Some remarks on X(6900)

The analysis of the LHCb data on found in the di- system was performed using a momentum-dependent Flatté-like parameterization. The use of the pole counting rule and spectral density function sum rule provides consistent evidence that both confining and molecular states are possible. Alternatively, the nature of cannot be distinguished if only the di- experimental data with current statistics are available. Nevertheless, we found that the lowest state in the di- system likely has the same quantum numbers as , and is probably not interpreted as a molecular state.


I. INTRODUCTION
Recently, LHCb Collaboration observed a structure around 6900 MeV/ , named as , in the diinvariant mass spectrum [1], with a signal statistical significance above . It is probably composed of four (anti)charm quarks ( ) and its widths [1] are determined to be and MeV in two fitting scenarios of Breit-Wigner parameterizations with constant widths. Additionally, a broad bump and a narrow bump exist in the low and high sides of the di-mass [1], respectively, where the former might be a result from a lower broad resonant state (or several lower states) or an interference effect, and the latter was found to be a hint of a state located at 7200 MeV, called .
This intriguing observation has aroused widespread concern in the physics community. In accordance with QCD sum rule, Ref. [2] pointed out that the lowest broad structure between 6200 and 6800 MeV can be regarded as an -wave tetraquark state with quantum numbers or , while can be considered as a -wave tetraquark with or . In the framework of a non-relativistic potential quark model (NRPQM) for a heavy quark system, Ref. [3] deemed that the lowest one can be interpreted as an -wave state at [cc] [cc] X(6900) X(6900) J/ψJ/ψ J/ψψ (2S ) approximately 6500 MeV, whereas can be interpreted as a -wave state. Moreover, in NRPQM, Ref. [4] takes as a candidate for the first radially excited tetraquarks with or , or the or -wave state, and considered that there exist two states below with exotic quantum numbers, and , and may decay into the -wave and dimodes, respectively. Ref. [5] indicated, in an extended relativistic quark model, that the lowest broad structure should contain one or more ground tetraquark states, while the narrow structure near 6900 MeV can be categorized as the first radial excitation of a system. Exploiting three potential models (a color-magnetic interaction model, a traditional constituent quark model, and a multiquark color flux-tube model), Ref. [6] systematically investigated the properties of the states : a broad structure ranging from 6200 to 6800 MeV can be described as the ground tetraquark state in the three models, while the narrow exhibits different properties in different potential models and more data associated with determination of quantum numbers are needed to shed light on the nature of these states. Ref. [7] argued that the structure can be well described within two variants of a unitary couplechannel approach: (i) with two channels, namely and , with energy-dependent interactions, or (ii) with three channels, namely , , and , with only constant contact interactions. They also predicted the existence of a near-threshold state [7] in the system with quantum numbers or . Similarly, in coupled-channel analyses, Ref. [8] identified as , and provided hints of the existence of other states: a , a , and a . Ref. [9] predicted a narrow resonance of molecular origin located below the threshold. Employing a contact-interaction effective field theory with heavy anti-quark di-quark symmetry, Ref. [10] implied that can be regarded as the fully heavy quark partner of . Ref. [11] showed that the structure , as a dynamically generated resonance pole, can arise from Pomeron exchanges and coupled-channel effects between , scatterings. Based on the perturbative QCD method, Ref. [12] found that there should exist another state near the resonance at approximately 6.9 GeV, and the ratio of production cross sections of to the undiscovered state is very sensitive to the nature of . Besides discussing the nature of , Ref. [13] studied the production of in reaction by using an effective Lagrangian approach and Breit-Wigner formula and predicted that it is feasible to find in the collision in D0 and forthcoming PADNA experiments.
Generally, a molecular state may locate near the threshold of two (or more) color singlet hadrons, such as deuteron, [14], [15][16][17], and [18,19]. We found that the state is close to the threshold of , , , and ; and the state is close to the threshold of and . In this study, inspired by this and on the assumption that couples to , , , , and processes (see Table 1), and couples to , , and (see Table 2; for details on the parameters of charmonia used in this analysis, see Table 3), a Flatté-like parameterization with momentum-dependent partial widths for the two resonances was used to fit the experimental data, and then the pole positions of the scattering amplitude in the complex plane were searched. For -wave coupling, the pole counting rule (PCR) [22], which has been applied to the studies of " " physics in Refs. [23][24][25][26], and the spectral density function sum rule (SDFSR) [25,[27][28][29][30] were employed to analyze the nature of both structures, i.e., whether they are more inclined to be confining states bound by color force, or loosely-bounded hadronic molecular states. We also examined the coupling with a threshold below 's mass of MeV. This threshold is far away from the mass of , and thus it does not resemble a molecular state; however, the process is easily accessible in experiments.

