$B^0\to K^{\ast 0}\mu^+\mu^-$ Decay in the Aligned Two-Higgs-Doublet Model

In the aligned two-Higgs-doublet model, we perform a complete one-loop computation of the short-distance Wilson coefficients $C_{7,9,10}^{(\prime)}$, which are the most relevant ones for $b\to s\ell^+\ell^-$ transitions. It is found that, when the model parameter $\left|\varsigma_u\right|$ is much smaller than $\left|\varsigma_d\right|$, the charged-scalar contributes mainly to chirality-flipped $C_{9,10}^\prime$, with the corresponding effects being proportional to $\left|\varsigma_d\right|^2$. Numerically, the charged-scalar effects fit into two categories: (A) $C_{7,9,10}^\mathrm{H^\pm}$ are sizable, but $C_{9,10}^{\prime\mathrm{H^\pm}}\simeq0$, corresponding to the (large $\left|\varsigma_u\right|$, small $\left|\varsigma_d\right|$) region; (B) $C_7^\mathrm{H^\pm}$ and $C_{9,10}^{\prime\mathrm{H^\pm}}$ are sizable, but $C_{9,10}^\mathrm{H^\pm}\simeq0$, corresponding to the (small $\left|\varsigma_u\right|$, large $\left|\varsigma_d\right|$) region. Taking into account phenomenological constraints from the inclusive radiative decay $B\to X_s\gamma$, as well as the latest model-independent global analysis of $b\to s\ell^+\ell^-$ data, we obtain the much restricted parameter space of the model. We then study the impact of the allowed model parameters on the angular observables $P_2$ and $P_5'$ of $B^0\to K^{\ast 0}\mu^+\mu^-$ decay, and find that $P_5'$ could be increased significantly to be consistent with the experimental data in case B.


Introduction
The rare B → K * + − decays, being the flavour-changing neutral-current (FCNC) processes, do not arise at tree level and are highly suppressed at higher orders within the Standard Model (SM), due to the Glashow-Iliopoulos-Maiani (GIM) mechanism [1]. However, new TeV-scale particles in many extensions of the SM could affect the decay amplitude at a similar level as the SM does. These decays play, therefore, a crucial role in testing the SM and probing various NP scenarios beyond it [2]. It is particularly interesting to note that, based on these decays, observables with a very limited sensitivity to hadronic uncertainties can be constructed, enhancing therefore the discovery potential for NP [3][4][5][6][7][8][9][10].
Experimentally, several interesting deviations from the SM predictions have been observed in these decays. In 2013, the form-factor-independent angular observable P 5 [8,9] of B 0 → K * 0 µ + µ − decay was measured by the LHCb collaboration [11], showing a 3.7σ disagreement with the SM expectation [12][13][14][15]. Recently, the LHCb collaboration has released new measurements of the angular observables in this decay, based on the dataset of 3 fb −1 of integrated luminosity, and still found a 3.4σ deviation for P 5 [16]. Moreover, being in agreement with the LHCb measurements, a deviation with a significance of 2.1σ was also reported by the Belle collaboration [17]. Besides the P 5 anomaly, there are some other slight deviations beyond the 2σ level, such as the observables P 2 in q 2 ∈ [2, 4.3] GeV 2 and P 4 in q 2 ∈ [14.18, 16] GeV 2 [18][19][20].
These anomalies have triggered lots of theoretical studies both within the SM and in various NP models [8-10, 12-15, 18-44].
As a minimal extension of the SM scalar sector, the two-Higgs-doublet model (2HDM) [45] can easily satisfy the electroweak (EW) precision data and, at the same time, lead to a very rich phenomenology [46]. The scalar spectrum consists of two charged scalars H ± and three neutral ones h, H, and A, one of which is to be identified with the SM-like Higgs boson found at the LHC [47,48]. The direct search for these additional scalar states would be an important task for high-energy colliders in the next few years. It should be noted that, complementary to the direct searches, indirect constraints on the 2HDM could also be obtained from the rare FCNC decays like B → K * + − , since these scalars can affect these processes through the penguin and box diagrams. These studies are also very helpful to gain further insights into the scalar sector of supersymmetry and other models that contain similar scalar contents [49][50][51].
In a generic 2HDM, the non-diagonal couplings of neutral scalars to fermions lead to tree-level FCNC interactions, which can be avoided by imposing on the Lagrangian an ad-hoc discrete Z 2 symmetry. Depending on the Z 2 charge assignments to the scalars and fermions, this results in four types of 2HDMs (types I, II, X, Y) [46,52] under the hypothesis of natural flavour conservation (NFC) [53]. In the aligned two-Higgs-doublet model (A2HDM) [54], on the other hand, the absence of tree-level FCNCs is automatically guaranteed by assuming the alignment in flavour space of the Yukawa matrices for each type of right-handed fermions.
In this paper, we will study the decay B 0 → K * 0 µ + µ − in the A2HDM. Our paper is organized as follows: In section 2, we give a brief overview of the A2HDM, focusing mainly on the scalar and Yukawa sectors. In section 3, a complete one-loop computation of the shortdistance (SD) Wilson coefficients C ( ) 7,9,10 is presented, and the final analytical expressions are given both within the SM and in the A2HDM. The angular observables of B 0 → K * 0 µ + µ − decay are also introduced in this section. In section 4, taking into account phenomenological constraints from the inclusive radiative decay B → X s γ and the latest model-independent global analysis of b → s + − data, we study the impact of the allowed model parameters on the angular observables P 2 and P 5 of B 0 → K * 0 µ + µ − decay. Finally, our conclusions are made in section 5. Some relevant functions for the Wilson coefficients are collected in the appendices.

