\(\varvec{B^0\rightarrow K^{*0}\mu ^+\mu ^}\) decay in the aligned twoHiggsdoublet model
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Abstract
In the aligned twoHiggsdoublet model, we perform a complete oneloop computation of the shortdistance Wilson coefficients \(C_{7,9,10}^{(\prime )}\), which are the most relevant ones for \(b\rightarrow s\ell ^+\ell ^\) 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 chiralityflipped \(C_{9,10}^\prime \), with the corresponding effects being proportional to \(\left \varsigma _{d}\right ^2\). Numerically, the chargedscalar effects fit into two categories: (A) \(C_{7,9,10}^\mathrm {H^\pm }\) are sizable, but \(C_{9,10}^{\prime \mathrm {H^\pm }}\simeq 0\), 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 \(C_{9,10}^\mathrm {H^\pm }\simeq 0\), corresponding to the (small \(\left \varsigma _{u}\right \), large \(\left \varsigma _{d}\right \)) region. Taking into account phenomenological constraints from the inclusive radiative decay \(B\rightarrow X_{s}{\gamma }\), as well as the latest modelindependent global analysis of \(b\rightarrow s\ell ^+\ell ^\) 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 \(B^0\rightarrow K^{*0}\mu ^+\mu ^\) decay, and we find that \(P_5'\) could be increased significantly to be consistent with the experimental data in case B.
1 Introduction
The rare \(B\rightarrow K^{*}\ell ^+\ell ^\) decays, being the flavorchanging neutralcurrent (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 TeVscale 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 formfactorindependent angular observable \(P'_5\) [8, 9] of \(B^0\rightarrow K^{*0}\mu ^+\mu ^\) decay was measured by the LHCb collaboration [11], showing a \(3.7\sigma \) 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 \(3~\mathrm {fb}^{1}\) of integrated luminosity, and still found a \(3.4\sigma \) deviation for \(P'_5\) [16]. Moreover, being in agreement with the LHCb measurements, a deviation with a significance of \(2.1\sigma \) was also reported by the Belle collaboration [17]. Besides the \(P'_5\) anomaly, there are some other slight deviations beyond the \(2\sigma \) level, such as the observables \(P_2\) in \(q^2\in [2,4.3]~\mathrm {GeV}^2\) and \(P'_4\) in \(q^2\in [14.18,16]~\mathrm {GeV}^2\) [18, 19, 20]. These anomalies have triggered lots of theoretical studies both within the SM and in various NP models [8, 9, 10, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44].
As a minimal extension of the SM scalar sector, the twoHiggsdoublet 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 \(H^\pm \) and three neutral ones \(h,\,H\), and A, one of which is to be identified with the SMlike Higgs boson found at the LHC [47, 48]. The direct search for these additional scalar states would be an important task for highenergy 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 \(B\rightarrow K^{*}\ell ^+\ell ^\), 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 have similar scalar contents [49, 50, 51].
In a generic 2HDM, the nondiagonal couplings of neutral scalars to fermions lead to treelevel FCNC interactions, which can be avoided by imposing on the Lagrangian an ad hoc discrete \(\mathcal {Z}_2\) symmetry. Depending on the \(\mathcal {Z}_2\) 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 flavor conservation (NFC) [53]. In the aligned twoHiggsdoublet model (A2HDM) [54], on the other hand, the absence of treelevel FCNCs is automatically guaranteed by assuming the alignment in flavor space of the Yukawa matrices for each type of righthanded fermions. Interestingly, the A2HDM can recover as particular cases all known specific implementations of the 2HDMs based on \(\mathcal {Z}_2\) symmetries. The model also features possible new sources of CP violation beyond that of the Cabibbo–Kobayashi–Maskawa (CKM) matrix [55, 56]. These features make the A2HDM very attracting both in highenergy collider physics [57, 58, 59, 60, 61, 62, 63] and in lowenergy flavor physics [64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74].
