Semileptonic decays of $B_c$ meson to $P$-wave charmonium states

Inspired by a series of unexpected measurements of semileptonic decays mediated via $b\rightarrow c $ charged current interactions, we explore semileptonic $B_c$ decays to the four lightest $P$-wave charmonium states, $\chi_{c0}, \chi_{c1}, \chi_{c2}, h_c$, by the recently developed improved perturbative QCD formalism, in which the charm quark mass effect is included both in the Sudakov factor and the hard kernels. We first directly evaluate the concerned transition form factors with vector and axial-vector currents in the region of small momentum transfer squared, and then recast them to the full kinematical region by adopting the exponential parametrization. The obtained form factors are used to evaluate the semileptonic decay branching ratios, which can reach the order of $10^{-3}$, letting the corresponding measurement appear feasible. For a better analysis, a comparison of our results with the predictions of other models is provided. We also present the ratios between the tau and light lepton branching ratios and the polarization contributions in the relevant processes, which still need experimental tests in the ongoing and forthcoming experiments. Any significant deviations from the Standard Model results may provide some hints of new physics effects.

Inspired by a series of unexpected measurements of semileptonic decays mediated via b → c charged current interactions, we explore semileptonic Bc decays to the four lightest P -wave charmonium states, χc0, χc1, χc2, hc, by the recently developed improved perturbative QCD formalism, in which the charm quark mass effect is included both in the Sudakov factor and the hard kernels. We first directly evaluate the concerned transition form factors with vector and axial-vector currents in the region of small momentum transfer squared and then recast them to the full kinematical region by adopting the exponential parametrization. The obtained form factors are used to evaluate the semileptonic decay branching ratios, which can reach the order of 10 −3 , letting the corresponding measurement appear feasible. For a better analysis, a comparison of our results with the predictions of other models is provided. We also present the ratios between the tau and light lepton branching ratios and the polarization contributions in the relevant processes, which still need experimental tests in the ongoing and forthcoming experiments. Any significant deviations from the Standard Model results may provide some hints of new physics effects.
The yield value lies at about 2σ above the range of existing predictions in the SM [15][16][17]. These ratios have been calculated to high precision due to the cancellation of numerous uncertainties common to the numerator and denominator. Within the SM, the deviation from unity is mainly caused by the massive τ lepton, which also increases the sensitivity to new physics (NP) in these decays. Then, the possible NP effects in the semileptonic decays have been discussed recently in several papers [18][19][20][21][22][23][24][25][26]. To maximize future sensitivity to NP contributions, measuring and understanding the semileptonic modes involving various P -wave orbitally excited charmonium X(X ∈ {χ c0 , χ c1 , χ c2 , h c }) in the final state for the same flavor content are important and necessary, not only as they can give additional and complementary information on the NP but also as they constitute backgrounds to the R(J/ψ) measurements. Experimentally, many nonleptonic decays with J/ψ or ψ(2S) as the final charmonium have been detected [27], and the first evidence for the decay B c → χ c0 π is found at 4.0σ significance by the LHCb experiment [28]. However, for the semileptonic decays, so far, only the B c → J/ψ transitions have recently been observed by the LHCb Collaboration [14,29]. As the LHC accumulates more and more data, the semileptonic B c decays to the P -wave charmonium will have more possibilities to be detected. Theoretically, essential to the study of the semileptonic decays is the calculation of the invariant form factors describing the corresponding hadronic transitions. In the literature, a wide range of various approaches has been used to compute the B c → X transition form factors, such as the QCD sum rules (QCDSR) [30,31], the covariant light-front quark model (LFQM) [32], the renormalization group method (RGM) [33], the relativistic constituent quark model (RCQM) [34], relativistic quark model (RQM) [35], the nonrelativistic quark model (NRQM) [36], the Bethe-Salpeter approach (BS) [37], the relativistic quark model based on the quasipotential approach (RQMQP) [38], and the Isgur-Scora-Grinstein-Wise II model (ISGW II) [39]. More recently, the relativistic corrections to the form factors of the B c meson into P -wave orbitally excited charmonium have been investigated using the nonrelativistic QCD effective theory (NRQCD) [40].
As a successive work of [15,41,42], in this paper, we do not attempt to resolve the R(J/ψ) anomaly beyond the SM, but provide more reliable calculations of those orbitally excited state modes within the SM. A future improvable measurement might reveal whether a similar anomaly also exists in R(X). In order to meet the measurements for charmonium B c decays with good precision, we adopt the so-called improved perturbative QCD formalism [43] recently developed by Xin Liu et al.. The charmonium B c decays are a multiscale process, which contain three scales: the bottom quark mass m b , the charm quark mass m c , and the QCD scale Λ QCD . Under the hierarchy of m b ≫ m c ≫ Λ QCD , the charm quark effect enters the Sudakov exponent through an additional large infrared logarithm log (m b /m c ), which should be resummed. For the detailed derivation of the k T resummation technique with the finite charm quark mass, the reader is referred to [43].
The outline of the paper is as follows: In Sec. II, we define kinematics and describe the meson distribution amplitude of the initial and final states. In Sec. III, we give the factorization formulas for the B c → X form factors in the PQCD approach. Subsequently, we present the general formalism for the semileptonic differential decay widths with the lepton-helicity states. Section. IV is devoted to the numerical analysis of the form factors, branching ratios, polarizations and comparison of our results with the other approaches. A summary is given in Sec. V.

