$B$-meson anomalies and Higgs physics in flavored $U(1)'$ model

We consider a simple extension of the Standard Model with flavor-dependent $U(1)'$, that has been proposed to explain some of $B$-meson anomalies recently reported at LHCb. The $U(1)'$ charge is chosen as a linear combination of anomaly-free $B_3-L_3$ and $L_\mu-L_\tau$. In this model, the flavor structure in the SM is restricted due to flavor-dependent $U(1)'$ charges, in particular, quark mixings are induced by a small vacuum expectation value of the extra Higgs doublet. As a result, it is natural to get sizable flavor-violating Yukawa couplings of heavy Higgs bosons involving the bottom quark. In this article, we focus on the phenomenology of the Higgs sector of the model including extra Higgs doublet and singlet scalars. We impose various bounds on the extended Higgs sector from Higgs and electroweak precision data, $B$-meson mixings and decays as well as unitarity and stability bounds, then discuss the productions and decays of heavy Higgs bosons at the LHC.


Introduction
The observed fermion masses and mixing angles are well parametrized by the Higgs Yukawa couplings in the Standard Model (SM). However, the neutrino masses and mixing angles call for the addition of right-handed (RH) neutrinos or physics beyond the SM and, moreover, the flavor structures of quarks and leptons are not understood yet. As there is no flavor changing neutral current at tree level in the SM due to the GIM mechanism, the observation of flavor violation is an important probe of new physics up to very high energy scales and it can be complementary to direct searches at the LHC. In particular, the violation of lepton flavor universality would be a strong hint at new physics.
Motivated by the B-anomalies R K ( * ) , some of the authors recently proposed a simple extension of the SM with extra U (1) gauge symmetry with flavor-dependent couplings [8]. The U (1) symmetry is taken as a linear combination of U (1) Lµ−Lτ and U (1) B 3 −L 3 , which might be a good symmetry at low energy and originated from enhanced gauge symmetries such as in the U (1) clockwork framework [9]. In this model, the quark mixings and neutrino masses/mixings require an extended Higgs sector, which has one extra Higgs doublet and multiple singlet scalars beyond the SM. As a result, nonzero off-diagonal components of quark mass matrices are obtained from the vacuum expectation value (VEV) of the extra Higgs doublet and correct electroweak symmetry breaking is ensured by the VEV of one of the singlet scalars.
In this paper, we study the phenomenology of the heavy Higgs bosons in the flavored U (1) model mentioned above. We first show that the correct flavor structure of the SM is well reproduced in the presence of the VEV of the extra Higgs doublet. In particular, in the case with a small VEV of the extra Higgs doublet or small tan β, we find that the heavy Higgs bosons have sizable flavor-violating couplings to the bottom quark and reduced flavor-conserving Yukawa couplings to the top quark such that LHC searches for heavy Higgs bosons can be affected by extra or modified production and decay channels. We also briefly mention the implication of our extended Higgs sector for R D ( * ) anomalies. We discuss various constraints on the extended Higgs sector from Higgs and electroweak precision data, flavor data such as the B-meson mixings and decays, as well as unitarity and stability bounds. For certain benchmark points that can evade such bounds, we study the productions and decays of the heavy Higgs bosons at the LHC and show distinct features of the model with flavor-violating interactions in the Higgs sector. This paper is organized as follows. First, we begin with a summary of the U (1) model with the extended Higgs sector and new interactions. The Higgs spectrum and Yukawa couplings for heavy Higgs bosons are presented in Sec. 3. We then discuss various theoretical and phenomenological constraints on the Higgs sector are studied in Sec. 4, and collider signatures of the heavy Higgs bosons at the LHC are studied in Sec. 5. Finally, conclusions are drawn. There are four appendices dealing with the extended Higgs sector, unitarity bounds, quark Yukawa couplings, and the U (1) interactions.

