Muon and electron g − 2 anomalies in a ﬂavor conserving 2HDM with an oblique view on the CDF M W value

We consider a type I or type X two Higgs doublets model with a modiﬁed lepton sector. The generalized lepton sector is also ﬂavor conserving but with the new Yukawa couplings completely decoupled from lepton mass proportionality. The model is one loop stable under renormalization group evolution and it allows to reproduce the g − 2 muon anomaly together with the diﬀerent scenarios one can consider for the electron g − 2 anomaly, related to the Cesium and/or to the Rubidium recoil measurements of the ﬁne structure constant. Thorough parameter space analyses are performed to constrain all the model parameters in the diﬀerent scenarios, either including or not including the recent CDF measurement of the W boson mass. For light new scalars with masses in the 0 . 2-1 . 0 TeV range, the muon anomaly receives dominant one loop contributions; it is for heavy new scalars with masses above 1 . 2 TeV that two loop Barr-Zee diagrams are needed. The electron g − 2 anomaly, if any, must always be obtained with the two loop contributions. The ﬁnal allowed regions are quite sensitive to the assumptions about perturbativity of Yukawa couplings, which inﬂuence unexpected observables like the allowed scalar mass ranges. On that respect, intermediate scalar masses, highly constrained by direct LHC searches, are allowed provided that the new lepton Yukawa couplings are fully scrutinized, including values up to 250 GeV. In the framework of a complete model, fully numerically analysed, we show the implications of the recent M W measurement


Introduction
In the search of Physics beyond the Standard Model (SM), disagreement between measurements and theoretical expectations, that is "anomalies", can play the role of beacons to guide our explorations.One longstanding anomaly concerns the anomalous magnetic moment of the muon a µ = gµ−2 2 .The Muon g-2 experiment at Brookhaven [1] and its successor at Fermilab [2,3] have produced the following result where a Exp µ is the experimental observation and a SM µ the SM theoretical expectation .Although there are unsettled discrepancies concerning Hadronic Vacuum Polarization (HVP) contributions to a SM µ [25][26][27], we interpret δa Exp µ in eq. ( 1) as a signal of New Physics (NP).5Besides the muon, recent results concerning the anomalous magnetic moment of the electron might also be interpreted as NP hints [32].On the one hand, perturbative calculations of a e = ge−2 2 , which have reached impressive levels [5,[33][34][35][36], yield a SM e as a series in powers of the fine structure constant α.On the other hand, we have precise measurements of a Exp e such as [37].In the past, such measurements were indeed used to infer values of α.On the contrary, measurements of atomic recoils [38] provide now more precise determinations of α, which give values of a SM from measurements with 133 Cs [39], and δa Exp,Rb e = (4.8± 3.0) × 10 −13 , from measurements with 87 Rb [40].
The present work extends and improves several aspects of [41].
• An improved numerical exploration of the parameter space shows that some unexpected regions of interest can be appropriately covered.
• Some theoretical assumptions like the perturbativity limits on Yukawa couplings had a significant impact on the analysis and were not fully considered.
• The latest Muon g-2 Fermilab result [2,3] consolidates the need of NP brought by the previous Brookhaven result.
• For a Exp e the situation is rather unclear: within the present scenario, accommodating the values in eq. ( 2) or in eq. ( 3) may have non-trivial consequences in the model, since they differ in size and in sign.
• The recent measurement of the W boson mass by the CDF collaboration [96], which disagrees with SM expectations [97], can also be addressed in this context.
All in all, we are entering an era of exclusion or discovery at the LHC and improved analyses of such potential NP hints are necessary.The manuscript is organized as follows.In section 2, the model is presented.Section 3 is devoted to a discussion of general constraints which apply regardless of δa .The new contributions to δa are analysed in section 4. The main aspects of the numerical analysis are introduced in section 5. Next, section 6 contains the results of the different analyses together with the corresponding discussions.Finally, the conclusions are presented in section 7. We relegate to the appendices some aspects concerning different sections.

