Electron EDM and muon anomalous magnetic moment in two-Higgs-doublet models

The CP violating two-Higgs doublet model of type-X may enhance significantly the electric and magnetic moment of leptons through two-loop Barr-Zee diagrams. We analyze the general parameter space of the type-X 2HDM consistent with the muon g − 2 and the electron EDM measurements to show how strongly the CP violating parameter is constrained in the region explaining the muon g − 2 anomaly.


Introduction
After the discovery of the 125 GeV Higgs boson [1,2], its nature has been found to follow very closely to the prediction of the Standard Model (SM) in terms of its production and decay properties [3] as well as the CP property [4]. Nonetheless, we expect that there must be new physics beyond the SM for various theoretical and phenomenological reasons. Extending the Higgs sector with an additional doublet is an interesting option which has potentially important implications on several new physics phenomena. A Two-Higgs Doublet Model (2HDM) is able to realize electroweak baryogenesis [5], and the relevant CP violation (CPV) may appear in electric dipole moments (EDMs) of fermions [6][7][8][9][10][11][12][13]. Such CP violation effect could be probed as well in the future collider experiments [14][15][16][17].
On the other hand, the observed deviation of the muon anomalous moment [18,19] can be explained in type-X 2HDM [20][21][22][23][24][25][26][27][28]. Both for EDM and the muon g − 2, sizable contributions come from two-loop Barr-Zee (BZ) diagrams [29]. However, the favored parameter spaces are largely orthogonal to each other. The (g − 2) µ data can be accommodated JHEP12(2019)068 only by type-X 2HDM with large tan β and a very light pseudoscalar, while strong phase transition for electroweak baryogenesis can be realized for a very heavy pseudoscalar Higgs boson preferring low tan β [30]. A recent study [28] showed that strongly first order phase transition could be obtained for some limited parameter region explaining the muon g − 2 deviation with a light pseudoscalar. It is interesting to see whether sizable CPV can be allowed for successful electroweak baryogenesis. Collider searches for such a light pseudoscalar at LHC were studied in [31][32][33]. However, more studies are needed to probe the whole parameter space favorable for the muon g − 2.
In this paper, we analyze the general (complex) parameter region of the type-X 2HDM compatible with the EDM and (g − 2) µ measurements. Since both quantities come from the same type of loop diagrams, only the electron EDM is highly enhanced in the region compatible with the muon g − 2 anomaly and thus the CP violating coupling is severely constrained by the electron EDM limit.
The contents of the paper are as follows. In section 2, we give a brief introduction to the general properties of the CP-violating type-X 2HDM, and define our input parameter set. In section 3, we describe all the theoretical and experimental bounds on the parameter space to find allowed region and the results of analysis are presented in section 4. Finally we conclude in section 5. In appendix, we collect all the necessary formula to compute EDM in 2HDM.

The CP violating type-X two-Higgs doublet model
In this section we will briefly summarize the CP violating two-Higgs doublet model. The model consists of two scalar doublets Φ 1 and Φ 2 of equal hypercharge. The general form of the scalar potential is given by In general fermions can couple to both the scalars; consequently flavor changing neutral current interactions (FCNCs) can appear at tree level. To avoid the problematic FCNCs we impose a Z 2 symmetry under which the scalars are oppositely charged, Φ 1 → −Φ 1 and Φ 2 → Φ 2 , and the fermions are also charged appropriately. This Z 2 charge assignment forbids the λ 6 and λ 7 term in the scalar potential in eq. (2.1). Here, we include the soft Z 2 breaking term m 2 12 which can generates CP violation in the scalar sector [34]. This term is also important to keep the quartic coupling λ 1 below perturbativity limit [35,36]. The complex parameters in the potential are m 2 12 and λ 5 while all other parameters are real. The scalars can be parameterized as

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where the vacuum expectation values (vevs) v 1 , v 2 are real and v 2 1 + v 2 2 = v 2 = (246GeV) 2 . In general one of the vevs can be complex. Under the global phase transformation the couplings m 2 12 and λ 5 can absorb the global phase. Thus v 1 , and v 2 are real parameters. Minimization of the scalar potential yields the following relations (we define tan β = v 2 /v 1 ), Im(m 2 12 ) = v 2 sin β cos β Im(λ 5 ) . (2.5) From eq. (2.5) it is evident that there is only one free CPV parameter as shown in [10,15] and from now on we will use Im(λ 5 ) as free parameter to quantify the CP violation.

