Discovery potential for the neutral charmoniumlike $Z^{0}(4200)$ by $\bar{p}p$ annihilation

Inspired by the observation of charmoniumlike $Z(4200)$, we explore the discovery potential of the neutral $Z^{0}(4200)$ production by antiproton-proton annihilation with an effective Lagrangian approach. By investigating the $\bar{p}p\rightarrow J/\psi \pi ^{0}$ process including the $Z^{0}(4200)$ signal and background contributions, it is found that the center of mass energy $E_{c.m.}\simeq 4.0-4.5$ GeV is the best energy window for searching the neutral $Z^{0}(4200)$, where the signal can be clearly distinguished from background. The relevant calculations is not only helpful to search for the neutral $Z^{0}(4200)$ in the future experiment, but also will promote the understanding the nature and production mechanism of neutral $Z^{0}(4200)$ better.

It is notice that the Z + (4200) has the largest width among those charmonium-like Z states which have been * Corresponding author: xywang@impcas.ac.cn observed in experiment. In Ref. [7], by analyzing all the available experimental information about charmoniumlike states within the framework of the color-magnetic interaction, the Z(4200) was supposed to be a very promising candidate of the lowest axial-vector hidden-charm tetraquark state and its dominant decay should be J/ψπ. In Ref. [8][9][10], the relevant results also support the tetraquark interpretation of Z(4200) in the frame of QCD sum rule. Moreover, with QCD sum rule approach, the Z(4200) was described as the molecule-like state in Ref. [11] These experimental and theoretical results indicate that the Z(4200) 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 γγ fusion process, e + e − annihilation, B meson decay and hidden-charm dipion decays of higher charmonia or charmonium-like states. Obviously, it is an important topic to study the the production of the charmonium-like state in different processes. As mentioned above, Z(4200) was only observed in the B meson decay. It is natural to ask whether Z(4200) can be found in other processes.
In addition, we note that searching for the neutral partner of the charmonium-like or bottomoniumlike state also aroused the great interest both in experiment and theory. The first observation of neutral Z c (3900) has been reported by CLEO-c [15]. Later, the neutral bottomoniumlike Z 0 b (10610) and charmonium-like Z 0 c (4020) have been observed by Belle and BESIII [16,17], respectively. In theory, the production of the neutral Z 0 (4430) inpp → ψ ′ (2s)π 0 reaction was investigated in our previous work [18]. These studies not only help in confirming these charmonium-like state, but also opens a window to investigate the nature and production mechanism of exotic state beyond the conventional qq states.
Since the Z(4200) was observed in J/ψπ channel [5] and its dominant decay mode is very likely to be J/ψπ [7], the neutral Z 0 (4200) should has a coupling with J/ψπ 0 . Besides, the tetraquark or molecule-like state can be re-gard as a general four-quark state [10], which means that the neutral Z 0 (4200) probably is composed by |ccuū or ccdd . According to the OZI rule [19], one can speculate that the partial decay width of Z 0 (4200) →pp may be larger than that of J/ψ →pp. Therefore, thē pp → J/ψπ 0 reaction is probably an ideal channel for searching and studying the neutral Z 0 (4200).
In this work, with an effective Lagrangian approach, the production of neutral Z 0 (4200) inpp → J/ψπ 0 reaction are investigated for the first time. Furthermore, in light of the situations of PANDA detector at FAIR@GSI [20][21][22], the feasibility of searching the neutral Z 0 (4200) bypp annihilation is discussed, which can provide valuable information to future experimental exploration of neutral Z 0 (4200). This paper is organized as follows. After an introduction, we present the investigate method and formalism for Z 0 (4200) production. In Sec. III, the background contributions to the J/ψπ 0 final states are discussed. The numerical result are given in Sec. IV. Finally, This paper ends with the discussion and conclusion.
II. THE Z 0 (4200) YIELD BY THE ANTIPROTON-PROTON SCATTERING It will be difficult to study Z 0 (4200) at quark-gluon level in the now energy range. Therefore, the effective Lagrangian method in terms of hadrons will be used in our research.

