-
Recently, LHCb Collaboration observed a structure around 6900 MeV/
c2 , named asX(6900) , in the di-J/ψ invariant mass spectrum [1], with a signal statistical significance above5σ . It is probably composed of four (anti)charm quarks (cˉccˉc ) and its widths [1] are determined to be80±19(stat.)±33(sys.) and168±33(stat.)± 69(sys.) 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-J/ψ 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, calledX(7200) .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
S -waveccˉcˉc tetraquark state with quantum numbersJPC=0++ or2++ , whileX(6900) can be considered as aP -waveccˉcˉc tetraquark withJPC=0−+ or1−+ . 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 anS -wave state at approximately 6500 MeV, whereasX(6900) can be interpreted as aP -waveccˉcˉc state. Moreover, in NRPQM, Ref. [4] takesX(6900) as a candidate for the first radially excited tetraquarks withJPC=0++ or2++ , or the1+− or2−+ P -wave state, and considered that there exist two states belowX(6900) with exotic quantum numbers,0^{–-} and1^{-+} , and may decay into theP - wave\eta_c J/\psi and di-J/\psi modes, respectively. Ref. [5] indicated, in an extended relativistic quark model, that the lowest broad structure should contain one or more groundcc\bar{c}\bar{c} tetraquark states, while the narrow structure near 6900 MeV can be categorized as the first radial excitation of acc\bar{c}\bar{c} 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[Q_1Q_2][\bar{Q}_3\bar{Q}_4] (Q = c,b) : a broad structure ranging from 6200 to 6800 MeV can be described as the ground tetraquark state[c\bar{c}][c\bar{c}] in the three models, while the narrowX(6900) 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 theX(6900) structure can be well described within two variants of a unitary couple-channel approach: (i) with two channels, namelyJ/\psi J/\psi andJ/\psi\psi(2S) , with energy-dependent interactions, or (ii) with three channels, namelyJ/\psi J/\psi ,J/\psi\psi(2S) , andJ/\psi\psi(3770) , with only constant contact interactions. They also predicted the existence of a near-threshold stateX(6200) [7] in theJ/\psi J/\psi system with quantum numbersJ^{PC} = 0^{++} or2^{++} . Similarly, in coupled-channel analyses, Ref. [8] identifiedX(6900) as2^{++} , and provided hints of the existence of other states: a0^{++} X(6200) , a2^{++} X(6680) , and a0^{++} X(7200) . Ref. [9] predicted a narrow resonanceX(6825) of molecular origin located below the\chi_{c0}\chi_{c0} threshold. Employing a contact-interaction effective field theory with heavy anti-quark di-quark symmetry, Ref. [10] implied thatX(7200) can be regarded as the fully heavy quark partner ofX(3872) . Ref. [11] showed that the structureX(6900) , as a dynamically generated resonance pole, can arise from Pomeron exchanges and coupled-channel effects betweenJ/\psi J/\psi ,J/\psi\psi(2S) 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 ofX(6900) to the undiscovered state is very sensitive to the nature ofX(6900) . Besides discussing the nature ofX(6900) , Ref. [13] studied the production ofX(6900) inp\bar{p}\to J/\psi J/\psi reaction by using an effective Lagrangian approach and Breit-Wigner formula and predicted that it is feasible to findX(6900) in thep\bar{p} 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,
Z_b(10610) [14],Z_c(3900) [15-17], andP_c(4470) [18,19]. We found that theX(6900) state is close to the threshold ofJ/\psi \psi(3770) ,J/\psi \psi_2(3823) ,J/\psi \psi_3(3842) , and\chi_{c0} \chi_{c1} ; and theX(7200) state is close to the threshold ofJ/\psi \psi(4160) and\chi_{c0} \chi_{c1}(3872) . In this study, inspired by this and on the assumption thatX(6900) couples toJ/\psi J/\psi ,J/\psi\psi(3770) ,J/\psi \psi_2(3823) ,J/\psi \psi_3(3842) , and\chi_{c0}\chi_{c1} processes (see Table 1), andX(7200) couples toJ/\psi J/\psi ,J/\psi\psi(4160) , and\chi_{c0}\chi_{c1}(3872) (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 complexs plane were searched. ForS -waveJ/\psi J/\psi coupling, the pole counting rule (PCR) [22], which has been applied to the studies of "XYZ " physics in Refs. [23-26], and the spectral density function sum rule (SDFSR) [25,27-30] 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 theX(6900)\to J/\psi\psi(2S) coupling with a threshold belowX(6900) 's mass of\sim 100 MeV. This threshold is far away from the mass ofX(6900) , and thus it does not resemble aJ/\psi-\psi(2S) molecular state; however, the process is easily accessible in experiments.J^{PC}\;\text{of di-}J/\psi Couple channels of X(6900) Threshold/MeV Couple channels of X(7200) Threshold/MeV 0^{++} J/\psi-\psi(2S) 6783.0 J/\psi-\psi(4160) 7287.9 J/\psi-\psi(3770) 6870.6 2^{++} J/\psi-\psi(2S) 6783.0 J/\psi-\psi(4160) 7287.9 J/\psi-\psi(3770) 6870.6 J/\psi-\psi_2(3823) 6919.1 J/\psi-\psi_3(3842) 6939.6 Table 1. Involved
S -wave couple channels except for di-J/\psi .J^{PC}\;\text{of di-}J/\psi Couple channels of X(6900) Threshold/MeV Couple channels of X(7200) Threshold/MeV 1^{-+} \chi_{c0}-\chi_{c1} 6925.4 \chi_{c0}-\chi_{c1}(3872) 7286.4 (0,1,2)^{-+} J/\psi-\psi(3770) 6870.6 J/\psi-\psi(4160) 7287.9 Table 2. Involved
P -wave couple channels except for di-J/\psi .J/\psi \chi_{c0} \chi_{c1} \psi(2S) \psi(3770) \psi_2(3823) \psi_3(3842) \chi_{c1}(3872) \psi(4160) J^{PC} 1^{--} 0^{++} 1^{++} 1^{--} 1^{--} 2^{--} 3^{--} 1^{++} 1^{--} mass/MeV 3096.9 3414.7 3510.7 3686.1 3773.7 3822.2 3842.7 3871.7 4191.0 n^{2S+1} L_{J} 1^{3}S_{1} 1^{3}P_{0} 1^{3}P_{1} 2^{3}S_1 1^{3}D_{1} 1^{3}D_2 1^{3}D_3 2^{3}P_{1} [21]2^{3}D_{1} Table 3. Parameters for the involved charmonium states [20].
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States
X(6900) andX(7200) are parameterized using a momentum-dependent Flatté-like formula. The non-resonance background shape is parametrized by the two-body phase space ofR\to J/\psi J/\psi times an exponential function. To better meet the di-J/\psi spectrum, a Flatté-like function that only considers theJ/\psi J/\psi 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 asX(6500) hereafter). The exclusion ofX(6500) from the fitting process led to divergence. It means that a state with the same quantum numbers asX(6900) 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\begin{aligned}[b] {\cal{M}}_{1} = &\frac{g_{1} n_{11}(s) {\rm e}^{{\rm i} \phi_{1}}}{s-M_{1}^{2}+{\rm i} M_{1} \Gamma_{11}(s)}, \\ {\cal{M}}_{i} =& \frac{g_{i}n_{i1}(s) {\rm e}^{{\rm i} \phi_{i}}}{s-M_{i}^{2}+{\rm i} M_{i} \sum\nolimits_{j = 1}^{2} \Gamma_{i j}(s)}, \\ {\cal{M}}_{\rm{NoR}} =& c_0 {\rm e}^{c_1(\sqrt{s}-2 m)} \sqrt{\frac{s-4 m^{2}}{s}}, \end{aligned}
(1) where
M_1\; (\Gamma_1) is the line-shape mass (width) forX(6500) ,m is theJ/\psi mass [20],M_i (i = 2,3 ) corresponds to the line-shape mass ofX(6900) andX(7200) , respectively;\Gamma_{ij} corresponds to the partial width of thej -th couple channel on thei -th pole;\phi_1 and\phi_i are interference phases;g_1 ,g_i ,c_0 , andc_1 are free constants;n_{ij}(s) combines the threshold and barrier factors; andj = 1 represents theJ/\psi J/\psi channel (throughout the analysis). Note thatn_{ij}(s) and\Gamma_{ij} can be expressed [20] asn_{ij}(s) = \left(\frac{p_{ij}}{p_0}\right)^l F_l(p_{ij}/p_0), \quad \Gamma_{ij}(s) = g_{ij}\rho_{ij}(s)n_{ij}^2(s),
(2) where,
l is the orbital angular momentum in channelj ,p_{ij} is the center-of-mass momentum of one daughter particle of channelj for two body decays①,p_0 denotes a momentum scale,g_{ij} is a coupling constant, and\rho_{ij}(s) = 2p_{ij}/\sqrt{s} is the phase space factor. The factorp^l guarantees the correct threshold behavior. The rapid growth of this factor for angular momental>0 is commonly compensated at higher energies by the phenomenological form factorF_l(p_{ij}/p_0) . The Blatt-Weisskopf form factors are usually employed, e.g. [31-33],F_{0}^{2}(z) = 1 ,F_{1}^{2}(z) = 1 /(1+z) ,F_{2}^{2}(z) = 1 /\left(9+3 z+z^{2}\right) withz = (p_{ij}/p_0)^2 . Refs. [34,35] setz = (p_{ij}R)^2 and found thatR , 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. Withp_0 andR being positive real values, it is easy to find thatp_{ij}/p_0 = p_{ij}R . Therefore,p_0 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.
