Implications of hidden gauged $U(1)$ model for $B$ anomalies

We propose a hidden gauged $U(1)_H$ $Z'$ model to explain deviations from the Standard Model (SM) values in lepton flavor universality known as $R_K$ and $R_D$ anomalies. The $Z'$ only interacts with the SM fermions via their mixing with vector-like doublet fermions after the $U(1)_H$ symmetry breaking, which leads to $b \to s \mu\mu$ transition through the $Z^{\prime}$ at tree level. Moreover, introducing an additional mediator, inert-Higgs doublet, yields $b\to c \tau \nu$ process via charged scalar contribution at tree level. Using flavio package, we scrutinize adequate sizes of the relevant Wilson coefficients to these two processes by taking various flavor observables into account. It is found that significant mixing between the vector-like and the second generation leptons is needed for the $R_K$ anomaly. A possible explanation of the $R_D$ anomaly can also be simultaneously addressed in a motivated situation, where a single scalar operator plays a dominant role, by the successful model parameters for the $R_K$ anomaly.

We propose a hidden gauged U (1) H model to explain the R K anomaly, in which a Z gauge boson interacts with the SM particles only through mediator particles. The mediators are vector-like doublet fermions, whose masses are assumed to be around O(1) TeV. The Z couplings to the SM particles appear with symmetry breaking of the hidden gauged U (1) H , which results in b → sµµ transition through the tree-level Z exchange. Furthermore, introducing an another mediator, we can have a relevant b → c transition to the R D anomaly.
The role of the additional mediator is played by an inert-Higgs doublet, and the charged Higgs component of the inert Higgs can induce the b → c transition at tree level. Since some essential couplings for the b → sµµ are also involved in b → cτ −ν , the former process can affect a possibility of explaining the R D anomaly.
Typically, the b → s transition is tightly constrained by various observables such as B s −B s , B s → µµ, etc. While in the lepton sector, the Z coupling to the muon contributes to neutrino trident production as discussed in [52,53]. At the same time, we carefully examine that there is no significant FCNC Z couplings, although the mixing between the mediator and the SM fermions could be significantly large. In addition to these constraints, we also address direct searches of the Z and the charged Higgs at collider experiments. To avoid flavor constraints from the neutral Higgses, we take degenerate masses on the inert Higgs spectra. We utilize flavio package [10] to perform a comprehensive analysis on various flavor observables, and obtain the global fit to the Wilson coefficients on the R K and R D anomalies. Applying these results to our hidden gauged model, we scrutinize the favored parameter space on explaining these anomalies.
The structure of this paper is as follows. We introduce the hidden gauged extension of the SM in Sec. II. In Sec. III, the mixing between the vector-like and SM fermions and the Z couplings are presented. Using the flavio, we perform a global fit on the relevant Wilson coefficients to the R K and R D anomalies in Sec. IV. Subsequently, various experimental searches and constraints are discussed in section VI. In Sec. V, we apply the obtained results in previous sections to the model parameters. Finally, Sec. VII is devoted to conclusions.

II. THE MODEL
We consider a hidden sector extension of the SM under the hidden gauged U (1) H symmetry [54]. Although all the SM particles are not charged under the U (1) H symmetry, the U (1) H gauge boson can couple to them through mediators, such as new vector-like fermions or scalars. The mediators have both the U (1) H and the SM gauge charges. The schematic framework of this model is described in Fig. 1.
For the R K anomaly, essential mediators are the vector-like fermions. These fermions could be SU (2) L singlet, doublet, or multiplets, with one or more generations. Here, we focus on one minimal fermion assignments: all the vector-like fermions are SU (2) L doublets, with only one generation 1 . To be free of gauge anomaly, the new vector-like fermions need to possess appropriate quantum numbers. We assign the SM quantum numbers on the 1 Another minimal choice is to introduce only vector-like singlet fermions with one generation. Since our purpose is to illustrate how the U (1) H works for the R K anomaly, the fermion assignment is not so important. We expect that the analysis is quite similar to different fermion assignments. vector-like fermions such that they mix with the SM fermions at tree level.
