Branching fractions of B ( c ) decays involving J / ψ and X ( 3872 ) *

We study two-body B(c) → Mc(π,K) and semileptonic Bc → Mcl −ν̄l decays with Mc = (J/ψ,X 0 c), where Xc ≡ X (3872) is regarded as the tetraquark state cc̄uū(dd̄). With the decay constant fX0c = (234±52) MeV determined from the data, we predict that B(B− →Xcπ −) = (11.5±5.7)×10, B(B̄ →XcK̄ ) = (2.1±1.0)×10, and B(B̄s →X 0 cK̄ )= (11.4±5.6)×10. With the form factors in QCD models, we calculate that B(B−c →X 0 cπ ,XcK −)= (6.0±2.6)×10 and (4.7±2.0)×10, and B(B−c → J/ψμ −ν̄μ,X 0 cμ −ν̄μ) = (2.3±0.6)×10 −2 and (1.35±0.18)×10, respectively, and extract the ratio of the fragmentation fractions to be fc/fu = (6.4±1.9)×10 .


Introduction
Through the b → ccd(s) transition at quark level, B decays are able to produce cc bound states like J/ψ; particularly, the hidden charm tetraquarks to consist of ccqq , such as X 0 (3872), Y(4140), and Z + c (4430), known as the XYZ states [1].For example, we have [2,3] where X 0 c ≡ X 0 (3872) is composed of ccuū(d d), measured to have the quantum numbers J P C = 1 ++ .On the other hand, the B − c decays from the b → cūd(s) transition can also be a relevant production mechanism for the cc and ccqq bound states.However, the current measurements have been done only for the ratios, given by [4,5] where f c,u are the fragmentation fractions defined by In addition, none of the XYZ states have been observed in the B c decays yet.
From Figs. 1(a) and 1(d), the B → M c M decays proceed by the B → M transition, which is followed by the recoiled M c = (J/ψ, X 0 c ) with J P C = (1 −−,++ ), respectively, presented as the matrix elements of M c |cγ µ (1 − γ 5 )c|0 .Unlike J/ψ, which is a genuine cc bound state, while the matrix element for the tetraquark production is in fact not computable, X 0 c is often taken as a charmonium state in the QCD models [6][7][8].In this study, we will extract respectively, where ), G F is the Fermi constant, and V ij are the CKM matrix elements.In the factorization approach, a 1(2) ≡ c eff 1(2) + c eff 2(1) /N c is composed of the effective Wilson coefficients in Ref. [9], with (c eff 1 , c eff 2 ) = (1.168,−0.365),where N c is the color number.In Eq. ( 3), the decay constant, four-momentum vector, and four polarization (f M (c) , q µ , ε µ * ) are defined by while the matrix elements of the B → (M, J/ψ, X 0 c ) transitions can be parametrized as [8] M|qγ respectively, where q = p B − p M (c) , t ≡ q 2 , and (F 1,2 , A (i) , V (i) ) with i = 0, 1, 2 are the form factors.
Table 2.The branching ratios of the Bc → J/ψ(M, lν l ) decays, where the first (second) errors of our results are from the form factors (a1).
decay modes our results QCD models B − c → J/ψπ − (10.9 ± 0.8 For the B → X 0 c (π, K) decays, the results are given in Table 3.While Table 3.The branching ratios for the B (c) → X 0 c M and Bc → X 0 c lν l decays.For our results, the first errors come from (f X 0 c , f (0)), and the second ones from (a1, a2). 1) We thank the authors in Ref. [8] for the useful communication.
For the semileptonic B − c → M c l − νl decays, B(B − c → J/ψeν e ) = B(B − c → J/ψµν µ ) = (1.94 ± 0.20) × 10 −2 is due to the both negligible electron and muon masses, of which the numerical value is close to those from Refs.[15,17] but 2 − 3 times smaller than those in Ref. [18], which calls for future experimental examination.Note that by taking B(B − c → J/ψπ − ) as the theoretical input in Eq. ( 2), we derive that which agrees with the above theoretical prediction.For the τ mode, which suppresses the phase space due to the heavy m τ , we obtain B(B − c → J/ψτ − ντ ) = (4.47 ± 0.48) × 10 −3 .The ratio of B(B − c → X 0 c e − νe )/B(B − c → X 0 c τ − ντ ) 1/20 is close to that in Ref. [19], but B(B − c → X 0 c e − νe ) = (1.35± 0.18) × 10 −3 is apparently 4-5 times smaller than that in Ref. [19], though with uncertainties the two results overlap with each other.With the spectra of B − c → (J/ψ, X 0 c )l − νl in Fig. 2, our results can be compared to the recent studies on the semileptonic B c cases in Refs.[20,21] for the XYZ states.

2 Formalism
and B0 s → X 0 c K − , of which the extraction allows X 0 c to be the tetraquark state.On the other hand, to calculate the B − c → (J/ψ, X 0 c )M decays in Figs.1(b) and 1(e) and the semileptonic B − c → (J/ψ, X 0 c )lν l decays in Figs.1(c) and 1(f), we use the B c → M c transition matrix elements from the QCD calculations.In terms of the effective Hamiltonians at quark level for the b → ccq, b → cūq, and b → clν l transitions in Fig. 1, the amplitudes of the B − c → M c M, B → M c M, and B − c → M c l − νl decays can be factorized as [9, 10]

Fig. 1 .
Fig. 1.Diagrams for the B and Bc decays with formation of the cc pair, where (a), (b) and (c) correspond to the B → X 0 c M, B − c → X 0 c M, and B − c → X 0 c lν l decays, while (d), (e) and (f) the B → J/ψM, B − c → J/ψM, and B − c → J/ψlν l decays, respectively.