Joint explanation ofW-mass and muong–2 in the 2HDM

The CDF collaboration presented their new result for theW-boson mass measurement as [1]

There is an approximately $ 7\sigma $ discrepancy between the experimental central value and the standard model (SM) prediction $ 80.357 \pm 0.006 $ GeV [2]. Moreover, there has been a long-standing discrepancy between the experimental value and the SM prediction for muon anomalous magnetic moment (muon $ g-2 $ ). The combined result of the FNAL [3] and BNL experiments [4] has a deviation of approximately $ 4.2\sigma $ from the SM prediction [5–7],

Both deviations strongly imply the existence of new physics beyond the SM, and plausible explanations have already been given for the CDFW-mass [8–41].

Among various new physics models, two-Higgs-doublet models (2HDMs) are simple extensions of the SM (for an example of a recent review, see [42]). The 2HDM introduces a second $ S U(2)_{\rm L} $ Higgs doublet and thus predicts two neutral CP-even Higgs bosonshandH, a neutral pseudoscalarA, and a pair of charged Higgs bosons $ H^\pm $ [43]. The 2HDM can offer additional corrections to the masses of gauge bosons via the exchange of extra Higgs fields in self-energy diagrams. In addition, if the extra Higgs bosons have appropriate couplings to leptons, muon $ g-2 $ can be explained simply. Because bothW-mass and muon $ g-2 $ can be affected by the mass splittings amongH,A, and $ H^\pm $ , we use a 2HDM withμ-τlepton flavor violation (LFV) interactions to study the possibility of a simultaneous explanation for both anomalies. In our analysis, we extensively examine the parameter space of this model by considering various relevant theoretical and experimental constraints. For the single explanation of muon $ g-2 $ using the Higgs doublet field withμ-τLFV interactions, see, for example, [44–62].

This paper is organized as follows: In Sec. II, we introduce the 2HDM withμ-τLFV interactions. In Sec. III and Sec. IV, we study theW-boson mass and muon $ g-2 $ after imposing relevant theoretical and experimental constraints. Finally, we give our conclusion in Sec. V.

The 2HDM withμ-τLFV interactions may be derived from a general 2HDM by taking specific parameters. It can also be naturally obtained by introducing an inert Higgs doublet $ \phi_2 $ under discrete $ Z_4 $ symmetry, and the $ Z_4 $ charge assignment is displaced inTable 1[53]. The Higgs potential with $ Z_4 $ symmetry is given by

	$\phi_1 $	$\phi_2 $	$Q_L^{i} $	$U_{\rm R}^i$	$D_{\rm R}^i$	$L_L^e $	$L_L^\mu $	$L_L^\tau $	$e_{\rm R} $	$\mu_{\rm R} $	$\tau_{\rm R} $
Z4	1	−1	1	1	1	1	i	−i	1	i	−i

Table 1.Assignment ofZ₄charge in the 2HDM withμ-τ-philic Higgs doublet.

Here, all parameters are real. Although $ \lambda_5 $ is the only possible complex parameter, it can be rendered real with phase redefinition of one of the two Higgs fields. The two Higgs doublets $ \phi_1 $ and $ \phi_2 $ are expressed by

The $ \phi_1 $ field produces a nonzero vacuum expectation value (VEV),v= 246 GeV, whereas the $ \phi_2 $ field has a zero VEV. The parameter $ Y_1 $ can be determined from the minimization condition of the Higgs potential,

The fields $ G^0 $ and $ G^+ $ indicate Nambu-Goldstone bosons, which are consumed by the gauge bosons. The fieldsAand $ H^+ $ represent the mass eigenstates of theCP-odd Higgs boson and charged Higgs boson, respectively, whose masses are written as

There is no mixing between the twoCP-even Higgs bosonshandH, and their masses are

We obtain the masses of fermions via Yukawa interactions with $ \phi_1 $ as

where $ \widetilde{\phi}_1 = {\rm i}\tau_2 \phi^*_1 $ , and $ E_{\rm R} $ , $ U_{\rm R} $ , and $ D_{\rm R} $ denote the three generations of right-handed fermion fields for charged leptons, up-type quarks, and down-type quarks, respectively. We define $ L_{\rm L} = (v_{{\rm L}_i},\ell_{{\rm L}_i})^{T} $ and $ Q_{\rm L} = (u_{{\rm L}_i},d_{{\rm L}_i})^T $ , withirepresenting generation indices. Under $ Z_4 $ symmetry, the lepton Yukawa matrix $ y_\ell $ is diagonal, and therefore the lepton fields ( $ L_{\rm L} $ , $ E_{\rm R} $ ) are mass eigenstates.