In this paper, we will study the decay \(B^0\rightarrow K^{*0}\mu ^+\mu ^\) in the A2HDM. Our paper is organized as follows: in Sect. 2, we give a brief overview of the A2HDM, focusing mainly on the scalar and Yukawa sectors. In Sect. 3, a complete oneloop computation of the shortdistance (SD) Wilson coefficients \(C_{7,9,10}^{(\prime )}\) is presented, and the final analytical expressions are given both within the SM and in the A2HDM. The angular observables of \(B^0\rightarrow K^{*0}\mu ^+\mu ^\) decay are also introduced in this section. In Sect. 4, taking into account phenomenological constraints from the inclusive radiative decay \(B\rightarrow X_s\gamma \) and the latest modelindependent global analysis of \(b\rightarrow s\ell ^+\ell ^\) data, we study the impact of the allowed model parameters on the angular observables \(P_2\) and \(P_5'\) of \(B^0\rightarrow K^{*0}\mu ^+\mu ^\) decay. Finally, our conclusions are drawn in Sect. 5. Some relevant functions for the Wilson coefficients are collected in the appendices.
2 The aligned twoHiggsdoublet model
2.1 Scalar sector
2.2 Yukawa sector
The onetoone correspondence between different specific choices of the couplings \(\varsigma _{f}\) and the 2HDMs based on discrete \(\mathcal {Z}_2\) symmetries
Model  \(\varsigma _{d}\)  \(\varsigma _{u}\)  \(\varsigma _{\ell }\) 

Type I  \(\cot {\beta }\)  \(\cot {\beta }\)  \(\cot {\beta }\) 
Type II  \(\tan {\beta }\)  \(\cot {\beta }\)  \(\tan {\beta }\) 
Type X  \(\cot {\beta }\)  \(\cot {\beta }\)  \(\tan {\beta }\) 
Type Y  \(\tan {\beta }\)  \(\cot {\beta }\)  \(\cot {\beta }\) 
Inert  0  0  0 
3 \(\varvec{B^0\rightarrow K^{*0}\mu ^+\mu ^}\) in the A2HDM
3.1 Effective weak Hamiltonian
Within the SM, the electromagnetic dipole operator \(O_7\) and the semileptonic operators \(O_{9,10}\) play the leading role in the decay \(B^0\rightarrow K^{*0}\mu ^+\mu ^\). Besides modifying the values of the SD Wilson coefficients \(C_{7,9,10}\), the chargedscalar contributions could also make the chiralityflipped operators \(O_{7,9,10}^{\prime }\) defined above to contribute in a significant manner, especially in some regions of the parameter spaces discussed later.
3.2 Wilson coefficients in the SM
3.3 Wilson coefficients in the A2HDM
Our results for the chiralityflipped Wilson coefficients \(C_{7,9,10}^{\prime \mathrm {H^\pm }}\) are presented for the first time in the A2HDM. In the particular cases of the \(\mathcal {Z}_2\) symmetric 2HDMs, our results agree with the ones calculated in Refs. [89, 90, 91, 92]. It is also noted that the nexttoleading order QCD corrections to \(C_{7,9,10}^\mathrm {H^\pm }\) in the supersymmetry and typeII 2HDM have already been calculated in Refs. [93, 94, 95, 96, 97].
3.4 Angular observables in \({B^0\rightarrow K^{*0}\mu ^+\mu ^}\) decay
4 Numerical results and discussions
4.1 Choice of the model parameters

The chargedscalar mass is assumed to lie in the range \(M_{H^{\pm }} \in [80,1000]~\mathrm {GeV}\), 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 \(\varsigma _{u}\) is assumed to lie in the range \(\varsigma _{u}\le 2\), to be compatible with the current data of loopinduced processes, such as \(Z\rightarrow b\bar{b}\), \(b\rightarrow s \gamma \), \(B_{s,d}^0\)–\(\bar{B}_{s,d}^0\) mixings, as well as the h(125) decays [62, 63, 65, 66, 67, 68, 69].

The alignment parameter \(\varsigma _{d}\) 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 \(\mathcal {Z}_2\) symmetries, the parameters \(\varsigma _{u}\) and \(\varsigma _{d}\) are not independent but are related to each other through the ratio of the VEVs \(\tan \beta =v_2/v_1\). The upper limit for \(\tan \beta \) also comes from the unitarity and stability of the scalar potential [102, 103, 104, 105]; we assume here \(\tan \beta \le 50\).
