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Study of Bc -->J/psi pi, etac pi decays with perturbative QCD approach

The Bc -->J/psi pi, etac pi decays are studied with the perturbative QCD approach. It is found that form factors and branching ratios are sensitive to the parameters w, v, f_J/psi and f_etac, where w and v are the parameters of the charmonium wave functions for Coulomb potential and harmonic oscillator potential, respectively, f_J/psi and f_etac are the decay constants of the J/psi and etac mesons, respectively. The large branching ratios and the clear signals of the final states make the Bc -->J/psi pi, etac pi decays to be the prospective channels for measurements at the hadron colliders

It is estimated that one could expect around 5 × 10 10 B c events per year at LHC [1]. The nonleptonic decays of the B c mesons have been studied in previous literature [1,2]. The theoretical status of the B c meson was reviewed in [1]. In this paper, we will concentrate on the B c → J/ψπ, η c π decays using the perturbative QCD approach. There are several reasons : (i) From the experimental point of view, the decay modes containing the signal of J/ψ meson are among the most easily reconstructible B c decay modes, due to the narrowpeak of J/ψ and the high purity J/ψ → ℓ + ℓ − . For example, the B c mesons are firstly discovered via B c → J/ψℓν by the CDF Collaboration in 1998 [3]. Recently the CDF and D0 Collaborations announced their accurate measurements on the B c mesons via B c → J/ψπ mode [4,5]. Especially, compared with the semi-leptonic decays where the neutrino momentum is not detected directly, all final-state particles are detectable for the B c → J/ψπ, η c π decays. It is estimated that the ATLAS detector would be able to record about 5600 events of B c → J/ψπ per year [1]. So B c → J/ψπ, η c π decays may be two of the most prospective channels for measurements.
(ii) From the phenomenological point of view: In recent years, several attractive methods have been proposed to study the nonleptonic B decays, such as the QCD factorization [6], perturbative QCD method (pQCD) [7,8,9], soft and collinear effective theory [10,11], and so on. The study of B c decays provides opportunities to test the k T and collinear factorizations, to check the various treatments for the entanglement of different energy modes, to deepen our understanding on perturbative and nonperturbative contributions. These methods developed recently are widely applied to the nonleptonic two-body B u,d,s decays in literature, but with very few application of these methods on the B c meson decays. The appealing feature of the pQCD factorization [7,8,9] is that form factors can be computed in terms of wave functions (nonperturbative contributions) and hard kernels (perturbative contributions arising from hard gluon exchange) assuming that additional soft contributions are suppressed by the Sudakov factor in the heavy quark limit. Although there is still some controversy about the pQCD method, for example, the problem of gauge invariant [12], the pQCD method has been extensively used in the past to study nonleptonic B decays with fairly good phenomenological results [13]. In this paper, we will take the B c → J/ψπ, η c π decays as examples to discuss the B c decays in the perturbative QCD method.
(iii) From the theoretical point of view: The B c → J/ψπ, η c π decays are similar to the q π (where q = u, d, s) decays with the "spectator quark" ansatz. The B q → D ( * ) q π decays have been studied with the pQCD method [15]. Compared with the B q → D ( * ) q π decays, the B c → J/ψπ, η c π decays are easy to deal with because that the B c meson and the J/ψ (or η c ) meson are heavy quarkonia and could be described approximatively by nonrelativistic dynamics. Given m Bc ≃ m b + m c , the wave function of the B c mesons would be close to δ(x − m c /m Bc ) in the nonrelativistic limit (where m Bc , m b , and m c are the masses of the B c mesons, b quark, and c quark, respectively; x denotes the momentum fraction of the c quark in the B c meson). The wave functions for pion are well-defined. The only parameter is the wave function of the J/ψ (or η c ) meson. So the B c → J/ψπ, η c π decays provide good platform to test quark potential models derived from QCD. This paper is organized as follows: In Section II, we discuss the theoretical framework and compute the decay amplitudes for B c → J/ψπ, η c π with the perturbative QCD approach.
The section III is devoted to the numerical results. Finally, we summarize in Section IV.

A. The effective Hamiltonian
Using the operator product expansion and renormalization group (RG) equation, the low energy effective Hamiltonian for B c → X cc π decay can be written as (where X cc = J/ψ, η c ): where V cb V * ud is the CKM factor accounting for the strengths of the concerned nonleptonic decay processes. The parameters C i (µ) are Wilson coefficients which have been evaluated to the next-to-leading order with the perturbation theory. The expressions of the local operators are where α, β are color indices. The essential problem obstructing the calculation of decay amplitude is how to evaluate the hadronic matrix elements of the local operators.

