# The Genuine Resonance of Full-Charm Tetraquarks

## Abstract

**:**

## 1. Introduction

## 2. Quark Model and Wave Functions

#### 2.1. The Chiral Quark Model

#### 2.2. The Wave Functions of $cc\overline{c}\overline{c}$ System

## 3. Results and Discussions

## 4. Summary

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Choi, S.-K.; Olsen S., L.; Abe, K.; Abe, T.; Adachi, I.; Ahn, B.; Aihara, H.; Akai, K.; Akatsu, H.; Akemoto, M.; et al. Observation of a narrow charmoniumlike state in exclusive B
^{±}→ K^{±}π^{+}π^{−}J/ψ decays. Phys. Rev. Lett.**2003**, 91, 262001. [Google Scholar] [CrossRef] [Green Version] - Aubert, B.; Barate1, R.; Boutigny, D.; Couderc, F.; Karyotakis, Y.; Lees, J.P.; Poireau, V.; Tisserand, V.; Zghiche, A.; Grauges, E.; et al. Observation of a broad structure in the π
^{+}π^{−}J/ψ mass spectrum around 4.26 GeV/c2. Phys. Rev. Lett.**2005**, 95, 142001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Yuan, C.Z.; Shen, C.P.; Wang, P.; McOnie, S.; Adachi, I.; Aihara, H.; Aulchenko, V.; Aushev, T.; Bahinipati, S.; Balagura, V.; et al. Measurement of the e
^{+}e^{−}→π^{+}π^{−}J/ψ cross section via initial-state radiation at Belle. Phys. Rev. Lett.**2007**, 99, 182004. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ablikim, M.; Achasov, M.N.; Ai, X.C.; Albayrak, O.; Ambrose, D.J.; An, F.F.; An, Q.; Bai, J.Z.; Ferroli, R.; Baldini; et al. Observation of a Charged Charmoniumlike Structure in e
^{+}e^{−}→π^{+}π^{−}J/ψ at s = 4.26 GeV. Phys. Rev. Lett.**2013**, 110, 252001. [Google Scholar] [CrossRef] [Green Version] - Liu, Z.Q.; Shen, C.P.; Yuan, C.Z.; Adachi, I.; Aihara, H.; Asner, D.M.; Aulchenko, V.; Aushev, T.; Aziz, T.; Bakich, A.M.; et al. Study of e
^{+}e^{−}→π^{+}π^{−}J/ψ and Observation of a Charged Charmoniumlike State at Belle. Phys. Rev. Lett.**2013**, 110, 252002. [Google Scholar] [CrossRef] [Green Version] - Xiao, T.; Dobbs, S.; Seth, K.K.; Tomaradze, A. Observation of the Charged Hadron Zc (3900) at sqrt (s) = 4170 MeV. Phys. Lett. B
**2013**, 727, 366. [Google Scholar] [CrossRef] [Green Version] - Ablikim, M.; Achasov, M.N.; Ai, X.C.; Albayrak, O.; Albrecht, M.; Ambrose, D.J.; Amoroso, A.; An, F.F.; An, Q.; Bai, J.Z.; et al. Observation of Zc (3900)
^{0}in e^{+}e^{−}→π^{0}π^{0}J/ψ. Phys. Rev. Lett.**2015**, 115, 112003. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bondar, A.; Garmash, A.; Mizuk, R.; Santel, D.; Kinoshita, K.; Adachi, I.; Aihara, H.; Arinstein, K.; Asner, D.M.; Aushev, T.; et al. Observation of Two Charged Bottomoniumlike Resonances in Υ (5 S) Decays. Phys. Rev. Lett.
