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In this work, we investigate the and decays into with the and assignments to and , respectively. The results show that the decay width of is at least one order smaller than that of . This observation explains why only one structure, , has been observed in the invariant mass spectrum for the process .
Subject IndexB62B69D32arxiv-idarXiv:1311.6274
The fusion process is an ideal platform to produce charmonium-like states. In the past, the Belle and BaBar experiments have reported many charmonium-like states in the fusion processes. Among these observations, has mass MeV and width MeV. Since was observed in the invariant mass spectrum of , the possible quantum number should be or , which results in the corresponding Belle measurement of eV or eV [1]. As a candidate for charmonium (), was first observed in the process [2]. The experimental information on gives MeV, MeV, and keV [2]. Later, the BaBar Collaboration also confirmed the observation of in [3].
In Ref. [4], the assignments of or as or charmonium states were proposed by analyzing the mass spectrum and calculating the strong decay of P-wave charmonium. Later, in Ref. [5], the BaBar Collaboration announced that the charmonium-like state had been confirmed in the process with a spin-parity [5], which is consistent with the prediction in Ref. [4].
If is a state, theoretically has the hidden-charm decay channel besides its observed open-charm decay . Hence, the signal of should appear in the same invariant mass spectrum as , which was observed by Belle [1]. However, the experimental data for the invariant mass spectrum show no evidence of . This fact urges us to explain why there only exists one signal, , observed in the process .
In this work, we dedicate ourselves to studying the and decays into under the and assignments to and , respectively. By this study, we wan to answer whether the decay is suppressed compared with under the P-wave charmonium assignments to and , which can shed light on the above puzzle.
As higher charmonia, the and decays into occur via hadronic loop effects with the open-charm decay channels as the intermediate state. This mechanism has been studied in Refs. [6–13] when calculating the hidden-charm and open-charm decays of charmonium and other charmonium-like states.
The and under discussion are candidates for the first radial excitations of and , respectively. Since the masses of and are above the thresholds of and and below the threshold, and dominantly decay into and , which contribute to the total widths of and (see Ref. [4] for more details). As the subordinate decay mode, is assumed from the rescattering contribution of the dominant decays , which is the reason why we only consider the intermediate and contributions in this work. Under the assignment to , the hidden-charm decay occurs through the intermediate states since with dominantly decays into , as indicated in Ref. [4]. The hadron level descriptions of are shown in Fig. 1(a). The expression for the decay amplitude of the hidden-charm decay reads
As the state, mainly decays into and [4]. Thus, its hidden-charm decay is shown in Figs. 1(b)–(d). The amplitude for the processes can be expressed as
where the factor 4 in Eqs. (1) and (2) results from the charge conjugate and isospin transformations.
Typical diagrams describing the (a) and ((b)–(d)) decays. After making the charge conjugate transformation () and the isospin transformation ( and ), one gets other diagrams.
To write out the amplitudes corresponding to the diagrams listed in Fig. 1, we adopt the effective Lagrangian approach. The effective Lagrangian expressing the interactions of with or is given by [14]
The couplings of charmed mesons with the light vector meson or charmonium are constructed in Refs. [15, 16], paying attention to the heavy quark symmetry and the chiral symmetry, and are given below:
where , , and . The light vector nonet meson can form the following matrix :
The coupling constants of and are obtained by fitting the total widths of and , which will be presented later. The coupling constants of interacting with a pair of charmed mesons and a coupling constant of charmed mesons interacting with a light vector meson are given in Table 1 [15–18].
The values of the coupling constants shown in Eqs. (3)–(5). Here, we take , , , MeV, , GeV, , and MeV [15–18].
Coupling
Expression
Value
Coupling
Expression
Value
7.71
3.71
3.98 GeV
4.64
7.71
2.31 GeV
3.71
The amplitudes for corresponding to Fig. 1(a) are given by
where and are the amplitudes corresponding to diagram (a) in Fig. 1 with the and meson exchanges, respectively. Similarly, we can easily write out the the expressions for the decay amplitudes of corresponding to Figs. 1(b)–(d), which are
with , where is the form factor, which is introduced not only to compensate the off-shell effects of the charmed meson but also to describe the structure effects of the vertex of a charmed meson pair interacting with or . In this work, we adopt the form factor in the form
where and are the momentum and the mass of the exchanged charmed meson, respectively. Furthermore, can be parameterized as with a dimensionless parameter and MeV. The parameter is of order unity and depends on the specific process [13, 18].
With the above elaborate expressions for the amplitudes, one can obtain the partial decay width for as
where the overline indicates the sum over the polarizations of the vector meson and tensor meson , and indicates the three-momentum of in the initial state at rest.
If is a state, is its dominant decay. Hence, we can use the experimental width of [1] to determine the coupling constant of interaction, i.e., GeV. However, for , there exist two main decay modes and Since experiments have so far not given the ratio of to , we must determine these corresponding coupling constants from the theoretical results estimated by the quark pair creation model. In Ref. [4], the wave functions of are simulated by a simple harmonic oscillator wave function with a parameter , which means a root-mean-square radius of the wave function. The partial and total decay widths of are dependent on this unique parameter . One can determine the parameter value in the spatial wave function from the partial decay width of under the assumption . With the parameter estimated by the center value of , we obtain and .
