Phenomenological discriminations of the Yukawa interactions in two-Higgs doublet models with $Z_2$ symmetry

There are four types of two-Higgs doublet models under a discrete $Z_2$ symmetry imposed to avoid tree-level flavour-changing neutral current, i.e. type-I, type-II, type-X and type-Y models. We investigate the possibility to discriminate the four models in the light of the flavour physics data, including $B_s-\bar B_s$ mixing, $B_{s,d} \to \mu^+ \mu^-$, $B\to \tau\nu$ and $\bar B \to X_s \gamma$ decays, the recent LHC Higgs data, the direct search for charged Higgs at LEP, and the constraints from perturbative unitarity and vacuum stability. After deriving the combined constraints on the Yukawa interaction parameters, we have shown that the correlation between the mass eigenstate rate asymmetry $A_{\Delta\Gamma}$ of $B_{s} \to \mu^+ \mu^-$ and the ratio $R={\cal B}(B_{s} \to \mu^+ \mu^-)_{exp}/ {\cal B}(B_{s} \to \mu^+ \mu^-)_{SM}$ could be sensitive probe to discriminate the four models with future precise measurements of the observables in the $B_{s} \to \mu^+ \mu^-$ decay at LHCb.


Introduction
Altough the Standard Model (SM) for particle physics has been successful for over three decades, it still possesses some problems which solutions could imply physics beyond its scope [1][2][3]. Recently, the ATLAS [4,5] and CMS [6,7] experiments at LHC have discovered a new neutral boson with properties consistent with those of the SM Higgs boson [8][9][10][11][12][13]. With the experimental progress at LHC, it is of great interest to confirm whether this boson is the only one fundamental scalar just as the SM, or belongs to an extended scalar sector responsible to the electroweak symmetry breaking (EWSB). The simplest scenario entertaining the latter possibility is provided by the two-Higgs doublet models (2HDM).
Besides the SM Higgs sector, an additional Higgs doublet is introduced in the 2HDMs.
This class of models can provide new source of CP violation beyond the SM [14], which are needed to explain the observed cosmic matter-antimatter asymmetry. The 2HDMs could also be understood as an effective theory for many natural EWSB scenarios, such as the Minimal Supersymmetric Standard Model (MSSM) [15].
However, unlike the SM, the tree-level flavour-changing neutral current (FCNC) transition in the 2HDM is not forbidden by the Glashow-Iliopoulos-Maiani (GIM) mechanism. These FCNCs can cause severe phenomenological difficulties [16][17][18]. Besides some other solutions [19][20][21][22][23], this problem can be addressed by imposing a discrete Z 2 symmetry [24]. According to different Z 2 charge assignments, there are four types of 2HDMs, referred respectively as the type-I, type-II, type-X and type-Y 2HDMs [25]. Therefore, phenomenologically distinguishing between these 2HDMs is an important issue and worthy of detailed investigation [26].
The 2HDMs present very interesting phenomena in both low-energy flavour transitions such as B → X s γ decay and B s −B s mixing, and high-energy collider processes such as various Higgs decay channels. At present, many analyses have been performed [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43], however most of them concentrate on the type-II 2HDM. In this work, we shall extend the previous analyses and study the possibility to discriminate the four different types of 2HDM in favor of experimental measurement. To constrain the model parameters, we shall consider the following constraints: • flavour processes: B s −B s mixing,B → X s γ, B → τ ν and B s,d → µ + µ − decays, • direct search for Higgs bosons at LEP, Tevatron, and LHC, • perturbative unitarity and vacuum stability.
For the B s → µ + µ − decay, there are several interesting observables very sensitive to new physics effects as suggested recently by De Bruyn et al. [44]. In this paper, we use these observables to probe the 2HDMs and find the correlation between the mass eigenstate rate asymmetry A ∆Γ and the ratio R = B(B s → µ + µ − ) exp /B(B s → µ + µ − ) SM , which could be used to discriminate the four models with future precise measurements of the observables in the B s → µ + µ − decay at LHCb.
Our paper is organized as follows: In the next section, we give a brief review on the 2HDM with the Z 2 symmetry. In section 3, the theoretical formalism for the flavour observables are presented. In section 4, we give our detailed numerical results and discuss the possibility of discriminating the four types of 2HDM. Conclusions are given in section 5. The relevant Wilson coefficients due to the contributions of 2HDMs are presented in the appendix A and B.