Model
The 2HDM is based on the SM gauge group with identical fermion matter content 6and an additional complex scalar doublet.Hence, we have Φ j (j = 1, 2) and their corresponding C-conjugate fields defined as Φj ≡ iσ 2 Φ * j , with opposite sign hypercharge.The most general scalar potential of 2HDMs can be written as with real µ 2 11 , µ 2 22 and λ i (i = 1 to 4), whereas µ 2 12 and λ j (j = 5 to 7) are complex in general.We assume that V( Φ 1 , Φ 2 ) has an appropriate minimum at being θ j and v j (v j ≥ 0) real numbers.Taking this into account, the Higgs doublets can be parametrized around the vacuum as GeV) 2 , one can perform a global SU (2) rotation in the scalar space and express the scalar doublets in the so-called Higgs basis [98][99][100] where only one linear combination of the scalar doublets, namely H 1 , has a non-zero vacuum expectation value (vev): The explicit degrees of freedom in this basis are defined by where As we can check, the would-be Goldstone bosons G 0 and G ± get isolated as components of the first Higgs doublet.Likewise, we already identify two charged physical scalars H ± and three neutral fields {H 0 , R 0 , I 0 } that are not, in general, the mass eigenstates.The latter are determined by the scalar potential, which generates their mass matrix M 2 0 .This can be diagonalized by a 3 × 3 real orthogonal transformation, R, as and thus the physical scalars {h, H, A} are given by Neglecting CP violation in the scalar sector, one has where s αβ ≡ sin(α + β) and c αβ ≡ cos(α + β), with π/2 − α being the mixing angle that parametrizes the change of basis from the fields in eq. ( 6) to the mass eigenstates in eq.(12).We should point out that different conventions for eq.( 13) can be found in the literature.
Regarding the Yukawa sector, it is extended to where the couplings Y dj , Y uj and Y j (j = 1, 2) are 3 × 3 complex matrices in flavor space.One should notice that there are only two flavor structures in the leptonic sector because we are not considering right-handed neutrinos.In the Higgs basis, the Yukawa Lagrangian takes the form It is then clear that the matrices M 0 f (f = d, u, ) are the non-diagonal fermion mass matrices since they are coupled to the only Higgs doublet that acquires a non-vanishing vev, i.e., H 1 .
The model we are considering in the quark sector is defined by which is equivalent to N In the leptonic sector, there exist two unitary matrices W L and W R such that both W † L Y i W R (i = 1, 2) get simultaneously diagonalized.It is well-known that the structure of the quark sector can be enforced through a Z 2 symmetry, but this is not the case in the lepton sector.Nevertheless, as it is shown in appendix A, the entire Yukawa structure is stable under one loop renormalization group evolution (RGE) and, therefore, the model is free from unwanted SFCNC.
Going to the fermion mass bases for our I-g FC model -type I in the quark sector and general flavor conserving in the lepton sector-, we get the relevant new Yukawa structures: with and M f (f = u, d, ) the corresponding diagonal fermion mass matrices.Note that the quark couplings N u and N d are those from 2HDMs of type I or X.On the other hand, the matrices N correspond to a general flavor conserving lepton sector.Therefore, they are diagonal, arbitrary and one loop stable under RGE, as it was shown in [43], meaning that they remain diagonal.
We must stress that it is the fact that n e and n µ are completely independent what implements the desired decoupling between electron and muon NP couplings in order to have enough freedom to address the corresponding (g − 2) anomalies.We assume that these couplings are real, i.e., Im(n ) = 0.This prevents us from dangerous contributions to electric dipole moments (EDMs), that are tightly constrained: |d e | < 1.1×10 −29 e•cm [101].
Furthermore, we consider an scalar potential shaped by a Z 2 symmetry that is softly broken by the term µ 2 12 = 0. Hence, we have to take λ 6 = λ 7 = 0 in eq. ( 4).We also assume that there is no CP violation in the scalar sector, so eq.( 13) is fulfilled.
Under these assumptions, the flavor conserving Yukawa interactions of neutral scalars read and those involving charged scalars are with i, j = 1, 2, 3 summing over generations.It is easy to check that h presents the same couplings as the SM Higgs boson when we take the scalar alignment limit, i.e., s αβ → 1.

General constraints
Before addressing the different contributions to the anomalous magnetic moments δa , we discuss in this section some general constraints which are relevant in the scenario under consideration.By "general" we mean that they do not depend specifically on the values of Re (n e ), Re (n µ ), δa e and δa µ .Furthermore, their effects can be understood in simple terms.
• Alignment.The couplings of the scalar h, assumed to be the SM-Higgs-like particle with m h = 125 GeV, deviate from SM values through the scalar mixing in eq. ( 13).Measurements of the signal strengths in the usual set of production mechanisms and decay channels impose c αβ 1. Concerning the scalar sector, we are thus in the alignment limit.
In 2HDMs, in the alignment limit mentioned above, one can observe that the corrections to S and T are kept under control when either m H ± m A or m H ± m H , as shown in figure 1a.Recently, the CDF collaboration announced a measurement of the W boson mass which disagrees with SM expectations [96].In fits of electroweak precision observables this disagreement can be translated into values of the oblique parameters (∆S, ∆T ) = (0, 0) [103,104] (although fits including ∆U have also been considered, we focus on the case ∆U = 0, appropriate here).In order to "explain" the CDF M W "anomaly" one can thus consider (∆S, ∆T ) constraints from [103,104] instead of eq. ( 22).We can consider, in particular, (i) the "conservative scenario" in [103] which combines the CDF with previous measurements and gives ∆S = 0.086 ± 0.077 , ∆T = 0.177 ± 0.070 , ρ = 0.89 , (ii) the results in [104] which solely use the CDF measurement and give ∆S = 0.15 ± 0.08 , ∆T = 0.27 ± 0.06 , ρ = 0.93 .
In the alignment limit, for m H ± = 1 TeV, eqs.( 23) and (24) give the allowed regions represented in figures 1b and 1c respectively.In sharp contrast with figure 1a, notice in figures 1b and 1c how near degeneracy of the three new scalars is excluded, and how even near degeneracies m H ± m A or m H ± m H are quite disfavored.Furthermore, notice that the 1σ region (2D-∆χ 2 ≤ 2.23) does not appear in figure 1c: contrary to eq. ( 23), with eq. ( 24) one cannot obtain the minimum χ 2 Min with m H ± = 1 TeV.• H ± -induced FCNC.The charged scalar H ± can contribute to ∆F = 1 and ∆F = 2 FCNC processes like b → sγ and B q -Bq mixings (for example, through SM-like box diagrams for B q -Bq in which W ± are replaced with H ± ).The dominant contributions involve virtual top quarks as in the SM, with couplings including now t −1 β factors.Keeping those contributions within experimental bounds only allows, roughly, the colored region in figure 2. For each value of m H ± there is a lower bound on t β .See [105][106][107] for further details.• Scalar sector perturbativity.Additional constraints on scalar masses vs. t β arise from perturbativity requirements on the quartic coefficients of the scalar potential and from perturbative unitarity of 2 → 2 scattering amplitudes [108][109][110][111][112][113][114].With a Z 2 symmetric potential, it is difficult to obtain masses above 1 TeV and values of t β larger than 8. Larger values of the masses and larger values of t β can be nevertheless obtained with the introduction of a soft symmetry breaking term µ 2 12 = 0 in eq. ( 4) [114,115].
• Gluon-gluon fusion production cross section.Let us consider the production cross section of H and A through the one loop gluon-gluon fusion process.In the scalar alignment limit, one can read from eq. ( 20) that the same t −1 β factor applies to both pure scalar H and pure pseudoscalar A couplings with the top quark in the triangle loop: The corresponding loop functions F H and F A [116][117][118][119][120][121] are different due to the scalar or pseudoscalar character: Figure 3 shows as a function of the scalar mass.It is clear that the pseudoscalar A has a larger gluon-gluon production cross section than the scalar H for m A = m H (up to a factor of 6 for m A = m H = 2m t ).Since dimuon searches [pp] ggF → S → µ + µ − at the LHC can be rather constraining for scalar masses m S < 1 TeV, one can expect that in that low mass region m A > m H .One could have worried about the validity of this expectation in case Br (A → µ + µ − ) Br (H → µ + µ − ), but the only way to achieve a suppression of Br (A → µ + µ − ) relative to Br (H → µ + µ − ) is through the existence of A → HZ decays, which are only available if m A > m H , and thus cannot change that expectation.
Figure 3: Loop functions controlling gluon-gluon production cross sections of scalars.
• e + e − → µ + µ − at LEP. Sizable n e and n µ are necessary ingredients for the contributions to a e and a µ involving the new scalars H, A and H ± .Data from LEP [122] on e + e − → µ + µ − with √ s up to 210 GeV are sensitive to s-channel H and A mediated contributions (contrary to the LHC gluon-gluon fusion process, being scalar or pseudoscalar does not change the sensitivity of LEP data).One can roughly expect that agreement with LEP data imposes m A , m H ≥ 210 GeV.

