Semileptonic decays $B_c \to (\eta_c,J/\psi) l \bar{\nu}_l $ in the"PQCD + Lattice"approach

In this paper, we studied the semileptonic decays B c → (ηc, J/ψ)lν̄l by employing the PQCD factorization approach, using the newly defined distribution amplitudes of theBc meson and also taking into account the Lattice NRQCD results about the relevant form factors. We found the following main results: (a) the PQCD predictions for the branching ratios will become smaller by about (10−50)% when the Lattice NRQCD results are taken into account in the extrapolation of the relevant form factors; (b) the PQCD predictions for the ratios Rηc , RJ/ψ and the longitudinal polarization Pτ are Rηc = 0.373 +0.003 −0.012, RJ/ψ = 0.300 +0.005 −0.004, P ηc τ = 0.356 +0.003 −0.005 and P J/ψ τ = −0.557 ± 0.002 ; and (c) after the inclusion of the Lattice NRQCD results the theoretical predictions changed moderately: Rηc = 0.300 +0.033 −0.031, RJ/ψ = 0.230 +0.041 −0.035, P ηc τ = 0.345 ± 0.010 and P J/ψ τ = −0.427 −0.093. The theoretical predictions for RJ/ψ agree with the measured one within errors, and other predictions could be tested in the future LHCb experiments.


I. INTRODUCTION
In the standard model (SM), all electroweak gauge bosons ( Z, γ and W ± ) have equivalent couplings to three generation leptons, and the only differences arise due to mass effects: m e < m µ ≪ m τ : this is the so-called Lepton flavor universality (LFU) in the SM . The B c meson can only decay through weak interactions because it is below the B-D threshold, it is therefore an ideal system to study the weak decays of heavy quarks. Since the rare semileptonic decays governed by the flavor-changing neutral currents (FCNC) are forbidden at the tree level in the SM, the precision measurements for such semileptonic B c decays can play an import role in testing the SM and in searching for the signal and/or evidence of NP beyond the SM. In recent years, the measured values of R(D) and R(D * ) [1] , defined as the ratios of the branching fractions B(B → D ( * ) τ ν τ ) and B(B → D ( * ) lν l )) , are clearly larger than the SM predictions: the combined deviation is about 3.8σ for R(D) − R(D * ) or 3σ for R(D * ) alone [1]. The semileptonic decays B → D ( * ) lν l with l = (e, µ, τ ) are therefore studied intensively by many authors in the framework of the SM [2][3][4][5][6], or in various new physics (NP) models beyond the SM for example in Refs. [5,[7][8][9].
If the above mentioned anomalies are indeed the first signal of the LFU violation (i.e. an indication of new physics ) in B u,d sector, it must be appeared in the similar semileptonic decays of B s and B c mesons. The B c (bc) meson, as a bound state of two heavy bottom and charm quarks, was firstly observed by the CDF Collaboration [10] and has been observed and studied at the Large Hadron Collider (LHC) experiments in recent years [11]. The properties of B c meson and the dynamics involved in B c decays could be fully exploited through the precision measurements at the LHC experiments, especially the measurements carried on by the LHCb Collaboration. Very recently, some hadronic and semileptonic B c meson decays have been measured by LHCb experiments [12,13]. Analogous to the cases for B decays, the generalization of the R(D ( * ) ) for the semileptonic B c decays are the ratio R ηc and R J/ψ : , f or X = (η c , J/ψ).
In a previous work [15] , we calculated the ratio R J/ψ and R ηc by employing the PQCD approach [44,45] and found the PQCD predictions [15]: which also agree well with the ones from the QCDSR or other different approaches in the frame work of the SM . In this paper, we will present a new systematic evaluation for the ratio R J/ψ and R ηc by using the PQCD factorization approach but with the following new improvements: (1) We here will use a newly developed form of φ Bc (x, b) as proposed very recently in Ref. [46]: instead of the simple δ-function as being used in Ref. [14,15]: (2) We will take into account the Lattice QCD results for the semileptonic form factors of the decays B c → (J/ψ, η c )lν, as reported by the HPQCD Collaboration [39][40][41], in the extrapolation of the relevant form factors from the low-q 2 region to the higher q 2 region. We will calculate the ratios R J/ψ and R ηc by using both the PQCD approach and the "PQCD+Lattice " method, and compare the resultant predictions.
