Singlet scalar and 2HDM extensions of the Standard Model: CP-violation and constraints from $(g-2)_\mu$ and $e$EDM

We study popular scalar extensions of the Standard Model, namely the singlet extension, the 2-Higgs doublet model (2HDM) and its extension by a singlet scalar. We focus on the contributions of the added scalars to the anomalous magnetic moment of the muon, $(g-2)_\mu$ in the presence of CP-violation, and the electric dipole moment of the electron ($e$EDM) in these models. In the absence of CP-violation, CP-even and CP-odd scalars contribute with an opposite sign to the anomalous magnetic moment of the muon and as a result these models generally require very light scalars to explain the observed discrepancy in $(g-2)_\mu$. We study the effect of CP-violation on the anomalous magnetic moment of the muon and its compatibility with the $e$EDM constraints. We show that given the current status of the global set of constraints applied on all values of $\cot\beta$, in the CP-violating scalar extensions, there exist no viable parameter space in agreement with both $a_\mu$ and $e$EDM bounds.


I. INTRODUCTION
While the Standard Model (SM) of particle physics agrees very well with data from high energy collider experiments, it still falls short on explaining several observed features of Nature. For example, SM does not provide sufficient amount of CP-violation to source the Baryon Asymmetry of the Universe (BAU) [1] and the scalar sector of the SM does not provide a first order phase transition [2], which would be needed to produce BAU at the electroweak transition. Another example is the need to understand the origin of neutrino masses and mixing patterns. One possible paradigm to address these issues is to enlarge the scalar sector of the SM. Many such extensions have been studied in the literature [3][4][5][6][7][8][9].
In addition to providing new sources for CP-violation, the extra scalars arising from such extensions could also help to explain the muon anomalous magnetic moment a µ = (g −2) µ /2 which deviates from the SM prediction by ∆a µ = a exp µ − a SM µ = (2.87 ± 0.8) × 10 −9 (3.6σ) (1) according to the most recent experiment done at BNL [10,11].
Finally, extended scalar sectors provide new scalar mass eigenstates which can, for example, provide a Dark Matter (DM) candidate. Typically their stability is guaranteed by an ad-hoc discrete symmetry. In this paper, we do not consider a DM candidate and therefore, in the models we study we try to avoid extra symmetries if possible.
We focus on the following well-known scalar extensions of the SM We show that in the CP-conserving limit, due to the cancelling effect of the CP-odd and CP-even scalars, one can not explain the excess in Eq. (2), unless very light scalars are present and tan β is very large. However, when CP-violation is introduced, we show that less dramatic values of tan β or scalar masses are required to produce the observed a µ .
Having introduced CP-violation, the parameter space of models under consideration is strongly constrained by the data from ACME collaboration on electron and neutron Electric Dipole Moment (EDM) [12]. The bounds on electron EDM (eEDM) with d e < 10.25 × 10 −29 e cm = 1.573 × 10 −15 GeV −1 , impose the strongest constraints on any Beyond Standard Model (BSM) scenario with CPviolation. In each of the models under consideration we study if the amount of CP-violation required to explain the a µ discrepancy can be accommodated within the limits imposed by Eq. (3). As CP-violation is one of the main ingredients of BAU, the identification of the surviving regions of the parameter space after imposing the eEDM bounds is a necessary prerequisite of BAU studies. The models we have listed above have appeared in the context of electroweak baryogenesis: for a singlet extension of the SM, see e.g. [13][14][15][16], for 2HDM see e.g. [17][18][19][20][21][22][23][24][25] and for a singlet extension of the 2HDM see [6,26,27]. In this paper, we show, for the first time, how to implement the constraints from eEDMs and from the muon anomalous moment systematically on these models. In particular we show how this allows to determine the experimentally favoured patterns of Yukawa interactions in these models.
The paper is organised as follows: For the reader's convenience, in Section II we review the computation of a µ and eEDM from a generic Lagrangian and show in detail the 1-loop and 2-loop calculations of such contributions. In Sections III, IV and V we present the scalar potential, theoretical and experimental constraints and a µ and d e contributions in the SM+RS, SM+CS, 2HDM and 2HDM+CS scenarios, respectively. In Section VI we draw our conclusions and present our outlook.

