Type-X two Higgs doublet model in light of the muon $\mathbf{g-2}$: confronting Higgs and collider data

The recent Fermilab measurement of the muon anomalous magnetic moment yields $4.2 \sigma$ deviations from the SM prediction when combined with the BNL E821 experiment results. In the Type-X two Higgs doublet model, we study the consequence of imposing the observed muon $g-2$, along with the constraints from theoretical stabilities, electroweak oblique parameters, Higgs precision data, and direct searches. For a comprehensive study, we scan the whole parameter space in two scenarios, the normal scenario where $h_{\rm SM} = h$ and the inverted scenario where $h_{\rm SM}=H$, where $h$ ($H$) is the light (heavy) CP-even Higgs boson. We found that large $\tan\beta$ (above 100) and light pseudoscalar mass $M_A$ are required to explain the muon $g-2$ anomaly. This breaks the theoretical stability unless the scalar masses satisfy $M_A^2 \simeq M_{H^\pm}^2 \simeq m_{12}^2 \tan\beta \approx M_{H/h}^2$. The direct search bounds at the LEP and LHC exclude the light $A$ window with $M_A \lesssim 62.5~$GeV. We also show that the observed electron anomalous magnetic moment is consistent with the model prediction, but the lepton flavor universality data in the $\tau$ and $Z$ decays are not. For a separate exploration of the model, we propose the golden mode $pp \to A h/AH \to 4 \tau$ at the HL-LHC.


I. Intoduction 2
II. Type-X 2HDM 4 III. ∆a µ in the Type-X 2HDM 6 IV. Theoretical and experimental constraints on the Type-X 2HDM 8 A. Scanning strategies in three steps 8 B. Results in the normal scenario 9 C. Results in the inverted scenario 11 V. Implications on the electron g − 2 and the LHC collider signatures 13 A. Electron anomalous magnetic moment 14 B. Production of the hadro-phobic new scalars at the LHC 14 VI. Lepton flavor universality data in the τ and Z decays 18

Acknowledgments 22
A. Used parameters in the τ ± and Z decays for the global χ 2 analysis 22 References 23

I. INTODUCTION
The recent measurement of the muon anomalous magnetic moment by the Fermilab National Accelerator Laboratory (FNAL) Muon g − 2 experiment [1, 2] achieved unprecedented precision. When combined with the old result of the Brookhaven National Laboratory (BNL) E821 measurement [3], it reads as where a µ = (g − 2) µ /2. As the experimental error is becoming comparable with the theoretical error, 1 reliable and accurate calculation of the SM prediction is more important than ever. The recent progress includes five loops in QED [4] and two loops in electroweak interactions [5,6]. Nevertheless, the most dominant contribution is from the strong interaction dynamics at O(1) GeV, which is categorized into the hadronic vacuum polarization (HVP) [7][8][9][10][11][12][13][14] and the hadronic light-by-light (HLbL) scattering [15][16][17][18][19][20][21][22][23][24][25][26][27]. These QCD corrections cannot be computed using perturbation theory. We have to resort to non-perturbative methods, either Lattice QCD or data-driven methods. On the Lattice side, the recent calculation of the leading order HVP (LO-HVP) contributions to a µ by the Budapest-Marseille-Wuppertal collaboration [28] yields a µ | LO−HVP = 707.5(5.3) × 10 −10 . If we take this result at face value, the Fermilab measurement of a µ is consistent with the SM prediction at ∼ 2σ. On the data-driven method side, however, the calculation of the HVP contribution [9,11,12] supports the long-standing discrepancy between the muon g − 2 experiment and the SM prediction, as ∆a obs µ = a exp µ − a SM µ = 251(59) × 10 −11 . (2) Which method is more appropriate needs further investigation. One checking point is the connection of the QCD corrections to electroweak precision fits [29][30][31][32], since some of the most important inputs to HVP and HLbL contributions come from measuring the R(s)-ratio in e + e − collisions. Lately, some tension was reported between the Lattice result and the electroweak data [30,31]. Another critical topic is how to combine the probability distribution functions with different errors.
In this paper, we take the 4.2σ deviation in Eq.