(3) Besides the ratios R ηc,J/ψ of the branching ratios, we here will calculate the longitudinal polarization P τ (η c ) and P τ (J/ψ) of the final state tau lepton, which was absent in Ref. [15]. Just like the polarization P D * τ firstly measured at Belle [47] , both P τ (η c ) and P τ (J/ψ) might be measured in the future LHCb experiment.
The paper is organized as follows: after this introduction, we give the distribution amplitudes of the B c meson and the final state η c and J/ψ mesons in Section 2. Based on the k T factorization formalism, we calculate and present the expressions for the B c → (η c , J/ψ) transition form factors in the large recoil regions in Section 3. The numerical results of the branching ratios, the ratios R M and the longitudinal polarization P (τ ) are given in Section 4. A short summary will be given in final section.ν FIG. 1. The charged current tree Feynman diagrams for the semileptonic decays B − c → Xl −ν l with X = (η c , J/ψ) and l = (e, µ, τ ) in the PQCD approach.

II. KINEMATICS AND THE WAVE FUNCTIONS
The lowest order Feynman diagrams for B c → Xlν are displayed in Fig. 1. The kinematics of these decays are discussed in the large-recoil (low q 2 ) region, where the PQCD factorization approach is applicable to the considered semileptonic decays involving η c or J/ψ as the final state meson [48]. In the B c meson rest frame, we define the B c meson momentum P 1 , and the final state meson momentum P 2 in the light-cone coordinates as [15,49] with where r = m M /m Bc is the mass ratio, and q = p 1 − p 2 is the momentum of the lepton pair. The longitudinal polarization vector ǫ L and transverse polarization vector ǫ T of the vector meson are defined in the same way as in Ref. [15]: The momentum k 1 and k 2 of the spectator quark in B c or in final state (J/ψ, η c ) are parameterized in the same way as in Ref. [15]. For the B c meson wave function, we make use of the same one as being used for example in Ref. [14,15], As for the distribution amplitude (DA) φ Bc (x, b), we here will use a new φ Bc (x, b) [46] as shown in Eq. (4) instead of the simple δ-function as given in Eq. (5). As usual, the normalization constant N Bc in Eq. (4) is fixed by the relation where the decay constant f Bc = 0.489 ± 0.005 GeV has been obtained in lattice QCD by the TWQCD Collaboration [50]. We will set the factor β Bc in Eq. (4) in the value of β Bc = 1.0 ± 0.1 GeV in order to estimate the uncertainty. For the pseudoscalar meson η c and the vector one J/Ψ, we use the same wave function as those in Ref. [15,16]: where the twist-2 asymptotic DAs are the same as those being used in Refs. [15,16].

III. THE FORM FACTORS AND DIFFERENTIAL DECAY WIDTHS
For the considered charged current B c → (η c , J/ψ)l −ν l decays, the quark level transition is the b → cl −ν l decay with the effective Hamiltonian where G F = 1.16637 × 10 −5 GeV −2 is the Fermi-coupling constant and V cb is the CKM matrix element. The differential decay widths of the semi-leptonic decays B − c → η c l −ν l can be written [15,18] in the following form: where m l is the mass of the charged leptons, Bc m 2 ηc is the phase space factor. In the PQCD factorization approach, the form factor F 0 (q 2 ) and F + (q 2 ) in Eq. (15) defined through the matrix element < η c (p 2 )|c(0)γ µ b(0)|B c (p 1 ) > [15,18] can be calculated and can be written as a summation of the auxiliary form factor f 1,2 (q 2 ): After making the analytical calculations in PQCD approach, one found the function f 1,2 (q 2 ): with the functions H i (t i ) in the following form where For B − c → J/ψl −ν l decays, the differential decay widths can be written in the following form [15,18]: The form factors V (q 2 ) and A 0,1,2 (q 2 ) can also be calculated in the framework of the PQCD factorization approach: where r = m J/ψ /m Bc , and the functions H i (t i ) are the same ones as those defined in Eq. (19).