The aligned two-Higgs doublet model
We consider the minimal version of 2HDM, which is invariant under the SM gauge group and includes, besides the SM matter and gauge fields, two complex scalar SU (2) L doublets,  [75][76][77], where the rotation angle (clockwise) tan β = v 2 /v 1 . In the new basis, only the scalar doublet Φ 1 and the two scalar doublets are now parametrized, respectively, by [54] where G ± and G 0 denote the massless Goldstone fields to be eaten by the W ± and Z 0 gauge bosons, respectively. The remaining five physical degrees of freedom are given by the two charged fields H ± (x) and the three neutral ones ϕ 0 where R is an orthogonal matrix obtained after diagonalizing the mass terms in the scalar potential.
Generally, none of these three neutral scalars can have a definite CP quantum number.
Explicitly, inserting Eq. (2.3) into Eq. (2.4) and imposing the minimization condition, one gets M 2 H ± = µ 2 + 1 2 λ 3 v 2 , and the mass-squared matrix M 2 of S 1,2,3 fields in terms of v and λ i . Using the orthogonal matrix R, one can then obtain the masses of the three neutral scalars, . In the CP-conserving limit, λ 5,6,7 are all real and the neutral scalars are CP eigenstates.
The CP-odd scalar A corresponds to S 3 , with the mass given by M 2 A = M 2 H ± + v 2 λ 4 2 − λ 5 , while the two CP-even scalars h and H are orthogonal combinations of S 1 and S 2 , where the mixing angleα is determined by (2.6) The masses of the two neutral scalars are given, respectively, by .
Here M h M H by convention and the SM limit is recovered whenα = 0.

Yukawa sector
The Yukawa Lagrangian of the 2HDM is most generally given by [46,54] whereφ a (x) ≡ iτ 2 φ * a (x) are the charge-conjugated fields with Y = − 1 2 ,Q L andL L are the lefthanded quark and lepton doublets, and u R , d R and R the corresponding right-handed singlets, in the weak-interaction basis. All fermionic fields are written as 3-dimensional vectors and the couplings Γ a , ∆ a and Π a are therefore 3 × 3 complex matrices in flavour space.
Transforming to the Higgs basis, Eq. (2.8) becomes where ξ f (ς f ) are arbitrary complex parameters and could introduce new sources of CP violation beyond that of the CKM matrix.
The interactions of the charged scalar with the fermion mass-eigenstate fields then read where P L(R) ≡ (1 ∓ γ 5 )/2 is the left (right)-handed chirality projector, and V CKM the CKM matrix [55,56]. Here we did not give the neutral scalar sector [54] in L Y or the FCNC local structures induced beyond tree-level (quantum corrections) [64], because their effects are highly suppressed by the muon mass in the decay B 0 → K * 0 µ + µ − . The usual NFC models [46,52], with discrete Z 2 symmetries, are recovered for particular values of ς f , as shown in Table 1.