2HDM under the Z symmetry
In the 2HDM, the two Higgs doublets Φ 1 and Φ 2 can be generally parameterized as For a CP-conserving Higgs potential, the two vacuum expectation values (vevs) v 1 and v 2 are real and positive [15]. They satisfy the relations v 1 = v cos β and v 2 = v sin β with v = 246 GeV.
The physical scalars can be obtained by the rotations where the rotation matrix is given by The mixing angles α and β are determined by the parameters of the Higgs potential. The physical Higgs spectrum consists of five degrees of freedom: two charged scalars H ± , two CPeven neutral scalars h and H, and one CP-odd neutral scalar A.  In the interaction basis, the Yukawa interactions of these Higgs bosons can be written as (2.4) whereΦ i = iσ 2 Φ * i with σ 2 the Pauli matrix, Q L and L L denote the left-handed quark and lepton doublets, u R , d R and e R are the right-handed up-type quark, down-type quark and lepton singlet, respectively. The Yukawa coupling matrices Y f i (f = u, d, ) are 3 × 3 complex matrices in flavour space.
In order to avoid tree-level FCNC, it is natural to introduce a discrete Z 2 symmetry [24].
All the possible nontrivial Z 2 charge assignments are listed in table 1, which define the four well-known types of 2HDM, i.e. type-I, type-II, type-X and type-Y. The Yukawa interaction in the four models are different. In the mass-eigenstate basis, they can be unified in the form

Theoretical formalism for flavour observables
In this section, we shall recapitulate the basic theoretical formulae for the relevant B-meson decay and mixing processes and discuss the contributions of the four types of 2HDMs.

B s −B s mixing
For the B s −B s mixing, the mass difference is defined as where H and L denote the heavy and light mass eigenstates. This quantity arises from W box diagrams in the SM and can receive contributions from Higgs box diagrams in 2HDM, as shown in figure 1. The theoretical prediction can be expressed as [45][46][47] ∆m with the definitions x t ≡ (m t (m t )) 2 /m 2 W and x H ± ≡ m 2 H ± /m 2 W . The long-distance QCD effects are contained in the bag factor B Bs (m b ) and the decay constant f Bs [45]. The short-distance contributions from the SM and 2HDM are encoded in the Inami-Lim function S(x t , x H ± ), with its explicit expression given in appendix A, and the QCD correction factorη Bs .

3.2B → X s γ decay
The effective Hamiltonian forB → X s γ at the scale µ b = O(m b ) is given as follows [48][49][50][51][52][53][54] where Q 1−6 are the four-fermion operators whose explicit expressions are given in ref. [51]. The remaining magnetic-penguin operators, which are characteristic for this decay, are defined as In the SM and the four types of 2HDM, analytic expressions for the Wilson coefficients up to the next to leading order (NLO) are given in refs. [52,53].
The branching ratio ofB → X s γ with an energy cut-off E 0 can be expressed as with the semi-leptonic factor The perturbative quantity P (E 0 ), which is expressed in terms of Wilson coefficients, and the non-perturbative correction N (E 0 ) can be found in ref. [52].

