$B_{c}$ ${\to}$ $BP$, $BV$ decays with the QCD factorization approach

We studied the nonleptonic $B_{c}$ ${\to}$ $BP$, $BV$ decay with the QCD factorization approach. It is found that the Cabibbo favored processes of $B_{c}$ ${\to}$ $B_{s}{\pi}$, $B_{s}{\rho}$, $B_{u}\bar{K}$ are the promising decay channels with branching ratio larger than 1\%, which should be observed earlier by the LHCb Collaboration.


I. INTRODUCTION
The B c meson is the ground pseudoscalar meson of thebc system [1]. Compared with the heavy unflavored charmonium cc and bottomonium bb, the B c meson is unique in some respects. (1) Heavy quarkonia could be created in the parton-parton process ij → QQ at the order of α 2 s (where ij = gg or qq, Q = b, c), while the production probability for the B c meson is at least at the order of α 4 s via ij → B ( * )+ c + bc where the gluon-gluon fusion mechanism is dominant at Tevatron and LHC [2]. The B c meson is difficult to produce experimentally, but it was observed for the first time via the semileptonic decay mode B c → J/ψℓν in pp collisions by the CDF collaboration in 1998 [3], which showed the realistic possibility of experimental study of the B c meson. One of the best measurements on the mass and lifetime of the B c meson is reported recently by the LHCb collaboration, m Bc = 6276.28±1.44±0. 36 MeV [4] and τ Bc = 513.4±11.0±5.7 fs [5]. With the running of the LHC, the B c meson has a promising prospect. It is estimated that one could expect some 10 10 B c events at the high-luminosity LHC experiments per year [6] . The studies on the B c meson have entered a new precision era.
(2) For charmonium and bottomonium, the strong and electromagnetic interactions are mainly responsible for annihilation of the QQ quark pair into final states. The B c meson, carrying nonzero flavor number B = C = ±1 and lying below the BD meson pair threshold, can decay only via the weak interaction, which offers an ideal sample to investigate the weak decay mechanism of heavy flavors that is inaccessible to both charmonium and bottomonium. The B c weak decay provides great opportunities to investigate the perturbative and nonperturbative QCD, final state interactions, etc.
With respect to the heavy-light B u,d,s mesons, the doubly heavy B c meson has rich decay channels because of its relatively large mass and that both b and c quarks can decay individually. The decay processes of the B c meson can be divided into three classes [2, [7][8][9]: (1) the c quark decays with the b quark as a spectator; (2) the b quark decays with the c quark as a spectator; (3) the b and c quarks annihilate into a virtual W boson, with the ratios of ∼ 70%, 20% and 10%, respectively [2]. Up to now, the experimental evidences of pure annihilation decay mode [class (3)] are still nothing. The b → c transition, belonging to the class (2), offers a well-constructed experimental structure of charmonium at the Tevatron and LHC. Although the detection of the c quark decay is very challenging to experimentalists, the clear signal of the B c → B s π decay is presented by the LHCb group using the B s → D s π and B s → J/ψφ channels with statistical significance of 7.7σ and 6.1σ, respectively [10].
Anticipating the forthcoming accurate measurements on the B c meson at hadron colliders and the lion's share of the B c decay width from the c quark decay [11], many theoretical papers have devoted to the study of the B c → BP , BV decays (where P and V denote the SU(3) ground pseudoscalar and vector mesons, respectively), such as Refs. [12][13][14][15] with the BSW model [16] or IGSW model [17], Refs. [18][19][20] based on the Bethe-Salpeter (BS) equation, Refs. [21][22][23][24][25] with potential models, Ref. [26] with constituent quark model, Ref. [27] with QCD sum rules, Ref. [28] with the quark diagram scheme, Ref. [29] with the perturbative QCD approach (pQCD) [30], and so on. The previous predictions on the branching ratios for the B c → BP , BV decays are collected in Table III. The discrepancies of previous investigations arise mainly from the different model assumptions. Recently, several phenomenological methods have been fully developed to cope with the hadronic matrix elements and successfully applied to the nonleptonic B decay, such as the pQCD approach [30] based on the k T factorization scheme, the soft-collinear effective theory [31] and the QCD-improved factorization (QCDF) approach [32] based on the collinear approximation and power countering rules in the heavy quark limits. In this paper, we will study the B c → BP , BV decay with the QCDF approach to provide a ready reference to the existing and and upcoming experiments.
This paper is organized as follows. In section II, we will present the theoretical framework and the amplitudes for the B c → BP , BV decays within the QCDF framework. The section III is devoted to numerical results and discussion. Finally, the section IV is our summation.

