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Wang, Xiao-Yun; Chen, Xu-Rong

doi:https://doi.org/10.1155/2015/918231

Advances in High Energy Physics

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Research Article | Open Access

Volume 2015 | Article ID 918231 | https://doi.org/10.1155/2015/918231

Discovery Potential for the Neutral Charmonium-Like by Annihilation

Xiao-Yun Wang^1,2,3and Xu-Rong Chen^1,3

Academic Editor: George Siopsis

Received07 Mar 2015

Accepted12 May 2015

Published25 May 2015

Abstract

Inspired by the observation of charmonium-like , we explore the discovery potential of the neutral production by antiproton-proton annihilation with an effective Lagrangian approach. By investigating the process including the signal and background contributions, it is found that the center of mass energy ≃ 4.0–4.5 GeV is the best energy window for searching the neutral , where the signal can be clearly distinguished from background. The relevant calculations not only are helpful to search for the neutral in the future experiment but also will promote the understanding of the nature and production mechanism of neutral better.

1. Introduction

In the conventional constituent quark model (CQM), all mesons can be described as quark-antiquark () states. Such as the charmonium state , it is composed of a quark pair. However, a series of charmonium-like states (including , , , , , , , , , and ) have been observed in experiments [1–5], many of which do not fit into the meson spectrum in the classical CQM. In particular, some charged charmonium-like states are considered to have a minimal quark content of () or (), which have attracted great interest in experimental and theoretical aspect [1–11]. These charmonium-like states are interpreted as a tetraquark, a hadron molecule or just a cusp effect, and so on [1, 7–11]. In addition, a series of hidden-charm baryons also have been investigated in [12–14], which were considered as the pentaquark candidates and composed of . On the one hand, these studies enriched the picture of exotic states. On the other hand, it should be noted that our understanding of these exotic states is far from being sufficient. Thus, more theoretical and experimental works are needed.

Recently, a new charged charmonium-like decaying to has been observed by the Belle Collaboration with a significance of [5]. The relevant experiment results show that the spin-parity quantum number of favors . Furthermore, its mass and width are measured to be [5] It is noticed that the has the largest width among those charmonium-like states which have been observed in experiment. In [7], by analyzing all the available experimental information about charmonium-like states within the framework of the color-magnetic interaction, the was supposed to be a very promising candidate of the lowest axial-vector hidden-charm tetraquark state and its dominant decay should be . In [8–10], the relevant results also support the tetraquark interpretation of in the frame of QCD sum rule. Moreover, with QCD sum rule approach, the was described as the molecule-like state in [11]. These experimental and theoretical results indicate that the is an ideal candidate for investigating and understanding the nature of exotic charmonium-like states.

As of now, the charmonium-like states are only observed in four ways including the fusion process, annihilation, meson decay, and hidden-charm dipion decays of higher charmonia or charmonium-like states. Obviously, it is an important topic to study the production of the charmonium-like state in different processes. As mentioned above, was only observed in the meson decay. It is natural to ask whether can be found in other processes.

In addition, we note that searching for the neutral partner of the charmonium-like or bottomonium-like state also aroused the great interest in both experiment and theory. The first observation of neutral has been reported by CLEO-c [15]. Later, the neutral bottomonium-like and charmonium-like have been observed by Belle and BESIII [16, 17], respectively. In theory, the production of the neutral in reaction was investigated in our previous work [18]. These studies not only help in confirming these charmonium-like states, but also open a window to investigate the nature and production mechanism of exotic state beyond the conventional states.

Since the was observed in channel [5] and its dominant decay mode is very likely to be [7], the neutral should has a coupling with . Besides, the tetraquark or molecule-like state can be regard as a general four-quark state [10], which means that the neutral probably is composed of or . According to the OZI rule [19–21], one can speculate that the partial decay width of may be larger than that of . Therefore, the reaction is probably an ideal channel for searching and studying the neutral .

In this work, with an effective Lagrangian approach, the production of neutral in reaction is investigated for the first time. Furthermore, in light of the situations of PANDA detector at FAIR@GSI [22–24], the feasibility of searching the neutral by annihilation is discussed, which can provide valuable information to future experimental exploration of neutral .

This paper is organized as follows. After an introduction, we present the investigated method and formalism for production. In Section 3, the background contributions to the final states are discussed. The numerical results are given in Section 4. Finally, This paper ends with the discussion and conclusion.

2. The Yield by the Antiproton-Proton Scattering

It will be difficult to study at quark-gluon level in the now energy range. Therefore, the effective Lagrangian method in terms of hadrons will be used in our research.