From this motivation, we study the CP invariant Type-X (lepton-specific) 2HDM in light of the muon g−2. In Type-X, the couplings of the new scalar bosons to the SM quarks are inversely proportional to tan β, the ratio of two vacuum expectation values of two Higgs doublet fields, but those to the charged leptons are linearly proportional to tan β. Large tan β can enhance the new contributions of extra Higgs bosons to ∆a µ , while suppressing the contributions to the hadron-related data such as B → Kµ + µ − and B s → µ + µ − [74]. Since the other three types (Type-I, Type-II, and Type-Y) cannot accommodate this feature, there have been extensive studies of Type-X for the muon anomalous magnetic moment [75][76][77][78][79][80][81][82].
A comprehensive study of Type-X for ∆a µ , including the LHC Run-1 and LEP results as well as the lepton flavor universality (LFU) data in the Z and τ decays, was first conducted in Ref. [78]. Partial updates have followed, focusing on the LFU data [79] or the LHC data [80][81][82]. We generalize the previous studies of the Type-X 2HDM both in theoretical setup and in data analysis. First, we take the general setting in the Higgs sector, by considering two scenarios, the "normal" scenario where the observed Higgs boson is the lighter CP -even scalar h and the "inverted" scenario where the heavier CP -even scalar H is the observed one. The inverted scenario with a new light CP -even scalar has recently drawn a lot of interest because of the 3σ excess in the diphoton invariant mass distribution at around 96 GeV [83], but has not been analyzed in the context of the muon g − 2. This scenario seems incompatible with the recent measurement of positive ∆a µ , because the dominant Barr-Zee contributions of the τ ± loop mediated by the light CP -even h are negative [84,85]. We need to answer whether the inverted scenario remains viable.
For the general data analysis, we will investigate all the latest data of the LHC Run-2, the electron anomalous magnetic moment [86,87], the LFU data (adopting the updated HFLAV global fit results [88] and Michel parameters [89] in the τ decay), as well as theoretical stabilities and electroweak oblique parameters. We will also include the correlations among the observables. The correlations are often neglected in the literature, but they play a vital role in constraining new physics models. To draw a general conclusion on the Type-X 2HDM, we will scan the whole parameter space without any extra assumption on the masses or the couplings. Furthermore, the tension in the Type-X when simultaneously explaining ∆a µ and LFU data shall be quantified through the global χ 2 fit. Finally, the customized search strategy for the viable parameter space at the HL-LHC is to be studied. These are our contributions to the phenomenology of the Type-X 2HDM in light of the new Fermilab measurement of ∆a µ .
The paper is organized in the following way. In Sec. II, we briefly review the Type-X 2HDM and describe the characteristics of the normal and inverted scenarios in the Higgs alignment limit. In Sec. III, we discuss the new contributions of the Type-X 2HDM to ∆a µ . Section IV describes our scanning strategies in three steps and shows the results of the allowed parameter space at each step. Section V deals with the electron anomalous magnetic moment and the LHC signatures. In Sec. VI, we check the consistency of the model with the LFU data in the τ and Z decays. Conclusions are given in Sec. VII.

II. TYPE-X 2HDM
The 2HDM accommodates two complex SU (2) L Higgs doublet scalar fields, Φ 1 and Φ 2 [90]: where v = v 2 1 + v 2 2 = 246 GeV. Using the simplified notation of s x = sin x, c x = cos x, and t x = tan x, we define t β = v 2 /v 1 . To prevent the tree-level flavor changing neutral currents, a discrete Z 2 symmetry is imposed as Φ 1 → Φ 1 and Φ 2 → −Φ 2 [91,92]. The most general, renormalizable, and CP conserving scalar potential with softly broken Z 2 symmetry is where the m 2 12 term softly breaks the Z 2 parity. There are five physical Higgs bosons, the light CP -even scalar h, the heavy CP -even scalar H, the CP -odd pseudoscalar A, and two charged Higgs bosons H ± . The relations of the physical Higgs bosons with the weak eigenstates in Eq. (3) via two mixing angles α and β are referred to Ref. [93,94]. Note that the SM Higgs boson is a linear combination of h and H, as The Yukawa couplings to the SM fermions are written by where P R,L = (1 ± γ 5 )/2 and = µ, τ . The observed Higgs boson at a mass of 125 GeV is similar to the SM Higgs boson, more strongly in Type-X with large t β [95]. Therefore, we take the Higgs alignment limit where one of the CP -even neutral Higgs bosons is the SM Higgs boson h SM [96][97][98][99][100]. There are two ways to realize the Higgs alignment limit, the "normal" and "inverted" scenarios. In the normal scenario, the observed Higgs boson is the lighter CP -even scalar h, i.e., s β−α = 1. In the inverted scenario, c β−α = 1 so that the heavier CP -even scalar H is observed while the lighter one is hidden [99,101]. The model has five independent parameters in the physical basis, where M 2 = m 2 12 /(s β c β ) and ϕ 0 is the new CP -even neutral Higgs boson, i.e., ϕ 0 = H in the normal scenario and ϕ 0 = h in the inverted scenario. The two scenarios are summarized as follows: In the Higgs alignment limit, the quartic couplings are [100] where m 125 = 125 GeV. As shall be shown in the next section, the observed ∆a µ requires large t β . Then, the t 2 β terms in λ 1 easily break the perturbativity of λ 1 unless m 2 ϕ 0 is extremely close to M 2 , which is to be denoted by m 2 ϕ 0 ≈ M 2 . When applying this approximate equality to the perturbativity of λ 3 , we should accommodate quasi-degeneracy between M 2 and M 2 H ± . The mass degeneracy is weaker because of the absence of t 2 β terms in λ 3 . We use the notation of M 2 M 2 H ± for the weak equality. The perturbativity of λ 4 and λ 5 finally yields M A M H ± . In summary, the perturbativity of the quartic couplings for large t β limits the masses as For light M A , the exotic Higgs decay of h SM → AA severely restricts the model. When writing L = (1/2)λ h SM AA h SM AA, the vertex is Because of the condition in Eq. (10), it is difficult to accommodate λ h SM AA = 0. 2 Since the Higgs precision measurement puts a strong bound on the exotic Higgs decay as B(h SM → XX) O(0.1) [102], the parameter region with M A ≤ m 125 /2 is highly disfavored.
where φ = {ϕ 0 , A, H ± }, ρ i j = m 2 i /m 2 j , and the expressions for the loop function f φ are referred to Ref. [80]. The numerical factor in the second equality of Eq. (12) implies that the observed ∆a µ requires light M φ and large y φ µ . Because ρ µ φ 1, the loop functions show the following asymptotic behaviors: It is clear to see that the one-loop contributions of the CP -even scalar ϕ 0 are positive while those of A and H ± are negative: ∆a 1−loop µ is proportional to the square of y φ µ . More significant contributions to ∆a µ are from the two-loop Barr-Zee type diagrams with heavy fermions in the loop [84]: where and N c f are the mass, electric charge and color factor of the fermion f , and the loop functions are For the top quark and τ ± loops, the factor ρ f φ 0 in Eq. (14) significantly enhances ∆a BZ µ with respect to ∆a 1−loop µ in Eq. (12). The usual conclusion that a CP -even scalar boson makes a negative contribution to ∆a BZ µ holds true when y ϕ 0 µ y ϕ 0 f > 0. As shown in Eq. (8), the top quark incorporates y ϕ 0 µ y ϕ 0 t < 0 in both scenarios and thus generates positive two-loop Barr-Zee contributions.
In Fig. 1, we show ∆a µ (A) (blue line), −∆a µ (ϕ 0 ) (red line), and ∆a µ (A) + ∆a µ (ϕ 0 ) (black line) as a function of t β with M A = m ϕ 0 = 100 GeV in the left panel and as a function of M A = m ϕ 0 with t β = 100 in the right panel. To show negative ∆a µ (ϕ 0 ) in the logarithmic scale, we present −∆a µ (ϕ 0 ). The horizontal green (yellow) area denotes the allowed region of ∆a µ at 1σ (2σ). The dominant contribution of A is from two-loop Barr-Zee diagrams, which is always positive. The sign of the ϕ 0 contribution depends on the value of t β . For very large t β , ∆a µ (ϕ 0 ) is negative since the contribution of the τ ± loop in the two-loop Barr-Zee diagram is dominant. If t β 17, however, ∆a µ (ϕ 0 ) becomes positive (see the small figure inside the left panel) because dominant is the top quark loop in the two-loop Barr-Zee diagram. Although the contributions from both A and ϕ 0 are positive for t β 17, the absolute value of ∆a µ is not large enough to explain ∆a obs µ . In the right panel, we show ∆a µ (A) and −∆a µ (ϕ 0 ) as a function of M A = m ϕ 0 by fixing t β = 100. ∆a µ increases rapidly with decreasing scalar masses. Since the negative contributions of the CP -even ϕ 0 become severe with decreasing m ϕ 0 , the inverted scenario receives a stronger constraint.