Under $ Z_4 $ symmetry, the $ \phi_2 $ doublet is allowed to haveμ-τinteractions [53].

The additional Higgs bosonsH,A, and $ H^\pm $ only haveμ-τLFV Yukawa couplings. However, the SM-like Higgs bosonhhas exactly the same couplings to the gauge bosons and fermions as the SM Higgs, with noμ-τLFV couplings at tree level.

The model can produce corrections for the masses of gauge bosons via the exchange of extra Higgs fields in self-energy diagrams. The oblique parameters $ (S,\; T,\; U) $ [63,64] represent radiative corrections to the two-point functions of gauge bosons. Most effects on precision measurements can be described by these parameters. Recently, Ref. [9] gave the values of these parameters from an analysis of precision electroweak data, including the new CDF result forW-mass,

The correlation coefficients are given by

TheW-boson mass can be inferred from the following relation [64]:

In our analysis, we adopt $ \textsf{2HDMC} $ [65] to calculate the 2HDM corrections to the $ S,\; T,\; U $ parameters and perform a global fit to the predictions of the $ S,\; T,\; U $ parameters. Because the global fit results are presented on two-dimensional planes, a limit of $ \chi^2 < \chi^2_{\rm{min}} + 6.18 $ is set to obtain $ 2\sigma $ favored regions, where $ \chi^2_{\rm{min}} $ is the minimum of the $ \chi^2 $ corresponding to the best fit point. In addition, we consider theoretical constraints from perturbativity, vacuum stability, and unitarity, which are described in detail in Appendix A.

We scan the $ m_H $ , $ m_A $ , and $ m_{H^\pm} $ parameters in the following ranges:

InFigs. 1and2, we show samples explaining the CDFW-boson mass measurement within a $ 2\sigma $ range while satisfying the constraints on the oblique parameters and theory. FromFigs. 1and2, we see thatHorAis disfavored to exactly degenerate in mass with $ H^\pm $ ; however, their masses are allowed to be degenerate. The mass splitting between $ H^\pm $ and $ H/A $ imposes upper and lower bounds and is required to be larger than approximately 10 GeV. When $ m_H $ or $ m_A $ is close to $ m_{H^\pm} $ , the other is allowed to have sizable deviation from $ m_{H^\pm} $ . $ m_{H^\pm} $ and $ m_{A} $ are favored to be smaller than 650 GeV for $ m_H<120 $ GeV (seeFig. 2) and allowed to have larger values with increasing $ m_H $ (seeFig. 1).

Figure 1.(color online) For $ m_H> $ 130 GeV, samples explaining the CDF II results ofW-mass within a $ 2\sigma $ range while satisfying the constraints of the oblique parameters and theoretical constraints. The varying colors in each panel indicate the values of $ \mid m_A-m_{H^\pm}\mid $ , $ \mid m_H-m_{H^\pm}\mid $ , and $ m_{H^\pm} $ .

Figure 2.(color online) Same asFig. 1but for $ m_H< $ 120 GeV.

Now, we analyze the reason. In the model, the correction to theTparameter is expressed as

where the $ \mathcal{F} $ function is [66–68]

The function $ \dfrac{\mathcal{F}(m_1^2,m_2^2)}{m_W^2} $ and factor $ \dfrac{1}{16\pi s_W^2} $ in theTparameter are usually larger than those of theSandUparameters. Therefore, in general, one expectsTto be dominant in oblique corrections. Detailed discussions can be found in Ref. [69]. The expressions in Eqs. (13) and (14) show that theTparameter is sensitive to the mass splittings among $ H,\; A $ , and $ H^{\pm} $ . TheTparameter will be zero for $ m_H = m_{H^\pm} $ or $ m_A = m_{H^\pm} $ but takes a non-zero value for $ m_A = m_{H} $ . Therefore, the corrections of the model to the oblique parameters tend to decrease as either $ m_H $ and $ m_{A} $ approaches $ m_{H^\pm} $ . However, to accommodate theW-mass reported by the CDF II collaboration, the model must produce an appropriate value ofT, which excludes $ m_H = m_{H^\pm} $ or $ m_A = m_{H^\pm} $ .

