Abstract
Inspired by the recent measurements on the , weak decays at BESIII and the potential prospects of charmonium at high-luminosity heavy-flavor experiments, we study and weak decays into final states including one charmed meson plus one light meson, considering QCD corrections to hadronic matrix elements with QCD factorization approach. It is found that the Cabibbo-favored , , decays have large branching ratios , which might be accessible at future experiments.
1. Introduction
More than forty years after the discovery of the meson, the properties of charmonium (bound state of ) continue to be the subject of intensive theoretical and experimental study. It is believed that charmonium, resembling bottomonium (bound state of ), plays the same role in exploring hadronic dynamics as positronium and/or the hydrogen atom 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 plentiful dedicated investigation from BES, CLEO-c, LHCb, and the studies via decays of the mesons at factories.
The and mesons are -wave charmonium states below open-charm kinematic threshold and have the well-established quantum numbers of and , respectively. They decay mainly through the strong and electromagnetic interactions. Because the -parity conserving hadronic decays , and are suppressed by the compact phase space of final states, and because the decays into light hadrons are suppressed by the phenomenological Okubo-Zweig-Iizuka (OZI) rules [2–4], the total widths of and are narrow (see Table 1), which might render the charmonium weak decay as a necessary supplement. Here, we will concentrate on the and weak decays into final states, where denotes the low-lying pseudoscalar and vector meson nonet. Our motivation is listed as follows.
From the experimental point of view, () some data samples have been collected by BESIII since 2009 [5]. It is inspiringly expected to have about billion and 3 billion events at BESIII experiment per year of data taking with the designed luminosity [6]: over at LHCb [7], ATLAS [8], and CMS [9] per data in 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. () Identification of the single meson would provide an unambiguous signature of the charmonium weak decay into 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 , decays has been performed at BESIII, although signals are unseen for the moment [10]. Of course, the branching ratios for the inclusive charmonium weak decay are tiny within the standard model, about and , where denotes the neutral charmed meson [11] and and stand for the total widths of and 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 [11].
From the theoretical point of view, () the charm quark weak decay is more favorable than the bottom quark weak decay, because the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements obey [12]. Penguin and annihilation contributions to nonleptonic charm quark weak decay, being proportional to the CKM factor with the Wolfenstein parameter [12], are highly suppressed and hence negligible relative to tree contributions. Both and 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. () There are few works devoted to nonleptonic weak decays in the past, such as [13] with the covariant light-cone quark model, [14] with QCD sum rules, and [15–17] with the Wirbel-Stech-Bauer (WSB) model [18]. Moreover, previous works of [13–17] concern mainly the weak transition form factors between the and charmed mesons. Fewer papers have been devoted to nonleptonic and 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 2, we will present the theoretical framework and the amplitudes for , decays. Section 3 is devoted to numerical results and discussion. Finally, Section 4 is our summation.
2. Theoretical Framework
2.1. The Effective Hamiltonian
Phenomenologically, the effective Hamiltonian responsible for charmonium weak decay into final states can be written as follows [25]: where [12] is the Fermi coupling constant; is the CKM factor with , ; the Wilson coefficients , which are independent of one particular process, summarize the physical contributions above the scale of . The expressions of the local tree four-quark operators are where and are color indices.
It is well known that the Wilson coefficients 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 can be evaluated with the renormalization group (RG) equation [25]: where is the RG evolution matrix which transforms the Wilson coefficients from scale of to . The expression for can be found in [25]. The numerical values of the leading-order (LO) and NLO in the naive dimensional regularization scheme are listed in Table 2. The values of coefficients in Table 2 agree well with those obtained with “effective” number of active flavors [25] rather than formula (3).
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.
2.2. Hadronic Matrix Elements
Analogous to the exclusive processes with perturbative QCD theory proposed by Lepage and Brodsky [26], 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 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 final states. It is commonly assumed that the virtuality of the gluon connecting to the heavy spectator is of order , where is the characteristic QCD scale. Hence, the transition form factors between charmonium and 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 decays, for example, HME can be written as where is the weak transition form factor and and are the decay constant and LCDA of the meson , respectively. The leading twist LCDA for the pseudoscalar and longitudinally polarized vector mesons can be expressed in terms of Gegenbauer polynomials [23, 24]: where ; is the Gegenbauer polynomial, is the Gegenbauer moment corresponding to the Gegenbauer polynomials ; for the asymptotic form; and for because of the -parity invariance of the , , , , meson distribution amplitudes. In this paper, to give a rough estimation, the contributions from higher-order Gegenbauer polynomials are not considered for the moment.
Hard scattering function in (4) is, in principle, calculable order by order with the perturbative QCD theory. At the order of , . 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 and higher orders, the renormalization scale dependence of hadronic matrix elements could be recuperated to partly cancel the -dependence of the Wilson coefficients. In addition, part of the strong phases could be reproduced from nonfactorizable contributions.
Within the QCDF framework, amplitudes for decays can be expressed as
In addition, the HME for the decays are conventionally expressed as the helicity amplitudes with the decomposition [27, 28], The relations among helicity amplitudes and invariant amplitudes , , are where three scalar amplitudes , , describe , , wave contributions, respectively.
The effective coefficient at the order of can be expressed as [19] where the color factor ; the color number . For the transversely polarized light vector meson, the factor in the helicity amplitudes beyond the leading twist contributions. With the leading twist LCDA for the pseudoscalar and longitudinally polarized vector mesons, the factor is written as [19]
From the numbers in Table 2, it is found that () the values of coefficients agree generally with those used in previous works [14–17, 20], () the strong phases appear by taking nonfactorizable corrections into account, which is necessary for violation, and () the strong phase of is small due to the suppression of and . The strong phase of is large due to the enhancement from the large Wilson coefficients .
