Charged charmoniumlike structures in the \(e^+ e^ \rightarrow \psi (3686) \pi ^+ \pi ^\) process based on the ISPE mechanism
Abstract
In 2017, the BESIII Collaboration announced the observation of a charged charmoniumlike structure in the \(\psi (3686)\pi ^\pm \) invariant mass spectrum of the \(e^+ e^ \rightarrow \psi (3686) \pi ^+ \pi ^\) process at different energy points, which enables us to perform a precise study of this process based on the initial single pion emission (ISPE) mechanism. In this work, we perform a combined fit to the experimental data of the cross section of \(e^+ e^ \rightarrow \psi (3686) \pi ^+ \pi ^\), and the corresponding \(\psi (3686)\pi ^\pm \) and dipion invariant mass spectra. Our result shows that the observed charged charmoniumlike structure in \(e^+ e^ \rightarrow \psi (3686) \pi ^+ \pi ^\) can be well reproduced based on the ISPE mechanism, and that the corresponding dipion invariant mass spectrum and cross section can be depicted with the same parameters. In fact, it provides strong evidence that the ISPE mechanism can be an underlying mechanism resulting in such novel a phenomenon.
1 Introduction
Since 2003, study on the charmoniumlike XYZ states has become a hot spot of hadron physics and particle physics. Since it has the close relation to nonperturbative behavior of quantum chromodynamics (QCD), the relevant investigation of the charmoniumlike XYZ states is helpful to deepen our understanding of strong interaction. In the past 16 years, there have been extensive discussions about this issue (see reviews [1, 2, 3, 4] for more details).
To solve the puzzling phenomena existing in the hiddenbottom dipion decays of \(\Upsilon (10860)\), the initial single pion emission (ISPE) mechanism was proposed in Refs. [5, 6, 7], where the charged \(Z_b(10610)\) and \(Z_b(10650)\) structures can be qualitatively reproduced. In 2012, considering the similarity between the bottomonium and charmonium families, the ISPE mechanism was applied to study the hiddencharm dipion decays of higher charmonia and charmoniumlike state Y(4260), where the predicted charged charmoniumlike structures near \(D\bar{D}^*\) or \(D^*\bar{D}^*\) threshold may exist in the corresponding \(J/\psi \pi ^+\), \(\psi (3686)\pi ^+\), and \(h_{c}\pi ^+\) invariant mass spectra [8]. These predictions given by the ISPE mechanism have provided valuable information to search for such charged charmoniumlike structures.
In 2013, the BESIII and Belle collaborations indeed discovered a charged charmoniumlike \(Z_c(3900)\) in the process \(e^+ e^ \rightarrow J/\psi \pi ^+ \pi ^\) around \(\sqrt{s}=4.26\ \mathrm {GeV}\) [9, 10], which was quickly and further confirmed by CLEOc in the same process but at \(\sqrt{s}=4.17\) GeV [11]. Later, another charged charmoniumlike structure named \(Z_c(4020)\) was observed in the \(h_c \pi ^+\) invariant mass spectrum of the \(e^+ e^ \rightarrow \pi ^+ \pi ^ h_c\) process [12]. Additionally, at the centerofmass energy of 4.26 GeV, the BESIII Collaboration reported two charged charmoniumlike structures, \(Z_c(3885)\) in the \((D^*\bar{D})^{\pm }\) invariant mass spectrum of \(e^+e^ \rightarrow \pi ^\mp (D^*\bar{D})^\pm \) process [13] and \(Z_c(4025)\) in the \((D^*\bar{D}^*)^\pm \) invariant mass spectrum of the \(e^+ e^ \rightarrow \pi ^\mp (D^*\bar{D}^*)^\pm \) process [14].
The BESIII’s experimental measurement of the \(e^+e^\rightarrow \psi (3686)\pi ^+\pi ^\) process forces us to carry out a further study of this process based on the ISPE mechanism. Before the present work, in Ref. [16], the similar idea was once adopted to study the \(e^+e^\rightarrow J/\psi \pi ^+\pi ^\) process combined with the BESIII data of the \(J/\psi \pi ^\pm \) invariant mass spectrum and the corresponding dipion invariant mass spectrum of \(e^+e^\rightarrow J/\psi \pi ^+\pi ^\) at one energy point \(\sqrt{s}=4.26\) GeV, which shows that the \(Z_c(3900)\) structure can be reproduced well by the ISPE mechanism. Different from Ref. [16], the present work will perform a combined fit to the released data of the \(\psi (3686)\pi ^\pm \) and \(\pi ^+\pi ^\) invariant mass spectrum from the \(e^+e^\rightarrow \psi (3686)\pi ^+\pi ^\) process at different energy points. Comparing with the former work [16], the present work is obviously a tough task and is full of challenge. In this work, we will reproduce the structures observed in the \(e^+e^\rightarrow \psi (3686)\pi ^+\pi ^\) process step by step based on the ISPE mechanism. It also reveals the relation between the observed charged charmoniumlike structures and the ISPE mechanism, which will be helpful to deepen our understanding on such a novel phenomenon.