**2012**, 108, 122001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Aaij, R.; Adeva, B.; Adinolfi, M.; Affolder, A.; Ajaltouni, Z.; Akar, S.; Albrecht, J.; Alessio, F.; Alexander, M.; Ali, S.; et al. Observation of J/ψ p Resonances Consistent with Pentaquark States in ${\Lambda}_{b}^{0}$ → J/ψ K- ρ Decays. Phys. Rev. Lett.
**2015**, 115, 072001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Aaij, R.; Adeva, B.; Adinolfi, M.; Adrover, C.; Affolder, A.; Ajaltouni, Z.; Albrecht, J.; Alessio, F.; Alexander, M.; Alkhazov, G.; et al. Observation of J/ψ-pair production in pp collisions at s= 7 TeV. Phys. Lett. B
**2012**, 707, 52. [Google Scholar] [CrossRef] - Aaij, R.; Adeva, B.; Adinolfi, M.; Ajaltouni, Z.; Akar, S.; Albrecht, J.; Alessio, F.; Alexander, M.; Ali, S.; Alkhazov, G.; et al. Measurement of the J/ψ pair production cross-section in pp collisions at s = 13⎷s = 13 TeV. J. High Energy Phys.
**2017**, 10, 068. [Google Scholar] [CrossRef] [Green Version] - Khachatryan, V.; Sirunyan, A.M.; Tumasyan, A.; Adam, W.; Bergauer, T.; Dragicevic, M.; Erö, J.; Fabjan, C.; Friedl, M.; Fruehwirth, R.; et al. Meaurement of prompt J/ψ pair production in pp colliion at ⎷s= 7 Te. J. High Energy Phys.
**2014**, 09, 094. [Google Scholar] [CrossRef] [Green Version] - Aaboud, M.; Aad, G.; Abbott, B.; Abdallah, J.; Abdinov, O.B.; Abeloos, B.; Abidi, S.H.; AbouZeid, H.; Abraham, N.L.; Abramowicz, H.; et al. Search for dark matter at ⎷s = 13 TeV in final states containing an energetic photon and large missing transverse momentum with the ATLAS detector. Eur. Phys. J. C
**2017**, 77, 76. [Google Scholar] [CrossRef] [Green Version] - Abe, K.; Abe, R.; Abe, T.; Adachi, I.; Ahn, B.S.; Aihara, H.; Akatsu, M.; Asano, Y.; Aso, T.; Aulchenko, V.; et al. Observation of Double c c Production in e
^{+}e^{−}Annihilation at ⎷s ≈ 10.6 GeV. Phys. Rev. Lett.**2002**, 89, 142001. [Google Scholar] [CrossRef] [Green Version] - Khachatryan, V.; Sirunyan, A.M.; Tumasyan, A.; Adam, W.; Asilar, E.; Bergauer, T.; Brandstetter, J.; Brondolin, E.; Dragicevic, M.; Erö, J.; et al. Observation of Y (1S) pair production in proton-proton collisions at ⎷s= 8 TeV. J. High Energy Phys.
**2017**, 1705, 013. [Google Scholar] [CrossRef] - Durgut, S. Search for Exotic Mesons at CMS. Available online: http://meetings.aps.org/Meeting/APR18/Session/U09.6 (accessed on 16 April 2018).
- Aaij, R.; Beteta, C.A. Observation of structure in the J/ψ -pair mass spectrum. Sci. Bull.
**2020**, 65, 1983–1993. [Google Scholar] - Zhu, R. Fully-heavy tetraquark spectra and production at hadron colliders. Nucl. Phys. B
**2021**, 966, 115393. [Google Scholar] - Zhang, J.R. 0
^{+}fully-charmed tetraquark states. Phys. Rev. D**2021**, 103, 014018. [Google Scholar] [CrossRef] - Chen, H.X.; Chen, W.; Liu, X.; Zhu, S.L. Strong decays of fully-charm tetraquarks into di-charmonia. Sci. Bull.
**2020**, 65, 1994–2000. [Google Scholar] [CrossRef] - Becchi, C.; Ferretti, J.; Giachino, A.; Maiani, L.; Santopinto, E. A study of ccc¯c¯ tetraquark decays in 4 muons and in D
^{(*)}D¯^{(*)}at LHC. Phys. Lett. B**2020**, 811, 135952. [Google Scholar] [CrossRef] - Weng, X.Z.; Chen, X.L.; Deng, W.Z.; Zhu, S.L. Systematics of fully heavy tetraquarks. Phys. Rev. D