II. CALCULATION OF a µ AND d e CONTRIBUTIONS
By definition, the a µ = (g − 2) µ /2 and eEDM contributions are where F µν is the electromagnetic field strength and σ µν = i[γ µ , γ ν ]/2. Therefore, the relevant parts of the Lagrangian are L ⊃ e m l 8π 2 c L (lσ µν P L l)F µν + c R (lσ µν P R l)F µν + h.c., (6) where l stands for the relevant lepton (e for eEDM calculations and µ for the muon anomalous magnetic moment). Expanding the Lagrangian for the explicit forms of the operators, leads to L ⊃ e m l 8π 2 (c L + c * R )(lσ µν P L l)F µν + e m l 8π 2 (c * L + c R )(lσ µν P R l)F µν = e m l 8π 2 Re(c L + c * R ) (lσ µν l)F µν − i e m l 8π 2 Im(c L + c * R ) (lσ µν γ 5 l)F µν (7) where P L = (1 − γ 5 )/2 and P R = (1 + γ 5 )/2 are the left and right projection operators. One, therefore, needs to explicitly calculate d e = e m e 4π 2 Im(c L + c * R ), (9) where c L and c R are the Wilson coefficients to be calculated for each loop diagram in Figure 1 separately.

A. 1-loop contributions
The digram contributing to the a µ and d e at 1-loop is shown in Figure 2, where h i are the neutral scalars in the model with their coupling to electrons and muons represented by Y h i ee and Y h i µµ , respectively. The charged scalar mediated version of this diagram is sub-dominant and is therefore neglected [28]. The mass of the charged scalar is set to be equal to the mass of the heaviest scalar to comply with the ElectroWeak precision data. Note also that in the models we study, we only extend the scalar sector of SM and do not add any extra vector or fermion fields, such as right-handed neutrinos. The Wilson coefficients are calculated to be where Y h i ll is the scalar h i 's coupling to ll and could in general be complex, The contribution from the 1-loop diagrams to a µ and d e are then caluclated to be where n is the number of the scalars mediating the loop in Fig. 2. Our formulas are in agreement with the known results in [7], [28]- [29].

B. 2-loop contributions
The main 2-loop contributions to a µ and d e , shown in Figure 3, arise from the Bar-Zee diagrams mediated by the scalar states. The diagrams with the Z boson in the loop (instead of γ) are suppressed by a factor of ( 1 4 − sin 2 θ W ), which makes their contribution almost two orders of magnitude smaller than diagrams with a photon in the loop. We therefore ignore such diagrams in the calculations that follow. Similarly, contributions from the charged scalars are ignored since they too are sub-dominant [30]. For our 2-loop calculations, we use the results of [31].
The contribution from 2-loop diagrams with heavy fermions (f = t, b, c, τ ) 1 and W to the a µ are For the SM-Higgs coupling to W W , we use the notation Y h SM W W which in the pure SM limit is 2m 2 and in the pure SM limit is m f /v.
The contribution from 2-loop diagrams to the d e from heavy fermions, f , and W loops are 1 The subscript f stands for fermion and is not to be confused with the loop function f (z).
The loop functions f (z), g(z) and h(z) appearing in Eqs. (14) and (16) and are presented in Appendix A.