2.1. Feynman Diagrams and Effective Interaction Lagrangian Densities

The basic tree level Feynman diagram for the production of in reaction through -channel is depicted in Figure 1. For the , its quantum number of spin-parities has been determined by Belle Collaboration to be [5]. Therefore, the relevant effective Lagrangian for the vertices of and (for simplicity, we use and to denote and , resp.) read as [25]:where , , and denote the fields of , , and nucleon, respectively. The and are the coupling constants. Considering the size of the hadrons, we introduce the general form factor for the intermediate as used in [26–28]:where , , and are the 4-momentum, mass, and cut-off parameters for the intermediate , respectively.

2.2. Coupling Constants and the OZI Analysis in the Process of

With the effective Lagrangians above, the coupling constants and can be determined by the partial decay widths and , respectively:withwhere is the Källen function with .

As of now, no relevant experiment data about and are available [29]. However, in [10, 11] the decay widths MeV or MeV were obtained with QCD sum rule by assuming that the is a tetraquark state or molecule-like state, respectively. Thus we get the coupling constants , , which correspond to the partial decay width , MeV.

For the partial decay width of , we try to obtain it by analyzing and comparing with the OZI suppressed process of . According to constituent quark model, the traditional charmonium is regarded as a pure state, while there are only up and down quarks (antiquarks) in the proton (antiproton). Thus the decay is actually a disconnected process and at least needs three gluons to connect it (as shown in Figure 2(a)). In the frame of the Okubo-Zweig-Iizuka (OZI) rule [19–21], the incidence of process is greatly suppressed. With the total decay width and branch ratios of listed in PDG [29], one get the partial decay width keV, which is indeed a small value and consistent with the prediction by OZI rule [19–21]. Since the neutral may have minimal quark content of () or (), as seen in Figure 2(b), the reaction is a connected whole and it is an OZI allowed process [19–21]. Therefore, in principle, the probability of decay to should be higher than , which may be part of reason that the total decay width of is 3 orders larger than the total width of . Accordingly we speculate that the partial width may well be at least three magnitudes larger than . Then we get by taking keV.

(a)

(b)

2.3. Amplitude

Following the Feynman rules and using the above Lagrangian densities, we can obtain the invariant amplitude for the reaction through -channel as shown in Figure 1:where are the propagators of the , taking the Breit-Wigner form [30]:Hereis the projection operator for the state with spin-1.

3. The Background Analysis and Cross Section

Figure 3 shows the process through -channel (a) and -channel (b) by exchanging a proton, which can be regarded as the main background contributions for the production of as described in Figure 1.

(a)

(b)

The Lagrangian densities for the vertices of and read as [31]where and denote the fields of and nucleon, respectively, while is Pauli matrix.

The coupling constant is adopted [32], while coupling constant is determined by partial decay widths:withThus we get , which is calculated by the measured branching fractions and total widths of (MeV and keV) [29].

The monopole form factors for the and vertices are introduced as the same as Bonn potential model [33]:where , , and ( and ) are the cut-off parameter, mass, and four momentums of the exchanged proton, respectively.

According to the Feynman rules and the above equations, the full invariant amplitude for the background as depicted in Figure 3 can be obtained:With the amplitudes listed in (6) and (13), we get the square of the total invariant amplitude (in principle, the interference between amplitudes for the signal and the nonresonant background should be considered; since we do not now have experimental data, we take the relative phase between different amplitudes as zero in the present work; thus the total cross section for the process obtained by us is an upper estimate):We define , and then the unpolarized differential cross section for the reaction at the center of mass (c.m.) frame is as follows:where and are the three momentums of initial antiproton and final , while denotes the angle of the outgoing meson relative to the antiproton beam direction in the c.m. frame. The total cross section can be easily obtained by integrating the above equation.

4. Numerical Results and Discussion

With the formalisms and equations determined above, we calculate the total and differential cross section including both signal and background contributions as presented in Figures 4–7.

(a)

(b)

(c)

(d)

In these calculations, we note that the cut-off parameter related to the form factor is the only free parameter. Therefore, first we need to discuss the effect of cut-off parameter on cross section of signal and background.

We present the variation of the cross section from the -channel signal contribution for reaction with different cut-off parameters as shown in Figure 4, where is taken as 1.0–3.0 GeV with the step of 1.0 GeV.

