Lepton-Specific Two Higgs Doublet Model as a Solution of Muon $g-2$ Anomaly

We discuss the Type-X (lepton-specific) two Higgs doublet model as a solution of the anomaly of the muon $g-2$. We consider various experimental constraints on the parameter space such as direct searches for extra Higgs bosons at the LEP II and the LHC Run-I, electroweak precision observables, the decay of $B_s \to \mu^+\mu^-$, and the leptonic decay of the tau lepton. We find that the measurement of the tau decay provides the most important constraint, which excludes the parameter region that can explain the muon $g-2$ anomaly at the 1$\sigma$ level. We then discuss the phenomenology of extra Higgs bosons and the standard model-like Higgs boson ($h$) to probe the scenario favored by the $g-2$ data at the collider experiments. We find that the $4\tau$, $3\tau$ and $4\tau + W/Z$ signatures are expected as the main signal of the extra Higgs bosons at the LHC. In addition, we clarify that the value of the $h\tau\tau$ coupling is predicted to be the standard model value times about $-1.6$ to $-1.0$, and the branching fraction of the $h\to \gamma\gamma$ mode deviates from the standard model prediction by $-30\%$ to $-15\%$. Furthermore, we find that the exotic decay mode, $h$ decaying into the $Z$ boson and a light CP-odd scalar boson, is allowed, and its branching fraction can be a few percent. These deviations in the property of $h$ will be tested by the precision measurements at future collider experiments.

As it has been well known that there is a discrepancy between the experimental value and the prediction of the standard model (SM). According to the calculation evaluated in Refs. [2,3] a exp µ − a SM µ = (28.7 ± 8.0) × 10 −10 , (Davier et. al.) a exp µ − a SM µ = (26.1 ± 8.0) × 10 −10 , (Hagiwara et. al.) the discrepancy is more than the 3σ level, which can be considered as an indirect evidence of the existence of a new physics model. This discrepancy will be further probed at Fermilab [4] and J-PARC [5] in the near future. Since the size of the deviation is the same order as the electroweak contribution a EW µ = 15.4 × 10 −10 [6], we expect that new physics exists at the electroweak scale if the strength of new interactions is as large as that of the weak interaction. In such a new physics scenario, new particles are expected to be light enough to be directly discovered at the LHC. Therefore, it is quite interesting to consider models beyond the SM as a solution of the muon g − 2 anomaly.
Among various models which can explain the anomaly (for a review, e.g., see Ref. [7]), two Higgs doublet models (2HDMs) give simple solutions. In 2HDMs, there are extra Higgs bosons (H, A, and H ± ) in addition to the SM-like Higgs boson (h), and they can give new contributions to a µ . Usually, a softly-broken discrete Z 2 symmetry is imposed [8] to avoid flavor changing neutral current (FCNC) processes at the tree level. Under the Z 2 symmetry, four independent types of Yukawa interactions are allowed depending on the assignment of the Z 2 charge to the SM fermion [9,10], which are called as Type-I, Type-II, Type-X (or lepton specific) and Type-Y (or flipped) [11]. In all the types of Yukawa interactions, the lepton couplings to the extra Higgs bosons can be sizable enough to explain a µ . In the Type-I and Type-Y 2HDMs, however, the top Yukawa coupling also becomes large together with the enhancement of the lepton couplings. This is disfavored from the view point of perturbativity. Thus, the Type-II and Type-X 2HDMs are suitable to solve the muon g − 2 anomaly.
In the early 2000s, this was calculated at the one-loop level in the Type-II 2HDM in Ref. [12]. After that, it was pointed out in Refs. [13,14] that the two-loop Barr-Zee type diagrams [23,24] give a significant contribution to a µ if a mass of A is O(10-100) GeV and if there is large Abb or Aτ + τ − couplings. In Ref. [19], the implication on collider signatures was discussed in the Type-X 2HDM, namely, the h → AA → 4τ process can be important in the favored parameter region by a µ . After the discovery of the Higgs boson at the LHC [25,26], the muon g − 2 has been reanalyzed by taking into account the Higgs boson search data in addition to the previous experimental constraints [20][21][22].
Furthermore, the recent observation of Br(B s → µ + µ − ) at the LHC [27] gives a new constraint on the parameter space of 2HDMs [21].
The difference between the Type-II and Type-X 2HDMs is the quark couplings to the extra Higgs bosons. In the Type-II 2HDM, both the lepton and down-type quark couplings are enhanced simultaneously, and thus the model is severely constrained by flavor physics and direct searches of the extra Higgs bosons. On the other hand, in the Type-X 2HDM, the quark couplings to the extra Higgs bosons are suppressed when the lepton couplings are enhanced. Thus, the constraints are weaker than those in the Type-II 2HDM. In fact, it was clarified in Refs. [20,21] that only the Type-X 2HDM can solve the muon g − 2 anomaly with satisfying the current experimental data.
Another important constraint comes from the lepton flavor physics. In the Type-X 2HDM, the constraint from the leptonic τ decay [11,[28][29][30] gives a severe constraint on the parameter space favored to explain the g − 2 anomaly because of the lepton coupling enhancements. However, this important constraint has not been included in the previous analyses. Therefore, in this paper, we calculate the leptonic τ decay and the Zτ τ vertex at the one-loop level in the Type-X 2HDM in order to compare the precise experimental measurements. We then investigate the favored parameter region by a µ under these constraints in addition to those already known. Furthermore, we evaluate the running of the scalar quartic couplings by renormalization group equations (RGEs), and require

