Study of the ${\psi}(1S,2S)$ and ${\eta}_{c}(1S,2S)$ weak decays into $DM$

Inspired by the recent measurements on the $J/{\psi}(1S)$ ${\to}$ $D_{s}{\rho}$, $D_{u}K^{\ast}$ weak decays at BESIII and the potential prospects of the charmonium at the high-luminosity heavy-flavor experiments, we study ${\psi}(1S,2S)$ and ${\eta}_{c}(1S,2S)$ weak decays into final states including one charmed meson plus one light meson, considering the QCD corrections to hadronic matrix elements with the QCD factorization. It is found that the Cabibbo favored ${\psi}(1S,2S)$ ${\to}$ $D_{s}^{-}{\rho}^{+}$, $D_{s}^{-}{\pi}^{+}$, $\overline{D}_{u}^{0}\overline{K}^{{\ast}0}$ decays have branching ratios ${\gtrsim}$ $10^{-10}$, which might be accessible at the future experiments.


I. INTRODUCTION
More than forty years after the discovery of the J/ψ meson, the properties of charmonium (bound state of cc) continue to be the subject of intensive theoretical and experimental study.
It is believed that charmonium, resembling bottomonium (bound state of bb), plays the same role in exploring hadronic dynamics as positronium and/or the hydrogen atom plays in understanding the atomic physics. Charmonium and bottomonium are good objects to test the basic ideas of QCD [1]. There is a renewed interest in charmonium due to the (OZI) rules [2][3][4], the total widths of ψ(1S, 2S) and η c (1S, 2S) are narrow (see Table I), which might render the charmonium weak decay as a necessary supplement. Here, we will concentrate on the ψ(1S, 2S) and η c (1S, 2S) weak decays into DM final states, where M denotes the low-lying SU(3) pseudoscalar and vector meson nonet. Our motivation is listed as follows.   [7]; over 10 10 J/ψ at LHCb [8], ATLAS [9] and CMS [10] per fb −1 data in pp collisions. A large amount of data sample offers a realistic possibility to explore experimentally the charmonium weak decays. Correspondingly, theoretical study is very necessary to provide a ready reference. (2) Identification of the single D meson would provide an unambiguous signature of the charmonium weak decay into DM states. With the improvements of experimental instrumentation and particle identification techniques, accurate measurements on the nonleptonic charmonium weak decay might be feasible. Recently, a search for the J/ψ → D s ρ, D u K * decays has been performed at BESIII, although signals are unseen for the moment [11]. Of course, the branching ratios for the inclusive charmonium weak decay is tiny within the standard model, about 2/(τ D Γ ψ ) ∼ 10 −8 and 2/(τ D Γ ηc ) ∼ 10 −10 , where D denotes the neutral charmed meson [12], Γ ψ and Γ ηc stand for the total widths of the ψ(1S, 2S) and η c (1S, 2S) resonances, respectively. Observation of an abnormally large production rate of single charmed mesons in the final state would be a hint of new physics beyond the standard model [12].
From the theoretical point of view: (1) The charm quark weak decay is more favorable than the bottom quark weak decay, because the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements obey |V cb | ≪ |V cs | [5]. Penguin and annihilation contributions to nonleptonic charm quark weak decay, being proportional to the CKM factor |V cb V ub | ∼ O(λ 5 ) with the Wolfenstein parameter λ ≃ 0.22 [5], are highly suppressed, and hence negligible relative to tree contributions. Both c andc quarks in charmonium can decay individually, which provides a good place to investigate the dynamical mechanism of heavy flavor weak decay and crosscheck model parameters obtained from the charmed hadron weak decays. (2) There are few works devoted to nonleptonic J/ψ weak decays in the past, such as Ref. [13] with the covariant light-cone quark model, Ref. [14] with QCD sum rules, and Refs. [15][16][17] with the Wirbel-Stech-Bauer (WSB) model [18]. Moreover, previous works of Refs. [13][14][15][16][17] concern mainly the weak transition form factors between the J/ψ and charmed mesons.
Fewer papers have been devoted to nonleptonic ψ(2S) and η c (1S, 2S) weak decays until now even though a rough estimate of branching ratios is unavailable. In this paper, we will estimate the branching ratios for nonleptonic two-body charmonium weak decay, taking the nonfactorizable contributions to hadronic matrix elements into account with the attractive QCD factorization (QCDF) approach [19].
This paper is organized as follows. In section II, we will present the theoretical framework and the amplitudes for the ψ(1S, 2S), η c (1S, 2S) → DM decays. Section III is devoted to numerical results and discussion. Finally, section IV is our summation.