B → τ ν decay
The tauonic decay B → τ ν is described as annihilation processes mediated by W boson in the SM and the charged Higgs boson in 2HDM, as shown in figure 3. Therefore, this process is very sensitive to the charged Higgs boson H ± and provides important constraint on the model parameters.
Within 2HDM, the decay width of this channel reads [25,55,56], where V ub is the CKM matrix element and f B denotes the B-meson decay constant. Figure 3: Tree-level diagrams contributing to B → τ ν τ in the SM and 2HDM.
In the SM, the B q → µ + µ − decays (q = d or s) arise from the W box and Z penguin diagrams at the quark level [57,58], as shown in figure 4. The helicity suppression in these decays may be relaxed by NP contributions, which can significantly enhance their branching ratios. Generally, the low-energy effective Hamiltonian for B q → µ + µ − decay can be written as [59] with s W ≡ sin θ W . The semi-leptonic operators are defined as Among the Wilson coefficients C S,P,A , only C A is non-zero in the SM. Its explicit expressions up to NLO can be found in refs. [60][61][62]. Recently, the NLO EW [63] and NNLO QCD [64] corrections have also been completed [65]. In the 2HDM, C A is not affected, whereas C S,P receive contributions from both charged and neutral Higgs bosons. At present, only the diagrams shown in figure 4 have been calculated in the type-II 2HDM with large tan β [59]. Based on these results, we give the Wilson coefficients C S,P corresponding to these diagrams in all the four types of 2HDM with arbitrary tan β in appendix B. It is noted that, contributions from other diagrams may be important for some specific values of tan β (large or small) and will become crucial with future high-precision measurement of B q → µ + µ − decays.
For B q → µ + µ − decays, one important observable is the CP averaging branching ratio, which reads with the definitions It is noted that the contributions of C S,P terms do not suffer helicity suppression, but are suppressed by the small leptonic Yukawa coupling in the 2HDMs. However, they may be enhanced by large tan β (or cot β) factor [57][58][59][60][61].
Recently, a sizable width difference ∆Γ s between the B s mass eigenstates has been measured at the LHCb [66] where Γ s denotes the inverse of the B s mean lifetime τ Bs . As pointed out in ref. [44], the measured branching ratio of B q → µ + µ − should be the time-integrated one, denoted by B(B q → µ + µ − ). For B s → µ + µ − decay, in order to compare with the experimental measurement, the sizable width difference effect should be taken into account in the theoretical prediction, and one has where A ∆Γ denotes the mass eigenstate rate asymmetry and can be expressed as The observable A ∆Γ is complementary to the branching ratio of B s → µ + µ − , offering independent information on the short-distance structure of this decay. It can be extracted from the time-dependent untagged decay rate [44,67]. In the SM, A ∆Γ = +1. In addition, since the finite works well.
Following the ref. [44], it is convenient to introduce the ratio where ϕ P and ϕ S denote the phase of the quantity P and S, respectively. In the four types of It is also useful to define the following quantity in which some uncertainties of input parameters are canceled out. For example, the f Bs /f B d in the above ratio can be directly determined by Lattice QCD and the corresponding theoretical uncertainty is significantly reduced [68,69].

Numerical analysis and discussions
With the theoretical framework presented in the previous sections, we proceed to present our numerical results and discussion in this section.   Recently, the LHC and Tevatron data collected so far [4][5][6][7]78] confirm the SM Higgs-like nature [8][9][10][11][12][13] of the new boson discovered at the LHC, with a spin/parity consistent with the SM 0 + assignment [79][80][81]. The observation of its γγ decay mode demonstrates that it is a boson with J = 1, while the J P = 0 − and 2 + hypotheses have been already excluded at about 99% CL, by analysing the distribution of its decay products. The masses measured by ATLAS and CMS are in good agreement, giving the average value [82] m h = (125.64 ± 0.35) GeV.
If the light neutral Higgs boson h in 2HDM is identified as the observed resonance at LHC, the decoupling limit sin(β − α) = 1 is needed to keep its Yukawa couplings SM-like [83,84].

Perturbative unitarity and vacuum stability
Besides the experimental constraints mentioned in previous sections, there are theoretical conditions which allow one to restrict the 2HDM parameter space [15,25,85,86]. The vacuum stability [87] arise from the requirement that the Higgs potential must have a minimum. The perturbative unitarity [88] is the condition that all the (tree-level) scalar-scalar scattering amplitudes must respect unitarity. From these conditions, the following bound can be obtained,

Procedure in numerical analysis
As shown in section 2, the relevant 2HDM parameters contain two angles α and β, and four  In the numerical analysis, we impose the experimental constraints in the following way: each point in the parameter space corresponds to a theoretical range, constructed from the prediction for the observable in that point together with the corresponding theoretical uncertainty. If this range overlaps with the 2σ range of the experimental measurement, this point is regarded as allowed. In this procedure, to be conservative, the theoretical uncertainty is taken as twice the one listed in table 4. Since the main theoretical uncertainties arise from hadronic inputs, common to both the SM and 2HDM, the relative theoretical uncertainty is assumed constant over the parameter space.