The 2HDMs
In this section, we define the Lagrangian of the 2HDM, in which the Higgs sector is composed of two SU (2) L doublet scalar fields H 1 and H 2 . To avoid the tree level FCNC, we impose a Z 2 symmetry in the Higgs sector which can be softly-broken in general. Under the Z 2 parity, four types of Yukawa interactions are defined depending on the assignment of Z 2 charge as listed in Table 1.
The most general Higgs potential with the softly-broken Z 2 parity is given as Throughout the paper, we consider the CP-conserving case of the Higgs sector for simplicity, so that the imaginary parts of m 2 12 and λ 5 are assumed to be zero. The Higgs fields are parametrized as where v 1 and v 2 are the VEVs of the Higgs doublets which are related to the Fermi The ratio of the two VEVs is parametrized The mass eigenstates of the scalar bosons are expressed by introducing the mixing angles α and β as where G 0 and G ± are the Nambu-Goldstone bosons which are absorbed by the Z and W bosons as the longitudinal component, respectively.
The squared masses for the physical Higgs bosons are given by where M 2 = m 2 12 /(sin β cos β) describes the breaking scale of the Z 2 symmetry, and M 2 ij are given by where λ 345 = λ 3 + λ 4 + λ 5 . The mixing angle α is also expressed in terms of M 2 ij as All the quartic coupling constants in the Higgs potential can be rewritten in terms of the physical parameters as where s β−α = sin(β − α) and c β−α = cos(β − α).
A size of some combinations of λ's in the Higgs potential is constrained by taking into account perturbative unitarity [31][32][33][34] and vacuum stability [35,36]. Through Eq. (12), such a constraint can be translated into a bound on the physical parameters; e.g., the masses of the scalar bosons. First, the condition for vacuum stability; i.e., the requirement for bounded from below in any direction of the Higgs potential with large scalar fields, is given by [35,36] Second, the perturbative unitarity bound is obtained by requiring that all the eigenvalues of the s-wave amplitude matrix a 0 i,± for the elastic scatterings of two body boson states are satisfied as All the independent eigenvalues a 0 i,± were derived in Refs. [32][33][34] as The Yukawa interaction terms are given by where , and V f f is the Cabibbo-Kobayashi-Maskawa matrix element. The ξ h f and ξ H f factors are defined by The ξ f factors in Eqs. (22) and (23) are given in Table 1. Figure 1: One-loop (left) and two-loop Barr-Zee (right) diagrams which give corrections to the muon g − 2.
From the kinetic terms of the scalar fields, the ratios of the coupling constant among the CP-even scalars and gauge bosons are extracted as As it is seen in Eqs. (22), (23) and (24), in the limit of sin(β − a) → 1, both hff and hV V couplings become the same as those in the SM, so that we can call this limit as the SM-like limit.

Constraints on the Type-X 2HDM
In the 2HDMs, the one-loop diagrams and the two-loop Barr-Zee type diagrams shown in Fig. 1 give dominant contributions to the muon g − 2. It has been known that the Barr-Zee type diagrams give a sizable positive contribution to a µ in the case of a large A + − coupling and a small m A as pointed it out in Refs. [13,14]. In the Type-X 2HDMs, a large A + − can be realized by taking tan β 1 since ξ = − tan β as shown in Table 1.
Typically, when tan β 40 and m A = O(10-100) GeV, the muon g − 2 anomaly can be explained in the Type-X 2HDM [20]. In this section, we focus on the Type-X 2HDM with the large tan β and small m A scenario to explain the g − 2 anomaly, and we discuss important experimental constraints in this situation.