4.2 Constraints on the model parameters
For the other input parameters, we take \(M_Z=91.1876~\mathrm {GeV}\), \(M_{W}=80.385~\mathrm {GeV}\), \(m_{t}=(174.2\pm 1.4)~\mathrm {GeV}\), \(m_{b}=(4.78\pm 0.06)~\mathrm {GeV}\), and \(\bar{m}_{s}(2~\mathrm {GeV})=(96^{+8}_{4})~\mathrm {MeV}\) [106]. Since \(C_7^{\prime \mathrm {H}^\pm }=\bar{m}_{s}/\bar{m}_{b} C_7^{\mathrm {H}^\pm }\) and \(\bar{m}_{s} \ll \bar{m}_{b}\), the contribution from \(O_7'\) will be safely neglected.
Under the constraint from Eq. (4.3), we show in Fig. 2 the allowed regions in the \(\varsigma _{u}\)–\(\varsigma _{d}\) plane (\(\varsigma _{d}>0\)), with three representative values of the chargedscalar mass, \(M_{H^\pm }=80\), 300 and \(500~\mathrm {GeV}\) as benchmarks. The case with \(\varsigma _{d}<0\) is obtained from Fig. 2 with the changes \(\varsigma _{u}\rightarrow \varsigma _{u}\) and \(\varsigma _{d}\rightarrow \varsigma _{d}\). It is observed that the allowed range of \(\varsigma _{d}\) becomes quite large when \(\varsigma _{u}\) tends to 0; particularly, when \(\varsigma _{u}=0\), no constraint on \(\varsigma _{d}\) is obtained, because in this limit the SM result is recovered. When \(\varsigma _{d}=0\), on the other hand, a bound on \(\varsigma _{u}\) can be set with the allowed range of \(\varsigma _{u}\) further strengthened for smaller values of the chargedscalar mass. These qualitative observations are consistent with those observed previously in Refs. [64, 65, 66]. However, the allowed regions for \(\varsigma _{u}\) and \(\varsigma _{d}\) 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 \(\left \varsigma _{u}\right =2\) is reached when \(\left \varsigma _{d}\right \) varies within a range away from 0, rather than at \(\varsigma _{d}=0\); for example, taking \(M_{H^\pm }=80~\mathrm {GeV}\), we find that \(\left \varsigma _{u}\right \) approaches 2 when \(0.6<\left \varsigma _{d}\right <0.8\). This novel observation motivates us to display the \(\varsigma _{d}\)axis in the logarithmic coordinate, making clear the correlation between \(\varsigma _{u}\) and \(\varsigma _{d}\) in the range \(\left \varsigma _{d}\right <1\). The inversely proportional and parabolic boundary curves in the first quadrant indicate that the NP contribution to \(C_{7}^\mathrm {H^\pm }\) (cf. Eq. (3.14)) is dominated by the \(\varsigma _{d}\varsigma _{u}^*\) and \(\left \varsigma _{u}\right ^2\) terms, respectively. As the large samesign solutions for \(\varsigma _{u}\) and \(\varsigma _{d}\) 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 \(B\rightarrow K^*\gamma \) decays [66, 116], they are not shown in Fig. 2.
It is also interesting to note that, under the constraint from Eq. (4.3) as well as the bounds on \(C_{9,10}^{\mathrm {H^\pm }}\) and \(C_{9,10}^{\prime \mathrm {H^\pm }}\) from Eqs. (4.4) and (4.5), we could obtain a bound on \(\varsigma _{d}\) even when \(\varsigma _{u}\) equals 0. Such a bound arises entirely from the information on \(C_{9,10}^{\prime \mathrm {H^\pm }}\) due to the \(\left \varsigma _{d}\right ^2\) terms in these two Wilson coefficients (cf. Eqs. (3.18) and (3.19)). For illustration, the allowed regions in the \(\varsigma _{d}\)–\(M_{H^\pm }\) plane when \(\varsigma _{u}=0\) and in the \(\varsigma _{u}\)–\(M_{H^\pm }\) plane when \(\varsigma _{d}=0\) are shown in Fig. 4. Numerically, we obtain \(\left \varsigma _{u}\right \le 0.506\), 0.763 and 0.990, and \(\left \varsigma _{d}\right \le 212\), 476 and 622, corresponding to \(M_{H^\pm }=80\), 300 and \(500~\mathrm {GeV}\), respectively. This means that the more accurate \(C_{9,10}^{\prime \mathrm{NP}}\) can be better used to restrict the parameter \(\varsigma _{d}\).