II. KINEMATICS AND MESON DISTRIBUTION AMPLITUDES
For simplicity we work in the rest frame of the B c meson and use light-cone coordinates. The momentum of the B c meson and charmonium can be denoted as [15,16,44] with the ratio r = m/M , and m(M ) is the mass of the charmonium (B c ) meson. The factors η ± = η ± η 2 − 1 are defined in terms of the velocity transfer η = v 1 · v 2 with v 1 = P 1 /M and v 2 = P 2 /m [44]. For the momentum transfer q = P 1 − P 2 , there exists η = 1+r 2 2r − q 2 2rM 2 . The momentum of the valence quarks k 1,2 , whose notation are displayed in Fig 1, are parametrized as where the k 1T,2T , x 1,2 represent the transverse momentum and longitudinal momentum fraction of the charm quark inside the meson, respectively.
As the direct analogue of the vector charmonium [15], for an axial-vector charmonium, the longitudinal (transverse) polarization vectors ǫ(0(±)) can be defined as where the abbreviations S, A, and T correspond to scalar, axial-vector, and tensor charmonium states, respectively. ψ v S , ψ L,T A , and ψ (T ) T are of twist-2, while ψ s S , ψ t,V A , and ψ t,V T are of twist-3. For their expressions, the same form and parameters are adopted as in [41].

III. FORM FACTORS IN THE PQCD APPROACH
Based on the k T factorization theorem, the transition form factors can be expressed as the convolution of a hard kernel with the distribution amplitudes of those mesons involved in the decays in the heavy-quark and large-recoil limits. For a review of this approach, please see Ref. [48]. The hard kernel can be treated by PQCD at the leading order in an α s expansion (single gluon exchange as depicted in Fig. 1). Below, we will derive the general formulas of the B c → S, A, T transition form factors in the PQCD approach.
The B c → χ c0 form factors are defined by [32,49] It is conventional to define two auxiliary form factors f 1 (q 2 ) and f 2 (q 2 ), which are related to F + (q 2 ) and F 0 (q 2 ) by After standard calculations, we obtain their factorization formulas as follows: with . α e and β a,b are the virtuality of the internal gluon and quark, respectively. Their expressions are where the explicit expressions of the functions S t , h, and the scales t a,b are referred to [50]. The modified Sudakov factor S ab , which includes the charm quark mass effect, can be found in [43].
B. Bc → χc1, hc form factors Following Ref. [15], the B c → χ c1 , h c transition induced by the vector and axial-vector currents is parametrized by where the convention ǫ 0123 = +1 is taken. Compared with the B c → J/ψ transition, here the behavior of the vector and axial-vector currents is interchanged, and the factor M + m is replaced by M − m. The relation 2rV is obtained to smear the singularity at q 2 = 0. The factorization formulas are acquired as It should be stressed that the nonlocal matrix element for the axial-vector and scalar charmonium meson in Eq. (11) can be related to the vector and pseudoscalar ones [15], respectively, by multiplying by the structure (−i)γ 5 from the left-hand side. The factorization formulas f 1,2 , V 0,1,2 , and A here are similar to the corresponding ones in [15] with the r c term flipping signs and the replacement 1 + r → 1 − r.

