Fermion mass hierarchies from vector-like families with an extended 2HDM and a possible explanation for the electron and muon anomalous magnetic moments

We study an extended 2 Higgs doublet model (2HDM) in which the Standard Model (SM) Yukawa interactions are forbidden due to a global $U(1)^\prime$ symmetry, but may arise via mixing with vector-like families. In this model, the hierarchical structure of Yukawa couplings of quarks and leptons in the SM arises from the heavy masses of the fourth and fifth vector-like families. Within this model, we consider various non-standard contributions to the electron and muon anomalous magnetic moments. We first consider the $W$ exchange at one-loop level, consistent with the $\mu \rightarrow e \gamma$ constraint, and show that it yields a negligible contribution to both electron and muon anomalous magnetic moments. We then consider Higgs scalar exchange, together with vector-like leptons, at one-loop level and show that it is possible to have non-standard contributions to the electron and muon anomalous magnetic moments within the $1\sigma$ constraint of certain experiments. We present some benchmark points for both the muon and the electron anomalies, together with some numerical scans around these points, which indicate the mass regions of the Higgs scalars of the 2HDM in this scenario.


I. INTRODUCTION
The Standard Model (SM) has made many successful predictions for the phenomenology of both quark and lepton sectors with very high accuracy. However there are long-established anomalies which are not addressed by the SM such as muon and electron anomalous magnetic moments a µ = (g − 2) µ /2, a e = (g − 2) e /2. The muon anomalous magnetic moment reported by the Brookhaven E821 experiment at BNL [1] and the electron anomaly have confirmed +3.5σ and −2.5σ deviations from the SM, respectively. Detailed data analysis of the Standard Model predictions for the muon anomalous magnetic moment are provided in [2][3][4][5][6]. The experimentally observed values for the muon and electron anomalies at 1σ of experimental error bars, respectively, read 1 : (1) When trying to explain both anomalies to within 1σ, a main difficulty arises from the sign of each anomaly: the muon anomaly requires positive definite non-standard contributions, whereas the electron anomaly requires such contributions to contribute with a negative sign [7]. Without loss of generality, the Feynman diagrams corresponding to the contributions for the muon and electron anomalies take the same internal structure at one-loop except from the fact that the external particles are different. The similar structure of the one-loop level contributions to the muon and electron anomalous magnetic moments might be able to be explained by the same new physics, but accounting for the relative negative sign is challenging. For example, considering the one-loop exchange of W or Z gauge bosons results in theoretical predictions for the muon and electron anomalies having the same sign.
In this paper we take the view that both anomalies should be explained to 1σ using the same internal structure at the one-loop level by some new physics which is capable of accounting for the correct signs of the anomalies. To explain the muon and electron anomalies, we focus on a well motivated model which is also capable of accounting for origin of Yukawa couplings and hierarchies in the SM. The model we consider will account for the Yukawa coupling constant for the top quark being nearly 1 while that for the electron is around 10 −6 , as well as all the other fermion hierarchies in between, as well as the neutrino masses and mixing. In order to achieve this we shall introduce vector-like particles, which are charged under a global U (1) symmetry. In a related previous work [9], with a gauged U (1) symmetry, the first family of quarks and leptons remained massless when only one vector-like family is included. Here we shall modify the model to include two vector-like families charged under a global U (1) to allow also the first family to be massive and avoid Z constraints. Then we shall apply the resulting model to the problem of muon and electron anomalous magnetic moments. The considered model is based on a 2 Higgs doublet model (2HDM) extension of the SM, supplemented by a global U (1) symmetry, where the particle spectrum is enlarged by the inclusion of two vector-like fermion families, as well as one singlet Higgs to break the U (1) symmetry 2 . The SM Yukawa interactions are forbidden, but the Yukawa interactions with vector-like families charged under the U (1) symmetry are allowed. Once the flavon develops a vev and the heavy vector-like fermions are integrated out, the effective SM Yukawa interactions are generated, as indicated in Figure 1. Furthermore, this model also highlights the shape of the 2HDM model type II, since in our proposed model, one Higgs doublet (which in the alignment limit corresponds to the SM Higgs doublet) couples with the up type quarks whereas the other one features Yukawa interactions with down type quarks and SM charged leptons.
Regarding the neutrino sector, since we consider the SM neutrinos as Majorana particles, we have that this sector requires another approach relying on the inclusion of a new five dimensional Weinberg-like operator, which is allowed in this model and which requires both SM Higgs doublets to be present, namely the so called Type Ib seesaw model [11].
We shall show that the heavy vector-like leptons are useful and necessary to explain the anomalous electron and muon magnetic moment deviations from the SM, of magnitude and opposite signs given in Equation 1. A study of such g − 2 anomalies in terms of New Physics and a possible UV complete explanation via vector-like leptons was performed in [12], although the model presented here is quite different, since our model is motivated by the requirement of accounting also for the fermion mass hierarchies. Other theories with extended symmetries and particle spectrum have also been proposed to find an explanation for the muon and electron anomalous magnetic moments . In the following we provide a brief comparison of our model to other works, starting with the model proposed in [49] where vector-like leptons are also present. The model of [49] corresponds to an extended type X lepton specific 2HDM model of [49] having a Z 2 discrete symmetry under which one of the scalar doublets and the leptonic fields are charged. In such model the vector-like leptons induces a one-loop level contribution to the electron anomalous magnetic moment whereas the muon anomalous magnetic moment is generated at two-loop via the exchange of a light pseudoscalar. On the other hand, in our proposed model a spontaneously broken global U (1) symmetry is considered instead of the Z 2 symmetry and the vector-like leptons generate one-loop level contributions to the muon and electron anomalous magnetic moments and at the same type produce the SM charged lepton masses, thus providing a connection of the charged lepton mass generation mechanism and the g − 2 anomalies, which is not given in the model of [49]. It is also worth emphasising that our model is very different from other models proposed in the literature based on the Universal Seesaw mechanism [59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76]. Universal Seesaw models are typically based on the left-right symmetric model with electroweak singlet fermions only, while our vector-like fermions involves complete families, including electroweak doublets which are typically the lightest ones. Some examples of theories relying on the Universal Seesaw mechanism to explain the SM charged fermion mass hierarchy are provided in [59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76].
In the approach followed in this paper the large third family quark and lepton Yukawa couplings are effectively generated via mixing with a vector-like fourth family of electroweak doublet fermions, which are assumed to be relatively light, with masses around the TeV scale. The smallness of the second family quark and lepton Yukawa couplings is due to their coupling to heavier vector-like fourth family electroweak singlet fermions. Similar considerations apply to the lightest first family quarks and leptons which couple to heavy fifth family vector-like fermions. It may seem that the problem of the hierarchies of SM fermions is not solved but simply reparameterised in terms of unknown vector-like fermion masses. However, there are four advantages to this approach. Firstly, the approach is dynamical, since the vector-like masses are new physical quantities which could in principle be determined by a future theory. Secondly, it has experimental consequences, since the new vector-like fermions can be discovered either directly, or (as in this paper) indirectly via their loop contributions. Thirdly, this approach can also account for small quark mixing angles [9], as well as large lepton mixing angles via the type Ib seesaw mechanism [11]. Fourthly, the effective Yukawa couplings are proportional to a product of two other dimensionless couplings, so a small hierarchy in those couplings can give a quadratically larger hierarchy in the effective couplings. For all these reasons, the approach we follow in this paper is both well motivated and interesting.
Returning to our proposed model framework, we first consider the contribution of W boson exchange with neutrinos to the electron and muon anomalous magnetic moments at the one-loop level. Since this model involves the vector-like neutrinos, the sensitivity of the branching ratio of µ → eγ decay can be enhanced with respect to the observable level and the muon and electron anomalous magnetic moments are studied while keeping the µ → eγ constraint. As a result, we find that the impact of our predictions with W exchange at one-loop level is negligible when compared to their experimental bound. We then consider the contributions from the 2HDM scalar exchange. To study the implications of the one-loop level scalar exchange in the muon and electron anomalous magnetic moments, we first construct a scalar potential and derive the mass squared matrix for CP-even, CP-odd and charged Higgses assuming there is no mixing between the SM Higgs h and two non-SM physical scalars H 1,2 . A diagonal Yukawa matrix for charged leptons implies the absence of mixing between charged leptons, resulting in vanishing branching ratio for the µ → eγ decay, which in turn leads to a fulfillment of the charged lepton flavor violating constraints in this scenario.
In such a framework we show that both anomalies can successfully explain both anomalies, including their opposite signs, at the 1σ level. We present some benchmark points for both the muon and the electron anomalies, together with some numerical scans around these points, which indicate the mass regions of the Higgs scalars of the 2HDM in this scenario. We also provide some analytic arguments to augment the numerical results. The layout of the remainder of the paper is as follows. In Section II we discuss the origin of Yukawa couplings from a fourth and fifth vector-like family, within a mass insertion formalism. In Section III we construct the effective Yukawa matrices using a more detailed mixing formalism which goes beyond the mass insertion formalism. In Section IV we consider W exchange contributions to (g − 2) µ , (g − 2) e and BR (µ → eγ) based on the type Ib seesaw mechanism within our model and show that the contributions are too small. In Section V we turn to Higgs scalar exchange contributions to (g − 2) µ , (g − 2) e and BR (µ → eγ), focussing on analytical formulae. Then in Section VI we give a full numerical analysis of such contributions, showing that they can successfully explain the anomalies, presenting some benchmark points for both the muon and the electron anomalies, together with some numerical scans around these points, which indicate the mass regions of the Higgs scalars of the 2HDM in this scenario. Section VII concludes the main body of the paper. Appendix A provides a discussion of the quark mass matrices in two bases. Appendix B includes a brief discussion of heavy scalar production at a proton-proton collider.