Moreover, if we introduce additional mediators such as a charged scalar, the R D anomaly could also be explained as a bonus. In this work, we add a second scalar doublet H as an inert type [54,55] to keep our setup simple.
and the Lagrangian is written as with the covariant derivative D µ Φ = (∂ µ + ig Z µ )Φ. Here, g is the gauge coupling of the U (1) H , and the gauge boson mass is given by m Z = g v Φ . In the above potential, H describes the SM SU (2) doublet, and the two Higgs doublets are parametrized by The potential V (H, Φ, H ) is not relevant to our flavor study, and for simplicity, it is taken which respects a Z 2 (H ) × Z 2 (Φ) symmetry. The µ 2 H parameter is positive, and thus H does not get the VEV.
The vector-like fermions are defined by and their Dirac masses are All the new fermions are charged under U (1) H , as shown in Table I. 2 The right-handed vector-like fermions can couple to the left-handed SM fermions through the Yukawa inter- 2 These new fermions contribute to the kinetic mixing between U (1) Y and U (1) H at the one loop order.
The mixing parameter is estimated to be roughly eg (4π) 2 ln m Q m L . Here for simplicity, we take small or zero mass splitting between the vector-like quarks and leptons, and thus the kinetic mixing is very tiny and negligible.
where the SU (2) symmetry imposes a relation 3 with a generation index i = 1, 2, 3, Cabbio-Kobayashi-Maskawa (CKM) matrix V CKM and Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix V PMNS . Similar situations are studied in [52,56] for the quark sectors and [57] for the lepton sectors. The interactions in Eq. (8) describe the mixing between the vector-like and SM fermions, which yieldssbZ andμµZ couplings after diagonalization of mass matrices. These Z couplings are originated from Apart from the right-handed vector-like fermions, the left-handed vector-like fermions have couplings to the right-handed SM fermions through the inert doublet To make it a dark matter candidate, we assign a Z 2 charge for the χ. Under U (1) H breaking down to the Z 2 symmetry, the χ is Z 2 -odd while all other mediators and the SM particles are Z 2 even.

III. MASS MATRICES AND COUPLINGS
The interactions in Eq. (8) imply off-diagonal parts in fermion mass matrices, which are described by . , 3 and a = 1, . . . , 4. y i is the SM Higgs Yukawa coupling, and its VEV is v 246 GeV. The matrix M f is diagonalized by which gives relationships between the original gauge eigenstate f a and new mass eigenstate This diagonalization can affect original gauge interactions such as the W and Z boson couplings. However, it should be noted that, since the left-handed vector-like fermions have the same SM charges as those of the SM fermions, the Z boson couplings to the left-handed fermions do not change. We will see this situation later.
In the charged-lepton sector, the mixing between the second generation lepton µ and vector-like leptonL is essential for the b → sµµ transition. Assuming there is only one mixing, namely, Y µL R = 0, the matrices U L and U R can be described by 4 where s α ≡ sin α and c α ≡ cos α. The mixing angle α L and α R are approximately given by For the down-quark sector, the Yukawa couplings Y bD R and Y sD R are necessary for the R K anomaly, which implies the nonzero values of Y tŨ R and Y cŨ R with the same order of magnitude. Although, apart from the charged-lepton sector, the diagonalization is somewhat complicated due to the two mixing parameters, we obtain similar situation Again, the deviations of the right-handed mixing matrices from identity are suppressed by order mq m Q . For the neutrino parts, its Lagrangian contains As explained in the previous section, the second term is related to that of the charged lepton due to the SU (2). As long as we keep only nonzero Y µL R , the coupling is given by For sake of simplicity, we assume that the V PMNS is a unit matrix, which results in only nonzero Y νµÑ R with more direct relationship of Y νµÑ R = Y µL R . As in the same way of the charged-lepton sector, if we rotate the neutrino fields with where the Lagrangian in Eq. (17) becomes We find that U ν L U L due to the approximate SU (2) symmetry, which deviates once the lepton masses are taken into account. After diagonalizing the mass matrices, the Z boson couplings to the lepton sector are where . It is seen that the left-handed Z couplings are diagonal, while the flavor-violating Z couplings to the right-handed charged leptons exist. However, the right-handed mixing angle is tiny, which results in the small flavor-violating couplings. Similarly, in the quark sectors, while the Z couplings to the left-handed quarks do not receive any corrections from the field definitions, the right-handed couplings have the flavor-violating parts but they are suppressed.