C. Bc → χc2 form factors
In analogy with B c → J/ψ form factors, we parametrize the B c → χ c2 form factors induced by the vector and axial-vector currents as Note that the structure of above form factors is analogous to the J/ψ case with the replacement ǫ → ǫ T . In addition, as mentioned before, the LCDAs of a tensor meson are also similar to the vector ones except that the ǫ is replaced by ǫ • . So, the factorization formulas here can be straightforwardly obtained by replacing the twist-2 or twist-3 LCDAs of the J/ψ with the corresponding twists of the χ c2 one in Eq. (11). After multiplying by the different definitions of the polarization vector, we have [47] D. The semileptonic differential decay rates As is well known, the above form factors are reliable only in the small q 2 region in the PQCD framework [47,51]. In order to estimate the semileptonic differential decay rates, we need to know the q 2 -dependent form factors in the full kinematical region. Our form factors are truncated at about q 2 = m 2 τ with m τ the mass of the τ lepton. We first perform the PQCD calculations on them in the range of 0 < q 2 < m 2 τ , while the momentum dependence of the form factors in the m 2 τ < q 2 < (M − m) 2 region is determined by fitting through a three-parameter function. The following fit parametrization is chosen for the form factors with respect to q 2 [15]: where F i denotes any one of the form factors, and a, b are the fitted parameters.
After integrating out the off-shell W boson, the effective Hamiltonian for the b → clν l transition is written as [52] where G F = 1.16637 × 10 −5 GeV −2 is the Fermi coupling constant and V cb is one of the CKM matrix elements. The differential decay rate of the exclusive processes B c → (S, A)lν can be expressed in terms of the form factors as [32] dΓ where m l is the lepton mass and λ(q 2 ) = (M 2 + m 2 − q 2 ) 2 − 4M 2 m 2 . The subscripts L, +, and − denote the longitudinal, positive, and negative polarizations of the final state, respectively. As stated before, the decay width of B c → χ c2 lν can be related to the J/ψ one [15] by making the following replacement: where the factors 2(η 2 −1) 3 and η 2 −1 2 come from Eq.(7). The total differential widths for the axial-vector and tensor charmonium modes can be written as