III. THE REAL SINGLET EXTENSION (SM+RS)
The real singlet model is often presented with a Z 2 symmetry imposed on the scalar potential in order to stabilise the singlet field as a viable DM candidate [4,[32][33][34][35]. As mentioned earlier, in this paper we shall not look into DM phenomenology and hence we consider the model in its general form with no extra symmetries.
The most general potential in this case has the following form Note that by a translation of S, the linear κ 1 term can be removed. The fields Φ and S are, respectively, the SM gauge doublet and singlet with Vacuum Expectation Values (VEVs) v and w. Their field decomposition is as follows, Since S is an SU (2) singlet, it has no direct couplings to the SM gauge bosons or fermions.
The field Φ plays the role of the SM Higgs doublet, therefore, G + and G 0 are the would-be Goldstone bosons which are "eaten" by the W ± and Z bosons.
The minimum of the potential requires The gauge eigenstates φ 1,2 are then rotated to the mass eigenstates h 1,2 with the rotation matrix R defined as where we take h 1 to be the observed 125 GeV Higgs boson. The mixing angle θ is calculated to be The value of sin θ is bounded by experimental [36] and theoretical [37] constraints to be | sin θ |< 0.33.
Throughout this paper we take the conservative limit of sin θ 0.3 into account.
Note that the two neutral scalar mass eigenstates, h 1,2 , are a mixture of φ 1,2 which are CP-even. Clearly there is no possibility of introducing CP-violation explicitly (through complex parameters in the potential) or spontaneously (through a complex VEV of the doublet and/or singlet). Hence, CP-violation is introduced through a higher dimension operator [5,38]. In the absence of a Z 2 symmetry, we take this to be the following dimension-5 operator, where is the complex CP-violating parameter, Λ is the scale of new physics generating the effective operator, Q L and t R are, respectively, the left-handed doublet and right-handed quarks of the SM. Note that the sole source of CP-violation here is the parameter η, which is only introduced for the top quark couplings.
We use the conventional SM Yukawa couplings as defined by the Lagrangian, where, as clarified before, φ 1 is the SM Higgs field.
To calculate a µ and d e discussed in Section II, one needs to identify the couplings of the mass eigenstates h 1,2 to leptons, quarks and the W boson. These are: where R ij are the components of the rotation matrix defined in Eq. (20). Note that the only Following from Eq. (12)- (16), one can see that since Im(Y h i ll ) = 0, only the imaginary part of η contributes to d e and only the real part to a µ , as it will be shown in detail in the next two subsections. We will therefore quantify our results in terms of the dimensionless quantities vRe(η)/(2Λ) for a µ and vIm(η)/(2Λ) for d e .
For the theoretical and experimental constraints, we have adopted the results in [37].
A. a µ in the SM+RS model As shown in detail in Section II, a µ is proportional to the real part of the fermion-scalar couplings. Hence, all 1-loop and 2-loop contributions are non-zero and calculated to be where R 11 = cos θ and R 12 = sin θ are the elements of the rotation matrix in Eq. (20).
We find that SM+RS model is incapable of explaining the muon anomalous moment for m h 2 of a few hundred GeV, even in the presence of a non-zero L CP V . In Figure 4, we show contours of a µ in the sin θ-vRe(η)/(2Λ)-plane for a representative value of m h 2 = 500 GeV.
The green region is where the model produces a µ within the observed window in Eq. (2).
Hence, one would need a very large, O(10 3 ), non-trivial coupling to top quark in Eq. (23).
Note that the green region shown in the plot is not affected by eEDM constraints, which are governed by different couplings. We will discuss these constraints in detail in the next section.
In Figure  The SM doublet Φ and SM singlet S are defined as: with v and w as VEVs of the doublet and the singlet, respectively. Similar to the SM+RS case, Φ is the SM Higgs doublet with G ± and G 0 as the Goldstone bosons.
The gauge eigenstates φ 1,2,3 are rotated to the mass eigenstates h 1,2,3 through Extending the SM with one extra scalar doublet with the same SM quantum numbers as the SM-Higgs doublet 3 , one arrives at the well-studied 2HDM [40][41][42][43][44][45][46]. The most general 2HDM potential can be written in the following form: In the general case, due to the existence of two scalar doublets to which fermions can couple, 2HDMs suffer from Flavour Changing Neutral Current interactions (FCNCs) at treelevel, which are strongly restricted experimentally. It is known that imposing a softly broken Z 2 symmetry on the scalar potential, and extending it to the fermion sector can forbid these FCNCs [47], [48]. Depending on the Z 2 charge assignment of the fermions, four independent types of Yukawa interactions are allowed, and these are known as Type-I, Type-II, Type-X (Lepton-specific) and Type-Y (Flipped) in the literature [41,42,49], and references therein.
These will be discussed in Section IV B. In what follows, the transformation of the scalar doublets under this Z 2 symmetry is fixed to be Φ 1 → +Φ 1 and Φ 2 → −Φ 2 .
Imposing the softly broken Z 2 symmetry forbids the λ 6,7 terms in the potential in Eq. (36), The rest of the parameters are real with the exception of µ 2 3 and λ 5 which are complex and defined as In this paper, we take the VEVs of the doublets to be real and positive and study explicit CP-violation which occurs when Im(λ * 5 [µ 2 3 ] 2 ) = 0 [50], through the complex parameters of the potential.
In general, the decomposition of the scalar doublets is as follows where v 1 and v 2 are taken to be real with v 2 = v 2 1 + v 2 2 = (246GeV) 2 and, as usual, we define tan β = v 2 /v 1 .

