Charged Higgs boson contribution to $B^-_{q} \to \ell \bar \nu$ and $\bar B\to (P, V) \ell \bar\nu$ in a generic two-Higgs doublet model

We comprehensively study the charged-Higgs contributions to the leptonic $B^-_q \to \ell \bar \nu$ ($q=u,c$) and semileptonic $\bar B \to X_q \ell \bar\nu$ ($X_u=\pi, \rho; X_c=D,D^*$) decays in the type-III two-Higgs-doublet model (2HDM). We employ the Cheng-Sher ansatz to suppress the tree-level flavor-changing neutral currents (FCNCs) in the quark sector. When the strict constraints from the $\Delta B=2$ and $b\to s \gamma$ processes are considered, parameters $\chi^u_{tq}$ from the quark couplings and $\chi^\ell_\ell$ from the lepton couplings dictate the leptonic and semileptonic $B$ decays. It is found that when the measured $B^-_u\to \tau \bar \nu$ and indirect bound of $B^-_c \to \tau \bar \nu$ obtained by LEP1 data are taken into account, $R(D)$ and $R(\pi)$ can have broadly allowed ranges; however, the values of $R(\rho)$ and $R(D^*)$ are limited to approximately the standard model (SM) results. We also find that the same behaviors also occur in the $\tau$-lepton polarizations and forward-backward asymmetries ($A^{X_q,\tau}_{FB}$) of the semileptonic decays, with the exception of $A^{D^*,\tau}_{FB}$, for which the deviation from the SM due to the charged-Higgs effect is still sizable. In addition, the $q^2$-dependent $A^{\pi,\tau}_{FB}$ and $A^{D,\tau}_{FB}$ can be very sensitive to the charged-Higgs effects and have completely different shapes from the SM.


I. INTRODUCTION
In spite of the success of the standard model (SM) in particle physics, we are still uncertain as to the solutions for baryongenesis, neutrino mass, and dark matter. It is believed that the SM is an effective theory at the electroweak scale, and thus there should be plenty of room to explore the new physics effects in theoretical and experimental high energy physics.
A known extension of the SM is the two-Higgs-doublet model (2HDM), where the model can be used to resolve weak and strong CP problems [1,2]. Due to the involvement of new scalars, such as one CP-even, one CP-odd, and two charged Higgses, despite its original motivation, the 2HDM provides rich phenomena in particle physics [3][4][5][6], especially, the charged-Higgs, which causes lots of interesting effects in flavor physics. According to the imposed symmetry (e.g., soft Z 2 symmetry) to the Lagrangian in the literature, the 2HDM is classified as type-I, type-II, lepton-specific, and flipped models, for which detailed introduction can be found in [7]. Among these 2HDM schemes, only the type-II model corresponds to the tree-level minimal supersymmetric standard model (MSSM) case.
Intriguingly, when the R(D) − R(D * ) correlation is taken into account, the deviation with respect to the SM prediction is 4.1σ. Based on these observations, possible extensions of the SM for explaining the excesses are studied in .
Moreover, when |V ub | ≈ 3.72×10 −3 is taken from the results of lattice QCD [22] and lightcone sum rules (LCSRs) [67,68], the SM result of BR(B − u → τν) SM ≈ 0.89 × 10 −4 is slightly smaller than the current measurement of BR(B − u → τν) exp = (1.09 ± 0.24) × 10 −4 [69]. In addition to the uncertainties of V ub and B-meson decay constant f B , the difference between the SM prediction and experimental data may raise from new charged current effects [71][72][73][74]. Since theB → D ( * ) τν and B − u → τν processes are associated with the W ± -mediated b → (u, c)τν decays in the SM, in this work, we study the charged-Higgs contributions to the decays in detail in the 2HDM framework.
The charged-Higgs can be naturally taken as the origin of a lepton-flavor universality violation because its Yukawa coupling to a lepton is usually proportional to the lepton mass. Due to the suppression of m ℓ /v (v ≈ 246 GeV), we thus need an extra factor in the coupling to enhance the charged-Higgs effect. In the 2HDM schemes mentioned above, it can be easily found that only the type-II model can have a tan 2 β enhancement in the Hamiltonian of b → (u, c)τν. However, the type-II 2HDM cannot resolve the excesses for the following reasons: (i) the sign of type-II contribution is always destructive to the SM contributions in b → (u, c)τν, and (ii) the lower bound of the charged-Higgs mass limited by b → sγ is now m H ± > 580 GeV [75], so that the change due to the charged-Higgs effect is only at a percentage level. Inevitably, we have to consider other schemes in the 2HDM that can retain the tan β enhancement, can be a constructive contribution to the SM, and can have a smaller m H ± .
The desired scheme can be achieved when the imposed symmetry is removed; that is, the two Higgs doublets can simultaneously couple to the up-and down-type quarks. This scheme is called the type-III 2HDM in the literature [5,24,29]. In such a scheme, unless an extra assumption is made [76], the flavor changing neutral currents (FCNCs) are generally induced at the tree level. In order to naturally suppress the tree-induced ∆F = 2 (F = K, B d(s) , D) processes, we can adopt the Cheng-Sher ansatz [77], where the FCNC effects are parametrized to be the square-root of the production involving flavor masses. We find that the same quark FCNC effects also appear in the charged-Higgs couplings to the quarks.
Using the Cheng-Sher ansatz, it is found that in addition to the achievements of the tan β enhancement factor and a smaller m H ± , new unsuppressed factors denoted by χ u tc(tu) occur at the vertices c(u)bH ± , which play an important role in B − u → τν and R(D ( * ) ). We note that the type-II 2HDM and MSSM can generate the similar Yukawa couplings of the type-III model through the Z 2 soft-breaking term, which is from the Higgs potential, when loop effects are considered. Due to loop suppression factor, the loop-induced effects from type-II 2HDM in our study are small. Although the loop effects in supersymmetric (SUSY) models could be sizable, since we focus on the non-SUSY models, the implications of loop-induced FCNCs in MSSM can be found in [78][79][80][81].
With the full Υ(4S) data set, Belle recently reported the measurement of B − u → µν with a 2.4σ significance, where the corresponding BR is BR(B − u → µν) exp = (6.46±2.22±1.60)× 10 −7 , and the SM result is BR(B − u → µν) SM = (3.8 ± 0.31) × 10 −7 [82]. The experimental measurement approaches the SM prediction, and it is expected that the improved measurement soon will be obtained at Belle II [83]. In other words, in addition to the B − u → τν channel, we can investigate the new charged current effect through a precise measurement on the B − u → µν decay. In order to comprehensively understand the charged-Higgs contributions to the b → (u, c)ℓν (ℓ = e, µ, τ ) in the type-III 2HDM, in addition to the chiral suppression channels B − u → (τ, µ)ν, we study various possible observables for the semileptonic processesB → (P, V )ℓν (P = π, D; V = ρ, D * ), which include BRs, R(P ), R(V ), lepton helicity asymmetry, and lepton forward-backward asymmetry. To constrain the free parameters, we not only study the constraints from the tree-and loop-induced ∆B = 2 processes, but also the b → sγ decay, which has arisen from the new neutral scalars and charged-Higgs. Although the neutral current contributions to b → sγ are much smaller than those from the charged-Higgs, for completeness, we also formulate their contributions in the paper. In addition, the upper bound of BR(B − c → τν) < 10% obtained in [66] is also taken into account when we investigate theB → D * τν decay. II.