The W couplings to the leptons are given by Because of the approximate SU (2) symmetry, the left-handed couplings have U † L diag(1, 1, 1, 1)U ν L 1. The right-handed interactions are also present with a suppression factor U † R . On the other hand, the quark sector has a somewhat different situation due to the CKM structure.
The model parameters Y d iDR , v Φ and m Q , which decide U u L,R and U d L,R , should be chosen to realize the measured CKM values. In our analysis, we use moderate values of the model parameters which satisfy the CKM values within error bars.
One of our main interests is the Z couplings, which are given by These interactions can be described by integrating out the heavy fermions and this situation is schematically drawn in Fig. 2. If we focus on only relevant Yukawa couplings Y bD R , Y sD R and Y µL R to the R K anomaly, they yield where proportional to the mixing angles, and we omit the prime in the fermion fields. The first two terms in Eq. (26) lead to the b → sµµ transition, and the three Yukawa couplings also produce the Z couplings to the up quark and neutrino sectors, such asttZ andν µ ν µ Z , due to the SU (2) symmetry. Note that the right-handed Z couplings are also present, however, they are induced by the inert scalar loop and are numerically suppressed. Therefore, our current study does not take them into account.
The other intriguing interactions are the Yukawa couplings between the inert-Higgs doublet and the SM fermions, in particular, the charged Higgs couplingscbH + andντ H + . Using mass-insertion method as in the Z coupling, we obtain the Yukawa couplings with These interactions are originated from the mixing between the vector-like and SM fermions as seen in Fig. 3. Their dependences on v Φ /m Q,L are different from those in Eq. (27). The neutral components in the inert Higgs also induce the Yukawa couplings as in Eq. (28). In our study, we assume that their masses are degenerate. This assumption produces a simple situation where some scalar operators disappear due to the degeneracy as mentioned later.
As we will see in the next section, while the explanation of R K anomaly needs nonzero g L sb and g L µµ , the R D anomaly requires nonzero Y cb and Y ντ . From the phenomenological point of view, we consider a minimal setup in which these relevant couplings are exclusively focused and We also assume that the Yukawa couplings are real, and a flavor hierarchy |Y b | |Y s |.

IV. RARE B DECAY ANOMALIES
The relevant effective Hamiltonians to b → sµ + µ − and b → cτ −ν are given by [58][59][60] The Wilson coefficients of C 9 and C 10 are induced by the Z interactions in Fig. 2. Integrating out the Z boson, the Wilson coefficients are given by 16π 2 . The final expression in Eq. (34) is obtained by using the couplings defined in Eq. (27). Although m Z dependence appears in the middle of the Eq. (34), it is canceled out by those in g L sb and g L µµ . Thus, v Φ is finally left only in the numerator. The For the R D anomaly, neglecting the neutrino flavor textures which is irrelevant to our study, we simply assume that the PMNS matrix is a unit matrix, which results in Y νµ = Y µ .
The Yukawa coupling allows the flavor-violating charged Higgs couplingτ P L ν µ H − . As seen in [29], such a new flavor-violating process would also contribute to the R D anomaly.
Therefore, the charged scalar interactions in Fig. 3 produce The third terms are written by Eq. (28). It is seen that the model parameters appearing in C 9 and C 10 in the Eq. (34) are correlated to C S and C S except m H and y b,c,τ . Therefore, in our model, the R D anomaly is related to the R K anomaly through the common parameters.
