Electron and muon $g-2$ anomalies in general flavour conserving two Higgs doublets models

In general two Higgs doublet models (2HDMs) without scalar flavour changing neutral couplings (SFCNC) in the lepton sector, the electron, muon and tau interactions can be decoupled in a robust framework, stable under renormalization group evolution. In this framework, the breaking of lepton flavour universality (LFU) goes beyond the mass proportionality, opening the possibility to accommodate a different behaviour among charged leptons. We analyze the electron and muon $(g-2)$ anomalies in the context of these general flavour conserving models in the leptonic sector (g$\ell$FC). We consider two different models, I-g$\ell$FC and II-g$\ell$FC, in which the quark Yukawa couplings coincide, respectively, with the ones in type I and in type II 2HDMs. We find two types of solutions that fully reproduce both $(g-2)$ anomalies, and which are compatible with experimental constraints from LEP and LHC, from LFU, from flavour and electroweak physics, and with theoretical constraints in the scalar sector. In the first type of solution, all the new scalars have masses in the 1--2.5 TeV range, the vacuum expectation values (vevs) of both doublets are quite similar in magnitude, and both anomalies are dominated by two loop Barr-Zee contributions. This solution appears in both models. In a second type of solution, one loop contributions are dominant in the muon anomaly, all new scalars have masses below 1 TeV, and the ratio of vevs is in the range 10--100. The second neutral scalar $H$ is the lighter among the new scalars, with a mass in the 210--390 GeV range while the pseudoscalar $A$ is the heavier, with a mass in the range 400--900 GeV. The new charged scalar $H^\pm$ is almost degenerate either with the scalar or with the pseudoscalar. This second type of solution only appears in the I-g$\ell$FC model. Both solutions require the soft breaking of the $\mathbb{Z}_{2}$ symmetry of the Higgs potential.


Introduction
After an improved determination of the fine structure constant [1], a new anomaly has emerged [2] concerning the anomalous magnetic moment of the electron a e = (g e −2)/2: there is a discrepancy among the experimental determination and the Standard Model (SM) prediction [3][4][5][6], δa e ≡ a Exp e − a SM e = −(8.7 ± 3.6) × 10 −13 . (1) Another well known and long standing anomaly concerns the anomalous magnetic moment of the muon [7][8][9], It is to be noticed that the anomalies in eqs. (1) and (2) have opposite sign. Because of this difference of sign, several New Physics solutions addressing eq. (2) tend to be eliminated as solutions to both eqs. (2) and (1). In particular, many popular models in which the anomaly scales with the square of the lepton mass [10] tend to generate too large δa e with the wrong sign. Some authors [11] argue that if the origin of both anomalies is beyond the SM, the corresponding model must incorporate some sort of effective decoupling between µ and e. Recent beyond-SM explanations of both anomalies can be found in [12][13][14][15][16][17][18][19][20][21][22]. A minimal extension of the SM is the two Higgs doublets model (2HDM) [23] which introduces, in general, a new set of flavour structures in the Yukawa sector. Those structures could implement the decoupling between µ and e required to explain δa µ and δa e . Of course, the most popular 2HDMs shaped by symmetries [24,25], the so-called 2HDMs of types I, II, X and Y [26][27][28], do not implement in a straightforward way this decoupling between µ and e, since the new Yukawa couplings in the lepton sector are proportional to the charged lepton mass matrix. Going one step further in generality, the so-called "Aligned" 2HDM (A2HDM) [29] gives up stability of the model under the renormalization group evolution (RGE) [30] (the model is not shaped by a symmetry). The A2HDM cannot, however, incorporate some effective decoupling between µ and e since the new Yukawa structures are still proportional to the fermion mass matrices. It is nevertheless interesting to note that the lepton sector of the A2HDM is stable under one loop RGE 4 [31][32][33]: scalar flavour changing neutral couplings (SFCNC), absent