A. Minimisation of the 2HDM potential
The minimization of the potential implies where s β and c β stand for sin β and cos β, respectively.
At this point, it is useful to rotate the doublets to the so called Higgs basis [51], where only one of the doublets has a VEV and one can separate the Goldstone bosons, G ± , G 0 , from the physical states. The mass of the charged Higgs is calculated to be The neutral mass-squared matrix, M 2 , shown in detail in Appendix B, is a 3 × 3 matrix which is diagonalised by the rotation matrix R, where we take h 1 to be the observed Higgs boson at the LHC with m h 1 = 125 GeV.
The rotation matrix, R, depends on the three mixing angles, θ 12 , θ 13 and θ 23 , where the latter two angles represent CP-violation and will vanish in the CP-conserving limit.
Therefore, we take these angles to be small since, as it will be shown later, they prove to be very small in the interesting and allowed regions of the parameter space. The angle θ 12 represents the mixing of the SM-like Higgs with the other CP-even state. As shown in Eq. (22), to agree with the observed Higgs data, we take θ 12 to be small.
With all mixing angles being small (cos θ i 1 and sin θ i θ i ), the rotation matrix, R, simplifies to the form With this simplified form, one can calculate the angles in terms of the parameters of the potential as shown in Appendix B.
After minimisation, the 9 independent parameters of the model, can be expressed in terms of which we take as input parameters for our numerical calculations.

B. Yukawa and gauge couplings
In the general 2HDM, interactions of the scalar sector with SM fermions are defined as where Φ u,d,e are Φ 1 and/or Φ 2 depending on the type of Yukawa interactions. This correspondence is determined according to Table I after the Z 2 charge assignments for fermions have been specified.
Starting from Eq. (48), one rotates Φ 1,2 to Φ 1,2 in the Higgs basis using Eq. (41). The primed fermion gauge doublets and singlets, will have to be written in terms of the unprimed mass eigenstates using the usual unitary matrices U L and U R , which also diagonalise the fermion mass and Yukawa matrices simultaneously. The Yukawa interactions can then be written in the following compact form where the R ij are the rotation matrix elements defined in Eq. (45) and the coefficients ξ i are Type-specific as defined in Table I. The scalar-gauge interactions are derived from the kinetic terms and are of the form where, again, R 1i are rotation matrix elements defined in Eq. (45). In all the results that follow, we take into account theoretical and experimental bounds as shown in detail in Appendix C. For our plots, we find it instructive to show a large region of cot β, and point out, in each subsection, the regions that are ruled out experimentally.
C. a µ and d e in 2HDMs 1. General Type-independent formulas for a µ and d e The contribution from 1-loop diagrams to a µ and d e are The 2-loop contributions from up-Type and down-Type quarks, leptons and the W boson to a µ are The 2-loop contributions from up-Type and down-Type quarks, leptons and the W boson Note that these results are type-independent: each type of 2HDM can be studied further numerically when the corresponding values of ξ l , ξ d and ξ u presented in Table I are implemented.
where the last two lines are the contributions of the down-Type quarks (mostly b) and charged leptons (mostly τ ) to the Barr-Zee diagrams which clearly are sub-dominant. Hence, Type-I and Type-Y (and similarly, Type-II and Type-X), whose only difference is in ξ d , contribute almost identically to d e and a µ , especially when ξ l,d are not very large as it will be clarified further here.
To see the exact difference between Type-I and Y (and similarly Type-II and X), we show the explicit numeric formulas in each case. For Type I, and for Type Y, which has a subdominant correction with respect to Type-I. Note that when tan β < 10 2 , this correction is negligibly small, as it is shown in Figure 6. From these equations we see that Type-I (and Y) are capable of producing a large enough a µ in the cot β 10 region.
Note that the difference between the d e contribution in Type-I and Y is proportional to tan β whose effect is visible in the low cot β region in Figure 6.
Superimposing the a µ and d e plots, one can see that with heavy scalars, it is not possible to have a large enough a µ contribution with the amount of CP-violation that is allowed by the eEDM data.
We emphasize that the above numerical formulas are presented for exemplary values of masses m h 2,3 = 200, 300 GeV. In the next subsection, we will analyse different mass hierarchies in more detail.