A. Scanning strategies in three steps
For the comprehensively study of the Type-X 2HDM in light of the muon g − 2, we perform the successive and cumulative scan of the model parameters in three steps.
Step I: We demand that the model explains ∆a obs µ at 2σ.
Step II: Among the parameters that survive Step I, we impose the constraints from theoretical stabilities and electroweak precision data, as detailed below.
Step III: For the parameters that survive Step II, we demand to satisfy the collider bounds.
1. Higgs precision data by using HiggsSignals [110,111]: The HiggsSignals-v2.2.0 [111] provides the χ 2 value for 107 Higgs observables. Since our model has five parameters, the number of degrees of freedom for the χ 2 analysis is 102. We require that the calculated Higgs signal strengths be consistent with the experimental measurements at 2σ.
For each scattering process, we compute the r 95% defined by where S 2HDM (S 95% obs ) is the predicted (observed) cross section. A point in the parameter space is excluded at the 95% confidence level if r 95% > 1.
In the normal scenario, we obtained 5 × 10 5 parameter sets that satisfy Step II.
Step III excludes about 80% of the parameter sets that survived Step II. The exclusion is more severe in the inverted scenario, for which we separately collected 5 × 10 5 parameter sets that pass Step II. Only ∼ 1.8% parameter sets survive at Step III. Let us go back to discussing the allowed (M A , t β ). At Step III, which additionally imposes the constraints from the collider data at the LEP, Tevatron, and LHC, a large portion of the parameter space is removed: see the right panel in Fig. 2. We found that the recent LHC theoretical stability, is easier to satisfy with light m ' 0 . At Step III (right panel), the collider constraints in the inverted scenario are stronger than in the normal scenario, leading to larger t as t & 120. As discussed in Sec. IV A, only ⇠ 1.8% of the allowed points at Step II survive Step III in the inverted scenario, compared with ⇠ 20% in the normal scenario. And at the final step, we see the correlation of a µ with M A and t , as suggested by the unblended colors. Figure 6 presents the smoking-gun processes which yield the largest deviation of the model theoretical stability, is easier to satisfy with light m ' 0 . At Step III (right panel), the collider constraints in the inverted scenario are stronger than in the normal scenario, leading to larger t as t & 120. As discussed in Sec. IV A, only ⇠ 1.8% of the allowed points at Step II survive Step III in the inverted scenario, compared with ⇠ 20% in the normal scenario. And at the final step, we see the correlation of a µ with M A and t , as suggested by the unblended colors. Figure 6 presents the smoking-gun processes which yield the largest deviation of the model data plays a crucial role in the curtailment. To demonstrate the role, we present the allowed parameter points in (M A , t β ) by the LHC data before 2015 (red) and those after 2015 (blue) in Fig. 4: 2015 is taken as the reference point in consideration of Ref. [78]. The new LHC data exclude the whole parameter space of M A < m 125 /2 and t β 90. The accumulation of the LHC null results in the NP searches gives a significant implication on the Type-X 2HDM in the context of muon g − 2.