Flavored U (1) model
We consider a simple extension of the SM with U (1) , where a new gauge boson Z couples specifically to heavy flavors. It is taken as a linear combination of U (1) Lµ−Lτ and U (1) for real parameters x and y [8]. 1 Introducing two Higgs doublets H 1,2 is necessary to have right quark masses and mixings. We add one complex singlet scalar S for a correct vacuum to break electroweak symmetry and U (1) . Moreover, in order to cancel the anomalies, the fermion sector is required to include at least two RH neutrinos ν iR (i = 2, 3). One more RH neutrino ν 1R with zero U (1) charge as well as extra singlet scalars, Φ a (a = 1, 2, 3), with U (1) charges of −y, x+y, x, respectively, are also necessary for neutrino masses and mixings. As L µ − L τ is extended to RH neutrinos, L µ − L τ and L 2 − L 3 can be used interchangeably in our model. The U (1) charge assignments are given in Table 1.
The Lagrangian of the model is given as where Z µν = ∂ µ Z ν − ∂ ν Z µ is the field strength of the U (1) gauge boson, sin ξ is the gauge kinetic mixing between U (1) and SM hypercharge, and The extended Higgs sector is presented in the next section and studied in more detail in Appendix A. For a set of quartic couplings for S and H 1,2 that are relevant for electroweak symmetry and U (1) breaking, we have collected unitarity bounds in Appendix B, which are used to constrain the parameter space of the Higgs sector in Sec. 4.
The Yukawa Lagrangian for quarks and leptons is given by After electroweak symmetry and U (1) are broken by the VEVs of scalar fields, 2, the quark and lepton mass terms are given as with the following flavor structure: Since the mass matrix for charged leptons is already diagonal, the lepton mixings come from the mass matrix of RH neutrinos. There are four other categories of neutrino mixing matrices [10], that are compatible with neutrino data. In all the cases, we need at least three complex scalar fields with different U (1) charges, similarly to the case given in (2.11). The quark Yukawa couplings to Higgs bosons are summarized in Appendix C.
We find the Z-like (Z 1 ) and Z -like (Z 2 ) masses as (2.14) Here s ϕ ≡ sin ϕ, c ϕ ≡ cos ϕ, and t ϕ ≡ tan ϕ. The modified Z boson mass can receive constraints from electroweak precision data, which is studied in Sec. 4. We note that for a small mass mixing, the Z -like mass is approximately given by m 2 Z 2 ≈ m 2 Z and we can treat m Z and g Z to be independent parameters due to the presence of nonzero ω i 's. The U (1) interactions are collected in Appendix D.

Higgs spectrum and Yukawa couplings
We here specify the Higgs spectrum of our model and identify the quark and lepton Yukawa couplings of neutral and charged Higgs bosons for studies in next sections. The expressions are based on results in Appendices A and C.

The Higgs spectrum
The Higgs sector of our model has two Higgs doublets, which are expressed in components as and the complex singlet scalar decomposed into S = (v s + S R + iS I ) / √ 2.
In the limit of negligible mixing with the CP -even singlet scalar, the mass eigenstates of CP -even neutral Higgs scalars, h and H, are given by h = − sin α ρ 1 + cos α ρ 2 , H = cos α ρ 1 + sin α ρ 2 . (3. 2) The general case where the CP -even part of the singlet scalar S mixes with the Higgs counterpart is considered in Appendix A. The mass eigenvalues of CP -even neutral Higgs scalars are denoted as and m s ≡ m h 3 , and there are three mixing angles, α 1,2,3 : α 1 = α in the limit of a decoupled CP -even singlet scalar, while α 2 and α 3 are mixing angles between ρ 1,2 and S R , respectively. For 2κ 1 v 1 v s ≈ µv 2 / √ 2 and 2κ 2 v 2 v s ≈ µv 1 / √ 2, the mixing between ρ 1,2 and S R can be neglected. For a later discussion, we focus mainly on this case.
The CP -odd parts of the singlet scalars, S and Φ a , can mix with the Higgs counterpart due to a nonzero U (1) charge of the second Higgs H 2 , but for a small x and small VEV of H 2 , the mixing effect is negligible. In this case, the neutral Goldstone boson G 0 and the CP -odd Higgs scalar A 0 are turned out to be with tan β ≡ v 2 /v 1 . The massless combination of η 1 and η 2 is eaten by the Z boson, while a linear combination of S I and other pseudoscalars of Φ a is eaten by the Z boson if the Z mass is determined dominantly by the VEV of S. The other combination of the CP -odd scalars from two Higgs doublets has the mass of On the other hand, the charged Goldstone bosons G + and charged Higgs scalar H + identified as with nonzero mass eigenvalue given by We remark that in the limit of µv s v 2 , the heavy scalars in the Higgs doublets become almost degenerate as m 2 A ≈ m 2 H ≈ m 2 H + ≈ µv s /( √ 2 sin β cos β) and m 2 s ≈ 2λ S v 2 s from Eqs. (3.4), (3.6) and (A.5). In this limit, the mixing angles between the SM-like Higgs and extra scalars can be negligibly small and the resulting Higgs spectrum is consistent with Higgs data and electroweak precision tests (EWPT) as will be discussed in Subsec 4.2. But, as µv s is constrained by perturbativity and unitarity bounds on the quartic couplings with Eq. (A.7) or (A.9), as will be discussed in Sec. 4, the extra scalars in our model remain nondecoupled. Since it is sufficient to take almost degenerate masses for two of m A , m H , and m H + for EWPT, we henceforth consider more general scalar masses but with small mixings between the SM-like Higgs and the extra neutral scalars.