at tree level, do not appear at one loop. A generalization of the A2HDM is the general flavour conserving (gFC) 2HDM where, at tree level, all Yukawa couplings are diagonal in the fermion mass basis [34,35]. As in the A2HDM, it has been shown that the charged lepton sector of the gFC-2HDM is one loop stable under RGE, in the sense that SFCNC, absent at tree level, are not generated at one loop. This implies that a well behaved and minimal 2HDM that can implement the effective decoupling among µ and e is a gFC-2HDM in the leptonic sector. Since this is all what is required to address the two anomalies in eqs. (1)-(2), we consider two minimal models in which the quark sector is a 2HDM of either type I or type II, while the lepton sector corresponds to a gFC-2HDM. We refer to them as models I-g FC and II-g FC respectively. Note that these models do not have SFCNC at tree level neither in the quark nor in the lepton sectors. Additionally, the new Yukawa couplings in the lepton sector are independent of the charged lepton mass matrix. In the appropriate limits, model I-g FC can reproduce 2HDMs of types I and X while, similarly, model II-g FC can reproduce 2HDMs of types II and Y. In this sense model I-g FC is a generalization of 2HDMs of types I and X, while model II-g FC is instead a generalization of 2HDMs of types II and Y. The convenience of adopting this kind of generalized flavour conserving 2HDMs for phenomenological analyses was advocated in [35]. The paper is organised as follows. In section 2 the models are presented in detail. In section 3, the one and two loop contributions to a are revisited. In a simplified analysis it is shown that, with dominating two loop contributions, a new simple scaling law follows: δa e δa µ = m e Re (n e ) m µ Re (n µ ) , with n e , n µ , the new Yukawa couplings of the charged leptons, in the lepton mass basis. In order to solve the discrepancies in eqs. Re (n e ) (4) in the framework of models I-g FC and II-g FC. Besides solutions with dominating two loop contributions, an additional possibility with relevant one loop contributions is also analysed (similarly to [2]). In section 4, a number of constraints, relevant for a full analysis, is addressed in detail. In section 5, the main results of such a full analysis are presented and discussed. Details concerning some aspects of the different sections are relegated to the appendices.

The I-g FC and II-g FC models
In 2HDMs, the Yukawa sector of the SM is extended to whereΦ j = iσ 2 Φ * j , and, as in the SM, neutrinos are massless (in the leptonic sector only two flavour structures are present). The vacuum expectation values v j of the scalar fields Φ j are in general non-vanishing; expanding around the vacuum appropriate for electroweak symmetry breaking, The so-called Higgs basis [36][37][38] is defined by in such a way that only one of the scalar doublets has a non-vanishing vacuum expectation value: . Expanding around the vacuum the would-be Goldstone bosons G 0 , G ± and the physical charged scalar H ± are already identified. The neutral scalars {H 0 , R 0 , I 0 } are not, in general, the mass eigenstates. It is in the Higgs basis where the Yukawa couplings have the simplest interpretation: Since only the neutral component of H 1 has a non-vanishing vacuum expectation value, the Yukawa couplings M 0 f , for all the fermions f = u, d, , will be the corresponding mass matrices. Going directly to the fermion mass bases, we obtain the relevant new Yukawa structures where M f are the diagonal fermion mass matrices for f = u, d, and N f are the new flavour structures that may be able to explain the electron and muon anomalies in eqs. (1)- (2). As motivated previously, we consider two models.
• Model I-g FC is defined by 5 The couplings N u , N d are the same as in 2HDMs of types I or X.
• Model II-g FC is defined by The couplings N u , N d are the same as in 2HDMs of types II or Y.
In both models N is diagonal, arbitrary and stable at one loop level under RGE, in the sense that it remains diagonal. Note that the effective decoupling among the new couplings of e and µ that is required in order to explain the g − 2 anomalies is simply obtained from the independence of n e and n µ .