The question that follows is which LHC processes exclude the region of M A < m 125 /2. In principle, multiple processes exclude one parameter set simultaneously. For efficient illustration, we present in Fig. 5 the smoking-gun process that has the largest deviation of the model prediction from the observation, r 95% in Eq. (17). The green points pass all the constraints. The orange points are rejected by the LHC bounds on h SM → AA → µ + µ − τ + τ − [116,149]. The red points are excluded by the combined LEP results of e + e − → H + H − including the decays of H + H − into cscs, csτ ν, τ ντ ν, W * Aτ ν, and W * AW * A [150]. The overlap of the allowed (green) and excluded (red) points is attributed to the projection of the five-dimensional hypervolume onto the two-dimensional (M A , t β ) plane. In summary, the normal scenario of the Type-X 2HDM in light of the muon g−2 is phenomenologically viable for

C. Results in the inverted scenario
In the inverted scenario, the pattern of the exclusion at Step I, Step II, and Step III is similar to that in the normal scenario: see Fig. 6. In the quantitative aspect, however, there are some differences. At Step I, the observed ∆a µ prefers lighter M A than in the normal scenario, as the light CP -even h makes a sizably negative contribution. The constraints at Step II are weaker than in the normal scenario. The perturbativity of λ 1 , the most critical factor for the theoretical stability, is easier to satisfy with light m h . At Step III (right panel), the collider constraints in the inverted scenario are stronger than in the normal scenario, leading to larger t β as t β 120. Figure 7 presents the collider smoking-gun processes in the inverted scenario. The green points are finally allowed. The orange and red points are excluded by h SM → AA [116,149]  c β−α , is maximal in the alignment limit of the inverted scenario. We found that the constraint from e + e − → Ah is so strong that only the kinematic ban of √ s ee < M A + M h saves the parameter point.
In Fig. 8 Upon obtaining the finally allowed parameter points of the Type-X 2HDM in light of the new muon g − 2, we investigate the phenomenological implications of the surviving parameters. First, we study the electron anomalous magnetic moment. For ∆a e , there is controversy over the value of the fine structure constant α. Therefore, we check the consistency of the surviving parameters with ∆a e rather than accept ∆a e as an observable. Second, we study the LHC phenomenology so to suggest the golden mode for the hadro-phobic scalar bosons. Since direct searches at high energy colliders provide independent information, the LHC exploration should continue.

A. Electron anomalous magnetic moment
As a flavor universal theory, Type-X 2HDM has the same contributions to ∆a e and ∆a µ except the differences of the electron and muon masses. Positive ∆a µ demands positive ∆a e . In the measurement, however, ∆a e has not been settled yet because of the discrepancy in the recent two experiments for the fine structure constant α, the most sensitive input to ∆a e . Depending on whether we take the data from 133 Cs [86] or from 87 Rb [87], the deviations of the electron g − 2 from the SM prediction [46,151,152]  At 2σ level, ∆a Cs e is negative while ∆a Rb e can be positive. In Fig. 9 The new scalar bosons with intermediate mass have escaped the LHC searches because of their hadro-phobic nature due to large t β . For the intermediate-mass H ± , the current LHC search depends on its production via the decay of a top quark into bH ± , followed by H ± → τ ν [147,148]. When t β 100, however, the H ± -t-b vertex is extremely small, suppressing the production of the charged Higgs boson. For the intermediate-mass A and ϕ 0 , the LHC searches resort to the gluon fusion production via top quark loops, which is also suppressed. These hadro-phobic new scalar bosons need different search strategies.
We study the branching ratios of A, ϕ 0 , and H ± in the viable parameter space. Both A and ϕ 0 dominantly decay into τ + τ − . The branching ratios of H → ZA and H → H ± W ∓ are below 10%. For the H ± decays, Fig. 10 presents the scatter plot of the branching ratios as a function of (M H ± − M A ) in the normal (left panel) and inverted (right panel) scenario. The color code indicates the value of t β . The primary decay channel of H ± is into τ ± ν. The second important mode is H ± → W ± A, which is sizable for larger (M H ± − M A ) and smaller t β . In the normal scenario, B(H ± → W ± A) can reach up to about 30%. In the inverted scenario, its maximum is only about 3%.
Based on these characteristics, we consider the following two channels: The process in Eq. (19) is efficient since the Z-A-ϕ 0 vertex has the maximal value in the alignment limit of both scenarios. In addition, m ϕ 0 can be measured through the τ + τ − invariant mass distribution, differentiating the normal scenario from the inverted scenario. The pair   Figure 11 shows the parton-level total cross sections of pp → Z * → AH/Ah → 4τ at the 14 TeV LHC, by scanning all the viable parameter points. In both scenarios, we see a strong anticorrelation of σ tot with M A + M H . In the normal scenario (left panel), the total cross section lies between ∼ 25 fb and ∼ 260 fb. In the inverted scenario, σ tot goes up to about 300 fb, larger than in the normal scenario. Considering the observed σ(pp → ZZ → 4τ ) 17 fb at the 13 TeV LHC [153,154], the process pp → Aϕ 0 → 4τ has a high potential to probe the model. In Fig. 12, we present the prediction of the viable parameters to the total cross sections of the process pp → H + H − → τ ντ ν at the 14 TeV LHC. The results in the normal (inverted) scenario are in the left (right) panels. Two upper (lower) panels present the total cross sections for qq → H + H − (gg → H + H − ). In most parameter spaces, the Drell-Yan production has a much larger signal rate, since the hadro-phobic nature of the charged Higgs boson suppresses the gluon fusion production mediated by the top quark loop. The irreducible backgrounds for the final state of τ + ντ − ν are pp → W + W − → τ + ντ − ν and pp → ZZ → τ + τ − νν. Considering σ SM tot (pp → W + W − → τ + ντ − ν) 1.7 pb [155] and σ SM tot (pp → ZZ → τ + τ − νν) 100 fb [153,154] at the 13 TeV LHC, there is a chance to see the process.