D. 2HDM Results
We divide this section into three subsections dealing with heavy (m h 2,3 m h 1 ), medium (m h 2,3 ≈ m h 1 ) and light (m h 2,3 m h 1 ) mass regions.

Heavy mass region
To investigate the effect of CP-violation more closely, in Figure 7 Aside from the eEDM constraints, note that large values of cot β lead to large scalarfermion couplings which are ruled out due to flavour and/or collider constraints. It has been shown in the CP-conserving limit in [29,[52][53][54] that due to these constraints only Type I and X models survive in the low cot β region.

Medium mass region
Next we turn to the medium mass region where all scalars have masses comparable with m h 1 . In Figure 9, Type Y is more constrained, and with Type II, X leaning towards the cot β ≈ 10 region.
To get a closer look at the effect of CP-violation, we present in Figure 10  increases and the surviving region of the parameter space shrinks. As mentioned before, the behaviour of Type Y and II are similar to Type I and X, respectively.
To see the effect of the scalar masses, in Figure

Light mass region
It has been shown [30,55] that in the CP-conserving limit, Type-X 2HDM can produce a large enough a µ due to the positive contribution from a very light CP-odd scalar, m A ≈ 30 GeV, and a large tan β ≈ 60. Our calculations, when taken to the CP-conserving limit (θ 13 = θ 23 = 0), confirm these results.
In Figure 13, we show the effect of CP-violation on the a µ contribution in different 2HDM Types for light scalars masses. We also show the d e contribution in different 2HDM Types in this mass region and the upper limit imposed by the eEDM experiments.
To clarify the effect of CP-violation, in Figure 14, we plot a µ and d e contributions in Type I and X for fixed scalar mass values while varying the CP-violating angles. The corresponding plots for Type Y and II are similar to Type I and X, respectively, as discussed in detail before.  Similar to the other mass regions, large cot β values are ruled out due to flavour and/or collider constraints in Type II and Y, and only Type I and X survive in the low cot β region.
To summarise the findings in these subsections: In all mass ranges, Type I and Y contribute efficiently to a µ in the large cot β region. In Type II and X, very small cot β values lead to the correct a µ values. On the other hand, bounds from d e experiments are satisfied in the small cot β region for Type I and Y with Type Y more constrained, and in the cot β 1 for Type II and X.
Super-imposing the d e and a µ plots reveal that only in Type I and Y and only in the medium mass region, one can simultaneously produce the observed contribution to a µ and remain compatible with the results from eEDM experiments. However, this happens only at relatively large values of cot β ruled out by current experimental data. In the low cot β region we find that only Type I and Type X models remain viable: Type X is a suitable choice for obtaining a µ contributions in the absence of CP-violation, while Type I has the smallest contribution to d e and therefore a suitable choice for a CP-violating model, but yielding only the SM contribution to a µ .