NUMERICAL RESULTS
In the numerical calculations we use the following input parameters (here masses and decay constants are in units of GeV) [1,11]: For the considered semileptonic B c meson decays, it is easy to see that the theoretical predictions for the differential decay rates and other physical observables strongly depend on the form factors F 0,+ (q 2 ) for B c → η c lν l decays , and the form factors V (q 2 ) and A 0,1,2 (q 2 ) for B c → J/ψlν l decays [15,18]. The value of these form factors at q 2 = 0 and their q 2 dependence in the whole range of 0 ≤ q 2 ≤ q 2 max contain a lot of information of the physical process. Up to now, these form factors have been calculated in many rather different methods, for example, in Refs. [17,21,22,24,25,27,28].
In Refs. [3,4,49,51], the authors examined the applicability of the PQCD approach to (B → D ( * ) ) transitions, and have shown that the PQCD approach with the inclusion of the Sudakov effects is applicable to the semileptonic decays B → D ( * ) lν l at the low q 2 region [3,4]. Since the PQCD predictions for the considered form factors are reliable only at the low q 2 region, we first calculate explicitly the values of the relevant form factors at the sixteen points in the lower TABLE I. The PQCD predictions for the form factors F 0,+ (q 2 ), V (q 2 ) and A 0,1,2 (q 2 ) at q 2 = 0, and the parametrization constants "a" and "b".  region 0 ≤ q 2 ≤ m 2 τ by using the expressions as given in Eqs. (17,18,(23)(24)(25)(26) and the definitions in Eq. (16).
In the large q 2 region m 2 τ ≤ q 2 ≤ q 2 max with q 2 max = (m Bc − m x ) 2 and x = (η c , J/ψ), however, one has to make an extrapolation for all relevant form factors from the lower q 2 region to larger q 2 region. In this work we will make the extrapolation by using two different methods.
(1) The first method is the same one as being used in Ref. [15]: using the parametrization formula [18,52] where F i (q 2 ) stands for the relevant form factors F 0,+ (q 2 ), V (q 2 ), A 0,1,2 (q 2 ), and a, b are the parameters to be determined by the fitting procedure. In Table I In Table II , as a comparison, we show the central values of all form factors F i (0) in the PQCD approach and some other typical approaches, such as the BSW [53], the NRQCD [34] and the LCSR [17]. One can see easily that the theoretical predictions from different approaches are indeed rather different in values.
(2) The second one is the " PQCD+Lattice" method, similar with what we did in Ref. [54]. The authors in HPQCD Collaboration [39,40] calculated the form factors f 0,+ (q 2 ) for B c → η c transition, and V (q 2 ) and A 1 (q 2 ) for B c → J/ψ transition by using the "Lattice NRQCD" at q 2 = 0 and several other points of q 2 . In order to reduce the theoretical uncertainty in the extrapolation of F i (q 2 ), we use their results at point q as the Lattice QCD input in the fitting process. At present no Lattice QCD results are available for other two B c → J/ψ form factors A 0,2 (q 2 ).
