Discovery potential for the LHCb fully-charm tetraquark $X(6900)$ state via $\bar{p}p$ annihilation reaction

Inspired by the observation of the fully-charm tetraquark $X(6900)$ state at LHCb, the production of $X(6900)$ in $\bar{p}p\rightarrow J/\psi J/\psi $ reaction is studied within an effective Lagrangian approach and Breit-Wigner formula. The numerical results show that the cross section of $X(6900)$ at the c.m. energy of 6.9 GeV is much larger than that from the background contribution. Moreover, we estimate dozens of signal events can be detected by D0 experiment, which indicates that searching for the $X(6900)$ via antiproton-proton scattering may be a very important and promising way. Therefore, related experiments are suggested to be carried out.


I. INTRODUCTION
In recent decades, more and more exotic hadron states have been observed [1]. These exotic hadron states are not only conducive to the development of hadron spectrum, but also provides an important opportunity for us to better understand the multiquark states and strong interactions [2,3]. Although most of these exotic states that have been observed are concentrated in the charm or bottom quark energy region, the exotic states composed entirely of heavy quarks are still very limited [1][2][3].
Very recently, a narrow X(6900) structure, which may consisting of four charm quarks, was observed in J/ψ pair invariant mass spectrum by LHCb experiment with more than 5σ of significance level [4]. The mass and width of the X(6900) resonance are measured to be either M = 6886 ± 11 ± 11 MeV, (1) Γ = 168 ± 33 ± 69 MeV, (2) based on the simple model with interference. Actually, based on various theoretical models, a lot of research has been conducted on fully-heavy tetraquark states , which are important to reveal the structure and properties of the fully-heavy tetraquark state. For example, the X(6900) was interpreted as a P−wave tetraquark state in a nonrelativistic quark model [27], or the first radial excitation of cccc in an extended relativized quark model [28]. In the QCD sum rule framework, the narrow structure X(6900) can be interpreted as a second radial excited S −wave tetraquark state [29] or a P−wave state * xywang@lut.edu.cn † qylin@jmu.edu.cn ‡ xuh2018@nwpu.edu.cn § xieyaping@impcas.ac.cn ¶ huangy2019@swjtu.edu.cn * * xchen@impcas.ac.cn [30] with J PC = 0 −+ , 1 −+ , respectively. In refs. [31,32], the theoretical results indicate that there may exist the resonance states, with masses range between 6.3 GeV to 7.4 GeV, and the quantum numbers J P = 0 + , 1 + and 2 + . In ref. [33], the spin parity of the state formed by the two-vector system was discussed, and a possible method for determining its quantum number was given.
In addition to the study on the structure and properties of X(6900), the research on the production of the X(6900) through different scattering reactions is also very important and will help us to determine its nature as genuine states. For example, in ref. [34], the production of the ground cccc with J PC = 0 ++ , 2 ++ in pp collisions was calculated, and the upper limit of cross section is about 40 fb for the 4 muons channel. In addition to pp collisions, determining the tetraquark state via the annihilation of positive and negative protons is usually an important and effective way [35][36][37][38][39][40][41]. In this work, the discovery potential for the tetraquark state X(6900) viapp annihilation reaction will be investigated. One will estimate the cross-section of X(6900) production viapp → J/ψJ/ψ reaction and analyze the corresponding background contribution. The theoretical results obtained will be an important theoretical basis for thepp annihilation experiment. This paper is organized as follows. After the introduction, one present the formalism for the production of X(6900) in Section II. The numerical results of the X(6900) production follow in Section III. Finally, the paper ends with a brief summary.
For the production of the resonance X(6900) in s-channel ofpp → ψψ reaction, the cross section can be calculated by the standard Breit-Wigner formula [1].
Where W is the c.m. energy, J is the spin of the resonance X(6900), and the S 1 and S 2 are the spin of initial anti-proton and proton, respectively. Moreover, k in is the c.m. momentum in the initial state, W 0 is the c.m. energy at mass of X(6900), and Γ is the full width of X(6900).
The production cross-section of the X(6900) structure relative to that of all J/ψ pair, times the branching fraction Br(X → ψψ), R, is determined as [2.6 ± 0.6 (stat) ± 0.8 (syst)] by LHCb experiment [4]. Therefore, in this work, we roughly take the value of the branching ratio Br(X → ψψ) ≃ 2.
Having fixed the branch ratio Br(X → ψψ), we turn to the value of Br(X →pp), which is indispensable in Eq. 3. However, it has not been well determined, neither experimentally nor theoretically. Meanwhile, we notice in refs. [36,37] that, the branching fraction of Br(Y(4260) →pp) and Br(ψ(4040) →pp) were estimated using the branching ratio of J/ψ state, multiplying by the ratio of the width of the Y(4260) or ψ(4040) to the width of J/ψ. Thus in this work, we will adopt the same method as in refs. [36,37] to naively determine the branching ratio of X decaying intopp. Due to the lack of deep understanding of the full-charm tetraquark X(6900), its inner structure and quantum numbers are still unknown. Different J PC assignments were assumed to evaluated the mass of a full-charm tetraquark, as discussed in Sec. I. To carry out the estimation of the cross section of the s-channel, we assume that the quantum number J PC of X(6900) is 0 ++ or 2 ++ , same as the assumption in Ref. [34]. In the known charmonium family, the spin-parity of χ c0 and χ c2 are 0 ++ and 2 ++ , respectively. Therefore, we plan to simply replace J/ψ with χ c0 and χ c2 to estimate the branching ratio of X →pp, namely, and Br(X →pp) ≃ Br(χ c2 →pp) × Γ χ c2 Γ X , for X with J PC = 2 ++ (5) where Γ χ c0 = 10.8 MeV, Γ χ c2 = 1.97 MeV and Γ X = 168 MeV are the total width of χ c0 , χ c2 and X(6900) state, respectively. By taking the branching ratio Br(χ c0 →pp) = 2.24 × 10 −4 and Br(χ c2 →pp) = 7.33 × 10 −5 , one get Br(X →pp) = 1.44×10 −5 for X with J PC = 0 ++ and Br(X →pp) = 8.6×10 −7 for X with J PC = 2 ++ . Here, it must be noted that if we use the branching ratio of J/ψ for estimation, Br(X →pp) = 1.17 × 10 −6 is obtained, which is just between the value estimated by the branching ratio of χ c0 and χ c2 .
B. The background analysis Fig. 1 (b)-(c) presents thepp → ψψ reaction with t and u channel by exchanging a nucleon, which can be considered as the main background contributions for the production of X(6900). By employing the effective Lagrangian approach, the cross section ofpp → ψψ reaction can be calculated.
The Lagrangian density for the vertice of ψNN is written as [38], where ψ and N denote the fields of J/ψ and nucleon, respectively. The values of coupling constant g ψNN can be derived from the corresponding decay width, Thus we get g ψNN ≃ 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→pp ≃ 0.197 keV) [1]. Based on the Lagrangians above, the scattering amplitude for the reactionspp → ψψ can be constructed as where u is the Dirac spinor of nucleon, and ǫ γ is the polarization vector of photon. The reduced amplitude A µν for the t and u channel background reads For two ψNN vertices, a general form factor F t (q 2 t ) = F u (q 2 u ) =(Λ 2 t/u −m 2 N )/(Λ 2 t/u −q 2 N ) is taken into account [38,40]. In refs. [38,40], it can be found that the cross section of pp → J/ψπ reaction with nucleon exchange were consistent with the E760 and E835 data by taking Λ t/u = 1.9 and 3.0 GeV, respectively. In the spirit of estimating the upper limit of background contribution, in this work, we take Λ t = Λ u = 3 GeV.
With the preparation in the previous section, the cross section of the reactionpp → ψψ can be calculated. The differential cross section in the center of mass (c.m.) frame is written as Here, s = (k 1 + p 1 ) 2 , and θ denotes the angle of the outgoing J/ψ meson relative top beam direction in the c.m. frame. k c.m. 1 and k c.m.
2 are the three-momenta of the initial photon beam and final J/ψ meson, respectively.