In order to determine the possible sizes of these Wilson coefficients, we use flavio 0.21 [10]: • C 9 and C 10 The observables and the corresponding experimental measurements listed in the Appendix in Ref. [9] are used, and Table IV in the last page summarizes them. We take into account all known correlations among observables and approximate the uncertainties as Gaussian. From Eq. (34), our model holds the relation C 9 = −C 10 , therefore essentially only one Wilson coefficient needs to be fitted. Assuming all the UV parameters are real, we define the following real parameters in the global fit: C 9,new = C 9 · exp(i arg (C bs SM )) = −C 10,new = −C 10 · exp(i arg(C bs SM )) The Bayesian method is employed in the global fit with following procedures: we first obtain the likelihood function with a single argument C 9,new from flavio FastFit class, and assume a uniform prior probability of C 9,new ranges from -3 to 3. We then use pymultinest [61] to implement a Monte Carlo sampling, and finally obtain the posterior probability shown in Fig. 4 with corresponding 1σ and 2σ uncertainties, which is plotted using Superplot [62]. In the figure, the best-fit point is indicated by a star, and 1σ and 2σ regions are represented by blue and green lines on the top of curve. The obtained values are in agreement with the current observed branching ratio for B 0,± → K * 0,± µµ, B s → φµµ, B → X s µµ and B s → µµ.
• C S and C S For the R D * , measurements in Ref. [16,18,63] are used, while those in Ref. [14,15] are taken into account for the R D . We use FastFit in flavio to obtain the 2-D global fit for C S and C S , which is shown in Fig. 5. The four blue regions are 1σ, 2σ and 3σ allowed regions by fitting with the R D and R D * measurements mentioned above. The light and dark red region are excluded by the current LHC limit BR(B − c → τ −ν ) < 30% [43], and recasted LEP limit BR(B − c → τ −ν ) < 10% [64], respectively. We find that among four favored regions, only two of them are more preferred after considering the constraints from the BR(B − c → τ −ν ) limits.
For later use, we project the two dimensional contour to a 1-dimensional fit by setting C S = 0. The Bayesian fit for C S alone is obtained using the same strategy as in the fitting of C 9,new . For this fit, we choose prior probability as uniformly distribute between −2 and 0, the fitting results is shown in Fig. 6. As in Fig. 4, the best fit value is represented by the star at the bottom, and 1σ and 2σ regions are blue and green lines above curve.
Finally, we summarize the obtained numerical values in Table. II and III, where those of C S and C S without the constraint from B − c → τ −ν transition are listed.

V. EXPERIMENTAL SEARCHES AND CONSTRAINTS
The hidden gauge boson Z encounters direct constraints from collider searches. Depending on the Z mass, m Z = g v Φ , there are different limits on the signal rate of the Z production. Here, we discuss the following cases: light Z case (m Z < m Z ) and heavy Z case (m Z > m Z ). • When the Z is lighter than the Z mass, some viable parameter regions exist. Since the Z has no coupling to the electron, LEP searches cannot provide direct constraint on the light Z . Furthermore, the Tevatron [65,66] and LHC [67,68] searches for Z to dilepton final state only apply to the case of m Z > 100 GeV. The relevant limit to the light Z case comes from the LHC searches at pp → Z → 4µ. Its typical SM process is through the off-shell Z mediated 2µ decay, while the light Z could be on-shell when m Z < m Z − 2m µ . A detailed analysis on how to recast the current LHC search limit has been done in Ref. [53]. Mapping their analyses to our model, we obtain the constraint shown in Fig. 7. In the figure, the orange line indicates the g coupling as a function of m Z with v Φ = 700 GeV. The excluded region corresponds to the blue region. For v Φ = 700 GeV, we find that the region of m Z 10 GeV and 50 GeV m Z still have some spaces for 10 −3 < g < 0.14. However, one should note that for m Z 10 GeV, we are not able to integrate out the Z particle when deriving the Wilson coefficients C 9 and C 10 in Eq.34.
• In the case of the heavy Z , the LHC searches in the di-muon final states put the tightest constraints on its mass. The current limit on m Z is around 3 ∼ 4 TeV [67,68]. In order to apply this limit to our case, we should first take into account the suppressed coupling to the first generation quarks by the small mixing with vector-like  fermions, which results in small production cross section. Moreover, although the LHC searches assume that the decay branching ratio to the muon or electron lepton is 100%, our Z can also decay into the SM quarks, and other light U (1) H charged particles.