Quark mass matrices
We now consider the quark mass matrices and their diagonalization. After two Higgs doublets develop VEVs, we obtain the quark mass matrices from Eqs. (2.7) and (2.8) as The quark mass matrices can be diagonalized by thus the CKM matrix is given as V CKM = U † L D L . We note that the Yukawa couplings of the second Higgs doublet are sources of flavor violation, which could be important in meson decays/mixings and collider searches for flavor-violating top decays and/or heavy Higgs bosons [11][12][13]. The detailed derivation of flavor-violating Higgs couplings is presented in the next section.
Since h u 31 and h u 32 correspond to rotations of right-handed up-type quarks, we can take U L = 1, so V CKM = D L . In this case, we have an approximate relation for the down-type quark mass matrix, M d ≈ V CKM M D d , up to m d,s /m b corrections. Then the Yukawa couplings between the third and first two generations are given as follows.
On the other hand, taking U L = 1 as above, we find another approximate relation for the up-type quark mass matrix: Then the rotation mass matrix for right-handed down-type quarks becomes U † R = (M D u ) −1 M u , which is given as From the unitarity condition of U R we further find the following constraints on the up-type quark Yukawa couplings:

Quark Yukawa couplings
Using the results in Appendix C, we get the Yukawa interactions for the SM-like Higgs boson h and heavy neutral Higgs bosons H, A as In this case, as compared to two-Higgs-doublet model type I, extra Yukawa couplings are given bỹ We find that the flavor-violating couplings for light up-type quarks vanish, while the top quark Yukawa can have a sizable modification due to nonzeroh u 33 . On the other hand, the flavor-violating couplings for down-type quarks can be large if tan β is small, even though the couplings have the suppression factors of CKM mixing and smallness of bottom quark mass. The couplings can be constrained by bounds from B-meson mixings and decays as is discussed in the next section. We note that the flavor-violating interactions of the SM-like Higgs boson are turned off in the alignment limit where α = β − π/2. The Yukawa terms of the charged Higgs boson are given as and

Lepton Yukawa couplings
As seen in (2.9), the mass matrix for charged leptons e j is already diagonal due to the U (1) symmetry. Thus, the lepton Yukawa couplings are in a flavor-diagonal form given by

Constraints on the Higgs sector
In this section we consider various phenomenological constraints on the model coming from B-meson mixings and decays as well as Higgs and electroweak precision data on top of unitarity and stability bounds on the Higgs sector. We also show how to explain the deficits in R K and R K * in the B-meson decays at LHCb in our model, and discuss the predictions for R D and R D * through the charged Higgs exchange.

Unitarity and stability bounds
Before considering the phenomenological constraints, we consider unitarity and stability bounds for the Higgs sector. As derived in Appendix B, the conditions for perturbativity and unitarity are  directions of large Higgs doublet and singlet scalar fields. Following Refs. [14][15][16], we obtain the stability conditions as follows: The stability conditions along the other scalar fields Φ a can be obtained in the similar way, but they are not relevant for our study because Φ a 's do not couple directly to Higgs doublets as long as the extra quartic couplings for Φ a are positive and large enough.
The unitarity and stability bounds are depicted in Figs. 1 and 2 for the parameter space in terms of m h 2 and tan β, or v s and µ, with assuming the alignment limit, cos(α −β) = 0.05, and zero mixing between heavy CP -even scalars. In each figure, the gray region corresponds to the parameter space excluded by the unitarity and stability conditions. In Fig. 1, we have taken the different choices of Higgs masses: m h 2 = m A and m H ± = 500 GeV in the left, while m h 2 = m H ± and m A = 140 GeV in the right panel. On the other hand, the parameter space in terms of v s and µ has been shown in Fig. 2 but taking different values of tan β. We note that the unitarity and stability bounds are sensitive to the choice of tan β, while insensitive to the mixing angle of heavy CP -even scalars, in constraining the mass parameters. The allowed parameter space for mass parameters becomes narrower as tan β is smaller.