S-S couplings, B.P-P couplings, and C.S-P andP-S couplings, where the former denotes the angular momentum of theJ/\psi J/\psi channel and the latter other channels listed in Tables 1-2. The corresponding pole positions ofX(6900) andX(7200) were determined. -
Constrained by the generalized bose symmetry for identical particles and
J^{PC} conservation, the quantum numbers of theS -waveJ/\psi J/\psi pair must be0^{++} or2^{++} . Based on the0^{++} or2^{++} assumption forX(6900) andX(7200) , the otherS -wave couple channels near the mass of the two states were considered, as summarized in Table 1. They could be divided into three cases forX(6900) decays:Case I:
J/\psi J/\psi andJ/\psi \psi(3770) ,Case II:
J/\psi J/\psi andJ/\psi \psi_2(3823) ,Case III:
J/\psi J/\psi andJ/\psi \psi_3(3842) .For
X(7200) ,J/\psi J/\psi and the near-thresholdJ/\psi \psi(4160) channels were used in the couple channel analysis. TheX(6500) state has the same quantum numbers asX(6900) (similarly hereafter), as previously mentioned. Thus, the total amplitude{\cal{M}} satisfies|{\cal{M}}|^2 = \Bigg| \sum\limits_{i = 1}^{3} {\cal{M}}_i + {\cal{M}}_{\rm{NoR}} \Bigg|^{2} + {\cal{B.G.}},
(3) where
{\cal{M}}_{\rm{NoR}} describes the coherent background (BG), and the incoherent background{\cal{B.G.}} takes a parameterization similar to{\cal{M}}_{\rm{NoR}} . 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.Case I Case II Case III Solution I Solution II Solution I Solution II Solution I Solution II \chi^2 /d.o.f.100.1/86 100.6/86 97.6/86 96.4/86 99.2/86 99.6/86 M_2 /MeV6883.3\pm100.3 6881.9\pm203.2 6921.5\pm147.6 6850.0\pm136.7 6829.8\pm113.6 6850.0\pm107.8 g_{21} /MeV338.8\pm25.8 1029.1\pm91.1 1000.7\pm19.3 1006.6\pm56.7 606.5\pm18.8 1005.9\pm54.1 g_{22} /MeV110.9\pm123.0 1644.1\pm244.4 645.9\pm56.9 1683.5\pm239.1 259.6\pm71.6 1661.9\pm245.5 \phi_2 /rad0.7\pm1.6 1.3\pm1.7 3.9\pm0.9 1.8\pm1.1 2.7\pm1.6 2.1\pm0.9 M_3 /MeV7195.1\pm212.8 7150.0\pm747.5 7221.1\pm172.4 7165.6\pm656.0 7222.2\pm182.9 7169.9\pm583.2 g_{31} /MeV68.5\pm12.4 151.5\pm48.6 120.0\pm31.0 130.4\pm52.3 110.0\pm32.2 127.2\pm57.7 g_{32} /MeV94.1\pm92.9 832.3\pm245.7 0.0004\pm151.9 774.5\pm262.8 0.0003\pm155.4 772.7\pm 251.7 \phi_3 /rad5.5\pm0.9 0.8\pm1.1 4.4\pm1.9 1.1\pm1.0 5.2\pm1.4 1.2\pm0.9 aThe remaining parameters which are not listed here are included in Table A1. Table 4. Summary of numerical results for
S-S couplings, whereM_2 andM_3 are the masses ofX(6900) andX(7200) , respectively;g_{21} andg_{31} are the coupling constants ofX(6900) andX(7200) decaying toJ/\psi J/\psi , respectively;g_{22} is the coupling constant ofX(6900)\to J/\psi \psi(3770) ,J/\psi \psi_2(3823) , andJ/\psi \psi_3(3842) in turn in the three cases,g_{32} is the coupling constant ofX(7200)\to J/\psi\psi(4160) ;\phi_i\; (i = 2,3) is the interference phase as expressed in Eq. (3)a.Figure 1. (color online) Fitting projections of two solutions for the