2 Mechanisms occurring in \(e^+ e^ \rightarrow \psi (3686) \pi ^+ \pi ^\)
Meson  Mass (GeV)  Width (MeV)  \(\Gamma _{e^+ e^}\) (keV) 

\(\psi (4160)\)  4.191  70  0.48 
\(\psi (4415)\)  4.421  62  0.58 
\(\psi (4220)\)  4.218  59  0.29 
\(\psi (4380)\)  4.384  84  0.26 
3 Numerical results and discussion
 \(a_R\):

a parameter in the resonance form factor,
 \(\phi _\mathrm{Dir}\):

a phase angle of the direct production amplitude,
 \(F_\mathrm{Dir},\ \kappa _\mathrm{Dir}\):

parameters in the direct production amplitude,
 \(\phi _\sigma \):

a phase angle of the \(\sigma \) meson production amplitude,
 \(f_\sigma , \ g_\sigma \):

S and D wave coupling constants in the \(\sigma \) meson production amplitude,
 \(\varphi _\sigma \):

a relative phase angle between S and D wave terms in the \(\sigma \) meson production amplitude,
 \(\phi _\mathrm{ISPE}\):

a phase angle of the \(\mathrm {ISPE}\) amplitude,
 \(g_{R D^{(*)} \bar{D}^{(*)} \pi }\):

coupling constants of the fourparticle interactions in the \(\mathrm {ISPE}\) mechanism.
In the following, we need to fit our results with the experimental data of \(e^+e^ \rightarrow \psi (3686) \pi ^+ \pi ^\), which include the measured total cross section, the \(\psi (3686) \pi ^\pm \), and \(\pi ^+\pi ^\) invariant mass spectra released in Ref. [15]. For the \(e^+e^ \rightarrow \psi (3686) \pi ^+ \pi ^\) process, there exist contributions from intermediate vector charmonia as shown in Fig. 2(2)–(4). Thus, it is a key point how to select intermediate vector charmonia. In the present work, we adopt two scenarios, which will be addressed in Sects. 3.1 and 3.2.
3.1 Threecharmonium scenario to \(e^+e^ \rightarrow \psi (3686) \pi ^+ \pi ^\)
When checking the vector charmonium states located in 4.1–4.6 GeV listed in particle data book [17], there exist only two observed charmonia, \(\psi (4160)\) and \(\psi (4415)\). Besides, we must mention that there is another wellestablished charmoniumlike state Y(4220) in this energy range. This state has been observed experimentally in the processes of \(e^+ e^ \rightarrow \chi _{c0} \omega \) [25], \(e^+ e^ \rightarrow \pi ^+ \pi ^ h_c\) [26], \(e^+ e^ \rightarrow \pi ^+ \pi ^ J/\psi \) [27], \(e^+ e^ \rightarrow \pi ^+ D^0 D^{*}\) [28], and \(e^+ e^ \rightarrow \pi ^+ \pi ^ \psi (3686)\) [15]. The mass of this state is consistent with the predictions of a missing higher charmonium in the \(J/\psi \) family (see the discussion in Refs. [18, 29, 30]). Thus, Y(4220) is replaced by the renamed \(\psi (4220)\) in the following discussion.
The fitted values of the parameters in threecharmonium scenario
Parameters  \(\psi (4160)\)  \(\psi (4220)\)  \(\psi (4415)\) 

\(a_{R}\) (\(\hbox {GeV}^{1}\))  6.0  5.0  4.0 
\(\phi _{\mathrm{Dir}}\)  \(\) 1.631  \(\) 1.506  1.762 
\(F_\mathrm{Dir}\)  \(\) 3.477  5.881  0.982 
\(\kappa _{Dir}\)  \(\) 1.829  \(\)0.025  \(\) 0.918 
\(\phi _{\sigma }\)  1.130  2.009  2.241 
\(f_{\sigma }\) (\(\hbox {GeV}^2\))  7.562  5.785  \(\) 3.971 
\(g_{\sigma }\)  77.498  21.803  \(\) 1.920 
\(\varphi _{\sigma }\)  1.186  \(\) 2.426  \(\) 1.923 
\(\phi _\mathrm{ISPE}\)  \(\) 1.320  2.980  \(\) 2.813 
\(g_{ R D \bar{D} \pi }\) (\(\hbox {GeV}^{3}\))  1.717  1.574  0.002 
\(g_{R D^*\bar{D} \pi }\)  \(\) 1.834  \(\) 0.670  0.025 
\(g_{R D^*\bar{D}^*\pi }\) (\(\hbox {GeV}^{1}\))  0.120  \(\) 0.096  0.105 
\(g_\mathrm{NoR}=18.940\) GeV  \(a_\mathrm{NoR}=4.702\hbox { GeV}^{2}\)  
\(\chi ^2/d.o.f.=3.704\) 
After integrating over \(m_{ \pi ^\pm \psi (3686)}\) or \(m_{\pi ^+ \pi ^}\) in Eq. (10) at each fixed \(\sqrt{s}\), one gets the differential cross sections of \(e^+ e^ \rightarrow \pi ^+ \pi ^ \psi (3686)\) for each invariant mass squared distribution. The cross section depending on \(\sqrt{s}\) can be obtained by integrating over both \(m_{ \psi (3686)\pi ^\pm }\) and \(m_{\pi ^+ \pi ^}\) in Eq. (10). Here, we perform a combined fit of the invariant mass squared dependent cross sections to the experimental data of the differential cross sections at a fixed \(\sqrt{s}\). In the threeresonance scenario, there are 38 parameters, which should be determined by the present fit. Our fitted values for all the parameters are presented in Table 2. With these parameters, \(\chi ^2/\mathrm{d.o.f}\) is estimated to be 3.704.