YUKAWA COUPLINGS IN THE GENERIC 2HDM
To study the charged-Higgs contributions to the b → qℓν (q = u, c) decays in the type-III 2HDM, we analyze the relevant Yukawa couplings in this section, especially, the charged-Higgs couplings to ub and cb, where they can make significant contributions to the leptonic and semileptonic B decays. The characteristics of new Yukawa couplings in the type-III model will be also discussed.

A. Formulation of H ± Yukawa couplings to the quarks and leptons
Since the charged-Higgs couplings to the quarks and the leptons in type-III 2HDM were derived before [5], we briefly introduce the relevant pieces in this section. We begin to write the Yukawa couplings in the type-III model as: where the flavor indices are suppressed; Q T L = (u, d) L and L T = (ν, ℓ) L are the SU(2) L quark and lepton doublets, respectively; f R (f = U, D, ℓ) is the singlet fermion; Y f 1,2 are the 3 × 3 Yukawa matrices, andH i = iτ 2 H * i with τ 2 being the Pauli matrix. The components of the Higgs doublets are taken as: and v i is the vacuum expectation value (VEV) of H i . We note that Eq. (2) can recover the type II 2HDM when Y u 1 , Y d 2 , and Y ℓ 2 vanish. The physical states for scalars can then be expressed as: where the mixing angles are defined as c α (s α ) = cos α(sin α), c β = cos β = v 1 /v, and In this work, h is the SM-like Higgs while H, A, and H ± are new scalar bosons.
The fermion mass matrix can be formulated as: Without assuming the relation between Y f 1 and Y f 2 , both Yukawa matrices cannot be simultaneously diagonalized [76]. Thus, the FCNCs mediated by scalar bosons are induced at the tree level. We introduce unitary matrices U f L and U f R to diagonalize the fermion mass matrices by following L,R denote the physical (weak) eigenstates. Then, the Yukawa couplings of H ± can be written as [5]: where denotes the Cabibbo-Kobayashi-Maskawa (CKM) matrix, and the Xs are defined as: X u,d are the sources of tree-level FCNCs in the type-III model. In order to accommodate the strict constraints from the ∆F = 2 processes, such as ∆m P (P = K, B d,s , D), we adopt the so-called Cheng-Sher ansatz [77] in the quark and lepton sectors, where X f is parametrized as: and χ f ij are the new free parameters. Using this ansatz, it can be seen that ∆m P arisen from the tree level is suppressed by m d m s /v 2 for K-meson, m d(s) m b /v 2 for B d(s) , and m u m c /v 2 for D-meson. Since we do not study the origin of neutrino mass, the neutrinos are taken as massless particles in this work. Nevertheless, even with a massive neutrino case, the influence on hadronic processes is small and negligible. In addition, to simplify the numerical analysis, in this work we use the scheme with as a result, the Yukawa couplings of H ± to the leptons can be expressed as: with P R(L) = (1 ± γ 5 )/2. The suppression factor m ℓ /v could be moderated using the scheme of large tan β.

B.
b-quark Yukawa couplings to H ± From Eq. (6), it can be seen that the coupling u iR b L H ± (u i = u, c) in the type-II 2HDM (i.e. X d,u = 0) is suppressed by m u i /(vt β )V u i b , and this effect can be neglected. However, the situation is changed in the type-III model. In addition to the disappearance of suppression factor 1/t β , the new effect X u accompanied with the CKM matrix in form of numerically plays the role of |V u i b |, and the magnitude of the coupling is dictated by the free parameter χ u u i t , which in principle is not suppressed. Additionally, the u iL b R H ± coupling is also remarkably modified. In order to more comprehend the influence of the new charged-Higgs couplings on the B decays, in the rest of this subsection, we discuss the u i bH ± coupling in detail. For convenience, we rewrite the H ± couplings to the b-quark and light up-type quarks as: where u j (d j ) indicates the sum of all possible up(down)-type quarks.
In the following, we analyze the characteristics of the C L u(c)b and C R u(c)b couplings in the type-III 2HDM with the Cheng-Sher ansatz. Due to can simplify the C L ub coupling as: With m u ∼ 5.4 MeV, m t ∼ 165 GeV, and v ≈ 246 GeV, it can be found that √ m u m t /v ≈ 3.84 × 10 −3 is very close to the value of |V ub |; therefore, C L ub can be read as Clearly, unlike the case in the type-II 2HDM, which is highly suppressed by m u /(vt β ), C L ub in the type-III model is still proportional to |V ub |, can be sizable, and is controlled by χ u * tu . For the C R ub coupling, the decomposition from Eq. (10) can be written as: The numerical values of the first two terms can be obtained as: V ud m d /m b ≈ 0.047 and V us m s /m b ≈ 0.032 ≫ |V ub |. Unless χ d db,sb are strictly constrained, each term with different CKM factors may be important and cannot be arbitrarily dropped. For clarity, we rewrite C R ub to be: Due to |V ub | ≪ V us,ud , the magnitude of χ R ub in principle can be of O(10), and the resulted C R ub is much larger than that in the type-II 2HDM. In order to avoid obtaining an C R ub that is too large, we can require a cancellation between V ud m d /m b χ d db and V us m s /m b χ d sb when χ d db,sb both are sizable. However, we will show that χ d db,sb indeed are constrained by the measured B d,s mixing parameters and that their magnitudes should be less than O(10 −2 ).