IV. NUMERICAL ANALYSIS AND DISCUSSION
For numerical evaluation, we collect the input parameters such as the masses and the meson decay constants in Table I, while the CKM matrix elements and B c lifetime are set as V cb = 0.0405 [27] and τ Bc = 0.507 ps [27], respectively. In the fitting procedure, the form factors in the lower region, namely, q 2 ∈ [0, m 2 τ ], are computed in the PQCD framework. The numerical results of the relevant form factors at the scale q 2 = 0 as well as the fitted parameters a and b are presented in Table II, and here the uncertainties for our results are estimated including three aspects. The first type of error comes from the shape parameter ω of the B c meson distribution amplitude; the second one is from the charm quark mass; the last one is caused by the decay constants of the charmonium states. In the  [43], while the decay constants of the P -wave charmonium states are adopted from the recent updated values evaluated from the QCD sum rules at the scales µ = mc [53]. Other parameters are from PDG 2016 [27].  evaluation, these uncertainties are obtained by simply taking a ±10% uncertainty on the central value. The combined uncertainties can reach 25%. In addition, the uncertainties from the CKM matrix elements and the hard scale t are very small and have been neglected.
It is found that the form factors of the P -wave modes are smaller than those of the S-wave ones in our previous study [15]. This phenomenon can be understood from the wave functions of the two states. The additional nodes in the wave functions of the orbital excited charmonium state cause the overlap between the initial and final state wave functions to become smaller. In addition, the smaller decay constants of P -wave charmonium states also suppress the corresponding values. Comparing the form factors of B c → χ c1 with B c → h c in Table. II, one can find the large differences between them. The main reason is the different DAs and the decay constants for the two kinds of axial-vector charmonium. Because of the G parity, the DAs for χ c1 and h c mesons exhibit the different asymptotic behaviors [41]. Moreover, the longitudinal and transverse decay constants (see Table I) in the two axial-vector mesons can also contribute to different values. The B c → T transition form factor is somewhat larger since the prefactor in Eq.(23) is roughly 2r/(1 − r 2 ) ≈ 1.7 at the maximally recoiling point, which enhanced the numbers accordingly.
So far, several authors have calculated the form factors of the concerned decays via different frameworks. To compare the results, we should rescale them according to the form factor definitions in Eqs. (12), (17), and (22). For example, comparing the definitions of the B c → T transition form factor of Ref. [32] with ours, we have the following relations at the maximal recoil point: where the values of h, k, and b + can be found in [32]. Note that we have dropped an overall phase factor i which is irrelevant for the calculation of the decay widths. Other results, such as QCDSR [31], ISGW II [39], and NRQCD [40], are also converted into the numbers according to our definitions in this paper and are listed in Table II. As indicated in Table II, the results evaluated in the different models are roughly comparable. Our results are generally close to those of the LFQM [32] and the QCDSR [31], while some of the results for the B c → χ c1 transition from the ISGW II model possess a sign that is the opposite of ours. The recent NRQCD predictions in [40] are obviously larger for most of the decay channels. Based on the values of the transition form factors at q 2 = 0 and the fit parameters a and b listed in Table II, we can plot the momentum transfer squared dependence of these form factors in Fig. 2 for the four processes in the whole accessible kinematical range. The difference of the curve behavior for the various P -wave charmonium states is the consequence of their different LCDAs. The form factors for the B c → χ c2 transition have a relatively stronger momentum dependence than others. The main reason is that the B c → χ c2 form factors received additional q 2 dependence as can be seen from the factorization formulas in Eq. (23), which provide an enhancement to the corresponding values with the increase of q 2 .
With the form factors at hand, one can directly obtain the partial decay width by integrating the corresponding differential decay rates over q 2 in Eqs. (26)- (29). We are now ready to calculate the respective semileptonic decay branching ratios. The numerical results are shown in Table III, together with the numbers obtained in other model calculations for comparison. In general, it is observed that the branching ratios have close values within the error bars in all models. In particular, our results match very well with those of QCDSR [31].
Since the electron and muon are very light compared with the heavy tau lepton, we neglect their masses in the calculations. It is seen that the semitauonic decays branching ratios fall short by a large factor compared with the corresponding values of the e and µ channels due to suppression from the phase space. In order to reduce the theoretical uncertainties from the hadronic parameters, we define four ratios between the branching fractions of semitauonic decays of B c mesons relative to the decays involving lighter lepton families, From the numbers in Table III, where all uncertainties are added in quadrature. The central values lie between 0.08 and 0.22, which are typically smaller than our previous prediction for that of J/ψ with R(J/ψ) = 0.29 [15] because the heavy P -wave charmonium states bring a smaller phase space than the S-wave ones. More recently, the LHCb Collaboration [14] published a measurement R(J/ψ) = 0.71 that shows the discrepancy with the prediction of the SM. It would be interesting to see whether the similar anomalies also exist independently in these P -wave charmonium modes. Therefore, the measurements of various ratios such as R(X) in the future will give an additional hint for the NP effect in the b → clν transition. Next, we made a comprehensive polarization analysis of the axial-vector and tensor channels. Since the initial state B c is a spinless particle, the final state axial-vector/tensor charmonium and lepton pair carry spin degrees of freedom. According to the angular momentum conservation, the semileptonic decays of B c → A/T lν l contain three different polarizations. It is meaningful to define three polarization fractions f L,± = Γ L,± /(Γ L + Γ + + Γ − ). Their individual polarization fractions are shown in Table IV, where the sources of the errors in the numerical estimates have the same origin as in Table II. We made the following observations. First, the minus polarization fractions have larger magnitudes in comparison to the plus components, and the latter are only at the percent level. From Table II, one can see the form factors A and V 1 have the same sign, which gives constructive contributions to the minus polarized decay width but destructive contributions to the plus partners as can be seen in Eq. (28). Second, for B c → χ c1 decays, the transverse polarization contributions dominated the branching ratio due to a destructive interference between V 1 and V 2 in the longitudinally polarized decay width. However, in the case of the B c → h c transition, the value of V 2 is a negative number, which reverses the constructive or destructive interference situation. The dramatically different polarization contributions between the two axial-vector decay channels are similar to the explanation in [32]. Finally, for each charmonium channel, the longitudinal, plus, and minus polarization fractions of the τ are roughly equal to the corresponding values of e, which reflects that the relative polarization contributions still favor the lepton flavor IV: The PQCD predictions for the polarization fractions. The errors are induced by the same sources as in Table II. universality. These results will be tested at the ongoing and forthcoming hadron colliders.

V. CONCLUSION
Semileptonic charmonium decays of B c mesons play a critical role in the determination of the magnitudes of the CKM matrix elements V cb , and in the test of the lepton flavor universality which is a basic assumption of the SM. The investigation of the corresponding P -wave charmonium modes is of special interest and further provides complementary information on physics beyond the SM. In this paper, we first calculated the B c → χ c0 , χ c1 , χ c2 , h c transition form factors at the small momentum region within the improved PQCD framework. By fitting an auxiliary three-parameter exponential function we obtained the momentum-squared-dependent form factors in the full kinematical region. We used them to estimate the branching ratios of the considered semileptonic. The order of branching ratios shows that these channels are accessible in the near future experiments. We also gave predictions on the ratio between the tau and light lepton branching ratio R(X), which are smaller than our previous calculation of R(J/ψ) due to the suppression from the phase space. A future improvable measurement might reveal whether a similar anomaly exists in these ratios. Three polarization contributions were also investigated in detail for the axial-vector and tensor modes. The approximately equal polarization fractions between the tau and light lepton with the same charmonium in the final states may indicate that the lepton flavor universality violation is negligible in the relative polarization contributions. These results and findings will be further tested by the LHCb and Belle II experiments in the near future.