Study of the Bc → J/ψ + D(*)q decays in covariant confined quark model

In this work we study the Bc → J/ψ + D(∗) q decays. It was defined ratios (RD∗s/Ds , RD∗/D, RD∗/D∗s and RD/Ds ) of nonleptonic branching ratios of Bc meson, which will be hopefully tested on LHC experiments. We compare the obtained results with available experimental data and with the results from other theoretical approaches. Introduction Recently the ATLAS Collaboration reported on the measurement of the various branching fractions of the decays Bc → J/ψDs and Bc → J/ψD∗+ s [1]. The first observations of these decays have been performed by the LHCb Collaboration [2]. In view of these developments, we decided to calculate ratios of branching fractions within the covariant confined quark model(CCQM). The decay properties of the above processes were studied in various theoretical approaches [3–12]. Effective Hamiltonian and matrix element The effective Hamiltonian describing the Bc nonleptonic decays into charmonium and D(Ds) meson is given by (see, Ref. [13]) Heff = − GF √ 2 VcbV cq 6 ∑ i=1 Ci Oi, O1 = (c̄a1 ba2 )V−A(q̄a2 ca1 )V−A, O2 = (c̄a1 ba1 )V−A, (q̄a2 ca2 )V−A, O3 = (q̄a1 ba1 )V−A(c̄a2 ca2 )V−A, O4 = (q̄a1 ba2 )V−A(c̄a2 ca1 )V−A, O5 = (q̄a1 ba1 )V−A(c̄a2 ca2 )V+A, O6 = (q̄a1 ba2 )V−A(c̄a2 ca1 )V+A, (1) where the subscript V −A refers to the usual left–chiral current O = γμ(1−γ5) and V +A to the usual right–chiral one Oμ+ = γ(1 + γ5). The ai denote the color indices. The quark q stands for either s or d. The numerical values of the Wilson coefficients are taken as in Ref.[14]. They are listed in Table 1. Since in NNLO the numerical values of the C5 and C6 coefficients are less to the order than the C3 e-mail: issadykov@jinr.ru © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). EPJ Web of Conferences 177, 09007 (2018) https://doi.org/10.1051/epjconf/201817709007 AYSS-2017


Introduction
Recently the ATLAS Collaboration reported on the measurement of the various branching fractions of the decays B + c → J/ψD + s and B + c → J/ψD * + s [1].The first observations of these decays have been performed by the LHCb Collaboration [2].In view of these developments, we decided to calculate ratios of branching fractions within the covariant confined quark model(CCQM).

Effective Hamiltonian and matrix element
The effective Hamiltonian describing the B c nonleptonic decays into charmonium and D(D s ) meson is given by (see, Ref. [13]) where the subscript V − A refers to the usual left-chiral current O µ = γ µ (1 − γ 5 ) and V + A to the usual right-chiral one O µ + = γ µ (1 + γ 5 ).The a i denote the color indices.The quark q stands for either s or d.The numerical values of the Wilson coefficients are taken as in Ref. [14].They are listed in Table 1.Since in NNLO the numerical values of the C 5 and C 6 coefficients are less to the order than the C 3 e-mail: issadykov@jinr.ru In the numerical calculations we set the colorsuppressed parameter ξ to zero.Then the Wilson coefficients are equal to which should be compared with the old ones a 1 = 1.14 and a 2 = −0.20 used in paper [3].

Invariant and helicity amplitudes
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 and q = p 1 − p 2 .Here p 1 is the momentum of the ingoing meson with a mass m 1 (B c ) and p 2 is the momentum of the outgoing meson with a mass m 2 .It is convenient to express all physical observables through the helicity form factors H m .The helicity form factors H m can be written in terms of the invariant form factors in the following way [15]: Spin S=0: Spin S=1: Here The nonleptonic B c decay widths in terms of the helicity amplitudes are given by where we use the short notation

Numerical results
All necessary details and values of the calculations of the leptonic decay constants and hadronic form factors may be found in recent publications [14,16].In Table 2 we show widths of the B c meson for general values of the Wilson coefficients a 1 and a 2 obtained in this work and in other works.In Table 3 we show the values of branching fractions obtained in this work for two different set of the Wilson coefficients ( a 1 = 0.93, a 2 = −0.27and a 1 = 1.14, a 2 = −0.20)and compare them with other theoretical approaches.Finally, we compare our results for ratios of nonleptonic branching fractions of B c meson with available experimental data and the results obtained in other approaches in Table 4, where

Table 1 .
Values of the Wilson coefficients in NNLO.Fierz transformation one can check that O 3 = O 1 and O 4 = O 2 .Then the calculation of the matrix elements describing the nonleptonic decays of the B c meson into charmonium and D(D s ) meson is straightforward.The combinations of the Wilson coefficients appear as

Table 2 .
Exclusive nonleptonic decay widths of the B c meson in units of 10 −15 GeV for general values of the Wilson coefficients a 1 and a 2 .

Table 3 .
Branching ratios (in %) of nonleptonic B c decays obtained in this work for two different set of the Wilson coefficients.

Table 4 .
Comparison of the results for the ratios of nonleptonic branching fractions of B c meson