II. THE ORIGIN OF YUKAWA COUPLINGS FROM A FOURTH AND FIFTH VECTOR-LIKE FAMILY
We start by asking a question: what is the origin of the SM Yukawa couplings? In addressing such question, we assume that the SM Yukawa Lagrangian is the low energy limit of an extended theory with enlarged symmetry and particle spectrum, and arises after the spontanous breaking of an U (1) global symmetry at an energy scale as low as TeV. Therefore, understanding the origin of the Yukawa interaction naturally leads to the presence of another Higgses whose masses are higher than the mass of the SM Higgs. Furthermore, the SM Yukawa interactions are forbidden by the global U (1) symmetry, however the Yukawa interaction with the vector-like particles are allowed. With these considerations in place, the possible diagrams generating the Yukawa interactions can be drawn as indicated in Figure  1.
There are two key features in Figure 1, one of which is the presence of the assumed flavon φ and the other one is the vector-like mass M . Once the flavon φ develops its vev, the effective Yukawa interactions ψ iL Hψ jR are generated with a coupling constant proportional to φ /M , assumed to be less than unity, which appears in front of the usual Yukawa constant. The proportional factor φ /M plays a crucial role in explaining why one Yukawa constant can be relatively smaller or bigger than the other ones since the magnitude of each Yukawa constant is accompanied by the mass of the vector-like particles. The effective Lagrangian in this diagram reads in the mass insertion formalism: where ψ, ψ = Q, u, d, L, e (neutrinos will be treated separately) and x is a Yukawa constant in the interaction with φ and y is in the interaction with H as per Figure 1. Throughout this work, we take a view that the Yukawa constant y can be ideally of order unity while the x is small compared to the y. We shall also use a mixing formalism rather than the mass insertion formalism. For an analysis of the phenomenology described above, we extend the SM fermion sector by adding two vector-like fermions, the SM gauge symmetry by including the global U (1) symmetry and the scalar sector of the 2HDM model is enlarged by considering a gauge scalar singlet, whose VEV triggers the spontaneous breaking of the U (1) symmetry. The scalar sector of the model is composed of by two SU (2) doublet scalars H u,d and one flavon φ. Our extended 2HDM with enlarged particle spectrum and symmetries has the interesting feature that the SM Yukawa interactions are forbidden due to the global U (1) symmetry whereas the Yukawa interactions of SM fermions with vector-like families are allowed. Furthermore, such vector-like families have mass terms which are allowed by the symmetry. Thus, the SM charged fermions masses are generated from a Universal Seesaw mechanism mediated by heavy vectorlike fermions. Unlike the U (1) model proposed in [77], we assume that the U (1) symmetry is global instead of local. This allows us more flexibility in the allowed range for the scale where the U (1) symmetry is broken. On top of that, the up-type quarks feature Yukawa interaction with the up-type Higgs whereas the down-type ones interact with down-type Higgs. In this BSM model, the SM particles are neutral under the U (1) symmetry, while the vector-like particles and all other scalars are charged under the symmetry. The particle content and symmetries of the model are shown in Table I Figure 2: Diagrams in this model which lead to the effective Yukawa interactions for the up quark sector(two above diagrams) and the down quark sector(two below diagrams) in mass insertion formalism, where i, j = 1, 2, 3 and k, l = 4, 5 and M lk is vector-like mass.

B. mass insertion formalism
The renormalizable Yukawa interactions and mass terms for both up and down quark sectors read: where i, j = 1, 2, 3, k, l = 4, 5 and H = iσ 2 H * . The possible diagrams contributing to the low energy quark Yukawa interaction are given in Figure 2: The above two diagrams correspond to the up-type quark sector whereas the below two diagrams correespond to the down-type quark sector. The model under consideration is an extended 2HDM where the up-type Higgs H u is relevant for the up-type quark sector whereas thedown-type Higgs H d is suitable for the down-type quark and charged lepton sectors. Like in the quark sector, the Yukawa interactions and mass terms for charged leptons can be written in a similar way: Then, the possible diagrams giving rise to the charged lepton Yukawa interactions are shown in Figure 3: As for the neutrinos, its behaviour is different as compared to the quarks or charged leptons since there exists only Majorana neutrinos in this model so initial and final neutrinos in mass insertion formalism diagrams must be same. The Yukawa interactions and mass terms for the neutrino sector are given by: Here, one important feature in Equation 5 is the presence of the vector-like mass M . From the two Yukawa interactions in Equation 5, it follows that both ν R and ν R have a lepton number +1 and they are different particles. And then taking a look at the vector-like mass term in Equation 5, it can be confirmed that the vector-like mass is not a strict Majorana mass because ν R and ν R are different particles but plays a role of Majorana mass since the mass term  Figure 4: Type Ib seesaw diagram [11] which leads to the effective Yukawa interactions for the Majorana neutrinos in mass insertion formalism, where i, j = 1, 2, 3 and k, l = 4, 5 and M lk is vector-like mass.
violates the lepton number conservation. The corresponding diagram for the neutrino sector in the mass insertion formalism is given in Figure 4. However for our calculations we use a mixing formalism (see next section).
The operator L i L j H u H d resulting from Figure 4 gives rise to the so called type Ib seesaw mechanism [11] which differs from the usual type Ia seesaw mechanism corresponding to the Weinberg operator L i L j H u H u and will be discussed later in detail.

III. EFFECTIVE YUKAWA MATRICES USING A MIXING FORMALISM
As seen from Equation 2, we need to mix Higgses with the flavon to generate the effective Yukawa Lagrangian required to produce the SM fermion mass hierarchy. Since there is no an extra symmetry or constraint to keep the mixing between Higgses and flavon from taking place, it is natural to assume their mixing.
A. The 7 × 7 matrix Consider the 7 × 7 mass matrix for Dirac fermions: with the coefficients y and x being Yukawa constants where the former is expected to be of order unity whereas the latter is smaller than y. Furthermore, the 125 GeV SM like Higgs boson H will corresponds to the lightest of the CP even neutral scalar states arising from H u , H d and φ, whereas M is the vector-like mass. The column vector located at the lower left block in Equation 6 consists of left-handed particles while the row vector at the upper right block are made up of right-handed particles. The zeros in the 3 × 3 upper block in Equation 6 mean that no SM Yukawa interactions take place due to charge conservation as well as zeros in two 2 × 2 blocks. Since we are interested in explaining the muon and electron anomalous magnetic moments in this model, we first focus on the lepton sector in the next subsection and the method used for obtaining the low energy SM Yukawa matrices in the lepton sector can be applied to the quark sector in the same way with a slight change so that the quark sector will be discussed in Appendix A.