**2021**, 103, 034001. [Google Scholar] [CrossRef] - Zhao, J.; Shi, S.; Zhuang, P. Fully-heavy tetraquarks in a strongly interacting medium. Phys. Rev. D
**2020**, 102, 114001. [Google Scholar] [CrossRef] - Lü, Q.F.; Chen, D.Y.; Dong, Y.B. Masses of fully heavy tetraquarks QQQ¯ Q¯ in an extended relativized quark model. Eur. Phys. J. C
**2020**, 80, 871. [Google Scholar] [CrossRef] - Bedolla, M.A.; Ferretti, J.; Roberts, C.D.; Santopinto, E. Spectrum of fully-heavy tetraquarks from a diquark+antidiquark perspective. Eur. Phys. J. C
**2020**, 80, 1004. [Google Scholar] [CrossRef] - Iwasaki, Y. A possible model for new resonances: Exotics and hidden charm. Prog. Theor. Phys.
**1975**, 54, 492. [Google Scholar] [CrossRef] [Green Version] - Heller, L.; Tjon, J.A. Bound states of heavy Q
^{2}Q^{−2}systems. Phys. Rev. D**1985**, 32, 755. [Google Scholar] [CrossRef] - Lloyd, R.J.; Vary, J.P. All-charm tetraquarks. Phys. Rev. D
**2004**, 70, 014009. [Google Scholar] [CrossRef] [Green Version] - Berezhnoy, A.V.; Luchinsky, A.V.; Novoselov, A.A. Heavy tetraquarks production at the LHC. Phys. Rev. D
**2012**, 86, 034004. [Google Scholar] [CrossRef] [Green Version] - Chen, W.; Chen, H.X.; Liu, X.; Steele, T.G.; Zhu, S.L. Hunting for exotic doubly hidden-charm/bottom tetraquark states. Phys. Lett. B
**2017**, 773, 247. [Google Scholar] [CrossRef] - Bai, Y.; Lu, S.; Osborne, J. Beauty-full tetraquarks. Phys. Lett. B
**2019**, 798, 134930. [Google Scholar] [CrossRef] - Esposito, A.; Polosa, A.D. A $bb\overline{b}\overline{b}$ di-bottomonium at the LHC? Eur. Phys. J. C
**2018**, 78, 782. [Google Scholar] [CrossRef] - Debastiani, V.R.; Navarra, F.S. A non-relativistic model for the tetraquark. Chin. Phys. C
**2019**, 43, 013105. [Google Scholar] [CrossRef] - Chao, K.T. The cccc (diquark-antidiquark) states in e
^{+}e^{−}annihilation. Z. Phys. C**1981**, 7, 317. [Google Scholar] [CrossRef] - Ader, J.P.; Richard, J.M.; Taxil, P. Do narrow heavy multiquark states exist? Phys. Rev. D
**1982**, 25, 2370. [Google Scholar] [CrossRef] [Green Version] - Barnea, N.; Vijande, J.; Valcarce, A. Four-quark spectroscopy within the hyperspherical formalism. Phys. Rev. D
**2006**, 73, 054004. [Google Scholar] [CrossRef] [Green Version] - Wang, Z.G. Analysis of the QQQ¯ Q¯ tetraquark states with QCD sum rules. Eur. Phys. J. C
**2017**, 77, 432. [Google Scholar] [CrossRef] - Karliner, M.; Nussinov, S.; Rosner, J.L. QQQ¯ Q¯ states: Masses, production, and decays. Phys. Rev. D
**2017**, 95, 034011. [Google Scholar] [CrossRef] [Green Version] - Wu, J.; Liu, Y.R.; Chen, K.; Liu, X.; Zhu, S.L. Heavy-flavored tetraquark states with the QQQ¯ Q¯ configuration. Phys. Rev. D
**2018**, 97, 094015. [Google Scholar] [CrossRef] [Green Version] - Liu, M.S.; Lü, Q.F.; Zhong, X.H.; Zhao, Q. All-heavy tetraquarks. Phys. Rev. D
**2019**, 100, 016006. [Google Scholar] [CrossRef] [Green Version] - Gordillo, M.C.; Soto, F.D.; Segovia, J. Diffusion Monte Carlo calculations of fully-heavy multiquark bound states. Phys. Rev. D
**2020**, 102, 114007. [Google Scholar] [CrossRef] - Yang, G.; Ping, J.L.; He, L.Y.; Wang, Q. A