Covariant Light-Front Approach for B c Decays into Charmonium: Implications on Form Factors and Branching Ratios

,


I. INTRODUCTION
During the period from the 1970s to 1980s, the light-front quark model (LFQM) was developed [1,2] to deal with nonperturbative physical quantities such as decay constants, transition form factors, and so on.This relativistic quark model is based on a light-front formalism [3] and quantum chromodynamics (QCD) light-front quantization [4].The LFQM can provide a relativistic treatment of the hadron momentum and fully treatment of the quark spin by using the so-called Melosh rotation.At the same time, the light-front wave functions are independent of the hadron momentum and thus are manifestly Lorentz invariant.Equipped with these advantages, the LFQM becomes a convenient approach and has been employed to calculate decay constants and form factors [5][6][7][8][9].While under the LFQM, the constituent quark and antiquark in a bound state are required to be on their mass shells, which makes degrees of freedom of light-front momentum become three and the Lorentz covariance of the matrix elements to be lost.The usual practice is only taking the plus component (µ = +) of the current matrix elements, which will miss the zero-mode contributions.However, lacking the zero-mode contributions sometimes affects the calculation accuracy.Unfortunately, such conventional LFQM approach with defect is powerless to calculate the zero-mode contributions.At the end of the twentieth century, Jaus put forward the covariant Light-Front quark model (CLFQM) [10].The previous LFQM is usually called the standard Light-Front quark model (SLFQM) [10].The CLFQM is more convincing than the SLFQM.In the CLFQM approach [10], when evaluating the light-front matrix element from the momentum loop integral by a lightfront decomposition to the internal momentum and carrying out the integration over the minus component (p − = p 0 − p 3 ) by means of contour methods, one will encounter additional spurious contribution proportional to the light-like vector ω µ = (0, 2, 0 ⊥ ), which violates the covariance.While this spurious contribution is just canceled by the zero-mode contribution, at the same time the covariance of the current matrix elements is restored, and all the problems can be resolved.Since the popular CLFQM was proposed, it has been widely used to study the form factors and the decay constants of the ground-state S-wave and low-lying P-wave mesons, and is further applied to phenomenological studies about B c decays [11][12][13][14][15]. Certainly, there still exist some discussions about the self-consistency of the CLFQM, for example, the decay constant of the vector meson, which is different as a result of extracting from different polarization (longitudinal and transverse) states [16][17][18][19].
B c meson decays have received extensive attention because of its unique structure in the Standard Model.The B c meson is the only heavy meson composed of two heavy quarks with different flavors (b and c), which cannot annihilate into gluons (photon) via strong (electromagnetic) interaction.Decays of the B c meson occur only via weak interaction, which includes three types at the quark level, the b → c(u), c → s(d) transitions, and the weak annihlilations.Although the phase space of the c quark decays is much smaller than that of the b quark decays, the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements are greatly in favor of the c quark decays (i.e., |V cs | ≫ |V cb |, |V cd | ≫ |V ub |), which provide about 70% of the B c decay width, while the b quark decays and the weak annihilations only amount to about 20% and 10%, respectively [20].On the experimental side, since the B c meson was first discovered by the Collider Detector at Fermilab (CDF) collaboration via the decay of B c → J/Ψlν in 1. 8 TeV pp collisions at the Fermilab Tevatron, many B c decay channels have been observed by the Large Hadron Collider beauty (LHCb) collaboration, TABLE I: Feynman rules for the vertices iΓ ′ M of the incoming meson-quark-antiquark, where p ′ 1 and p 2 are the quark and antiquark momenta, respectively.
Under the covariant light-front quark model, the light-front coordinates of a momentum p are used, p = (p − , p + , p ⊥ ), with p ± = p 0 ± p z , and p 2 = p + p − − p 2 ⊥ .The Feynman diagrams for B c meson decay and transition amplitudes are shown in Fig. 1.The incoming (outgoing) meson has mass M ′ (M ′′ ) with the momentum P ′ = p ′ 1 +p 2 (P ′′ = p ′′ 1 +p 2 ), where p terms of the internal variables (x i , p ′ ⊥ ) as with x 1 + x 2 = 1.Using these internal variables, we can define some quantities for the incoming meson which will be used in the following calculations: where M ′ 0 is the kinetic invariant mass of the incoming meson and can be expressed as the energies of the quark and the antiquark e is the quark momentum, p 2 is the antiquark momentum and X denotes the vector or axial vector transition vertex.
To calculate the amplitudes for the transition form factors, we need the Feynman rules for the meson-quark-antiquark vertices (iΓ ′ M (M = P, V, A, S)), which are listed in Tab.I.It is noted that for the outgoing meson, we should use i(γ 0 Γ ′ † M γ 0 ) for the relevant vertices.The B c → M (M denotes a pseudoscalar (P), a vector (V), an axial-vector (A) or a scalar (S) meson) form factors induced by vector and aixal-vector currents are defined as 1 A (P ′′ , ε) |A µ | B c (P ′ ) = − q 1 A (q 2 )ǫ µναβ ε * ν P α q β , (6) In calculations, the Bauer-Stech-Wirbel (BSW) [37] form factors for the B c → M transition are more frequently used and defined by where P = P ′ + P ′′ , q = P ′ − P ′′ , and the convention ǫ 0123 = 1 is adopted.
To smear the singularity at q 2 = 0 in Eq.( 13) and ( 14), the relations V BcA These two kinds of form factors are related to each other via