States
and are parameterized using a momentum-dependent Flatté-like formula. The non-resonance background shape is parametrized by the twobody phase space of times an exponential function. To better meet the di-spectrum, a Flattélike function that only considers the channel for the structure below 6800 MeV was employed in the fitting process. Given that we did not identify the lowest state that contributes to the peak at approximately 6500 MeV or that corresponds to the dip (caused by destructive interference) below 6800 MeV, we did not analyze the nature of the lowest state (named as hereafter). The exclusion of from the fitting process led to divergence. It means that a state with the same quantum numbers as is essential to describe the extremely deep dip below 6800 MeV by destructive interference. As mentioned above, the components of the fit can be written as is the line-shape mass (width) for , is the mass [20], ( ) corresponds to the line-shape mass of and , respectively; corresponds to the partial width of the -th couple channel on the -th pole; and are interference phases; , , , and are free constants; combines the threshold and barrier factors; and represents the channel (throughout the analysis). Note that and can be expressed [20] as where, is the orbital angular momentum in channel , is the center-of-mass momentum of one daughter particle of channel for two body decays 1) , denotes a momentum scale, is a coupling constant, and 1) For tow-body final states and , ; for , is done using analytic continuation . 103102-2 is the phase space factor. The factor guarantees the correct threshold behavior. The rapid growth of this factor for angular momenta is commonly compensated at higher energies by the phenomenological form factor . The Blatt-Weisskopf form factors are usually employed, e.g. [31][32][33], , , with . Refs. [34,35] set and found that , varying between 0.1 GeV -1 and 10 GeV -1 , is a phenomenological factor (generally representing the "radius" of a particle [34]) with little sensitivity to the partial width. With and being positive real values, it is easy to find that . Therefore, varies between 0.1 GeV and 10 GeV, and was set as 2 GeV in this analysis.
Owing to limited data statistics, only two-channel couplings were investigated, that is, A.
couplings, and C. and couplings, where the former denotes the angular momentum of the channel and the latter other channels listed in Tables 1-2. The corresponding pole positions of and were determined.
Constrained by the generalized bose symmetry for identical particles and conservation, the quantum numbers of the -wave pair must be or . Based on the or assumption for and , the other -wave couple channels near the mass of the two states were considered, as summarized in Table 1. They could be divided into three cases for decays: Case I: and , Case II: and , Case III: and .

For
, and the near-threshold channels were used in the couple channel analysis. The state has the same quantum numbers as (similarly hereafter), as previously mentioned. Thus, the total amplitude satisfies describes the coherent background (BG), and the incoherent background takes a parameterization similar to . This background parameterization is similar to the LHCb experiment [1]. Interestingly, two sets of solutions with almost equivalent goodness of fit were found in the three cases, one of which favors that both states are confining states whereas the other supports that they are molecular bound states, using both PCR and SDFSR mentioned above. The fitting results are summarized in Table 4. Given that the fitting curves of the three cases look very similar, we only drew the fitting projections of two solutions in case I, shown in Fig. 1. Each set of the parameters can be used to determine whether the resonance structure studied in this paper is a confining state or a molecular state. The definition of Riemann sheets for two channels is listed in Table 5. The pole positions in the plane obtained by using parameters in Table 4 for all cases are summarized in Table 6. For Solution I, the fact that pole positions of on the second and third sheets are equal in Case I and Case II indicates the state hardly couples to the channel, whereas its pole positions in Case III manifests that it tends to be a confining state. Furthermore, it is evident for this solution in each case that the co-existence of two poles near the threshold in- dicates that it might be a confining state for -wave couplings. For Solution II in each case, the fact that only one pole is found on sheet II near the second threshold demonstrates that the two states tend to be molecular states. Thus, in case of assuming and to be and considering the couple channels listed in Table 1, different conclusions concerning the goodness of fit being almost equivalent are drawn. This is mainly caused by low statistics and unavailable information on other channels. Consequently, it is impossible to distinguish whether the two states are confining or molecular states under the current situation. More experimental measurements in the coupling channels, , , , , and , , are therefore in urgent need to clarify their nature.
At last, we also tested the situation in which couples to and . A solution that favors as a confining state was found. Meanwhile, we could not find a good solution in favor of a molecular state interpretation of .  Table 2. From this table, the couplings can be divided into two cases: Table 3. Parameters for the involved charmonium states [20].      Fig. 2, and the corresponding numerical results are listed in Table 7. Note that the parameterization with thewave coupling assumption can also meet the experimental data well, with almost equivalent goodness of fit in the -wave couplings. The pole positions in the complex plane are listed in Table 8. Note that the method adopted in this study cannot distinguish a -wave confining state from a -wave molecule, given that they both contribute two pair of poles near the threshold. -wave for and .
The following total amplitude is applicable for both cases, n i j (s)