Therefore, the branching ratio could be much smaller than the assumed value. We estimate pp → Z production cross section with Madgraph [69], and demonstrate the constraint on σ(pp → Z → µµ) in Fig. 8   curve. According to the figure, the Z mass can be as low as 1.5 TeV. Furthermore, the branching ratio can be reduced if additional decay channels come from unknown hidden gauge sector particles, which are irrelevant to our flavor study. However, if we allow opening additional decay channels, the constraint could be significantly relaxed.
For example, taking the branching ratio to be BR(Z → µµ) = 1/20, which is shown in the green curve, the Z mass can be lower than 750 GeV.
Next, we briefly comment on the constraints on the mediators. In [70], the vector-like quark search is discussed with its decay into W b, and the current limit is m Q > 1.1 TeV.
On the other hand, studies about the multi-lepton final state [71] show that the limit on the vector-like lepton is m L > 500 GeV. In our numerical analysis, we take the benchmark point m Q = m L = 1.4 TeV, which is consistent with these limits. Regarding the inert-Higgs doublet, the charged scalar receives the tightest bounds. The LHC searches for H ± are all associated with the top quark production through tbH ± coupling. Therefore, we can evade any constraints from the LHC by setting y t = y b = 0 such that the tbH ± coupling vanishes, and it follows that C S = 0. Actually, we investigate the scenario that y t and y b are nonzero and calculate the production cross section for the process pp →tbH + with Madgraph for a benchmark point: We calculate the allowed region of this operator using the measurements of ∆M s for B s −B s mixing in Ref. [74]. In our analysis, the Bayesian fit is also employed. The likelihood function is approximated by Gaussian probability distribution function with mean as difference of the theoretical prediction and experimental central value, and the variance as the the quadratic sum of the theoretical uncertainty and the experimental uncertainty. The prior probability distribution is taken as uniform distribution between −3 × 10 −10 to 3 × 10 −10 . The fitting results are shown in Fig. 9, and it is found that |C LL V | < 1.29 × 10 −11 at 1 σ. Note that, although scalar operators O LL S and O RR S listed in [75] are caused by neutral and CP-odd scalars in the inert-Higgs doublets, they vanish as long as the two scalar masses are degenerate. Also, we do not consider an operator O LR S since its Wilson coefficient is proportional to y s that is irrelevant to our current study.

• B → Kνν
In the same way as b → sµµ process, b → sνν transition is also induced as shown in Fig. 10. The effective Hamiltonian is [29]  Our current setup produces only the 2nd generation neutrino at final states, therefore the Wilson coefficient is described by with g L νµνµ = g L µµ . Following the bound on the Wilson coefficient obtained in [29], we find that −13Re The Yukawa couplings in Eq. (28) contribute to b → sγ process through the charged scalar loop. In Fig. 11, two possible diagrams are drawn. It is seen that while the left diagram contains the product y b y c related to C S and C S , the diagram on the right is proportional to y c y s . Therefore, we include only the left diagram which contains two relevant parameters to b → c transition. The effective operators are given by [79] with The Wilson coefficients in our model are where x c = m 2 c /m 2 H and the loop functions C 0 7,8 are listed in [79]. The Wilson coefficients C 7 and C 8 are proportional to a product of Y s Y c y b y c . While Y s and Y c are predicted by the explanation of the R K anomaly, y b and y c are related to C S and C S in Eqs. (35) and (36). According to a global fit of the Wilson coefficients in the b → s transition in [3], contributions from new physics to C 7 (µ b = 5 GeV) are allowed between −0.04 ≤ C 7 (µ b ) ≤ 0.0. Taking that the charged scalar mass is 300 GeV, we find that the loop functions are roughly O(10 −5 ∼ 10 −6 ). In addition, the Wilson coefficients receive suppression proportional to Y s Y c ∼ 10 −4 . Later, we will check this constraint numerically. It should be noted that, as in the B s −B s mixing, the neutral scalars do not contribute to the process due to cancellations between H and A contributions for the degenerate mass spectra in the inert Higgs.