Branching ratios of \(\psi _R \rightarrow \pi ^+ \pi ^ \psi (3686) \) obtained from the present fit in the unit of \(10^{3}\)
\(\psi (4160)\)  \(\psi (4220)\)  \(\psi (4415)\)  \(\psi (4380)\)  

3R Fit  15.711  8.101  9.654  – 
4R Fit  4.389  7.370  5.368  17.285 
The same as Table 2 but for the fourcharmonium scenario
Parameter  \(\psi (4160)\)  \(\psi (4220)\)  \(\psi (4415)\)  \(\psi (4380)\) 

\(a_{R}\) (\(\hbox {GeV}^{1}\))  5.5  4.0  3.3  3.0 
\(\phi _\mathrm{Dir}\)  \(\) 0.146  1.661  \(\) 0.423  \(\) 0.345 
\(F_\mathrm{Dir}\)  \(\) 12.243  3.361  0.504  1.523 
\(\kappa _\mathrm{Dir}\)  \(\) 0.151  \(\) 0.215  \(\)1.190  \(\) 0.463 
\(\phi _{\sigma }\)  0.517  \(\) 0.950  2.910  \(\) 0.328 
\(f_{\sigma }\) (\(\hbox {GeV}^2\))  11.953  1.813  1.832  \(\) 5.940 
\(\varphi _{\sigma }\)  \(\) 1.680  0.287  \(\) 0.838  \(\) 1.455 
\(g_{\sigma }\)  58.717  \(\) 6.612  \(\) 7.864  \(\) 25.906 
\(\phi _{ \mathrm {ISPE}}\)  \(\) 1.759  2.875  1.396  \(\) 5.475 
\(g_{R D \bar{D} \pi }\) (\(\hbox {GeV}^{3}\))  \(\) 2.223  \(\) 1.736  \(\) 0.257  0.276 
\(g_{R D^*\bar{D} \pi }\)  1.646  0.353  \(\) 0.470  1.249 
\(g_{R D^*\bar{D}^*\pi }\) (\(\hbox {GeV}^{1}\))  0.323  0.131  0.133  \(\)0.022 
\(g_\mathrm{NoR}=4.847\,\hbox { GeV}\)  \(a_\mathrm{NoR}=2.058\,\hbox {GeV}^{2}\)  
\(\chi ^2/d.o.f.=2.767\) 
3.2 Fourcharmonium scenario to \(e^+e^ \rightarrow \psi (3686) \pi ^+ \pi ^\)
The fitted parameters are presented in Table 4. With the center values of these parameters, one can get the differential cross sections for each invariant mass squared distribution and the cross section depending on \(\sqrt{s}\), which are presented in Figs. 4 (red curves) and 5 (red curve), respectively. After including \(\psi (4380)\), one finds that almost all the differential cross sections can be quantitatively fitted better than the three resonancescenario. Especially, the jump near 15 \(\hbox {GeV}^2\) in \(m^2_{\pi ^\pm \psi (3686) }\) invariant mass spectrum at \(\sqrt{s}=4.258\) GeV can be well reproduced and \(m^2_{\pi ^+\pi ^}\) invariant mass spectrum at this energy point is well fitted. The fits to differential cross sections at \(\sqrt{s}=4.358\) GeV are also much better than the one in the threeresonance scenario. As for the \(\sqrt{s}\) dependent cross section, the fourresonance scenario can well reproduce experimental data, especially the data near at 4.3 GeV. Quantitatively, the \(\chi ^2/\mathrm{d.o.f}\) is reduced to 2.77 in the fourcharmonium scenario. Similar to the case of the threecharmonium scenario, we present the individual contributions of the nonresonance background and resonances in Fig. 7. One can find that the resonance contributions are smaller than the corresponding one in the threecharmonium scenario. The fitted branching ratios of \(\psi _R\rightarrow \pi ^+ \pi ^ \psi (3686)\) are presented in Table 3, where the central values of these branching ratios for \(\psi (4160)\), \(\psi (4220)\), and \(\psi (4415)\) are of the order of \(\sim 10^{3}\), while the one for \(\psi (4380)\) is of the order of \(\sim 10^{2}\).