2.3. Form Factors
The weak transition form factors between charmonium and a charmed meson are defined as follows [18]:where ; denotes the ’s polarization vector. The form factors and are required compulsorily to cancel singularities at the pole of . There is a relation among these form factors:
There are four independent transition form factors, , , and , at the pole of . They could be written as the overlap integrals of wave functions [18]: where is the Pauli matrix acting on the spin indices of the decaying charm quark; and denote the fraction of the longitudinal momentum and the transverse momentum of the nonspectator quark, respectively.
With the separation of the spin and spatial variables, wave functions can be written as where the total angular momentum because the orbital angular momentum between the valence quarks in , , mesons in question have ; denote the spins of valence quarks in meson; and for the and mesons, respectively.
The charm quark in the charmonium state is nearly nonrelativistic with an average velocity based on arguments of nonrelativistic quantum chromodynamics (NRQCD) [29–31]. For the meson, the valence quarks are also nonrelativistic due to , where the light quark mass MeV and MeV [32]. Here, we will take the solution of the Schrödinger equation with a scalar harmonic oscillator potential as the wave functions of the charmonium and mesons: where the parameter determines the average transverse quark momentum, . With the NRQCD power counting rules [29], for heavy quarkonium. Hence, parameter is approximately taken as in our calculation.
Using the substitution ansatz [33], one can obtain where the parameters and are the normalization coefficients satisfying the normalization condition,
The numerical values of transition form factors at are listed in Table 3. It is found that () the model dependence of form factors is large; () isospin-breaking effects are negligible and flavor breaking effects are small; and () as stated in [18] holds within collinear symmetry.
3. 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 and are collected in Appendices A and B, respectively.
In our calculation, we assume that the light vector mesons are ideally mixed; that is, and . For the mixing of pseudoscalar and meson, we will adopt the quark-flavor basis description proposed in [22] and neglect the contributions from possible gluonium compositions; that is, where and ; the mixing angle [22]. The mass relations are
The input parameters, including the CKM Wolfenstein parameters, decay constants, and Gegenbauer moments, are collected in Table 4. If not specified explicitly, we will take their central values as the default inputs. Our numerical results on branching ratios for the nonleptonic two-body , weak decays are displayed in Tables 5 and 6, where the uncertainties of this work come from the CKM parameters, the renormalization scale , and hadronic parameters including decay constants and Gegenbauer moments, respectively. For comparison, previous results on weak decays [14, 16, 17] with parameters and are also listed in Table 5. The following are some comments.(1)There are some differences among the estimates of branching ratios for weak decays (see the numbers in Table 5). These inconsistencies among previous works, although the same values of parameters are used, come principally from different values of form factors. Our results are generally in line with the numbers in columns “A” and “B” which are favored by [17].(2)Branching ratios for weak decay are about two or more times as large as those for decay into the same final states, because the decay width of is about three times as large as that of .(3)Due to the relatively small decay width and relatively large space phases for decay, branching ratios for weak decay are some five (ten) or more times as large as those for weak decay into the same () final states.(4)Among and mesons, has a maximal decay width and a minimal mass resulting in a small phase space, while has a minimal decay width. These facts lead to the smallest [or the largest] branching ratio for [or ] weak decay among , weak decays into the same final states.(5)Compared with decays, the corresponding decays, where and have the same flavor structures, are suppressed by the orbital angular momentum and so have relatively small branching ratios. There are some approximative relations and .(6)According to the CKM factors and parameters , nonleptonic charmonium weak decays could be subdivided into six cases (see Table 7). Case “i-a” is the Cabibbo-favored one, so it generally has large branching ratios relative to cases “i-b” and “i-c.” The -dominated charmonium weak decays are suppressed by a color factor relative to -dominated ones. Hence, the charmonium weak decays into and final states belonging to case “1-a” usually have relatively large branching ratios; the charmonium weak decays into the 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 .(7)Branching ratios for the Cabibbo-favored , , decays can reach up to , which might be measurable in the forthcoming days. For example, production cross section can reach up to a few with the LHCb and ALICE detectors at LHC [7, 8]. Therefore, over samples are in principle available per 100 data collected by LHCb and ALICE, corresponding to a few tens of , , events for about 10% reconstruction efficiency.(8)There is a large cancellation between the CKM factors and , which results in a very small branching ratio for charmonium weak decays into state.(9)There are many uncertainties in our results. The first uncertainty from the CKM factors is small due to high precision on the Wolfenstein parameter with only 0.3% relative errors now [12]. The second uncertainty from the renormalization scale could, in principle, be reduced by the inclusion of higher order corrections. For example, it has been shown [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 Tables 5 and 6 just provide an order of magnitude estimate. Many other factors, such as the final state interactions and dependence of form factors, which are not considered here, deserve many dedicated studies.
4. Summary
With the anticipation of abundant data samples on charmonium at high-luminosity heavy-flavor experiments, we studied the nonleptonic two-body and weak decays into one ground-state charmed meson plus one ground-state light meson based on the low energy effective Hamiltonian. By considering QCD radiative corrections to hadronic matrix elements of tree operators, we got the effective coefficients containing partial information of strong phases. The magnitude of agrees comfortably with those used in previous works [14–17]. The transition form factors between the charmonium and charmed meson are calculated by using the nonrelativistic wave functions with isotropic harmonic oscillator potential. Branching ratios for , decays are estimated roughly. It is found that the Cabibbo-favored , , decays have large branching ratios , which are promisingly detected in the forthcoming years.
Appendices
A. The Amplitudes for Decays
Consider
B. The Amplitudes for the Decays
Consider
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The work is supported by the National Natural Science Foundation of China (Grants nos. 11547014, 11275057, 11475055, U1232101, and U1332103).