\(B_c\rightarrow B_{sJ}\) form factors and \(B_c\) decays into \(B_{sJ}\) in covariant light-front approach

The European Physical Journal C, Oct 2016

We suggest to study the \(B_{s}\) and its excitations \(B_{sJ}\) in the \(B_c\) decays. We calculate the \(B_c\rightarrow B_{sJ}\) and \(B_c\rightarrow B_{J}\) form factors within the covariant light-front quark model, where the \(B_{sJ}\) and \(B_{J}\) denote an s-wave or p-wave \(\bar{b}s\) and \(\bar{b}d\) meson, respectively. The form factors at \(q^2=0\) are directly computed while their \(q^2\)-distributions are obtained by extrapolation. The derived form factors are then used to study semileptonic \(B_c\rightarrow (B_{sJ},B_{J})\bar{\ell }\nu \) decays, and nonleptonic \(B_c\rightarrow B_{sJ}\pi \). Branching fractions and polarizations are predicted in the standard model. We find that the branching fractions are sizable and might be accessible at the LHC experiment and future high-energy \(e^+e^-\) colliders with a high luminosity at the Z-pole. The future experimental measurements are helpful to study the nonperturbative QCD dynamics in the presence of a heavy spectator and also of great value for the study of spectroscopy.

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\(B_c\rightarrow B_{sJ}\) form factors and \(B_c\) decays into \(B_{sJ}\) in covariant light-front approach

Eur. Phys. J. C Bc → Bs J form factors and Bc decays into Bs J in covariant light-front approach Yu-Ji Shi 1 Wei Wang 0 1 Zhen-Xing Zhao 1 0 State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences , Beijing 100190 , China 1 INPAC, Shanghai Key Laboratory for Particle Physics and Cosmology, Department of Physics and Astronomy, Shanghai Jiao-Tong University , Shanghai 200240 , China We suggest to study the Bs and its excitations Bs J in the Bc decays. We calculate the Bc → Bs J and Bc → BJ form factors within the covariant light-front quark model, where the Bs J and BJ denote an s-wave or p-wave b¯s and b¯d meson, respectively. The form factors at q2 = 0 are directly computed while their q2-distributions are obtained by extrapolation. The derived form factors are then used to study semileptonic Bc → (Bs J , BJ ) ¯ν decays, and nonleptonic Bc → Bs J π . Branching fractions and polarizations are predicted in the standard model. We find that the branching fractions are sizable and might be accessible at the LHC experiment and future high-energy e+e− colliders with a high luminosity at the Z -pole. The future experimental measurements are helpful to study the nonperturbative QCD dynamics in the presence of a heavy spectator and also of great value for the study of spectroscopy. 1 Introduction In the past decades, there has been a lot of progress in hadron spectroscopy, thanks to the well-operating experiments including the e+e− colliders and hadron colliders. The immense interest in spectroscopy is not only due to the fact that one is able to find many missing hadrons to complete the quark model, but more importantly due to the observations of states that are unexpected in the simple quark model. The latter ones are generally called hadron exotics. A milestone in the exploration of the exotics is the discovery of X (3872), first in B decays by Belle Collaboration [1] and subsequently confirmed in many distinct processes in different experiments [2–4]. It was found the propa e-mail: b e-mail: c e-mail: erties of this meson are peculiar. Since then the identification of multiquark hadrons has become a focus topic in hadron physics. Inspired by the discovery of X (3872), a number of new interesting structures were discovered in the mass region of heavy quarkonium. Refer to Refs. [5–8] for recent reviews. On the theoretical side, deciphering the underlying dynamics of these multiquark states is a formidable challenge, and is often based on explicit and distinct assumptions. In many assumptions, the quarkonium-like states are usually composed of a pair of heavy constituents, which makes it vital to study