Higgs and electroweak precision data
Provided that the Higgs mixings with the singlet scalar are small, the mixing angle α between CP -even Higgs scalars are constrained by Higgs precision data [17][18][19][20][21]. The parameter space for sin α and tan β allowed by the Higgs data is shown in Fig. 3. We take the (33) component of the up-type Higgs Yukawa coupling to be y u 33 = y SM t in the left, and y u 33 = y SM / cos β in the right panel. For illustration, we have also imposed unitarity and stability bounds discussed in the previous subsection for m h 2 = m H ± = 450 GeV, m A = 140 GeV and v s = 1 TeV. As a result, we find a wide parameter space close to the line of the alignment, α = β − π/2, that is consistent with both the Higgs data and unitarity/stability bounds for tan β 0.1. Thus, henceforth, for the phenomenology of the extra Higgs scalars, we focus on the parameter space near the alignment limit, cos(α − β) ∼ 0.
To see bounds from electroweak precision data, we obtain effective Lagrangian after integrating out W and Z bosons as follows [22,23]: where J µ Z = J µ 3 −sin 2 θ * J µ EM with θ * being the modified Weinberg angle. Here the non-oblique terms, a and b, are determined at tree level as From the Z-boson like mass given in Eq. (D.6) and the Z-Z mixing angle in Eq. (D.7), we find the correction to the ρ parameter as where we assumed that tan 2ζ 2m 2 12 /m 2 Z 2 1. Taking the limit of zero gauge kinetic mixing, i.e. sin ξ = 0, we have which is consistent with the result in Ref. [8]. Therefore, for tan β 1, g Z 1, and x 0.05, Z with the mass m Z 400 GeV is consistent with electroweak precision data. The mass splittings between extra Higgs scalars can also be constrained by the electroweak precision data, but it can be easily satisfied if we take m h 2 = m H ± or m h 2 = m A , and a small mixing between CP -even scalars.

B-meson anomalies from Z
Before considering constraints from B-meson mixings and decays, we show how to explain the B-meson anomalies in our model and identify the relevant parameter space for that. This section is based on the detailed results on U (1) interactions presented in Appendix D and phenomenological findings in Ref. [8].
From the relevant Z interactions for B-meson anomalies and the Z mass term, we get the classical equation of motion for Z as Then, by integrating out the Z gauge boson, we obtain the effective four-fermion interaction forb →sµ + µ − as follows.
Consequently, as compared to the effective Hamiltonian with the SM normalization, with O µ 9 ≡ (sγ µ P L b)(μγ µ µ) and α em being the electromagnetic coupling, we obtain new physics contribution to the Wilson coefficient, with α Z ≡ g 2 Z /(4π), and vanishing contributions to other operators, C µ,NP 10 = C µ,NP 9 = C µ,NP 10 = 0. We note that xy > 0 is chosen for a negative sign of C µ 9 , being consistent with B-meson anomalies. Requiring the best-fit value, C µ, NP Therefore, m Z 1 TeV for xy 1 and α Z α em . For values of xy less than unity or α Z α em , Z can be even lighter.
Various phenomenological constraints on the Z interactions coming from dimuon resonance searches, other meson decays and mixing, tau lepton decays and neutrino scattering have been studied in Ref. [8], leading to the conclusion that the region of xg Z 0.05 for yg Z 1 and m Z 1 TeV is consistent with the parameter space for which the B-meson anomalies can be explained.