Direct searches for the extra Higgs bosons
There has been no signal of the extra Higgs bosons at any collider experiments. This gives lower limits on the masses of the extra Higgs bosons depending on the magnitude of couplings with SM particles. We first summarize the current bounds from the LEP II experiment, and we also review those from the LHC Run-I.  Similar to the neutral Higgs boson productions, the cross section of the H ± production such as gb → H ± t is also suppressed by cot 2 β in the Type-X 2HDM. If m H ± +m b < m t , the has been driven to be between 0.23% and 1.3% at 95% C.L. for m H ± in the range of 80 GeV to 160 GeV [40].
This gives the bounds, for example, tan β 6 and 15 for m H ± = 100 and 150 GeV at 95% C.L. in the Type-X 2HDM using 0.23% of the product of the branching fractions.
Apart from the production processes via Yukawa couplings, one must take care of the h → AA decay in the case of m A < m h /2. In the Type-X 2HDM, this typically gives the four τ final state, because the A → τ τ decay can be the main decay mode as explained in Sec. 3.1.1. In Ref. [41], the upper bound on Br(h → AA → 4τ ) is given to be about 0.2 for m A > 30 GeV and 0.2-0.5 for 15 < m A < 30 GeV. In the 2HDMs, the branching fraction is determined by the dimensionless hAA coupling λ hAA defined as the coefficient of the hAA vertex in the Lagrangian; i.e., L = vλ hAA hAA + · · · which is given by The partial decay width of h → AA is then expressed by where Γ SM = 4.41 MeV is the total decay width of the SM Higgs boson for m h = 125 GeV [42]. Therefore, to satisfy Br(h → AA)< 0.2, λ hAA < ∼ 6.7 × 10 −3 is required.
We can simply take λ hAA = 0 by setting an appropriate value of β − α from Eq. (25) as In the case of tan β From the above expressions, we find that the SM-like behavior of h is realized by taking tan β 1, because of sin(β − α) 1.

Electroweak precision observables
The extra Higgs bosons can modify the electroweak precision observables from the SM prediction via the loop effects. Such an effect can be used as an indirect search for the extra Higgs bosons and also used to constrain parameter space in the 2HDM. In this subsection, we discuss the constraints from the oblique parameters and the Z boson decay.

Oblique parameters
The electroweak oblique S, T and U parameters are introduced by Peskin and Takeuchi [43] which parametrize new physics effects on the gauge boson two point functions. In the SM-like limit sin(β − α) → 1, these parameters are given to be the same formulae as those given in the inert doublet model [44]. For the case of m A m Z m H ± m H , the contribution from the additional scalar bosons is given by ∆T We also find that ∆U is the same order as ∆S in our setup for large ∆T regime. If we take m H = m H ± and sin(β − α) = 1, the Higgs potential respects the custodial SU (2) V symmetry [45], which makes ∆T = 0. The S and T parameters driven by the Gfitter group [46] are with the reference values of m h = 125 GeV and m t = 173 GeV. The prediction of ∆S parameter is inside the 1σ error of the measured value, and the T parameter constrains on the mass splitting |m H − m H ± | = O(10) GeV. Hence we take m H = m H ± to avoid the constraint on the oblique parameters in the following analysis.