Contributions to δa
The complete prediction of the anomalous magnetic moment a Th , = e, µ, is where a SM is the SM contribution and δa the NP correction.The anomalies in eqs.( 1)-( 3) are "solved" for δa e = δa Exp e and δa µ = δa Exp µ .We introduce for convenience ∆ such that For while for δa e one needs ∆ Cs e −16 , where the superscript corresponds to the different values in eqs.( 2) and ( 3).
In the model considered here, it is well known that both one loop and two loop (of Barr-Zee type) contributions can be dominant.In this section we analyse both types of contributions in the scalar alignment limit s αβ → 1 and keeping only leading terms in

S
, S = H, A, H ± .Full results, used for instance in the numerical analyses, can be found in appendix B.

One loop contributions to δa
The one loop result ∆ (1) has contributions from H, A and H ± .With the approximations mentioned above and the couplings in eqs.( 20) and ( 21), we have where The range of interest in our analyses will be m S ∈ [0.2; 2.5] TeV, in which case while In eq. ( 31), the H contribution is positive, the A contribution is negative and the H ± contribution is negligible.One can then anticipate the following.
• The muon anomaly ∆ µ 1 can only be explained with the one loop H contribution and provided Considering |n µ | < 250 GeV, a priori there could be a one loop explanation of δa µ for m H < 1 TeV.Since the A contribution has opposite sign, if m A ∼ m H a substantial cancellation would occur.As discussed in section 3, it is precisely for light H that one expects m A > m H , in which case that cancellation is largely avoided and a one loop H explanation viable.For heavier m H , the muon anomaly needs other contributions.
• For the electron Cs anomaly, ∆ Cs e −16 can only be explained with the one loop A contribution provided For |n e | < 250 GeV, this would require the pseudoscalar A to be rather light, m A < 300 GeV.On the other hand,

Two loop contributions to δa
The dominant two loop contributions are the Barr-Zee ones.Diagrammatically they correspond to contributions where a closed fermion loop is attached to the external lepton through two propagators: one photon and one of the new scalars H, A. In the scalar alignment limit, It is important to notice that these contributions are linear in n .Detailed expressions are provided in appendix B. In eq. ( 38) we have 2α The function F depends on the masses of the fermions in the closed loop, their couplings to H and A, and on m A and m H . Considering the dominant contributions from top and bottom quarks, and also from tau and muon leptons since n τ and n µ are free parameters, with The functions f (z) and g(z) are defined in appendix B. It is to be noticed that (i) f (z) ∼ g(z) in the range of interest, (ii) larger values correspond to heavier fermions, (iii) for the top quark loop, f and g vary between 0.08 and 1 in the relevant range of scalar masses, m S ∈ [0.2; 2.5] TeV.
• If the electron anomaly is to be obtained through the two loop contributions, and thus from ∆ Cs e , Re (n e ) F 1.8 GeV , from ∆ Rb e , Re (n e ) F −1.0 GeV . ( The sign and the magnitude of F is fixed by the Re (n e ) value to fix δa e .
• For m H > 1 TeV, two loop contributions are necessary to explain the muon anomaly, in which case If follows that, for m H > 1 TeV, for ∆ Cs e and ∆ µ , Re (n µ ) ∼ −13Re (n e ) , for ∆ Rb e and ∆ µ , Re (n µ ) ∼ 23Re (n e ) . ( These correlations show that, in the present framework, the independence of n e and n µ is essential to explain the different sign of ∆ Cs e and ∆ µ .This sign difference is challenging for many scenarios addressing simultaneously both anomalies.In this sense, addressing ∆ Rb e and ∆ µ is less challenging.