For the processes dictated by the
the H ± Yukawa coupling of C L cb can be simplified as: where m c /t β term has been ignored due to the use of large t β scheme, and the factor in parentheses can be numerically estimated to be 2.19χ u * tc . This behavior is similar to C L ub , but it is χ u * tc that controls the magnitude. Clearly, if χ u * tc is not suppressed, it can make a signifiant contribution to the b → c transition. Using the fact that we can formulate the C R cb coupling as: Since C R cb has the t β enhancement, its magnitude is comparable with the SM W -gauge coupling of gV cb / √ 2. For comparison, we also show the tbH ± couplings as: where the small effects related to V ub,cb and V ts,td have been dropped. Although there is a m t enhancement in the first term of C L tb , 1/t β will reduce its contribution when a large tan β value is taken; therefore comparing with χ u * tt /s β , this term can be ignored, i.e., C L tb ≈ −m t V tb χ u * tt /s β . From the above analysis, it can be seen that C L,R ub,cb,tb are different from those in the type-II model not only in magnitude but also in sign. For completeness, the other Yukawa couplings of H ± to the quarks are shown in detail in the Appendix.

III. PHENOMENOLOGICAL ANALYSIS
The charged current interactions in this model arise from the SM W -gauge and the charged-Higgs bosons. Based on the Yukawa couplings in Eqs. (9) and (10), the effective Hamiltonian for b → qℓν can be written as: where the fermionic currents are defined as , and the dimensionless coefficients for the b → u and b → c decays are given as: Based on the interactions shown in Eqs. (19) and (20), we investigate the charged-Higgs influence on the leptonic and semileptonic B decays in the type-III 2HDM.
The hadronic effect in a leptonic B decay is the B-meson decay constant. The decay constant associated with an axial-vector current for the B q -meson is defined as: Using the equation of motion, the decay constant associated with pseudoscalar current is given by: From the effective interactions in Eq. (19), the decay rate for B − q → ℓν can be formed as: Since a leptonic meson decay is a chirality-suppressed process, the decay rate in Eq. (24) is proportional to m 2 ℓ . From Eq. (20a) to Eq. (20d), it can be seen that in the type-II 2HDM, C L ub ∼ C L cb ∼ 0 and C R ub,cb are negative in sign; therefore, the H ± contribution to the B − q → ℓν decay is always destructive. The magnitude and the sign of C R,L qb in the type-III can be changed due to the new effects of χ u,d ij and χ ℓ ℓ ,. Before doing a detailed numerical analysis, we can numerically understand the impact of 2HDM on the B − q → ℓν decay as follows: taking t β = 50 and m H ± = 300 GeV, we can see that the charged-Higgs contributions to the b → u and b → c decays are respectively given as: where the sign can be positive when the parameters of χ u * tu,tc and χ ℓ ℓ are properly taken, and φ 3 is the phase in V ub . We note that the Yukawa coupling of the charged-Higgs to lepton is proportional to the lepton mass; therefore, the ratio in Eq. (25) does not depend on m ℓ .
The lepton-flavor dependent effect is dictated by the χ ℓ ℓ parameter.
Since the semileptonic B decays involve the hadronic QCD effects, in order to formulate the decays, we parametrize the form factors for a B decay to a pseudoscalar (P) meson as: where P = p 1 + p 2 and q = p 1 − p 2 . The form factors for a B decay to a vector (V) meson is defined as: With the equation of motion, the form factors of f BP S and f BV P can be obtained as: Using the interactions in Eq. (19) and the form factors defined above, we can obtain the transition matrix elements forB → (P, V )ℓν as: where q 2 -dependence in the form factors are hidden, and M L V and M T V are the longitudinal and transverse V -meson components, respectively. From the formulations, we see that the charged Higgs only affects M P and the longitudinal part of the V -meson.