VI. LEPTON FLAVOR UNIVERSALITY DATA IN THE τ AND Z DECAYS
In Sec. IV, we found that the Type-X 2HDM as a solution to the muon g − 2 does not allow decoupling of any new Higgs boson. The generically flavor-universal model may yield excessive violation of the LFU in the τ and Z decays, through the loop contributions mediated by new Higgs bosons. For the rigorous analysis, we first categorize the LFU data as follows: (i) HFLAV global fit results in the τ decay: To parameterize the LFU in the τ decays, we introduce the coupling ratios defined by The second factors in the right-hand sides of Eq. (21) cancel the mass differences of the charged leptons. Including the hadronic decays of τ → πν/Kν → µνν, we consider where R τ 1,···5 SM = 1. Since R τ 2 /R τ 1 = R τ 3 , only four in Eq. (23) are independent. We should remove one redundant degree of freedom that has a zero eigenvalue in the covariance matrix.
(ii) Michel parameters: In the decay of τ − → − νν τ , the energy and angular distribution of − provides valuable information on the LFU. The distribution is written in terms of the Michel parameters ρ, η, ξ, and δ, as [156,157] where x = 2E /m τ , x 0 = 2m /m τ , P τ is the τ − polarization, and θ * is the angle between the − momentum and the τ − spin quantization axis. In the SM [158], they are 3 Including the leptonic and hadronic decays of τ − , we consider the Michel parameters of (iii) LFU in the Z decay: From the partial decay rates of the leptonic Z decays, we take two ratios of [161] R where R Z 1,2 The Type-X 2HDM makes two sorts of contributions to the observables in the τ decays, the tree-level contributions (mediated by the charged Higgs boson) and the one-loop level contributions. We parameterize them by Here τ tree is where g(x) = 1 + 9x − 9x 2 − x 3 + 6x(1 + x) ln x and f (x) is in Eq. (22). For the Michel parameters, only the η µ , (ξδ) µ , and ξ µ are modified as The corrections to ρ e , (ξδ) e , and ξ e are suppressed by the small electron mass. For the hadronic τ decays, the corrections are independent of t β , which is much smaller in the large t β limit than the t 2 β corrections in the leptonic τ decays. For R Z i 's, new contributions are written as where g SM L = s 2 W − 1/2, g SM R = s 2 W , and the full expressions for δg µ,τ L/R at one-loop level are referred to Ref. [80].
Rough estimation of new contributions is useful. Since δ tree δ loop and δ tree τ tree , the dominant contribution is While δ tree is positive so that η µ < 0 in the model, the ALEPH result is η ALEPH µ = 0.160 ± 0.150 [89]. The observed η µ threatens the consistency of the Type-X 2HDM with the LFU data. Now we perform the global χ 2 fit of the Type-X 2HDM to The experimental results of R's and the correlation matrices are summarized in Appendix A.
Altogether we have 17 independent observables, N obs = 17, since we removed one redundant degree of freedom in R τ 1,··· ,5 . In the SM where the number of degree of freedom is N dof = 17, χ 2 min and p value are χ 2 min (SM) = 37.3, p(SM) = 0.003.