4 Summary
In the past years, the BESIII and Belle Collaborations have made a great progress on finding charged charmoniumlike XYZ states. A series of charged charmoniumlike structures, \(Z_c(3900)\) [9], \(Z_c(3885)\) [13], \(Z_c(4020)\) [12], and \(Z_c(4025)\) [14], were reported one by one. These experimental results have stimulated a great interest of both theorists and experimentalists since the charged property of these charmoniumlike states indicates these states should contain at least four constituent quarks, thus they could be good candidates of tetraquark states. The estimates in frameworks of the QCD sum rule [31, 32, 33, 34, 35, 36] and potential model [37, 38, 39, 40, 41] indicate that both \(Z_c(3900)/Z_c(3885)\) and \(Z_c(4020)/Z_c(4025)\) could be explained as tetraquark states with different configurations. Furthermore, the observed masses of \(Z_c(3900)/Z_c(3885)\) and \(Z_c(4020)/Z_c(4025)\) are very close to the thresholds of \(D^*\bar{D}\) and \(D^*\bar{D}^*\), thus, these charmoniumlike states have been extensively investigated in deutronlike hadronic molecular scenarios. Based on the molecular assumption, mass spectrum, decay properties, and productions of these charmoniumlike states have been investigated in different models, such as QCD sum rule [42, 43, 44, 45], potential model [46, 47, 48, 49], and some other phenomenological Lagrangian approaches [50, 51, 52, 53, 54, 55, 56, 57, 58].
However, before definitely identifying these charged charmomiumlike structures as exotic tetraquark states, we need to exhaust all the possibilities of explaining them in the conventional theoretical framework. Here, we must mention our previous prediction of charged charmoniumlike structures in our theoretical work [8], where these predicted charged charmoniumlike structures by the ISPE mechanism may exist in the \(J/\psi \pi ^\pm \), \(h_c\pi ^\pm \), and \(\psi (3686)\pi ^\pm \) invariant mass spectra of higher charmonia, and in the charmoniumlike state Y(4260) decays into \(J/\psi \pi ^+\pi ^\), \(h_c\pi ^+\pi ^\) and \(\psi (3686)\pi ^+\pi ^\) [8]. After two years of our paper, the BESIII and Belle collaborations discovered a charged charmoniumlike \(Z_c(3900)\) in the process \(e^+ e^ \rightarrow J/\psi \pi ^+ \pi ^\) around \(\sqrt{s}=4.26\ \mathrm {GeV}\) [9, 10], and CLEOc confirmed it in the same process but at \(\sqrt{s}=4.17\) GeV [11]. In 2013, BESIII observed another charged charmoniumlike structure named as \(Z_c(4020)\) in the \(h_c \pi ^+\) invariant mass spectrum of the \(e^+ e^ \rightarrow \pi ^+ \pi ^ h_c\) process [12]. These experimental observations make us believe that the ISPE mechanism indeed plays an important role in producing these novel phenomena. What is more important is that these measured experimental data make us possible to study them with higher precision. In Ref. [17], we fitted the experimental result of the charged \(Z_c(3900)\) observed in the \(J/\psi \pi ^\pm \) invariant mass spectrum [9, 10] and indicated that the charged \(Z_c(3900)\) structure can be well established based on the ISPE mechanism. This study further enforces theorist’s confidence to explain these experimental results in the conventional theoretical framework. Similar studies were proposed in Refs. [59, 60, 61, 62, 63]. Later, the Lattice QCD calculation [64, 65] also supports such a scenario.
In 2017, the BESIII Collaboration again observed charged charmoniumlike structures in the \(\psi (3686)\pi ^\pm \) invariant mass spectrum of the \(e^+e^\rightarrow \psi (3686)\pi ^+\pi ^\) process at different energy points \(\sqrt{s}=4.226,\,4.258,\,4.358,\,4.387, 4.416,\,4.600\) GeV [15]. This new observation inspires theorist great interest in the nature of charged \(Z_c\) states. In Ref. [66], by using the dispersion theory and taking into account the effects of the charged \(Z_c\) state, the cross sections and the invariant mass distributions of dipion and \(\pi \psi (3686)\) could be reproduced. Moreover, the success of ISPE mechanism stimulate our interest in further testing the ISPE mechanism by using the present precise experimental data. A crucial task of the present work is to examine whether the cross section of the \(e^+e^\rightarrow \psi (3686)\pi ^+\pi ^\) process, and the corresponding \(\psi (3686)\pi ^\pm \) and \(\pi ^+\pi ^\) invariant mass spectra at different \(\sqrt{s}\) values can be reproduced by the ISPE mechanism. As illustrated by our calculation, we find that the reported charged charmoniumlike structures in the \(\psi (3686)\pi ^\pm \) invariant mass spectrum of the \(e^+e^\rightarrow \psi (3686)\pi ^+\pi ^\) process at different energy points can be depicted in a unified framework, which sheds light on the properties of the observed charged charmoniumlike structures. We need to emphasize that our study also further supports the existence of a new higher charmonium \(\psi (4380)\) which was predicted in Ref. [18], since the fitting result under the fourcharmonium scenario is better than that under the threecharmonium scenario. We strongly suggest the BESIII and BelleII collaborations to search for this predicted missing charmonium \(\psi (4380)\), especially by analyzing the \(e^+e^\rightarrow \psi (3686)\pi ^+\pi ^\) process in near future.
In the following years, studies on charmoniumlike XYZ states will still be an interesting research topic at the typical facilities like BESIII, LHCb, and BelleII. Since the observed charmoniumlike XYZ states can be a good candidate of an exotic tetraquark matter, it is a main task for both experimentalists and theorists how to identify them as a genuine exotic multiquark state. To achieve this aim, we need to check whether the charmoniumlike XYZ states can be assigned into a conventional charmonium family or can be settled down in a conventional theoretical framework. The present work is the effort along this line. Besides theoretical investigation, the Lattice QCD study about charmoniumlike XYZ will provide valuable information to shed light on the underlying properties of the XYZ states. As emphasized in a recent review article [3], a joint effort from phenomenological method, experimental analysis, and Lattice QCD calculation should be paid more attentions on, which must promote our knowledge of how to form charmoniumlike XYZ states.