Decay amplitudes in helicity basis
To derive the angular differential decay rate, we take the coordinates of the kinematic variables in the rest frame of the ℓν invariant mass as: where M denotes P -and V -meson; θ ℓ is the polar angle of a neutrino with respect to the moving direction of M meson in the q 2 rest frame, and the components of p ℓ can be obtained from p ν by using π − θ ℓ and φ + π instead of θ ℓ and φ.
The solutions of the Dirac equation for positive and negative energy can be expressed as: where the ± indices in χ are the eigenvalues of σ · p/| p|, and +(−) denotes the left(right)-handed state. If the spatial momentum of a particle is taken as p = p(sin θ cos φ, sin θ sin φ, cos θ), the eigenstates of σ · p can be found as: With the Pauli-Dirac representation of γ-matrices, which are defined as: , γ µ , σ µν }, and ℓ u ± denote the charged-lepton in u ± states. Since we take neutrinos as massless particles, the neutrino states are always left-handed, i.e.,l u ± [.
With the chosen coordinates and the spinors in Eqs. (32) and (33), the leptonic current in lepton helicity basis for theB → P ℓν decay can be derived as: where β ℓ = 1 − m 2 ℓ /q 2 , and the auxiliary polarization vector e X is defined as: In order to include the V -meson polarizations in theB → V ℓν decay, we separate a lepton current in the lepton helicity basis into longitudinal and transverse parts, where the longitudinal part of the V -meson is given as: while the two transverse parts of the V -meson are respectively given as: The auxiliary polarizations e Z and e V (T ) are defined as: Using the helicity basis and the lepton currents discussed before, theB → P ℓν decay amplitudes with the charged-lepton positive and negative helicity are respectively obtained as: As mentioned earlier, since the V -meson carries spin degrees of freedom, we separate each lepton helicity amplitude into longitudinal (L) and transverse (T) parts to show the Vmeson polarization effects. Therefore, we write the helicity amplitudes ofB → V ℓν for the longitudinal polarization of the V -meson as: It can be seen that the formulae for M L,h=± V are similar to those for M h=± P . The helicity amplitudes for the transverse polarizations of V -meson can be derived as: Since the charged-Higgs only affects the longitudinal part, M T =±,h=± V are dictated by the SM. From these obtained helicity amplitudes, it can be seen that due to angular-momentum conservation, M h=+ P and M L(T ),h=+ V , which come fromlγ µ (1 − γ 5 )ν, are chirality-suppressed and proportional to m ℓ . However, the charged lepton inl(1 − γ 5 )ν, which arises from the charged-Higgs interaction, prefers the h = + state, and the associated contribution in principle exhibits no chiral suppression factor. Nevertheless, the m ℓ factor indeed exists in our case due to the Cheng-Sher ansatz.
2. Angular differential decay rate, lepton helicity asymmetry, and forward-backward asymmetry When the three-body phase space is included, the differential decay rates with lepton helicity and V polarization as a function of q 2 and cos θ ℓ can be obtained as: Using Eq. (48), we can investigate various interesting physical quantities, such as BR, leptonhelicity asymmetry, lepton forward-backward asymmetry (FBA), and polarization distributions of V -meson. We thus introduce these observables in the following discussions.
When the polar angle is integrated out, the differential decay rate with each lepton helicity as a function of q 2 can be obtained as follows: For theB → P ℓν decay, they can be expressed as: and for theB → V ℓν decay, they are shown as: Accordingly, the partial decay rates forB → (P, V )ℓν can be directly obtained as: Moreover, the q 2 -dependent longitudinal polarization and transverse polarization fractions can be defined as: Based on Eqs. (49) and (50), we define the q 2 -dependent lepton helicity asymmetry as: where the sum of V polarizations is indicated in dΓ h=± V ℓ . Thus, the results for the pseudoscalar and vector meson processes can be respectively formulated as: In addition, using the helicity decay rates, the q 2 -independent lepton helicity asymmetry can be defined as [25,28,54,92,93]: where the formulations forB → (P, V )ℓν with charged Higgs effects can be found as: From the angular differential decay rates shown in Eq. (48), the lepton FBA can be defined as: where z = cos θ ℓ and dΓ M ℓ /(dq 2 dz) have included all possible lepton helicities and polarizations of the V -meson. The FBAs mediated by the charged Higgs and W -boson in B → (P, V )ℓν ℓ are obtained as: From the above equations, it can be seen that A P,ℓ F B and the longitudinal part of A V,ℓ F B depend on m ℓ and are chiral suppressed. Since m τ /m b ∼ 0.4 is not highly suppressed, it can be expected thatB → P τν can have a sizable FBA. A V,ℓ F B does not vanish in the chiral limit; therefore, it can be sizable for a light lepton.
The observations of the tau polarization and FBA rely on tau-lepton reconstruction. Due to the involvement of one invisible neutrino in the final state, it is experimentally challenging to measure these observables. As an alternative to the τ reconstruction, the extraction of τ polarization and FBA through an angular asymmetry of visible particles in a tau decay was recently proposed in [94,95], where the τ → πν τ decay is the most sensitive channel. Using this approach, a statistical precision of 10% can be reached at Belle II with an integrated luminosity of 50 ab −1 . The detailed study can be found in [95].
It is known that tree-level FCNCs can occur in the generic 2HDM; therefore, the measured mass difference ∆M q ′ (q ′ = d, s) of neutral B q ′ -meson will give a strict limit on the parameters X d q ′ b,bq ′ . In our approach, due to the Cheng-Sher ansatz, the ∆B = 2 process, mediated by the neutral scalars at the tree level, is proportional to Although the treelevel effect has a suppression factor m q ′ /v, the factor t 2 β can largely enhance its contribution; hence, ∆M q ′ will severely bound the χ d q ′ b,bq ′ parameters. In addition to the tree-level effects, we find through box diagrams that the charged-Higgs contributions to ∆B = 2 can be significant when t β is large, and χ u tt,ct and χ d bb are of O(0.1)-O(1). The same charged-Higgs effects also contribute to the radiative b → s(d)γ decay via penguin diagrams. Since b → sγ is measured well in experiments, in this study, we only focus on the b → sγ decay. It is of interest to investigate whether the sizable new parameters χ u tt,ct and χ d bb in the generic 2HDM can accommodate the ∆M q ′ and b → sγ data. Hence, in this section, we formulate the contributions of charged-Higgs and neutral Higgses to the B d,s -B d,s mixings and b → sγ process.

A.
Charged-Higgs contributions to the ∆M q ′ We first consider the charged-Higgs contributions to the ∆B = 2 processes, where the typical Feynman diagrams mediated by W + -H + , G + -H + , and H + -H + are sketched in Fig. 1, and G + is the charged Goldstone boson. Since the Yukawa couplings of H ± to the quarks are associated with the quark masses, the vertices that involve heavy quarks can enhance the loop H ± effects. Thus, we only consider the top-quark loop contributions in the B-meson system. Accordingly, the relevant charged-Higgs interactions are shown as: where the parameters ζ f ij are defined as: Detailed discussions for the couplings of tq ′ H ± can be found in the Appendix. From Eqs. (62) and (63), when χ f ij = 0, the vertices in the type-II 2HDM are reproduced. Unlike the type-II model, where ζ u tt,tq ′ ≪ 1 for t β ∼ m t /m b , ζ u tt,tq ′ in the type-III model can be of order unity even at small t β . We will show the impacts of these new 2HDM parameters on the flavor physics in the following analysis.
Based on the convention in [98], the effective Hamiltonian for B q ′ -B q ′ mixing can be written as: where the effective operators with the color indices α, β are given as: The operatorsÕ j can be obtained from O j using P R instead of P L . The Wilson coefficients at the scale µ = m b = 4.6 GeV can be related to those at µ H scale and are given as [98]: where can be found in [98]. To obtain C j (µ H ), we adopt the 't Hooft-Feynman gauge for the propagator of W -gauge boson; therefore, the charged Goldstone G ± boson effects have to be taken into account. To show the results of the box diagrams, we define some useful parameters as: Thus, the effective Wilson coefficients at µ H scale can be formulated as: where the loop integral functions are defined as: The effective Wilson coefficients for theÕ 1,2 operators at µ H scale are given as: We have checked that our results are the same as those obtained in [99] when y b = χ u,d ij = 0. Using Eq. (66) and the magic numbers shown in [98], we obtain the Wilson coefficients The matrix elements of the renormalized operators for ∆B = 2 are defined as [98]: where B iq ′ denote the nonperturbative QCD bag parameters, and the mixing matrix elements in the SM are related to B 1q ′ . Using the results obtained by HPQCD [100], FNAL-MILC [101], and RBC-UKQCD [102] collaborations, the lattice QCD results with N f = 2+1 averaged by the flavor lattice averaging group (FLAG) can be found as B 1d ≈ 0.80 and B 1s ≈ 0.84 [103]. In our numerical calculations, the quark masses and B iq ′ parameters at the m b scale in the Landau RI-MOM scheme [98,[104][105][106] and the decay constants of B q ′ are shown in Table I, where for self-consistency, all B iq values are quoted from [106]. Due eff |B q ′ can be written as: The SM result and the charged-Higgs contributions can be formulated as: , [107];η iB are the QCD corrections, and their values are shown in Table I. Accordingly, the mass difference between the physical B q ′ states can be obtained by: where the B iq results are quoted from [106]. The decay constants of the B d,s mesons are from [96], and f Bc is from [97].    [96], in which the next-to-leading order (NLO) QCD corrections [108][109][110] and the uncertainties from various parameters, such as CKM matrix elements, decay constants, and top-quark mass, are taken into account. Hence, from Eq. (74), the bounds from ∆B = 2 can be used as:

B. ∆M q ′ from the tree FCNCs
To formulate the scalar boson contributions to ∆M q ′ at the tree level, we write the Yukawa couplings of scalars H and A to the quarks with Cheng-Sher ansatz as [5]: The effective Hamiltonian for ∆B = 2 process mediated by the neutral scalar bosons H and A at µ H scale can then be straightforwardly obtained as: It can be seen that when m H = m A , the contributions from the operators Q 2 andQ 2 vanish.

We note that the box diagrams, mediated by Z-H(A), G 0 -H(A), and H(A)-H(A), involve
the q i -b-H(A) FCNC couplings, which are the same as the tree contributions. Thus, it is expected that the box contributions will be smaller than the tree; therefore, we do not further discuss such box diagrams and neglect their contributions.
Using Eq. (66) and the hadronic matrix elements shown in Eq. (71), the ∆M q ′ , which combines the SM and S = H + A effects, can be found as: where the H and A contributions are expressed as: x b(q ′ ) = m 2 b(q ′ ) /m 2 W , theη iB are the QCD factors as shown in Table I, and the factors C S 2 , C S 2 , and C S 4 are defined as: Since Eq. (79) is directly related to χ d bq ′ ,q ′ b , in order to show the ∆M q ′ constraint on the different parameters, here we do not combine the neutral scalar with the charged-Higgs contributions. According to Eq. (76), the bounds on ∆ S d,s can be given as: C. Charged-Higgs contributions to the b → sγ process In addition to the ∆B = 2 processes, the penguin induced b → sγ decay is also sensitive to new physics. The current experimental value is BR(B → X s γ) exp = (3.32 ± 0.15) × 10 −4 for E γ > 1.6 GeV [15], and the SM prediction with next-to-next-to-leading oder (NNLO) QCD corrections is BR(B → X s γ) SM = (3.36 ± 0.23) × 10 −4 [111,112]. Since the SM result is close to the experimental data, we can use theB → X s γ decay to give a strict bound on the new physics effects. The effective Hamiltonian arisen from the W ± and H ± bosons for b → sγ at µ H scale can be written as: where the electromagnetic and gluonic dipole operators are given as: and the Q ′ 7γ,8G operators can be obtained from the unprimed operator using P L instead of P R . We note that C ′ 7γ,8G from the SM contributions are suppressed by m s and are negligible; therefore, the main primed operators are from the new physics effects.
According to the charged-Higgs interactions in Eq. (62), the relevant Feynman diagrams for b → s(γ, g) are sketched in Fig. 2, and the H ± contributions to C H ± 7γ,8G at µ H scale can be derived as : where the loop integral functions are defined as: ,LL (y t ), and C ′H ± 7(8),LR (y t ) = −C H ± 7(8),RL (y t ). From Eq. (85), we can easily understand the effects of the type-II 2HDM as follows: taking χ u tt,ct = χ d bb,sb = 0 in Eq. (85), (ζ u * ts ζ u tt ) type−II is suppressed by 1/t 2 β , and (ζ u bb ζ u * ts ) type−II = 1 becomes t βindependence. As a result, the mass of charged-Higgs in type-II 2HDM is limited to be m H ± > 580 GeV at 95% confidence level (CL) when NNLO QCD corrections are taken into account [75]. In the generic 2HDM, since the new parameters χ u tt,ct /c β and χ d bb /s β are involved in Eq. (85), we have more degrees of freedom to reduce ζ u bb ζ u * ts away from unity; thus, the charged-Higgs mass can be lighter than 580 GeV.
To calculate the BR ofB → X s γ, we employ the results in [113,114], which are shown as: where N(E γ ) = (3.6 ± 0.6) × 10 −3 denotes a nonperturbative effect; C SM 7γ ≈ −0.364 at µ b ≈ 2.5 GeV. The NLO [115][116][117] and NNLO [118] QCD corrections to the C 7γ (µ b ) in the 2HDM have been calculated. In this study, the charged-Higgs effects with RG running are taken from [113,114], and they are written as: where κ 7,8 are the LO QCD effects, for which their values with different values of µ H can be found in [113,114].

D. H/A contributions to the b → sγ process
In addition to the charged currents, the b → sγ process can be generated through the FCNCs in the type-III 2HDM, where the corresponding Feynman diagrams for b → s(γ, g) are shown in Fig. 3. From the diagrams, it can be seen that unlike the m 2 t /m 2 H ± result from the H ± and top-quark loops, the b-quark loops are suppressed by m 2 b /m 2 H,A . Therefore, it is expected that the radiative b decay induced by the neutral currents will be much smaller than the charged currents.
Using the Yukawa couplings in Eq. (77), we can derive the Wilson coefficients of C 7γ and C ′ 7γ at the µ H scale, defined in Eq. (83), as: where the superscript S denotes the scalar contributions; Q b = −1/3 is the electric charge of b-quark, and the functions J 1,2 are defined as: The contributions of H and A bosons to the chromomagnetic dipole operators can be related to the electromagnetic dipole operators, and the relations can be easily found as C We can apply the result in Eq. (88) to get the Wilson coefficients at µ b scale as: Using Eq. (87), we can directly obtain the S-mediated BR(B → X s γ).