first the heavy–light hadron. In the system with one heavy charm quark, a series of important results have started with the discoveries of the narrow states Ds (2317) in the Ds+π 0 final state and Ds (2460) in the Ds∗π 0 and Ds γ final state [9,10]. Along this line, a few other new states, such as Ds1(2536), Ds2(2573), Ds (2710), have been observed at the B factory and other facilities [11]. Bottomed hadrons are related to charmed mesons by heavy quark symmetry. But compared to the charm sector, there is less progress for the bottomed hadrons. In experiment, only a few bottom–strange mesons are observed, most of which are believed to be in agreement with the quark model. In this paper, we propose to use the Bc decays and study the spectrum of the Bs J . We observe a few advantages. First of all, the large production rate of the Bc is in expectation, in particular the LHCb will produce a number of Bc events and thus the Bc → Bs J decays will have a large potential to be observed. Second, the scale over the mW can be computed in the perturbation theory and the QCD evolution between the mW and the low energy scale mc is well organized by making use of the renormalization group improved perturbation theory. Consequently, the Bc decays into Bs and other excited states have received some theoretical attention [12–30]. In the following we will address the investigation of the production rates of Bs J meson (an s-wave or p-wave b¯s hadron) in semileptonic and nonleptonic Bc meson decays in the framework of the covariant light-front quark model (LFQM) [31]. In the Bc → Bs J decays, the quark level transition is c → s in which the heavy bottom quark acts as a spectator. Since most of the momentum of the hadron is carried by the spectator, there is no large momentum transfer and the transition is dominated by the soft mechanism. A form factor can then be expressed as an overlap of the wave functions of the initial and final state hadrons. Treatments in quark models like the LFQM are of this type. As pointed out in Ref. [32], the light-front approach shows some unique features which are suitable to handle a hadronic bound state. The LFQM [33–36] can provide a relativistic treatment of moving hadrons and give a full treatment of hadron spins in terms of the Melosh rotation. Lightfront wave functions, which characterize the hadron in terms of their fundamental quark and gluon degrees of freedom, are independent of hadron momentum and thus are Lorentz invariant. Moreover, in covariant LFQM [31], the spurious contribution which depends on the orientation of the light front is elegantly eliminated by including zero-mode contributions. This covariant model has been successfully extended to study the decay constants and form factors of various mesons [37–49]. Through this study of Bc → Bs J in LFQM, we believe that one will not only obtain the information as regards the decay dynamics in the presence of a heavy spectator but will also provide a side-check for the classification of the heavy–light mesons. It is also helpful toward the establishment of a global picture of the heavy–light spectroscopy including the exotic spectrum. The rest of this paper is organized as follows. In Sect. 2, we will give a brief description of the parametrization of form factors, the framework of covariant LFQM, and the form factor calculation in this model. We present our numerical results for various transitions in Sect. 3. In Sect. 4, we use the form factors to study semileptonic and nonleptonic Bc decays. In this section, we will present our predictions for branching fractions and polarizations. The last section contains a brief summary. 