Bounds from B-meson mixings and decays
We now consider the bounds from B-meson mixings and decays. After integrating out the heavy Higgs bosons, the effective Lagrangian for B s(d) → µ + µ − from the flavor-violating Yukawa interactions in (3.18) is The extra contributions to the effective Hamiltonian for B s → µ + µ − are thus In the alignment limit with α = β − π/2 and m A m H , the Wilson coefficients become identical and suppressed for a small tan β. The effective Hamiltonian in the above leads to the corrections of the branching ratio for B s → µ + µ − as follows [25]: where m Bs , f Bs , and τ Bs are mass, decay constant, and lifetime of B s -meson, respectively. C , respectively. We note that there is no contribution from Z interactions to B s → µ + µ − since the muon couplings to Z are vectorlike. On the other hand, in the alignment limit the bounds obtained from B s,d → µ + µ − in Ref. [25]  This leads to tan β < 5.4 for m H,A = 500 GeV.
The flavor-violating Yukawa couplings of heavy Higgs bosons as well as Z interactions [8] can modify the B s -B s mixing. The additional effective Hamiltonian relevant for the mixing is given by (4.21) The mass difference in the B s system becomes where B s 123 (µ) is a combination of bag-parameters [26] and C SM V LL 4.95 [27]. The SM prediction and the experimental values of ∆M s are given by (∆M Bs ) SM = (17.4 ± 2.6) ps −1 [27] and (∆M Bs ) exp = (17.757 ± 0.021) ps −1 [28], respectively. Then, taking into account the SM uncertainties, we obtain the bounds on ∆M Bs as 16 (13) ps −1 < ∆M Bs < 21 (23) ps −1 or (∆M Bs ) BSM < 3.0 (5.6) ps −1 at 1σ (2σ) level for new physics. We also note that the most recent lattice calculations show considerably large values for the bag parameters, leading to (∆M Bs ) SM = (20.01 ± 1.25) ps −1 [29]. It needs an independent confirmation, but if it is true, the new physics contributions coming from the heavy Higgs bosons and Z would be constrained more tightly.
Taking the SM prediction as (∆M Bs ) SM = (17.4 ± 2.6) ps −1 [27], from Eq. (4.23) Here, since we need to choose xg Z 0.05 for m Z 1 TeV to satisfy the B-meson anomalies and the LHC dimuon bounds at the same time as discussed in the previous section, we can safely ignore the contribution of Z interactions to the B s -B s mixing on the right-hand side of Eq. (4.23). Furthermore, with the Z contribution ignored, the B d -B d mixing leads to a similar bound [26]: Comparing to the bounds from B s → µ + µ − in (4.17), the B-B mixings could lead to tighter constraints on the flavor-violating Yukawa couplings for down-type quarks unless m H and m A are almost degenerate. The upper frames of Fig. 4 show that a wide range of heavy Higgs masses up to 600-700 GeV are allowed for m h 2 = m A and tan β = O(1). On the other hand, for tan β = 0.5, the neutral Higgs boson can be as heavy as 400 GeV, but the charged Higgs mass is constrained as 240 GeV m H ± 650 GeV. For illustration, the case with m h 2 = m H ± has also been shown in the lower frames of Fig. 4, where the narrower region is allowed as compared with the case with m h 2 = m A .
Another important bound comes from the inclusive radiative decay, B → X s γ. The effective Hamiltonian relevant for the b → sγ transition is The charged Higgs contributions to the Wilson coefficients are given by [13,30] 7 (x t ), Here λ H − t L,R are given by Eqs.     whereas the experimentally measured value of B(B → X s γ) from HFAG is [28] B(B → X s γ) = (3.43 ± 0.21 ± 0.07) × 10 −4 . As a result, the SM prediction for B → X s γ is consistent with experiments, so we obtain the bounds on the modified Wilson coefficients as −0.032 < C BSM 7 (µ b ) < 0.027 at 2σ level [33]. This constrains tan β in terms of charged Higgs mass as shown in Fig. 5, where unitarity and stability bounds are displayed as well. We also find that the case with y u 33 = y SM t / cos β has been excluded by B → X s γ, hence the case with y u 33 = y SM t is considered in Figs. 4 and 5 and collider studies in the next section.

Predictions for R D and R D *
We briefly discuss the implications of flavor-violating couplings with charged Higgs on R D and R D * . The effective Hamiltonian relevant for B → D ( * ) τ ν in our model is given as follows: where the Wilson coefficient in the SM is C cb SM = 2V cb /v 2 , and the new Wilson coefficients generated by charged Higgs exchanges are The SM expectations are R D = 0.300 ± 0.008 and R D * = 0.252 ± 0.003 [34], but the experimental results for R D ( * ) are deviated from the SM values by more than 2σ [4][5][6][7]. Including the additional contributions from charged Higgs exchanges, we find the simplified forms for R D and R D * as follows [30,35]: (4.34) As can be seen in Fig. 6, a light charged Higgs is necessary to have large deviations of R D and R D * . However, it is excluded by B → X s γ. [See Fig. 5.] Therefore, our model cannot explain the experimental results for R D ( * ) simultaneously with the other bounds.

Productions and decays of heavy Higgs bosons at the LHC
We investigate the main production channels for heavy Higgs bosons at the LHC, including the contributions from flavor-violating interactions of quarks. The decay modes of the heavy Higgs bosons for some benchmark points are also studied, and we discuss smoking gun signals for heavy Higgs searches at the LHC. In this section, mixings with singlet scalar have been neglected and the heavy neutral Higgs boson H denotes h 2 . h ≡ h 1 is the SM-like Higgs with m h = 125 GeV.