Z boson decay
The property of the Z boson such as the mass, the total width, and the decay branching ratios were precisely measured at the LEP experiment. If new physics particles modify such a precisely measured quantity, their masses and/or couplings are severely constrained.
In our scenario, the Zτ + τ − vertex can be significantly deviated from the SM prediction by loop effects of the extra Higgs bosons, because they strongly interact with charged leptons in the large tan β case. In order to discuss how the modified vertex affects the observables, we define the effective Zff vertex as where g Z = g/ cos θ W and θ W being the weak mixing angle. Although there are several definitions for sin 2 θ W , we here use the on-shell definition [47] of it which is determined by using W and Z boson masses, i.e., The effective vector couplinĝ v f and axial vector couplingâ f can be separately written by the contributions from the tree level and from the one-loop level aŝ where the tree level contributions are expressed as After imposing the on-shell renormalization conditions, the counter term contribution is expressed by [48] δv where Π 1PI f f,X are the 1PI diagram contributions to the fermion two point functions defined as In the SM-like limit sin(β − α) = 1, the deviation in v loop τ and a loop τ purely comes from the extra Higgs boson loop diagrams. In this case for f = τ , we obtain where the loop functions are given as In the above expressions, we neglect the mass of the tau lepton in the loop functions. We note that the Let us apply the modified Zτ + τ − vertex to the leptonic partial decay width of the Z boson Z → : We define the ratio of the partial width of Z → τ τ to that of Z → ee as The deviation in the ratio from the SM predictions are then given by The SM prediction is given by The measured values of the leptonic decay width and R τ /e are given by [52] Γ(Z → ) exp = 83.984 ± 0.086 MeV, R exp τ /e = 1.0019 ± 0.0032.
We find that tan β > 50 (70) is excluded for m A = 10 (50) GeV when m H = m H ± = 300 GeV. The bound becomes weaker for the larger m H ± . We will combine the constraint from the Z → decay with the muon g − 2 result in Sec. 4.

Flavor experiments
Effects of the extra Higgs bosons can appear in various observables measured at flavor experiments. Therefore, similar to the electroweak precision measurements, flavor measurements can be used to constrain the parameter space in the 2HDM. In this subsection, we discuss B s → µ + µ − and the leptonic decay of τ . For the calculation of B s → µ + µ − in the 2HDM, we use the formulae given in Ref. [50].
We show the constraint from B s → µ + µ − on the parameter space of 2HDM in Sec. 4.

Leptonic τ decay at the tree level
In the SM, the leptonic τ decay is caused by the W boson exchange diagram at tree level.
In the 2HDM, the H ± mediated diagram also contributes to the leptonic τ decay. The effect of H ± contribution on the partial decay width of τ was calculated in Refs. [11,29], and that on the Michel parameters, which is defined just below, in Ref. [30].
The differential decay rate of τ → µν µ ν τ is given in terms of the Michel parameters (ρ, η, δ and ξ) andĜ µτ defined in Eqs. (52) and (53) as [51] where ω ≡ (m 2 τ + m 2 µ )/2m τ , x ≡ E µ /ω and x 0 ≡ m µ /ω with E µ being the muon energy. P τ is the polarization of the tau, and θ is the angle between the polarization and the momentum direction of the muon. The functions F (x) and A(x) are defined as By using z ≡ m µ m τ tan 2 β/m 2 H + , we find 1Ĝ We see that ρ and δ are equal to the SM values at the tree level. The observed Michel parameters of the τ decay are η = 0.013 ± 0.020 and ξ = 0.985 ± 0.030 [52]. The ratio of the decay rate in the 2HDM to that in the SM prediction is given as [11,29] where the phase functions f (x) and g(x) are given by f ( G eµ and G eτ . Since m e , m µ m τ , the corresponding terms to the rightest term in Eq. (54) for G eµ and G eτ are 1, and thus G eµ = G eτ = G F in 2HDM. There are constraints on the lepton universality given by HFAG group [53] and their correlation coefficient is 0.48. Since G eµ = G eτ = G F in the present scenario, by combining the above two values, we find and thus we find We use this bound and Eq. (54) to make constraint on 2HDM.
In Fig. 2, we show the z dependence of the ratio of the decay rate given in Eq. (54) (upper two panels) and the Michel parameters η (lower left) and ξ (lower right). First, from the upper panels we can see that the allowed ranges of z are found to be z 0.003 and 0.50 z 0.57. Second, from the lower left panel, z 0.05 is excluded by the measurement of η. The constraints from the ξ parameter is weaker than that from η.
Therefore, by combining the first and the second statements, the allowed region of z is restricted to be z 0.003 By using z 1.88 × 10 −3 × (tan β/30) 2 × (300 GeV/m H ± ) 2 , we find that tan β 70 is excluded for m H ± = 300 GeV.