B. Wave functions and decay constants
In order to calculate the form factors, we need to specify the light-front wave functions.In principle, one can obtain them by solving the relativistic Schrödinger equation.But it is difficult to obtain the exact solution in many cases.Therefore, phenomenological wave functions are usually employed to describe the hadronic structure.In the present work, we shall use the phenomenological Gaussian-type wave functions where the parameter β ′ describes the momentum distribution and is approximately of order Λ QCD .It is usually determined by the decay constants through the analytic expressions in the conventional light-front approach, which are given as follows [10,11]: where m ′ 1 and m 2 are the constituent quarks of meson M(M = P, V, 3 A, 1 A, S).By the way, a tensor meson ( 3 P 2 state) cannot be produced through (V ± A) or tensor current, so we should not define its decay constant.The explicit forms of h ′ M are given by [11] h It is easy to see that the decay constants of the scalar meson and 1 A type of axial meson are zero for m ′ 1 = m 2 , which satisfy the SU(N) flavor constrain.The other nontrivial decay constants can be obtained through the experimental results for the purely leptonic decays or the lattice QCD calculations.The constituent quark masses used in the calculations will be listed in the next section.

C. Form factors
One important difference between the conventional light-front quark approach and the covariant one lies in the treatment of the constituent quarks.In the conventional lightfront framework, the constituent quarks are required to be on their mass shells, and the physical quantities, such as decay constant and form factor, can be extracted from the plus component of the corresponding current matrix elements.However, this framework misses the zero-mode contributions and renders the matrix elements non-covariant.In order to resolve this problem, the covariant light-front approach was proposed by Jaus [10], which provides a systematical way to deal with the zero-mode contributions by including the so-called Z-diagram contributions.Then physical quantities can be calculated in terms of Feynman momentum loop integrals in a manifestly covariant way.As a result, the constituent quarks of the meson will be off-shell.For the general B c → P transition, the decay amplitude for the lowest order is where arise from the quark propagators, and the trace S BcP µ can be directly obtained by using the Lorentz contraction, In practice, we use the light-front decomposition of the Feynman loop momentum and integrate out the minus component through the contour method.If the covariant vertex functions are not singular when performing integration, the transition amplitudes will pick up the singularities in the antiquark propagators.The integration then leads to with p ′′ ⊥ = p ′ ⊥ − x 2 q ⊥ , and M in the subscript and superscript denotes a pseudoscalar (P), a vector (V), an axial-vector (A) or a scalar (S) meson.The explicit forms of h ′(′′) M have been given in Eq.(31)-Eq.(33).For the B c → V, A transitions, the ω ′′ M (M = V, A) in the corresponding vertex operators listed in Tab.I are given as where After performing the integration with the contour method, we will be confronted with additional spurious contributions proportional to the light-like four-vector ω = (0, 2, 0 ⊥ ).These undesired spurious contributions can be eliminated by inclusion of the zero-mode contributions, which amount to performing the p − integration in a proper way.The specific rules under the p − integration have been derived in Refs.[10,11], and the relevant ones are collected in the Appendix.
Using Eqs.( 35)- (37) and taking the integration rules given in Refs [10,11], we obtain the B c → P form factors, It is similar for the B c → V transition amplitudes, which are given by [11] B BcV µ where From the above equation, we can get the expressions for B c → V form factors defined in Eqs.( 4) and ( 5) [11] g(q where the functions A (1) 2 , A 3 , A 4 and Z 2 are listed in the Appendix.and the physical form factors V BcV (q 2 ), A BcV 0 (q 2 ), A BcV 1 (q 2 ), and A BcV 2 (q 2 ) can be related to the above formulae through Eqs.(20) and (21).
The extension to B c → A transitions is straightforward and their form factors have similar expressions as those in the B c → V transitions case.The B c → 3 A, 1 A transition amplitudes are defined as [11] where the traces S Bc i A µν (i = 1, 3) By comparing Eq.( 42) and Eqs.( 49),(50), we have ) form factors can be related to the B c → V form factors through the following replacements: where the replacement of m ′′ 1 → −m ′′ 1 is not applied to m ′′ 1 in w ′′ and h ′′ , because they arise from the propagator and quark-antiquark-meson coupling vertex.The physical form factors can be related to the above formulae through Eqs.( 22) and ( 23).
we finally turn to the B c → S transition amplitude, which is given as [11] where the trace Using the formulae above and the integration rules obtained in Refs.[10,11], we have the B c → S form factors