Analysis
In section 3 we have discussed some general constraints that apply without regard to the values of n e and n µ of interest to reproduce the δa anomalies; in section 4 we have explored the obtention of the δa anomalies through one and two loop contributions.It is now time to present the main aspects of our detailed numerical analyses.The goal of the numerical analyses is to explore the parameter space of the model and map the different regions where a chosen set of relevant constraints is satisfied and the δa anomalies are explained in terms of the new contributions.The independent parameters of the model are 2  12 , c αβ } control the scalar sector (together with v and m h ) while {Re (n e ) , Re (n µ ) , Re (n τ )} give the lepton Yukawa couplings (quark Yukawa couplings are fixed by t β ).The set of relevant constraints includes the following.
• Boundedness from below of the scalar potential [123], perturbativity of quartic couplings and perturbative unitarity of high energy 2 → 2 scattering in the scalar sector [110].
• Corrections to the oblique parameters S and T in agreement with electroweak precision data [97,102].
For additional details on the different constraints we refer to [41].The constraints are typically modelled with a gaussian likelihood or an equivalent χ 2 term, the overall likelihood is sampled over parameter space using Markov chain Monte Carlo techniques in order to obtain the regions where (best) agreement with the constraints is obtained.There are two final aspects of central importance which require a specific discussion: (i) how are the anomalies included in the analyses, (ii) what ranges are considered for the n parameters.
Concerning the a anomalies, the situation for δa Exp µ is clear: one should consider eq. ( 1).On the contrary, for δa Exp e the situation is not settled: we have eq.( 2) and eq.( 3), which are rather incompatible.In order to have a complete picture, we analyse both cases separately.Furthermore, we also consider two additional possibilities concerning δa Exp e : • despite the marginal compatibility of δa We will refer to these separate analyses as "a Cs e ", "a Rb e ", "a Avg e ", "a Bound e ".For their implementation in the analyses, we assign a joint χ 2 contribution (corresponding to a gaussian factor in the likelihood) where c is the experimental central value and s is the experimental uncertainty divided by 4. The scope of this choice -dividing the experimental uncertainty by 4 instead of simply using the experimental uncertainty -is to show clearly that the model can reproduce easily and simultaneously both the muon and the electron anomalies, and to guarantee that we are definitely reproducing a sizable deviation from the SM both in a µ and in all cases for a e , except the "a Bound e " analysis where there is no δa e term in eq.(47) and |δa e | ≤ 20 × 10 −13 is imposed.As a summary, all four selected cases of δa Exp

Results
In the next subsections we discuss the most relevant results of the analyses done following the lines of the previous section.In subsection 6.1 we consider the scenario "a Cs e " when |n | ≤ 100 GeV is imposed.The implications of changing this last assumption to |n | ≤ 250 GeV are addressed in subsection 6.2.The implications of the different assumptions for the electron anomaly, that is scenarios "a Rb e ", "a Avg e " and "a Bound e " are explored in subsection 6.3.The impact of the recent measurement of M W by the CDF collaboration is considered in subsection 6.4.Finally, to further illustrate these discussions, a few complete example cases are shown in subsection 6.5.