III. NUMERICAL RESULTS
After the above preparations, one calculated the total cross section for the reactionpp → ψψ from threshold to 12 GeV of the center of mass energy, as depicted in Fig. 2. As can be seen from the Fig. 2, the cross section from the X(6900) contribution have a distinct peak near the center of mass energy of 6.9 GeV. And when the spin-parity quantum number of X is 0 ++ or 2 ++ , the cross section of X production through the s channel can reach dozens of pb, which is much larger than the cross section from the background contribution. In addition, we also calculated the background cross section without the from factor (abbreviated as FF). Although it is more than an order of magnitude higher than the background cross section with the FF added, it is still much lower than the cross section of the X production. The main reason for the very small background term is that the g ψNN coupling constant is very small.
The above results indicate that the best energy window for searching for the fully-charm tetraquark state X(6900) via thē pp → ψψ process is near the c.m. energy 6.9 GeV, in which the signal can be clearly distinguished from background. The D0 [41] and the forthcoming PANDA [36] experiments are ideal platforms to study new hadronic states viapp reactions, of which the designed c.m. energy is below the double J/ψ threshold for the latter one. By taking the cross section of X(6900) production calculated above, one finds that the number of events of X(6900) can reach as many as dozens if taking an integrated luminosity of 10.4 fb −1 collected with the D0 detector, when a 50% selection efficiency is adopted. Considering that the background is very clean, it should be possible for X(6900) to be observed by the D0 experiment with a high confidence level. Moreover, in this work the cross section of X(6900) is obtained when the quantum number is assumed to be 0 ++ or 2 ++ . In fact, the quantum number of X(6900) is currently undecided. But according to our estimation, even if X(6900) takes other quantum numbers, its cross section is about the same order as the cross section we have obtained so far. This seems to mean that no matter what the quantum number of X is, it may be an effective way to find or observe X(6900) through thepp annihilation.

IV. SUMMARY AND DISCUSSION
In this work, the X(6900) production in thepp → ψψ reaction is investigated by employing the effective field theory and the Breit-Wigner formula. The numerical results show that the cross section of X production via the s channel can reach dozens of pb at the best energy window W = 6.9 GeV, which the signal can be clearly distinguished from background. In addition, according to our estimation, dozens of X(6900) can be detected with a data sample of 10.4 fb −1 collected with the D0 detector, which indicates that it is feasible to find X(6900) through thepp → ψψ reaction. Hence, an experimental study of fully charm tetraquark X(6900) viapp annihilation is suggested, which will be of great significance to clarify the production mechanism and nature of X(6900).
It should be noted that the pp will be annihilated into gluons first, then those gluons will be converted into the di-J/ψ final state. The production of the full-charm tetraquarks at LHC was discussed and the relative cross section for the process gg → J/ψJ/ψ was estimated [42,43]. The cross sections performed are around 10 pb, which have the same order of magnitude with the results exhibited here.