B s −B s mixing within 2HDM
The mixing parameter ∆m Bs is proportional to the Inami-Lim function S(x t , x H ± ). In the leading order (LO) approximation and taking m H ± = 500 GeV, we have numerically From these results, we make the following observations: • For the four different 2HDMs, the dominant effect is proportional to cot β. They always work constructively with the SM contribution, even when the charged Higgs mass m H ± is not fixed but larger than about 90 GeV.
• Since ∆m Bs is only affected by charged Higgs, the contributions from type-I and -X (type-II and -Y) 2HDMs are the same. The type-I and -X Yukawa couplings of downtype quarks are different from the type-II and -Y ones. Thus, there is additional term proportional to tan β in the latter two 2HDMs, however, suffered from down-type quark mass suppression.
In figure 5, the constraints on the parameter space (tan β, m H ± ) from ∆m Bs are shown. As expected, the allowed parameter space in type-I, -X 2HDMs and type-II, -Y 2HDMs are almost the same, in which the regions with small tan β are excluded. The difference appears in the region with large tan β. However, the allowed charged Higgs mass in this region is below the LEP lower limit.

4.4B → X s γ decay within 2HDM
The branching ratio ofB → X s γ decay is proportional to |C eff 7γ (µ b )| 2 in the LO approximation. In 2HDM, the Wilson coefficient C eff 7γ (µ b ) reads numerically at m H ± = 500 GeV in the LO, From these numerical results, we make the following observations: • In the type-I and -X models, the 2HDM effect is proportional to cot β and destructive with the SM contribution.
• In the type-II and -Y models, the 2HDM contribution works constructively with the SM one. Besides the tan β terms, there are also β-independent terms, which dominate the 2HDM contribution for large tan β.  In figure 6, the constraints on the parameter space (tan β, m H ± ) from B(B → X s γ) are shown.
The regions with small tan β are largely excluded in all the four types. However, there is still one solution in the type-I and -X 2HDMs, where the destructive interference between the SM and 2HDM contributions makes the coefficient C eff 7γ sign-flipped. For the type-II and -Y 2HDMs, the charged Higgs mass is strongly bounded, m H ± ≥ 259 GeV, which mainly arises from the β-independent terms. This lower limit is stronger than the LEP bound.

B → τ ν decay within 2HDM
For B → τ ν decay, the numerical expressions of the branching ratio read, in which the second equality in each line holds for m H ± = 500 GeV. Here, the 2HDM effects arise from tree-level charged Higgs with leptonic couplings, which make the following features: • In all the four types, the 2HDM effects are largely suppressed by the charged Higgs mass. • In the type-II (-I) model, the 2HDM effect is constructive with the SM one and proportional to tan β (cot β). The large (small) tan β can compensate the mass suppression.
• In the type-X and -Y model, the 2HDM contribution is β-independent and proportional to 1/m H ± . Thus, small m H ± is expected to be strongly bounded.
The constraints on the parameter space (tan β, m H ± ) from B(B → τ ν) are shown in figure 7.
As expected, in the type-I (II) 2HDM, excluded regions mainly arise from the parameter space with small (large) tan β. There also exists one solution (narrow band in figure 7), where the sign of the SM contribution is flipped by the 2HDMs. For the type-X and -Y 2HDMs, a βindependent bound on the charged Higgs mass is obtained, m H ± ≥ 5 GeV. However, this lower limit is much weaker than the LEP bound.
Since B(B → τ ν) is independent of tan β in type-X and type-Y, we also present its theoretical prediction as a function of m H ± in figure 8, which may be helpful for understanding these two models with reduced experimental and theoretical uncertainties in the future.
Here, both the charged and the neutral Higgs bosons are involved, which results in the following features: • In all the four types, the 2HDM effects are strongly suppressed by the large mass of CPeven Higgs m H and small leptonic Yukawa coupling, and could be enhanced by the small mass of CP-odd Higgs m A .
• In the type-II (-I and -X) models, the suppressed 2HDM contributions can be compensated by large tan β (cot β).