|n | ≤ 100 GeV
Here we present the results of the analysis "a Cs e " with the perturbativity constraint |n | ≤ 100 GeV.This serves to revisit the main results of [41] and as a reference for the analysis with |n | ≤ 250 GeV addressed in the following subsection.The perturbativity   36)) which is not allowed by e + e − → µ + µ − LEP data.On that respect, lepton flavor universality constraints also limit the possibility of a one loop explanation for the electron anomaly, as discussed later.This leaves us with two scenarios, one where both anomalies are explained via the two loop contribution, following the scaling law in eq. ( 45), and another where the muon anomaly is one loop dominated while the electron one is still generated at two loops.
In figure 5a the allowed regions for Re (n µ ) are presented as a function of m H . Three disjoint regions in the scalar mass can be seen: two in the 200-400 GeV range and the other above 1.2 TeV.The low mass regions belong to the scenario where the muon anomaly is obtained through the one loop contribution in agreement with the relation in eq.(35).Note that this contribution depends on the absolute value of the coupling, so both signs are allowed for Re (n µ ).In the large mass region both leptonic anomalies are two loop dominated.Figure 5b shows m H vs. t β .It contains two separate allowed regions again: in the t β ∼ 1 regime only scalar masses above 1.2 TeV are allowed; conversely for t β larger than 10, m H lies in the 200-400 GeV interval.To complement the previous two plots, in figure 5c the relation between the masses m H and m H ± is shown.In the low mass region we can clearly distinguish two scenarios.One where m H ± m H and another where m H ± > m H ; in the latter, m H ± m A .The degeneracy of H ± with either H or A arises from the oblique parameters constraint, as mentioned in section 3.In the large mass region the mass differences do not exceed ±300 GeV.  with respect to the scalar mass m S for S = H, A. The black line corresponds to the limit observed by CMS [140].Although LHC direct searches are already constraining the allowed regions, there is ample room for extra scalars that can explain both g − 2 anomalies simultaneously.
Let us now discuss some results concerning Re (n e ) and Re (n τ ).With a two loop   38) and ( 40) (see appendix B for further details) that one could have expected that both the coupling Re (n e ) and the deviation δa e have opposite sign: this is confirmed in figure 7 in the 1σ region.However, this figure also contains regions where Re (n e ) is negative.This behavior might be understood by analysing with some detail the two loop contribution to δa e : F in eq. ( 40) can be decomposed as F = F q + F τ + F µ , where F f is the contribution with fermion f running in the closed loop.One can estimate the importance of the different contributions for different t β , m H and m A ranges.
• For t β ∼ 1 and m H , m A > 1.2 TeV, eq. ( 40) gives It is clear that in this region the quark-induced contribution F t is (i) necessarily dominant and (ii) it requires Re (n e ) ∼ 4 − 10 GeV, as figure 7a illustrates, in order to reproduce ∆ Cs e −16.
(49) In this case, large values of Re (n τ ) ±100 GeV give τ -induced contributions at the same level of, or even larger than, the quark-induced contribution.This occurs despite some cancellation among the τ H and τ A contributions in eq. ( 40).This scenario would require Re (n e ) −15 GeV or Re (n e ) 7 GeV, as shown in figure 7a, in order to reproduce ∆ Cs e −16.
From this simple estimates one can conclude that, besides the expected regions where δa e arises from quark-induced two loop contributions, regions where the τ -induced contributions have an important role might be present.For this to occur, one might expect some peculiarities: besides light H and large t β , large values of both |Re (n τ ) | and |Re (n e ) |, with Re (n τ ) and Re (n e ) having the same sign, are required.Contrary to the case with dominating quark induced contributions, one might then have allowed regions where Re (n e ) < 0. This is illustrated in figures 7a and 7b where one can observe how allowed Re (n e ) < 0 only appear for a light H, and how the regions with large ±Re (n e ) correspond to large ±Re (n τ ).
To close this subsection, it is worth analysing in detail the role of the lepton flavor universality constraints mentioned in section 5.As justified later, we focus on observables involving only µ's and e's.For the ratios the current constraints are [97] In the present scenario, and thus, for ∆ P 1, The presence of M 2 P and the lepton masses allows us to concentrate on R K µe and neglect the n µ contribution.Therefore from eq. ( 51) we get the constraint Re (n e ) < 5 Then, • for t β 1 and m H ± 2 TeV, Re (n e ) < 20 GeV, • while for t β 10 2 and m H ± 0.5 TeV, Re (n e ) < 125 GeV.
From muon decay constraints on the H ± mediated contributions we also have a t β independent constraint (since the process is purely leptonic) [97,107]: This constraint is relevant for the low mass region: for Re (n µ ) 100 GeV, we can rewrite which is more restrictive than the bound from R K µe above.Concerning other observables involving τ leptons, semileptonic processes are not sensitive to n τ due to me mτ and mµ mτ suppressions, while purely leptonic decays have looser bounds than eq.( 55).This simple numerical exercise confirms that δa Exp e cannot be explained through one loop contributions.

|n | ≤ 250 GeV
As previously motivated, perturbativity bounds on the Yukawa couplings should be studied in detail.Here we explore higher scales in n , namely changing from |n | ≤ 100 GeV to |n | ≤ 250 GeV while maintaining the same constraints of the previous section.Conversely to what one would naively expect, it is not just the allowed regions in the different n that might change, but it has direct consequences on other physical observables such as the scalar masses and t β , among others.
Figure 8a shows results for Re (n µ ) vs. m H .It is clear that the allowed regions in parameter space are notably enlarged with respect to those in figure 5a   Figure 8c shows correlations among the scalar masses m H and m H ± .Concerning the low mass regions where H ± is degenerate either with H or A, already mentioned in figure 5c, it can be observed that enlarging perturbativity bounds pushes the upper limit of these regions in such a way that m H ± ∈ [0.2; 1.2] TeV for m H ± m H and m H ± ∈ [0.4; 1.2] TeV for m H ± m A , to a high degree of accuracy.Figures 9a and 9b complete the results for the scalar masses.For instance, it is still true that m A > m H in the low mass region, according to the general constraints presented in section 3.  TeV, opens the possibility to detect a sizeable signal in that range of scalar masses that was not contemplated in figure 6.Moreover, it is clear that increasing |n | up to |n | ≤ 250 GeV modifies our expectations for Br (S → + − ) and, in particular, enlarges the allowed parameter space, as one can easily check.Finally, we should stress some aspects concerning Re (n e ) and Re (n µ ) from figure 11.In spite of increasing our perturbativity bound up to |n | ≤ 250 GeV, it still seems difficult to obtain a one loop explanation for the electron anomaly since it requires quite large couplings, namely |n e | > 160 GeV in the Cs case. Figure 11b shows that |n e | < 150 GeV in the relevant range of scalar masses, thus indicating that δa Exp,Cs e is mainly explained at two loops.This agrees with the discussion on universality constraints closing subsection 6.1.
As it was already explained in the discussion of figure 7a, now in figure 11b and for large scalar masses, one can easily check that the electron coupling is positive and lies in the range Re (n e ) ∼ 4 − 20 GeV.Furthermore, according to eq. ( 45), there exists a linear relation between Re (n µ ) and Re (n e ) for m H > 1.2 TeV, which implies that they have opposite sign in the Cs case and therefore Re (n µ ) should be negative in this region.The region Re (n µ ) = −13Re (n e ) can be seen in the lower part of figure 11a inside the 1σ region as it should.Departure from this straight line introduces an important one loop contribution to the muon anomaly lowering also the scalar masses ranges.On the other hand, for light scalar masses, Re (n e ) might be either positive or negative by the same arguments discussed in section 6.1.It is also important to recall that, in this low mass region, ∆ µ receives dominant one loop contributions and thus Re (n µ ) could naturally appear with both signs.From figure 11a, one may notice as well that |n µ | is in general larger than |n e | in the whole parameter space.