A. Numerical and theoretical inputs
In addition to the parameter values shown in Table I, the values of the CKM matrix elements used in the following analysis are taken as [15]: To study the semileptonicB → (P, V )ℓν decays, we need the information for theB → (P, V ) transition form factors. For theB → π decay, we use the results obtained by the LCSRs and express them as [67,68]: where we take f 1 (0) = 0.245, α BZ = 0.40, and r BZ = 0.64. It is worth mentioning that lattice QCD results with N f = 2 + 1 for theB → π form factors, calculated by HPQCD [119], FNAL-MILC [22], and RBC-UKQCD [120] collaborations, recently have significant progress.
The detailed summary of the lattice QCD results can be found in [103]. We checked that the results of LCSRs are consistent with the values of Table IV in [120]. For theB → ρ decay, the form factors based on the LCSRs are given as [121]: Recently, the B → D ( * ) form factors associated with various types of currents, which are formulated in the heavy quark effective theory (HQET) [122], were studied up to O(Λ QCD /m b,c ) and O(α s ) in [18], where several fit scenarios were shown. We summarize the relevant results of Ref. [18] with "th:L w≥1 +SR" scenario in the appendix, where the "th:L w≥1 +SR" scenario combines the QCD sum rule constraints and the QCD lattice data [16]. The parametrizations of HQET form factors are different from those shown in Eqs. (26) and (27), and their relations can be straightforwardly found as follows: For B → D, they are: while for B → D * , they can be written as: where w = (m 2 B + m 2 D ( * ) − q 2 )/(2m B m D ( * ) ), and the h i functions and their relations to the leading and subleading Isgur-Wise functions can be found in the Appendix.

B.
Case with χ d bq ′ = 0 and χ u tt,ct = χ d bb = χ ℓ ℓ = 0 The free parameters involved in this study are: χ u tt , χ u ct,tc , χ u ut,tu , χ d bb , χ d bs,sb , χ d bd,db , t β , and the scalar masses m H,A,H ± . To reduce the number of free parameters without loss of generality, we adopt χ q ij = χ q ji and take the new free parameters to be real numbers with the exception of χ u tu,ut . Thus, the parameters χ d db,sb and χ u tc in leptonic B − q → ℓν become correlated to χ d bd,bs and χ u ct in the ∆B = 2 and b → sγ processes. According to Eq. (78), it can be seen that the involving parameters in S-mediated ∆B = 2 processes are only related to χ d bs and χ d bd . To understand how strict the experimental bounds on the χ d bq ′ are, we first discuss the simple situation with χ u tt,ct = χ d bb = 0. Thus, the contours of of C R,ℓ qb relies on the magnitude of χ R qb ; however, the feasibility is excluded by the ∆M q ′ constraint due to the result of χ d bq ′ ∼ O(10 −2 ). Hence, in such cases, the charged-Higgs effect in the type-III model is also destructive to the SM result. To illustrate the H ± influence on the leptonic decays, we show the contours of BR(B − u → τν) (dot-dashed lines) in units of 10 −4 in Fig. 4(a) and (b). Since χ d bd and χ d bs both appear in χ R ub , as shown in Eq. (14), when we focus on one of them, the other is set to vanish. From the plot, it can be seen that BR(B − u → τν) is always smaller than the SM result: In addition, the resulted BR(B − u → τν) is even smaller than the experimental lower bound of 1σ errors. Since similar behavior also occurs in B − c → τν, here, we just show the B − u → τν decay. Hence, only considering the χ d bq ′ effect will not cause interesting implications in the leptonic B − q decay. The χ d bs also affects the radiative b → sγ decay through the intermediates of H ± and S shown in Figs is of O(10 −4 ) and is thus negligible. According to Eq. (85), the χ d bs of the H ± contribution only appears in C ′H ± 7γ(8G) and shows up by means of ζ d * ts ζ d bb and ζ d * ts ζ u tt . Although the former has a t 2 β enhancement, due to the m 2 b /m 2 t suppression in C H ± 7(8),RR , the associated contribution is much smaller than the latter, which is insensitive to t β . We find that with |χ d bs | = 0.02, the result is |C ′H ± 7γ | ≈ 0.012 and is still much less than |C SM 7γ |. We note that the situation with χ u tt,ct = χ d bb = 0 is similar to the type-II model; therefore, with |χ d bs | < O(0.1), the charged-Higgs effect on b → sγ is insensitive to t β and χ d bs , but is sensitive to m H ± . To numerically show the result, we plot the contours of BR(B → X s γ) in units of 10 −4 in Fig. 5, where the dashed line denotes the 2σ upper limit of experimental data, and the lower bound on the charged-Higgs mass is given by m H ± > 580 GeV. According to above analysis, we learn that when χ u tt,ct = χ d bb = 0 is taken in the type-III 2HDM, due to the strict limits of ∆M d and ∆M s , the χ d bd and χ d bs effects contributing to B − q → ℓν and b → sγ are small and have no interesting implications on the phenomena of interest. For simplicity, we thus take χ d bd = χ d bs = 0 in the following analysis; that is, we only consider the charged-Higgs contributions.

C. Correlation with the constraint from the H/A → τ + τ − limits
In the 2HDM, m H ± indeed correlates with m H(A) . According to the study in [5], the allowed mass difference can be m H −m H ± ∼ 100 GeV if m A = m H is used. Since m H ± = 300 GeV is taken in our numerical analysis, the effects arisen from m S ≡ m H(A) ∼ 400 GeV in the 2HDM cannot be arbitrarily dropped. Using this correlation, it was pointed out that the upper limit of tau-pair production through the pp(bb) → H/A → τ + τ − processes measured in the LHC can give a strict bound on the parameter space, which is used to explain the R(D ( * ) ) anomalies [125].
In order to understand how strict the constraint from the LHC data is, we now write the scalar Yukawa couplings to the quarks, proposed in [125], as: where , and j denotes the flavor index. It can be seen that the parameters shown in the bb → H/A → τ + τ − processes are associated with Y b and Y τ . In our model, the parameters Y b,τ are given as: Comparing with Eq. (9), it can be seen that the lepton couplings to H(A) are the same as those to H ± . Due to the FCNC and CKM matrix effects, the H ± c L b R coupling shown in Eq. (10) is generally different from Y b ; however, when we take χ d bb = χ d sb = χ d db = 0, they become the same and are According to the ATLAS search for the τ -pair production through the resonant scalar decays, in which the result was measured at √ s = 13 TeV with a luminosity of 3.2 fb −1 , it was shown in [125] that the allowed values of Y b and Y τ in Eq. (99) should satisfy |Y b Y τ |v 2 /m 2 S < 0.3 for m S = 400 GeV. Thus, using t β = 50, we can obtain the limit from Eq. (100) as: where m b (m S ) = 3.18 GeV and m τ = 1.78 GeV are applied. Hence, we will take Eq. (101) as an input to bound the χ ℓ τ and χ d bb parameters.