B. A convenient basis for charged leptons
From Equation 6, we can take a specified basis by rotating some fields as below: where v d = H 0 d and ν φ = φ . We start by pointing out the reason why we take this specific basis for the charged leptons. The reason is that the strong hierarchical structure of the SM fermion Yukawa couplings can be implemented by the rotations with a simple assumption in this model to be specified below. In order to arrive from Equation 6 to Equation 7, we rotate the leptonic fields L 4L and L 5L to turn off M L 54 and rotate e 4R and e 5R to turn off M e 45 . Then, we can rotate L 1L and L 3L to set x L 14 v φ to zero and then rotate L 2L and L 3L to set x L 24 v φ to zero. The same rotation can be applied to e 1R,2R,3R to set y e 41,42 v d to zero. Finally, we can further rotate L 1L and L 2L to switch off y e 14 v d and this rotation also goes for e 1R,2R to switch off x e 41 v φ . The above given mass matrix includes three distinct mass scales which are the vev v d of the neutral component of the Higgs doublet H d , the vev v φ of the flavon φ and the vector-like masses M , whose orders of magnitude can be in principle be different. Therefore, the mass matrix will be diagonalized by the seesaw mechanism step-by-step instead of diagonalising it at once. This mechanism is also known as Universal Seesaw, and was proposed for the first time, in the context of a left-right symmetric model in [59].

C. A basis for decoupling heavy fourth and fifth vector-like family
As mentioned in the previous section III B, the mass matrix in Equation 7 involves three distinct mass scales v d , v φ and M so it is possible to split this whole mass matrix by partial blocks to group mass terms with vev of H d as in Equation 8 and then elements of the blocks involving φ can be rotated away to make those zeros by unitary mixing matrices of Equation 10 as per Equation 9: where the indices α, β run from 1 to 5, and tilde, primes repeated in the mass matrix mean that the parameters are rotated. The unitary 5 × 5 matrices are defined to be where each of the unitary matrices V i4,5 are parameterized by a single angle θ i4,5 describing the mixing between the ith chiral family and the 4, 5th vector-like family. The 5 × 5 Yukawa constant matrix in a mass basis (primed) can be diagonalized by the unitary rotation matrices as below: From Equation 7, we can read off the 5 × 5 upper block and confirm that the (3,4), (1,5), (2,5), (3,5) mixings in the L sector and (2, 4), (3,4), (1,5), (2,5), (3,5) mixings in the e sector are required to go to the decoupling basis. The unitary matrices of Equation 10 and mixing angles appearing in the unitary matrices are parameterized by Given the above unitary rotations, the 5 × 5 Yukawa matrices are computed in terms of the mixing angles and the upper 3 × 3 block would be the effective SM Yukawa matrix. Assuming all cos θ to be 1 and neglecting order of θ square or more than that, we have a simple 3 × 3 Yukawa matrix of Equation 14.

