The decays $B_{c}\to J/\psi+\bar\ell\nu_\ell$ and $B_{c}\to J/\psi + \pi(K)$ in covariant confined quark model

In the wake of the recent measurements of the decays $B_c\to J/\psi\,\pi(K)$ and $B_c\to J/\psi\,\ell\nu_\ell$ reported by the LHCb Collaboration we calculate the form factors for the $B_c\to J/\psi$ and $B_c\to \eta_{c}$ transitions in full kinematical region within covariant confined quark model. Then we use the calculated form factors to evaluate the partial decay widths of the above-mentioned semileptonic and nonleptonic decays of the $B_{c}$ meson. We find that the theoretical predictions on the ratios of $\mathcal{R}_{K^{+}/ \pi^{+}}$ and $\mathcal{R_{\pi^{+}/ \mu^{+} \nu}}$ are in good agreement with last LHCb-data. However, the prediction for the $\mathcal{R}_{J/\psi}$ is found to be underestimated.

Recently, LHCb collaboration reported about measurement of the ratio of semileptonic branching fractions R J/ψ [15]: This result lies within 2 standard deviations above the predictions obtained in several theoretical models Refs. [6,13,[16][17][18][19] based on the Standard Model. Note that the semileptonic B c decays provide an excellent laboratory to measure the CKM-matrix elements: V cb , V ub , V cs and V cd . The theoretical description of semileptonic and nonleptonic decays is, however, nontrivial problem because one needs to know the transition form factors characterizing the strong transition of the B c to the charmonium.
In this paper we use the form factors for the B c → J/ψ and B c → η c transitions calculated in the full kinematical region within the covariant confined quark model [20,21]. We evaluate the partial decay widths of the above-mentioned semileptonic and nonleptonic decays of the B c meson and compare our predictions for the ratios R π + /µ + ν , R K + /π + , R J/ψ with available experimental data given by Eqs.
The paper is organized in the following manner. In Sec. II we define the hadronic matrix elements in terms of the invariant and helicity form factors. We calculate the form factors within the covariant confined quark model (CCQM) in the full kinematical region of the momentum transfer squared with all model parameters to be fixed.
In Sec. III we present our numerical results for the semileptonic and nonleptonic branching ratios of the B c meson calculated within our model and compare them with those calculated in other approaches. Finally, we briefly conclude in Sec. IV.

II. FORM FACTORS, HELICITY AMPLITUDES AND DECAY WIDTHS
The form factors of the transitions of the B c into charmonia η c and J/ψ have been calculated in our recent papers [20,21] in the framework of the covariant confined quark model. The accuracy of calculation was estimated of 10%.
The behavior of the calculated form factors in the kinematical region 0 ≤ q 2 ≤ q 2 max is shown in Fig. 1. Note that q 2 max = (m Bc − m ηc ) 2 = 10.9 GeV 2 for the η c and q 2 max = (m Bc − m J/ψ ) 2 = 10.1 GeV 2 for the J/ψ. The results of our numerical calculations can be approximated with high accuracy by the parameterization the relative error of the approximation is less than 1%. The values of F (0), a, and b are listed in Table I.  The invariant form factors for the semileptonic B c -decay into the hadron with spin S = 0, 1 are defined by where P = p 1 + p 2 , q = p 1 − p 2 and ε P qµν ≡ ε αβµν P α q β . It is convenient to express all physical observables through the helicity form factors H m . The helicity form factors H m can be expressed in terms of the invariant form factors in the following way [22]: (a) Spin S = 0: (b) Spin S = 1: where |p 2 | = λ 1/2 (m 2 1 , m 2 2 , q 2 )/(2 m 1 ) is the momentum of the outgoing particles in the B c rest frame. are written down as The effective Hamiltonian which is needed to describe the nonleptonic decays B c → M cc + π(K) is given by where the subscript V −A refers to the usual left-chiral current O µ = γ µ (1−γ 5 ) and q = d, s.
The Feynman diagram describing such decays is shown at the right panel in Fig. 2. The nonleptonic B c -decay widths in terms of the helicity amplitudes are given by (P + = π + , K + , and q = d, s, respectively), (V + = ρ + , K * + , and q = d, s, respectively), where a 1 = C 2 + ξ C 1 with ξ = 1/N c . The leptonic decay constants are also calculated f P (V ) in the framework of the CCQM.  .
We will use the numerical values of the Wilson coefficients from [24] obtained at the scale µ = 4 GeV at leading order with Λ (5) M S = 225 MeV. One has C 2 = 1.141 and C 2 = −0.310 that gives a 1 = C 2 + ξC 1 = 1.038. Note that this value has been also used in the paper [25].
It differs from the most old papers where the color-suppressed factor ξ was set to zero.
The results of theoretical predictions of the branching ratios of the semileptonic B c decays and ratio R J/ψ in comparison with LHCb data [15] are shown in Table II and Table III in units of a 2 1 · 10 −15 GeV obtained in our approach, whereas the absolute values of the branching fractions in % are displayed in Table IV.
The values of the ratios branching fractions obtained by LHCb and calculated in several theoretical approaches are given in Table V.

IV. SUMMARY AND DISCUSSION
We have calculated the semileptonic and nonleptonic decays of the B c meson within CCQM. We have found that the ratios of the branching fractions R π + /µ + ν and R K + /π + are in good agreement with the LHCb data and other theoretical approaches.At the same time the theoretical predictions for the ratio R J/ψ are more than 2 σ less than the experimental data. This may indicate on the possibility of New physics effects in this decay. The possible influence of such effects or some physical observables in the decays mediated b → c l ν 0.189 ± 0.037 0.518 ± 0.104 0.015 ± 0.003 0.029 ± 0.006 0.101 ± 0.02 0.334 ± 0.067 0.008 ± 0.002 0.019 ± 0.004 transitions have been discussed in [17,18,28].

V. ACKNOWLEDGMENTS
The work has been carried out under financial support of the Program of the Ministry of Education and Science of the Republic of Kazakhstan IRN number AP05132978. Author