III. NUMERICAL RESULTS AND DISCUSSIONS
Equipped with explicit expressions of the form factors for B c → S transitions, we now proceed to perform numerical studies using the CLFQM.In the earlier works [12,13], the form factors of B c decays into the ground-state charmonia and charmed mesons were calculated.In this work, besides updating the transition form factors of B c decays to these ground-state charmonia and charmed mesons, we also study the results of B c transitions to some excited-state charmonia.With these form factors, we then calculate the branching ratios of 80 B c decays with a charmonium involved in each channel.
As mentioned earlier, the shape parameter β ′ in the wave function describes the momentum distribution and can be calculated using the meson's decay constant under the CLFQM.The analytic expressions for the calculations are listed in Sect.2.2.The decay constant for the B c meson is employed by the result provided by the lattice QCD [38] which is larger than the value used in Refs.[12,13].The decay constant of J/Ψ can be determined by the leptonic decay width with the electric charge of the charm quark Q c = 2 3 , α em being a fine-structure constant.Using the updated measured result for the electronic width of J/Ψ given in PDG22 [39] Γ ee = (5.53± 0.10) keV, one can obtain the decay constant of J/Ψ which is different from the previous value f J/Ψ = (416 ± 5) MeV [12].Similarly, using the measured result Γ(ψ(2S) → e + e − ) = 2.36 ± 0.04 keV, we obtain the decay constant of the radially excited meson ψ(2S), f ψ(2S) = 296 +3 −2 MeV.The decay constant of the radially excited state ψ(3S) is determined as f ψ(3S) = (187 ± 8) MeV by using the data Γ ψ(3S)→ee = (5.53± 0.10) keV.As for the decay constant f ηc , we use the the lattice QCD result given in Ref. [40] which is a little larger than the value f ηc = 340.9+16.3  −16.6 MeV extracted from the data of η c → γγ decay.The decay constant f ηc(2S) can be determined by the double photon decay of η c (2S) as By using the measured results of the branching ratio Br(η c (2S) → γγ) = (1.9 ± 1.3) × 10 −4 and Γ ηc(2S) = 11.3 +3.2 −2.9 MeV [39], we can obtain the decay constant However, there is no calculation for the decay constant of η c (3S) or the data on η c (3S) → γγ decay used to extract it from experiment.We can fix the decay constant f ηc(3S) through the assumption f J/Ψ [41,42] and obtain it as To determine the shape parameter of χ c1 , we use the decay constant f χ c1 = 185 MeV evaluated from the light-cone QCD sum rules at the scale µ = m c [43].This value is much smaller than f χ c1 = 340 +119 −101 MeV given in Ref. [13].So the corresponding shape parameter β ′ χ c1 = (0.536 ± 0.023) GeV is smaller than the value β ′ χ c1 = (0.7 ± 0.1) GeV obtained in Ref. [13].For the charmonia χ c0 and h c , we will assume the same values and introduce an uncertainty of 10% to the shape parameters to compensate the different Lorentz structures, that is β ′ χ c0 = β ′ hc = (0.536 ± 0.023) GeV.The decay constant of X(3872) is determined using the branching fractions Br(B − → J/ΨK − ) = (1.026± 0.031) × 10 −3 and Br(B − → X(3872)K − ) = (2.3 ± 0.9) × 10 −4 and is obtained as  III: The shape parameters β ′ (in units of GeV) in the Gaussian-type light-front wave functions defined in Eq.( 25), and the uncertainties are from the decay constants.
0.382 +0.045 −0.054 0.536 ± 0.023 0.536 ± 0.023 0.536 ± 0.023 0.62 +0.057 −0.064 which is lower than f X(3872) = 329 +111 −95 MeV used in the previous CLFQM calculations [14].The experimental results for the decay constants of charmed mesons are given as [39] f D = (204.6± 5.0) MeV, f Ds = (257.5 ± 4.6) MeV.(66) As for the decay constants of the vector charmed meson D * and D * s , we used the lattice QCD results MeV [44]. 1 Using these decay constants and the masses of the constituent quarks and mesons given in Tab.II, we can obtain the values of the shape parameters β ′ for our considered mesons which are listed in Tab.III.
From Tab.IV, we can find that the form factors of B c transitions to charmed mesons (D, D * , D s , D * s ) at the maximally recoiling point (q 2 = 0) are smaller than those of B c transitions to ground-state charmonia.This is because the initial charm quark in the B c decays to charmed mesons is almost at rest, and its momentum is of order m c , while the charmed mesons in the final states move very fast, and the final charm quark tends to have a very large momentum of order m b .So the overlaps of the initial and final states' light-front wave functions in these transitions are limited, which induce small values for 0.17 +0.00+0.00−0.00−0.000.12 +0.00+0.00−0.00−0.00−1.53   63) are very large, so there are relevant large uncertainties in the B c → η c (2S) transition form factors.It is noted that if evaluating the form factors at q 2 > 0 region in the frame of q ⊥ = 0, we must include the non-valence configuration (the so-called Z-graph contribution) arising from quark pair creation from the vacuum, which is difficult for us to calculate reliably.While if one calculates in the frame of q + = 0, such non-valence contribution vanishes automatically.Because of the condition q + = 0 imposed in the course of calculation, the form factors are obtained only for space-like momentum transfer q 2 = −q 2 ⊥ ≤ 0, while the physical transition processes are relevant for the time-like form factors.Many authors [10][11][12] have proposed parametrization of form factors by using some explicit functions of q 2 in the space-like region, then extending them to the time-like region.Here, we will adopt the parametrization form given in Ref. [12]:
and A 2 (q 2 ) for the B c → D * s transition.
The parameters a and b will be fitted in the space-like region (−10 GeV 2 ≤ q 2 ≤ 0).The q 2 -dependence of form factors in the time-like region are plotted in Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11.In general the slope parameters a, b are very sensitive to the values of β ′ , while the form factors at q 2 = 0 are less sensitive to the variation in β ′ values.→ η c and B c → J/Ψ transition form factors at q 2 = 0 between this work and other literature.Here the results with ω = 0.8 GeV are quoted. 2The results out (in) the brackets are evaluated in sum rules (potential) model.FIG.7: Form factors V (q 2 ), A 0 (q 2 ), A 1 (q 2 ) and A 2 (q 2 ) for the B c → J/Ψ (left), B c → ψ(2S) (center) and B c → ψ(3S) (right) transitions.
In Tab.VI, we compare our results of the B c → D, D * , D s , D * s transition form factors at q 2 = 0 with other calculations.One can find that our results are consistent with those calculated using the relativistic quark model (RQM) [57,58], while they are too large for the results given by the relativistic constituent quark model (RCQM) [52], the light-cone sum rule (LCSR) [53,54], and the QCD sum rules [55,56].In Refs.[55,56], the form factors have a threefold enhancement by including the Coulomb-like α s /v corrections for the heavy quarkonium B c .It seems too small for the values of F BcD 0,1 predicted using the Bauer-Stech-Wirbel (BSW) relativistic quark model [48].Compared with the previous CLFQM calculations [12], our predictions for the form factors V BcD * , A BcD * 0,1,2 have a significant enhancement by using a larger decay constant f D * , while the influence from the difference values for the decay constant f Bc is small.
For the B c transition to a ground-state vector meson, which is either a charmed meson or a charmonium, the form factor V is the largest one, and A 0,1,2 are close to each other.It is easy to find this character in Figs. 4 and 5 and the first pannel of Fig. 7. On the other hand, if the final-state meson is a radially excited meson ψ(2S) or ψ(3S), the form factors show a hierarchy V > A 2 > A 1 > A 0 , which can be found in the last two pannels of Fig. 7. Nevertheless, V is always the largest one among the form factors of B c transition to either a ground state or a radially excited charmonium.There also exits another hierarchy for the B c → η c , η c (2S), η c (3S) transitions, . The q 2 dependence of the B c → χ c1 transition is shown in Fig. 9, where V 1 is much larger than other form factors A, V 0,2 .It is very like the case of the B c → X(3872) transition shown in Fig. 10.Thus it is a natural assignment of this state as the first radial excitation of a 1P charmonium state χ c1 .Because both χ c1 and X(3872) have the same quantum numbers J P C = 1 ++ , they should have similar properties in B c decays, while it is very different for the B c transition to another type of axial vector meson h c with J P C = 1 +− , where the value of V 0 is large and close to that of V 1 as shown in Fig. 11.Certainly, the values of the form factor V 1 for both of B c transitions to these two types of axial vector charmonia are large.By comparing with Fig. 10 and 11, one can find that both of the values of V 2 in the B c → X(3873) and B c → h c transition form factors are the smallest; especially, V 2 for the B c → h c transition becomes negative.As we know, there is sill no definite answer about the internal properties of the X(3872).From Fig. 10, one can find that the form factors of the B c → X(3872) transition are almost flat in their q 2 behaviors except for A BcX(3872) .A comparison of these values with experimental measurements for the B c → X(3872) transition form factors will provide unique insight into the mysterious inner structure of X(3872).The form factors for the B c to these P-wave charmonium transitions are listed in Tab.VII.
From Tab.VIII, we can find that the form factors of B c → η c (2S, 3S), ψ(2S, 3S) transitions calculated in the PQCD approach [29] may be more than twice as larger as those predicted in the CLFQM, which will induce large differences for the branching  A Bchc 0.06 +0.00+0.00−0.00−0.000.01 +0.00+0.00−0.00−0.00−6.24 ratios of some correlative decay channels given by these two approaches.In the PQCD appraoch, the form factors are sensitive to the formulae of the B c wave functions.In Ref. [29], the authors argued that the B c wave function in the light-cone formula is broader in shape than that of the traditional zero-point one, which is ∝ δ(x − r c ), so the overlap between the initial and final states' wave functions becomes larger by using the light-cone wave function for the B c meson, which induces larger form factors.Our predictions for the B c to vector charmonium J/Ψ, ψ(2S) transition form factors are close to the results given by the LFQM calculations [15] except for A Bcψ(2S) 2 . For the B c to the axial vector charmonium transition form factors, our results are also consistent with the previous CLQM calculations [13,14].