S-S couplings in Case I withX(6900)\to J/\psi J/\psi andJ/\psi \psi(3770) , andX(7200)\to J/\psi J/\psi andJ/\psi \psi(4160) couples, where the dots with error bars are the LHCb data [1], the red lines are the best fit, the green dashed lines show the coherent BG, the purple dashed lines are the contribution of the incoherent BG, and the vertical lines indicate the corresponding mass thresholds ofX(6900) andX(7200) .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
{s} plane obtained by using parameters in Table 4 for all cases are summarized in Table 6. For Solution I, the fact that pole positions ofX(7200) on the second and third sheets are equal in Case I and Case II indicates theX(7200) state hardly couples to theJ/\psi \psi(4160) 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 theX(6900) threshold indicates that it might be a confining state forS -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 assumingX(6900) andX(7200) to beJ^{PC} = (0,2)^{++} 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,X(6900)\to J/\psi \psi(3770) ,J/\psi \psi_2(3823) ,J/\psi \psi_3(3842) ,\chi_{c0} \chi_{c1} , andX(7200) \to J/\psi\psi(4160) ,\chi_{c0} \chi_{c1} (3872) , are therefore in urgent need to clarify their nature.I II III IV \rho_{i1} + − − + \rho_{i2} + + − − Table 5. Definition of Riemann sheets (
i = 2,3 ).Case State Sheet II Sheet III Sol. I I X(6900) 6885.4-68.0 i 6874.4-80.0 i X(7200) 7202.2-16.6 i 7187.1-18.0 i II X(6900) 6947.6-172.0 i 6810.4-274.0 i X(7200) 7220.8-31.0 i 7220.8-31.0 i III X(6900) 6845.2-117.0 i 6789.2-138.0 i X(7200) 7221.9-28.0 i 7221.9-28.0 i Sol. II I X(6900) 6937.9-97.0 i 6527.3-323.0 i X(7200) 7210.7-27.5 i 7037.3-47.5 i II X(6900) 6933.9-111.0 i 6443.8-275. 0i X(7200) 7218.9-24.0 i 7067.9-41.5 i III X(6900) 6933.3-113.0 i 6452.3-275.0 i X(7200) 7221.9-23.0 i 7073.7-41.0 i Table 6. Summary of pole positions obtained using central values of parameters for
S-S couplings. Here, the symbol "Sol." denotes "Solution" throughout the analysis.At last, we also tested the situation in which
X(6900) couples toJ/\psi J/\psi andJ/\psi\psi(2S) . A solution that favorsX(6900) as a confining state was found. Meanwhile, we could not find a good solution in favor of a molecular state interpretation ofX(6900) . -
With the quantum numbers of the
P -waveJ/\psi J/\psi pair being(0,1,2)^{-+} , coupling channel thresholds close to the two states are summarized in Table 2. From this table, the couplings can be divided into two cases:Case I:
X(6900)\to J/\psi J/\psi ,\chi_{c0} \chi_{c1} ;X(7200)\to J/\psi J/\psi ,\chi_{c0}\chi_{c1}(3872) .Case II:
X(6900)\to J/\psi J/\psi ,J/\psi \psi(3770) ;X(7200) \to J/\psi J/\psi andJ/\psi \psi(4160) .By employing Eq. (3) with the threshold and barrier factors
n_{ij}(s) included, the fitting projections are shown in Fig. 2, and the corresponding numerical results are listed in Table 7. Note that the parameterization with theP -wave coupling assumption can also meet the experimental data well, with almost equivalent goodness of fit in theS -wave couplings. The pole positions in the complexs plane are listed in Table 8. Note that the method adopted in this study cannot distinguish aP -wave confining state from aP -wave molecule, given that they both contribute two pair of poles near the threshold.Figure 2. (color online) Fitting projections for the