In Fig. 2 and 3 , we show the theoretical predictions for the q 2 −dependence of the six relevant form factors for B c → (η c , J/ψ) transitions, obtained by using the PQCD approach and the "PQCD+ Lattice" approach. The theoretical predictions based on the LFQM [18] are also shown in these two figures as a comparison. In these two figures, the blue dashed curves show the theoretical predictions for the q 2 -dependence of f 0,+ (q 2 ) and (V (q 2 ), A 0,1,2 (q 2 )) in the pQCD approach only [15], the green dot-dashed curves denote the form factors evaluated by using the LFQM as given for example in Ref. [18]. In Fig. 2 and 3 (a,c), the red solid curves show the four form factor (f 0,+ (q 2 ), V (q 2 ), A 1 (q 2 )) obtained by using the "PQCD+Lattice" approach. One can see from the theoretical predictions as illustrated in Figs. 2 and 3 that (a) the theoretical predictions for the form factors from three different methods are very similar with each other at the low q 2 region, but become rather different at large q 2 region for the results from the PQCD and the LFQM method; (b) the difference between the theoretical predictions from the "PQCD+Lattice" method and the LFQM remain small even in the large region of q 2 . From the formulae of the differential decay rates as given in Eqs. (15,22), it is straightforward to make the integration over the range of m 2 where the major theoretical errors come from the uncertainties of the input parameters β Bc = 1.0 ± 0.1 GeV, |V cb | = (40.5 ± 1.5) × 10 −3 and m c = 1.27 ± 0.03 GeV.  In Table III, we list the central values of the theoretical predictions (in unit of 10 −3 ) for the branching ratios of the decays B c → (η c , J/ψ)l −ν l with l = (µ, τ ), obtained in this paper, or from the previous PQCD work [15], and from other different models or approaches [17,18,34,38]. One can see that the difference between different theoretical predictions can reaches a factor of two for the same decay mode. In Table IV, we show the theoretical predictions for the ratios R ηc and R J/ψ of the branching ratios for the considered semileptonic B c decays, as defined in Eq. (1) and evaluated in this paper, a previous PQCD work [15], and other three models [17,18,34,38].
From the theoretical predictions for the branching ratios and the ratios R x of the branching ratios as listed in Eqs. (30)(31)(32)(33) and Table III and IV, we find the following points: (1) Because of the phase space suppression, the branching ratios of the decay modes with a final τ lepton are smaller than those decay modes with a final µ lepton. The PQCD predictions for the branching ratios become smaller by a degree of (10 − 50)% when the Lattice NRQCD results are taken into account in the extrapolation of the relevant form factors.
(2) The central values of the ratio R ηc and R J/ψ are around 0.23 − 0.37 in all considered models or approaches, while R J/ψ is a little smaller than R ηc due to the mass suppression of the phase space. The theoretical predictions for R J/ψ in both PQCD and PQCD+Lattice methods are smaller than the measured value as given in Eq. (2) , but still agree with it because of still large errors of the experimental measurements. These ratios could be measured in high precision at the future LHCb experiment and can help us to test the theoretical models or approaches.
(3) The dominant theoretical error comes from the uncertainty of the factor β Bc in the B c meson distribution amplitude φ Bc (x, b). The theoretical errors of the PQCD or PQCD+Lattice predictions for the branching ratios are still large, say around (10 − 30)% in magnitude as can be seen from the numerical results in Eqs. (30)(31)(32)(33). For the ratios R ηc and R J/ψ , however, the theoretical errors are largely cancelled in these ratios of the branching ratios. One can see from the numerical results as listed in Table IV that the theoretical error of R ηc and R J/ψ is now about (3 − 10)% in size.
For both kinds of the semileptonic decays B → D ( * ) l −ν l and B − c → (η c , J/ψ)l −ν l , their quark level weak decays are indeed the same charged current tree transitions: b → cl −ν l with l = (e, µ, τ ). The only difference between them are the spectator quark: one is the charm quark, another is the up or down quark. Consequently, it is reasonable to assume that the dynamics for these two kinds of semileptonic decays are similar in nature, we therefore can use similar method to study these two kinds of semileptonic decays.
For B → D ( * ) τν τ decays, besides the decay rates and the ratios R(D ( * ) ), the longitudinal polarization P τ (D ( * ) ) of the tau lepton and the fraction of D * longitudinal polarization F D * L are also the additional physical observables and are sensitive to some kinds of new physics [55][56][57][58] .