In addition to the above constraints, lepton-flavor-violating processes, such as τ → 3µ and τ → µγ can also be induced. However, these processes also need irrelevant parameters to the B anomalies, we do not deal with them here.
Finally, we briefly discuss the hidden particles which are only charged under the U (1) H group. Although these hidden particles are not relevant to the B physics observables, it provides additional signatures. We assume that there is a WIMP dark matter candidate in the hidden sector, and the dark matter particle χ should annihilate into SM particles during freeze-out at early universe. In the non-relativistic approximation, the annihilation cross section times relative velocity σv can be decomposed as where a and b are the s-wave and p-wave cross sections. We only take the dominant contributions into account. When m χ < m Z , the dominant annihilation channel is through the s-channel processχχ → ff , where f = µ, ν µ , b, t: where g Z f f denotes the Z coupling to the fermion f . When m χ > m Z , additional t-channel χχ → Z Z → ff ff appears. The annihilation cross section is The thermal relic density is written as where s 0 is the entropy density of the present universe, ρ crit the critical density, n χ the dark matter number, h the hubble parameter, g eff the effective degree of freedom during freezeout, and I( , with x f = m χ /T f and T f the temperature during freeze-out.
There are also constraints from the direct detection measurements. However, we expect the limit is not so tight for the light Z , because the size of the gauge coupling g is typically smaller than 0.1. Furthermore, in our model the dark matter only dominantly couples to the third generation quarks inside nucleons. This kinds of scenarios have been studied in Ref. [80]. From their study, we find that for the light Z , if g ∼ 0.05, the direct detection constraints could be escaped. We will perform detailed study on the dark matter sector in future.

VI. RESULTS
We first consider the effective operators for the R K anomaly in terms of With these expressions, the Eq. (37) is described by This expression consists of only relevant model parameters through the combinations of X q and X µ that are related to the mixing angles in the quark and lepton sectors. It should be noted that the value of C 9,new in Table. III is positive, therefore either X b or X s should be negative to compensate for the minus sign in the above expression. The left plot in Fig. 12 shows 1σ and 2σ regions with black solid and dashed lines in (−X b X s , X 2 µ ) plane with the fixed value of v Φ = 700 GeV. The best fitted value is drawn by a green line, and the blue region is excluded by B s −B s mixing. We also consider the constraint from the B → Kνν process, however, the present parameter region can evade the restriction. It is seen that X b X s is tightly constrained by the B s −B s mixing and the allowed size is roughly O(10 −3 ) at most. And, it follows that a somewhat large value of X 2 µ ∼ O(0.1) is necessary to achieve the required Wilson coefficients for the R K . Such a large value can be obtained by taking a large Y µ or a small m L . Here, we fix the mass m L at 1.4 TeV and see how large Yukawa coupling, Y µ , is needed at one benchmark region of −X b X s = 2.3 × 10 −3 . The region is indicated by a red line in the left panel in Fig. 12, and the benchmark value results in and Y s = 0.0184. In the right plot of Fig. 12, the size of C 9,new is shown as a function of Y µ .
Here, instead of the mass-insertion method, we use g L sb and g L µµ obtained after diagonalization of the mass matrices. All lines in the right figure correspond to those in the left one. The 2σ region requires 1.2 ≤ Y µ ≤ 1.9, and the best fitted value stays around Y µ ∼ 1.5. The values of Y b = −1, Y s = 0.0184 and Y µ = 1.5 yield g L sb /g = −2.17 × 10 −3 and g L µµ /g = 0.22, which leads to C 9,new = 0.6. The current size of g L µµ /m Z gives the cross section of the neutrino trident production in Eq. (39), σ SM+NP /σ SM = 1.0, which is within the 1σ error. Later, we also use these Yukawa couplings as the benchmark values for the discussion of the R D anomaly. Figure 13 shows the experimental data and theoretical predictions on the B decay branching ratios in different energy bins for dBR(B 0 → K 0 µ + µ − )/dq 2 (upper left), dBR(B 0 → K * 0 µ + µ − )/dq 2 (upper right), dBR(B ± → K ± µ + µ − )/dq 2 (lower left), dBR(B + → K * + µ + µ − )/dq 2 (lower right). The experimental results obtained by CDF [81], LHCb [82], CMS [83,84] are plotted. The purple bands correspond to the SM predictions with uncertainties obtained by varying the input parameters with the Gaussian distribution. The green bands represent the best fitted value of C 9,new , which is indicated by the green line in Fig. 12, with the uncertainties obtained by the same way as the purple bands. The width of the purple and green bands are not the same, and their ratios are roughly proportional to |C bs SM /(C bs SM + C i )| 2 (i = 9, 10). It is seen that our model provides a better fit on the data than the SM.