4.3 \(P_2\) and \(P_5'\) in the A2HDM
In this subsection, with the constrained parameter space for \(\varsigma _{u}\) and \(\varsigma _{d}\), we investigate the impact of A2HDM on the angular observables \(P_2\) and \(P_5'\) in the decay \(B^0\rightarrow K^{*0}\mu ^+\mu ^\). As there are involved only three model parameters \(\varsigma _{u}\), \(\varsigma _{d}\) and \(M_{H^\pm }\) in Eqs. (3.14)–(3.19), the five Wilson coefficients (\(C_7^{\prime \mathrm {H^\pm }}\) is neglected because \(\bar{m}_{s} \ll \bar{m}_{b}\)) are expected to be highly correlated with each other. Using the allowed values of \(\varsigma _{u}\) and \(\varsigma _{d}\) with three benchmark values of chargedscalar mass obtained in the previous subsection, we show in Fig. 5 the correlations among these five Wilson coefficients. One can see that, while \(C_7^\mathrm {H^\pm }\) is hardly correlated with the other four Wilson coefficients (Fig. 5a–d), \(C_9^\mathrm {H^\pm }\) and \(C_{10}^\mathrm {H^\pm }\) are obviously linearly correlated with each other and the slope depends only on the chargedscalar mass \(M_{H^\pm }\) (Fig. 5e), with the blue, red, and green lines obtained with \(M_{H^\pm }=80\), 300, and \(500~\mathrm {GeV}\), respectively. In addition, \(C_9^{\prime \mathrm {H^\pm }}\) and \(C_{10}^{\prime \mathrm {H^\pm }}\) are found to be approximately linearly correlated with each other (Fig. 5f), and the slope starts to be nearly a constant when \(M_{H^\pm }\ge 250~\mathrm {GeV}\), which explains why the two lines with \(M_{H^\pm }=300\) and \(500~\mathrm {GeV}\) almost overlap completely in Fig. 5f. In fact, from the analytic expressions for these Wilson coefficients (cf. Eqs. (3.15)–(3.16) and (3.18)–(3.19), together with (B.3)–(B.10)), we find that \(C_{9}^\mathrm {H^\pm }/C_{10}^\mathrm {H^\pm }\rightarrow 1+4\sin ^{2}\theta _{W}\,\left[ 1+4/(9x_{t})\right] \) and \(C_{9}^{\prime \mathrm {H^\pm }}/C_{10}^{\prime \mathrm {H^\pm }}\rightarrow 1+4\sin ^{2}\theta _{W}\) when \(M_{H^\pm }\) goes to infinity. This explains why the lines shown in Fig. 5e, f get closer to each other with larger \(M_{H^\pm }\).
The zerocrossing points of \(P_2\) (nonzero one) and \(P_5'\) both within the SM and in the A2HDM
SM  Case A  Case B  

\(q^2_0(P_2)\)  \(3.43^{+0.33}_{0.32}\)  \((3.02,\,3.90)\)  \((3.02,\,4.79)\) 
\(q^2_0(P_5')\)  \(2.02^{+0.19}_{0.15}\)  \((1.77,\,2.32)\)  \((1.79,\,4.85)\) 
4.4 2HDMs with \(\mathcal {Z}_2\) symmetries
In the generic 2HDMs with discrete \(\mathcal {Z}_2\) symmetries, the three alignment parameters \(\varsigma _{f}\) will be reduced to a single parameter \(\tan \beta =v_2/v_1\ge 0\), as indicated in Table 1. There are, therefore, only two model parameters, \(\tan \beta \) and \(M_{H^\pm }\), in the Wilson coefficients \(C_{7,9,10}^{(\prime )\mathrm {H^\pm }}\). We show in Fig. 7 the allowed regions in the \(\tan \beta \)–\(M_{H^\pm }\) plane corresponding to the four different types of 2HDMs with \(\mathcal {Z}_2\) symmetries. As \(C_{7,9,10}^{(\prime )\mathrm {H^\pm }}\) do not depend on the parameter \(\varsigma _\ell \), 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 Fig. 7, the bound \(M_{H^\pm }>432\,\mathrm{GeV}\) is obtained for types II and Y 2HDMs, while there is no further bound found for \(M_{H^\pm }\) in types I and X 2HDMs with sizable \(\tan \beta \).