V. 2HDM + SINGLET EXTENSION
The singlet extension of the 2HDM is a relatively popular model [6,26,27] for several reasons: First, the scalar sector of 2HDM+S resembles that of the Next to Minimal Supersymmetric SM (NMSSM). Second, it is understood that the singlet extension of the SM and the 2HDM are incapable of providing a viable DM candidate and allow for CPviolation simultaneously [56]. Therefore, going beyond the simplest extensions of the SM seems inevitable.
In general, the decomposition of the scalar multiplets is as follows.
Here, we discuss directly the complex singlet extension of 2HDM. in the calculation of the d e and a µ , the sums will be over i = 1, . . . , 4, corresponding to the four scalar mass eigenstates, h 1,2,3,4 .
The most general 2HDM+CS potential has the the form Similarly as in the case of SM+RS, the linear term κ 1 can be removed by a translation of S.
Similar to the 2HDM, the 2HDM+CS suffers from tree-level FCNCs due to the existence of more than one scalar doublet which in general could couple to fermions. As in 2HDM, this can be alleviated by imposing a Z 2 symmetry on the scalar sector and extending it to the fermion sector in a similar manner. The transformation of the scalar multiplets under this Z 2 symmetry is fixed to be The fermionic Z 2 charges are as shown in Table I which define the 2HDM type of the model.
We take the VEVs, v 1 , v 2 and w to be real.
The minimisation conditions are presented in Appendix D under which the minimum of the potential is realised at Similar to the 2HDM, it is useful to rotate the doublets to the Higgs basis while the singlet remains unchanged, with tan β = v 2 /v 1 . Then, only one of the doublets has a VEV, and one can separate the Goldstone bosons, G ± , G 0 , from the physical states. The charged Higgs mass is calculated to be The neutral mass-squared matrix, M 2 , shown in detail in Appendix D, is a 5 × 5 matrix which is diagonalised by the rotation matrix R, where, as before, we take h 1 to be the observed Higgs boson at the LHC.
The rotation matrix, R, contains ten mixing angles, θ 12−15 , θ 23−25 , θ 34,35 and θ 45 among which five represent CP-violation, namely θ 13,15,23,25,34 , and will vanish in the CP-conserving limit. We, therefore, take these angles to be small since, as it will be shown later, they prove to be very small in the interesting and allowed regions of the parameter space. Of the remaining angles θ 12 and θ 14 represent the mixing of the SM-like Higgs with the other CP-even states. To keep this state mostly doublet-like and to agree with the observed Higgs data, we take these angles to be small.
The remaining angles, θ 24 , θ 35 and θ 45 do not contribute to the observables in which we are interested here, and therefore, to simplify the analysis, we assume all mixing angles to be small (cos θ i 1 and sin θ i θ i ). As a result, the rotation matrix, R, simplifies to the With this simplified form, one can calculate the angles in terms of the parameters of the potential as shown in Appendix D.
After minimisation, the 32 independent parameters of the model, which we take as input parameters for our numerical calculations. In all the results that follow, we take into account the same theoretical and experimental bounds as in section IV translated to fit the 2HDM+CS model accordingly. able to produce the observed value of a µ while remaining consistent with the d e constraints as shown in Figure 16. However, as mentioned before, such large values of cot β are ruled out due to flavour and/or collider constraints.