2 Transition form factors in the covariant LFQM 2.1 Bc → Bs J form factors The effective electroweak Hamiltonian for the Bc → Bs J l¯ν reads where G F and Vcs are the Fermi constant and Cabibbo– Kobayashi–Maskawa matrix element, respectively. Leptonic parts can be computed in perturbation theory while hadronic contributions are parametrized in terms of form factors. An s-wave meson corresponds to a pseudoscalar meson or a vector meson, abbreviated as P and V , respectively. For a p-wave meson, the involved state is a scalar S, an axialvector A or a tensor meson T . In the following we introduce the abbreviations P = P + P , q = P − P , and we adopt the convention of 0123 = 1. The Bc → P, V form factors can be defined as follows: 1 V ( P , ε )|Vμ|Bc( P ) = − m Bc + mV × μναβ ε ∗ν Pαqβ V BcV (q2), V ( P , ε )| Aμ|Bc( P ) = 2i mV +i (m Bc + mV ) A1BcV (q2) εμ∗ − ε ∗ · q qμ In analogy with Bc → V form factors, we parametrize the Bc → T form factors as 2V BcT (q2) μνρσ (εT∗ )ν ( P )ρ ( P )σ , = − m Bc + mT T ( P , ε )| Aμ|Bc( P ) = 2i mT εT∗q 2· q qμ A0BcT (q2) +i (m Bc + mT ) A1BcT (q2) εT∗ μ − εT∗ · q qμ The Bc → S, A form factors can be defined by exchanging the vector and axial-vector current: S( P )| Aμ|Bc( P ) = −i A( P , ε )|Vμ|Bc( P ) = −2m A −(m Bc + m A)V1Bc A(q2) εμ∗ − ε ∗ · q qμ V2Bc A(q2) Pμ − m2B − m2A qμ , 1 A( P , ε )| Aμ|Bc( P ) = − m Bc − m A i × μναβ ε ∗ν Pαqβ ABc A(q2). The spin-2 polarization tensor can be constructed using the standard polarization vector ε: εμν ( P , ±2) = εμ(±)εν (±), 1 εμν ( P , ±1) = √ [εμ(±)εν (0) + εν (±)εμ(0)], 2 It is symmetric and traceless, and εμν P ν = 0. If the recoiling meson is moving on the plus direction of the z axis, their explicit structures are chosen as where ET and | pT | are the energy and the momentum magnitude of the tensor meson in the Bc rest frame, respectively. 2.2 Covariant light-front approach In the covariant LFQM, it is convenient to use the light-front decomposition of the momentum P = ( P −, P +, P ), with P ± = P 0 ± P 3, and thus P 2 = P + P − − P⊥⊥2. The incoming (outgoing) meson has the momentum P = p1+ p2 ( P = p1 + p2) and the mass M (M ). The quark and antiquark inside the incoming (outgoing) meson have the mass m ( ) and m2, respectively. Their momenta are denoted 1 as p1( ) and p2 respectively. In particular these momenta can be written in terms of the internal variables (xi , p ) by ⊥ with the momentum fractions x1 + x2 = 1. With these internal variables, one can define some useful quantities for both incoming and outgoing mesons: M02 = (e1 + e2)2 = M0 = mi( )2 + p⊥2 + pz2, pz = p⊥2 + m12 p⊥2 + m22 , − 2x2 M0 Table 1 Meson–quark–antiquark vertices used in the covariant LFQM. In the case of the outgoing meson, one should use instead i(γ0ΓM†γ0) for the corresponding vertices Pseudoscalar ( 1 S0) Scalar ( 3 P0) Vector ( 3 S1) Axial ( 3 P1) Axial ( 1 P1) Tensor ( 3 P2) i 21 HT [γμ − W1T ( p1 − p2)μ]( p1 − p2)ν −p2 Fig. 1 Feynman diagram for transition form factors, where the cross symbol in the diagram denotes the transition current The Feynman rules for meson–quark–antiquark vertices can be derived using the conventional light-front approach, whose forms for the s-wave and p-wave states are collected in Table 1 [31,37]. An extension to the d-wave vertices has been conducted in Ref. [50]. In the following we will take the Bc → Bs transition as the example and illustrate the calculation. To do so, we will consider the matrix element P( P )|Vμ| P( P ) ≡ BμP P , whose Feynman diagram is shown in Fig. 1. It is straightforward to obtain 1 N1 N1 N2 SVPμP = Tr[γ5( /p1 + m1)γμ( /p1 + m1)γ5( /p2 − m2)], N1( ) = p1( )2 − m1( )2 + i , Here we consider the q+ = 0 frame. The p1− integration picks up the pole p2 = pˆ2 = [( p22⊥ + m22)/ p2+, p2+, p2⊥] and leads to M0 2 = p⊥2 + m12 p⊥2 + m22 , with p⊥ = p⊥ − x2q⊥. The explicit form of h P has been derived in Refs. [31,37] h P = (M 2 − M02) Nc √2M0 where ϕ is the light-front momentum distribution amplitude for s-wave meson. In practice, the following Gaussian-type wave function can be adopted [31,37]: ϕ = ϕ (x2, p⊥) = 4 As shown in Refs. [31,37], the inclusion of the so-called zero-mode contribution in the above matrix elements in practice amounts to the replacements pˆ1μ =. Pμ A(11) + qμ A(21), . where the symbol = in the above equation reminds us that it is true only in the