Lepton universality at the one-loop level
As we discussed in Sec. 3.3.2, the typical size of the H ± contribution to the ratio of the tau decay is O(10 −2 ) at the tree level as it is seen in Fig. 2. However, the SM prediction is given at almost the lower edge of the experimental bound (see Eq. (57)), so that the negative contribution to G µτ /G F of order 10 −4 is constrained. Thus, we focus on the quantum corrections to the process via W exchange diagram. Figure 3: The leading one-loop diagrams for the leptonic tau decay process.
The dominant contribution arises from the diagrams with picking up two tau Yukawa couplings which are proportional to (m 2 τ /v 2 ) tan 2 β. We show the diagrams which give the dominant contributions to the process at the one-loop level in Fig. 3. Other diagrams, such as box diagrams, are smaller than these contributions and we ignore them in this analysis. The quantum correction is flavor dependent, and there is no flavor dependent counter terms in this model, so the correction is finite. We find the contributions from where Finally, we find where ω and x 0 are defined in Sec

Triviality bound
In order to avoid the constraints from the various observables, we need to take large mass differences between A and H ± , and A and H. As a result, the Higgs quartic couplings are as large as O(1). Such a large coupling can be grown up in a certain energy scale, and it becomes too strong to rely on the perturbative calculation. We thus take into account the triviality bound in which we require that all the Higgs quartic couplings do not exceed a certain value until a given energy scale.
We calculate the β-functions up to the two loop level for the RGE by using SARAH [54], and run the couplings to higher energies. We treat the coupling values at the tree level The result for Λ = 10 TeV with requiring λ max < 4π is shown in the left panel in Fig. 5. We find that m H ± 370 GeV for m A 20 GeV is required for Λ 10 TeV. This constraint on m H ± is stronger than the one from the perturbative unitarity bound using Eqs. (14)- (20), i.e., m H ± 700 GeV. We check that this result is consistent with that given in [55]. If we require λ max < √ 4π instead of λ max < 4π, then the bound becomes stronger. In such a case, we find m H ± 260 GeV is required. Here we take tan β = 30 at µ = m t as the input value, but the result is insensitive for tan β. We also plot the case for Λ = 100 TeV in the right panel in Fig. 5. The bound is stronger than Λ = 10 TeV case. We find m H ± 310 GeV (240 GeV) for λ max = 4π ( √ 4π). Our parameter choice here is the same as in Sec. 4.

Muon g − 2
We show the numerical results for the muon g − 2 under all the constraints discussed in the previous section. We calculate the muon g − 2 by using 2HDMC 1.6.4 [56] which contains the one-loop diagrams [12] and the two-loop Barr-Zee diagrams [13] as shown in Fig. 1 gives the constructive contribution, and it makes the discrepancy small. Therefore, A is required to be lighter than H and H ± in order to solve the muon g − 2 anomaly.
The Higgs sector in the 2HDM has eight parameters. Two of them are fixed to reproduce the SM parameters, i.e., G F = 2 −1/2 v −2 = 1.166379 × 10 −5 GeV −2 and m h = 125 GeV. To suppress Br(h → AA), we set λ hAA = 0. In addition, we take m H = m H ± to avoid a large contribution to ∆T . Furthermore, we fix λ 1 = 0.1. This λ 1 value is realized by taking M 2 m 2 H in the large tan β case, and the fixing the value of λ 1 is not significant to the result in the following. Therefore, we have three remaining parameters which can be expressed as tan β, m A and m H ± . We note that in this parametrization, sin(β − α) is given as the output parameter, which is determined via Eq. (25). As it is seen in Eq. (28), 1 − sin(β − α) is suppressed by 1/ tan 2 β in the large tan β case, so that h behaves as the SM-like Higgs boson.
In Fig. 6 Consequently, the parameter region which can explain the g − 2 anomaly at the 1σ level is excluded by the measurement of the tau decay at the 95% C.L., and at best we can explain the anomaly at the 2σ level. The typical parameters to explain muon g − 2 anomaly at the 2σ level is 10 < ∼ m A < ∼ 30 GeV, 200 < ∼ m H ± < ∼ 350 GeV and 30 < ∼ tan β 50.
In this parameter space, however, the region with m H ± 270 GeV has tension with the signal strength for the h → τ τ mode measured at the LHC as we will see in the next section.

Impact on the Higgs phenomenology at collider experiments
In the previous section, we have seen that the relatively light extra Higgs bosons and large tan β are favored to explain the muon g − 2 anomaly. Such a light particle can be directly discovered at the LHC Run-II and the International Linear Collider (ILC). Furthermore, the precise measurement of the property of the SM-like Higgs boson h will give an indirect probe of this scenario. In this section, we first discuss the decay and production of the extra Higgs bosons at the LHC, and then we investigate how the property of the SM-like Higgs boson is modified in the favored parameter set indicated by the muon g − 2 in the Type-X 2HDM.
Throughout this section, we consider the case of In addition, the following values of the SM parameters are used [52,57]: Other parameters are taken from Ref. [52]. The mass observed Higgs boson is taken to be 125 GeV.