A. The effective Hamiltonian
Phenomenologically, the effective Hamiltonian responsible for charmonium weak decay into DM final states can be written as [20]: where G F = 1.166×10 −5 GeV −2 [5] is the Fermi coupling constant; V * cq 1 V uq 2 is the CKM factor with q 1,2 = d, s; The Wilson coefficients C 1,2 (µ), which are independent of one particular process, summarize the physical contributions above the scale of µ. The expressions of the local tree four-quark operators are where α and β are color indices.
It is well known that the Wilson coefficients C i could be systematically calculated with perturbation theory and have properly been evaluated to the next-to-leading order (NLO).
Their values at the scale of µ ∼ O(m c ) can be evaluated with the renormalization group (RG) equation [20] where U f (µ f , µ i ) is the RG evolution matrix which transforms the Wilson coefficients from scale of µ i to µ f . The expression for U f (µ f , µ i ) can be found in Ref. [20]. The numerical values of the leading-order (LO) and NLO C 1,2 in the naive dimensional regularization scheme are listed in Table II. The values of coefficients C 1,2 in Table II agree well with those obtained with "effective" number of active flavors f = 4.15 [20] rather than formula Eq.(4).
To obtain the decay amplitudes and branching ratios, the remaining works are to evaluate accurately the hadronic matrix elements (HME) where the local operators are sandwiched between the charmonium and final states, which is also the most intricate work in dealing with the weak decay of heavy hadrons by now.

B. Hadronic matrix elements
Analogous to the exclusive processes with perturbative QCD theory proposed by Lepage and Brodsky [21], the QCDF approach is developed by Beneke et al. [19] to deal with HME based on the collinear factorization approximation and power counting rules in the heavy quark limit, and has been extensively used for B meson decays. Using the QCDF master formula, HME of nonleptonic decays could be written as the convolution integrals of the process-dependent hard scattering kernels and universal light-cone distribution amplitudes (LCDA) of participating hadrons.
The spectator quark is the heavy-flavor charm quark for charmonium weak decays into DM final states. It is commonly assumed that the virtuality of the gluon connecting to the heavy spectator is of order Λ 2 QCD , where Λ QCD is the characteristic QCD scale. Hence, the transition form factors between charmonium and D mesons are assumed to be dominated by the soft and nonperturbative contributions, and the amplitudes of the spectator rescattering subprocess are power-suppressed [19]. Taking the η c → DM decays for example, HME can be written as where F ηc→D i is the weak transition form factor, f M and Φ M (x) are the decay constant and LCDA of the meson M, respectively. The leading twist LCDA for the pseudoscalar and longitudinally polarized vector mesons can be expressed in terms of Gegenbauer polynomials [22,23]: a M n is the Gegenbauer moment corresponding to the Gegenbauer polynomials C 3/2 n (z); a M 0 ≡ 1 for the asymptotic form; a n = 0 for n = 1, 3, 5, · · · because of the G-parity invariance of the π, η (′) , ρ, ω, φ meson distribution amplitudes. In this paper, to give a rough estimation, the contributions from higher-order n ≥ 3 Gegenbauer polynomials are not considered for the moment.
Hard scattering function H i (x) in Eq. (5) is, in principle, calculable order by order with the perturbative QCD theory. At the order of α 0 s , H i (x) = 1. This is the simplest scenario, and one goes back to the naive factorization where there is no information about the strong phases and the renormalization scale hidden in the HME. At the order of α s and higher orders, the renormalization scale dependence of hadronic matrix elements could be recuperated to partly cancel the µ-dependence of the Wilson coefficients. In addition, part of the strong phases could be reproduced from nonfactorizable contributions.
Within the QCDF framework, amplitudes for η c → DM decays can be expressed as: In addition, the HME for the ψ(1S, 2S) → DV decays are conventionally expressed as the helicity amplitudes with the decomposition [24,25], The relations among helicity amplitudes and invariant amplitudes a, b, c are where three scalar amplitudes a, b, c describe the s, d, p wave contributions, respectively.
The effective coefficient a i at the order of α s can be expressed as [19]: where the color factor C F = 4/3; the color number N c = 3. For the transversely polarized light vector meson, the factor V = 0 in the helicity H ± amplitudes beyond the leading twist contributions. With the leading twist LCDA for the pseudoscalar and longitudinally polarized vector mesons, the factor V is written as [19]: From the numbers in Table. II, it is found that (1) the values of coefficients a 1,2 agree generally with those used in previous works [14 -17, 26]. (2) The strong phases appear by taking nonfactorizable corrections into account, which is necessary for CP violation. (3) The strong phase of a 1 is small due to the suppression of α s and 1/N c . The strong phase of a 2 is large due to the enhancement from the large Wilson coefficients C 1 .  [5], where a 1,2 in Ref. [26] is used in the D meson weak decay.