P-P couplings, where Fig. (a) showsX(6900) decaying toJ/\psi J/\psi and\chi_{c0} \chi_{c1} , andX(7200) toJ/\psi J/\psi and\chi_{c0}\chi_{c1}(3872) , and Fig. (b) illustratesX(6900)\to J/\psi J/\psi andJ/\psi \psi(3770) ,X(7200)\to J/\psi J/\psi andJ/\psi \psi(4160) . Here, the descriptions of the components of the figures are similar to those of Fig. 1.Case I Case II \chi^2 /d.o.f.95.7/86 95.6/86 M_2 /MeV6866.9\pm51.4 6869.6\pm60.2 g_{21} /MeV4679.6\pm128.6 4679.2\pm136.3 g_{22} /MeV1010.1\pm1520.1 2585.2\pm1828.7 \phi_2 /rad1.7\pm1.4 1.7\pm1.6 M_3 /MeV7228.2\pm50.1 7229.7\pm50.7 g_{31} /MeV568.8\pm60.6 569.3\pm59.8 g_{32} /MeV7105.6\pm5964.6 8158.6\pm5610.5 \phi_3 /rad5.7\pm0.6 5.7\pm0.6 aThe remaining parameters which are not listed here are displayed in the Table A2. Table 7. Summary of numerical results for
P-P couplings, whereM_2 andM_3 are the masses ofX(6900) andX(7200) , respectively;g_{21} andg_{31} are the coupling constants ofX(6900) andX(7200) decaying toJ/\psi J/\psi , respectively;g_{22} is the coupling constant ofX(6900)\to\chi_{c0}\chi_{c1} andJ/\psi \psi(3770) in the two cases,g_{32} is the coupling constant ofX(7200)\to J/\psi\psi(4160) ;\phi_i\; (i = 2,3) is the interference phase as expressed in Eq. (3)a.State Sheet II Sheet III Case I X(6900) 6838.7-119.0 i 6840.9-113.0 i X(7200) 7220.8-31.0 i 7232.5-23.0 i Caes II X(6900) 6844.1-122.0 i 6841.2-110.5 i X(7200) 7221.4-32.0 i 7234.4-22.0 i Table 8. Summary of pole positions obtained using central values of parameters for the
P-P couplings. -
Owing to the limited statistics but multiple states, only two cases were considered in the analysis for the
S-P andP-S couplings:Case I (
S-P ):S -wave forX(6900)\to J/\psi J/\psi ,J/\psi \psi(3770) ;P -wave forX(7200) \to J/\psi J/\psi andJ/\psi \psi(4160) .Case II (
P-S ):P -wave forX(6900)\to J/\psi J/\psi ,J/\psi \psi(3770) ;S -wave forX(7200) \to J/\psi J/\psi andJ/\psi \psi(4160) .The following total amplitude is applicable for both cases,
|{\cal{M}}|^2 = \Bigg| \sum\limits_{i = 1}^{2} {\cal{M}}_i + {\cal{M}}_{\rm{NoR}} \Bigg|^{2} + |{\cal{M}}_3|^{2} + {\cal{B.G.}}
(4) By employing Eq. (4) with the respective threshold and barrier factors,
n_{ij}(s) 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 ofX(7200) was found in Case II. The pole positions in the complexs plane are summarized in Table 9. Based on PCR, it may be concluded that Solution I favorsX(6900) as a confining state and Solution II supports that it is a molecular bound state.Sol. State Sheet II Sheet III Case I I X(6900) 6901.0-32.6 i 6884.4-61.7 i X(7200) 7196.2 -19.5 i 7200.8-17.4 i II X(6900) 6894.8-65.3 i - X(7200) 7097.8-17.6 i 7128.1-14.0 i Case II X(6900) 6900.5-14.5 i 6900.3-15.2 i X(7200) 7362.2-67.9 i - Table 9. Summary of pole positions obtained using central values of parameters for
S-P andP-S couplings. -
Concerning