VI. CONCLUSIONS AND OUTLOOK
We have studied popular scalar extensions of the Standard Model and their contributions to the muon (g − 2), and electric dipole moment of the electron, eEDM. Concretely, we studied first the real and complex singlet extension of the SM, and second, as the main part of our analysis, we considered the 2-Higgs doublet model (2HDM) of Types I, II, X and Y and the 2HDM extended with the inclusion of a singlet scalar.
In the singlet extension of the SM, CP-violation is introduced by a non-trivial higher dimensional operator connecting the singlet scalar with the Higgs field and the top quark.
We found that while the imaginary part of this coupling can be compatible with the eEDM bounds, a very large real part must be introduced in order to explain a µ .
In the CP-conserving limit, extensive (g − 2) µ studies have been done in different types of 2HDM with varying scalar masses, mixing angles and tan βs in the literature. It has been shown that at 1-loop level contributions to a µ are positive for the CP-even scalars and negative for the CP-odd scalars (and the charged scalar whose negative effect is negligible).
The dominant contribution is from the 2-loop processes to which CP-even scalars contribute negatively and CP-odd scalars contribute possitively. As a result, very light CP-odd scalars have been shown to produce a large enough a µ in large tan β regions for Type II,X and large cot β regons in Type I,Y. It is, therefore, intuitive to expect that by introducing CPviolation, less dramatic values of tan β/cot β or scalar masses are required to produce an adequate a µ contribution.
All our calculations, confirm the known results in the CP-conserving limit and show the well-understood (g − 2) µ behaviour of different 2HDM types for varying masses in a wide range of tan βs. Incremental changes in the scalar masses do not change the a µ drastically.
By introducing CP-violation, which only adds the two CP-violating angles θ 13 and θ 23 to the known input parameters, we show that indeed the value of a µ is affected. This effect, however small, is enhanced by increasing the amount of CP-violation (manifested in the values of θ 13 , θ 23 angles). On the other hand, θ 13 , θ 23 angles are strongly constrained by eEDM experiments which only leave a small window in the parameter space to be explored.
We provide the detailed formulas for a µ and d e contributions to show the subdominant effect of the parameters for which an exemplary value has been chosen in the plots.
In particular, we found that only when all scalars are relatively close in mass, CP-violating Type I and Y 2HDMs explain the muon anomalous magnetic moment in cot β ∼ 10 which is also allowed by the eEDM constraints. However, such large values of cot β are already ruled out by flavour/collider experiments. Therefore, given the current status of the global set of constraints applied on all values of cot β, in the CP-violating 2HDM, there exist no viable parameter space in agreement with both a µ and eEDM bounds. In the low cot β region, Type X remains the only 2HDM type which has a large enough contribution to a µ , while Type I is the preferred type when introducing CP-violation as it contributes minimally to d e .
In the singlet extension of the 2HDM, we show that the 2HDM behaviour is repeated and the model is capable of explaining the (g − 2) µ within the d e bounds when all scalars are relatively close in mass and the Yukawa interactions are of Type I and Y. However, this only occurs in the cot β ≈ 10 region which, again, is ruled out by flavour and collider experiments.
We have presented a robust way to implement the constraints on the muon anomalous magnetic moment and electric dipole moment of the electron on these models. Our central finding is that this allows one to categorically exclude different types of 2HDMs and 2HDM+CS and, consequently, identify most viable Yukawa interaction patterns which would be useful for more general model building based e.g. on the paradigm of minimal flavour violation.
The scalar extensions we have studied are applied in attempts to explain the BAU via electroweak baryogenesis. Since then also CP violation in the new sector needs to be introduced, our results help in establishing phenomenological viability of these models.

Stability of the potential
The scalar potential stability requires the potential to be bounded from below in any direction of the scalar space whose necessary and sufficient conditions are [57] λ 1 > 0, λ 2 > 0, 2. Positive-definiteness of the Hessian -positivity of mass eigenvalues to be a minimum of the potential, the second order derivative matrix must have a positive definite determinant. Similar constrains are achieved by requiring the mass eigenvalues to be positive.

Perturbative unitarity
S-matrix unitarity for 2 to 2 elastic scattering, constrains the value of combinations of λs in the potential [58], [59].
Recently, the BaBar Collaboration has reported a measured ratios BR(B → D * τ ν)/BR(B → D * ν) and BR(B → Dτ ν)/BR(B → D ν) ( = e, µ) to deviate from the SM predictions by 2.7 σ and 2.0 σ, respectively, and their combined deviation is 3.4 σ [68]. Note that these deviations cannot be simultaneously explained by a Z 2 symmetric 2HDM which is flavour conserving, with or withour CP-violation.

A → Zh searches
Using LHC Run-I data, an upper limit on the σ(gg → A)×BR(A → Zh)×BR(h → ff ) has been given [74] for m A = 220-1000 GeV. The upper limit for f = τ (b) is measured to be 0.098 − 0.013 pb (0.57 − 0.014 pb). Our typical gg → H, A cross section is 1 pb for m H,A = 200 GeV and tan β 2, and the A → Zh branching ratio is 10 −2 .
Considering that the decay rate of the SM-like Higgs boson does not change much from the SM prediction, the h → τ τ (bb) branching ratio is ∼ 7%(60%), meaning that our cross section is well below the upper limit.

Gauge bosons width
The contribution of the extra scalars to the total gauge bosons widths [109] constrain the scalar masses:

Direct searches for charged scalars and their lifetime
Moreover, to satisfy the bounds from long-lived charged particle searches, an upper limit is set on their lifetime to be τ H ± ≤ 10 −7 s, to guarantee their decay within the detector, which translates to an upper bound on their total decay width Γ tot H ± ≥ 6.58 × 10 −18 GeV.
The zero entries in the mass-squared matrix are the result of the imposed Z 2 symmetry and vanishing parameters in Eq. (78).
The angles defining the rotation matrix, R in Eq. (84), can be calculated in the general case, however the expressions are too lengthy to present here. We only show the values of