D. A convenient basis for neutrinos
The relevant Yukawa and mass terms of the neutrino sector give rise to the following neutrino mass matrix: Here, the zeros in the upper 3 × 3 block of Equation 15 mean that neutrinos remain massless in the SM. Therefore, the SM neutrinos can be massive via the inclusion of two vector-like families. In order to make this mass matrix as simple as possible, the only choice left is to rotate ν 4R and ν 5R to turn off M ν 45 since rotations between L 1L,2L,3L are already used in the charged lepton sector.
Within the framework of our proposed model, we start by investigating the muon and electron anomalous magnetic moments with W boson exchange first. Given that such W boson exchange contribution also involves virtual neutrinos The left is the Weinberg operator(or type 1a seesaw mechanism) in which mass M is Majorana mass and the right is Weinberg-like operator(or type 1b seesaw mechanism) in which mass M is vector-like mass.
in the internal lines of the loop, we revisit the mass matrix for neutrinos. In this mass matrix, we remove fifth vectorlike neutrinos ν 5R and ν 5R since they are too heavy to contribute to the phenomenology under study. As mentioned in the previous section, we stick to a condition where the coefficient y is expected to be of order unity, whereas the coupling x is expected to be smaller than y. Such condition can be easily seen by substituting the coefficients y ν i4 by y ν i and the coefficients x L i4 by y ν i where is a suppression factor. Putting all these considerations together, the mass matrix for neutrinos in Equation 15 after electroweak symmetry breaking takes the form: where A. Type 1b seesaw mechanism Now that we constructed the neutrino mass matrix for this task, the next step is to read off the operator which gives rise to the neutrino mass from the mass matrix. Generally, the well-known operator for neutrino mass is the Weinberg operator(type 1a seesaw mechanism) 1 Λ L i L j HH. A main feature of the Weinberg operator is the same SM Higgs should be repeated in the operator, however that property is not present in our model since the Higgs doublets H u,d are negatively charged under the U (1) symmetry, which implies the corresponding Weinberg operator having such fields will not be invariant under the U (1) unless an insertion of a quadratic power of the gauge singlet scalar φ is considered. However we do not consider the operators 1 in the neutrino sector, since they are very subleading and thus will give a tiny contribution to the light active neutrino masses. Instead of relying on a seven dimensional Weinberg to generate the tiny masses for the light active neutrinos, we take another approach named type 1b seesaw mechanism (we call the Weinberg operator "type 1a seesaw mechanism" to differentiate with) where the mixing of different SU (2) Higgs doublets can appear satisfying charge conservation. Diagrams for the operators are given in Figure 5 for comparison: The diagrams in Figure 5 clearly tell the difference between Majorana mass and vector-like mass. They share a common property that they violate the lepton number conservation, whereas the particles appearing in a Majorana mass term are same but those ones involved in vector-like mass terms are different. As the type 1b seesaw mechanism only works in this model, we make use of this seesaw mechanism for the analysis of neutrinos. With the operator, the renormalizable Lagrangian for neutrinos can be written as: where i = 1, 2, 3 and k = 4. The renormalizable Lagrangian of Equation 17 above the electroweak scale generates an effective Lagrangian after decoupling the heavy vector-like neutrinos, which is suitable for study of low energy neutrino phenomenology. The effective Lagrangian for neutrino at electroweak scale is given by [11] where the coefficient c d=5 ij is suppressed by a factor of the vector-like mass M . The neutrino mass matrix of Equation 16 can be diagonalized by the unitary matrix U as below: where m diag ν is a diagonal matrix for the light left-handed neutrinos ν iL and M diag N is that for the heavy vector-like neutrinos ν 4R , ν 4R . Here, the unitary mixing matrix U is defined by multiplication of two unitary matrices which we call U A and U B , respectively [78]: The unitary matrix U PMNS in U B is the well-known Pontecorvo-Maki-Nakagawa-Sakata matrix and is parameterized by [11,79] U where the Majorana phase α is set to zero in this model. The mixing matrices U A,B are unitary, however the 3 × 3 upper block of the unitary matrix U is not unitary due to the factor I − ΘΘ † /2 for the light neutrinos. An interesting feature of the unitary matrix U is it is unitary globally, but non-unitary locally and this non-unitarity contributes to explain muon and electron anomalous magnetic moments. Replacing the unitary matrices in Equation 20 back to Equation 19, the result is simplified with the assumption M N m D to the conventional seesaw mechanism: where m is the effective mass matrix resulted from Equation 16.
Therefore, smallness of the light neutrino masses can be understood not only from mass of vector-like mass M ν 44 but also from the suppression factor and the presence of allows more flexibility in the allowed mass values of the vector-like neutrinos. Revisiting non-unitarity part for the light neutrinos from the unitary matrix U [78,80], it reads: The non-unitarity η is associated with the presence of the heavy vector-like neutrinos and can be derived from a coefficient of the effective Lagrangian at dimension 6 [81]: Once the SM Higgs doublets in Equation 25 develop its vev, the Lagrangian at dimension 6 causes non-diagonal kinetic terms for the light neutrinos and it gives rise to deviations of unitarity when it is diagonalized. The deviations of unitarity can be expressed in terms of the coefficient at From the fourth term in Equation 26, the term with 2 can be safely ignored due to both relative smallness of v d and the suppression factor . Thus, the deviation of unitarity η consists of the vector-like mass M ν 44 and the Yukawa couplings y ν i,j . As an interesting example, it is possible that the Yukawa couplings y ν i,j can be obtained from the observables such as the PMNS mixing matrix and two mass squared splitting, ∆m 2 sol and ∆m 2 atm , in the neutrino oscillation experiments. Since the hierarchy between the light neutrinos is not yet determined, there are two possible scenarios, normal hierarchy(NH) and inverted hierarchy(IH), and the lightest neutrino remains massless, whereas two other neutrinos get massive. The Yukawa couplings y ν,ν i for the NH(m 1 = 0) are determined by where y and y are real numbers and ρ = (1 − √ r)/(1 + √ r) with r ≡ |∆m 2 sol |/|∆m 2 atm | = ∆m 2 21 /∆m 2 31 , whereas the Yukawa couplings y ν,ν i for the IN(m 3 = 0) are B. The charged lepton flavour violation(CLFV) µ → eγ decay Consider the three light neutrinos in the SM for the CLFV µ → eγ decay first. In this case, the unitary mixing matrix becomes just the PMNS mixing matrix and the GIM mechanism which suppresses flavour-changing process works, therefore it leads quite suppressed sensitivity for BR (µ → eγ) about 10 −55 [82], which is impossible to observe with the current sensitivity of µ → eγ decay. This impractical sensitivity can be enhanced to the observable level by introducing the heavy vector-like neutrinos which give rise to deviation of unitarity. With the presence of heavy vector-like neutrinos, the GIM mechanism is gone and the factor suppressed by GIM mechanism can survive with a factor of deviation of unitarity, which plays a crucial role to increase significantly order of theoretical prediction for µ → eγ decay [83]. Therefore, the strongest constraint for deviation of unitarity in the modified PMNS mixing matrix comes from CLFV µ → eγ decay. The possible one-loop diagrams for the CLFV µ → eγ with all neutrinos in this model are given in Figure 6. The amplitude from above diagrams in Figure 6 reads [82]: Figure 6: Diagrams for CLFV µ → eγ decay with all neutrinos. Here n = 1, 2, 3, 4, 5.
where u is Dirac spinor for the muon and electron, q is four momentum of an outgoing photon, F 1,2 are form factors, A L,R are left-and right-handed amplitude defined to be A L,R = F 1 ± F 2 and lastly P L,R are projection operators. From the amplitude, the helicity flip between initial particle and final particle should arise and this makes the helicity flip process takes place on one of external legs since the W gauge boson couples only to left-handed fields. Comparing the left diagram with the right, the left is proportional to the muon mass, while the right is proportional to the electron mass, which means that impact of the right is ignorable. The unpolarized squared amplitude |M| 2 takes the form: Then, the decay rate is given by where A R is expressed by [82,84] Taking the unitary gauge into account, ξ → ∞, the additional ξ-dependent terms in A R all are cancelled by contribution of Goldstone bosons so A R is gauge invariant. Substituting the gauge invariant A R back into the decay rate of Equation 31 and dividing the expanded decay rate by the total muon decay rate Γ (µ → eνν) = G 2 F m 5 µ /192π 3 , we have the prediction for µ → eγ decay [11,82]: Figure 7: Diagrams for muon anomalous magnetic moment with all neutrinos. Here n = 1, 2, 3, 4, 5.
Numerator in Equation 33 can be simplified by separating the light neutrinos and heavy vector-like neutrinos as below(Contribution of the fifth neutrino ν 4R is safely ignored both by the suppression factor and by relative smallness of v d compared to v u ): The final form for the CLFV µ → eγ decay in this model reads: where α em is the fine structure constant. We find that our theoretical prediction for the µ → eγ decay can be expressed in terms of the deviation of unitarity η 21 .
C. The anomalous muon magnetic moment g − 2 We derive our prediction for the muon anomalous magnetic moment in this section and confirm the derived expression can be consistent with an expression of the theoretical prediction for µ → eγ in references [82,85]. Consider two possible diagrams for muon anomalous magnetic moment at one-loop level in Figure 7.
The amplitude for the muon anomalous magnetic moment at one-loop level is: Unlike the CLFV µ → eγ decay, muon anomaly diagrams have the same structure for helicity flip process. So we conclude A R is equal to A L and can make use of other expression of this amplitude to derive our own expression for ∆a µ [85].
Comparing Equation 37 with Equation 38, we confirm that Here, we can use the condition that A R = A L identified in Figure 7 and can rearrange A L,R in terms of A M,E µµ , which are essential to derive our theoretical muon anomaly prediction. Then, we find our desirable form A M,E µµ for the muon anomalous magnetic moment.
Using the definition for both the muon anomalous magnetic moment and branching ratio of µ → eγ decay in [85], we can check our analytic argument for the observable and constraint are correct.
One difference between A M,E µµ and A L,R is thatA M,E µµ is only determined by the internal structure of the loop in Figure  7, whereas A L,R is the extended factor by multiplying A M,E µµ by the helicity flip mass in one of the external legs. Therefore, it is natural to think A M,E µµ is the same as A M,E µe since their internal structure of loop are exactly same 4 . The muon anomalous magnetic moment and the branching ratio of µ → eγ take the form: where the α W is the weak coupling constant. As for the branching ratio of µ → eγ in Equation 42, we showed that substituting A µe back into the branching ratio in Equation 41 is exactly consistent with the one in Equation 33.
Expanding the unitary mixing matrices in the muon anomaly prediction in Equation 42, yields the following relation: Looking at Equation 43, it is clear that the SM part which is without η and the BSM having η are entangled together. We arrive at the right prediction for the muon anomaly at one-loop by removing the SM part from Equation 43 Similarly to the branching ratio of µ → eγ decay, it can be confirmed that the prediction for the muon anomaly also consists of the factor of deviation of unitarity η. D. The anomalous electron magnetic moment g − 2 As in the muon anomalous magnetic moment, the same diagrams with external particles replaced by electrons can be generated in Figure 8.
Using the complete form of the muon anomaly prediction in Equation 44, we can derive the right prediction for the electron anomalous magnetic moment with slight modifications m µ → m e , η 22 → η 11 .