B. Branching ratios
Besides the masses of the constituent quarks and mesons listed in Table.II, other inputs, such as the B c meson lifetime τ Bc , the Wilson coefficients a 1 , a 2 , and the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, are listed as [39,62] τ Bc = (0.510 ± 0.009) × 10 −12 s, a 1 = 1.07, a 2 = 0.234, (68) First, we consider the branching ratios of the decays B c → η c (J/Ψ)P (V ), which can be calculated through the formula This work 0.57 0.41 0.35 0.17 [15] 0.525 0.452 0.335 0.102 [29] 1.71 0.80 0.87 1.22 This work 0.46 0.31 0.27 0.14 [29] 1.07 0.41 0.41 0.66 where the decay width Γ(B c → η c (J/Ψ)P (V )) for each channel is given as following with the subscript q = d(s) in the CKM element V uq for the decays with π, ρ(K, K * ) involved.For the decays B c → J/ΨV , the corresponding decay width is the summation of the three polarizations where p is the three-momentum of either of the two final states in the B c rest frame and the three polarization amplitudes A L , A N , andA T are given as − 4r 2 J/Ψ .From Tab.IX, one can find that our predictions are consistent with the results given by the QCD sum rules [32], the relativistic constituent quark model (RCQM) [34] and the Bethe-Salpeter equation approach under the so-called instantaneous nonrelativistic approximation [33].In Ref. [36], the authors calculated these decays in the nonrelativistic QCD (NRQCD) approach at the next-to-leading-order (NLO) in the QCD coupling α s .It is interesting that the leading-order (LO) results for these channels, except for the decay B + c → J/Ψρ + are in agreement with our predictions, while the branching ratios obtain substantial enhancement after including the NLO QCD correction, which provides a large factor K. We wonder whether these results will still be stable with the higher order corrections, such as the next-to-next-to-leading-order (NNLO) contributions, involved.In Ref. [63], the branching ratios are calculated in the relativistic quark model using v/c expansion for B c and the charmonium, and the obtained results are smaller than most of other predictions, including ours; for example, their results are only about one third of our predictions in most cases.Certainly, the results given in the QCD relativistic (potential) models [66,67] are also small.As mentioned earlier, the results calculated using the PQCD approach are sensitive to the types of wave functions for B c meson (the traditional zero-point wave function and the light-cone wave function).For example, if taking the light-cone wave function for the B c meson, the branching ratio of the decay B + c → η c π + will reach (5.2 +2.6 −1.4 ) × 10 −3 [29], which is much larger than (2.98 +1.24 −1.05 ) × 10 −3 obtained using the traditional zero-point one.From Tab.IX, one can find that the ratio of the branching fractions R K/π ≡ Br(B + c →J/ΨK + ) Br(B + c →J/Ψπ + ) = 0.081 ± 0.011, which is consistent with the value R K/π = 0.079 ± 0.007 ± 0.003 given by the LHCb collaboration [69].
If replacing P (V ) with D(D * ), the branching ratios of the corresponding decays B c → η c (J/Ψ)D(D * ) can be obtained by their decay widths: For the decay B c → J/ΨD * , the corresponding decay width is the summation of the three polarizations: where the three polarization amplitudes A L , A N and A T are given as As for the decay widths of the decays B c → η c (J/Ψ)D s (D * s ), they can be obtained by performing the replacements 79)− (85).The calculation results are listed in Tab.X.One can find that our predictions are a little larger than most of other results, but are smaller the the PQCD calculations.The branching ratios of the decays with D ( * ) s involved are at least one order larger than those of the corresponding decays with D ( * ) involved.This is because the CKM matrix element V cs associated with the former is much larger than V cd associated with the latter.All of these decays with the ground-state S-wave charmonia involved have large branching ratios, which lie in the range of 10 −4 ∼ 10 −3 and can be detected by the present LHCb experiments.In Ref. [24], if assuming that the spectator diagram dominates and that factorization holds, one can obtain the approximations which were measured as R D + s /π + = 2.90 ± 0.57 ± 0.24 and R D * + s /D + s = 2.37 ± 0.56 ± 0.10 by the LHCb collaboration [24], and given as R D + s /π + = 2.76±0.47 and R D * + s /D + s = 1.93±0.26by ATLAS [71].From our calculations, these two ratios are obtained as where the value of R D + s /π + is consistent with the measurements given by LHCb and ATLAS, while R D * + s /D + s can explain the ATLAS result within errors.
Next, we consider the decays with the P-wave charmonia involved in the final states.The P-wave charmonium can be χ c0 , χ c1 or h c .The decay widths of the decays B c → χ c0(1) P (V ), h c P (V ) are given as follows where the subscript q = d(s) in the CKM element V uq for the decays with π, ρ(K, K * ) involved.For the decays B c → χ c1 V , the corresponding decay widths are the summation of the three polarizations where It is noted that the analytic formulae of the decay widths between the decays B c → h c P (V ) and B c → χ 1c P (V ) are similar.Summing the branching fractions of the these decays in Tab.XI, we find that the results of the decays with π + (ρ + ) involved are about one order of magnitude larger compared with those of the decays with K + (K * + ) involved.The difference mainly comes from the the CKM matrix elements: the former involve a larger factor V ud ∼ 1, while the latter is associated with a smaller factor V us = λ ∼ 0.225.Our predictions are comparable to most other theoretical results, such as the QCD-motivated RQM based on the quasi-potential approach [72], the NRQM [59], the RCQM [73].The branching ratios of the decays B c → χ c0(1) P (V ) predicted by most works have a common property: Br(B c → χ c0 P (V )) are much larger than Br(B c → χ c1 P (V )).This characteristic can be tested by the present LHCb experiments.
If replacing P (V ) with D(D * ) in the upper decays, the branching ratios of the corre- sponding decays B c → χ c0(1) (h c )D(D * ) can be obtained by their decay widths: For the decay B c → χ c1 D * , the corresponding decay width is the summation of the three polarizations: where the three polarization amplitudes A L , A N and A T are given as with As to the decay widths of the channels B c → χ c0 (χ c1 )D s (D * s ), they can be obtained by performing the replacements D → D s , D * → D * s , V cd → V cs in Eqs. ( 96)− (102).The branching ratios of the these decays are given in Tab.XII, where we also list the results given by the Salpeter method [78].This method is the relativistic instantaneous approximation of the original Bethe-Salpeter equation.