potential model prediction of fully-heavy tetraquarks QQQ¯ Q¯ (Q = c, b). arXiv
**2006**, arXiv:2006.13756v3. [Google Scholar] - Yang, G.; Ping, J.L.; Segovia, J. Tetra- and Penta-Quark Structures in the Constituent Quark Model. Symmetry
**2020**, 12, 1869. [Google Scholar] [CrossRef] - Chen, X. Analysis of hidden-bottom $bb\overline{b}\overline{b}$ states. Eur. Phys. J. A
**2019**, 55, 106. [Google Scholar] [CrossRef] - Jin, X.; Xue, Y.Y.; Huang, H.X.; Ping, J.L. Full-heavy tetraquarks in constituent quark models. arXiv
**2006**, arXiv:2006.13745. [Google Scholar] - Simon, J. Resonance state lifetimes from stabilization graphs. J. Chem. Phys.
**1981**, 75, 2465. [Google Scholar] [CrossRef] - Hiyama, E.; Kamimura, M.; Hosaka, A.; Toki, H.; Yahiro, M. Five-body calculation of resonance and scattering states of pentaquark system. Phys. Lett. B
**2006**, 633, 237. [Google Scholar] [CrossRef] [Green Version] - Hiyama, E.; Hosaka, A.; Oka, M.; Richard, J. Quark model estimate of hidden-charm pentaquark resonances. Phys. Rev. C
**2018**, 98, 045208. [Google Scholar] [CrossRef] [Green Version] - Myo, T.; Kikuchi, Y.; Masui, H.; Kato, K. Scattering phase shifts in the complex scaling method. Prog. Part. Nucl. Phys.
**2014**, 79, 1. [Google Scholar] [CrossRef] [Green Version] - Kurtz, H.A.; Öhrn, Y. Calculation of
^{2}p shape resonances in Be and Mg. Phys. Rev. A**1979**, 19, 43. [Google Scholar] [CrossRef] - Huang, H.X.; Ping, J.L. Investigating the hidden-charm and hidden-bottom pentaquark resonances in scattering process. Phys. Rev. D
**2019**, 99, 014010. [Google Scholar] [CrossRef] [Green Version] - Chen, X.Y.; Ping, J.L.; Roberts, C.D.; Segovia, J. Light-meson masses in an unquenched quark model. Phys. Rev. D
**2018**, 97, 094016. [Google Scholar] [CrossRef] [Green Version] - Vijande, J.; Fernández, F.; Valcarce, A. Constituent quark model study of the meson spectra. J. Phys. G.
**2005**, 31, 481. [Google Scholar] [CrossRef] [Green Version] - Chen, X.Y.; Ping, J.L. Looking for a $\mathrm{u}\overline{d}\overline{s}\overline{b}$ bound state in the chiral quark model. Phy. Rev. D
**2018**, 98, 054022. [Google Scholar] [CrossRef] [Green Version] - Hiyama, E.; Kino, Y.; Kamimura, M. Gaussian expansion method for few-body systems. Prog. Part. Nucl. Phys.
**2003**, 51, 223. [Google Scholar] [CrossRef] - Abazov, V.M.; Abbott, B.; Acharya, B.S.; Adams, M.; Adams, T.; Agnew, J.P.; Alexeev, G.D.; Alkhazov, G.; Alton, A.; Askew, A.; et al. Evidence for simultaneous production of J/ψ and Υ mesons. Phys. Rev. Lett.
**2016**, 116, 082002. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**The stabilization plots for the energies of ${\left({\eta}_{c}{\eta}_{c}\right)}^{0}$ state of ${J}^{PC}={0}^{++}$ with respect to the scaling factor $\alpha $. The horizontal axis shows the value of $\alpha $ and the vertical axis represents the energy of the states, which is the same for the following figures.