BμP P integration. A(ji) and Z2, which are functions of x1,2, p 2, p⊥ · q⊥, and q2, are listed in Appendix ⊥ A. After the replacements, we arrive at −x1(m1 − m2)2 − x1(m1 − m2)2 , ⊥ x2 Nˆ 1 Nˆ 1 ×(x2m1 + x1m2) + 2 q · P q2 M 2 − x2(q2 + q · P) − (x2 − x1)M 2 +2x1 M02 − 2(m1 − m2)(m1 + m1) . Finally we get the form factors through the relations Similarly, one can derive the other form factors, whose expressions are collected in Appendix A. Before closing this section, it is worth mentioning that the axial-vector mesons may not be classified as 3 P1 or 1 P1 state. In the quark limit with m Q → ∞, the QCD interaction is independent of the heavy quark spin and thus it will decouple with the light system. A consequence of this decoupling is that heavy mesons are classified into multiplets labeled by the total angular momentum of the light degrees of freedom. The s-wave pseudoscalar and vector states are in the same multiplets denoted as sl = 1/2. For the p-wave states, two kinds of axial-vector mesons P13/2 and P1/2 are mixtures of 1 3 P1 or 1 P1: Since the form factors involving P3/2 and P1/2 can be 1 1 straightforwardly obtained by the linear combination for those given above, we shall calculate the form factors using the 2S+1 L J basis in the following analysis. 3 Numerical results for form factors 3.1 Input parameters In the covariant LFQM, the constituent quark masses are used as (in units of GeV): which have been widely used in various B and Bc decays [42– 49]. The masses of the Bc and Bs J are taken from the PDG (in units of GeV) [11]: m Bs2 = 5.840, while for the Bs0 and Bs1, we quote the results [51,52] (see also estimates in Refs. [53–55]): Since the masses of the P1/2 and P3/2 are close to the 1 1 observed state Bs1(5830) [11], m Bs1(5830) = 5.829 GeV, we use the same value for both the 3 P1 and the 1 P1 state. The parameter β, characterizing the momentum distribution, is usually determined by fitting the meson decay constant. For instance, in this approach the pseudoscalar and vector meson’s decay constants read F (q2) = F (q2) = 1 − a mq22H + b( mq22H )2 1.04∗ 0.37∗ 0.75∗ 0.95∗ 3.24∗ 9.56∗ BJ form factors, for Table 2 Bc → Bs , Bs∗, Bs0, Bs1, Bs1, and Bs2 form factors in the lightfront quark model, which are fitted using Eq. (32), while for the form factors with an asterisk, the parametrization in Eq. (36) is adopted A1Bc Bs∗ V Bc Bs2 18.60 2.28∗ 2.08∗ A2Bc Bs∗ We will also calculate the Bc → which we use the masses [51,52] The above result is derived from the decay constant result [60], f B = (193 ± 6) MeV. 3.2 Form factors and momentum transfer distribution With the inputs in the previous subsection, we can predict the Bc → Bs , Bs∗, Bs0, Bs1, Bs1, and Bs2 form factors in the LFQM and we show our results in Table 2. In order to access the q2 distribution, one may adopt the fit formula: h P ⊥ x1x2(M 2 − M02) 4(m1x2 +m2x1), × x1 M02 − m1(m1 − m2) − p 2 ⊥ + For the Bc meson, the decay constant can in principle be determined by leptonic and radiative-leptonic decays [56– 59], both of which show lack of experimental data yet. Twoloop contributions in the NRQCD framework have been calculated in Ref. [58] and the authors have found f Bc = 398 MeV. We will adopt this result, but it is necessary to note that the above value is smaller than the lattice QCD result by approximately 2σ : f Bc = (434 ± 15) MeV. We use the recent lattice QCD result for the Bs decay constant with N f = 2 + 1 + 1 [60], f Bs = (229 ± 5) MeV. This is close to the previous lattice QCD result [61,62]: f Bs = (224 ± 5) MeV. Using the decay constants, the shape parameters are fixed as βBc = 0.886 GeV, βBs = 0.623 GeV, and we assume that the values of β for the other Bs J mesons are approximately equal to that for the Bs , that is, βBs∗ = βBs0 = βBs1 = βBs1 = βBs2 = 0.623 GeV. In the literature, the dipole form has been used to parametrize the q2 distribution: with the m H = m D for D decays and m H = m B for B decays. This parametrization is inspired by analyticity. Taking