Finally, we discuss the possibility of explaining the R D anomaly using the parameter space which could explain the R K anomaly. Once Y b and Y s are fixed, Y c is also given due to the SU (2) symmetry. Using the benchmark values, Y b = −1, Y s = 0.018, we find that As seen in Fig. 5, positive C S is disfavored by the upper limit on the B − c → τ −ν process, and the absolute values of C S and C S should be O(0.1). In the current setup, the negative C S can be realized by Y b = −1. However, a naive estimation implies that C S C S since C S is proportional to Y c which is two orders of magnitude smaller than Y b . Therefore, we exclusively focus on a specific situation with C S = 0 but C S = 0 as listed in Table III. Figure 14 shows the 1σ and 2σ regions of C S in (y c , y τ ) plane. Some of the model parameters are correlated with the R K anomaly, and they are fixed at Y b = −1, Y s = 0.018, Y µ = 1.5, v φ = 700 GeV and m Q = m L = 1.4 TeV. The charged scalar mass is taken as 300 GeV. As we can expect from the required value of C S , it is found that the large Yukawa couplings y c and y τ are needed for the explanation of the R D anomaly. As long as we focus on only C S , the size of y b is not predicted. Therefore, our current setup does not affect b → sγ process. However, it is found that, even if we take y c = 2 and y b = 1, C 7 (µ b ) roughly becomes 1.6 × 10 −3 which is consistent with the current observed value.

VII. CONCLUSION
We have studied a hidden gauged U ( In order to determine desired sizes of the Wilson coefficients, we utilized the flavio package and performed a global Bayesian fit including several observables, B 0,± → K * 0,± µµ, B s → φµµ, B s → X s µµ and B s → µµ. At the same time, we also considered various flavor constraints, B s −B s mixing, B → Kνν, neutrino trident productions, b → sγ, and B − c → τ −ν . It is found that the absolute values of C 9 (= −C 10 ), C S and C S are required to be O(0.5) roughly. Also, it turned out that the positive values of C S are almost excluded by B − c → τ −ν if we impose its severe limit on the branching ratio, BR(B − c → τ −ν ) < 10%. The required sizes of C 9 and C 10 can be achieved by somewhat large Y µ , which describes the mixing parameter between the vector-like and SM leptons, since those in the quark sector Y b and Y s are highly constrained by the B s −B s mixing. Our benchmark point with m Q = m L = 1.4 TeV, v Φ = 700 GeV and Y b Y s = −0.018 needs Y µ ∼ 1.5 for the successful scenario of the R K anomaly. This benchmark leads to a consistent situation in which C S is suppressed due to tiny Y c and negative C S is obtained by taking Y b < 0. The possible region of C S requires O(1) couplings of y c and y τ .
Since this model contains new particles, there are promising collider and astrophysical signatures. Current LHC searches on pp → µμ and pp → 4µ put strong constraints on the Z masses, although the light region of M Z < 100 GeV is still allowed. Furthermore, since the U (1) gauge symmetry is hidden, it is likely that other hidden particles also exist, such as dark matter candidate, in the hidden sector. Although the hidden sector particles do not contribute to the B physics observables, they are able to contribute to the dark matter relic density and possible indirect detections, and would relax the constraints on the heavy Z .
On the other hand, the charged Higgs is not tightly constrained due to its Yukawa texture.
We anticipate that the future collider searches will explore larger mass regions for the Z and charged Higgs, and these collider signatures and potential dark matter signatures could be able to validate this model.