To complete the definition of the model, in accordance with the fact that the quark sector is a type I or type II 2HDM, we adopt a Z 2 symmetric scalar potential For µ 2 12 = 0, the Z 2 symmetry is softly broken. This potential generates the mass matrix of the neutral scalars M 2 0 , which is diagonalised by a 3 × 3 real orthogonal matrix R The physical neutral scalars {h, H, A} are: The Yukawa couplings of the neutral scalars are flavour conserving 6 : In the following we focus on a simplified case: we assume that (i) there is no CP violation in the scalar sector and (ii) the new Yukawa couplings do not introduce further sources of CP violation, that is N = N † , Im (n ) = 0. In the scalar sector, this corresponds to with s αβ ≡ sin(α−β) and c αβ ≡ cos(α−β), where α− π 2 is the mixing angle parametrizing the change of basis from the fields in eq. (6) to the mass eigenstates in eq. (15). The alignment limit, in which h has the same couplings of the SM Higgs, corresponds to s αβ → 1. Table 1 collects the Yukawa couplings, as expressed in eq. (16), in both models I-g FC and II-g FC. The absence of CP violation is clear from the exact relation a S f b S f = 0 [39]; one important consequence of this simplification is the absence of new contributions generating electric dipole moments (EDMs), in particular contributions to the electron EDM d e , which is quite constrained: |d e | < 1.1 × 10 −29 e·cm [40,41]. The Yukawa couplings of H ± are of the form 6 The general form of the Yukawa couplings is given, for completeness, in appendix A. where q + 1 2 ,j = u j , q − 1 2 ,j = d j , l + 1 2 ,j = ν j , l − 1 2 ,j = j , and the corresponding couplings are given in Table 2. Note that the Yukawa couplings of the charged leptons in Tables 1 and 2 are the same in both models I-g FC and II-g FC.

The new contributions to δa
The full prediction a Th of the anomalous magnetic moments of = e, µ has the form with a SM the SM contribution and δa the corrections due to the model. To solve the discrepancies in eqs. (1)-(2), the aim is to obtain δa e δa Exp e and δa µ δa Exp µ within models I-g FC and II-g FC. It is convenient to introduce ∆ following The quantities K collect the typical factors arising in one loop contributions; since K e 5.5 × 10 −14 and K µ 2.3 × 10 −9 , in order to reproduce the anomalies we roughly need ∆ e −16 , It is well known that in the type of models considered here, both one loop [42] or two loop Barr-Zee contributions [43][44][45][46][47] can be dominant. Complete expressions used in the full analyses of section 5, can be found in appendix B. For the moment, we consider in this section two approximations: we only keep leading terms in a (m /m S ) 2 expansion (for the different scalars S = h, H, A), and the alignment limit s αβ → 1. With these approximations, the one loop contribution to ∆ in eq. (20) is where Equation (22) and thus the dominant contributions to ∆ (1) in eq. (22) are the logarithmically enhanced contributions from H and A. Then, ∆ e −16 can only arise from the A contribution because of the sign: this would require [Re (n e )] 2 ∼ m 2 A , which can easily violate perturbativity requirements in the Yukawa sector or contraints from resonant dilepton searches. Consequently, we do not expect an explanation of δa e in terms of one loop contributions. For δa µ , any relevant one loop contribution in eq. (22) should arise from the H contribution attending, again, to the required sign. Such a contribution needs [Re (n µ )] 2 ∼ [m H /4] 2 , that is a not too heavy H (in order to have reasonably perturbative n µ ) and m A > m H in order to avoid cancellations with wrong sign contributions. In the same approximation (leading m /m S terms and s αβ → 1), the two loop contributions are dominated by Barr-Zee diagrams in which the internal fermion loop is connected with the external lepton via one virtual photon and one virtual neutral scalar H or A. The leading contribution to ∆ in eq. (20) is (for detailed expressions, see appendix B) The factor F depends on the masses of the fermions in the closed loop, on the couplings of those fermions to H and