Different δa e
As commented in section 5, the situation concerning a e is to some extent unclear.In this section we discuss the implications of different assumptions for the value of δa Exp e , that is, in terms of the model, the implications of requiring different values of the new contributions δa e .The ultimate answer is definitely provided by repeating detailed numerical analyses under the different assumptions δa Exp e .However, one can anticipate part of the answer with simple considerations.As analysed in section 4, δa e arises from two loop contributions proportional to Re (n e ): this fact, together with the results of section 6.2 corresponding to δa e δa Exp,Cs e , can give us a first insight.Consider for example an allowed point in parameter space (i.e. a point respecting all imposed constraints) which gives δa e δa Exp,Cs e .This point has a certain Re (n e ) = Re (n e ) Cs ; it is straightforward that changing Re (n e ) → Re (n e ) = Re (n e ) Rb = Re (n e ) Cs × δa Exp,Rb e δa Exp,Cs e and no other parameter, one would obtain δa e δa Exp,Rb e .The question is, of course, if such a change in Re (n e ) alone still gives an allowed point.On that respect, one needs to analyse which observables constrain Re (n e ) and how those constraints work.These are the ones related to lepton flavor universality in leptonic decays i → j ν ν and in semileptonic decays involving kaons and pions, analysed in section 6.1.In particular, attending to δa Exp,Cs when no other parameter is changed.There are two different aspects:  56) is necessarily less restrictive when eq. ( 57) is considered; 2. besides the uncertainty in R K µe in eq. ( 51), as discussed previously, there is a "sign" question concerning the deviation, at the same 5 × 10 −3 level of the uncertainty, from R K µe = 1.In order to obtain δa e δa Exp,Cs e < 0, the expectation is Re (n e ) > 0, and that produces R K µe − 1 > 0 in eq. ( 53), which goes "in the wrong direction".For both cases in eq. ( 57), that problem is alleviated.
It is then clear that the analysis with δa Exp,Cs e is somehow a "worst case" scenario in terms of the dependence of the constraints on Re (n e ): besides the naive mapping of allowed regions expected from eq. ( 57), one might then expect larger allowed regions not only for Re (n e ) but also for other quantities of interest.As mentioned in section 5, we also perform an analysis where |δa e | ≤ 20 × 10 −13 is imposed (instead of requiring some specific value, as summarized in figure 4).This serves a double purpose: identifying which allowed regions are necessary in order to obtain an appropriate δa µ without regard to δa e , and identifying which regions are absolutely excluded for any value of δa e reasonably compatible with δa Exp,Cs e or δa Exp,Rb e that one could consider.In figures 12, 13 and 14, the color coding follows figure 4. Figure 12 shows Re (n µ ) vs. Re (n e ) and Re (n e ) vs. m H allowed regions: comparison with figures 11a and 11b confirms the simple expectations of the previous discussion in terms of the position of the allowed regions and their extension.The same applies to figure 13, which shows Re (n µ ) vs. m H (to be compared with figure 8a).In particular it is clear from figure 13c that once δa µ δa Exp µ is imposed, the allowed regions for some parameters (besides Re (n µ ), obviously) are coarsely determined and the sensitivity of the analysis on the requirement for δa e only concerns a finer level of detail.There is a final point that the analysis with |δa e | ≤ 20 × 10 −13 confirms.Figure 14 shows δa e vs. Re (n e ): under the simple expectations for the two loop contributions discussed in section 4.2, one would have Re (n e )×δa e < 0.Besides that expected region, one can observe smaller allowed regions where Re (n e ) × δa e > 0: they correspond to the unexpected situation in which the two loop contributions are dominated by virtual τ 's in the fermion loop, and furthermore it is clear that the values of δa e that can be obtained in this manner are more restricted, with |δa e | < 10