**Figure 2.**The stabilization plots for the energies of the ${\left({\eta}_{c}^{8}{\eta}_{c}^{8}\right)}^{0}$ states for ${J}^{PC}={0}^{++}$ with respect to the scaling factor $\alpha $.

**Figure 3.**The stabilization plots for the energies of the ${(J/\psi J/\psi )}^{0}$ states for ${J}^{PC}={0}^{++}$ with respect to the scaling factor $\alpha $.

**Figure 4.**The stabilization plots for the energies of the states for ${J}^{PC}={0}^{++}$ in the full hidden-color channels coupling (${\left({\eta}_{c}^{8}{\eta}_{c}^{8}\right)}^{0}$+${(J/{\psi}^{8}J/{\psi}^{8})}^{0}$+${\left(DA\right)}_{1}^{0}$+${\left(DA\right)}_{2}^{0}$) with respect to the scaling factor $\alpha $.

**Figure 5.**The stabilization plots for the energies of the states for ${J}^{PC}={0}^{++}$ obtained by considering the coupling between hidden-color channels ${\left({\eta}_{c}^{8}{\eta}_{c}^{8}\right)}^{0}$+${(J/{\psi}^{8}J/{\psi}^{8})}^{0}$ and the two scattering states ${\left({\eta}_{c}{\eta}_{c}\right)}^{0}$+${(J/\psi J/\psi )}^{0}$ with respect to the scaling factor $\alpha $. The red horizontal line represents the resonance states and the blue horizontal line represents the threshold, which is the same for the following figures.

**Figure 6.**The stabilization plots for the energies of the states for ${J}^{PC}={0}^{++}$ obtained by considering the coupling between two diquark–antidiquark structures ${\left(DA\right)}_{1}^{0}$+${\left(DA\right)}_{2}^{0}$ and the two scattering states ${\left({\eta}_{c}{\eta}_{c}\right)}^{0}$+${(J/\psi J/\psi )}^{0}$ with respect to the scaling factor $\alpha $.

**Figure 7.**The stabilization plots for the energies of the states for ${J}^{PC}={0}^{++}$ obtained by considering the coupling between all hidden-color channels and the color-singlet channel ${\left({\eta}_{c}{\eta}_{c}\right)}^{0}$ with respect to the scaling factor $\alpha $.

**Figure 8.**The stabilization plots for the energies of the states for ${J}^{PC}={0}^{++}$ obtained by considering the coupling between all hidden-color channels and the color-singlet channel ${(J/\psi J/\psi )}^{0}$ with respect to the scaling factor $\alpha $.

**Figure 9.**The stabilization plots for the energies of the states for ${J}^{PC}={0}^{++}$ obtained by considering the full-channel couplings with respect to the scaling factor $\alpha $.

**Figure 10.**The stabilization plots for the energies of the states for ${J}^{PC}={1}^{+-}$ obtained by considering the full hidden-channels coupling with respect to the scaling factor $\alpha $.

**Figure 11.**The stabilization plots for the energies of the states for ${J}^{PC}={1}^{+-}$ obtained by considering the full-channel couplings with respect to the scaling factor $\alpha $. Upper for energies 6000–7000 MeV and down for energies 7000–7300 MeV.

**Figure 12.**The stabilization plots for the energies of the states for ${J}^{PC}={2}^{++}$ obtained by considering the full hidden-channels coupling with respect to the scaling factor $\alpha $.

**Figure 13.**The stabilization plots for the energies of the states for ${J}^{PC}={2}^{++}$ obtained by considering the full-channel couplings with respect to the scaling factor $\alpha $. Upper for energies 6100–7000 MeV and down for energies 7000–7300 MeV.

Quark masses | ${m}_{u}={m}_{d}$ | 313 |

(MeV) | ${m}_{s}$ | 536 |

${m}_{c}$ | 1728 | |

${m}_{b}$ | 5112 | |

Confinement | ${a}_{c}$ (MeV fm${}^{-2}$) | 101 |

$\Delta $ (MeV) | −78.3 | |

OGE | ${\alpha}_{0}$ | 3.67 |

${\Lambda}_{0}\left({\mathrm{fm}}^{-1}\right)$ | 0.033 | |

${\mu}_{0}$ (MeV) | 36.98 | |

${s}_{0}$ (MeV) | 28.17 |

**Table 2.**The masses of some heavy mesons (unit: MeV). ${M}_{cal}$ and ${M}_{exp}$ represent the theoretical and experimental masses, respectively.