the F1B→π as an example, we consider the timelike matrix element Fig. 2 The Bc → Bs form factors and their q2-dependence parameterized by Eqs. (32) and (36) 0|u¯γ μb|π(− pπ )B( pB ) ∼ For the ABc Bs∗ , F0Bc Bs0 , V0B,1c Bs1 , and ABc Bs2 , we found that 2 2 the fitted values for the m2fit are negative, and thus we use the following formula: where the one-particle contribution has been singled out. The lowest resonance that can contribute is the vector B∗. This leads to the pole structure at large q2: F (q2) = F1(0) F1B→π (q2) ∼ 1 − q2/m2B∗ . Except the pole at m B∗ , there are residual dependences on q2, which can be effectively incorporated into the a, b of the dipole parametrization as shown in Eq. (33). However, for the Bc → Bs J transition, one cannot simply apply Eq. (33), since the contributing states are the Ds resonances. Using the m H = m Bc will not only disguise the genuine poles, but also lead to irrationally large results for parameters a and b. So in order to avoid this problem, we have adopted the parametrization in Eq. (32). From the results in Table 2, one can see that the mfit for most form factors is between 1.5 and 2.0 GeV, close to the mass of a Ds J resonance. This has validated our parametrization. The q2-dependent form factors of Bc → Bs are shown in Fig. 2. From this figure, we can see that except for the Bc → Bs1 transition, most form factors are rather stable against the variation of q2. This is partly because of the limited phase space. This will also lead to a reliable prediction for the branching fractions given in the next section. 4 Phenomenological applications 4.1 Semileptonic Bc decays The decay width for semileptonic decays of Bc → Ml¯ν, where M = P, V , S, A, T , can be derived by dividing the = e 1.51 × 10−2 1.96 × 10−2 6.58 × 10−4 8.31 × 10−5 5.38 × 10−4 2.98 × 10−5 1.43 × 10−2 1.83 × 10−2 5.23 × 10−4 6.33 × 10−5 3.98 × 10−4 1.97 × 10−5 τBc = (0.452 × 10−12) s, decay amplitude into hadronic part and leptonic part, both of which are Lorentz invariant so that can be readily evaluated. Then the differential decay widths for Bc → Pl¯ν and Bc → V l¯ν turn out to be The differential decay widths for Bc → Sl¯ν and Bc → Al¯ν can be obtained by making the following replacements in the above expressions for Bc → Pl¯ν and Bc → V l¯ν: = 1−mˆ l2 2 +3mˆ l2(m2Bc − m2P )2 F02(q2) , F Bc P (q2) → F Bc S(q2), i = 0, 1, m Bc + mV → m Bc − m A, V BcV (q2) → ABc A(q2), AiBcV (q2) → ViBc A(q2), i = 0, 1, 2, = 1−mˆ l2 2 (m2Bc − m2V − q2)(m Bc + mV ) A1(q2) = 1−mˆ l2 2 × m Bc + mV ∓ where the superscript +(−) denotes the right-handed (lefthanded) polarizations of the vector mesons. We have λ(m2Bc , mi2, q2) = (m2Bc + m2 i − q2)2 − 4m2Bc mi2 with i = P, V . mˆ l = ml / q2. The combined transverse and total differential decay widths are given by is given by Eq. (38) multiplied by ( 23 |mpTT| )2 and Eq. (39) multiplied by ( √1 | pT | )2, respectively. Here the pT denotes 2 mT the momentum of the tensor meson in the Bc rest frame and mT is mass of the tensor meson. For the Bs J final state, the inputs are form factors given in Table 2 and the masses of Bc and Bs J s given in Eq. (21). The other input parameters are given as follows [11]: G F = 1.166 × 10−5 GeV−2, |Vcs | = 0.973, and our predictions for the branching fractions are given in Table 3. It should be mentioned that in the above calculation we have considered Bs1(Bs1) and B1(B1) to be in the 3 P1(1 P1) eigenstates. For the BJ final state, we need to evaluate the form factors for Bc → BJ by following the same method, and our results are given in Table 4. The masses of Bc and BJ s are also given in Eqs. (21) and (29). The other inputs are the same as Eq. (44) but with |Vcs | = 0.973 replaced by |Vcd | = 0.225 [11]. With these inputs, our results for the branching fractions and ratios are given in Table 5. From these tables, we can see that the branching fractions for Bc → Bs ¯ν and Bc → Bs∗ ¯ν are at the percent level, while those for the Bc → B ¯ν and Bc → B∗ ¯ν are