The model gives additional corrections to the muon $ g-2 $ anomaly ( $ \Delta a_{\mu} $ ) via one-loop diagrams involving theμ-τLFV couplings ofHandA[44–46].

Here, we find that $ \Delta a_{\mu}>0 $ for $ m_A>m_H $ .

Because the extra Higgs bosons haveμ-τLFV interactions, the model can affect lepton flavor universality (LFU) inτlepton decays. The HFAG collaboration tested LFU from the ratios of the partial widths of a heavier lepton. They obtained [70]

using pure leptonic processes, namely,

with $ \overline{\Gamma} $ representing the partial width, which is normalized by the corresponding SM value. $ g_{e\; (\mu,\tau)} $ denote the effective couplings betweene $ (\mu,\tau) $ and $ \nu_e $ $ (\nu_\mu,\nu_\tau) $ . With the two semi-hadronic processes,

wherehindicatesπorK, they measure

The statistical correlation matrix for the five fitted coupling ratios is

In this model, we can calculate

The tree-level correction $ \delta_{\rm{tree}} $ is from the contribution of $ H^\pm $ to $ \tau\to \mu\nu\overline{\nu} $ ,

which can give a positive correction. $ \delta_{\rm{loop}}^\mu $ and $ \delta_{\rm{loop}}^\tau $ are the corrections to the vertices $ W\bar{\nu_{\mu}}\mu $ and $ W\bar{\nu_{\tau}}\tau $ from one-loop diagrams containingA,H, and $ H^\pm $ . Because we assume $ \rho_{\mu\tau} = \rho_{\tau\mu} $ in the lepton Yukawa matrix, the two corrections are identical [53,71,72].

where $ H(x_\phi) \equiv \ln(x_\phi) (1+x_\phi)/(1-x_\phi) $ with $ x_\phi = m_\phi^2/m_{H^{\pm}}^2 $ . Meanwhile, for the semi-hadronic processes, we have

In our study, we perform a global fit to the predictions of these five ratios. Note that there is a vanishing eigenvalue in the covariance matrix constructed from Eqs. (16), (21), and (22), and therefore such a degree of freedom is removed. Using the remaining four degrees of freedom, the $ 2\sigma $ confidence level region is obtained by adopting a limit of $ \chi^2_\tau<9.72 $ . Thus, the surviving samples are considerably more consistent with the experimental results than that of the SM, which has a $ \chi^2_\tau $ of $ 12.25 $ . Furthermore, the model can affect LFU inZ-boson decays [73], and the constraints fromZ-boson decays are generally weaker than those fromτdecays.

In the model, the fieldsH,A, and $ H^{\pm} $ have no couplings to quarks; therefore, they are produced at the LHC mainly via electroweak processes, $ pp\to W^{\pm *} \to H^\pm A/H $ , $ pp\to Z^* \to HA $ , and $ pp\to Z^*/\gamma^* \to H^+H^- $ . The final state signal mainly includes multi-leptons, and therefore multi-lepton event searches at the LHC can impose stringent constraints that require $ m_H $ to be larger than 560 GeV [55]. Moreover, a very lightHmay escape the constraints of direct searches at the LHC [56]. In this case, the explanation of the muon $ g-2 $ anomaly requires a smallρ, which leads to the contributions of the model toτdecays being too small to explain the data of LFU in theτdecays. Therefore, in this study, we discuss the scenario for $ m_H>560 $ GeV. In addition, to respect perturbativity, we choose the Yukawa coupling parameter $ \rho<1 $ .

At tree level, the 125 GeV Higgs has the same couplings to SM particles as in the SM. The $ h\to \gamma\gamma $ decay will be corrected by the one-loop diagram of the charged Higgs [74]. We consider the bound of the diphoton signal strength [2] to be

InFig. 3, we project the surviving samples explaining the muon $ g-2 $ anomaly and LFU inτ-decays within $ 2\sigma $ ranges, satisfying other relevant constraints from theory, the oblique parameters, the CDF IIW-mass, and the diphoton signal data of the 125 GeV Higgs. Eq. (15) shows thatHandAgive positive and negative contributions to muon $ g-2 $ , respectively, and their contributions are suppressed by their masses. Therefore, the explanation of muon $ g-2 $ requires $ m_A > m_H $ andρto increase with increasing $ m_H $ and decreasing $ (m_A-m_H) $ (see the upper-left panel).