Heavy neutral Higgs boson
The main channels for neutral Higgs productions are the gluon fusion gg → H, bottom-quark fusion bb → H, and additional productions through the flavor-violating interactions for the bottom quark, bd i → H and d ib → H, where d i denotes light down-type quarks, d i = d, s.
There are bottom quark associated productions, bg → bH and d i g → bH, as well.
The leading-order cross section for the gluon fusion process at parton level iŝ where τ q = m 2 H /(4m 2 q ). The loop function A H 1/2 (τ ) is given in Ref. [36].ŝ is the partonic center-of-mass energy. Here the contributions of only top and bottom quarks have been taken into account. Note that the top quark contribution is vanishing if one takes y u 33 = y SM t and the alignment limit as can be seen in Eq. (3.36). The parton-level cross section for bottom-quark fusion bb → H iŝ There are other single Higgs production channels through the flavor-violating interactions, bd i → H and d ib → H. The corresponding cross section is given bŷ The bottom quark associated production of the Higgs boson can occur by initial states with a bottom quark, that is, bg → bH, through the flavor-conserving interactions or initial states with a light down-type quark, d i g → bH, via the flavor-violating interactions. The former is nonvanishing even if all the components ofh d are zero. The diagrams of the bottom quark associated production are shown in Fig. 7. The differential cross section for bg → bH at parton level is and λ H b is given in (3.21). For the d i g → bH process, it is with We perform the integration by using the Monte Carlo method to obtain the production cross sections at proton-proton collisions of 14 TeV and employ the NNPDF2.3 parton distribution function (PDF) set [37] via the LHAPDF 6 library [38]. The renormalization and factorization scales are set to m H , and m b = 4.7 GeV. The resulting production cross sections as a function of m H are shown in Fig. 8. In all frames we set cos(α − β) = 0.05, close to the alignment limit, and y u 33 = y SM t . A constant K-factor of 2.5 has been multiplied to the gluon fusion production cross section, while the leading-order expressions have been used for the other production channels.
In the alignment limit, the neutral Higgs coupling to the top quarks λ H t is vanishing as can be seen in Eq. (3.36). In this case, the single Higgs production through the gluon fusion process is suppressed compared to the SM case, though nonvanishing due to the bottom quarks in the loop. Still, the gluon fusion production convoluted with PDF is the most dominant channel for the single Higgs production and bb → H is the subdominant one for tan β O(0.1). On the other hand, for smaller tan β, the flavor-violating Higgs couplings to light quarks become larger and contributions from the initial states with the light down-type quarks d ib → H is subdominant, and become even the most dominant channel in the case of very small tan β = O(0.01). However, since we find that such scenarios with very small tan β have been excluded by bounds from the experimental results on Bmeson mixings and decays, particularly by B → X s γ as seen in the previous section, we have chosen tan β = 1 and 0.5 as benchmarks for this study. For m H = 200 GeV and tan β = 1 (0.5), σ pp→H 225.2 (110.5) fb, and (σ bd i →H + σ d ib →H )/σ gg→H = 0.62% (1.6%), while (σ bd i →H + σ d ib →H )/σ bb→H 1.6% (10.9%) at the LHC. As the neutral Higgs gets heavier, the production cross sections rapidly decreases. For m H = 400 GeV and tan β = 1 (0.5), σ pp→H 38.4 (31.7) fb.
As can be seen in Fig. 8, the production cross section of the bottom quark associated We now turn to the decay widths of the neutral Higgs bosons and obtain their branching ratios. Ignoring the mixing among the SM-like Higgs and singlet scalar, the partial decay widths to quarks are where q = t, b, c. λ H b and λ H t are given in (3.21) and (3.22), and On the other hand, the Higgs interactions to the charged leptons are flavor-conserving and the corresponding decay width is given as The partial widths to electroweak gauge bosons V = W , Z are given as where δ W = 2 and δ Z = 1. These partial widths are vanishing in the alignment limit. If m H > 2m Z , the decay mode of H → Z Z opens. Ignoring the small mixing with the Z boson, the decay width is However, we find that this decay mode is almost negligible for small g Z x O(0.05) and m Z 400 GeV, which would be necessary to evade constraints from the Z searches at the LHC.
The neutral Higgs boson can also decay into γγ and gg through fermion or gauge boson loops. At leading order, the decay widths are given as where Q q is the electric charge of the quark and τ i ≡ m 2 H /(4m 2 i ). The loop functions A H