E. Numerical analysis of W exchange contributions
The presence of heavy vector-like neutrinos leads to the deviation of unitarity and the observables ∆a µ,e and constraint BR (µ → eγ) can be written in terms of the factor of non-unitarity η.
1. The branching ratio of µ → eγ decay We consider the branching ratio of µ → eγ decay first. Since we assume that mass of heavy vector-like neutrinos are heavier than 1 TeV, the value of F (0) for the light neutrinos converges to approximately 3.3, while that of F (x 4 ) for the heavy vector-like neutrino converges to 1.3. Therefore, the branching ratio of µ → eγ decay can be reduced to [11] BR (µ → eγ) = 3α em 8π The non-unitarity η of Equation 26 consists of four free parameters: mass of heavy vector-like neutrinos M ν 44 , a real number y, a CP violation phase δ, and a Majorana phase α. The experimental branching ratio of µ → eγ decay constrains the minimal parameter space in terms of M ν 44 and y, while setting up two phases δ, α which maximize or minimize the branching ratio of µ → eγ [11], and the minimal parameter space is shown in Figure 9.
There are two interesting features in the above prediction for the muon and electron anomalous magnetic moments.
One feature is sign of each prediction. As mentioned in the introduction, this prediction with the W exchange can not flip the sign of each anomaly. In order to explain both anomalies at 1σ, the prediction for both anomalies with W exchange requires additional contributions such as Z or scalar exchange. Another feature is magnitude of each prediction. For the muon anomaly, the experimental order of magnitude at 1σ is about 10 −9 , however our prediction is much smaller than that of the experimental bound as well as the electron anomaly, which means the non-unitarity derived from the presence of heavy vector-like neutrino can not bring the anomalies to the observable level. This inadequate prediction with W exchange has been a good motivation to search for another possibility such as scalar exchange.
V. HIGGS EXCHANGE CONTRIBUTIONS TO (g − 2) µ , (g − 2) e AND BR (µ → eγ) The relevant sector for the muon and electron anomalous magnetic moments with scalar exchange is the charged lepton Yukawa matrix which can be expressed in the mass insertion formalism as, The effective Yukawa matrix of Equation 51 in the mass basis is diagonalized by the universal seesaw mechanism due to involving a few of different mass scales. Therefore, the only diagonal components should alive in the mass matrix.
In order to make the mass matrix diagonal, we assume that  x L 34 φ /M L 44 and the diagonal elements from top-left to bottomright should be responsible for electron, muon and tau Yukawa constants, respectively. After removing all irrelevant terms to both anomalies and applying the assumption, the 7 × 7 mass matrix in the interaction basis is also reduced to as below: The reduced charged lepton mass matrix of Equation 53 clearly tells that no mixing between charged leptons arise so the branching ratio of µ → eγ is naturally satisfied under this scenario. The scalar exchange for both anomalies can be realized by closing the Higgs sectors in Figure 3 as per Figure 10. In Figure 10, the CP-even non-SM scalars H 1,2 and CP-odd scalars A 1,2 appear as a result of mixing between Higgses H u , H d and φ in the interaction basis. The Higgs sector in the interaction basis is defined by For consistency, we equate v u , v d and v φ to v 1 , v 2 and v 3 , respectively. The scalar potential of the model under consideation takes the form: where the λ i (i = 1, 2, · · · , 8) are dimensionless parameters whereas the µ j (j = 1, 2, 3) are dimensionful parameters and µ sb is a dimensionfull soft-breaking parameter. We consider the U (1) symmetry as global in this model so our model does not feature Z boson and the scalar potential requires the inclusion of the soft-breaking mass term −µ 2 sb φ 2 + (φ * ) 2 in order to prevent the appearance of a massless scalar state arising from the imaginary part of φ.
The minimization conditions of the scalar potential yield the following relations: B. Mass matrix for CP-even, CP-odd neutral and charged scalars The squared mass matrix for the CP-even scalars in the basis Re H 0 u , Re H 0 d , Re φ takes the form: From the mass matrix given above, we find that the CP-even scalar spectrum is composed of the 125 GeV SM-like Higgs h and two non-SM CP-even Higgses H 1,2 . Furthermore, we assume that no mixing between the SM physical Higgs h and the two non-SM CP-even Higgses H 1,2 arise and this assumption constrains the (1, 2), (1, 3), (2, 1) and (3, 1) elements of CP-even mass matrix of Equation 57. The constraints are given by the following decoupling limit scenario and then the CP-even mass matrix of Equation 57 with the constraints is simplified to In the above given decoupling limit scenario, chosen in order to simplify our analysis, the CP-even neutral scalar states contained in the SU (2) doublet H u will not mix with the CP-even neutral ones contained in H d . In such limit, the neutral CP-even states of H u will not feature mixing with the gauge singlet scalar φ. Thus, the lightest 125 GeV CP-even scalar of our model will have couplings to the SM particles close to the SM expectation, which is consistent with the current experimental data. Diagonalizing the simplified CP-even mass matrix, it reveals masses of the physical SM Higgs h and non-SM CP-even scalars H 1,2 in the physical basis (h, H 1 , H 2 ) The SM Higgs h is appeared as Re H 0 u itself and the non-SM CP-even scalars H 1,2 are the states which Re H 0 d is mixed with Re φ. Regarding the CP-odd scalar sector, we find that the squared mass matrix for the CP-odd scalars in the basis Im H 0 u , Im H 0 d , Im φ is given by: The squared CP-odd mass matrix is diagonalized in the same way as in the CP-even mass matrix and the CP-odd physical basis is given by (G Z , A 1 , A 2 ) where G Z is the massless Goldstone bosons associated with the longitudinal components of the Z gauge boson, whereas A 1 and A 2 are massive non-SM CP-odd scalars Furthermore, the squared mass matrix for the electrically charged scalars is given by: The charged scalar mass matrix can be diagonalized in the basis H ± 1 , H ± 2 as in CP-even or -odd mass matrix: Then, the electrically charged scalar sector contains the massive scalars H ± and the massless electrically charged scalars G ± W which correspond to the Goldstone bosons associated with the longitudinal components of the W ± gauge bosons. In the following sections we will analyze the phenomenological implications of our model in the Higgs diphoton decay as well as in the muon and electron anomalous magnetic moments.

C. The Higgs diphoton signal strength
The rate for the h → γγ decay is given by: where ρ i are the mass ratios ρ i = α em is the fine structure constant; N C is the color factor (N C = 1 for leptons and N C = 3 for quarks) and Q f is the electric charge of the fermion in the loop. From the fermion-loop contributions we only consider the dominant top quark term. Furthermore, C hH ± H ∓ is the trilinear coupling between the SM-like Higgs and a pair of charged Higges, whereas a htt and a hW W are the deviation factors from the SM Higgs-top quark coupling and the SM Higgs-W gauge boson coupling, respectively (in the SM these factors are unity). Such deviation factors are close to unity in our model and they are defined as below: Furthermore, F 1/2 (z) and F 1 (z) are the dimensionless loop factors for spin-1/2 and spin-1 particles running in the internal lines of the loops. These loop factors take the form: with In order to study the implications of our model in the decay of the 125 GeV Higgs into a photon pair, one introduces the Higgs diphoton signal strength R γγ , which is defined as: That Higgs diphoton signal strength, normalizes the γγ signal predicted by our model in relation to the one given by the SM. Here we have used the fact that in our model, single Higgs production is also dominated by gluon fusion as in the Standard Model. The ratio R γγ has been measured by CMS and ATLAS collaborations with the best fit signals [88,89]: R CMS γγ = 1.18 +0.17 −0.14 and R ATLAS γγ = 0.96 ± 0.14.
As it will be shown in the next subsection, the constraints arising from the Higgs diphoton decay rate will be considered in our numerical analysis.

D. The muon and electron anomalous magnetic moments
The Yukawa interactions relevant for the computation of the muon anomalous magnetic moment are: where the Yukawa coupling constants y e 24 , x e 42 are assumed to be real, the scalar fieds have been expanded by their real and imaginary parts and the properties of the projection operators P L,R acting on the charged leptonic fields have been used. By expressing the scalar fields in the interaction basis in terms of the scalar fields in the physical basis, the charged lepton Yukawa interactions relevant for the computation of the g − 2 anomalies take the form: where we are using the unitary gauge where the contributions arising from unphysical Goldstone bosons to the muon anomaly are excluded and we shorten the notations R CP by R e(o) . Here R e and R o are the rotation matrices that diagonalize the squared mass matrices for the CP even and CP odd scalars, respectively. Then, it follows that the muon and electron anomalous magnetic moments in the scenario of diagonal SM charged lepton mass matrix take the form: where the loop integrals are given by [85,[90][91][92][93]: and S(P ) means scalar (pseudoscalar) and E 4,5 stands for the vector-like family. It is worth mentioning that E 4 and E 5 only contribute to the muon and electron anomalous magnetic moments, respectively.

VI. NUMERICAL ANALYSIS OF THE HIGGS EXCHANGE CONTRIBUTIONS
For the sake of simplicity, we consider the scenario of absence of mixing between SM charged leptons, which automatically prevents charged lepton flavour violating decays. In our numerical analysis we have found that the non-SM CP-even scalar mass can reach values around 200 GeV. Despite the fact that the non SM CP-even scalar is quite light and can have a sizeable decay mode into a bottom-anti bottom quark pair, its single LHC production via gluon fusion mechanism is strongly suppressed since it is dominated by the triangular bottom quark loop. Such non SM CP-even scalar H can also be produced by vector boson fusion but such production is expected to have a low total cross section due to small HW W and HZZ couplings, which are proportional to v d . In this section we will discuss the implications of our model in the muon and electron anomalous magnetic moments.
A. The fitting function χ 2 and free parameter setup For an initial scan, we set up the starting parameter region as below:   Figure 9 2. For λ 1 , we fixed mass of the SM physical Higgs h to be 125 GeV to save time and to make the calculation faster. For λ 5,7 , the assumption that no mixing between the SM Higgs h and non-SM Higgses H 1,2 arise is reflected on these parameters. All quartic coupling constants λ 1,··· ,8 are set up not to go over 4π for perturbativity. After saturating value of the χ 2 function less than or nearly 2 which we believe it is converged enough, we find a best peaked value for each free parameter. For the given parameters, we rename them by adding an index "p" to the end of subscript of each parameter like tan β p and then the expansion factor κ is multiplied to find a correlation between the observables and the mass parameters. Then, the parameter region is refreshed by both the specific value of each parameter and the expansion factor κ as per Table III.  , which is consistent with the assumption made in section I, regarding the fact that the corresponding expansion parameter v 3 /M ψ is less than unity.
Here, we put two constraints on the lightest vector-like mass and the lightest non-SM scalar mass; the vector-like mass should be greater than 200 GeV as well as the non-SM scalar mass [94,95]. After we carry out second parameter scan based on the first scan result of Table IV, range of the parameters are given in Table V  are non SM CP-odd scalars and H ± stand for non SM charged scalars in this model. All data of ∆a µ,e are collected within the 1σ constraint of each anomaly.