A, and, of course, on m H and m A ; it is consequently different in models I-g FC and II-g FC: where and m H m A , it is easy to realize that for m H ∼ 1 − 2 TeV, δa e can be explained with Yukawa couplings Re (n e ) ∼ 3 − 7 GeV (Re (n e ) > 0 gives the right sign of δa e ). If we assume that δa µ must also be explained by the same kind of dominant Barr-Zee two loop contributions, which are independent of the specific charged lepton, it is straightforward that With this relation, the origin of the different signs of δa e and δa µ relies on the freedom to have Re (n e ) and Re (n µ ) with opposite signs, as anticipated in eq. (2). In terms of Re (n µ ), with the same assumptions ( The previous arguments apply to both models, I-g FC and II-g FC, since 4(f tH + g tA ) is the dominant term in both F I and F II . Attending to the flavour constraints discussed in section 4 (B d and B s meson mixings, b → sγ radiative decays), t β 1 are excluded in 2HDMs of types I and II, and thus also in I-g FC and II-g FC models, there is no need to discuss the t β 1 regime. Let us now analyse the two loop Barr-Zee contributions in eq. (26) for large values of t β . As a reference, consider the analysis above with t β ∼ 1 and m A ∼ m H ∼ 1 − 2 TeV; for definiteness we now take t β = 50. For large t β , it is clear that these contributions in models I-g FC and II-g FC are quite different. Starting with model I-g FC, in order to maintain the right value of δa e , the t β suppression in Re (n e ) t −1 β (f tH + g tA ) can be compensated with smaller m H , m A , and larger Re (n e ). For example, m A ∼ m H ∼ 200 GeV gives an increase of the loop functions by a factor of 10 with respect to m A ∼ m H ∼ 1 − 2 TeV; increasing then Re (n e ) by a factor of 5, the suppression t −1 β = 1/50 is compensated. Therefore, the discrepancy in δa e can be explained in the I-g FC model through two loop contributions, for large values of t β and Re (n e ) ∼ 15 − 35 GeV. The question now is if one can explain, with the two loop contributions, the muon anomaly δa µ . Attending to eq. (28), one would need Re (n µ ) ∈ −[225; 505] GeV, which would be in conflict with perturbativity requirements in the Yukawa sector. However, as the discussion on one loop contributions after eq. (22) shows, for light m H , e.g. m H ∈ [200; 400] GeV, δa µ can be obtained with H-mediated one loop contributions, and m A > m H to avoid cancellations. One needs |Re (n µ ) | ∼ m H /4, in which case |Re (n µ ) | ∈ [50; 100] GeV is acceptable from the perturbativity point of view. Summarizing the previous discussion, we envisage, at least, two kinds of solutions: • The first is realized with scalars having masses in the 1-2 TeV range, t β ∼ 1, and both anomalies produced by two loop Barr-Zee contributions. The coupling of electrons to the new scalar and pseudoscalar, Re (n e ), should be in the few GeV range. Following eq. (28), the corresponding muon coupling is larger. This first solution can appear, a priori, in both I-g FC and II-g FC models. In section 5 we refer to this first type of solution as "solution [A]".
• The second solution corresponds to a lighter H, m H ∈ [200; 400] GeV and a heavier A; the required values of t β are larger, t β 1. In this second solution, the electron anomaly is obtained with two loop contributions while the muon anomaly is one loop controlled; contrary to the first solution, there is no linear relation among Re (n µ ) and Re (n e ). This second kind of solution can clearly appear in the I-g FC model, but in this simplified analysis it cannot be elucidated if this possibility is also open in the II-g FC model. Anticipating the results of the complete numerical analyses of section 5, this will not be the case: within the II-g FC model there is no solution with large t β and relatively light H. In section 5 we refer to this second type of solution as "solution [B]". Notice also that, a priori, this second kind of solution might be obtained with both signs of Re (n µ ).

Constraints
In this section we discuss the different constraints that can play a relevant role in the detailed analyses of section 5.