A. The effective Hamiltonian
The low energy effective Hamiltonian responsible for the nonleptonic bottom-conserving B c → BP , BV decays constructed by means of the operator product expansion and the renormalization group (RG) method is usually written in terms of the four-quark interactions [33].
where the Fermi coupling constant G F ≃ 1.166 × 10 −5 GeV −2 [1]; Q 1,2 , Q a 1,2 and Q 3,···10 are the relevant local tree, annihilation and penguin four-quark operators, respectively, which govern the decays in question. The Cabibbo-Kobayashi-Maskawa (CKM) factor V uq i V * cq j and Wilson coefficients C i describe the coupling strength for a given operator.
Using the unitarity of the CKM matrix, there is a large cancellation of the CKM factors where the Wolfenstein parameter λ = sinθ c = 0.225 37 (61) [1] and θ c is the Cabibbo angle.
Hence, compared with the tree contributions, the contributions of annihilation and penguin operators are strongly suppressed by the CKM factor. If the CP -violating asymmetries that are expected to be very small due to the small weak phase difference for c quark decay are prescinded from the present consideration, then the penguin and annihilation contributions could be safely neglected. The local tree operators Q 1,2 in Eq.
(1) are expressed as follows, where α and β are the SU(3) color indices.
The Wilson coefficients C i (µ) summarize the physics contributions from scales higher than µ. They are calculable with the RG improved perturbation theory and have properly been evaluated to the next-to-leading order (NLO) [33]. They can be evolved from a higher scale µ ∼ O(m W ) down to a characteristic scale µ ∼ O(m c ) with the functions including the flavor thresholds [33], where U f (µ f , µ i ) is the RG evolution matrix converting coefficients from the scale µ i to µ f , and M(µ) is the quark threshold matching matrix. The expressions of U f (µ f , µ i ) and M(µ) can be found in Ref. [33]. The numerical values of LO and NLO C 1,2 with the naive dimensional regularization scheme are listed in Table I. The values of NLO Wilson coefficients in Table I are consistent with those given by Ref. [33] where a trick with "effective" number of active flavors f = 4.15 rather than formula Eq. (5) is used.
To obtain the decay amplitudes, the remaining work is how to accurately evaluate the hadronic matrix elements BM|Q i (µ)|B c which summarize the physics contributions from scales lower than µ. Since the hadronic matrix elements involve long distance contributions, one is forced to use either nonperturbative methods such as lattice calculations and QCD sum rules or phenomenological models relying on some assumptions. Consequently, it is very unfortunate that hadronic matrix elements cannot be reliably calculated at present, and that the most intricate part and the dominant theoretical uncertainties in the decay amplitudes reside in the hadronic matrix elements.

B. Hadronic matrix elements
Phenomenologically, based on the power counting rules in the heavy quark limit, M.
Beneke et al. proposed that the hadronic matrix elements could be written as the convolution integrals of hard scattering kernels and the light cone distribution amplitudes with the QCDF master formula [32]. The QCDF approach is widely applied to nonleptonic B decays and it works well [34][35][36][37], which encourage us to apply the QCDF approach to the study of B c → BP , BV decay. Since the spectator is the heavy b quark who is almost always on shell, the virtuality of the gluon linked with the spectator should be ∼ O(Λ 2 QCD ). The dominant behavior of the B c → B transition form factors and the contributions of hard spectator scattering interactions are governed by soft processes. According to the basic idea of the QCDF approach [35], the hard and soft contributions to the form factors entangle with each other and cannot be identified reasonably, so the physical form factors are used as the inputs.
The hard spectator scattering contributions are power suppressed in the heavy quark limit.
Finally, the hadronic matrix elements can be written as, where ing order α 0 s , H i (x) = 1, i.e., the convolution integral of Eq. (6) results in the meson decay constant. The hadronic matrix elements are parameterized by the product of form factors and decay constants, which are real and renormalization scale independent. One goes back to the simple "naive factorization" (NF) scenario. At the order α s and higher orders, the information of strong phases and the renormalization scale dependence of hadronic matrix elements could be partly recuperated. Combined the nonfactorizable contributions with the Wilson coefficients, the scale independent effective coefficients at the order α s can be obtained [32] as follows: where the expression of vertex corrections are [32]: with the twist-2 quark-antiquark distribution amplitudes of pseudoscalar P and longitudinally polarized vector V meson in terms of Gegenbauer polynomials [38]: wherex = 1 − x; a M n is the Gegenbauer moment and a M 0 ≡ 1. From the numbers in Table I, it is found that (1) for the coefficient a 1 , the nonfactorizable contributions accompanied by the Wilson coefficient C 2 can provide ≥ 10% enhancement compared with the NF's result, and a relatively small strong phase ≤ 5 • ; (2) for the coefficient a 2 , the nonfactorizable contributions assisted with the large Wilson coefficient C 1 are significant. In addition, a relatively large strong phase ∼ −155 • is obtained; (3) the QCDF's values of a 1,2 agree basically with the real coefficients a 1 ≃ 1.20 and a 2 ≃ −0.317 which are used by previous studies on the B c → BP , BV decays in Refs. [12-15, 18, 21-27], but with more information on the strong phases.

C. Decay amplitudes
Within the QCDF framework, the amplitudes for B c → BM decays are expressed as: The matrix elements of current operators are defined as: where f P and f V are the decay constants of pseudoscalar P and vector V mesons, respectively; m V and ǫ denote the mass and polarization of vector meson, respectively.
The transition form factors are defined as [16]: where q = p − k, and the condition of should be close to the result using the nonrelativistic harmonic oscillator wave functions with the BSW model [12], The flavor symmetry breaking effects on the form factors are neglectable in Eq. (16). For simplification, we take F Bc→B u,d,s 0,1 = 1.0 in our numerical calculation to give a rough estimation.