1/2
and A H 1 can be found in Ref. [36]. If m H > 2m h , the heavy neutral Higgs can decay into a pair of SM-like Higgs bosons. 2 The triple interaction comes from the scalar potential in (2.3), Hhh, (5.12) where g Hhh = 3(λ 1 sin α cos β + λ 2 cos α sin β) sin(2α) The decay width for the H → hh process is given as By combining all the decay widths, we obtain the branching ratio of each decay mode. Fig. 9 shows the branching ratios of the neutral Higgs boson H for cos(α − β) = 0.05 and v s = 1 TeV, but with different values of µ to satisfy the unitary and stability bounds studied in Subsec. 4.1. We observe that H → bd i /d ib is the predominant decay mode if m H < 2m h , whereas the di-Higgs mode H → hh becomes the most important if the mode is kinematically allowed, irrespective of tan β. In practice, the branching ratio of di-Higgs mode B(H → hh) depends on the choice of µv s value. If we take a smaller µv s value, for instance, µ = 200 GeV and v s = 500 GeV with tan β = 1, we find that H → bd i /d ib is always the most dominant decay mode. The dip near m H = 580 GeV in the left panel of Fig. 9 is due to the accidental cancellation in the Higgs triple coupling (5.13). The position of dip also depends on the value of µv s for given tan β and cos(α − β). On the other hand, the bb mode and diboson modes such as W W/ZZ are subdominant.
From these observations, we expect that the search strategies would be different depending on the mass of the heavy Higgs boson. For m H < 2m h , pp → H → bd i /d ib , i.e., dijet final states containing one b jet is the most important, but for m H > 2m h , the di-Higgs channel, and possibly in conjunction with the dijet channel with one b jet, is important to search the heavy neutral Higgs boson at the LHC. Thus, the neutral Higgs boson with m H < 250 GeV can receive constraints from dijet searches [40]. Although the dijet channel has typically been used to seek for heavy resonances in a few TeV scales, it can probe lower scales if it is associated with a hard photon or jet from initial state radiations. The ATLAS collaboration has searched light resonance with dijet invariant mass down to 200 GeV in the final states of dijet in association with a photon [41]. In our case, gluon fusion production is the most dominant channel and it is not associated with a hard photon. It can have a hard jet from the gluons in the initial states, but the mass region below 250 GeV has not been searched yet in the final states of dijet in association with a hard jet. For m H > 250 GeV, bounds from di-Higgs searches can be imposed, but we find that they do not have enough sensitivities for heavy neutral Higgs bosons in our model yet [42].

Heavy charged Higgs boson
One of the conventional search channels for the heavy charged Higgs with m H ± > m t at hadron colliders is the top quark associated production, bg → tH − , by the similar diagrams as bg → bH. Since the charged Higgs boson can have enhanced couplings with the light uptype quarks due to nonzero components ofh d , we can also have a sizable production cross section of the bottom quark associated process from the initial states with light up-type quarks, u i g → bH + where u i = u, c. 3 The differential cross section for bg → tH − at parton level is where Since the diagrams contributing to bottom quark associated processes has the same Lorentz structure as those for bg → tH − , we can obtain their parton-level cross sections by replacing λ H − t L,R with λ H − u iL,R , m b with m u i 0, and m t with m b . They are given as quite sensitive to tan β. For tan β = 1, the top quark associated production, pp → H ± t, is the dominant channel, while the bottom quark associated production, pp → H ± b, which is the characteristic channel of our model, can also be served as a good channel to search the charged Higgs boson at the LHC. On the other hand, for smaller tan β, the bottom quark associated production becomes the dominant channel due to the enhanced charged-Higgs couplings with light up-type quarks. The suppression of top quark associated production is also due to the partial cancellation of two terms in λ H − t L . Concerning the decays of charged Higgs, the most important fermionic decay mode is H + → tb. The decay width is By replacing m t with m c or m u and λ H − t L,R with λ H − c L,R or λ H − u L,R , one can obtain the decay widths of H + → cb and H + → ub. The other fermionic decay modes are H + → cs and cd, whose decay widths are proportional to tan 2 β|V cs | 2 and tan 2 β|V cd | 2 , respectively. The decay widths of leptonic decay modes are given as Meanwhile, if H + → W + A and W + H are kinematically forbidden, the only non-fermionic decay mode is H + → W + h. The decay width is By combining all the decay modes in the above we obtain the branching ratios of the heavy charged Higgs, which are shown in Fig. 11. Interestingly, the dominant decay mode of the charged Higgs boson is H + → W + h if it is kinematically allowed, although we have taken the alignment limit. H + → tb is subdominant. Together with the production, we expect that pp → H ± b → W ± h + b can be served as the important process to probe the charged Higgs boson at the LHC and future hadron colliders. Most LHC searches for W + h have been dedicated to heavy resonances [43] that decay directly into W + h, so our model is not constrained by W + h at the moment. On the other hand, the tb mode is next-todominant and this is not constrained by the current LHC data [44], because the production cross section for the heavy charged Higgs in our model is less than 10 fb in most of the parameter space.