Phenomenology of the extra Higgs bosons
First, we discuss the branching fraction of the extra Higgs bosons in the parameter set given in Eq. (62). For the CP-odd Higgs boson A, only the A → τ τ and A → bb modes are allowed at the tree level. Since the decay rate of the former (latter) channel is enhanced (suppressed) by tan 2 β (cot 2 β) in the Type-X 2HDM, the branching fraction of A → τ τ becomes almost 100% in our scenario. Similar enhancement happens in the decay rates of H → τ τ and H ± → τ ± ν. At the same time, the other modes H → AZ and H ± → AW ± are also important because of the large mass difference between H/H ± and A. Therefore, the following decay modes should be taken into account: The formulae for the decay rates are given in Appendix A. We here note that the H → AA and H → hh decays also open whose decay rates are determined by the trilinear HAA and Hhh couplings, respectively. However, these couplings are proportional to cos(β −α), and its magnitude is suppressed by cot β in the large tan β case. Thus, these decay modes are not important in our scenario.
In Fig. 7 Next, we discuss the production process of the extra Higgs bosons at the LHC. As we mentioned in Sec. 3.1.2, the production processes via the Yukawa interaction cannot be used in the large tan β case in the Type-X 2HDM. Therefore, the following electroweak processes give the dominant production mode: The analytic expressions for the parton level cross section are given in Appendix B. We find that the cross sections are determined by the masses of the extra Higgs bosons and sin(β − α), and they do not depend on the type of Yukawa interactions. By using CalcHEP [58] with CTEQ6L [59] parton distribution functions, the cross sections are calculated in Table 2.
In this calculation, we neglect the small deviation in sin(β − α) from unity. Because of the small m A , the cross sections of pp → H ± A and pp → HA are relatively large as compared to those of pp → H + H − and pp → H ± H. We note that the cross section for H + H/H + A is about twice larger than that for H − H/H − A, because the parton luminosity of ud in the initial proton is larger than that ofūd.
Combining the discussions of the decay and the production of the extra Higgs bosons, we can consider the following processes: The H + H − production may not be so useful for the feasibility study of the extra Higgs bosons as compared to the above processes because of the small cross section as seen in Table 2. The cross sections of 4τ (σ 4τ ), 3τ (σ 3τ ), 4τ plus W (σ 4τ W ) and 4τ plus Z (σ 2τ Z ) can be estimated as follows: where we used Br(A → τ τ ) 100%. The cross sections of the above processes are also shown in Table 2 in the case of tan β = 35.
The signal and background for the four and three tau final states (without a gauge boson) were studied in Ref. [60] in the Type-X 2HDM. It was clarified that the main background for these processes can be significantly reduced by requiring the high multiplicity of charged leptons and tau-jets with appropriate kinematical cuts in the final state.
In this paper, we only show the signal cross sections of the above mentioned processes as given in Table 2. Although the detailed background simulation must be necessary to clarify the feasibility to detect the signal events, such a study is beyond the scope of this paper.

