the direct detection cross section has no dependence on the Y 1i , Y 3i and Y 4i couplings, which directly govern the (g − 2) and cLFV phenomenology. Therefore, under the variation of different Yukawa couplings (as was done in Fig. 16 (b)), the σ SI remains mostly unchanged.

VII. CONCLUSION
In this paper, we have studied a simple extension where SM is augmented with a pair of vector like lepton doublets L 1 and L 2 , a SU (2) doublet scalar η in particular. Similarly, singlet-like states including a scalar S and a singlet fermion ψ are also considered for specific purposes. An additional Z 2 × Z 2 symmetry has been imposed under which all the SM fields are even while the new fields may be odd under the transformation. Adopting a bottom-up approach, in this paper, we systematically scrutinize the parameter space in terms of the allowed couplings and masses to obtain: (i) the Dirac masses for the SM neutrinos and mixings through a radiative mechanism, (ii) electron and muon (g − 2) discrepancy simultaneously while considering the cLFV and EWPO constraints and finally (iii) a viable DM candidate, consistent with direct detection observations so far.
We start with our proposed model where the new interactions have been introduced. Subsequently we discuss about the relevant constraints on the new parameters by reviewing the different experimental constraints related to the lepton (g − 2) observations, cLFV bounds, vacuum stability conditions, electroweak precision constraints and collider observables. In our model, L 1 and ψ may mix to produce the physical states, and the lightest state χ 0 can be regarded as the dark matter. Electroweak precision parameters and, more importantly, the null results from the dark matter direct detection experiments require a small mixing between L 1 and ψ; thus we choose sin θ = 0.01.
We have shown that in the absence of a tree-level neutrino mass (being forbidden due to the imposed symmetry), one can generate the correct neutrino mass matrix at one-loop level if the Z 2 is allowed to break softly. The masses and mixings may be controlled by two free parameters Y 2(1i) and Y 6(1i) which do not have any effect on the charged lepton flavor processes, e.g., (g − 2) µ/e or different cLFV processes like α → β γ and α → 3 β . We have performed a comprehensive study to show the interplay between different charged and neutral vector like leptons for satisfying (g − 2) e and (g − 2) µ bounds simultaneously.
A moderately large coupling Y 3µ is required to tune (g − 2) µ while ∆a e can easily be controlled with other O(1) couplings. Further, the same diagrams are able to generate α → β γ processes when α = β.  In this appendix we list all the Feynman rules required for our calculation. These rules have been expressed in physical eigen basis for particles: Neutral scalar s X = (η 0 , S), charged scalar η ± , neutral VL fermions χ a (a = 1, 0) and charged VL fermions L − a (a = 1, 2).

Scalar interactions
The Feynman rules for scalar interactions are given by,

Z boson interactions
The Feynman rules governing the Z boson interactions are given by, = − g cos θ W sin 2 θ W are the left and right chiral couplings among two SM leptons and Z boson respectively, g being the SU (2) L coupling constant.

APPENDIX C
In this section, we present the general and explicit results for the on-shell and off-shell decays of the charged leptons.
A. α → β γ The on-shell amplitude, mediated by the dipole operators, can be expressed as, Here e is the electric charge, q is the photon momentum, P L,R = 1 2 (1 ∓ γ 5 ) are the usual chirality projectors and the lepton spinors are denoted by u α,β , where α, β stand for the flavor indices. The coefficients in The amplitude for such a process like − α (p) → − β (p 1 ) − β (p 2 ) + β (p 3 ) can be decomposed into three major contributions given by, In general there should be a contribution from Higgs penguin diagrams (i.e. M H ) as well, but one can neglect it in most cases, in comparison to the other three contributions of Eq. (85). Different contributions can be expressed as follows: • Photon penguin contribution: The monopole and dipole contributions can be calculated from, M γ =ū β (p 1 ) q 2 γ µ (A L 1 P L + A R 1 P R ) + im α σ µν q ν (A L 2 P L + A R 2 P R ) u α (p) × e 2 q 2ū β (p 2 )γ µ v β (p 3 ) − (p 1 ↔ p 2 ).
The explicit form of the Wilson coefficients A L 2 and A R 2 are already described in Eqs. (81)− (84). The coefficients associated with the monopole operator can be calculated as, • Z penguin contribution: Feynman diagrams are shown in Fig. 7. We have calculated the coefficients as follows: Zū β (p 1 ) γ µ (F L P L + F R P R ) u α (p)ū β (p 2 ) γ µ (g ( ) where, as before, F L,R = F L,R . The expressions for these form factors are given below: • Box diagram contribution: Leading contributions are shown in Fig. 8. can be calculated as, where, And for the charged fermions (L ± 1 , L ± 2 ), where, with M 1 = M η and M 2 = M S . The generic functional forms for these D 0 andD 0 are again available in Appendix B.
The decay width for − α → − β − β + β can be obtained by considering all the possible contributions coming from photon and Z penguins in addition to the box diagrams and can be expressed as [135,136], where, The corresponding branching ratio can be directly calculated as Br , τ α being the lifetime of α .