A. Feynman diagrams and effective interaction Lagrangian densities
The basic tree level Feynman diagram for the production of Z 0 (4200) inpp → J/ψπ 0 reaction through s-channel is depicted in Fig. 1. For the Z(4200), its quantum number of spin-parity has been determined by Belle Collaboration to be J P = 1 + [5]. Therefore, the relevant effective Lagrangian for the vertices of Zpp and  Zψπ 1 read as [23], where Z, ψ and φ denote the fields of Z(4200), J/ψ and nucleon, respectively. The g Zpp and g Zψπ are the coupling constants. Considering the size of the hadrons, we introduce the general form factor for the intermediate Z 0 (4200) as used in Refs. [24][25][26], where q, M Z , and Λ Z are the 4-momentum, mass, and cut-off parameters for the intermediate Z 0 (4200), respectively.
B. Coupling constants and the OZI analysis in the process of Z 0 (4200) →pp With the effective Lagrangians above, the coupling constant g Zpp and g Zψπ can be determined by the partial decay widths Γ Z 0 (4200)→pp and Γ Z 0 (4200)→J/ψπ 0 , respectively, with | p c.m.
For the partial decay width of Z 0 (4200) →pp, we try to obtain it by analyzing and comparing with the OZI suppressed process of J/ψ →pp. According to constituent quark model, the traditional charmonium J/ψ is regarded as a pure cc state, while there are only up and down quarks (antiquarks) in the proton (antiproton). Thus the J/ψ →pp decay is actually a disconnected process and at least need three gluons to connect it (as shown in Fig. 2(a)). In the frame of the Okubo-Zweig-Iizuka (OZI) rule [19], the incidence of J/ψ →pp process is greatly suppressed. With the total decay width and branch ratios of J/ψ listed in PDG [27], one get the partial decay width Γ J/ψ→pp ≃ 0.2 keV, which is indeed a small value and consistent with the prediction by OZI rule [19]. Since the neutral Z 0 (4200) may be have minimal quark content of (ccuu) or (ccdd), as seen in Fig. 2(b), the Z 0 (4200) →pp reaction is a connected whole and it is an OZI allowed process [19]. Therefore, in principle, the probability of Z 0 (4200) decay tō pp should be higher than J/ψ →pp, which may be part of reason that the total decay width of Z(4200) is 3 order larger than the total width of J/ψ. Accordingly we speculate that the partial width Γ Z 0 (4200)→pp may well be at least three magnitude larger than Γ J/ψ→pp . Then we get g Zpp ≃ 0.05 by taking Γ Z 0 (4200)→pp = 200 keV.

C. Amplitude
Following the Feynman rules and using above Lagrangian densities, we can obtain the invariant ampli-p tude M signal Z for thep(p 1 )p(p 2 ) → J/ψ(p 3 )π 0 (p 4 ) reaction through s-channel as shown in Fig. 1, where G µα Z are the propagators of the Z 0 (4200), taking the Breit-Wigner form [28], Here is the projection operator for the state with spin-1. Fig. 3 show thep(p 1 )p(p 2 ) → J/ψ(p 3 )π 0 (p 4 ) process through t-channel (a) and u-channel (b) by exchanging a proton, which can be regarded as the main background contributions for the production of Z 0 (4200) as described in Fig. 1.