suppressed by one order of magnitude. This is consistent with the results in the literature [12–20]. Branching fractions for channels with p-wave bottomed mesons in the final state Table 4 Bc → B, B∗, B0, B1, B1, and B2 form factors in the covariant LFQM fitted through Eq. (32), except for the form factors with an asterisk, which are fitted using Eq. (36) 3.14∗ 6.49∗ 2.48∗ 51.50∗ range from 10−4 to 10−6. In decays with large phase space, the electron and muon masses can introduce about a few percents to branching ratios. While for those with limited phase space like Bc → Bs2 ¯ν, the effects due to the lepton mass difference can reach 30 %. We hope these predictions can be examined in the future on the experimental side. 4.2 Nonleptonic Bc decays Since the main purpose of this work is to investigate the production of Bs J , we will focus on the decay modes which can be controlled under the factorization approach. Such decay modes are usually dominated by tree operators with effective Hamiltonian G F Heff (c → sud¯) = √2 Vc∗s Vud × C1[s¯αγ μ(1 − γ5)cβ ][u¯β γμ(1 − γ5)dα] +C2[s¯αγ μ(1 − γ5)cα][u¯β γμ(1 − γ5)dβ ] , where C1 and C2 are the Wilson coefficients, and α and β denote the color indices. With the definitions of the decay constants, π +( p)|u¯γμγ5d|0 = −i fπ pμ, one can expect the factorization formula to have the following forms: i M(Bc+ → Bs π +) = N m2Bc (1 − r B2s )F0Bc Bs (m2π ), where N = = C1/Nc(Nc = 3). The particle decay width for Bc → Bs J π is given as G F /√2Vc∗s Vud a1 fπ , with a1 with | p1| being the magnitude of three-momentum of Bs J or π meson in the final state in the Bc rest frame. We use the transition form factors given in Table 2 and the masses of Bc and Bs J s given in Eqs. (21), (22), and (23) and the other inputs which are given as follows [11,44]: τBc = (0.452 × 10−12) s, mπ = 0.140 GeV, |Vcs | = 0.973, |Vud | = 0.974, fπ = 130.4 MeV, a1 = 1.07, where fπ can be extracted from π − → −ν¯ data and a1 is evaluated at the typical factorization scale μ ∼ mc [63]. Then our theoretical results for the Bc → Bs J π branching ratios turn out to be as follows: → Bs1π +) = 0.36 %, → Bs2π +) = 0.023 %. Using the 1 fb−1 data of proton–proton collisions collected at the center-of-mass energy of 7 TeV and 2 fb−1 data accumulated at 8 TeV, the LHCb Collaboration has observed the decay Bc → Bs π + [64]: = (2.37 ± 0.31 ± 0.11+−00..1173) × 10−3. The first uncertainty is statistical, the second is systematic, and the third arises from the uncertainty on the Bc+ lifetime. The ratio of cross sections σ ( Bc+)/σ ( Bs0) depends significantly on the kinematics, and a rough estimate has lead to the branching ratio for Bc+ → Bs0π + of about 10 % [64]. The estimated branching fraction is somewhat larger than but still at the same magnitude as our result. Moreover, our results have indicated that the LHCb Collaboration might be able to discover other channels with similar branching fractions like the Bc → Bs∗π . 5 Conclusions To understand the structure of the heavy–light mesons, especially the newly observed states, and to establish an overview of the spectroscopy, a lot of effort is required on both the experiment and the theory sides. One particular remark is the classification of these states. In the heavy quark limit, the charm quark will decouple with the light degree of freedom and acts as a static color source. Strong interactions will be independent of the heavy flavor and spin. In this case, heavy mesons, the eigenstates of the QCD Lagrangian in the heavy quark limit, can be labeled according to the total angular momentum sl of the light degree of freedom. The heavy mesons with the same angular momentum sl but different orientations of the heavy quark spin degenerate. One consequence is that heavy mesons can be classified by the multiplets characterized by sl instead of the usual scheme using the 2S+1 L J . In this work, we have suggested to study the Bs and its excitations Bs J in the Bc decays. We have calculated the Bc → Bs J and Bc → BJ form factors within the covariant light-front quark model, where the Bs J and BJ denotes an swave or p-wave b¯s and b¯d meson, respectively. The form factors at q2 = 0 are directly calculated while the q2-distribution is obtained by the extrapolation. The derived form factors are then used to study semileptonic Bc → ( Bs J , BJ ) ¯ν decays, and nonleptonic Bc → Bs J π decays. Branching fractions and polarizations are predicted, through which we find that the predicted branching fractions are sizable, especially at the LHC experiment and future high-energy e+e− colliders with a high luminosity at the Z -pole. The future experimental measurements are helpful to study the nonperturbative QCD dynamics in the presence of a heavy spectator and also they are of great value for the study of spectroscopy. Acknowledgments This work is supported in part by National Natural Science Foundation of China under Grant No. 11575110, Natural Science Foundation of Shanghai under Grant No. 15DZ2272100 and No. 15ZR1423100, by the Open Project Program of State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, China (No. Y5KF111CJ1), and by Scientific Research Foundation for Returned Overseas Chinese Scholars, State Education Ministry. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP3. Appendix A: Explicit expressions for form factors In the LFQM, it is more convenient to adopt a new set of parametrization of form factors, with the relations V Bc V (q2) = −(m Bc + m V )g(q2), = − m Bc + m V − 2m V a−(q2), − q · P u−(q2), The analytic expressions for P → P transition form factorsinthecovariantLFQMhavebeengiveninEq.(17),while for the P → V transition, they are given as follows [31,37]: −m2BmcqT2b−(q2). [ 3A,1A(q2),q 3A,1A(q2),c±3A,1A(q2)] +2(m1 − m2)(m1 + m1)2 −2(m1 − m1)(m1 − m2) , +(x2 − x1)m1] p⊥q·2q⊥ − 2x2q2 + p⊥ · q⊥[p⊥ · p⊥ x2q2wV +(x1m2 + x2m1)(x1m2 − x2m1)] , a−(q2) = 16Nπc3 dx2d2p⊥ x2hNPˆ1hNVˆ1 2(2x1 − 3) p 2 ×(x2m1 + x1m2) − 8(m1 − m2) q⊥2 + 2(p⊥q·4q⊥)2 4 +wV [M 2 + M 2 − q2 + 2(m1 − m1)(m1 − m2)] −2m1(m1 + m2) − 2m2(m1 − m2)](A(11) + A(21) − 1) +q · P pq⊥22 + (p⊥q·4q⊥) (4A(21) − 3) . 2 The explicit expressions for P → S and P → A transitions can be readily obtained by making the following replacements [37]: where only the 1/W terms in P → 1A form factors are kept. It should be cautious that the replacement of m1 → −m1 should not be applied to m1 in w and h . The P → T transition form factors are calculated [37] Nc h(q2) = −g(q2)|hV→hT + 16π3 dx2d2p ⊥ x2Nˆ1Nˆ1 × (m1 − m1)(A(32) + A(42)) ×[m2(q2 − Nˆ1 − Nˆ1 − m12 − m12) −m1(M 2 − Nˆ1 − m12 − m22) ×(2A(13) + 2A(23) − A(12)) q · P −4 A(23)Z2 + 3q2 (A(12))2 + 2A(12)Z2 , (A10) × 8(m2 − m1)( A(33) + 2 A(43) + A(53)) −2m1( A(11) + A(21))( A2(2) + A(32)) +2(m1 + m1)( A(22) + 2 A(32) + A(42)) ×( A(33) + 2 A(43) + A(3) 2 − A(32)) 5 − A(2) +[q2 − Nˆ 1 − Nˆ 1 − (m1 + m1)2] ×( A(22) + 2 A(32) + A(2) 1 − A(21))] , 4 − A(1) Nc b−(q2) = −a−(q2)|hV →hT + 16π 3 ⊥ x2 Nˆ 1 Nˆ 1 × 8(m2 − m1)( A(43) + 2 A(53) + A(63)) −2 A(63) − A(21)) ×( A(43) + 2 A(53) + A(3) 3 − A(42)) + 2Z2(3 A(42) 6 − A(2) 6 A(21) A(2) 2 ×( A(22) + 2 A(32) + A(2) 1 − A(21)) . 4 − A(1) The A(ji) in the above equations are given as follows: p⊥ · q⊥ , A(22) = ( A(11))2, A(32) = A(11) A(21), A(2) 1 4 = ( A(21))2 − q2 A(12), A(33) = A(11) A(22), A(43) = A(21) A(22), A(3) 2 6 = A(21) A(2) 4 − q2 A(21) A(12), Z2 = Nˆ 1 + m12 − m22 + (1 − 2x1)M 2 +(q2 + q · P) p⊥ · q⊥ . q2 The explicit forms of h M and wM are given by [37] h P = hV = (M 2 − M02) h S = h 3 A = (M 2 − M02) h 1 A = hT = (M 2 − M02) 1 − m2 Nc √ 1 M˜ 02 , w 1 A = 2, where ϕ and ϕ p are the light-front momentum distribution amplitudes for s-wave and p-wave mesons, respectively [37]: ϕ p = ϕ p(x2, p⊥) = dx2 = x1x2 M0 1. 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Yu-Ji Shi, Wei Wang, Zhen-Xing Zhao. \(B_c\rightarrow B_{sJ}\) form factors and \(B_c\) decays into \(B_{sJ}\) in covariant light-front approach, The European Physical Journal C, 2016, 555, DOI: 10.1140/epjc/s10052-016-4405-1