−
Though the X(3872) has been confirmed by many experimental collaborations, such as CDF [80], D0 [81], Babar [82] and LHCb [83], with quantum numbers J P C = 1 ++ and isospin I = 0, there are still many uncertainties.Though many different exotic hadron state interpretations, such as a loosely bound molecular state [84][85][86][87][88], a compact tetraquark state [89][90][91][92], ccg hybrid meson [93,94], glueball [94], have been put forward, the first raidal excitation of 1P charmonium state χ c1 (1P ) as the most natural assignment has not been ruled out [96][97][98].By assuming the X(3872) as a 1 ++ charmonium state, we calculate the branching ratios of the decays B + c → X(3872)M (here, M represents a light pseudoscalar, a vector meson, or a charmed meson).The analytic expressions of the corresponding decay widths are similar to those of the decays B c → χ c1 M listed in Eqs.(91), (92), (98) and (99).In Tab.XIII, we list the branching fractions of the decays B c → X(3872)M.One can find that our predictions for the decays B + c → X(3872)π + (K + ) are consistent with the results given in the PQCD approach [79], while they are much larger than those calculated in the generalized factorization (GF) approach [30].Certainly, some of these decays studied using the CLFQM about 15 years ago [14], the differences between our predictions and the previous calculations are induced by taking different values for some parameters.
Lastly, we turn to the branching ratios of the decays with the radially excited S-wave charmonia, such as η c (2S, 3S) and ψ(3S, 3S), involved in the final states.The correspond decay widths are similar to those of the decays B c → η c M, J/ΨM, where M represents a light pseudoscalar, a vector meson, or a charmed meson (D ( * ) , D ( * ) s ).As we know, in order to compare with experiments, the ratios are often used.If we still employ the traditional light-front wave functions for the radially excited chuarmonia given in Eq.( 25), we will get larger branching ratios than most other theoretical predictions; even worse, the obtained value R ψ(2S)/J/Ψ = 0.467 is much larger than the experimental data R ψ(2S)/J/Ψ = 0.268 ± 0.032 ± 0.007 ± 0.006 given by PDG [39].There exists a similar case in Ref. [15], so we follow the same strategy by choosing the modified harmonic oscillator wave functions: ).( 106) In order to keep the orthogonality and normalization for the wave functions of these radially excited states, one needs to introduce a factor n δ into the exponential functions in these wave functions, which can be determined by fitting the data of the corresponding decay constants.Similarly, there exists a 1/n exponential dependence factor in the wave functions of the hydrogen-like atoms, which are obtained by solving the Schrödinger equation.In Ref. [15], the authors supposed that the parameters shown in Eqs. ( 105) and ( 106) are the same as those for Υ(2S) and Υ(3S) under the heavy quark effective theory, which are given as [105] It is noted that these parameters are determined by assuming that the Υ(iS)(i = 2, 3) mesons have the same β ′ values as that of β ′ Υ for Υ(1S).In fact, under this assumption, once the value of δ is fixed, these parameters given in Eq.( 107) can be determined using the orthogonality and normalization for the wave functions of these ground and radially excited states.That is to say, if we only replace β ′ Υ with β ′ J/Ψ , the values of parameters a 2,3 , b 2,3 , c 3 are not changed.We call this case as scenario I (SI).As another possibility, we also assume here that each value of β ′ in the wave functions of J/Ψ and ψ(2S, 3S) is different but with δ = 1/1.82fixed, then we can get another group of values for these parameters: which are called as scenario II (SII).In this work, we calculate in these two scenarios for the B c decays with ψ(2S) or ψ(3S) involved in the final states.By using these modified wave functions for ψ(2S), one can obtained that R ψ(2S)/J/Ψ = 0.212 ± 0.071 in SI, which is consistent with the data.At the same time, the tensions between our predictions with other theoretical results are greatly reduced.For example, the branching ratio Br(B + c → η c (2S)π + ) = (7.70+0.11+0.03+0.84−0.12−0.04−0.52 ) × 10 −4 using the traditional light-front wave function for η c (2S), while Br(B + c → η c (2S)π + ) = (3.35+0.06+0.02+0.89−0.06−0.02−1.20 ) × 10 −4 by replacing with the modified wave function in SI, which are close to the results given by Refs.[33,99,101,102].This is similar for the decay B + c → ψ(2S)π + .The branching ratios of the decays B + c → η c (2S)π + and B + c → ψ(2S)π + in SI are close to each other; this is also supported by most of the other theoretical predictions shown in Table XIV.Furthermore, the differences of the branching ratios of the decays B c → J/Ψ(2S)P (V ) between these two scenarios are not large, while they are very different for the decays with η c (2S) involved.So one can use the decay channels B c → η c (2S)P (V ) to check which scenario is more accurate by comparing with the future experimental data.XIV: CLFQM predictions for branching ratios (×10 −4 ) of the decays B c → ψ(2S)P (V ), η c (2S)P (V ) with P (V ) representing a light pseudoscalar (vector) meson.For each decay channel, we calculate both in scenario I (upper line) and scenario II (lower line).The first two errors for these entries correspond to the uncertainties from the lifetime and the decay constant of the initial meson B c , respectively.The last one is from the parameter δ in the modified wave functions of the radially excited charmonium ψ(2S) or η c (2S). ).From our calculations and the numerical results, we find the following points: 1.The branching ratios of the decays with η c (2S) involved are more sensitive to the shape parameter β ′ .For example, for the decays B + c → η c (2S)D + (s) , η c (2S)D * + (s) , their branching ratios in SI are about five times larger than those in SII, while for the decays B + c → ψ(2S)D + (s) , ψ(2S)D * + (s) , the differences of the results between these two scenarios are less than two times.