Scalar sector
For the scalar sector, we require the potential to be bounded from below [48], we also require the quartic parameters to respect perturbativity and perturbative unitarity in 2 → 2 scattering [49][50][51][52] (see also [53][54][55]), and finally the corrections to the oblique parameters S and T have to be in agreement with electroweak precision data [56]. In the fermion sector, in order to forbid too large values of the new couplings n which would conflict with perturbativity requirements, we include a contribution of the following form to the χ 2 function driving the numerical analyses We choose n 0 = 95 GeV and σ n 0 = 1 GeV. One could have adopted a crude requirement such as imposing for example |n | ≤ 100 GeV with a sharp cut: eq. (29) is simply a smooth version (more convenient for numerical purposes) of that kind of requirement.

Higgs signal strengths
Concerning the 125 GeV Higgs-like scalar, agreement with the observed production × decay signal strengths of the usual channels is also imposed [57][58][59][60]. The measured signal strengths, with uncertainties reaching the 10% level, tend to favour the alignment limit in the scalar sector; it is to be noticed that since the models require |Re (n e ) | m e and |Re (n µ ) | m µ , the Higgs measurements in the µ + µ − channel such as [61] and [62] are even more effective in forcing that alignment limit. Constraints on the total width Γ h , arising from off-shell (ggF+VBF)→ h ( * ) → W W ( * ) [63], are also included [64,65], even if in the models considered here their effect is negligible in the alignment limit. For additional details, see [35,66,67].

H ± mediated contributions
Flavour transitions mediated by W ± can receive new contributions where W ± → H ± . For tree level processes involving leptons, one refers to "Lepton Flavour Universality" constraints; we also consider constraints at the loop level in the quark sector. One may also worry about too large H ± -mediated contributions to processes like j → k γ: since in the present models we are considering massless neutrinos, lepton family numbers are conserved -i.e. there is a [U (1)] 3 symmetry -, and such processes are absent.

Lepton Flavour Universality
Contributions mediated by H ± modify the leptonic decays j → k νν: where f(x) and g(x) are the usual phase space integrals 7 [68], x kj ≡ (m k /m j ) 2 ) and ∆ j k RC correspond to QED radiative corrections and most importantly, The notation g S,RR j→k reflects the fact that in the present models the new contributions only affect, in an effective description, the operatorν L jR¯ kR ν L . The ratios give the following constraints [69]: In addition, measurements of decay spectra with polarized leptons impose g S,RR µ→e < 0.035 at 90% CL, g S,RR τ →µ < 0.72 at 95% CL, g S,RR τ →e < 0.7 at 95% CL. (34) Besides purely leptonic decays j → kν ν, leptonic decay modes like K, π → eν, µν and τ → Kν, πν, provide additional constraints on the different n (together with the t β dependence of the quark couplings with H ± ). In particular, we consider ratios where the quark content of P + is u idj and Notice the enhanced sensitivity of these observables due to the n a m a factor: unlike the SM amplitude, the new H ± -mediated amplitude is not helicity suppressed. For ratios involving τ + → P + ν decays, the expressions are unchanged. The actual constraints [69][70][71] read All these LFU violating effects scale with 1/m 2 H ± and therefore one expects that in both models, I-g FC and II-g FC, the effects for large m H ± are much more suppressed, including in particular the solution [A] region introduced in section 3. This is quite clear in the pure leptonic decays, where the most relevant constraints, eq. (33) and g S,RR µ→e in eq. (34), can be comfortably satisfied, giving a contribution to the corresponding χ 2 at a level similar to the SM. Since solution [A] corresponds to t β ∼ 1, the effects in semileptonic processes are similar in both models, with the effects in kaons larger by a factor of 10 than the effects in pions. The leading contribution to R K µe −1 is of the order of the uncertainty: since in that channel there is essentially a change of sign between the contributions in models I-g FC and II-g FC, it turns out that in the II-g FC case the corresponding χ 2 value can improve over the SM one, while in the I-g FC case it is the other way around. In any case, for solution [A], these differences are small. For solution [B], the situation is different since we have: solution [B], model II-g FC,  Table 1: altogether, one expects that these two constraints tend to disfavour t β 1 and light H ± . We refer to [73][74][75] for further details.