Effective weak Hamiltonian
The rare decay B 0 → K * 0 µ + µ − proceeds through the loop diagrams both within the SM and in the A2HDM. When the heavy degrees of freedom, including the top quark, the weak gauge bosons, as well as the charged scalars, have been integrated out, we obtain the low-energy effective weak Hamiltonian governing the decay [6,78]: where G F is the Fermi coupling constant. Here we neglect the doubly Cabibbo-suppressed (proportional to V ub V * us ) contributions to Eq. (3.1), and focus only on the operators [6]: Within the SM, the electromagnetic dipole operator O 7 and the semileptonic operators O 9,10 play the leading role in the decay B 0 → K * 0 µ + µ − . Besides modifying the values of the SD Wilson coefficients C 7,9,10 , the charged-scalar contributions could also make the chirality-flipped operators O 7,9,10 defined above to contribute in a significant manner, especially in some regions of the parameter spaces discussed later.
The SD Wilson coefficients C i (µ) and C i (µ) can be obtained firstly at the matching scale µ W ∼ M W perturbatively, by requiring equality of the one-particle irreducible Green functions calculated in the full and in the effective theory [78]. Using the renormalization group equation, one can then get C i (µ) and C i (µ) at the lower scale µ b ∼ m b . During the calculation, the limit m u,c → 0 and the unitarity of the CKM matrix have been used. For simplicity, we introduce the mass ratios: Details of the computational method could be found, for example, in refs. [70,78].

Wilson coefficients in the SM
In the SM, the one-loop penguin and box diagrams have been calculated both in the Feynman (ξ = 1) and in the unitary (ξ = ∞) gauge [79][80][81][82][83][84][85][86][87], denoted by the subscript 'F' and 'U', respectively. The different contributions to C SM i (µ W ) can be split into the following forms: where the corresponding parts resulting from the W -box, Z-penguin and γ-penguin diagrams are given, respectively, by where θ W is the weak mixing angle, and the Inami-Lim functions [79] are defined as in the Feynman gauge, and in the unitary gauge. Here we introduce the notation L ≡ 1 +log

Wilson coefficients in the A2HDM
In the A2HDM, the charged-scalar exchanges lead to additional contributions to C 7,9,10 and could also make the chirality-flipped operators O 7,9,10 to contribute in a significant manner, through the Z 0 -and γ-penguin diagrams shown in Figure 1. Since we have neglected the light lepton mass, there is no contribution from the SM W -box diagrams with the W ± bosons replaced by the charged scalars H ± .
For each Feynman diagram shown in Figure 1, the contributions are identical in the two gauges. The total Wilson coefficients C 7,9,10 are split into two parts, one is from the SM contributions C SM 7,9,10 , and the other from the charged-scalar ones C H ± 7, 9,10 . For the chiralityflipped operators, C 7,9,10 = C H ± 7,9,10 , because the SM contributions are well suppressed by the factorm s /m b . For convenience, we decompose these new contributions in such a way to render explicit their dependence on the couplings ς u and ς d : where the coefficients of the different combinations of the couplings ς u and ς d are given by Eqs. (B.1)-(B.10). In the particular cases of type II and type Y 2HDMs with large tan β, the only terms enhanced by a factor tan 2 β originate from the |ς d | 2 part contributing only to C H ± 9,10 . The Wilson coefficients C ( )H ± 7,9,10 are found to be invariant under a global U(1) transformation, ς u → e iχ ς u and ς d → e iχ ς d . This invariance is well anticipated since it corresponds to an unphysical phase transformation of the second Higgs doublet, Φ 2 → e iχ Φ 2 , a leftover freedom in the Higgs basis [75,76]. There is an implicit µ W dependence via the s, b, t-quark masses, which depend on the precise definitions and have to be specified when going beyond the leading logarithm (LL). As we evaluate C ( )H ± 7,9,10 only at the leading order (LO) in α s , whether the running massesm q (µ W ) or the pole masses m q are used does not matter too much. As a consequence, we choose the pole masses m q as input in Eqs. (3.17)- (3.19).
Our results for the chirality-flipped Wilson coefficients C H ± 7,9,10 are presented for the first time in the A2HDM. In the particular cases of the Z 2 symmetric 2HDMs, our results agree with the ones calculated in refs. [89][90][91][92]. It is also noted that the next-to-leading order QCD corrections to C H ± 7,9,10 in the supersymmetry and type-II 2HDM have already been calculated in refs. [93][94][95][96][97].