C. Form factors
The weak transition form factors between charmonium and a charmed meson are defined as follows [18]: where q = p 1 − p 2 ; ǫ ψ denotes the ψ's polarization vector. The form factors F 0 (0) = F 1 (0) and A 0 (0) = A 3 (0) are required compulsorily to cancel singularities at the pole of q 2 = 0.
There is a relation among these form factors There are four independent transition form factors, F 0 (0), A 0,1 (0) and V (0), at the pole of q 2 = 0. They could be written as the overlap integrals of wave functions [18].
where σ y,z is the Pauli matrix acting on the spin indices of the decaying charm quark; x and k ⊥ denote the fraction of the longitudinal momentum and the transverse momentum of the nonspectator quark, respectively.
With the separation of the spin and spacial variables, wave functions can be written as where the parameter α determines the average transverse quark momentum, φ 1S | k 2 ⊥ |φ 1S = α 2 . With the NRQCD power counting rules [27], | k ⊥ | ∼ mv ∼ mα s for heavy quarkonium.
Hence, parameter α is approximately taken as mα s in our calculation.
Using the substitution ansatz [31], one can obtain where the parameters A and B are the normalization coefficients satisfying with the normalization condition, The numerical values of transition form factors at q 2 = 0 are listed in

III. NUMERICAL RESULTS AND DISCUSSION
In the charmonium center-of-mass frame, the branching ratio for the charmonium weak decay can be written as where the common momentum of final states is The decay amplitudes for A(ψ→DM) and A(η c →DM) are collected in Appendix A and B, respectively.
The following are some comments.
(1) There are some differences among the estimates of branching ratios for the J/ψ → DM weak decays (see the numbers in Table V) (6) According to the CKM factors and parameters a 1,2 , nonleptonic charmonium weak decays could be subdivided into six cases (see Table VII). Case "i-a" is the Cabibbo-favored one, so it generally has large branching ratios relative to case "i-b" and "i-c". The a 2dominated charmonium weak decays are suppressed by a color factor relative to the a 1dominated ones. Hence, the charmonium weak decays into the D s ρ and D s π final states belonging to case "1-a" usually have relatively large branching ratios; the charmonium weak decays into the D 0 u K * 0 final states belonging to case "2-c" usually have relatively small branching ratios. In addition, the branching ratio of case "2-a" (or "2-b") is usually larger than that of case "1-b" (or "1-c") due to |a 2 /a 1 | ≥ λ. can reach up to 10 −10 , which might be measurable in the forthcoming days. For example, the J/ψ production cross section can reach up to a few µb with the LHCb and ALICE detectors at LHC [8,9]. Therefore, over 10 12 J/ψ samples are in principle available per 100 fb −1 data collected by LHCb and ALICE, corresponding to a few tens of J/ψ → D − s ρ + , D − s π + , D 0 u K * 0 events for about 10% reconstruction efficiency.
(8) There is a large cancellation between the CKM factor V ud V * cd and V us V * cs , which results in a very small branching ratio for charmonium weak decays into D u η ′ state. (9) There are many uncertainties on our results. The first uncertainty from the CKM factors is small due to high precision on the Wolfenstein parameter λ with only 0.3% relative errors now [5]. The second uncertainty from the renormalization scale µ could, in principle, be reduced by the inclusion of higher order α s corrections. For example, it has been showed [34] that tree amplitudes incorporating with the NNLO corrections are relatively less sensitive to the renormalization scale than the NLO amplitudes. The third uncertainty comes from hadronic parameters, which is expected to be cancelled or reduced with the relative ratio of branching ratios.
(10) The numbers in Table V  11547014, 11275057, 11475055, U1232101 and U1332103). We thank the referees for their helpful comments.