4.4 e + e − → µ + µ − , τ + τ − at LEP LEP measured e + e − → µ + µ − , τ + τ − with center-of-mass energies up to √ s = 208 GeV: although s-channel contributions with virtual H and A do not interfere with SM γ and Z mediated contributions, for light H, A, the resonant enhancement together with the large couplings to leptons might give predictions in conflict with data (e.g. [76]). The effect of these LEP constraints is, essentially, to forbid values of m H , m A below 210 − 215 GeV.
For production, the narrow width approximation (NWA) is considered; the widths of H, A and H ± can reach ∼ 10% of their respective masses: if one incorporates finite width effects through the convolution of the cross section computed in the NWA with a (relativistic) Breit-Wigner distribution for the scalars, the computed signal would be partially "diluted". In this sense, using the NWA is conservative since it gives stronger pointwise bounds. Production cross sections incorporate corrections associated to the modified fermion-scalar vertices in the following manner. For generic interaction terms the gluon-gluon fusion production cross section reads with x q ≡ (m q /m S ) 2 , and F (x) andF (x) the loop functions corresponding to scalar or pseudoscalar couplings, respectively; σ[pp → S] SM-like [ggF] can be found in [88][89][90][91]. This simple recipe also gives sufficiently good agreement with results for a SM-Higgs-like neutral pseudoscalar, which can be found in [91][92][93][94]. The couplings in eq. (40) for S = H, A in each model can be read in Table 1. Similarly, for the production cross sections pp → H ± tb (i.e. H ± in association with tb), we refer to [95,96], which provide results, labeled here σ [Ref] , for a type II 2HDM with t β = 1. For arbitrary values of t β , we use As an additional check, (i) the previous cross sections and (ii) the computations of the decay branching ratios of the scalars, have been compared with the results of MadGraph5 aMC@NLO [97] at leading order. With FeynRules [98] and NLOCT [99,100], the needed universal Feynrules Output at NLO of the I-g FC and II-g FC models is produced. A good agreement in the gluon-gluon fusion production cross section is found, given the fact that the MadGraph5 aMC@NLO calculation is at leading order (one loop in this case). For the branching ratios, there is complete agreement.

δa constraints for the numerical analyses
The main motivation of this work is to accommodate the departures from SM expectations in the anomalous magnetic moments of both electron and muon. We now discuss how these departures are implemented as constraints in the analyses presented in section 5. The g − 2 anomalies δa Exp = a Exp − a SM in eqs.
The theoretical prediction in the present models is a Th = δa + a SM and thus a simple and natural measure of their ability to accommodate the experimental results is a χ 2 function where δa Exp = c ± σ in eq. (43). The interest in explanations of the experimental results in terms of non-SM contributions is due to the 3 − 4σ deviation χ 2 0 (0, 0) 15. For the numerical exploration of the regions in parameter space which could provide such an explanation, rather than including a contribution corresponding to eq. (44) in the likelihood function of the models, we impose a stronger requirement: instead of χ 2 0 (δa e , δa µ ) we use in order to guarantee that the models reproduce both anomalies simultaneously within less than 1 2 σ of the central values. This approach is adopted in order to ensure that, when representing allowed regions at a given confidence level in the next section, they do not include regions where one or both anomalies are only partially reproduced. For illustration, Figure 2 shows the allowed region obtained in the complete numerical analyses (which is identical in both models); that is, in the results of section 5, within all the represented allowed regions, the values of δa e and δa µ belong to the allowed region of Figure 2. Notice, finally, that the SM prediction a SM includes Higgs-mediated contributions: since these are just the h mediated contributions for exact alignment s αβ = 1, they have to be subtracted from the New Physics contributions to δa mediated by h (quantitatively, however, this subtlety is rather irrelevant).