Scalar spectrum
After the electroweak symmetry breaking (EWSB) we obtain one physical charged scalar and one Goldstone boson, and mass of the charged scalar reads as, where we define, Similarly, mixing between the two CP-odd components of the scalar doublets generates a Goldstone and a CP-odd scalar, (2.8) In the case of CP conserving 2HDM model the combination A 0 is the mass eigenstate.
Since we have explicit CP violation, A 0 will further mix with φ 0 r 1 and φ 0 r 2 to produce three neutral scalar mass eigenstates. The mass matrix is given by, The neutral mass squared matrix is, where c β = cos β, s β = sin β and λ 345 = (λ 3 + λ 4 + Re(λ 5 )). The mass matrix can be diagonalized by a rotation matrix R such that M diag = RM 2 R T and we get, and h 1 is the Higgs boson observed at the LHC with mass m h 1 = 125 GeV. As we will see that allowed CPV is small, the mass eigenstate h 3 is predominantly CP-odd whereas h 1 and h 2 are mostly CP-even. The rotation matrix R depends on three mixing angles, α, α b and α c where the last two angles stem from CP violation. The matrix R can be parameterized as There are ten real parameters in the scalar potential which are, It is possible to re-express the parameters of the potential in terms of the following phenomenological parameters, (2.14) We can express the quartic couplings in terms of the the phenomenological parameters [10]: Note that all the parameters in eq. (2.14) are not independent since there is only one independent CP violating parameter as indicated by eq. (2.5). For simplicity we have used Im(λ 5 ) as the measure of CP violation in our model and computed α b and α c accordingly. Moreover, the angle α is re-expressed by using cos(β − α) where the angle (β − α) diagonalizes the CP even neutral scalar mass matrix in the CP conserving 2HDM JHEP12(2019)068 scenario. Consequently, the phenomenological parameter set we have used for parameter scan is: To compute the quartic couplings in eq. (2.15) we need to recast the angles α b and α c in terms of the parameters as shown in eq. (2.16). From the diagonalization of the neutral scalar mass matrix we have, M 2 = R T M 2 diag R and using eq. (2.10) we get the following relations, Assuming small CP violation we can solve the above two equations under the assumption that sin 2 α b , sin 2 α c → 0. We obtain the following expression up to leading order of Im(λ 5 ), These expressions are valid as long as the mass of h 3 is not very close to any of the other two scalars. For our numerical computation we have used the full expressions without any assumptions. Now using our phenomenological parameter set we can compute all the scalar mixing and quartic couplings.

Yukawa and gauge interactions of the neutral scalars
The interaction of the fermions with the two scalar doublets depends on the Z 2 symmetry we invoked earlier to remove the tree level FCNCs. Depending on the Z 2 charge assignment for fermions, four independent types of Yukawa interactions are allowed [37]. We are interested in the scenario where the leptons couple to one doublet (Φ 1 ) and the quarks couple to the other doublet (Φ 2 ). This model is known as type-X 2HDM or leptophilic 2HDM. The Yukawa terms read as, The above Lagrangian can be realized if R is odd under the Z 2 symmetry while the other fermions are even under it. After the EWSB the general form of the Yukawa interaction is, where c f,i andc f,i are the Yukawa modifiers. We can express the Yukawa modifiers in terms of mixing matrix(R) and tan β as shown in table 1 where c u,i is the universal modifier for all generations of up type quarks and same is true for down type quarks (c d,i ), and leptons (c e,i ). In the limit cos(β − α) → 0, the modifiers to the SM like Higgs c f,1 goes to +1 essentially restoring the SM Yukawa coupling. This is called right sign (RS) Yukawa limit. However, if cos(β − α) takes the particular value 2/ tan β then the lepton Yukawa modifier c e,1 becomes '-1' and is known as wrong sign (WS) Yukawa limit. The couplings between neutral Higgs bosons and the gauge bosons can be written as,

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where the expression for a i is shown in table 1.

Perturbativity and vacuum stability
The quartic couplings have to satisfy the theoretical conditions for vacuum stability and perturbativity. For perturbativity we ensure that |λ i | < 4π. The vacuum stability conditions are [36], The limit on the quartic coupling and consequently on the scalar masses depends on the parameter cos(β − α). To satisfy the perturbativity and vacuum stability constraints the heavy scalar mass need to obey the following limits [20,23], We will appropriately choose the value of cos(β − α) and ν depending on the charged Higgs mass to satisfy these conditions.

Precision and collider bounds on mass spectrum
Constraints on 2HDM mass spectrum from the electroweak precision test (EWPT) was studied in detail in [20,38]. EWPT dictates that the charged higgs mass has to be almost degenerate with either h 2 or h 3 [39]. We assume in our analysis that h ± and dominantly CP even h 2 are degenerate while mass of h 3 can be anything. Bounds on the heavy scalar searches at LHC are very weak for type-X 2HDM with large tan β since the heavy scalars are hadrophobic and gluon fusion is not an effective way to produce them. The strongest limit on new scalars comes from LEP [40]: m h 2 + m h 3 > 185 GeV which is always satisfied on our analysis. Due to the hadrophbic nature of the new scalars b → sγ limit is satisfied for tan β > 2 with m h ± >200 GeV [38] and B s → µ + µ − restricts m h 3 to be greater than 15 GeV [23,25,31]. There is limit on the SM higgs decay to light pseudoscalar BR(h 1 → h 3 h 3 ) < 3% [41] and it can be easily satisfied by varying cos(β − α) within the allowed range as shown previously in [31]. GFitter collaboration has done the most recent global analysis of the Higgs coupling strengths for different 2HDM scenarios and for large tan β values the wrong sign limit is allowed [38].