Notes
Acknowledgements
This work is partly supported by the China National Funds for Distinguished Young Scientists under Grant No. 11825503, the National Natural Science Foundation of China under Grant No. 11775050, National Program for Support of Topnotch Young Professionals, and the Fundamental Re search Funds for the Central Universities.
References
 1.X. Liu, An overview of \(XYZ\) new particles. Chin. Sci. Bull. 59, 3815 (2014). arXiv:1312.7408 [hepph]CrossRefGoogle Scholar
 2.H.X. Chen, W. Chen, X. Liu, S.L. Zhu, The hiddencharm pentaquark and tetraquark states. Phys. Rep. 639, 1 (2016). arXiv:1601.02092 [hepph]MathSciNetADSCrossRefGoogle Scholar
 3.Y.R. Liu, H.X. Chen, W. Chen, X. Liu, S.L. Zhu, Pentaquark and tetraquark states. arXiv:1903.11976 [hepph]
 4.F.K. Guo, C. Hanhart, U.G. Meisner, Q. Wang, Q. Zhao, B.S. Zou, Hadronic molecules. Rev. Mod. Phys. 90(1), 015004 (2018). arXiv:1705.00141 [hepph]ADSCrossRefGoogle Scholar
 5.D.Y. Chen, X. Liu, \(Z_b(10610)\) and \(Z_b(10650)\) structures produced by the initial single pion emission in the \(\Upsilon (5S)\) decays. Phys. Rev. D 84, 094003 (2011). arXiv:1106.3798 [hepph]ADSCrossRefGoogle Scholar
 6.X. Liu, D.Y. Chen, Charged bottomoniumlike structures \(Z_b(10610)\) and \(Z_b(10650)\). Few Body Syst. 54, 165 (2013). arXiv:1110.4433 [hepph]ADSCrossRefGoogle Scholar
 7.D.Y. Chen, X. Liu, T. Matsuki, Interpretation of \(Z_b\)(10610) and \(Z_b\)(10650) in the ISPE mechanism and the charmonium counterpart. Chin. Phys. C 38, 053102 (2014). arXiv:1208.2411 [hepph]ADSCrossRefGoogle Scholar
 8.D.Y. Chen, X. Liu, Predicted charged charmoniumlike structures in the hiddencharm dipion decay of higher charmonia. Phys. Rev. D 84, 034032 (2011). arXiv:1106.5290 [hepph]ADSCrossRefGoogle Scholar
 9.M. Ablikim et al. [BESIII Collaboration], Observation of a charged charmoniumlike structure in \(e^+e^ \rightarrow J/\psi \pi ^+ \pi ^\) at \(\sqrt{s}\) =4.26 GeV. Phys. Rev. Lett. 110, 252001 (2013). arXiv:1303.5949 [hepex]
 10.Z.Q. Liu et al. [Belle Collaboration], Study of \(e^+e^ \rightarrow \pi ^+ \pi ^ J/\psi \) and observation of a charged charmoniumlike state at belle. Phys. Rev. Lett. 110, 252002 (2013). arXiv:1304.0121 [hepex]
 11.T. Xiao, S. Dobbs, A. Tomaradze, K.K. Seth, Observation of the charged hadron \(Z_c^{\pm }(3900)\) and evidence for the neutral \(Z_c^0(3900)\) in \(e^+e^\rightarrow \pi \pi J/\psi \) at \(\sqrt{s}=4170\) MeV. Phys. Lett. B 727, 366 (2013). arXiv:1304.3036 [hepex]ADSCrossRefGoogle Scholar
 12.M. Ablikim et al. [BESIII Collaboration], Observation of a charged charmoniumlike structure \(Z_c\)(4020) and search for the \(Z_c\)(3900) in \(e^+e^ \rightarrow \pi ^+ \pi ^ h_c\). Phys. Rev. Lett. 111(24), 242001 (2013). arXiv:1309.1896 [hepex]
 13.M. Ablikim et al. [BESIII Collaboration], Observation of a charged \((D\bar{D}^{*})^\pm \) mass peak in \(e^{+}e^{} \rightarrow \pi D\bar{D}^{*}\) at \(\sqrt{s} = 4.26 \text{GeV}\). Phys. Rev. Lett. 112(2), 022001 (2014). arXiv:1310.1163 [hepex]
 14.M. Ablikim et al. [BESIII Collaboration], Observation of a charged charmoniumlike structure in \(e^+e^ \rightarrow (D^{*} \bar{D}^{*})^{\pm } \pi ^\mp \) at \(\sqrt{s}=4.26\) GeV. Phys. Rev. Lett. 112(13), 132001 (2014). arXiv:1308.2760 [hepex]
 15.M. Ablikim et al. [BESIII Collaboration], Measurement of \(e^{+}e^{}\rightarrow \pi ^{+}\pi ^{}\psi (3686)\) from 4.008 to 4.600 GeV and observation of a charged structure in the \(\pi ^{\pm }\psi (3686)\) mass spectrum. Phys. Rev. D 96(3), 032004 (2017). arXiv:1703.08787 [hepex]