The branching ratios of the decays with D +
s or D * + s involved are at least one order larger than those of the corresponding decays with D + or D * + involved.It is because the former (the latter) are suppressed (enhanced) by the CKM matrix elements.
3. On the whole, the predictions in SII are closer to other theoretical results than those in SI, which supports that taking different value of the shape parameter β ′ for each radially excited charmonium is more reasonable.
At present, only a few papers have studied B c decays with ψ(3S) or η c (3S) involved, which are listed in Tables XVI and XVII.Most theoretical predictions show that the branching ratios of these decays are about or less than 10 −4 .Meanwhile, for the decay B + c → η c (3S)π + , its branching ratio was predicted as 1.4 × 10 −3 in the PQCD approach [29], where the authors obtained that the branching ratios of the decays B c → η c (2S)π + TABLE XVI: CLFQM predictions for branching ratios (×10 −5 ) of the decays B c → ψ(3S)P (V ), η c (3S)P (V ), with P (V ) representing a light pseudoscalar (vector) meson.For each decay channel, we calculate both in SI (upper line) and SII (lower line).The errors for these entries are the same with those in Table XIV.0.79(0.81)0.225 − 0.044 +0.001+0.000+0.017−0.001−0.000−0.042also researched by using the improved Bethe-Salpeter method [103], where the branching ratios of the decays B c → ψ(3S)D * + , η c (3S)D * + s are consistent with our predictions in SI.Whereas, the branching ratio of the decay B + c → ψ(3S)D + is predicted as 3.62×10 −8 and much smaller than our result.We predict that some of the decays with η c (3S) or ψ(3S) involved, such as B c → η c (3S)ρ, B c → ψ(3S)D * s , might have larger branching ratios (up to 10 −4 ) and may be accessible at the High Luminosity Large Hadron Collider in the near future.
Comparing Tables IX, XIV, and XVI, one can find that there is a hierarchy for these decays: Br(B c → J/ΨP (V )) > Br(B c → ψ(2S)P (V )) > Br(B c → ψ(3S)P (V )), (109) Br(B c → η c P (V )) > Br(B c → η c (2S)P (V )) > Br(B c → η c (3S)P (V )), (110) where P (V ) represents a light pseudoscalar (vector) meson.This is because for the decays with the higher excited charmonia involved, the phase spaces are tighter, and the form factors are smaller and less sensitive to the change of the momentum transfer q 2 .