Meson | ${\mathit{\eta}}_{\mathit{c}}$ | $\mathit{J}/\mathit{\psi}$ | ${\mathit{h}}_{\mathit{c}}$ | ${\mathit{\chi}}_{\mathit{c}0}$ | ${\mathit{\chi}}_{\mathit{c}1}$ | ${\mathit{\chi}}_{\mathit{c}2}$ |
---|---|---|---|---|---|---|

${M}_{cal}$ | 2986.3 | 3096.4 | 3417.3 | 3416.4 | 3416.4 | 3416.4 |

${M}_{exp}$ | $2983.4$ | $3096.9$ | $3525.4$ | $3414.8$ | $3510.7$ | $3556.2$ |

**Table 3.**The possible channels and corresponding wave functions for $cc\overline{c}\overline{c}$ tetraquarks with quantum numbers ${J}^{PC}={0}^{++},{1}^{+-},{2}^{++}$. The channels $\left(DA\right)$ represent the diquark–antidiquark structures and the superscripts of the channels represent the total angular momentum J.

${\mathit{J}}^{\mathit{PC}}={0}^{++}$ | ${\mathit{J}}^{\mathit{PC}}={1}^{+-}$ | ${\mathit{J}}^{\mathit{PC}}={2}^{++}$ | |||
---|---|---|---|---|---|

Channels | Wave Functions | Channels | Wave Functions | Channels | Wave Functions |

${\left({\eta}_{c}{\eta}_{c}\right)}^{0}$ | ${\chi}_{00}^{\sigma 1}{\chi}_{m0}^{f}{\chi}_{m}^{c1}$ | ${({\eta}_{c}J/\psi )}^{1}$ | ${\chi}_{11}^{\sigma 3}{\chi}_{m0}^{f}{\chi}_{m}^{c1}$ | ${(J/\psi J/\psi )}^{2}$ | ${\chi}_{22}^{\sigma 6}{\chi}_{m0}^{f}{\chi}_{m}^{c1}$ |

${\left({\eta}_{c}^{8}{\eta}_{c}^{8}\right)}^{0}$ | ${\chi}_{00}^{\sigma 1}{\chi}_{m0}^{f}{\chi}_{m}^{c2}$ | ${({\eta}_{c}^{8}J/{\psi}^{8})}^{1}$ | ${\chi}_{11}^{\sigma 3}{\chi}_{m0}^{f}{\chi}_{m}^{c2}$ | ${(J/{\psi}^{8}J/{\psi}^{8})}^{2}$ | ${\chi}_{22}^{\sigma 6}{\chi}_{m0}^{f}{\chi}_{m}^{c2}$ |

${(J/\psi J/\psi )}^{0}$ | ${\chi}_{00}^{\sigma 2}{\chi}_{m0}^{f}{\chi}_{m}^{c1}$ | ${(J/\psi {\eta}_{c})}^{1}$ | ${\chi}_{11}^{\sigma 4}{\chi}_{m0}^{f}{\chi}_{m}^{c1}$ | ${\left(DA\right)}^{2}$ | ${\chi}_{22}^{\sigma 6}{\chi}_{d0}^{f}{\chi}_{d}^{c1}$ |

${(J/{\psi}^{8}J/{\psi}^{8})}^{0}$ | ${\chi}_{00}^{\sigma 2}{\chi}_{m0}^{f}{\chi}_{m}^{c2}$ | ${(J/{\psi}^{8}{\eta}_{c}^{8})}^{1}$ | ${\chi}_{11}^{\sigma 4}{\chi}_{m0}^{f}{\chi}_{m}^{c2}$ | ||

${\left(DA\right)}_{1}^{0}$ | ${\chi}_{00}^{\sigma 2}{\chi}_{d0}^{f}{\chi}_{d}^{c1}$ | ${\left(DA\right)}^{1}$ | ${\chi}_{11}^{\sigma 5}{\chi}_{d0}^{f}{\chi}_{d}^{c1}$ | ||