D.
Constraints of b → sγ and B q ′ mixings From Eq. (85), there are two terms contributing to C H ± 7γ(8G) , where the associated charged-Higgs effects are ζ u * ts ζ u tt and ζ u * ts ζ d bb . Using the definitions in Eq. (63), it can be seen that the new factor χ L * ts χ u * tt /s 2 β in the first term is insensitive to t β > 10; however, ζ u * ts ζ d bb ∝ 1 − t β (χ L * ts /s β )(1 − χ d bb /s β ) ( unity denotes the result of type-II model) formed in the 2nd term can be largely changed by a large t β . In addition, we see that C H ± 7(8),LL and C H ± 7(8),RL are negative values, and the magnitude of the former is approximately one order smaller than that of the latter; that is, ζ u * ts ζ d bb indeed dominates. Due to the negative loop integral value, it can be understood that the Wilson coefficient C H ± 7γ (µ b ) in the type-II model is the same sign as C SM 7γ (µ b ); thus, m H ± is severely limited and the low bound is m H ± > 580 GeV, as shown in [75] and confirmed in Fig. 5.
Due to new Yukawa couplings involved in the type-III model, e.g. χ u tt,ct and χ d bb , the b → sγ constraint on m H ± can be relaxed. To see the b → sγ constraint, we scan the parameters with the sampling points of 5 × 10 5 , for which the results are shown in Fig. 6  We now know that H ± can be as light as a few hundred GeV in the type-III model.
In order to include the contributions of all χ d tt,ct and χ d bb with large t β and combine the constraints from the ∆B = 2 processes shown in Eq. (76) altogether, we fix t β = 50 and m H ± = 300 GeV and use the sampling points of 5 × 10 5 to scan the involving parameters.
The allowed parameter spaces, which only consider theB → X s γ constraint, are shown in Fig. 7(a), and those of combining theB → X s γ and ∆M d,s constraints are given in Fig. 7 where |χ u tt,ct | ≤ 1, |χ d bb | ≤ 1, and |χ ℓ τ | ≤ 2 have been used. Comparing Fig. 7(a) and 7(b), it can be obviously seen that ∆B = 2 processes can further exclude some free parameter spaces.

E.
Charged-Higgs on the leptonic B − q → ℓν decays After analyzing the b → sγ and ∆B = 2 constraints, we study the charged-Higgs contributions to the leptonic and semileptonic B decays in the remaining part of the paper. In order to focus on the χ u tc,tu and χ ℓ ℓ effects, we fix χ d bb = χ d db,sb = 0, t β = 50, and m H ± = 300 in the following numerical analyses, unless stated otherwise. With the numerical inputs, the BRs of leptonic B − u,c decays in the SM are estimated as: From Eqs. (23) and (25) [66] can constrain the parameters to be a small region, the constraint from the pp → H/A → τ + τ − processes further excludes the region of χ ℓ τ < −0.7. If BR(B − u → µν) can be measured at Belle II, the χ ℓ µ parameter can be further constrained.

F.
Charged-Higgs on theB d → (π + , ρ + )ℓν decays Compared to the charged B-meson decays,B d → (π + , ρ + )ℓν have larger BRs; thus, we discuss the neutral B-meson decays. With the LCSR form factors, the BRs of these decays in the SM are given in Table II, where the current measurements of light lepton channels are also shown. From the table, we can see that the BRs forB d → (π + , ρ + )ℓν (here ℓ = e, µ) in the SM are close to the observed values. Due to the H ± Yukawa coupling to the lepton being proportional to t β m ℓ /v, the charged-Higgs contributions to the light lepton channels are small. Thus, we can conclude that the consistency between the data and the SM verifies the reliability of the LCSR form factors in theB → (π, ρ) transitions. In the following analysis, we study the charged-Higgs influence on the τ -lepton modes and their associated observables. From Table II, the ratios of branching fractions forB d → (π + , ρ + )ℓν in the SM can be estimated as: Using Eq. (51), the contours for R(π) and R(ρ) as a function of χ u tu and χ ℓ τ are shown in Fig. 9(a) and (b), respectively, where the hatched regions denote BR(B − u → τν) exp with 2σ errors. According to the results, it can be found that due to the constraint of B − u → τν, the allowed R(ρ) is limited to being a very narrow range of ∼ (0.58, 0.60). From Fig. 9 Although it is difficult to measure the lepton polarization in theB d → (π + , ρ + )ℓν, we theoretically investigate the charged-Higgs contributions to the semileptonic B decays. Using Eqs. (58) and (59), the lepton helicity asymmetries in the SM can be found as: Due to the fact that the helicity asymmetry is strongly dependent on m ℓ , it can be understood that only τν modes can be away from unity. All lepton polarizations show negative values because the V − A current in the SM dominates. The sign of τ -lepton polarization in B → Dτν can be flipped to be a positive sign. In order to show the H ± influence, the contours for P τ π and P τ ρ as a function of χ u tu and χ ℓ τ are given in Fig. 10, where the constraint from pp → H/A → τ + τ − (dot-dashed) with χ d bb = 0 is also shown. With the B − u → τν constraint, the allowed values of P τ ρ are limited in a narrow region around the SM value. However, the allowed values of P τ π are wider and can have both negative and positive signs. The lepton FBAs are also interesting observables in the semileptonic B decays. Following the formulae in Eq. (61), we show the FBAs ofB d → π + τν andB d → ρ + τν as a function of q 2 in Fig. 11(a) and (b), respectively, where the solid line is the SM and the dashed line is the type-II model. For the type-III 2HDM, we select two benchmarks that obey the B − u → τν constraint as follows: the dotted line is χ u tu = −0.3 and χ ℓ τ = 1.37, which lead to R(π) ≈ 0.855 and R(ρ) ≈ 0.595; and the dot-dashed line denotes χ u tu = −0.8 and χ ℓ τ = −0.60, which lead to R(π) ≈ 0.550 and R(ρ) ≈ 0.577. From plot (a), we can see that A π,τ F B can be largely changed by the charged-Higgs effect; in other words, a zero-point can occur in A π,τ F B , where the zero point usually occurs in the ρ + channel, as shown in plot (b). Hence, we can use the characteristics of FBA to test the SM by examining the shape of A π,τ F B . From the plot (b), due to the strict limit of B − u → τν, the shape change of A ρ,τ F B in the type-III model is small.