C. The muon and electron anomalous magnetic moments
In order to confirm that our theoretical prediction for both anomalies can accommodate their constraints at 1σ and to analyze correlations between both anomalies and mass parameters, we consider cases B and E in Table IV since their benchmark point have relative lower values of the χ 2 function when compared to other cases. The reason that the cases B and E have the lower values of the χ 2 function arises from the obtained value of the electron anomaly, which is very close to the central experimental value. All cases reveal nearly central value of muon anomaly constraint at 1σ, whereas the other cases except B and E reveal nearly edge value of electron anomaly constraint at 1σ. Therefore, the reason why the cases B and E are more converged is related to whether our theoretical prediction for both anomalies can gain access to their central value of each anomaly constraint at 1σ. More importantly, the case E is only one satisfying vacuum stability conditions and a detailed investigation for the vacuum stability of each case will be studied in a subsection. For these reasons, we take the case E in Table V to study the correlations. The relevant parameter spaces are listed in Figure 11 and 12.
To begin with, we consider the parameter spaces for the muon anomaly versus electron anomaly with a mass parameter which attends both anomalies (H 1,2 , A 1,2 ) and does not (H ± ) in Figure 11. Even thought the non-SM charged scalar does not attend both anomalies, the similar pattern which the other scalars implement in Figure 11 is also appeared. We confirmed that mass of H 2 is nearly proportional to that of H ± , which causes the correlation identified in plots of the other non-SM scalars in Figure 11 is still maintained for the non-SM charged scalar. Interestingly, the cases A, B and C in Table V reported m H2 is nearly proportional to m H ± one-to-one ratio, whereas the cases D and E revealed a fat proportion between them and still maintained the correlation. As mentioned at the beginning of this section, we take the case E for the plots in Figure 11 and 12 and a main distinction between the case E and others arises from the value of electron anomaly. If we take other cases instead of the case E to investigate the parameter spaces, the parameter region appeared in top-left plot of Figure 11 will be  shifted upward by locating at the value of −5 or −6 × 10 −13 for the electron anomaly. In other words, the whole colored region in Figure 11 is shifted upwards to meet the scanned value of electron anomaly constraint at 1σ, holding the correlations. Therefore, the white region appeared in Figure 11 is not strictly excluded region and affected by how well a benchmark point is converged and by a factor of κ. However, these plots still tells a correlation between both anomalies and a tendency that the lighter mass of H 1 is located at edge region of the parameter space. Mass of the lightest non-SM scalar H 1 implied in top-left plot of Figure 11 is ranged from 200 to 220 GeV [95] and the cross section for this light non-SM scalar will be compared to that for SM Higgs in appendix. As for mass range of the other non-SM scalars confirmed in rest of other plots in Figure 11, they all implied heavier mass than that of H 1 which can be flexible depending on how the parameters are converged as seen in each case of Table V. We investigate a correlation for an anomaly versus a relevant mass parameter with another anomaly in bar in Figure 12. Note that the fourth vector-like mass is relevant only for the muon anomaly, whereas the fifth is only for the electron anomaly. Even though the fourth (fifth) is irrelevant to the electron (muon) anomaly, it is good to express them together since we rearrange the mass parameters and the anomalies in bar for comparison. The top-left plot in Figure 12 just fills in whole parameter region, thus no any correlation between the fourth vector-like mass and the muon anomaly is identified. After we rearranged the order of m e4 and ∆a µ,e from the top-left plot, we can confirm the similar correlation identified in Figure 11 from the top-right plot in Figure 12. The bottom-left plot identifies some correlation between the fifth vector-like mass and the electron anomaly contrary to the top-left plot. For the fifth vector-like mass, we put the constraint that the lightest vector-like mass should be greater than 200 GeV [94] and the mass region below 200 GeV is all excluded. After rearranging the order of m e5 and ∆a µ as in the above plot, we confirmed the similar correlation appears in the bottom-right plot. Interestingly, the top-right and the bottom-right plots check the similar correlation.
We confirmed that the muon and electron anomalous magnetic moments with vector-like particles can be explained to within 1σ constraint of each anomaly in a unified way, which is based on two attributes; the first one is the extended scalar sector and the second one is related with the contributions of the vector-like leptons. The first one which is reflected in our prediction for both anomalies, consists of four non-SM scalars and these contributions play a crucial role for determining the magnitude of each anomaly. The second one is seen by two vertices of both anomaly diagrams. The other Yukawa interactions can take place at each vertex since the vector-like leptons come in the loop, which is differentiated by the case where the normal SM particles enter in the loop. To be more specific, the helicity flip mass caused by the vector-like fermions in the CP-even and CP-odd basis couples the initial particle inside the loop to another particle of different chirality, thus allowing different interactions at each vertex. This means that the different sign problem can be solved by only considering multiplication of the Yukawa constants of each vertex and this property will be covered in detail in next subsection.