${\left(DA\right)}_{2}^{0}$ | ${\chi}_{00}^{\sigma 1}{\chi}_{d0}^{f}{\chi}_{d}^{c2}$ |

**Table 4.**The low-lying energies of the $cc\overline{c}\overline{c}$ states with ${J}^{PC}={0}^{++}$, ${1}^{+-}$, ${2}^{++}$ in single channel and full-channel coupling calculations. ${E}_{\mathrm{th}}^{\mathrm{theo}}$ and ${E}_{\mathrm{th}}^{\mathrm{exp}}$ represent the theoretical and experimental thresholds, respectively (unit: MeV).

Channel | E | ${\mathit{E}}_{\mathit{th}}^{\mathbf{theo}}$ | ${\mathit{E}}_{\mathit{th}}^{\mathbf{exp}}$ |
---|---|---|---|

${J}^{PC}={0}^{++}$ | |||

${\left({\eta}_{c}{\eta}_{c}\right)}^{0}$ | 5973.4 | 5972.6 | 5966.8 |

${\left({\eta}_{c}^{8}{\eta}_{c}^{8}\right)}^{0}$ | 6373.2 | ||

${(J/\psi J/\psi )}^{0}$ | 6193.7 | 6192.8 | 6193.8 |

${(J/{\psi}^{8}J/{\psi}^{8})}^{0}$ | 6356.9 | ||

${\left(DA\right)}_{1}^{0}$ | 6360.2 | ||

${\left(DA\right)}_{2}^{0}$ | 6390.9 | ||

full-channel coupling | 5973.4 | 5972.6 | 5966.8 |

${J}^{PC}={1}^{+-}$ | |||

${({\eta}_{c}J/\psi )}^{1}$ | 6083.6 | 6082.7 | 6080.3 |

${({\eta}_{c}^{8}J/{\psi}^{8})}^{1}$ | 6349.8 | ||

${(J/\psi {\eta}_{c})}^{1}$ | 6083.6 | 6082.7 | 6080.3 |

${(J/{\psi}^{8}{\eta}_{c}^{8})}^{1}$ | 6349.8 | ||

${\left(DA\right)}^{1}$ | 6397.6 | ||

full-channel coupling | 6083.6 | 6082.7 | 6080.3 |

${J}^{PC}={2}^{++}$ | |||

${(J/\psi J/\psi )}^{2}$ | 6193.7 | 6192.8 | 6193.8 |

${(J/{\psi}^{8}J/{\psi}^{8})}^{2}$ | 6365.3 | ||

${\left(DA\right)}^{2}$ | 6410.4 | ||

full-channel coupling | 6193.7 | 6192.8 | 6193.8 |

**Table 5.**The energy spectra for the $cc\overline{c}\overline{c}$ states with ${J}^{PC}={0}^{++}$ in the single hidden-color channel and hidden-color channels coupling (unit: MeV).

Channel | ${\left({\mathit{\eta}}_{\mathit{c}}^{8}{\mathit{\eta}}_{\mathit{c}}^{8}\right)}^{0}$ | ${(\mathit{J}/{\mathit{\psi}}^{8}\mathit{J}/{\mathit{\psi}}^{8})}^{0}$ | ${\left({\mathit{\eta}}_{\mathit{c}}^{8}{\mathit{\eta}}_{\mathit{c}}^{8}\right)}^{0}$+${(\mathit{J}/{\mathit{\psi}}^{8}\mathit{J}/{\mathit{\psi}}^{8})}^{0}$ | ${\left(\mathbf{DA}\right)}_{1}^{0}$ | ${\left(\mathbf{DA}\right)}_{2}^{0}$ | ${\left(\mathbf{DA}\right)}_{1}^{0}$+${\left(\mathbf{DA}\right)}_{2}^{0}$ | ${\left({\mathit{\eta}}_{\mathit{c}}^{8}{\mathit{\eta}}_{\mathit{c}}^{8}\right)}^{0}$+${(\mathit{J}/{\mathit{\psi}}^{8}\mathit{J}/{\mathit{\psi}}^{8})}^{0}$+${\left(\mathbf{DA}\right)}_{1}^{0}$+${\left(\mathbf{DA}\right)}_{2}^{0}$ |
---|---|---|---|---|---|---|---|