G.
Charged-Higgs on the B − u → (D 0 , D * 0 )ℓν decays From Eq. (51) and the HQET form factors introduced previously, the BRs for the B − u → (D 0 , D * 0 )ℓν decays in the SM can be estimated, as shown in Table III, where the current experimental results are also included [69]. It can be seen that the BRs of the light lepton channels in the SM are consistent with the experimental data; however, the τν mode results are somewhat smaller than those in the current data. The ratios of branching fractions are obtained as R(D) SM ≈ 0.309 and R(D * ) SM ≈ 0.257, which are consistent with the results obtained in the literature.  Model As discussed before, the H ± contributions to B − u → D 0 ℓν and B − u → D * 0 ℓν are associated with C R,ℓ cb +C L,ℓ cb and C R,ℓ cb −C L,ℓ cb , respectively, and the same factor C R,ℓ cb −C L,ℓ cb also appears in the B − c → ℓν decay; that is, R(D * ) and BR(B − c → τν) have a strong correlation [43,51,66]. Although there is no direct measurement of the B − c → τν decay, the indirect upper limit on the BR(B − c → τν) can be obtained by the lifetime of B c with a result of 30% [51] and the LEP1 data [66] with a result of 10%. We show R(D) and R(D * ) as a function of χ u tc and χ ℓ τ in Fig. 12  According to Eqs. (58) and (59), it is expected that the helicity asymmetry of a light lepton will negatively approach unity, and that only τ -lepton polarizations can significantly where the Belle's current measurement is P τ D * = −0.38 ± 0.51 +0.21 −0.16 [12]. Intriguingly, the sign of P τ D is opposite to that of P τ D * , and the situation is different from the negative sign in P τ π . We find that the origin of the difference in sign between P τ π and P τ D is from the meson mass. Due to m D ≫ m π , the positive helicity becomes dominant in B − u → D 0 τν. To see the influence of the charged-Higgs on the τ polarizations, we show the contours for P τ D and P τ D * in the right-panel of Fig. 12. With the limit of BR(B − c → τν) < 10%, it is found that P τ D can be largely changed by the charged-Higgs effect, and the allowed range of P τ D * is narrow and can be changed by ∼ 10%, where the change in R(D * ) from the same H ± effects is only ∼ 3%.

VI. CONCLUSION
We studied the constraints of the b → sγ and ∆B = 2 processes in the type-III 2HDM with the Cheng-Sher ansatz, where the detailed analyses included the neutral scalars H and A (tree + loop) and charged-Higgs (loop) effects. It was found that the tree-induced ∆B = 2 processes produce strong constraints on the parameters χ d db and χ d sb , and due to the m b /m H(A) suppression, the loop-induced b → sγ process by the same H, A effects is small.
When we ignore the χ d db,sb effects, the dominant contributions to the rare processes are the charged-Higgs.
We demonstrated that due to the new parameters involved, i.e. χ u tt,tc and χ d bb , the mass of charged-Higgs in the type-III model can be much lighter than that in the type-II model when the b → sγ constraint is satisfied. Taking m H ± = 300 GeV and tan β = 50, we comprehensively studied the charged-Higgs contributions to the leptonic B − u,c → ℓν and semileptonicB u,d → (P, V )ℓν (P = π + , D 0 ; V = ρ + , D * 0 ) decays in the generic 2HDM.
In addition to the constraints from the low energy flavor physics, such as B d,s -B d,s mixings and B s → X s γ, we also consider the constraint from the upper limit of pp → H/A → τ + τ − measured in LHC. It was found that the tau-pair production cross section can further constrain the χ ℓ τ parameter to be |1 − χ ℓ τ | < 1.70 with χ d bb = 0. The main difference in the b → (u, c)ℓν decays between type-II and type-III is that the former is always destructive to the SM results, and the latter can make the situation constructive. Therefore, BR(B − u → (µ, τ )ν) can be enhanced from the SM results to the current experimental observations. Although B − c → τν has not yet been observed, the charged-Higgs can also enhance its branching ratio from 2% to the upper limit of 10%, where the upper limit is obtained from the LEP1 data.
Since heavy lepton can be significantly affected by the charged-Higgs, we analyzed the potential observables in theB d → (π + , ρ + )τν and B − u → (D 0 , D * 0 )τν decays. It was shown that since B − u(c) → τν and B − u → ρ + (D * 0 )τν are strongly correlated to the same charged-Higgs effects, the allowed R(ρ + ), R(D * ), P τ ρ , P τ D * , and A ρ,τ F B are very limited in terms of deviating from the SM. Although the change in A D * ,τ F B is not large, the deviation is still sizable. In contrast, the observables in the π + and D 0 channels are sensitive to the charged-Higgs effects and exhibit significant changes. be: where √ 2C L us /v is around 10 −4 and is thus negligible. Although √ 2m c /vV cs ∼ 7.4×10 −4 , it is still two orders smaller than the gauge coupling in the SM. In the phenomenological analysis, the C L cs effect can be neglected. Similarly, the C R us and C R cs couplings can be simplified as: tsH + vertex: using m t V ts ∼ 6.72 GeV < √ m c m t V cs ∼ 14.8 GeV and m s V ts ≪ √ m s m b V tb ∼ 0.66 GeV, we can simplify C L,R ts to be: It can be seen that due to the new factor χ L ts , √ 2C L ts /v can be comparable with the SM coupling of gV ts / √ 2 without relying on the large t β scheme.
Thus, theB → D form factors can be defined as: while the form factors forB → D * are: where h − , h A 2 , and h T 2,3 vanish in the heavy quark limit, and the remaining form factors are equal to the leading order Isgur-Wise function ξ(w).
We take the parametrization of leading order Isgur-Wise function as [18,123]: where V 21 = 57.0, V 20 = 7.5; z * and a are defined as [123]: We take the results using the fit scenario of "th:L w≥1 +SR" shown in [18]. In addition tō ρ 2 * = 1.24 ± 0.08, the values of sub-leading Isgur-Wise functions at w = 1 are given in Table IV. Using these results, the correction of O(e b,c ) and O(α s ) can be obtained as: ∆(e b , e c , α s ) ≈ 0.582 ± 0.298 , where we adopt the 1S scheme for m b and use the value of m 1S b = 4.71 ± 0.05 GeV [18]. In addition, δm bc = m b − m c = 3.40 ± 0.02 GeV andΛ = 0.45 GeV are used. The results of sub-leading Isgur-Wise functions using the fit scenario of "th:L w≥1 +SR".