Vacuum stability
An important feature of our extended 2HDM theory is that it predicts large values for the Yukawa coupling constants y 2,1 , x 2,1 which can be ideally order of unity in our model. If the Yukawa coupling constants are much lower than unity, which means y 2,1 , x 2, 1 1, it will not cause any problem for stabilization of the scalar potential. However, large values of the leptonic Yukawa couplings are required in our model to successfully explain both g − 2 anomalies within the 1σ experimentally allowed range and since they are somehow related with the electroweak sector parameters, it might be able to destabilize the Higgs potential. As previously mentioned, in our analysis of the scalar sector and g −2 anomalies we are restricting to the scenario of decoupling limit, which implies that the large values of the leptonic Yukawa couplings will have a very small impact in the stability of up-type Higgs H u potential, whereas the conditons for the stability of the down type Higgs H d potential need to be determined. To discuss the stability of the scalar potential, one has to analyze its quartic terms because they will dominate the behaviour of the scalar potential in the region of very large values of the field components. To this end, the quartic terms of the scalar potential are written in terms of the Hermitian bilinear combination of the scalar fields. To simplify our analysis, we discuss the stability conditions of the resulting 2HDM scalar potential arising after the gauge singlet scalar field φ acquires a vacuum expectation value. Such stability conditions have been analyzed in detail in the framework of 2HDM in [96,97]. In order to analyze the stability of the H d potential, what we need to check if the quartic scalar couplings in each case of Table IV fullfill the stability conditions to be determined below. Given that our Higgs potential corresponds to the one of an extended 2HDM with the flavon field φ, in order to apply the stability conditions used in the reference [96] to our Higgs potential, we need to reduce the number of scalar degrees of freedom by considering the resulting 2HDM scalar potential arising after the gauge singlet scalar field φ is integrated out. From the scalar potential it follows that the relevant quartic coupling constant λ 6 must be positive, otherwise the vev v 3 would fall into negative infinity when the field φ value increases. For the same reason, the quartic coupling constants λ 1,2 must also be positive. From the aforementioned stability conditions we conclude that the cases A and C must be excluded since their corresponding quartic coupling constants λ 2 are negative. Assuming the flavon field φ develops its vev v 3 , we can rewrite the Higgs potential in terms of H u and H d fields as follows: Dropping all numbers and combining same order terms, the Higgs potential becomes much simpler as follows: where it is worth mentioning that the λ 4 term can be safely removed in the Higgs potential since it does not play a role in the CP-even, odd but charged mass matrix (Now our focus is the neutral scalar sectors). Here, we can impose one extra condition for the stabilization check, which is that the redefined mass terms must be negative, otherwise we get zero vev as a global minimum.
We have used the decoupling limit of Equation 58 at the first equality of Equation 79. From this equation, it is possible to determine the appropriate sign for the quartic coupling constant λ 3 . In our numerical analysis, the vev v 1 is much dominant than the vev v 2 so it leads to a negative sign for the quartic coupling constant λ 3 , otherwise the below equation of Equation 79 would become positive. The sign of the quartic coupling constant λ 3 also determines the one of λ 5,7 in the decoupling scenario, which means that λ 5,7 must also be negative. On top of that, the large Yukawa coupling constants y, x can be understood in connection with the vev v 3 . To this end, we consider the definition for the Yukawa coupling constants x 1 and x 2 , which are given by: where in order to successfully explain both g −2 anomalies within the 1σ experimentally allowed range, one has to rely on small values of v 3 , which are O(10 GeV), and the small values of v 3 do not significantly spoil the down-type Higgs H d potential as seen in Equation 79. In other words, the mass parameters µ 2 1,2 are much larger than the parameters λ 7,8 v 2 3 /2, thus allowing more freedom in the sign of λ 8 . Then, we are now ready to match our simplified Higgs potential with the one given in the reference [96]. Taking into consideration that our Higgs alignment is different than the one of [96], our mass parameters can be redefined as follows: Then, following [96,97], it is found that the scalar potential is stable, when the following relations are fullfilled: The last stability condition can be rewritten as shown on the right side since the β 4,5 are zero in our Higgs potential and the cases B and D must be excluded by this last condition shown in Equation 84. The conditions given in Eqs. (83) and (84) are crucial to guarantee the stability of the electroweak vacuum. Furthermore, one has to require that the squared masses for the physical scalars are positive. Besides that, according to [96], the minimum of the scalar potential is a global minumum when the following condition is fulfilled: where the latter condition on the left hand side is always successfully fulfilled for all cases, so we can simply drop off the condition as shown on the right side. Then, it is enough to confirm whether each case satisfies the reduced global minimum condition and the case E successfully fulfills that requirement as shown below: Thus, we have numerically checked that the best fit point corresponding to the case E obtained in the numerical analysis of the scalar potential and g − 2 muon and electron anomalies is consistent with the above given stability conditions of the scalar potential and at the same time ensure positive values for the squared masses of the physical scalars, consistent with the current experimental data. Finally, to close this section, it is worth mentioning that the large Yukawa coupling constants y, x involve the small vev v 3 in our model and this ensures that not only the H u potential is stable in the decoupling scenario but also the H d potential successfully fullfill the requirements of vacuum stability for both the small vev v 3 and appropriate values of the quartic scalar couplings.
2. How is the scalar exchange possible to accommodate both anomalies at 1σ constraint analytically?
In order to analyze how the scalar exchange is able to explain both anomalies within the 1σ range, we revisit the analytic expressions for both muon and electron anomalous magnetic moments: where I (e,µ) with S(P ) corresponding to scalar (pseudoscalar) and E 4,5 standing for the vector-like family. Furthermore, E 4 and E 5 only contribute to the muon and electron anomalous magnetic moments, respectively. First of all, we focus on the sign of each anomaly. The different signs of each anomaly indicated by the 1σ experimentally allowed range can be understood at the level of Yukawa constants apart from the loop structures. As seen in Table II, the Yukawa coefficient y can be either positive or negative, while x only remains positive since we take the absolute value to the x. We also considered the case where the coefficients x, y are purely positive, assuming v 3 is positive, without taking absolute value and the multiplication of the Yukawa coefficients x × y cannot change the sign of each anomaly since the denominator of x includes y and they are cancel out. Then, the sign problem depends on summing over loop functions and we found that the order of the muon anomaly prediction is suitable, whereas the corresponding to the electron anomaly is about 10 −16 which is too small to be accommodated within the 1σ experimentally allowed range. Therefore, we found that taking an absolute value to one of the Yukawa coefficients is an appropriate strategy for the sign and allows to reproduce the correct order of magnitude of each anomaly allowed by the 1σ experimentally allowed range, for an appropiate choice of the model parameters. This feature is a crucial difference compared with the W or Z gauge boson exchange [33]. The W gauge boson exchange covered in the main body of this work keeps the same coupling constant at each vertex, therefore it is completelly different from the scalar exchange with vector-like leptons. For the Z exchange covered in [33], it has the common property that the coupling constant of each vertex is different to each other, whereas the coupling constants of the Z are more constrained by the mixing angle between ith chiral family and fourth vector-like family, so it is impossible to explain both anomalies at the same time. As a result, allowing different Yukawa constants with appropiate signs enables both anomalies to be explained in a unified way.
Next we turn our attention to the order of magnitude of our predictions for both anomalies. Considering that the sign problem is solved by having each Yukawa constant y either positive or negative, it can be easily understood that inside the structure in parentheses of Equation 88 should imply the same direction, which is is determined by the contribution of all loop functions in parentheses. Since the mass difference among non-SM scalars and vector-like particles is not so big, we have to consider their masses in the computation of muon and electron anomalous magnetic moments, as follows from Equation 88. For an easy analysis, we take the case E reported in Table IV and suppose that where c 1,2 are arbitrary constant between 0 and 1 either positive or negative and mass ordering among d(e) i , (i = 1, 2, 3, 4) can be easily understood by considering mass difference between non-SM scalars and vector-like particles. The muon and electron anomaly prediction can be rewritten in terms of these redefined constants: VII. CONCLUSION We have proposed a model to account for the hierarchical structure of the SM Yukawa couplings. In our approach the SM is an effective theory arising from a theory with extended particle spectrum and symmetries. The considered model includes an extension of the 2HDM where the particle spectrum is enlarged by the inclusion of two vector-like fermion families, right handed Majorana neutrinos and a gauge singlet scalar field, together with the inclusion of a global U (1) symmetry spontaneously broken at the TeV scale. Since the U (1) symmetry is global, this model does not feature a Z boson and it is softly broken in the 2HDM potential to avoid a Goldstone boson. Its main effect is to forbid SM Yukawa interactions due to the U (1) charge conservation. Besides that, this model has the property of the 2HDM type II where one Higgs doublet couples with the up-type fermions whereas the remaining one has Yukawa interactions with down-type fermions, where such couplings are allowed between chiral fermions and vector-like fermions due to the choice of U (1) charges (chiral fermions having zero charges while vector-like fermions, Higgs and flavons have charges ±1). Below the mass scale of the vector-like fermions, such couplings result in effective Yukawa couplings suppressed by a factor φ /M where the numerator is the vev of the flavon and the denominator is the vector-like mass. This factor naturally determines the magnitude of SM interactions and the mass scale for the vector-like fermions under a suitable choice of the flavon vev. We have developed a mixing formalism based on 7×7 mass matrices to describe the mixing of the three chiral families with the two vector-like families.
Within the above proposed model, we have focused on accommodating the long-established muon and less established electron anomalous magnetic moments at one-loop level. A main difficulty arises from the sign of each anomalous deviation of the experimental value from its SM prediction. Generally, the Feynman diagrams for the muon and electron anomalous magnetic moments have the same structure except from the fact that the external particles are different, which makes it difficult to flip the sign of each contribution. Specifically we have required that both deviations in Equation 1) at one-loop should be accommodated within the 1σ experimentally allowed range, which is a challenging requirement.
We first considered in detail the W boson exchange contributions to the muon and electron anomalous magnetic moments at one-loop. The relevant sector for the W boson exchange is that of the neutrino and we analyzed a novel operator that generates the masses of the light active neutrinos in this model. The well-known five dimensional Weinberg operator which we refer as type Ia seesaw mechanism does not work in this model since it is forbiden by the U (1) symmetry due to the fact that both SU (2) scalar doublets are negatively charged under this symmetry. For this reason, we made use of the Weinberg-like operator known as type Ib seesaw mechanism allowed in this model. With the type Ib seesaw mechanism, we built the neutrino mass matrix with two vector-like neutrinos and ignored fifth vector-like neutrinos since they are too heavy to contribute to the phenomenology. The deviation of unitarity η derived from the heavy vector-like neutrinos plays a crucial role for enhancing the sensitivity of the CLFV µ → eγ decay to the observable level. Furthermore, the Yukawa constants of Dirac neutrino mass matrix can be connected to the observables measured in neutrino oscillation experiments. One of the neutrino Yukawa constants is defined with a suppression factor . Therefore, the effective 3 × 3 neutrino mass matrix tells that the tiny masses of the light active neutrinos depend on the mass scale of vector-like neutrinos as well as on the suppression factor . This implies that mass scale of vector-like neutrinos is not required to be of the order of 10 14 GeV, as in the conventional type Ia seesaw mechanism. In our proposed model, the vector-like neutrinos can have masses at the TeV scale, thus allowing to test our model at colliders. Those vector-like neutrinos can be pair produced at the LHC via Drell-Yan annihilation mediated by a virtual Z gauge boson. They can also be produced in association with a SM charged lepton via Drell-Yan annihilation mediated by a W gauge boson. These heavy vector like sterile neutrinos can decay into a SM charged lepton and light active neutrinos. Thus, the heavy neutrino pair production at a proton-proton collider will give rise to an opposite sign dilepton final state, which implies that the observation of an excess of events in this final state over the SM background can be a smoking gun signature of this model, whose observation will be crucial to assess its viability. It is confirmed that the branching ratio of µ → eγ decay can be expressed in terms of the deviation of unitarity η as shown in [11,82] and our prediction for the muon and electron anomalous magnetic moments can also be written in terms of non-unitarity. We derived the analytic expression for the anomalies and found that the order of magnitude of these predictions is too small to accommodate the experimental bound within the 1σ range and the sign of each prediction also points out in the same direction. Therefore, we concluded that the W boson exchange at one-loop is not enough to explain both anomalies at 1σ and this conclusion has been a good motivation to search for another possibility such as scalar exchange, which is one of the main purposes of this work.
We then turned our attention to the 2HDM contributions (inclusion also of the singlet scalar φ) to the muon and electron anomalous magnetic moments, assuming by a choice of parameters a diagonal charged lepton mass matrix to suppress the branching ratio of µ → eγ. In our analysis we considered in detail the scalar sector of our model, which is composed of two SU (2) scalar doublets H u and H d and one electrically neutral complex scalar φ by studying the corresponding scalar potential, deriving the squared mass matrices for the CP-even, CP-odd neutral and electrically charged scalars and determining the resulting scalar mass spectrum. We have restricted to the scenario corresponding to the decoupling limit where no mixing between the physical SM Higgs h and the physical non-SM scalars H 1,2 arise and within this scenario we have imposed the restrictions arising from the Higgs diphoton decay rate, the hW W coupling, the 125 GeV mass of the SM-like Higgs and the experimental lower bounds on non SM scalar masses, to determine the allowed parameter space consistent with the muon and electron anomalous magnetic moments. To this end, we have constructed a χ 2 fitting function, which measures the deviation of the values of the physical observables obtained in the model, i.e., (g − 2) e,µ , the 125 GeV SM-like Higgs mass, the Higgs diphoton signal strength, the hW W coupling, with respect to their experimental values. Its minimization allows to determine the values of the model parameters consistent with the measured experimental values of these observables. After saturating the χ 2 value less than or nearly 2, we obtained five independent benchmark points and carried out second scan with the benchmark points to find a correlation between observables and mass parameters.
For the plots, we took an appropriate case which is more converged when compared to other ones and satisfying the vacuum stability conditions. We found that our prediction for both anomalies can be explained within the 1σ constraint of each anomaly and a correlation proportional for muon versus electron anomaly is appeared in Figure  11 and 12. Here, we put two constraints on mass of the lightest non-SM scalar and of the lightest vector-like family; m H1 , m e5 > 200 GeV based on references. The second scan result tells that the available parameter space is not significantly constrained by current experimental results on non-SM scalar mass and vector-like mass, while keeping perturbativity for quartic couplings and Yukawa constants. An important feature of our BSM model is it predicts the large Yukawa coupling constants y, x, which might be able to destabilize the Higgs potential. The up-type Higgs H u potential is not significantly affected by the large Yukawa coupling constants in the decoupling scenario, whereas there is no safe condition for the down-type Higgs H d potential which can be worsen by mixing with the flavon field φ. The large Yukawa coupling constants x introduces small values for the vev v 3 in the definition of x and the energy scale is confirmed by order of 10 GeV in our numerical analysis. On top of that, we also identified the appropriate sign of quartic coupling constants can make the Higgs potential stable. Therefore, the down type H d Higgs potential is stable by both the small vev v 3 and the appropriate quartic coupling constants in our BSM model. Lastly, we discussed how we were able to explain both (g − 2) e,µ anomalies at 1σ constraint and impact of the light non-SM scalar H 1 . For the former, we first simplified the prediction for both anomalies and used some numerical values at the stage where no more analytic simplication is possible. For the latter, we compared the cross section for the SM process pp → h and BSM process pp → H 1 and included this comparison in Appendix B.
We conclude that the proposed model of fermion mass hierarchies is able to successfully accommodate both the muon and electron anomalous magnetic moments within the 1σ experimentally allowed ranges, with the dominant contributions arising from one loop diagrams involving the 2HDM scalars and vector-like leptons. The resulting model parameter space consistent with the (g − 2) e,µ anomalies requires masses of non-SM scalars and vector-like particles in the sub TeV and TeV ranges, thus making these particles accessible at the LHC and future colliders.
Notice that the same rotations operated in the lepton sector is applied to both up-and down-type quark sector except for y d 14 since quark doublet rotation is already used in the up-type quark sector. These two mass matrices clearly tells that this model is an extended 2HDM in that the up-type SM Higgs H u corresponds to up-type quark sector, while the down-type SM Higgs H d corresponds for down-type quark sector.