IV. SUMMARY
In this work we study the form factors of B c decays into charmonia in the coari-  ant light-front quark model.Here, the charmonia refer to the S-wave mesons, such as J/Ψ, η c , the corresponding radially excited states, such as ψ(2S, 3S), η c (2S, 3S), and the P-wave mesons, such as χ c0 , χ c1 , h c and X(3872).Certainly, the form factors of the B c → D ( * ) , D ( * ) s transitions are also considered for the purpose of the branching ratio calculation.We find that the analytic expressions for B c → S, A transition form factors can be obtained from those of B c → P, V analytical expressions by some simple replacements.The form factor F Bcηc (V BcJ/Ψ ) has been calculated by many approaches, most results of which lie in the range of 0.5 ∼ 0.7 (0.5 ∼ 1.0).We obtain a moderate value F Bcηc = 0.6(V BcJ/Ψ = 0.76).This can be used to check which method is more favored by comparing to the future experimental data.Compared with the form factors of B c transitions to these two ground-state S-wave charmonia, those of B c transitions to the radially excited S-wave charmonia, P-wave charmonia and charmed mesons are smaller.Except for each form factor at the zero recoiling point, we also calculate the corresponding one at the maximally recoiling point.Furthermore, we plot the q 2 -dependence for each transition form factor. Then we calculate the branching ratios of 80 B c decays with a charmonium involved in each channel.We find that the decays B + c → J/Ψ(η c )π + (ρ + ) and B + c → J/Ψ(η c )D + s (D * + s ) have larger branching ratios, which can reach the order of 10 −3 , while most other decay channels have smaller branching ratios, which are suppressed by 1 ∼ 3 orders.These predictions will be tested in the future by the LHCb experiments.