Energy | 6371 | 6355 | 6328 | 6388 | 6358 | 6344 | 6253 |

6708 | 6717 | 6402 | 6756 | 6690 | 6400 | 6316 | |

6973 | 6951 | 6657 | 6841 | 6763 | 6683 | 6528 | |

7036 | 7065 | 6765 | 6848 | 6974 | 6759 | 6577 | |

7140 | 7142 | 6944 | 7149 | 7022 | 6763 | 6714 | |

7301 | 7258 | 6978 | 7196 | 7063 | 6840 | 6763 | |

7357 | 7393 | 6987 | 7200 | 7102 | 6849 | 6805 | |

7448 | 7453 | 7110 | 7262 | 7268 | 6972 | 6849 | |

7507 | 7490 | 7130 | 7266 | 7326 | 7021 | 6877 | |

7613 | 7577 | 7154 | 7281 | 7329 | 7063 | 6886 |

**Table 6.**The resonance states with quantum numbers ${J}^{PC}={0}^{++},{1}^{+-},{2}^{++}$ for the $cc\overline{c}\overline{c}$ system (unit: MeV).

State | ${0}^{++}$ | ${1}^{+-}$ | ${2}^{++}$ |
---|---|---|---|

Energy | 6763 | 6383 | 6855 |

6849 | 6749 | 7085 | |

6884 | 6793 | 7139 | |

7063 | 6828 | 7204 | |

7098 | 6835 | 7270 | |

7138 | 7100 | ||

7156 | 7112 | ||

7200 | 7173 | ||

7185 | |||

7247 | |||

7249 | |||

7264 |

**Table 7.**The distances between c and $c\left(\overline{c}\right)$ quark for the possible resonance states of the $cc\overline{c}\overline{c}$ system.

State | Resonance (MeV) | ${\mathit{R}}_{\mathit{c}\overline{\mathit{c}}}$ (fm) | ${\mathit{R}}_{\mathit{cc}}$ (fm) |
---|---|---|---|

${0}^{++}$ | 6763 | 0.68 | 0.82 |

6849 | 0.66 | 0.70 | |

6884 | 0.80 | 0.85 | |

7063 | 0.80 | 1.01 | |

7098 | 0.84 | 1.03 | |

7138 | 0.96 | 0.98 | |

7156 | 0.88 | 0.81 | |

7200 | 0.83 | 0.73 | |

${1}^{+-}$ | 6383 | 0.85 | 0.88 |

6749 | 1.20 | 1.39 | |

6793 | 0.52 | 0.48 | |

6828 | 0.75 | 0.97 | |

6835 | 1.35 | 1.31 | |

7100 | 1.75 | 2.42 | |

7112 | 1.00 | 1.26 | |

7173 | 1.13 | 1.51 | |

7185 | 1.55 | 1.66 | |

7247 | 1.59 | 1.54 | |

7249 | 1.60 | 1.50 | |

7264 | 1.58 | 1.49 | |

${2}^{++}$ | 6855 | 0.66 | 0.71 |

7085 | 0.94 | 0.74 | |

7139 | 0.88 | 0.80 | |

7204 | 0.84 | 0.72 | |

7270 | 0.76 | 0.85 |

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**MDPI and ACS Style**

Chen, X.
The Genuine Resonance of Full-Charm Tetraquarks. *Universe* **2021**, *7*, 155.
https://doi.org/10.3390/universe7050155

**AMA Style**

Chen X.
The Genuine Resonance of Full-Charm Tetraquarks. *Universe*. 2021; 7(5):155.
https://doi.org/10.3390/universe7050155

**Chicago/Turabian Style**

Chen, Xiaoyun.
2021. "The Genuine Resonance of Full-Charm Tetraquarks" *Universe* 7, no. 5: 155.
https://doi.org/10.3390/universe7050155