For with the assignment of , it dominantly decays into and The absolute values of the coupling constants between and the charmed meson pairs are evaluated by the quark pair creation model. However, the relative sign of the coupling constants and in Eq. (3) can be either positive or negative, which corresponds to the subscripts and shown in Fig. 2, respectively. Thus, we discuss two cases for .
dependence of the ratio of the width of to that of . Here, we define and . In addition, we use the superscripts “Monopole” and “Dipole” to distinguish different results by taking monopole and dipole form factors in the calculation, respectively. The calculated results are –1400, –558, –130, and –59.
In Fig. 2, we give the ratio of the width of to that of . This result shows that the width of is at least one order of magnitude larger than that of in two different cases (see Fig. 2 for more details). Although the decay width for calculated in this work strongly depends on the parameter , the ratio of the width of to that of has a very large value and is weakly dependent on the parameter , as shown in Fig. 2. Such a large ratio could explain why Belle only reported one enhancement structure, , in the invariant mass spectrum of the process.
In addition, the dependence of the partial decay widths is presented in Fig. 3. Here, we take the monopole form factor as an example. As one finds in Fig. 3, the partial decay widths are strongly dependent on the parameter . As for , the partial decay width varies from to 0.15 MeV in the range , while, for , the partial decay width of varies from MeV to MeV or MeV to MeV depending on the relative sign between and . In the present calculations, the loop integrals in Eqs. (7) and (8) are evaluated by the Cutkosky cutting rules, where only the imaginary part of the amplitudes is considered. As for the case of the dipole form factor, the magnitudes of the partial decay widths are at least one order smaller than the corresponding ones estimated by the monopole form factor. The calculations in Ref. [26] also indicate that the monopole form factor is more suitable to estimate the partial decay widths of the process.
dependence of the partial decay widths of , where and are assigned as and , respectively. The loop integrals in Eqs. (7) and (8) are evaluated by the Cutkosky cutting rules. The subscripts and are the same as those in Fig. 2.
As indicated above, the absolute decay widths of and are strongly dependent on the parameter , which means that there exists uncertainty in the prediction of these decay widths. In addition, we notice that extracting the decay widths of from that of via the experimental data depends on our understanding of the two-photon decay width of and , where its predicted two-photon decay width varies with different models. In Refs. [20–23], the two-photon decay width is about 1–2 keV in the relativistic quark model, while the Salpeter method indicates that the decay width for can be larger than 3 keV in a relativistic form and about 5.47 keV in a non-relativistic form [24]. If the center values of the total decay width of and the measured branching ratio are adopted, the partial decay width of can be less than two hundred keV to 1 MeV, depending on the choice of . In the present work, the evaluated partial decay width can reach 150 keV for , which is consistent with the experimental measurements [1].
In summary, , reported by the Belle Collaboration, is the second enhancement observed in the fusion process. As indicated in Ref. [4], is a good candidate for , i.e., the first radial excitation of . Besides its open-charm decay, study of the hidden-charm decay of will provide a key hint to understanding the properties of and further test the P-wave charmonium explanation of in Ref. [4]. Since the mass of is above the threshold of and dominantly decays into , hadronic loop effects [6–13] will play an important role in the hidden-charm decay , which, in fact, results from the coupled-channel effects. In this work, we have performed the calculation of the processes.
Before the observation of , Belle reported a state named in fusion [2, 3], which is also a P-wave charmonium state of the first radial excitation. should decay into , which seems to indicate that there should exist two peaks close to each other in the invariant mass spectrum given by Belle [1]. However, currently, only one structure corresponding to has been observed [1]. In order to explain this contradiction, in this work we have further studied by the intermediate states and The results illustrated in Fig. 2 show that the partial decay width of is suppressed when compared with that of , which explains why cannot be observed in the invariant mass spectrum.
As more charmonium-like states are observed in the fusion process [1–3, 25], they provide us with a better chance to explore the properties of these states, especially P-wave charmonium states [4]. The study of the hidden-charm decay of in this work supports the proposal of assignment to in Ref. [4]. Besides applying the hidden-charm and open-charm decays of to test the assignment to , we suggest that an angular distribution analysis of in future experiments will be valuable to test the explanation of , since the quantum number of must be . Although is well established as a state [2, 3], its hidden-charm decay behavior was unclear before this work. Performing the calculation of by the hadronic loop mechanism, we further learn that the branching ratio of is at least one order smaller than that of , which not only successfully explains the appearance of only one enhancement, , in the invariant mass spectrum but also tests the hadronic loop effects, which is an important non-perturbative mechanism in the decays of charmonium or charmonium-like states [6–13].
Funding
Open Access funding: SCOAP3.
Acknowledgements
This project is supported by the National Natural Science Foundation of China under Grant Nos. 11175073, 11005129, 11375240, 11035006, the Ministry of Education of China (FANEDD under Grant No. 200924, DPFIHE under Grant No. 20090211120029, NCET, the Fundamental Research Funds for the Central Universities), the Fok Ying Tung Education Foundation (No. 131006), and the West Doctoral Project of the Chinese Academy of Sciences.