After making the proper integrations over q 2 , we found the theoretical predictions for the longitudinal polarization P τ for the considered semileptonic B c decays : in the PQCD approach, and P τ (η c ) = 0.345 ± 0.010, in the " PQCD + Lattice" approach. The dominant errors come from the uncertainty of β Bc and m c . It is easy to see that the theoretical uncertainties of polarization P τ (η c , J/ψ) as given in Eq. (42) are much larger than those in Eq. (41). The two reasons are the following: (1) For B c → η c transition, we know the Lattice NRQCD inputs f +,0 (8.72) simultaneously, the two relevant form factors f +,0 (q 2 ) show a similar q 2 −dependence in the large q 2 region after the inclusion of the Lattice NRQCD input. The PQCD predictions for both P τ (η c ) and P τ (J/ψ) have a very small theoretical error due to the consistent cancelation of the errors.
(2) For the four relevant form factors of B c → J/ψ transition, however, we know only the Lattice NRQCD inputs of V (5.44) and A 1 (5.44), but no Lattice NRQCD results for A 0 (q 2 ) and A 2 (q 2 ) available now. The partial Lattice NRQCD input leads to different high q 2 behaviour of the four form factors F i (q 2 ), as illustrated clearly in Fig. (3). Which results in the inconsistency in some degree and a relatively large theoretical error. We are waiting for further progress of the Lattice NRQCD results for all the four form factors.
As listed in Eq. (34), the longitudinal polarization P τ (D * ) for B → D * τ ν τ decays has been measured by Belle Collaboration [47]. The similar longitudinal polarization P τ (η c ) and P τ (J/ψ) of B c → (η c , J/ψ)τ ν τ decays could be measured in the future LHCb experiment when enough amount of B c decay events are collected.

SUMMARY
In this paper, we studied the semileptonic decays B c → (η c , J/ψ)lν by employing the pQCD factorization approach, using the newly defined distribution amplitudes of the B c meson and also taking into account the Lattice NRQCD results about the relevant form factors. We calculate the form factors f 0,+ (q 2 ), V (q 2 ) and A 0,1,2 (q 2 ) of the B c → (η c , J/ψ) transitions, present the predictions for the branching ratios B(B c → (η c , J/ψ)lν l ) , the ratios R ηc,J/ψ and the longitudinal polarization P τ (η c , J/ψ) of the final τ lepton.
From the numerical calculations and phenomenological analysis we found the following points: (1) The PQCD predictions for the branching ratios of B c → (η c , J/ψ)lν decays agree well with other SM predictions but based on rather different approaches or models. The theoretical predictions for the branching ratios will become smaller by about (10 − 50)% when the Lattice NRQCD results about the form factors are taken into account in the extrapolation of the relevant form factors.
(3) The theoretical predictions for the longitudinal polarization P (τ ) of the tau lepton are the following: These predictions could be tested in the future LHCb experiments.

ACKNOWLEDGMENTS
We wish to thank Wen-Fei Wang and Ying-Ying Fan for valuable discussions. This work was supported by the National Natural Science Foundation of China under Grant No. 11775117 and 11235005.
The factor exp[−S ab (t)] in Eq. (19) contains the Sudakov logarithmic corrections and the renormalization group evolution effects of both the wave functions and the hard scattering amplitude with S ab (t) = S Bc (t) + S X (t) as given in Ref. [46] S Bc = s c x 1 √ 2 m Bc , b 1 + 5 3 t mc dμ µ γ q (α s (μ)), where η + is defined in Eq. (7), while the hard scale t and the quark anomalous dimension γ q = −α s /π, which governs the aforementioned renormalization group evolution. The Sudakov exponent s c (Q, b) for an energetic charm quark is expressed [46] as the difference The hard scales t i are chosen as the largest scale of the virtuality of the internal particles in the hard b-quark decay diagram,