III. NUMERICAL RESULTS AND DISCUSSIONS
The branching ratios of nonleptonic two-body B c decays in the rest frame of the B c meson can be written as: where the lifetime of the B c meson τ Bc = 513.4±11.0±5.7 fs [5] and p is the common momentum of final particles. The decay amplitudes A(B c →BM) are listed in Appendix A.
The input parameters in our calculation, including the CKM Wolfenstein parameters, decay constants, Gegenbauer moments of distribution amplitudes in Eq. (10), are collected in Table II. If not specified explicitly, we will take their central values as the default inputs.
Our numerical results on the CP -averaged branching ratios are presented in Table III, where theoretical uncertainties of the "QCDF" column come from the CKM parameters, the renormalization scale µ = (1±0.2)m c , decay constants and Gegenbauer moments, respectively.
For comparison, previous results calculated with the fixed coefficients a 1 ≃ 1.22 and a 2 ≃ −0.4 are also listed. There are some comments on the branching ratios.
(1) From the numbers in Table III, it is seen that different branching ratios for the B c → BP , BV decays were obtained with different approaches in previous works, although the same coefficients a 1,2 are used. Much of the discrepancy comes from the different values of the transition form factors. If the same value of the form factor is used, then the disparities on branching ratios for the a 1 -dominated B c decays will be highly alleviated. For example, all previous predictions on Br(B c →B s π) will be about 10% with the same form factor F Bc→Bs 0 ≃ 1.0, which is generally in line with the QCDF estimation within uncertainties and also agrees with the recent LHCb measurement [10].
(2) There is a hierarchical structure between the QCDF's results on branching ratios for the B c → BP and BV decays with the same final B q meson, for example, which differs from the previous results. There are two decisive factors. One is the kinematic factor. The phase space for the B c → BV decays is more compressed than that for the B c → BP decays, because the mass of the light pseudoscalar meson (except for the exotic η ′ meson) is generally less than the mass of the corresponding vector meson with the same valence quark components. The other is the dynamical factor. The orbital angular momentum for the BP final states is ℓ BP = 0, while the orbital angular momentum ℓ BV = 1 for the BV final states.
(3) According to the CKM factors and the coefficients a 1,2 , there is another hierarchy of amplitudes among the QCDF's branching ratios for the B c decays, which could be subdivided into different cases as below. The CKM-favored a 1 -dominated B c → B s π decay are expected to have the largest branching ratio, ∼ 10%, within the QCDF framework. In addition, the branching ratios for the Cabibbo favored B + c → B 0 s ρ + , B + u K 0 decays are also larger than 1%, which might be promisingly detected at experiments.

Case
Coefficient CKM factor Branching ratio Decay modes (4) There are many uncertainties on the QCDF's results. The first uncertainty from the CKM factors is small due to the high precision on Wolfenstein parameter λ with only 0.3% relative errors [1]. Large uncertainty comes from the renormalization scale, especially for the a 2 dominated B c → B u P , B u V decays. In principle, the second uncertainty could be reduced by the inclusion of higher order α s corrections to hadronic matrix elements. It has been showed [40] that tree amplitudes incorporating with the NNLO corrections are relatively less sensitive to the choice of scale than the NLO amplitudes. As aforementioned, large uncertainty mainly comes from hadron parameters, such as the transition form factors, which is expected to be cancelled from the rate of branching ratios. For example, Especially, the relation of Eq. (22) might be used to give some information on the coefficients a 1,2 and to provide an interesting feasibility research on the validity of the QCDF approach for the charm quark decay. Finally, we would like to point out that many uncertainties from other factors, such as the final state interactions, which deserve the dedicated study, are not considered here. So one should not be too serious about the numbers in Table III. Despite this, our results will still provide some useful information to experimental physicists, i.e., the Cabibbo favored B c → B s π, B s ρ, B u K decays have large branching ratios > ∼ 1%, which could be detected earlier.

IV. SUMMARY
In prospects of the potential B c meson at the LHCb experiments, accurate and thorough studies of the B c decays will be accessible very soon. The carefully theoretical study on the B c decays is urgently desiderated. In this paper, we concentrated on the nonfactorizable contributions to hadronic matrix elements within the QCDF framework, while the transition form factors are taken as nonperturbative inputs, which is different from previous studies. It is found that the branching ratios for the Cabibbo favored B c → B s π, B s ρ, B u K decays are very large and could be measured earlier by the running LHCb experiment in the forthcoming years.

Acknowledgments
The work is supported by the National Natural Science Foundation of China (Grant Nos.
g It is estimated with the form factors F Bc→Bs 0 = 1.03, F Bc→B 0 = 1.01 in the relativistic independent quark model [25]. h It is estimated within a relativistic constituent quark model [26].
i It is estimated with the form factors F Bc→Bs 0 = 1.3, F Bc→B 0 = 1.27 in the QCD sum rules [27]. j It is estimated with the perturbative QCD approach based on the k T factorization scheme [29].