 16.D.Y. Chen, X. Liu, T. Matsuki, Reproducing the \(Z_c(3900)\) structure through the initialsinglepionemission mechanism. Phys. Rev. D 88(3), 036008 (2013). arXiv:1304.5845 [hepph]ADSCrossRefGoogle Scholar
 17.M. Tanabashi et al. [Particle Data Group], Review of particle physics. Phys. Rev. D 98(3), 030001 (2018)Google Scholar
 18.J.Z. Wang, D.Y. Chen, X. Liu, T. Matsuki, Constructing \(J/\psi \) family with updated data of charmonium like \(Y\) states. arXiv:1903.07115 [hepph]
 19.Q.Y. Lin, X. Liu, H.S. Xu, Charged charmoniumlike state \(Z_c(3900)^\pm \) via meson photoproduction. Phys. Rev. D 88, 114009 (2013). arXiv:1308.6345 [hepph]ADSCrossRefGoogle Scholar
 20.H. Xu, J.J. Xie, X. Liu, Implication of the observed \(e^{+}e^{}\rightarrow p{\bar{p}}\pi ^0\) for studying the \(p{\bar{p}}\rightarrow \psi (3770)\pi ^0\) process. Eur. Phys. J. C 76(4), 192 (2016). arXiv:1505.04571 [hepph]ADSCrossRefGoogle Scholar
 21.V.A. Novikov, M.A. Shifman, Comment on the \(\psi ^\prime \rightarrow J/\psi \pi \pi \) decay. Z. Phys. C 8, 43 (1981)ADSCrossRefGoogle Scholar
 22.D.Y. Chen, J. He, X.Q. Li, X. Liu, Dipion invariant mass distribution of the anomalous \(\Upsilon (1S) \pi ^{+} \pi ^{}\) and \(\Upsilon (2S) \pi ^{+} \pi ^{}\) production near the peak of \(\Upsilon (10860)\). Phys. Rev. D 84, 074006 (2011). arXiv:1105.1672 [hepph]ADSCrossRefGoogle Scholar
 23.D.Y. Chen, X. Liu, X.Q. Li, Anomalous dipion invariant mass distribution of the \(\Upsilon (4S)\) decays into \(\Upsilon (1S) \pi ^{+} \pi ^{}\) and \(\Upsilon (2S) \pi ^{+} \pi ^{}\). Eur. Phys. J. C 71, 1808 (2011). arXiv:1109.1406 [hepph]ADSCrossRefGoogle Scholar
 24.J.J. Xie, Y.B. Dong, X. Cao, Role of the \(\Lambda ^+_c(2940)\) in the \(\pi ^ p \rightarrow D^ D^0 p\) reaction close to threshold. Phys. Rev. D 92(3), 034029 (2015). arXiv:1506.01133 [hepph]ADSCrossRefGoogle Scholar
 25.M. Ablikim et al. [BESIII Collaboration], Study of \(e^+e^\rightarrow \omega \chi _{cJ}\) at centerofmass energies from 4.21 to 4.42 GeV. Phys. Rev. Lett. 114, 092003 (2015). arXiv:1410.6538 [hepex]
 26.M. Ablikim et al. [BESIII Collaboration], Evidence of two resonant structures in \(e^+ e^ \rightarrow \pi ^+ \pi ^ h_c\). Phys. Rev. Lett. 118, 092002 (2017). arXiv:1610.07044 [hepex]
 27.M. Ablikim et al. [BESIII Collaboration], Precise measurement of the \(e^+e^\rightarrow \pi ^+\pi ^J/\psi \) cross section at centerofmass energies from 3.77 to 4.60 GeV. Phys. Rev. Lett. 118, 092001 (2017). arXiv:1611.01317 [hepex]
 28.M. Ablikim et al. [BESIII Collaboration], Evidence of a resonant structure in the \(e^+e^\rightarrow \pi ^+D^0D^{*}\) cross section between 4.05 and 4.60 GeV. arXiv:1808.02847 [hepex]
 29.L.P. He, D.Y. Chen, X. Liu, T. Matsuki, Prediction of a missing higher charmonium around 4.26 GeV in \(J/\psi \) family. Eur. Phys. J. C 74(12), 3208 (2014). arXiv:1405.3831 [hepph]CrossRefGoogle Scholar
 30.D.Y. Chen, X. Liu, T. Matsuki, Observation of \(e^+e^\rightarrow \chi _{c0}\omega \) and missing higher charmonium \(\psi (4S)\). Phys. Rev. D 91(9), 094023 (2015). arXiv:1411.5136 [hepph]ADSCrossRefGoogle Scholar
 31.C.F. Qiao, L. Tang, Interpretation of \(Z_c(4025)\) as the hidden charm tetraquark states via QCD sum rules. Eur. Phys. J. C 74, 2810 (2014). arXiv:1308.3439 [hepph]ADSCrossRefGoogle Scholar
 32.Z.G. Wang, T. Huang, Analysis of the \(X(3872)\), \(Z_c(3900)\) and \(Z_c(3885)\) as axialvector tetraquark states with QCD sum rules. Phys. Rev. D 89(5), 054019 (2014). arXiv:1310.2422 [hepph]ADSCrossRefGoogle Scholar