One notices that there is an obvious peak structure at center-mass energy GeV which is near the threshold of . Moreover, the cross section of signal increases with the increasing of cut-off parameter , but at a modest rate. In particular, in the range of , it is found that the cross sections from signal contributions are not sensitive to the cut-off parameter . We take typical value GeV in the following, which can ensure the cross sections of signal are limited to a smaller value.

In Figure 5, we illustrate the proton exchange contributions with different cut-off parameters , which is obvious that the cross sections from background contributions are sensitive to the values of the cut-off . Fortunately, the reaction has been measured by the E760 and E835 experiment at low energy [34–36], which can help us to constrain the cut-off parameter . From Figure 5 it can be found that the numerical results from the proton exchange contributions are consistent with the E760 and E835 data by taking and GeV, respectively. Based on the consideration of seeking a larger limit for the cross section of background, we take GeV in the next calculation. Besides, we notice that the amplitude estimate of is about 0.3 nb at GeV in [37]. This value is closer to the E835 data [35, 36] if we consider its uncertainty. Thus we taking GeV in our calculation should be reasonable.

Figure 6 shows the total cross sections for reaction including both signal and background contributions by taking GeV and GeV. We notice that the cross section of production goes up very rapidly and has a peak around GeV. Besides, it is found that the contributions from the signal are dominant in the region of GeV GeV. Naturally, we can conclude that GeV GeV is the best energy window for searching the neutral charmonium-like in experiment, in which the signal can be clearly distinguished from background. Around the center of mass GeV, the total cross section from signal and background contributions is on the order of 0.14 μb and 0.04 μb, which correspond to the decay width MeV and MeV, respectively.

As mentioned above, the PANDA detector at FAIR [22–25] is an ideal platform searching for the by collision. With a beam of 15 GeV [22–25] one has GeV, which allows one to observe charmonium-like state in production up to a mass GeV. Assuming the integrated luminosity of PANDA can reach up to 1.5 fb⁻¹ per year [22–25], taking , one can expect about events per year for the production of at GeV, which are enough to meet the requirement of the experiment. At present, except for the , other charmonium-like states (such as , , and ) were also observed in the invariant mass. Since all of them have probably the identical quantum numbers , the interference of each other is possible. The above situation indicates that the contributions from other may be significant for the process. However, in this work, we focus only on the state and do not consider the contributions from other states.

Figure 7 shows the differential cross section including both signal and background contributions at the center of mass energy , 4.2, 4.5, and 5 GeV. We notice that the line shape of total differential cross section is less affected by background and almost coincident with the line shape of signal differential cross section at GeV, which are consistent with the calculations as presented in Figure 6. In comparison, it is found that the shapes of total angular distributions are different from the shapes of signal angular distributions at GeV, which due to the background has a strong effect on the total cross section at GeV. These predictions can be checked by the future experiment.

5. Discussion and Conclusion

In this work, we investigate the neutral production in reaction with an effective Lagrangian approach. Our numerical result indicates that the is very likely an ideal channel to study and search for the neutral hidden-charm . Furthermore, it is found that the center of mass energy GeV is the best energy window for searching the neutral , in which the signal can be easily distinguished from background. Moreover, according to our estimation, enough events near GeV can be produced at PANDA, which indicate that searching for the neutral by annihilation at PANDA is feasible. Besides, since our calculations are carried out in the premise of assuming that the has a coupling with and , the near future experiments at LHC and BelleII will be able to check our predictions on the respective coupling strengths of the .

It should be mentioned that the value of coupling constant is determined by analyzing and comparing the degree of OZI suppressed in the process of and , which is based on the assumption of the neutral which may be composed of or . According our estimate, even if taking keV (this value is one order smaller than that which we used in the above calculations) and MeV, the cross section from signal contributions by annihilation at GeV could still reach up to the level of 4 nb which is at least one order higher than that of background. This means that an obvious bump at GeV can be expected to appear in the process. Thus, we strongly suggest that the relevant experiment can be carried out, which will not only be conducive to verify the existence of but also enable us to have a more comprehensive understanding of the nature of exotic states and OZI rule.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to acknowledge Ju-Jun Xie for useful comments. Xiao-Yun Wang is grateful to Dr. Qing-yong Lin for valuable discussions and help. This project is partially supported by the National Basic Research Program (973 Program Grant no. 2014CB845406), the National Natural Science Foundation of China (Grant no. 11175220), and the One Hundred Person Project of Chinese Academy of Sciences (Grant no. Y101020BR0).

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Copyright

Copyright © 2015 Xiao-Yun Wang and Xu-Rong Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

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