With the constrained model parameters shown in Fig. 7, we then show in Fig. 8 the \(q^2\) dependence of \(P_2\) and \(P_5'\) in the four different types of 2HDMs with \(\mathcal {Z}_2\) symmetries. One can see that, compared with 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 chargedscalar effect on the left and righthanded semileptonic operators is controlled by the same parameter \(\tan \beta \) and, under the constraint shown in Fig. 7, sizable \(C_{9,10}^{\prime \mathrm {H^\pm }}\) are not allowed in these models. It is, therefore, concluded the 2HDMs with \(\mathcal {Z}_2\) symmetries cannot explain the socalled \(P_5'\) anomaly.
5 Conclusions
In this paper, we have presented a complete oneloop calculation of the SD Wilson coefficients \(C_{7,9,10}^{(\prime )\mathrm {H^\pm }}\) due to the chargedscalar exchanges through the \(Z^0\) and \(\gamma \)penguin diagrams within the A2HDM. For \(C_{9,10}^{\prime \mathrm {H^\pm }}\), although being suppressed by the factor \(\bar{m}_{b}\,\bar{m}_{s}/M_{W}^2\), they could play an important role in interpreting the observed \(P_5'\) anomaly in the decay \(B^0\rightarrow K^{*0}\mu ^+\mu ^\), when the model parameter \(\varsigma _{d}\) is large.
Under the constraints from the branching ratio \(\mathcal {B}(B\rightarrow X_{s}\gamma )\) and the recent global fit results of \(b\rightarrow s\ell ^+\ell ^\) data, we have obtained the allowed parameter spaces in the \(\varsigma _{u}\)–\(\varsigma _{d}\) plane, corresponding to three representative chargedscalar masses. We found that \(C_{9,10}^\mathrm {H^\pm }\) play a major role in the small \(\left \varsigma _{d}\right \) region (\(\left \varsigma _{d}\right <1\)), while \(C_{9,10}^{\prime \mathrm {H^\pm }}\) are most important when the model parameter \(\varsigma _{u}\) approaches 0. When \(\varsigma _{u}\) is far away from 0 and \(\left \varsigma _{d}\right \ge 1\), on the other hand, the impact of \(C_{7}^\mathrm {H^\pm }\) 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) \(C_{7,9,10}^\mathrm {H^\pm }\) are sizable, but \(C_{9,10}^{\prime \mathrm {H^\pm }}\simeq 0\), 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 \(C_{9,10}^\mathrm {H^\pm }\simeq 0\), corresponding to the (small \(\left \varsigma _{u}\right \), large \(\left \varsigma _{d}\right \)) region. We have then discussed their impacts on the angular observables \(P_2\) and \(P_5'\) in the decay \(B^0\rightarrow K^{*0}\mu ^+\mu ^\). It is found that there is only a small impact on \(P_2\) and \(P_5'\) in case A, while 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 zerocrossing point.
Finally, we have explored the constraints on \(\tan \beta \) and \(M_{H^\pm }\) in four types of \(\mathcal {Z}_2\)symmetric 2HDMs. The role of chiralityflipped operators \(O_{9,10}'\) becomes much more important for large values of \(\tan \beta \). Even with the current data, the types I and X and types II and Y could be clearly distinguished from each other. However, the chargedscalar effect on \(P_2\) and \(P_5'\) in these models is found to be small and does not help to explain the socalled \(P_5'\) anomaly.
Future precise measurements of the angular observables in \(b\rightarrow s\ell ^+\ell ^\) 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.
Notes
Acknowledgements
The work is supported by the National Natural Science Foundation of China (NSFC) under contract Nos. 11675061, 11435003, 11225523 and 11521064. Q.H. is supported by the Excellent Doctorial Dissertation Cultivation Grant from CCNU, under contract number 2013YBZD19.
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