 33.Z.G. Wang, Reanalysis of the \(Z_c(4020)\), \(Z_c(4025)\), \(Z(4050)\) and \(Z(4250)\) as tetraquark states with QCD sum rules. Commun. Theor. Phys. 63(4), 466 (2015). arXiv:1312.1537 [hepph]ADSCrossRefGoogle Scholar
 34.Z.G. Wang, Analysis of the \(Z_c(4020)\), \(Z_c(4025)\), \(Y(4360)\) and \(Y(4660)\) as vector tetraquark states with QCD sum rules. Eur. Phys. J. C 74(5), 2874 (2014). arXiv:1311.1046 [hepph]ADSCrossRefGoogle Scholar
 35.S.S. Agaev, K. Azizi, H. Sundu, Treating \(Z_c(3900)\) and \(Z(4430)\) as the groundstate and first radially excited tetraquarks. Phys. Rev. D 96(3), 034026 (2017). arXiv:1706.01216 [hepph]ADSCrossRefGoogle Scholar
 36.L. Zhao, W.Z. Deng, S.L. Zhu, Hiddencharm tetraquarks and charged \(Z_c\) states. Phys. Rev. D 90(9), 094031 (2014). arXiv:1408.3924 [hepph]ADSCrossRefGoogle Scholar
 37.M.Z. Liu, D.J. Jia, D.Y. Chen, Possible hadronic molecular states composed of \(S\)wave heavylight mesons. Chin. Phys. C 41(5), 053105 (2017). arXiv:1702.04440 [hepph]ADSCrossRefGoogle Scholar
 38.R. Zhu, Hidden charm octet tetraquarks from a diquarkantidiquark model. Phys. Rev. D 94(5), 054009 (2016). arXiv:1607.02799 [hepph]ADSCrossRefGoogle Scholar
 39.C. Deng, J. Ping, F. Wang, Interpreting \(Z_c(3900)\) and \(Z_c(4025)/Z_c(4020)\) as charged tetraquark states. Phys. Rev. D 90, 054009 (2014). arXiv:1402.0777 [hepph]ADSCrossRefGoogle Scholar
 40.M.N. Anwar, J. Ferretti, E. Santopinto, Spectroscopy of the hiddencharm \([qc][\bar{q} \bar{c}]\) and \([sc][\bar{s} \bar{c}]\) tetraquarks in the relativized diquark model. Phys. Rev. D 98(9), 094015 (2018). arXiv:1805.06276 [hepph]ADSCrossRefGoogle Scholar
 41.S. Patel, M. Shah, P.C. Vinodkumar, Mass spectra of fourquark states in the hidden charm sector. Eur. Phys. J. A 50, 131 (2014). arXiv:1402.3974 [hepph]ADSCrossRefGoogle Scholar
 42.C.Y. Cui, Y.L. Liu, W.B. Chen, M.Q. Huang, Could \(Z_{c}(3900)\) be a \(I^{G}J^{P}=1^{+}1^{+}\) \(D^{*}\bar{D}\) molecular state? J. Phys. G 41, 075003 (2014). arXiv:1304.1850 [hepph]ADSCrossRefGoogle Scholar
 43.J.R. Zhang, Improved QCD sum rule study of \(Z_{c}(3900)\) as a \(\bar{D}D^{*}\) molecular state. Phys. Rev. D 87(11), 116004 (2013). arXiv:1304.5748 [hepph]ADSCrossRefGoogle Scholar
 44.C.Y. Cui, Y.L. Liu, M.Q. Huang, Could \(Z_c\)(4025) be a \(J^P\) = \(1^+ D^* \bar{D^*}\) molecular state? Eur. Phys. J. C 73(12), 2661 (2013). arXiv:1308.3625 [hepph]ADSCrossRefGoogle Scholar
 45.Z.G. Wang, T. Huang, Possible assignments of the \(X(3872)\), \(Z_c(3900)\) and \(Z_b(10610)\) as axialvector molecular states. Eur. Phys. J. C 74(5), 2891 (2014). arXiv:1312.7489 [hepph]ADSCrossRefGoogle Scholar
 46.X. Liu, Z.G. Luo, Y.R. Liu, S.L. Zhu, X(3872) and other possible heavy molecular states. Eur. Phys. J. C 61, 411 (2009). arXiv:0808.0073 [hepph]ADSCrossRefGoogle Scholar
 47.Y.R. Liu, X. Liu, W.Z. Deng, S.L. Zhu, Is \(X(3872) \) really a molecular state? Eur. Phys. J. C 56, 63 (2008). arXiv:0801.3540 [hepph]ADSCrossRefGoogle Scholar
 48.Z.F. Sun, Z.G. Luo, J. He, X. Liu, S.L. Zhu, A note on the \(B^\ast \bar{B}\), \(B^\ast \bar{B}^\ast \), \(D^\ast \bar{D}\), \(D^\ast \bar{D}^\ast \), molecular states. Chin. Phys. C 36, 194 (2012)CrossRefGoogle Scholar
 49.J. He, X. Liu, Z.F. Sun, S.L. Zhu, \(Z_c(4025)\) as the hadronic molecule with hidden charm. Eur. Phys. J. C 73(11), 2635 (2013). arXiv:1308.2999 [hepph]ADSCrossRefGoogle Scholar