B. Hadronic matrix elements
The calculation of the hadronic matrix elements is difficult due to the nonperturbative effects arising from the strong interactions. Phenomenologically, the simplest approach to hadronic matrix elements is the Bauer-Stech-Wirbel (BSW) model [14] based on color transparency and naive factorization hypothesis, where the hadronic matrix elements are parameterized into the product of the decay constants and the transition form factors.
One defect of the rough BSW method is that the hadronic matrix elements cannot cancel the renormalization scheme-and scale-dependence of the Wilson coefficients. To remedy this problem, the "nonfactorizable" contributions must be taken into account. Using the Brodsky-Lepage approach [16], the hadronic matrix elements can be written as the convolution of a hard-scattering amplitude, including some perturbative QCD contributions, and meson wave functions.
Recently, a modified perturbative QCD formalism has been proposed under the k T factorization framework [7,8,9]. The Sudakov effects are introduced to modify the endpoint behavior. The decay amplitudes are factorized into three convolution factors: the "harder" functions, the heavy quark decay subamplitudes, and the nonperturbative meson wave functions, which are characterized by the W ± boson mass m W , the typical scale t of the decay processes, and the hadronic scale Λ QCD , respectively. Using the resummation technique and the RG treatment, the final decay amplitudes can be expressed as where the Wilson coefficient C(t) is calculated in perturbative theory at scale of m W and evolved down to the typical scale t using the RG equations, ⊗ denotes the convolution over parton kinematic variables, H(t) is the hard-scattering subamplitude, the wave functions Φ(x, b) absorb nonperturbative long-distance dynamics, x is the longitudinal momentum fraction of the valence quark of the meson, b is the conjugate variable of the transverse momentum of the valence quark of the meson. According the arguments in [7,8,9], the amplitude of Eq.(3) is free from the renormalization scale dependence.

C. Kinematic variables
For convenience, the kinematics variables are described in the terms of the light cone coordinate. The momenta of the valence quarks and hadrons in the rest frame of the B c meson are defined by where the notation of momenta of p i and k i are displayed in FIG.1. The null vectors n 2 and n 3 are the plus and minus directions, respectively. The mass of the π meson is neglected.
The momentum of the π meson is chosen to be parallel to the null vector n 3 . The mass

D. Bilinear operator matrix elements for mesons
In terms of the notation in [17], the nonlocal bilinear-quark operator matrix elements associated with the B c meson, π meson, the longitudinally polarized J/ψ meson, η c meson are decomposed into [17,18] where the wave functions φ a

. Their expressions are collected in APPENDIX A and B.
For the wave function φ Bc , we will take the nonrelativistic approximation as stated in the introduction, i.e.
where N c is the color number, f Bc is the decay constant of the B c meson.
The B c → X cc form factors are defined as [14,19]: where q = p 1 − p 2 , ǫ * denotes the polarization vector of the J/ψ meson. are the transition form factors. In addition, at large recoil limit, q 2 = 0, we have In the perturbative QCD approach, these form factors can be generally written as where the CKM matrix elements

III. NUMERICAL RESULTS AND DISCUSSIONS
The branching ratios in the B c meson rest frame can be written as: where the common momentum |p| = (m 2 Bc − m 2 Xcc )/2m Bc , the lifetime and mass of the B c meson are m Bc = 6.276 ± 0.004 GeV and τ Bc = 0.46±0.07 ps [20] . With some appropriate parameters, our results b on the form factors F 0 and A 0,1 are in agreement with those in the previous works [22,25,26,27,28,29,31,32]. Our results on the form factors A 2 are smaller than those in the previous works [25,26,27,28,29,31,32]. According to the "spectator quark" ansatz, there might be F Bc→ηc (or A B→D * 0,1,2 ) ≈ 0.6 by intuition. So maybe the results based on the three-point QCD sum rules [23,24] are small.
Our numerical results on the amplitudes and branching ratios for B c → J/ψπ, η c π decays a Here, we think that theoretical prediction on input parameters, such as ω and v, relies on our educated guesswork. All values within allowed ranges should be treated on an equal footing, irrespective of how close they are from the edges of the allowed range. For example, we cannot say that the probability of ω = 0.5 GeV is less than that of ω = 0.6 GeV, while the error means the usual one standard deviation in the form of A +δA −δA (such as the expression of m Bc = 6.276 ± 0.004 GeV). So our numerical results had better to be given by a range to show the theoretical uncertainties, rather than the form of A +δA  3(a,b) and the branching ratios decrease with the increasing parameters ω and v.
Besides the large uncertainties from the parameter ω and v, the uncertainties of the decay constants f J/ψ and f ηc will bring ∼ 7% and ∼ 24% uncertainties to the branching ratio for B c → J/ψπ and η c π decays, respectively. Considered the uncertainties from the input parameters, our results on the branching ratios are basically consistent with those in previous works [31,36,37,38,41,42,43] (see the numbers in TABLE.IV).
Compared with the results in [27,29] where small form factors are used (see the numbers in TABLE.I), we find that our predictions are large. If with the same factor factors, our results generally agree with those in [27,29]. The large predictions in [40] are obtained by the relations among the amplitudes under the quark diagram scheme, i.e. A(B c →η c π) = V cb /V ub A(B→D s π) and A(B c →J/ψπ) = V cb /V ub A(B→D s ρ). Intuitively, the distribution amplitudes of the heavy quarkonia (such as B c , J/ψ and η c mesons) should be narrower than those of the "heavy-light" systems (such as B and D mesons). So, the superposition among the B c − J/ψ(η c ) − π systems might be less than those among the B − D s − π(ρ) systems, i.e. there might be A(B c →η c π) < ∼ V cb /V ub A(B→D s π) and A(B c →J/ψπ) < ∼ V cb /V ub A(B→D s ρ). If this argument or/and assumption is true, then it is expected that the results in [40] would become smaller and be consistent with ours.
(iii) The dominating amplitudes A FIG.3(a,b) and the branching ratios subjected to the char-monium wave function for Coulomb potential are larger than those for harmonic oscillator potential. For a fixed value of parameter ω or v, the relation of the branching ratios is BR(B c →η c π) > ∼ BR(B c →J/ψπ). There are at least two reasons, one is that the phase spaces for B c → η c π decay is larger than those for B c → J/ψπ decay, the other is (see the numbers in TABLE.I). The relation of the branching ratios is in agreement with the previous predictions [22,27,29,37,38,40,41,42,43].
The signal of B c → J/ψπ decay has been identified by the detectors at hadron collider Tevatron. It is eagerly expected that the signal of B c → η c π decay is at the near corner for Tevatron and LHC.