CROSS SECTION
The Lagrangian densities for the vertices ofppπ 0 and J/ψpp read as [29] where ψ and φ denote the fields of J/ψ and nucleon, respectively, while τ is Pauli matrix. The coupling constant gp pπ = 13.5 is adopted [30], while coupling constant gp pψ is determined by partial decay widths: with Thus we get gp pψ ≃ 1.6 × 10 −3 , which is calculated by the measured branching fractions and total widths of J/ψ (m ψ = 3096.916 MeV and Γ ψ = 92.9 keV) [27]. The monopole form factors for theppπ 0 and J/ψpp vertices are introduced as the same as Bonn potential model [31]: where Λ N , m p and q i (q t = (p 3 − p 1 ) and q u = (p 4 − p 1 )) are the cut-off parameter, mass and four-momentum of the exchanged proton, respectively. According to the Feynman rules and above equations, the full invariant amplitude M N = M t N + M u N for the background as depicted in Fig. 3 can be obtained, With the amplitudes listed in Eqs. (9) and (17), we get the square of the total invariant amplitude 2 We define s = q 2 = (p 1 + p 2 ) 2 , then the unpolarized differential cross section for the reactionp(p 1 )p(p 2 ) → J/ψ(p 3 )π 0 (p 4 ) at the center of mass (c.m.) frame is as follows: where p c.m. 1 and p c.m.

3
are the three-momentum of initial anti-proton and final J/ψ, while θ denotes the angle of the outgoing J/ψ meson relative to the anti-proton beam direction in the c.m. frame. The total cross section can be easily obtained by integrating the above equation.
2 In principle, the interference between amplitudes for the signal and the non-resonant 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 thepp → J/ψπ 0 process obtained by us is an upper estimate.

IV. 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  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 s-channel signal contribution forpp → J/ψπ 0 reaction with different cut-off parameters Λ Z as shown in Fig. 4, where Λ Z is taken as 1.0-3.0 GeV with the step of 1. GeV. One notice that there is a obvious peak structure at center-mass energy E c.m. ≃ 4.2 GeV which is near the threshold of Z(4200). Moreover, the cross section of signal increases with the increasing of cut-off parameter Λ Z , but at a modest rate. Especially in the range of 4.0 GeV E c.m. 4.5 GeV , it is found that the cross sections from signal contributions are not sensitive to the cut-off parameter Λ Z . We take typical value Λ Z = 1.0 GeV in the following, which can ensure the cross section of signal are limited to a smaller value.
In Fig. 5, we illustrate the proton exchange contributions with different cut-off parameters Λ N , which is obvious that the cross sections from background contributions are sensitive to the values of the cutoff Λ N . Fortunately, the reactionpp → J/ψπ 0 has been measured by the E760 and E835 experiment at low energy [32,33], which can help us to constrain the cut-off parameter Λ N . From Fig. 5 it can be found that the numerical results from the proton exchange contributions are consistent with the E760 and E835 datas by taking Λ N = 1.9 and 3.0 GeV, respectively. Based on the consideration of seeking a larger limit for the cross section of background, we take Λ N = 3.0 GeV in the next calculation. Besides, we notice that the amplitude estimate ofpp → J/ψπ 0 is about 0.3 nb at E cm = 3.5 − 3.6 GeV in ref. [34]. This value is closer to the E835 data [33] if we consider its uncertainty. Thus we taking Λ N = 3.0 GeV in our calculation should be reasonable. Fig. 6 show the total cross sections forpp → J/ψπ 0 reaction including both signal and background contributions by taking Λ Z = 1.0 GeV and Λ N = 3.0 GeV. We notice that the cross section of Z 0 (4200) production goes up very rapidly and has a peak around E cm ≃ 4.2 GeV. Besides, it is found that the contributions from the sig-nal are dominant in the region of 4.0 GeV E c.m. 4.5 GeV. Naturally, we can conclude that 4.0 GeV E cm 4.5 GeV is the best energy window for searching the neutral charmoniumlike Z 0 (4200) in experiment, which the signal can be clearly distinguished from background. Around the center of mass E cm ≃ 4.2 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 Γ Z 0 (44200)→J/ψπ 0 = 87.3 MeV and Γ Z 0 (44200)→J/ψπ 0 = 24.6 MeV, respectively.
As mentioned above, the PANDA detector at FAIR [20][21][22][23] is an ideal platform searching for the Z 0 (4200) bypp collision. With ap beam of 15 GeV/c [20][21][22][23] one has E c.m. = 5.47 GeV, which allows one to observe charmoniumlike Z 0 (4200) state in J/ψπ 0 production up to a mass M Z ≃ 4.2 GeV. Assuming the integrated luminosity of PANDA can reach up to 1.5 fb −1 per year [20][21][22][23], taking σ total ≈ 0.04 − 0.14 µb, one can expect about 6 × 10 7 − 2.1 × 10 8 events per year for the production of J/ψπ 0 at E c.m. ≃ 4.2 GeV, which are enough to meet the requirement of the experiment. At present, except for the Z(4200), other charmonium-like Z states (such as Z(3900), Z(4025) and Z(4430) etc.) were also observed in the J/ψπ invariant mass. Since all of them have probably the identical quantum numbers J P = 1 + , the interference of each other is possible. The above situation indicate that the contributions from other Z 0 maybe be significant for thepp → J/ψπ 0 process. However, in this work, we focus only on the Z 0 (4200) state and do not consider the contributions from other Z 0 states. Fig. 7 show the differential cross section including both signal and background contributions at the center of mass energy E c.m. = 4.0, 4.2, 4.5 and 5 GeV. We notice that the line shape of total differential cross section are less affect by background and almost coincident with the line shape of signal differential cross section at E c.m. = 4.0 − 4.5 GeV, which are consistent with the calculations as presented in Fig. 6. In comparison, it is found that the shapes of total angular distributions is different from the shapes of signal angular distributions at E c.m. = 5 GeV, which due to the background has a strong effect on the total cross section at E c.m. = 5 GeV. These predictions can be checked by the future experiment.