 50.T. Gutsche, M. Kesenheimer, V.E. Lyubovitskij, Radiative and dilepton decays of the hadronic molecule \(Z_c^+\)(3900). Phys. Rev. D 90(9), 094013 (2014). arXiv:1410.0259 [hepph]ADSCrossRefGoogle Scholar
 51.D.Y. Chen, Y.B. Dong, Radiative decays of the neutral \(Z_c(3900)\). Phys. Rev. D 93(1), 014003 (2016). arXiv:1510.00829 [hepph]ADSCrossRefGoogle Scholar
 52.G. Li, Hiddencharmonium decays of \(Z_c(3900)\) and \(Z_c(4025)\) in intermediate meson loops model. Eur. Phys. J. C 73(11), 2621 (2013). arXiv:1304.4458 [hepph]ADSCrossRefGoogle Scholar
 53.Q. Wu, G. Li, F. Shao, R. Wang, Investigations on the charmless decay modes of Zc(3900) and Zc(4025). Phys. Rev. D 94(1), 014015 (2016)ADSCrossRefGoogle Scholar
 54.G. Li, X.H. Liu, Z. Zhou, More hidden heavy quarkonium molecules and their discovery decay modes. Phys. Rev. D 90(5), 054006 (2014). arXiv:1409.0754 [hepph]ADSCrossRefGoogle Scholar
 55.G. Li, X.H. Liu, Investigating possible decay modes of Y(4260) under \(D_1(2420)\bar{D}\) + c.c. molecular state ansatz. Phys. Rev. D 88(9), 094008 (2013). arXiv:1307.2622 [hepph]ADSCrossRefGoogle Scholar
 56.D.Y. Chen, Y.B. Dong, M.T. Li, W.L. Wang, Pionic transition from Y(4260) to \({\rm Z}_{c}\)(3900) in a hadronic molecular scenario. Eur. Phys. J. A 52(10), 310 (2016)ADSCrossRefGoogle Scholar
 57.C.J. Xiao, D.Y. Chen, Y.B. Dong, W. Zuo, T. Matsuki, Understanding the \(\eta _c\rho \) decay mode of \(Z_c^{(\prime )}\) via final state interactions. arXiv:1811.04688 [hepph]
 58.Q. Wu, D.Y. Chen, X.J. Fan, G. Li, Eur. Phys. J. C 79(3), 265 (2019). arXiv:1902.05737 [hepph]ADSCrossRefGoogle Scholar
 59.E.S. Swanson, \(Z_b\) and \(Z_c\) exotic states as coupled channel cusps. Phys. Rev. D 91(3), 034009 (2015). arXiv:1409.3291 [hepph]ADSCrossRefGoogle Scholar
 60.A.P. Szczepaniak, Triangle singularities and XYZ quarkonium peaks. Phys. Lett. B 747, 410 (2015). arXiv:1501.01691 [hepph]ADSCrossRefGoogle Scholar
 61.E.S. Swanson, Cusps and exotic charmonia. Int. J. Mod. Phys. E 25(07), 1642010 (2016). arXiv:1504.07952 [hepph]ADSCrossRefGoogle Scholar
 62.D.V. Bugg, An explanation of belle states \(Z_b(10610)\) and \(Z_b(10650)\). EPL 96(1), 11002 (2011). arXiv:1105.5492 [hepph]ADSCrossRefGoogle Scholar
 63.X.H. Liu, M. Oka, Q. Zhao, Searching for observable effects induced by anomalous triangle singularities. Phys. Lett. B 753, 297 (2016). arXiv:1507.01674 [hepph]ADSCrossRefGoogle Scholar
 64.Y. Ikeda [HAL QCD Collaboration], The tetraquark candidate \(Z_c\)(3900) from dynamical lattice QCD simulations. J. Phys. G 45(2), 024002 (2018). arXiv:1706.07300 [heplat]
 65.Y. Ikeda et al. [HAL QCD Collaboration], Fate of the tetraquark candidate \(Z_c\)(3900) from lattice QCD. Phys. Rev. Lett. 117(24), 242001 (2016). arXiv:1602.03465 [heplat]
 66.D.A.S. Molnar, I. Danilkin, M. Vanderhaeghen, arXiv:1903.08458 [hepph]
 67.Y.S. Oh, T. Song, S.H. Lee, \(J / \psi \) absorption by pi and rho mesons in meson exchange model with anomalous parity interactions. Phys. Rev. C 63, 034901 (2001). arXiv:nuclth/0010064 ADSCrossRefGoogle Scholar
 68.R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio, G. Nardulli, Phenomenology of heavy meson chiral Lagrangians. Phys. Rep. 281, 145 (1997). arXiv:hepph/9605342 ADSCrossRefGoogle Scholar
 69.P. Colangelo, F. De Fazio, T.N. Pham, Nonfactorizable contributions in B decays to charmonium: the case of \(B^ \rightarrow K^ h_c\). Phys. Rev. D 69, 054023 (2004). arXiv:hepph/0310084 ADSCrossRefGoogle Scholar
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