A basis for decoupling heavy fourth and fifth vector-like family
In this section, we treat the decoupling basis with quarks holding an assumption φ ≈ M Q 44 . As in the charged lepton mass matrix, we can obtain the Yukawa matrix from the 5 × 5 upper blocks of Equation A1, where α and β run from 1 to 5. The Yukawa matrices y u,d αβ can be diagonalized by the unitary rotations V where each of the unitary matrices V i4,5 are parameterized by a single angle θ i4,5 featuring the mixing between the ith SM chiral quark and the 4, 5th vector-like quark. In the rotated mass matrix, we need (3,4), (1,5), (2,5), (3,5) mixing in the Q sector and (2, 4), (3,4), (1,5), (2,5), (3,5) mixing in the u, d sectors to go to the decoupling basis therefore the unitary mixing matrices V are defined to be (A5) With the defined unitary mixing matrices in place, the 5×5 Yukawa matrices in a mass basis (primed) are transformed by where tilde with prime means interaction basis whereas tilde alone corresponds to the mass basis. Appendix B: Heavy scalar production at a proton-proton collider We have confirmed that the mass of the non-SM CP even scalar H 1 is ranged from 200 to 240 GeV in Table V and this light mass of H 1 has not been observed at CERN or other experiments so far. In order to see how big an impact of H 1 is when compared to that of SM Higgs h, we studied a total cross section for the SM process pp → h and for BSM process pp → H 1 . The SM cross section for pp → h process is α S is the strong coupling constant, v is the conventional SM Higgs vev 246.22 GeV, m h is the Higgs mass 125 GeV, m t is the top quark mass 173 GeV, S is the squared LHC center of mass energy (14 TeV) 2 , PDF corresponds to the parton distribution function where 0 means 0th parton -gluon, x is the momentum fraction of the proton carried out by the gluon. Here the factorization scale has been taken to be equal to the SM like Higgs boson mass m h and x 1,2 (y) are defined as follows: where m H1 is mass of non-SM CP even scalar H 1 , and x 1,2 are defined in a similar way: x 1 (y) = m 2 H1 /S S exp(y), x 2 (y) = m 2 H1 /S S exp(−y) (B6) One main distinction between Equation B1 and Equation B5 is the non-SM scalar H 1 only interacts with down-type quark pair bb since it is a mixed state between h 0 d and φ while the SM Higgs h can interact with top-quark pair tt. According to the mass range of H 1 reported in Table V, the total cross section for pp → H 1 is given in Figure 13. The total cross section for pp → H 1 runs from nearly 8 pb at 200 GeV to smaller values as mass of H 1 increases. The order of magnitude of this cross section for pp → H 1 is compatible to that of the SM process pp → h, however the BSM process is strongly suppressed since its single LHC production via gluon fusion mechanism is dominated by the triangular bottom quark loop as mentioned in Section VI. Therefore, our prediction with the light non-SM scalar H 1 is possible to accommodate each anomaly constraint at 1σ.