S -waves, SDFSR can be utilized to provide insights into the nature of theX(6900) state. Ref. [27] pointed out that the spectrum density function\omega(E) near threshold can be calculated by using the non-relativisticS -wave Flatté parameterization, and the renormalization constant{\cal{Z}} can be obtained, which represents the probability of finding the confining particle in the continuous spectrum: the greater the tendency of{\cal{Z}} to1 , the more confining is the state. On the other hand, if{\cal{Z}} tends to 0, the state tends to be molecular.Using a form similar to that in Refs. [25,27-30], the spectrum density function of a near-threshold channel can be expressed as
\omega(E) = \frac{1}{2\pi}\frac{\tilde{g}\sqrt{2\mu E}\theta(E)+\tilde{\Gamma}_{0}}{\left|E-E_{f}+\dfrac{\rm i}{2}\tilde{g}\sqrt{2\mu E}\theta(E)+\dfrac{\rm i}{2}\tilde{\Gamma}_0\right|^2},
(5) where
E\; (E_f) = \sqrt{s}\; (M)-m_{th} is the energy difference between the center-of-mass energy (resonant state) and the open-channel threshold,\mu is the reduced mass of the two-body final states of the channel,\theta is the step function,\tilde{g} = 2g/m_{th} is the dimensionless coupling constant of the concerned coupling mode, and\tilde{\Gamma_0} is the constant partial width for the remaining couplings, which mainly contains the distant channels (theJ/\psi J/\psi process in this analysis).By integrating Eq. (5), the probability of finding an "elementary" particle in the continuous spectrum can be obtained:
{\cal{Z}} = \int_{E_{\min}}^{E_{\max}}\omega(E){\rm d}E.
(6) The integral interval takes
E_f 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 theX(7200) state is of no significance (\sim3\; \sigma reported in the LHCb experiment [1]), only the{\cal{Z}} values ofX(6900) , as listed in Table 1, are calculated. Expanding\omega(E) near the threshold of each channel, and bringingX(6900) 's massM^{\rm pole}_2 and width\Gamma^{\rm pole}_2 extracted from the second Riemann sheet in Table 6, one can obtain the corresponding{\cal{Z}} value, where\tilde{\Gamma_0} = \Gamma_{21}(M^{\rm{pole}}_2) (see also Eq. (2)). The numerical{\cal{Z}} values are summarized in Table 10, where the interval[E_f-\Gamma,E_f+\Gamma] covers all thresholds of the calculated channels. The{\cal{Z}} values in Solution I are all slightly less than 50% in this interval, but rapidly exceeds 50% in larger integral intervals. HenceX(6900) may be considered as a confining state in Solution I. For Solution II, the{\cal{Z}} values are much smaller than 50% in the interval[E_f-\Gamma,E_f+\Gamma] , and also less than 50% in the interval[E_f-2\Gamma,E_f+2\Gamma] . This suggests that theX(6900) state is more likely a molecular state in Solution II. As a conclusion, the nature ofX(6900) 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.Case [E_f-\Gamma,E_f+\Gamma] [E_f-2\Gamma,E_f+2\Gamma] Sol. I I 0.459 0.671 II 0.379 0.592 III 0.468 0.681 Sol. II I 0.184 0.344 II 0.243 0.418 III 0.259 0.438 Table 10. Summary of
{\cal{Z}} values for theS-S couplings ofX(6900) . -
The remaining parameters which are not listed in Sec. II are presented below. The parameter values for