Phenomenology of the SM-like Higgs boson
Another important impact on the Higgs phenomenology in our scenario is found in the property of the SM-like Higgs boson h. Because the properties of h; e.g., the width, the branching fractions, and the couplings will be precisely measured at future collider experiments such as the LHC Run-II, the high luminosity LHC, and the ILC [61], it must   be quite important to study the deviation in the property of h from the SM prediction.
In particular, studying the pattern of the deviation in the various h couplings can be a powerful tool to determine the structure of the Higgs sector 3 .
As we discussed in Sec. 2, the value of sin(β −α) describes "SM-like ness" of h, namely, all the h couplings to the SM particles become the same as those in the SM prediction in the limit of sin(β − α) → 1. In other words, once sin(β − α) = 1 is given, both the hV V and hff couplings deviate from those of the SM values. In our scenario, the value of sin(β − α) is determined from Eq. (62). Thus, a small but non-zero deviation from the SM-like limit is given.
In order to describe the deviation in the h couplings, we introduce the so-called scaling factors defined as κ X = g hXX /g SM hXX and its deviation from unity; i.e, ∆κ X = κ X −1. From Eqs. (23) and (24) and the approximate formulae given in Eqs. (28) and (29), we obtain In the upper panels of Fig. 8   find that its magnitude is maximally about 1.6 times larger than the SM prediction, and its sign is opposite to the SM one [21]. From the measurement of the signal strength of the h → τ τ channel, i.e., µ τ τ at the LHC, the magnitude of κ is constrained. The definition of the signal strength is given as where σ h and Br(h → XY ) are respectively the production cross section of the SMlike Higgs boson h and the decay branching fraction of the h → XY mode. In our parameter set, σ h is almost the same as that in the SM, because of the small ∆κ q and ∆κ V , so that the signal strength is simply given as the ratio of the branching fraction as The ATLAS and CMS collaborations report the signal strength as µ τ τ = 1.43 +0.43 −0.37 [64] and µ τ τ = 0.91 ± 0.28 [65], respectively. By taking a naive combination of them 4 , we obtain µ τ τ = 1.08 ± 0.23.
Thus, the region with |κ | > 1.27 is excluded at 2σ level, which corresponds to the constraint of m H ± 270 GeV as seen in Fig. 8.
The deviation in the h couplings makes a difference in the branching fraction of h from the SM prediction. In Fig. 9, we show the branching fraction of h as a function of m H ± .
Because only the magnitude of the h coupling can be larger than the SM prediction, It is important to mention here that there appears the exotic decay mode h → AZ in our parameter set as seen in Fig. 9. Although the coupling constant of the hZA interaction is proportional to cos(β − α) which is suppressed by cot β as seen in Eq. bound Br(h → AZ) 14-28% in the Type-X 2HDM. The typical size of Br(h → AZ) is below the upper bound as explained in the above. In addition to this channel, e and µ are produced from the leptonic decay of τ . Thus, the ZA → τ τ → 4 + E T / channel can also contribute to the four lepton channel even though the invariant mass distribution of the four lepton system is different from that by ZZ * → 4 . This will be a subject of a future work.
Next, we discuss the one-loop induced h → γγ decay mode. Because of the H ± contribution, the decay rate can be significantly modified even if the h couplings are not changed so much from the SM prediction. We note that the deviation in the h coupling can be neglected in the decay rate of the h → γγ mode, because its effect appears in the tau loop contribution, but the tau Yukawa coupling is too small as compared to the top Yukawa coupling. The decay rate of the diphoton mode is given as The ATLAS and CMS collaborations report the signal strength as µ γγ = 1.17±0.27 [66] and and µ γγ = 1.12 ± 0.24 [65], respectively. By taking naive combination of them, we obtain µ γγ = 1.14 ± 0.18.
In Fig. 10, we show the ratio of the branching fraction of the h → γγ mode as a function of m H ± with tan β = 35. We find that the deviation in the branching fraction from the SM prediction is obtained in the range of −30% to −15%. The expected accuracy for the measurement of the decay rate of the diphoton mode is around −10% at the LHC 14 TeV 300 fb −1 and 5% at the ILC [67]. Therefore, our scenario is also probed by detecting the deviation in the h → γγ decay rate in addition to the extra Higgs boson searches discussed in Sec. 5.1 and the measurement of κ . By looking at the horizontal line representing the bound from µ γγ , we see that m H ± 220 GeV is excluded in the case of m A = 20 GeV, which weaker than the constraint of κ .

Conclusion
In this paper, we have explored the possibility to explain the muon g − 2 anomaly in the Type-X 2HDM. We have shown in Fig. 6 that the measurement of the leptonic tau decay gives an important constraint on the parameter space. As a result, the region which can explain the discrepancy in the muon g − 2 at the 1σ level is excluded by the constraint from the tau decay, and that at the 2σ level is allowed. We have found that the parameter space with 10 < ∼ m A < ∼ 30 GeV, 200 < ∼ m H,H ± < ∼ 350 GeV and 30 < ∼ tan β 50 is favored by the explanation of the anomaly at the 2σ level.
After finding the viable parameter region for the muon g − 2, we have discussed the implication of the favored parameter region to the collider phenomenology. In our scenario, the 4τ , 3τ , 4τ + Z and 4τ + W signatures are expected from the electroweak productions of the extra Higgs bosons at the LHC. The cross sections of these signals are shown in Table 2.