Muon anomalous magnetic moment
The muon anomalous magnetic moment is a long standing puzzle with disagreement between the SM prediction and the experimental determination. The hadronic contributions to muon magnetic moment including the running of the QED coupling have kept updating [42][43][44][45][46]. Taking into account all theoretical corrections, the recent estimate of theoretical [46] and experimental [18,19] value for a µ is: This gives us a 3.7σ deviation: The Fermilab Muon g − 2 experiment (E989) is aiming to measure a µ with a relative uncertainty of 140 ppb which has the potential to confirm the discrepancy with a 7σ significance [47] assuming that the central values of a th µ and a ex µ remain the same. In the CP-conserving type-X 2HDM, such a deviation can be explained by enhanced contribution coming from the two loop Barr-Zee diagram for large tan β and low pseudoscalar mass. In the presence of CP violation, there will be extra contributions from the CPV Yukawa couplings. However the additional contribution is negligible as we will see that the EDM bound strongly suppresses the CPV coupling (Im(λ 5 )).

Electron electric dipole moment
In the type-X 2HDM all the quark CPV couplings are inversely proportional to tan β while lepton CPV couplings are proportional to tan β. Consequently, in this model the light quark Chromo-EDM will not be enhanced as opposed to type-II 2HDM scenario. Hence limits on the CPV coupling coming from neutron EDM or diamagnetic atmoms like Hg will be suppressed. Hence we do not discuss these contributions. However, the type-X 2HDM provides tan β enhancement in the electron EDM contribution. We will now focus on electron EDM in the type-X 2HDM.
EDMs are very sensitive probes of new physics that contains CP-violating phases. In our model the complex quartic coupling in the Higgs sector results in CP violating phases in Yukawa couplings as shown in eq. (2.19). Note that electron EDM can come from both one loop and two loop diagrams. Contribution from 1-loop diagrams is induced by the neutral Higgs bosons or the charged Higgs. This contribution is proportional to the third power of the electron Yukawa coupling and is negligible. The other contribution originates from the so called Barr-Zee diagrams as shown in figure 1. This might provide sizable EDM since they are proportional to one power of the electron Yukawa coupling.
The effective dipole-moment operator can be written as where we have used the Higgs vacuum expectation value v = 246 GeV as the cut-off scale for new physics. We have summarized all the 2-loop contributions to the coefficient δ e to compute the EDM in appendix A. The current upper bound on electron EDM is reported by the ACME Collaboration [48]: |d e | < 1.1 × 10 −29 e cm, (3.6) at 90% confidence level. We will use this value to constrain the amount of CPV allowed in this model.

Lepton universality constraints
The test of the lepton universality has been evaluated from Z decay and µ/τ decays at the level of 0.1% [49,50]. In the type-X 2HDM, large loop corrections to the lepton universality can arise due to the enhanced leptonic couplings of extra Higgs bosons at large tan β. Such effects were well studied in refs. [21,51,52]. Let us first consider the test of the lepton universality by SLD and LEP experimental data [49]. The obtained Z-decay data can be converted into the leptonic branching ratios which are given by

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with a positive correlation of 0.63. For each lepton flavor, we can calculate different quantities which can be parameterized as In the limit of large tan β with m µ → 0, the correction on δ τ τ only exists.
The other lepton universality test taken by HFAG has been measured through the pure leptonic and semi-hadronic decay processes [50]: .0010 ± 0.0015, g τ g e = 1.0029 ± 0.0015, g µ g e = 1.0019 ± 0.0014, (3.10) In the large tan β limit of the type-X 2HDM, two crucial corrections to τ decays can arise. One correction comes from the tree-level contribution of the heavy charged Higgs boson and the other stems from the one-loop corrections of the extra Higgs bosons. Following the χ 2 −analysis in ref. [21], we calculate the theoretical bounds on the lepton universality.