V. DISCUSSION AND CONCLUSION
In this work, we investigate the neutral Z 0 (4200) production inpp → J/ψπ 0 reaction with an effective Lagrangian approach. Our numerical result indicate that thepp → J/ψπ 0 is very likely an ideal channel to study and search for the neutral hidden charm Z 0 (4200). Furthermore, it is found that the center of mass energy E c.m. ≃ 4.0 − 4.5 GeV is the best energy window for searching the neutral Z 0 (4200), which the signal can be easily distinguished from background. Moreover, According to our estimation, enough Z 0 (4200) events near E c.m. ≃ 4.2 GeV can be produced at PANDA, which indicate that searching for the neutral Z 0 (4200) bypp annihilation at PANDA is feasible. Besides, since our calculations are carried out in the premise of assuming that the Z 0 (4200) has a coupling withpp and J/ψπ 0 , the near future experiments at LHC and BelleII will be able to check our predictions on the respective coupling strengths of the Z 0 (4200). It should be mentioned that the value of coupling constant g Zpp is determined by analyzing and comparing the degree of OZI suppressed in the process of Z 0 (4200) →pp and J/ψ →pp, which is based on the assumption of the neutral Z 0 (4200) may be composed by |ccuū or ccdd . According our estimate, even if taking Γ Z 0 (4200)→pp = 20 keV (this value is one order smaller than that we used in the above calculations ) and Γ Z 0 (44200)→J/ψπ 0 = 24.6 MeV, the cross section from signal Z 0 (4200) contributions bypp annihilation at E c.m. ≃ 4.2 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 E c.m. ≃ 4.2 GeV can be expected to be ap-pear in thepp → J/ψπ 0 process. Thus, we strong suggest the relevant experiment can be carried out, which will not only be conducive to verify the existence of Z(4200) but enable us to have a more comprehensive understanding of the nature of exotic states and OZI rule.

VI. ACKNOWLEDGMENTS
The authors would like to acknowledge Ju-Jun Xie for useful comments.
The author X. Y. Wang is grateful Dr.Qing-yong Lin for valuable discussions and help. This project is partially supported by the Na-