Results
As discussed in section 3, we expect, at least, two different types of solution to the δa anomalies. In the following we refer to them, as anticipated, as solutions  [55,101,102], i.e. in that case one cannot have scalars heavier than ∼ 1 TeV (without violating requirements such as perturbativity). On the other hand, concerning solution [B], the exact Z 2 symmetry does not allow large t β . Introducing µ 2 12 = 0 removes both obstacles. In the plots to follow, the results from the "No LHC" analysis correspond to lighter red regions while the results from the full analysis correspond to darker blue regions. The regions represented are allowed at 2σ (for a 2D − χ 2 distribution); the χ 2 or likelihood function used in the numerical analysis implements the constraints of section 4. In Figure 3 we have Re (n µ ) versus Re (n e ); the full analysis shows, clearly, three disjoint regions. As indicated in the figure, the bottom left small region corresponds to solution [A], and reproduces the linear relation of eq. (28), arising from the explanation of both anomalies through two loop Barr-Zee contributions. The largest blue region to the bottom right corresponds to solution [B] with Re (n µ ) < 0, where δa e is two loop dominated while δa µ also receives significant one loop contributions. In this region there is no linear relation among Re (n e ) and Re (n µ ). For Re (n µ ) > 0, solution [B] corresponds to the top blue region (the subindex ± in [B ± ] refers to the sign of Re (n µ )). It is clear, from the underlying red region, that excluding LHC searches, there is a smooth transition between solutions [A] and [B − ] where all kinds of contributions must be considered: we recall that the numerical analyses incorporate the complete expressions of appendix B, which consider one and two loop contributions with all possible fermions in the fermion loop of Barr-Zee terms. It is important to stress that, since the lepton couplings to H and A can be quite large, it is mandatory to include all leptons in the computation of Barr-Zee terms. (ii) in all cases, m H < m A . This last inequality, as analysed later, must allow the decay A → HZ (additionally, either H ± → HW ± or A → H ± W ∓ would also be allowed); together with the electroweak precision constraints (in particular the oblique parameter T ), this forces either m A = m H ± or m H = m H ± . These two results match nicely with the need for H to be as light as possible (LEP constraints will force in any case m H ≥ 210 GeV) in order to produce the main contribution (at one loop) to δa µ . Figure 6 shows the resonant [pp] ggF → S → µ + µ − cross sections with respect to m S for S = H, A. The black line shows the LHC bounds included in the full analysis. In gluon-gluon fusion production, for the same scalar mass, the gluon-gluonpseudoscalar amplitude is 2-6 times larger than the corresponding gluon-gluon-scalar amplitude (that is 2 2 −6 2 larger pseudoscalar vs. scalar production cross sections). One could have expected, attending to this fact, that the constraints from LHC searches tan β
[B] shows it is rather σ(pp → H) ggF × Br (H → µ + µ − ) which shows how the bounds from LHC searches separate the solutions by excluding m H ∈ [380; 1200] GeV (i.e. eliminating the red region "bridge" connecting the blue regions). Comparing the shape of the allowed regions in Figures 6a and 6b it is also clear that, besides the production cross section, the branching ratios Br (H, A → µ + µ − ) may play an important role.