Results
We are now ready to discuss how the parameter space of type-X 2HDM is limited by the constraints described in the previous section. To satisfy the electroweak precision observable like isospin violation we assume degeneracy between the heavy scalar h 2 and the charged Higgs h ± mass keeping m h 3 free.
In figure 2 we depict absolute contribution to electron EDM coming from different BZ diagrams in this model for both RS (left panel) and WS scenario(right panel). We have fixed m h 3 = 40 GeV, m h 2 = m h ± = 200 GeV and Im(λ 5 ) = 10 −5 . The total contribution is shown in black curve where dominant contribution is coming from a light h 3 mediated diagram with an internal photon line and a tau loop as shown in top left panel of figure 1. This is due to the fact that h 3 is light and its coupling to leptons is tan β enhanced. Contribution coming from the heavy scalar, h 2 is suppressed due to its mass, whereas contributions originate from the BZ diagrams with an internal Z boson are order of magnitude small since Z boson is heavy and the Zff vector coupling is small. The h ± W ± γ diagrams, as shown in lower panel of figure 1 yield sub dominant contributions since both W and h ± are relatively heavy. The SM Higgs contribution comes from sin α b which is JHEP12(2019)068 proportional to cos(β + α). Hence the SM Higgs contribution goes as 1/ tan β (1/ tan 3 β) in RS(WS) scenario for large tan β. However, for RS scenario there is no tan β dependence since the dominant contribution is generated by the couplingc e,1 which is proportional to tan β(see table 1) and cancels the tan β dependency. The tan β dependency in WS limit is obvious. The contribution remains sub dominant due to absence of any tan β enhancement in the SM Higgs mediated BZ diagrams. Since we have plotted the absolute value of EDM, the spike appears when contribution from a particular diagram changes sign. The purple horizontal bar depicts the present limit on EDM as reported by the ACME Collaboration.
In m h 3 − tan β plane we present our main result where we show the parameter space compatible with (g − 2) µ , electron EDM and lepton universality constraints coming from Z → and τ decay measurements. In figure 3(a) the light and dark brown region can explain the present (g − 2) µ anomaly (eq. (3.4)) at 1σ and 2σ respectively. The region in green(yellow) depicts parameter space which will be able the explain the anomaly at 1σ(2σ) after full Fermilab data assuming that the central values remain the same. The parameter space right or below to the dark blue and orange lines are allowed at 2σ by lepton universality in Z decays and τ decays. The constraints coming from the electron EDM is shown by the red and purple curves which depends on the value of CPV coupling Im(λ 5 ). For a given Im(λ 5 ), along the red or purple curve the electron EDM is 1.1 × 10 −29 e-cm and anything right to that curve is allowed by the present EDM limit. For a light m h 3 the BZ contribution is very large and consequently a very small CPV coupling is allowed. On the other hand when h 3 is relatively heavy, say 150 GeV then relatively large Im(λ 5 ) is allowed. Similarly for large tan β the EDM contribution enhanced by tan 2 β and for a given m h 3 if we increase tan β the limit on CPV coupling becomes stronger. The red(purple) curves in both the RS and WS Yukawa couplings are allowed. The vertical gray bar in figure 3 depicts the region where h 3 and the the SM Higgs h 1 becomes degenerate and the approximate expressions for mixing angles as in eq. (2.18) breaks down. In figure 3(b),(c) we have shown the allowed parameter space for m h 2 = m ± h = 300 and 400 GeV and for simplicity the future Fermilab limit is not shown. The light(dark) brown region explain the present (g − 2) µ anomaly at 2 σ(1 σ). As we increase m h 2 the mixing angle sin α c decreases(see eq. (2.18)) and comparatively large Im(λ 5 ) is allowed for a given m h 3 and tan β. Note that for m h 2 ≥ 250 GeV, RS Yukawa couplings are disfavored because they cannot satisfy the perturbativity and vacuum stability bounds as in eq. (3.1). Hence for heavier m h 2 we have only WS Yukawa limit.

Conclusion and remarks
We explored the parameter space of the CP violating type-X 2HDM in which sizable enhancement of electric and magnetic moment of leptons can arise through two-loop Barr-

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Zee diagrams. Figure 3 summarizes our main results on the limits from the muon g − 2, electron EDM, and lepton universality determinations in the plane of (m h 3 , tan β). For this, we imposed the theoretical constraints of vacuum stability and perturbativity, and assumed degenerate masses for the charged Higgs boson and the heaviest neutral Higgs boson to be consistent with the electroweak precision test.
Let us remark that the region explaining the muon g − 2 anomaly is more tightly constrained by the lepton universality conditions compared with the previous studies, e.g., in [21]. This is because we used the new theoretical value [46] which increased the deviation a bit (eq. (3.4)). Future collider experiments improving the precision of lepton universality would be useful to test the type-X 2HDM as an explanation to the muon g − 2 deviation. In the parameter region explaining the muon g − 2 anomaly, the electron EDM is also enhanced and thus one can see that the CPV quartic coupling has to be smaller than about a few times 10 −5 to be compatible with the muon g − 2 explanation. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.