The CDF M W anomaly
As mentioned in section 3, one can use deviations from the SM in the oblique parameters (∆S, ∆T ) = (0, 0) in order to "explain" the CDF measurement of M W in [96]: this subsection is devoted to that "explanation".Figures 15 and 16 show results analogous to the ones in section 6.2 -which use eq.( 22)-, except for a different (∆S, ∆T ) constraint. Figure 15 is obtained with eq. ( 23) (the "conservative" average of [103]) and figure 16 is obtained with eq. ( 24) (the results in [104]).The coloring of the allowed regions corresponds, darker to lighter, to 1, 2, 3σ levels of a 2D-∆χ 2 .For ∆χ 2 we use the χ 2  Min value of the analysis in section 6.2 (that is, with the constraint in eq. ( 22) for ∆S, ∆T ).A few comments are in order.
• Besides the absence of degeneracies m H ± m H or m H ± m A , masses of the new scalars larger than 2 TeV are more difficult to accommodate.This can be understood attending to the clash between the mass differences discussed in section 3 that eqs.(23) or (24) require, and the need of near degenerate scalars that the perturbativity requirements on the scalar potential impose for new scalar masses much larger than v.
• Overall agreement with the imposed constraints is worse in several regions in figures 15 and 16 than it was in the analyses of section 6.2 (figures 8, 9 and 11).This is more dramatic in figure 16, where the agreement with constraints is worse than in figure 15 to the point that several regions are beyond the represented contour levels.
Despite these changes, the main characteristics of the allowed regions discussed in the previous sections still apply and are clearly identified in both figures 15 and 16. 15: Results with (∆S, ∆T ) in eq. ( 23), "conservative" case in [103].24), from [104].
Finally, since the oblique parameters S and T play an important role, figure 17 shows allowed regions for ∆S vs. ∆T in the two scenarios considered for the CDF M W "explanation", together with the imposed (∆T, ∆S) constraint in each case.As anticipated, the constraint in eq. ( 24) appears to be more difficult to accommodate than the constraint in eq. ( 23).In fact, despite the different position of the ellipses corresponding to the (∆T, ∆S) constraints in figures 17a and 17b, the allowed regions are quite similar in both cases, that is, the model appears to be unable to accommodate values ∆T > 0.22 together with ∆S > 0.02.Other possible explanations of the CDF M W anomaly have been addressed in [148][149][150][151][152][153][154][155].

Example points
In this section, some example points of the allowed parameter space are presented in order to specify the behavior pointed out in the previous plots.For the sake of clarity, we only focus on the analysis with "a Cs e " concerning the electron anomaly: other cases do not change substantially beyond the differences already mentioned in section 6.3.
From table 1, it is clear that points 1-2 correspond to the solution with small values of t β and large scalar masses: all scalars are above 1.2 TeV and their mass differences do not exceed ±200 GeV.In this region, both anomalies are explained at two loops through the top quark terms, as one can easily check in tables 2 and 3, where all loop contributions are normalized to the total δa in such a way that their sum must be 1.One may also notice that δa µ receives a subdominant one loop contribution.The lepton couplings Re (n e ) and Re (n µ ) have opposite sign and they roughly satisfy the linear relation in eq. ( 45) for the Cs case.
Regarding the appearance of the intermediate values of the scalar masses and t β previously commented in section 6.2, our point 3 gives a perfect example of that behavior.It is important to realize that large values of |Re (n µ ) | are required in this region; in fact, they are almost reaching the perturbativity upper bound |n | ≤ 250 GeV.On the other hand, although the top dominance still holds at two loops in the electron anomaly, the corresponding tau contributions begin to play a relevant role.This trend will continue as t β grows and the quark contributions are more suppressed.
Finally, points 4-9 belong to the low mass region corresponding to a wide range of t β 1 values.As we have stressed before, two possible scenarios arise: one where m H ± m A (points 4-6) and another where m H ± m H (points 7-9).In all cases, the scalar masses are below 1 TeV and m A > m H , as anticipated.Taking into account the large values of t β , the two loop contribution that dominates δa e is generated by the tau loop.This confirms our expectation for Re (n e ): its sign is not fixed and it could be either positive or negative (point 9).Furthermore, in this region the muon anomaly is clearly one loop dominated, albeit there exists a subdominant contribution from the tau loop as well.This in turn means that Re (n µ ) can take both signs, as one can easily check.
For completeness, the last two points have been included to give an example of the allowed parameter space in subsection 6. 4

Conclusions
The experimental determinations of the muon and the electron anomalous magnetic moment point towards the necessity of lepton flavor non-universal New Physics.Aiming to address both leptonic anomalies simultaneously, we have considered a type I or type X 2HDM with a general flavor conserving lepton sector, one loop stable under renormalization, in which the new Yukawa couplings are completely decoupled from lepton mass proportionality.The latter turns out to be crucial in order to reproduce the g − 2 muon anomaly together with the different scenarios one can consider for the g − 2 electron anomaly, related to the Cs and/or the Rb recoil measurements of the fine structure constant.A thorough analysis of the parameter space of the model has been performed including all relevant theoretical and experimental constraints.The results show that the muon anomaly receives dominant one loop contributions for light new scalar masses in the 0.2-1.0TeV range together with a significant hierarchy in the vacuum expectation values of the scalars, that is t β 1, while two loop Barr-Zee diagrams are also needed for heavy new scalars with masses above 1.2 TeV together with t β ∼ 1.On the other hand, the electron anomaly receives dominant two loop contributions in the whole range of scalar masses.Furthermore, we have analysed how the perturbativity assumptions on the lepton Yukawa couplings have direct impact on relevant physical observables: intermediate values of the scalar masses and t β only arise when the upper bound on n reaches the electroweak scale.This might be relevant since we are entering an era of exclusion or discovery at the LHC, so that the allowed parameter space of the model must be fully scrutinized.The disagreement between the recent CDF measurement of M W and the SM expectations for electroweak precision results can be translated into deviations (∆S, ∆T ) = (0, 0) of the oblique parameters.We have considered two different scenarios for (∆S, ∆T ) values which "explain" the CDF disagreement.Both scenarios require a scalar spectrum where near degeneracies m H ± m H or m H ± m A are now disfavored, and where masses larger than 2 TeV are more difficult to accommodate.However, concerning the n couplings and t β , the allowed regions have the same characteristics as in the analyses compatible with (∆S, ∆T ) = (0, 0).