Figure 3.(color online) Bullets explain muon $ g-2 $ within a $ 2\sigma $ range, whereas the squares explain muon $ g-2 $ andτ-decays within $ 2\sigma $ ranges while satisfying the constraints ofZ-decays. Other relevant constraints from theory, the oblique parameters, the CDF IIW-mass, and the diphoton signal data of the 125 GeV Higgs are also satisfied. The varying colors in each panel indicate the values of $ m_A-m_{H} $ ,ρ, $ m_{H^\pm}-m_H $ , and $ m_{H^\pm}-m_A $ .

The ratio $ \left( {g_\tau} / {g_e }\right) $ inτdecays has a deviation of approximately $ 2\sigma $ from the SM. Therefore, enhancing $ \Gamma(\tau\to \mu \nu\bar{\nu}) $ may give a better fit to the data of LFU in theτdecays. The decay $ \tau\to \mu\nu\nu $ obtains a positive correction from the $ \delta_{\rm{tree}} $ term of Eq. (23). Such a term is from the tree-level diagram mediated by the charged Higgs and proportional to $ \rho^4/m^2_{H^\pm} $ . Therefore, the upper-right panel ofFig. 3shows that the value of $ \chi_\tau^2 $ tends to increase with increasing $ m_{H^\pm} $ and decreasingρ. In addition, because of the constraints of the oblique parameters andW-boson mass, the upper-left panel ofFig. 1shows that $ m_{H^\pm} $ tends to increase with $ m_H $ , especially for large $ m_H $ , which implies that the value of $ \chi_\tau^2 $ increases with increasing $ m_H $ . Therefore, the upper-left panel ofFig. 3shows that a simultaneous explanation of muon $ g-2 $ and LFU inτdecays favors a largeρthat increases with $ m_H $ .

From the lower panel ofFig. 3, we see that the mass splittings amongH,A, and $ H^\pm $ are stringently constrained in the region simultaneously explainingW-mass, muon $ g-2 $ , and LFU inτdecays, that is, 1 $ 75 GeV, 65 $ 100 GeV, and 85 $ 125 GeV (–150 $ –85 GeV, and –105 $ –55 GeV).

In the model, pair production of extra Higgs bosons via electroweak processes at the LHC leads to detectable multi-lepton signals containingμandτ. The current lower bound of 560 GeV on $ m_H $ is mainly obtained from the CMS search for the electroweak production of charginos and neutralinos in multilepton final states [75]. By normalizing the number of background and signal events, the lower bound on $ m_H $ can be increased to 700 GeV for 300 fb $ ^{-1} $ of integrated luminosity data. Adopting the same procedure, we estimate that more than 3000 fb $ ^{-1} $ of integrated luminosity data is required to cover the entire surviving parameter, namely, $ m_H<950 $ GeV. However, this CMS search was originally designed to search for SUSY particles. A dedicated study focusing on $ \mu\tau $ final states and improving signal regions for the high mass region can greatly reduce the required integrated luminosity to an acceptable level, which is beyond the scope of this paper.

We examine the CDF IIW-boson mass and FNAL muon $ g-2 $ in the 2HDM in which the extra Higgs doublet hasμ-τLFV interactions. Imposing theoretical constraints, we find that the CDF IIW-boson mass disfavorsHorAdegenerating in mass with $ H^\pm $ , and the mass splitting between $ H^\pm $ and $ H/A $ is favored to be larger than 10 GeV. $ m_{H^\pm} $ and $ m_{A} $ are favored to be smaller than 650 GeV for $ m_H<120 $ GeV and allowed to have larger values with increasing $ m_H $ . Considering other relevant experimental constraints, we find that the mass splittings among $ H,\; A $ , and $ H^\pm $ are stringently restricted in the parameter space, which can simultaneously explain the CDF IIW-mass, the FNAL muon $ g-2 $ , and LFU data inτdecays.

We thank Lei Wu for helpful discussions.

The quartic couplings of the scalar potential in Eq. (3) cannot be too large individually otherwise the theory will no longer be perturbative. Thus, we require

Vacuum stability requires the potential to be bound from below and remain positive for arbitrarily large values of the fields. This requirement leads to restrictions on the parameters of the model.

The amplitudes for the scalar-scalar scattering $ s_1 s_2 \to s_3 s_4 $ at high energies respect unitarity, which leads to the following bounds on the parameters of the model [76,77]:

with