Angular observables in
The angular distribution of the B 0 → K * 0 (→ K + π − )µ + µ − decay is described by the dimuon invariant mass squared q 2 as well as the three angles θ , θ K * and φ, where θ is defined as the angle between the flight direction of the µ + (µ − ) and the opposite direction of the B 0 (B 0 ) in the rest frame of the dimuon system, and θ K * the angle between the flight direction of the K + (K − ) and that of the B 0 (B 0 ) in the K * 0 (K * 0 ) rest frame, while φ is the angle between the plane containing the dimuon pair and the plane containing K + and π − mesons in the B 0 (B 0 ) rest frame. In terms of these four kinematic variables, the full angular decay distribution of the decay is then given by [6,98] +Ī 3 sin 2 θ K * sin 2 θ cos 2φ +Ī 4 sin 2θ K * sin 2θ cos φ +Ī 5 sin 2θ K * sin θ cos φ +Ī s 6 sin 2 θ K * cos θ +Ī 7 sin 2θ K * sin θ sin φ +Ī 8 sin 2θ K * sin 2θ sin φ +Ī 9 sin 2 θ K * sin 2 θ sin 2φ , (3.20) where the angular coefficientsĪ  The self-tagging property of the decay B 0 → K * 0 µ + µ − makes it possible to determine both the CP-averaged and the CP-asymmetric quantities defined, respectively, by [ The previously studied observables, such as the q 2 distributions of the forward-backward asymmetry A F B and the CP asymmetry A CP , can be expressed in terms of these angular observables.
With the structure of the amplitudes at large recoil, it is possible to build clean observables whose sensitivity to the B → K * transition form factors is suppressed by α s or Λ QCD /m b [9].
These include the so-called P i and P i observables defined by [9, 99, 100] (3.23) The numerical impact of charged-scalar contributions to some of these observables will be discussed in the next section.
4 Numerical results and discussions

Choice of the model parameters
For the considered decay B 0 → K * 0 µ + µ − , only three model parameters, the charged-scalar mass M H ± and the two alignment parameters ς u and ς d , are involved. In the following we assume the parameters ς u,d to be real, indicating that the only source of CP violation in the A2HDM is still due to the CKM matrix. Following the previous studies, we give below the preset ranges of these model parameters: • The charged-scalar mass is assumed to lie in the range M H ± ∈ [80, 1000] GeV, where the lower bound comes from the LEP direct search [101], while the upper bound from the unitarity and stability of the scalar potential [102][103][104][105].
• The alignment parameter ς d is only mildly constrained through phenomenological requirements that involve additionally other model parameters. So we let it to be a free parameter.
• In the 2HDMs with discrete Z 2 symmetries, the parameters ς u and ς d are not independent but are related to each other through the ratio of the VEVs tan β = v 2 /v 1 . The upper limit for tan β also comes from the unitarity and stability of the scalar potential [102][103][104][105]; we assume here tan β ≤ 50.  and C H ± 8 should fulfill the constraint [115]:

Constraints on the model parameters
where C H ± 8 = |ς u | 2 C 8, uu + ς d ς * u C 8, ud [89], with the functions C 8, uu and C 8, ud given, respectively, by Eqs. and 500 GeV as benchmarks. The case with ς d < 0 is obtained from Figure 2 with the changes ς u → −ς u and ς d → −ς d . It is observed that the allowed range of ς d becomes quite large when ς u tends to zero; particularly, when ς u = 0, no constraint on ς d is obtained, because in this limit the SM result is recovered. When ς d = 0, on the other hand, a bound on ς u can be set with the allowed range of |ς u | further strengthened for smaller values of the charged-scalar mass.
However, the allowed regions for ς u and ς d are further reduced compared to those obtained in refs. [64][65][66], because the updated SM prediction (cf. Eq. (4.2)) becomes now more compatible with the current experimental data (cf. Eq. (4.1)). It is also found that the preset maximum value |ς u | = 2 is reached when |ς d | varies within a range away from zero, rather than at ς d = 0; for example, taking M H ± = 80 GeV, we find that |ς u | approaches to 2 when 0.6 < |ς d | < 0.8.
This novel observation motivates us to display the ς d -axis in the logarithmic coordinate, making clear the correlation between ς u and ς d in the range |ς d | < 1. The inversely-proportional and parabolic boundary curves in the first quadrant indicate that the NP contribution to C H ± 7 (cf. Eq. (3.14)) is dominated by the ς d ς * u and |ς u | 2 terms, respectively. As the large same-sign solutions for ς u and ς d obtained in refs. [64,65], corresponding to the case when the NP influence is about twice the size of the SM contribution but with an opposite sign, are already excluded by the isospin asymmetry of B → K * γ decays [66,116], they are not shown in Figure 2.  [18][19][20]29]. We use two of these global fit results to further constrain the A2HDM parameters. One is obtained from the combined fit to the b → s (µ + µ − , γ) mesonic decays (at µ b = 4.8 GeV) [19]: given at the 3σ level. This fit includes the branching ratios and optimized angular observables of B → K * µ + µ − and B s → φµ + µ − , the branching ratios of B → Kµ + µ − , the branching ratios of B → X s µ + µ − (restricted only to the range 1 GeV 2 ≤ q 2 ≤ 6 GeV 2 ) and B → X s γ, the branching ratio of B s → µ + µ − , as well as the isospin asymmetry and the time-dependent CP asymmetry of B → K * γ. Furthermore, both the large-and low-recoil data is included for the exclusive b → sµ + µ − decays, resulting in nearly a hundred observables in total in at the 1σ level. It is interesting to note that the latter prefers a shift to C 9 that is opposite in sign compared to the former [29]. Since the Wilson coefficients C H ± 9,10 (µ W ) and C H ± 9,10 (µ W ) are calculated only at the LO, they should be evolved to the lower scale µ b at the LL approximation, which means that they are actually not running [117]. Thus, we can apply directly the bounds given by Eqs. (4.4) and (4.5) to C H ± 9,10 and C H ± 9,10 . To be more conservative, we require each of these coefficients to lie within the smaller lower and bigger upper bounds of these two global fits.
Using these bounds as well as the constraint from Eq. (4.3), we find that the allowed parameter space in the ς u − ς d plane are significantly reduced, especially for the model parameter ς u , as shown in Figure 3. This means that C H ± 9,10 play a major role in the small |ς d | region (|ς d | < 1) and C H ± 9,10 can be quite sizable when ς u approaches to zero. It is also interesting to note that, under the constraint from Eq. on C H ± 9,10 and C H ± 9,10 from Eqs. (4.4) and (4.5), we could obtain a bound on ς d even when ς u equals to zero. Such a bound arises entirely from the information on C H ± 9,10 due to the |ς d | 2 terms in these two Wilson coefficients (cf. Eqs. (3.18) and (3.19)). For illustration, the allowed regions in the ς d − M H ± plane when ς u = 0 and in the ς u − M H ± plane when ς d = 0 are shown in Figure 4. Numerically, we obtain |ς u | ≤ 0.506, 0.763 and 0.990, and |ς d | ≤ 212, 476 and 622, corresponding to M H ± = 80, 300 and 500 GeV, respectively. This means that the more accurate C NP 9,10 can be better used to restrict the parameter ς d .

P 2 and P 5 in the A2HDM
In this subsection, with the constrained parameter space for ς u and ς d , we investigate the impact of A2HDM on the angular observables P 2 and P 5 in the decay B 0 → K * 0 µ + µ − . As there involve only three model parameters ς u , ς d and M H ± in Eqs. and C H ± 10 are obviously linearly correlated with each other and the slope depends only on the charged-scalar mass M H ± (Figure 5(e)), with the blue, red, and green lines obtained with M H ± = 80, 300, and 500 GeV, respectively. In addition, and C H ± 10 are found to be approximately linearly correlated with each other (Figure 5(f)), and C H ± 9,10 are sizable, but C H ± 9,10 0. (4.7) They are associated to the (large |ς u |, small |ς d |) and (small |ς u |, large |ς d |) regions, respectively.
In Figure 6, we show our predictions for the two angular observables P 2 and P 5 at large recoil both within the SM and in the A2HDM, with the Wilson coefficients obtained in the above two cases, together with the experimental data from the LHCb [11,16], Belle [17] and BaBar [118] collaborations. Here we follow closely the method used in refs. [6,13,18]: Firstly, we take as input the combined LCSR-lattice fit results for the B → K * transition form factors provided in ref. [13], which allow us to retain all the correlated uncertainties among these form factors.
Secondly, we have included the hadronic uncertainties due to non-factorizable power corrections associated with the non-perturbative charm loops [13,30], the latest discussions of which could be found in refs. [119,120]. Finally, these two angular observables are computed within the SM, with their respective uncertainties obtained by adding in quadrature the individual uncertainty LHCb2013 Belle2016 Figure 6: The q 2 dependence of the angular observables P 2 and P 5 , both within the SM (central value by a red curve and its uncertainty by a yellow band ) and in the A2HDM (the green and blue bands correspond to the case A and case B, respectively). The experimental data from the LHCb [11,16], Belle [17] and BaBar [118] collaborations are represented by the corresponding error bars in different q 2 bins. due to the B → K * form factors, the non-factorizable charm-loop contributions, and the parametric input (mainly fromm b (m b ) = 4.18 +0.04 −0.03 GeV and m c = 1.4 ± 0.2 GeV). For the NP contributions, however, we consider only the uncertainties of the model parameters and perform a random flat scan within their allowed regions. One can see clearly that there is only a small impact on P 2 and P 5 in case A, where the chirality-flipped operators O 9,10 are absent, while in case B P 5 could be increased significantly to be consistent with the experimental data and reduce P 2 when the dimuon invariant mass squared q 2 is higher than the zero-crossing point q 2 0 . Numerical results for the zero-crossing points of P 2 (nonzero one) and P 5 are given in Table 2, both within the SM and in the A2HDM. It is observed that the impact on q 2 0 in case B is more pronounced than in case A.