On this respect, let us start by observing that, since values of |Re (n µ ) | larger  Figure 5e it is clear that large m H > 1 TeV requires t β ∼ 1, while m H < 500 GeV is compatible with a broad range t β ∈ [1; 10 2 ]. This is the last ingredient necessary to interpret the shape of Figure 6b. For m H < 500 GeV, without constraints from LHC searches, the broad range of t β values gives a broad range for σ(pp → H) ggF : since the gluon-gluon fusion production cross section is proportional to t −2 β , and thus for solution [B] there is a substantial suppression of σ(pp → H) ggF due to t β 1. Due to the larger production cross section of a pseudoscalar, despite the t −2 β suppression, LHC searches might rule out pp → A → µ + µ − predictions for solution [B]: as Figure 6a shows, that is not the case. This is clearly achieved through a reduction of Br (A → µ + µ − ); Figures 8f and 8e show that A → HZ contributes decisively to reduce Br (A → µ + µ − ), evade LHC bounds and obtain a viable solution [B]. For this reason, as anticipated, m A > m H + M Z . For the charged scalar H ± , the behaviour of the most relevant decay channels H + → µ + ν, τ + ν, tb, HW ± mirrors the corresponding A → µ + µ − , τ + τ − , tt, HZ, as Figures 8j-8i show. The only minor difference arises for solution [B] in the small region where m H ± m H : in that region, (i) H ± → HW ± is forbidden and (ii) in addition to A → HZ, also A → H ± W ∓ (not shown) has a large branching ratio. Figure 7 shows that resonant τ + τ − searches are less constraining than the corresponding µ + µ − searches in Figure 6. Concerning production of H ± , Figure 9 shows that mA (TeV)

Conclusions
General 2HDMs without SFCNC in the lepton sector are a robust framework, stable under renormalization group evolution, in which the possibility of decoupling the electron, muon and tau interactions is open. In this context, lepton flavour universality is broken beyond the mass proportionality, and a different behaviour among charged leptons can be accommodated in a simple way, without introducing highly constrained SFCNC. We have considered two of these general flavour conserving models in the leptonic sector, to address simultaneously the electron and muon (g − 2) anomalies. These two models, I-g FC and II-g FC, differ in the quark Yukawa couplings, which coincide, respectively, with the ones in type I and in type II 2HDMs. There are two types of solutions that fully reproduce both the muon and electron (g − 2) anomalies, while remaining in agreement with constraints from LEP and LHC, from LFU, from flavour and electroweak physics, and theoretical requirements in the scalar sector. In one solution, all the new scalars have masses in the 1-2.5 TeV range, the vevs of both doublets are quite similar and both anomalies are dominated by two loop Bar-Zee contributions. This solution arises in both models, I-and II-g FC. There is a second type of solution, where one loop contributions are dominant in the muon anomaly, the new scalars have masses below 1 TeV, and the vevs quite different, with a ratio in the range 10-100. Among the new scalars, the second neutral one H is the lighter, with a mass in the range 210-390 GeV, while the pseudoscalar A is the heavier, with a mass in the range 400-900 GeV. The new charged scalar H ± is almost degenerate either with the scalar or with the pseudoscalar. This solution is only available in the I-g FC model. In both solutions, soft breaking of the Z 2 symmetry of the Higgs potential is required, together with lepton Yukawa couplings with values from 1 to 100 GeV. These results imply for LHC searches, in the light scalar solution, that it should be easier to find both charged and neutral Higgses in the muonic channel. The heavy channels, like the top quark channels, are more suited to searches addressing the heavy scalars solution.

A Yukawa couplings
For completeness we show in this appendix the form of the Yukawa couplings in the general case with arbitrary scalar mixing R and couplings N d , N u , N . For neutral scalars they read 8 (46) where s = 1, 2, 3 in correspondence with S = h, H, A; f = u, d, , and, in terms proportional to R 3s , (d) = ( ) = − (u) = 1. The Yukawa couplings of H ± read and V and U are, respectively, the CKM and PMNS mixing matrices 9 .

B.1 One loop contributions
Yukawa interactions (of neutral scalars S) of the form give one loop contributions to the anomalous magnetic moment of lepton of the form with x S ≡ m 2 /m 2 S and For x 1, Yukawa interactions (of charged scalars C ± ) of the form give one loop contributions to the anomalous magnetic moment of lepton of the form where x C = m 2 /m 2 C ± , and