X(7200) s X(6900)
By employing Eq. (4) with the respective threshold and barrier factors, included, the fitting projections are similar to Fig. 1. Two sets of solutions with almost equivalent goodness of fit were found in Case I. In comparison, only one solution in favor of a molecular interpretation of was found in Case II. The pole positions in the complex plane are summarized in Table 9. Based on PCR, it may be concluded that Solution I favors as a confining state and Solution II supports that it is a molecular bound state. Concerning -waves, SDFSR can be utilized to provide insights into the nature of the state. Ref. [27] pointed out that the spectrum density function near threshold can be calculated by using the non-relativistic -wave Flatté parameterization, and the renormalization constant can be obtained, which represents the probability of finding the confining particle in the con-Z 1 Z tinuous spectrum: the greater the tendency of to , the more confining is the state. On the other hand, if tends to 0, the state tends to be molecular.
Using a form similar to that in Refs. [25,[27][28][29][30], the spectrum density function of a near-threshold channel can be expressed as where is the energy difference between the center-of-mass energy (resonant state) and the open-channel threshold, is the reduced mass of the two-body final states of the channel, is the step function, is the dimensionless coupling constant of the concerned coupling mode, and is the constant partial width for the remaining couplings, which mainly contains the distant channels (the process in this analysis).
By integrating Eq. (5), the probability of finding an "elementary" particle in the continuous spectrum can be obtained: The integral interval takes as the central value. It is pointed out in Ref. [30] that the integration interval needs to cover the threshold of the coupling channel. Given that the state is of no significance ( reported in the LHCb experiment [1]), only the values of , as listed in Table 1, are calculated. Expanding near the threshold of each channel, and bringing 's mass and width extracted from the second Riemann sheet in Table 6, one can obtain the corresponding value, where (see also Eq. (2)). The numerical values are summarized in Table 10, where the interval covers all thresholds of the calculated channels. The values in Solution I are all slightly less than 50% in this interval, but rapidly ex- ceeds 50% in larger integral intervals. Hence may be considered as a confining state in Solution I. For Solution II, the values are much smaller than 50% in the interval , and also less than 50% in the interval . This suggests that the state is more likely a molecular state in Solution II. As a conclusion, the nature of is consistently drawn from both PCR and SDFSR based on the current limited data. Hence, we were not able to distinguish whether it is a confining state or a molecular bound state.

IV. CONCLUSION
J/ψψ(3770) J/ψψ 2 (3823) J/ψψ 3 (3842) In this analysis, the channels , , , and [ and ] close to the threshold of [ ] were selected to study their couplings. Fitting to recent LHCb data through Flatté-like parameterization with momentum-dependent partial widths, we found that the lowest state in the di-mass spectrum with the same quantum numbers as is essential to describe the extremely deep dip below 6800 MeV by destructive interference. The amplitude poles in the complex plane were obtained. For the -wave couplings, PCR and SDFSR were imposed to determine whether the structures are confining states (bound by color force) or molecular states. The two approaches provide consistent conclusions: both confining and molecular states are possible, or the nature of the two states cannot be distinguished if only the diexperimental data with current statistics are available. It is also discussed in Ref. [12] that the current experimental data are not enough to provide a definitive conclusion on the nature of . In addition, the coupling with the threshold far away from 's mass was taken into account. Our results do not favor the structure as a molecular state. In the future, we are hoping to obtaine more experimental data and more decay channels to clarify the nature of and as well as determine their quantum numbers. Reasonably, we suggest that experiments measure , , , and ; and , decays, which are expected to be available in LHCb, Belle-II, CMS and other (future) experiments.

APPENDIX A: VALUES OF PARAMETERS
S − S P − P The remaining parameters which are not listed in Sec. II are presented below. The parameter values for and couplings are displayed in Table A1 and   In addition, the parameters for both coherent and incoherent background terms, which were obtained from fitting to the mass spectrum without the signal amplitude, were fixed throughout default fitting to reduce the uncer- tainty of the multiple interference. Regarding the coherent background, and (MeV −1 ). Concerning the incoherent background, which takes a form similar to the coherent background, it has two parameters: , (MeV −1 ).