2HDMs with Z 2 symmetries
In the generic 2HDMs with discrete Z 2 symmetries, the three alignment parameters ς f will be reduced to a single parameter tan β = v 2 /v 1 ≥ 0, as indicated in Table 1. There are, therefore,  only two model parameters tan β and M H ± in the Wilson coefficients C ( )H ± 7,9,10 . We show in Figure 7 the allowed regions in the tan β − M H ± plane corresponding to the four different types of 2HDMs with Z 2 symmetries. As C ( )H ± 7,9,10 do not depend on the parameter ς , the type I (II) and type X (Y) models are indistinguishable from each other. However, one can clearly distinguish types I and X from types II and Y models. As shown in Figure 7, the bound M H ± > 432 GeV is obtained for types II and Y 2HDMs, while there is no further bound found for M H ± in types I and X 2HDMs with sizable tan β. Figure 8 the q 2 dependence of P 2 and P 5 in the four different types of 2HDMs with Z 2 symmetries. One can see that, compared to the SM predictions, both P 2 and P 5 are reduced in the types I and X (the green band), but increased in the types II and Y (the blue band) 2HDMs, only by a small amount. This is because the charged-scalar effect on the left-and right-handed semileptonic operators is controlled by the same parameter tan β and, under the constraint shown in Figures 7, sizable C H ± 9,10 are not allowed in these models. It is, therefore, concluded the 2HDMs with Z 2 symmetries can not explain the so-called P 5 anomaly.

Conclusions
In this paper, we have presented a complete one-loop calculation of the SD Wilson coefficients C ( )H ± 7,9,10 due to the charged-scalar exchanges through the Z 0 -and γ-penguin diagrams within the A2HDM. For C H ± 9,10 , although being suppressed by the factorm bms /M 2 W , they could play an important role in interpreting the observed P 5 anomaly in the decay B 0 → K * 0 µ + µ − , when the model parameter |ς d | is large.
Under the constraints from the branching ratio B(B → X s γ) and the recent global fit results of b → s + − data, we have obtained the allowed parameter spaces in the ς u − ς d plane, corresponding to three representative charged-scalar masses. We found that C H ± 9,10 play a major role in the small |ς d | region (|ς d | < 1), while C H ± 9,10 are most important when the model parameter ς u approaches to zero. When ς u is far away from zero and |ς d | ≥ 1, on the other hand, the impact of C H ± 7 will become more significant. Within the constrained parameter space, numerically, the effects of these NP Wilson coefficients can be divided into the following two cases: (A) C H ± 7,9,10 are sizable, but C H ± 9,10 0, corresponding to the (large |ς u |, small |ς d |) region; (B) C H ± 7 and C H ± 9,10 are sizable, but C H ± 9,10 0, corresponding to the (small |ς u |, large |ς d |) region. We have then discussed their impacts on the angular observables P 2 and P 5 in the decay B 0 → K * 0 µ + µ − .
It is found that there is only a small impact on P 2 and P 5 in case A, while the case B could obviously increase P 5 to be consistent with the experimental data and reduce P 2 when the dimuon invariant mass squared q 2 is higher than the zero-crossing point.
Finally, we have explored the constraints on tan β and M H ± in four types of Z 2 -symmetric 2HDMs. The role of chirality-flipped operators O 9,10 becomes much more important for large values of tan β. Even with the current data, the types I and X and types II and Y could be clearly distinguished from each other. However, the charged-scalar effect on P 2 and P 5 in these models is found to be small and does not help to explain the so-called P 5 anomaly.
Future precise measurements of the angular observables in b → s + − decays, especially with a finer binning of q 2 , would be very helpful to provide a more definite answer concerning the observed anomalies by the LHCb and Belle collaborations, restricting further or even deciphering the NP models.