3 (
+0.01+0.00−0.01−0.000.60 +0.05+0.06−0.03−0.042.85 +0.02+0.19−0.02−0.100.67 +0.00+0.06−0.00−0.04 the form factors.In the B c transitions to charmonia, both the spectator charm quark and the charm antiquark generated from the weak vertex are heavy, and the light-front wave functions of the charmonia have a maximum near E ∼ m c .It is expected that the overlaps of the B c and charmonium's light-front wave functions become large, which induce larger form factors. Thus it is easy to understand that for the B c decays to the charmonium and charmed meson, for example B c → J/ΨD, the Feynman amplitudes associated with B c transitions to charmonia are much more important than that associated with B c transitions to charmed mesons.Furthermore, the SU(3) symmetry breaking effects between the form factors of B c → D and B c → D s transitions are large, since the decay constant of D s meson is larger than that of the D meson.It is similar between the form factors of B c → D * and B c → D * s transitions.These can be checked by future experiments.The uncertainties from the decay constant of η c (2S) shown in Eq.(

TABLE VI :
Comparison of the B c → D, D * , D s , D * s transition form factors at q 2 = 0 between this work and other literature.

TABLE VII :
Results for the B c → χ c0 , χ c1 , h c , X(3872) transition form factors and the fitted parameters a and b.The uncertainties are from the decay constants of B c and final state mesons.

TABLE IX :
The CLFQM predictions for branching ratios (10 −3 ) of B c decays to final states containing a ground-state S-wave charmonium (η c or J/Ψ) and a light pseudoscalar or vector meson.The first error is induced by the B c meson life time, and the second and third uncertainties are from the decay constants of B c and charmonia.

TABLE X :
CLFQM predictions for branching ratios (10 −3 ) of B c decays to final states containing a ground-state S-wave charmonium (η c or J/Ψ) and a charmed meson.The errors are induced by the same sources as in TableIX.

TABLE XI :
The CLFQM predictions for branching ratios (10 −3 ) of B c decays to final states containing a P-wave charmonium and a light pseudoscalar or vector meson.The errors are induced by the same sources as in TableIX.

TABLE XII :
CLFQM predictions for branching ratios (10 −3 ) of B c decays to final states containing a P-wave charmonium (χ c0 , χ c1 or h c ) and a light pseudoscalar or vector meson.The errors are induced by the same sources as in TableIX.

TABLE XIII :
CLFQM predictions for branching ratios (×10 −3 ) of the decays B c → X(3872)M , where M represents a light pseudoscalar, a vector meson, or a charmed meson.The errors are induced by the same sources as in TableIX.

TABLE XVII :
CLFQM predictions for branching ratios (×10 −5 ) of the decays B + ), η c (3S)D + (s) (D * + (s) ).For each decay channel, we calculate in scenario I (upper line) and scenario II (lower line).The errors for these entries are the same as those in TableXIV.