Conclusions
We have considered an extra local U (1) with flavor-dependent couplings as a linear combination of B 3 − L 3 and L µ − L τ , that has been recently proposed to explain the B-meson anomalies. In our model, we have reproduced the correct flavor structure of the quark sector due to the VEV of the second Higgs doublet, at the expense of new flavor violating couplings for quarks and the violation of lepton universality.
The extra gauge boson leads to flavor violating interactions for down-type quarks appropriate for explaining B-meson anomalies in R K ( * ) whereas heavy Higgs bosons render up-type quarks have modified flavor-conserving Yukawa couplings and down-type quarks receive flavor-violating Yukawa couplings. We also found that the B-meson anomalies in R D ( * ) cannot be explained by the charged Higgs boson in our model, due to small flavor-violating couplings.
We showed how the extended Higgs sector can be constrained by unitarity and stability, Higgs and electroweak precision data, B-meson decays/mixings. Taking the alignment limit of heavy Higgs bosons from Higgs precision data, we also investigated the production of heavy Higgs bosons at the LHC. We found that there are reductions in the cross sections of the usual production channels in 2HDM, such as pp → H and pp → H ± t at the LHC. In addition, new production channels such as pp → Hb and pp → H ± b become important for tan β 1. Decay products of heavy Higgs bosons lead to interesting collider signatures due to large branching fractions of bd + bs modes for neutral Higgs bosons and W ± h mode for charged Higgs boson if kinematically allowed, thus requiring a more dedicated analysis for the LHC. the mass matrix for CP -even scalars can be written as We introduce a rotation matrix R to change the interaction basis (ρ 1 , ρ 2 , S R ) to the physical mass eigenstates, h 1 , h 2 and h 3 as The mass matrix M S can be then diagonalized as We use a convention such that the mass eigenstates are ordered as m h 1 < m h 2 < m h 3 . Here, the orthogonal matrix R is parametrized in terms of the mixing angles α 1 to α 3 as where s α i ≡ sin α i and c α i ≡ cos α i . Without loss of generality the angles can be chosen in the range of − π 2 ≤ α 1,2,3 < π 2 .
In the text we focus mainly on the situation where mixings between ρ 1,2 and S R are small.
The mass eigenvalues of CP -even neutral scalars are given by We can trade off quartic couplings, λ 1,2,3,4 and κ 1,2 , for mixing angles and Higgs masses.
In the case when the Higgs mixings with the singlet scalar are negligible, α 2 α 3 0, the rotation matrix can be simplified as where α = α 1 . Then the Higgs quartic couplings are given by .
For U L = 1, we haveh u = h u U R . As a result, where use is made of Eqs. (3.14), (3.16), and (3.17). Other components ofh u are vanishing. Moreover, withh d = V † CKM h d and using Eq. (3.9) for h d 13 and h d 23 , we obtain nonzero components ofh u as where ζ is the mass mixing angle and s W ≡ sin θ W , c W ≡ cos θ W , etc, we obtain the mass eigenvalues for massive gauge bosons: Here m 2 Z ≡ (g 2 + g 2 Y )v 2 /4 and We can rewrite the Z-boson like mass in terms of the heavy Z mass and the mixing angle ζ as , (D. 6) and the mixing angle as tan 2ζ = 2m 2 12 (m 2 (D.7) We note that the modified Z-boson mass is constrained by electroweak precision data, in particular, ∆ρ or T parameter.
The current interactions including Z are given by Here Q f is the electric charge and Q f is the U (1) charge of fermion f . For a small gauge kinetic mixing and/or the mass mixing ζ, the Z -like gauge boson Z 2µ couples to the electromagnetic current with the overall coefficient of ε = t ξ c ζ c W .
Ignoring the Z-Z mixing, the interaction terms for Z interactions is Now we change the basis into the one with mass eigenstates by d R = D R d R , u R = U R u R , d L = D L d L and u L = U L u L such that V CKM = U † L D L . Taking D R = U L = 1 and D L = V CKM , the above Z interactions become L Z = g Z Z µ 1 3 xt γ µ P L t + 1 3 x v 2 cos 2 β|y u 33 | 2 2m 2 tt γ µ P R t + 1 3 xd i γ µ Γ dL ij P L d j + 1 3 xb γ µ P R b + yμγ µ µ − (x + y)τ γ µ τ + yν µ γ µ P L ν µ − (x + y)ν τ γ µ P L ν τ + yν 2R γ µ P R ν 2R − (x + y)ν 3R γ µ P R ν 3R , Considering the general mixing of CP -even scalars while ignoring the Z-Z mixing, we obtain the interaction between neutral massive electroweak gauge bosons (V = W , Z) and Z as For a negligible mixing with the singlet scalar, the above couplings become One can see that in the alignment limit with α = β − π/2 the gauge interactions of h are the same as for the SM Higgs while the triple couplings of heavy Higgs boson to gauge bosons vanish.