Appendices A One loop stability under RGE
The evolution of the Yukawa couplings under one loop RGE [156][157][158][159][160] is given by: where D ≡ 16π together with the existence of two unitary matrices W L,R in the lepton sector such that are diagonal, guarantee the absence of SFCNC at tree level.
In order to ensure that eqs.( 62) and ( 63) in the quark sector hold at one loop, it is sufficient to impose [161] DY d2 = D(d)Y d1 + dDY d1 , or equivalently where the proportionality constants are precisely the running of the parameters d and u in eqs.( 62) and (63).It is easy to check that it is clear that as it should.Hence, the quark sector is stable under RGE.Concerning the lepton sector, one loop stability requires that remain simultaneously diagonal.In this sense, the only apparently problematic term in D(Y i ) has the structure Y a Y † b Y c , but that is obviously diagonal [43].Therefore, the lepton sector is also stable under RGE.

B.2 Two loop contributions
Together with eq. ( 77), the interactions generate two loop Barr-Zee contributions to the anomalous magnetic moment of lepton : δa where N f c and Q f are the number of colours and the electric charge of the fermion running in the closed loop of figure 18c, respectively, and z f S = m 2 f /m 2 S .The two loop functions f (z) and g(z) are We refer to [162] to see other two loop contributions.

Figure 1 :
Figure 1: Oblique parameters: allowed regions in m A − m H ± vs. m H − m H ± .Darker to lighter colors correspond to 2D-∆χ 2 1, 2 and 3σ regions.The plot corresponds to m H ± = 1 TeV and scalar alignment.

Figure 2 :
Figure 2: H ± FCNC: m H ± vs. t β allowed region when contributions of H ± to B q -Bq are below experimental uncertainty in ∆M Bq .

|n e | 2 m 2 HI
m A > 200 GeV would require |n e | > 160 GeV: besides perturbativity concerns, such values of |n e | might be hard to reconcile with other constraints.More importantly, since we expect m H < m A for light A, we also expect a sizable cancellation among H and A contributions.From this simple analysis, obtaining ∆ Cs e −16 with one loop contributions does not appear to be feasible.• For the electron Rb anomaly, ∆ Rb e 9 can only be explained with the one loop H contribution and provided 9 eH ⇒ |n e | ∼ 3 5 m H . (37) For m H > 200 GeV, this would require |n e | > 120 GeV.If the same concerns on the values of |n e | mentioned for ∆ Cs e −16 apply here, obtaining ∆ Rb e 9 does not seem to be feasible neither; otherwise ∆ Rb e 9 would be "easier" to accommodate with one loop contributions than ∆ Cs e −16 because of the sign difference and the smaller absolute value.

Figure 6
illustrates the allowed regions for the resonant process [pp] ggF → S → µ + µ − mH (TeV) For S = A.
, which are completely embedded in the ones of this new analysis, as one could have expected.On that respect, one may realize of the appearance of a new set of intermediate values for the scalar mass, m H ∈ [0.4; 1.2] TeV, when increasing our perturbativity upper bound.It can be easily understood by tracing an horizontal line at Re (n µ ) 0 = −100 GeV: we eliminate the blue region "bridge" connecting the low and high mass solutions.Therefore, this new range of scalar masses requires large values of |Re (n µ ) |.
m A vs. m H . m H ± (TeV) m A vs. m H ± .
For S = A.
Re (n e ) vs. m H .

e,
δa Exp,Rb e and δa Exp,Avg e in eqs.(2), (3) and (46), one is interested in the effect on those constraints of Re (n e ) = Re (n e ) Cs → Re (n e ) = Re (n e ) Rb = −0.55Re(n e ) Cs , Re (n e ) = Re (n e ) Cs → Re (n e ) = Re (n e ) Avg = 0.23Re (n e ) Cs ,

Figure 18 :
Figure 18: Illustrative one and two loop contributions to δa .
• a conservative approach in which we only assume that |δa e | ≤ 20 × 10 −13 .Rather than targeting a specific value, this analysis may help to single out regions of parameter space where one cannot reproduce δa Exp e.

Table 1 :
considering the CDF M W anomaly.It is clear that point 10 mimics the behavior of points 1-2, while point 11 presents the same features as points 4-6.Example points, masses and Re (n )'s in GeV.

Table 2 :
Example points, δa e values; columns 3 to 9 show the relative contributions of the different one and two loop terms to the value of δa e in the second column.

Table 3 :
Example points, δa µ values; columns 3 to 9 show the relative contributions of the different one and two loop terms to the value of δa µ in the second column.
2 dd ln µ , µ is the renormalization scale and , g, g the corresponding gauge coupling constants of SU (3) c , SU (2) L and U (1) Y , respectively.The alignment condition in the quark sector