Combined analysis and discrimination between the four 2HDMs
Combining all the constraints mentioned in the previous sections, we obtain the survived parameter space as shown in figure 10. From this plot, the following observations are made: • For small tan β, the most stringent constraints come from ∆m Bs and B(B → X s γ) in all the four types of 2HDM.
• For large tan β, the flavour observables put almost no constraints in the type-I and -X models. The LEP bound on m H ± is still the most strongest. For type-II and -Y models, the constraints mainly come from B(B → X s γ). The B s,d → µ + µ − decays exclude one additional parameter space of the type-II 2HDM.
• When m H ± become large, the combined constraints from flavour observables are almost the same for all the four 2HDMs.
• The allowed region of the type-II model is contained in the one of the type-Y model, which stay in the survived parameter space of the type-I and -X 2HDMs. Therefore, the 0.6 0. 8   From these plots, we make the following observations: • For the type-II 2HDM, large derivations from the SM predictions for both A ∆Γ and R are allowed, since both the Wilson coefficients C S and C P can be significantly enhanced by large tan β. It is also noted that the observable R always decreases.
• For the type-I and -X 2HDMs, only large regions for R are allowed. The reason is that, the coefficient C S can not be enhanced by small tan β, which has been excluded by the combined constraints discussed in section 4.7.
• For the type-Y 2HDM, as expected, its effect on both A ∆Γ and R are small.
• The observables A ∆Γ and R show the potential to discriminate the four types of 2HDM.
For the type-I, -II and -X models, there always exists allowed region for only one of them in the R − A ∆Γ plane. Interestingly, the allowed region of the type-Y is located in the intersection of the regions of the other 2HDMs. With refined measurement of A ∆Γ and R, one could distinguish one type 2HDM from others, or exclude all the four types.
• At present, due to large uncertainties, the observable R sd can not provide further information to distinguish between the four 2HDMs. It is also noted that R sd alway decreases in the type-II 2HDM.
It is concluded that the observables A ∆Γ , R and R sd in B s,d → µ + µ − decays show high sensitivity to the Yukawa structure of the 2HDMs. Improved experimental measurements and theoretical predictions will make these observables more powerful to distinguish between the four types of 2HDM.

Conclusions
In this paper, we have studied the possibility to discriminate the four types of 2HDM in the light of recent flavour physics data, including the B s −B s mixing, the leptonic B-meson decays B s,d → µ + µ − and B → τ ν, and the inclusive radiative decayB → X s γ, together with the experimental data from the direct search for Higgs bosons at LEP, Tevatron and LHC [4-7, 77, 78] and the constraints from perturbative unitarity [88] and vacuum stability [87]. The outcomes of this combined analysis are summarized as follows: • The flavour observables exhibit different dependence on the Yukawa couplings in the four types of 2HDMs. With the current experimental data, the allowed region of the type-II model is contained in the one of the type-Y model, which stay in the survived parameter space of the type-I and -X 2HDMs.
• The observables A ∆Γ and R in the B s → µ + µ − decay, which arise from the sizable B s width difference, are investigated. The correlation between these two observables is found to be sensitive probe to the Yukawa structure of 2HDM.
With the experimental progresses expected from the LHC and the future SuperKEKB, as well as the theoretical improvements, the constraints shown here are expected to be refined, which are helpful to discriminate the Yukawa structure if extended Higgs sector is discovered in the future.

A The Inami-Lim function S(x t , x H ±)
Within the four types of 2HDM discussed in section 2, the Inami-Lim function appearing in B s −B s mixing is given as [47] S(x t , x H ± ) = S W W (x t ) + 2S W H (x t , x H ± ) + S HH (x t , x H ± ), (A.1) where the basic functions S W W and S W H,HH correspond to the W box and the Higgs box diagrams shown in figure 1, respectively. For convenience, their explicit expressions are given below S W W (x t )

B The Wilson coefficients C S and C P
Within the four types of 2HDM discussed in section 2, the Wilson coefficients C S and C P appearing in the effective Hamiltonian of B s,d → µ + µ − are given as [59] C S = C box S + C peng S + C self S , where the functions C box S,P , C peng S,P and C self S,P correspond to the box, penguin and self-energy diagrams associated with Higgs bosons in figure 4. Based on the results in ref. [59], their explicit expressions are given below,