IV. SUMMARY AND CONCLUSION
In this paper, the B c → J/ψπ, η c π decays are studied with the perturbative QCD approach. It is found that the form factors A The distribution amplitude φ a π for the twist-2 wave function and the distribution amplitude φ p π and φ t π for the twist-3 wave functions are [17] φ a π (x) = with the decay constant f π = 130 MeV. The Gegenbauer polynomials are defined by

APPENDIX B: WAVE FUNCTIONS OF THE J/ψ AND η c MESONS
The heavy quarkonium, such as cc, similar to diatomic molecules, might be amenable to a Born-Oppenheimer treatment c [1]. Following the prescription in [18,33], two forms of the wave functions corresponding to two different nonrelativistic potentials will be derived.

wave functions for harmonic oscillator potential
In the nuclear shell model, a more realistic description of the nucleons inside the atomic nucleus is given by the Woods-Saxon potential. The Schrödinger equation subjected to the Woods-Saxon potential cannot be solved analytically, and must be treated numerically, but the energy levels as well as other properties can be arrived at by approximating the model with a three-dimensional harmonic oscillator. The spectroscopy of the heavy quarkonium cc can be treated by this model. The quantum number n r L for the J/ψ and η c mesons is 1S, c The heavy quark-antiquark pair is bound by the gluon and light-quark clouds. The heavy quarks correspond to the nuclei in diatomic molecules. The gluon and light-quark fields correspond to the electrons, and provide adiabatic potentials [1].
where n r and L are the radial quantum number and the orbital angular momentum quantum number, respectively. (note : the energy spectrum of a three-dimensional harmonic oscillator is given by E nrL = {2(n r − 1) + L + 3 2 }ω.) The radial wave function of the corresponding Schrödinger state is given by where α 2 = m c ω/2, ω is the frequency of oscillations or the quantum of energy.
Applying the Fourier transform, the state Eq.(B1) is replaced by the mapping representation on the momentum space, Employing the substitution ansatz [18,33]: wherex = 1 − x, and x is the longitudinal momentum fraction of the valence quark of the meson, the wave function can be taken as Applying the Fourier transform to replace the transverse momentum k ⊥ with its conjugate variable b, the 1S-oscillator wave function can be taken as The modified wave functions can be written as with Φ asy (x) being set to the asymptotic models of the corresponding twists for light mesons, which have been given in [33]. Therefore, we can obtain the wave functions of the J/ψ and η c mesons in Eq.(6) and Eq.(7) where N c is the color number, N L,t ψ , N v,s ηc are the normalization constants. All wave function in Eqs.(B7)-(B10) are symmetric under x ↔x and normalized : The parameter ω ≈ m ψ(2S) − m J/ψ(1S) ≈ m ηc(2S) − m ηc(1S) ≈ 0.6 GeV.

wave functions for Coulomb potential
In the static QCD potential, the interactions between heavy quarkonium can be param- where C F = 4/3 is the SU(3) colour factor.
The radial wave function of the corresponding Schrödinger state is given by (note : the principle quantum number n associated with Coulomb potential is given by n = (n r − 1) + where q B = C F µ c α s is the Bohr momentum, µ c = m c /2 is the reduced mass of the c-quark.
Analogous to the treatment for the case of harmonic oscillator discussed above, we can get where the typical velocity of the quarks in charmonium v = q B /m c = 2α s /3 ∼ 0.3 [34]. The wave functions of the J/ψ and η c mesons can be written as The normalization conditions are the same as those of Eq.(B11) and Eq.(B12).

APPENDIX C: FORM FACTORS IN THE PERTURBATIVE QCD APPROACH
The quark anomalous dimension γ q = −α s /π. The explicit expression of s(Q, b) appearing in Sudakov form factor can be found in [35]. The hard functions H are APPENDIX D: THE DECAY AMPLITUDES 1. The amplitudes for B c → J/ψπ decay with the perturbative QCD approach where C 1,2 are the Wilson coefficients. The definitions of other parameters are the same as those in APPENDIX C.
2. The amplitudes for B c → η c π decay with the perturbative QCD approach