The ∆a obs µ with the LFU data calls for NP. For the Type-X 2HDM, we address two issues. The first is the number of degrees of freedom, N dof = N obs − N par , where N par is the number of free parameters. We subtract N par under the assumption that we use one free parameter to explain one observable. But our hypothesis model is not a free Type-X 2HDM. It is the model severely limited by the theoretical and experimental constraints. In favor of the Type-X 2HDM, we take N dof = 17. The second issue is the range of the model parameters in the global χ 2 fit. When finding χ 2 min , we may scan either the whole parameter space without imposing other constraints or only the parameter space consistent with all the constraints. In the two cases, p-values show big differences as follows: p(NS: Step I) = 0.58, p(NS: Step III) = 0.02, (36) p(IS: Step I) = 0.059, p(IS: Step III) = 0.02.
Without the LHC data, the Type-X 2HDM in both scenarios well explains the ∆a µ and LFU data. With the combination of the LHC data and LFU data, however, the model is excluded as a solution to the new Fermilab measurement of the muon g − 2.

VII. CONCLUSION
In light of the recent measurement of the muon anomalous magnetic moment by Fermilab Muon g − 2 experiment, we comprehensively study the Type-X (Lepton-specific) two Higgs doublet model (2HDM). Beyond explaining only the observed ∆a µ , we included the theoretical stability conditions and almost all the available experimental results in the analysis. Since the Higgs precision data prefers the SM-like Higgs boson, more strongly for large t β , we assumed the Higgs alignment. Two possible scenarios are studied, the normal scenario where the lighter CP -even h becomes h SM and the inverted scenario where the heavier CP -even H is h SM . The model has five parameters, m ϕ 0 , M A , M H ± , M 2 , and t β , where ϕ 0 = H in the normal scenario and ϕ 0 = h in the inverted scenario.
Various phenomenological conditions cause a chain reaction of constraining the model parameters. First, the large and positive ∆a obs µ requires large t β and light M A . The dominant contribution is from the τ ± loop mediated by A in the two-loop Barr-Zee diagram. Unwanted is the negative contribution of ϕ 0 to the τ ± loop in the Barr-Zee diagram. But decoupling of ϕ 0 conflicts with the theoretical stability because of large t β . The Higgs quartic coupling λ 1 has t 2 β terms, which can easily break the perturbativity of λ 1 . Requiring the t 2 β terms to vanish yields m 2 ϕ 0 ≈ M 2 . Perturbativity of other quartic couplings subsequently demands M A M H ± M ≈ m ϕ 0 . Decoupling of any new Higgs boson is not possible. The direct search bounds at the LEP and LHC exclude a large portion of the parameter space: pp → h SM → AA in the normal scenario and e + e − → Z * → Ah in the inverted scenario are the smoking signals. Only the region with M A > m h SM /2 survives. In turn, ∆a obs µ demands t β 100. Through random scanning without any prior assumptions on the masses and couplings, we obtained the parameter points consistent with the muon g −2, theoretical stabilities, S/T /U parameters, Higgs precision data, and direct search results. We also studied the phenomenological implications of the allowed parameter space. The model prediction to the electron anomalous magnetic moment is consistent with the observation, ∆a Cs e (using the fine structure constant α from 133 Cs) at 3σ, and ∆a Rb e (using α from 87 Rb) at 2σ. For the HL-LHC searches, we calculated the total cross sections for the hadro-phobic new scalar bosons in two processes, pp → Aϕ 0 → 4τ and pp → H + H − → τ ντ ν. In particular, pp → Aϕ 0 → 4τ has the total cross section around 25 ∼ 260 fb in the normal scenario and 180 ∼ 300 fb in the inverted scenario. The model has a high potential to be probed at the LHC. As the final check of the model, we studied the lepton flavor universality in the τ and Z decays. Through a global χ 2 fit to 16 LFU data and ∆a obs µ , we showed that the combination of the LHC results and the LFU data excludes the Type-X 2HDM as a solution to the muon g − 2.
The confirmed deviation of the muon g − 2 from the SM prediction by the recent Fermilab experiment indicates the dawn of a new physics era. The Type-X 2HDM that explains ∆a µ is consistent with the LEP and LHC data in limited parameter space, but not with the LFU data in the τ and Z decays. The future LHC searches targeting the specific parameters shall provide a valuable and independent probe of the model, which we strongly support.

Acknowledgments
We would like to thank Kingman Cheung and Chih-Ting Lu for useful discussions. This work is supported by the National Research Foundation of Korea, Grant No. NRF-2019R1A2C1009419.