S-S andP-P couplings are displayed in Table A1 and Table A2, respectively.Case I Case II Case III Solution I Solution II Solution I Solution II Solution I Solution II g_1 /MeV2(34.5\pm 3.5)\times 10^6 (56.2\pm 5.6)\times 10^6 (35.0\pm 3.6)\times 10^6 (37.2\pm 4.3)\times 10^6 (57.8\pm 4.0)\times 10^6 (28.2\pm 3.8)\times 10^6 \phi_1 /rad3.6\pm 0.1 3.6\pm 0.1 5.8\pm 0.1 3.8\pm 0.1 4.8\pm 0.1 3.9\pm 0.1 M_1 /MeV6621.6\pm 60.0 6726.8\pm 164.0 6741.0\pm 59.1 6683.3\pm 155.9 6737.0\pm 78.5 6671.0\pm 101.6 g_{11} /MeV540.9\pm 55.6 719.7\pm 66.0 296.4\pm 22.2 543.3\pm 53.8 412.1\pm 33.4 450.0\pm 51.2 g_{2} /MeV2(11.9\pm 0.8)\times 10^6 (40.2\pm 5.6)\times 10^6 (71.3\pm 4.7)\times 10^6 (32.4\pm 3.3)\times 10^6 (35.0\pm 5.6)\times 10^6 (29.0\pm 2.9)\times 10^6 g_{3} /MeV2(2.3\pm 0.3)\times 10^6 (2.0\pm 0.6)\times 10^6 (1.8\pm 0.4)\times 10^6 (1.6\pm 0.5)\times 10^6 (1.6\pm 0.4)\times 10^6 (1.6\pm 0.5)\times 10^6 Table A1. Parameter values for
S-S coupling. Other parameter values are listed in Table 4.Case I Case II g_1 /MeV2(41.7\pm 3.1)\times 10^6 (41.7\pm 3.1)\times 10^6 \phi_1 /rad4.1\pm 0.1 4.0\pm 0.1 M_1 /MeV6753.8\pm 60.7 6748.7\pm 85.5 g_{11} /MeV402.8\pm 33.7 414.6\pm 34.1 g_2 /MeV2(74.3\pm 8.9)\times 10^6 (72.2\pm 8.4)\times 10^6 g_3 /MeV2(8.0\pm 0.6)\times 10^6 (7.9\pm 0.6)\times 10^6 Table A2. Parameter values for
P-P coupling. Other parameter values are listed in Table 7.For
S-P coupling, in whichX(6900) couples toS -wave di-J/\psi andJ/\psi\psi(3770) , andX(7200) couples toP -wave di-J/\psi andJ/\psi\psi(4160) , the parameter values are listed in Table A3.Sol. I Sol. II \chi^2/d.o.f. 98.2/87 103.3/87 g_1 /MeV2(19.9\pm 4.2)\times 10^{6} (41.8\pm 5.4)\times 10^6 \phi_1 /rad3.4\pm 0.1 3.3\pm 0.1 M_1 /MeV6599.5\pm 24.4 6670.7\pm 33.9 g_{11} /MeV462.4\pm 91.2 752.9\pm 91.7 g_2 /MeV2(50.0\pm 5.5)\times 10^6 (25.0\pm 1.3)\times 10^6 \phi_2 /rad2.0\pm 0.5 1.1\pm0.2 M_2 /MeV6896.1\pm 27.4 6800.0\pm 28.7 g_{21} /MeV215.8\pm 48.2 1000.0\pm 41.0 g_{22} /MeV250.0\pm 117.9 2550.0\pm 587.5 g_3 /MeV2(7.6\pm 1.6)\times 10^6 (7.4\pm 2.9)\times 10^6 M_3 /MeV7199.0 \pm 102.9 7115.3\pm 263.8 g_{31} /MeV400.0\pm 106.3 400.0\pm 127.5 g_{32} /MeV1463.7\pm 257.6 3424.8\pm 351.1 Table A3. Parameter values for
S-P coupling.For
P-S coupling, in whichX(6900) couples toP -wave di-J/\psi andJ/\psi\psi(3770) , andX(7200) couples toS -wave di-J/\psi andJ/\psi\psi(4160) , the parameter values are shown in Table A4.\chi^2/d.o.f. 97.23/87 g_1 /MeV2(29.5\pm 2.8)\times 10^{6} \phi_1 /rad4.4\pm 0.2 M_1 /MeV6742.8\pm 17.9 g_{11} /MeV1000.0\pm 80.8 g_2 /MeV2(20.0\pm 2.5)\times 10^6 \phi_2 /rad2.3\pm 0.5 M_2 /MeV6900.8\pm 25.9 g_{21} /MeV550.0\pm 31.6 g_{22} /MeV250.0\pm 40.6 g_3 /MeV2(31.2\pm 10.2)\times 10^6 M_3 /MeV7255.8 \pm 115.4 g_{31} /MeV1377.9\pm 505.0 g_{32} /MeV3999.7\pm 2352.5 Table A4. Parameter values for
P-S coupling.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 uncertainty of the multiple interference. Regarding the coherent background,
c_0 = 20.9 andc_1 = -6.9\times 10^{-4} (MeV−1). Concerning the incoherent background, which takes a form similar to the coherent background, it has two parameters:a_0 = 240.7 ,a_1 = -4.5\times 10^{-4} (MeV−1).
Some remarks on X(6900)
- Received Date: 2021-05-21
- Available Online: 2021-10-15
Abstract: The analysis of the LHCb data on