Where are the hidden-charm hexaquarks?

Zhe Liu^1,2 [email protected] Hong-Tao An^1,2 [email protected] Zhan-Wei Liu^1,2,3 [email protected] Xiang Liu^1,2,3 [email protected] ¹School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
²Research Center for Hadron and CSR Physics, Lanzhou University and Institute of Modern Physics of CAS, Lanzhou 730000, China
³Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, and Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, China

Abstract

In this work, we carry out the study of hidden-charm hexaquark states with the typical configurations $qqc\bar{q}\bar{q}\bar{c}$ ( $q=u,d,s$ ). The mass spectra of hidden-charm hexaquark states are obtained within the chromo-magnetic interaction model. In addition to the mass spectra analysis, we further illustrate their two-body strong decay behaviors. There exist some compact bound states which cannot decay through the strong interaction. Hopefully our results will help to search for such types of the exotic states in the future experiments.

I Introduction

With the improvement of the luminosity and precision in experiment, more and more charmonium-like $XYZ$ states and $P_{c}$ states have been observed Choi:2003ue ; Acosta:2003zx ; Abazov:2004kp ; Aaij:2014jqa ; Ablikim:2016qzw ; Ablikim:2017oaf ; Ablikim:2020hsk ; BESIII:2016adj ; Aaij:2015tga ; Aaij:2016phn ; Aaij:2019vzc . The present situation of hadronic states is far beyond the conventional quark model. The first doubly charm tetraquark $T_{\rm cc}^{+}$ with the configuration $cc\bar{u}\bar{d}$ was observed by the LHCb Collaboration Franz:2021talk , and this newly discovered particle is explicitly an exotic state which cannot be classified into the conventional mesons.

The hexaquark states were proposed and the spectra of light-flavored hexaquarks were dynamically investigated very early after the birth of quark model. The $d^{*}(2380)$ resonance with $I(J^{P})=0(3^{+})$ has been reported by CELSIUS/WASA and WASA-at-COSY Collaborations Faldt:2011zv ; Adlarson:2011bh ; Adlarson:2012fe , and it is expected to be a dibaryon which contains 6 constituent quarks. The deuteron is also a dibaryon. Jaffe firstly found the $H$ particle whose hyperfine interaction is much larger than that for two separated $\Lambda$ baryons within the chromo-magnetic interaction model Jaffe:1976yi , and this dibaryon $uuddss$ was also studied within other framework Mackenzie:1985vv ; Aerts:1984vv ; Balachandran:1983dj ; Straub:1988mz ; Paganis:1999ux ; Yost:1985mj ; Rosner:1985yh ; Karl:1987cg . Moreover, the heavy dibaryons ( $qqqqqQ$ ) Oka:2013iua ; Gerasyuta:2011yg ; Oka:2019mrd ; Liu:2012zzo ; Pepin:1998ih , doubly-heavy dibaryons ( $qqqqQQ$ ) Liu:2012zzo ; Vijande:2016nzk ; Wang:2017sto ; Meng:2017fwb ; Meguro:2011nr ; Li:2012bt ; Leandri:1995zm , triply-heavy dibaryons ( $qqqQQQ$ ) Wang:2020jqu ; Chen:2018pzd ; Richard:2020zxb , the other fully light dibaryons ( $qqqqqq$ ) Zhang:1997ny ; Gerasyuta:2010hn ; SilvestreBrac:1992yg ; Park:2015nha ; Oka:1988yq ; Chen:2019vdh , and even fully heavy dibaryons ( $QQQQQQ$ ) Huang:2020bmb were also proposed and discussed.

The hadronic states composed of three quarks and three antiquarks are another class of heaxquarks. The hidden-charm and hidden-bottom hexaquarks are especially focused on since they have much larger masses and thus are more easily distinguished from the ordinary mesons. With the hidden-charm tetraquark and pentaquark states observed in experiment, the discovery of hidden-charm hexaquarks would also come true in future.

Very recently, BESIII collaboration measured the cross section of the process $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}\psi(3686)$ and further confirms the existence of three charmonium-like states wherein $Y(4660)$ is closed to the threshold of $\Lambda_{c}$ - $\bar{\Lambda}_{c}$ systems BESIII:2021njb . Before this, the structure $Y(4660)$ has been observed in the process of $e^{+}e^{-}\rightarrow\gamma_{\text{ISR}}\pi^{+}\pi^{-}\psi(3686)$ in the Belle and BarBar experiments Belle:2007umv ; Belle:2014wyt ; BaBar:2012hpr . $Y(4660)$ was interpreted as a higher charmonium in Ref. Wang:2020prx and a hexaquark state configured by the triquark-antitriquark clusters in Ref. Qiao:2007ce . The charmonium states can very likely be bound inside light hadronic matters, and such hadro-charmonium may explain the properties of the $Y(4660)$ peak Dubynskiy:2008mq . G. Cotugno et al. suggested that the two observations of $Y(4660)$ and $Y(4630)$ are likely to be due to the same state constituted by four quarks in Ref. Cotugno:2009ys .

The $\Lambda_{c}$ - $\bar{\Lambda}_{c}$ structure was introduced to explain the production and decays of $Y(4260)$ in Refs. Qiao:2007ce ; Qiao:2005av ; Chen:2011cta . $Y(4630)$ was observed in process $e^{+}e^{-}\rightarrow\Lambda_{c}\bar{\Lambda}_{c}$ in the Belle experiments Belle:2008xmh and is considered as a candidate of $\Lambda_{c}\bar{\Lambda}_{c}$ bound state Lee:2011rka . Especially, heavy baryon chiral perturbation theory was applied to systemically study the $\Lambda_{c}$ - $\bar{\Lambda}_{c}$ , $\Sigma_{c}$ - $\bar{\Sigma}_{c}$ , and $\Lambda_{b}$ - $\bar{\Lambda}_{b}$ systems Chen:2013sba , and the results suggest that $Y(4260)$ and $Y(4360)$ could be $\Lambda_{c}$ - $\bar{\Lambda}_{c}$ baryonia. The two states are also suggested to be a mixture, with mixing close to maximal, of two states of hadrochamonium Li:2013ssa .

The masses of baryonia with the open and hidden charm, bottomness and strangeness are studied in the framework of dispersion relation technique in Refs. Gerasyuta:2013esc ; Gerasyuta:2020gyy ; Gerasyuta:2020fii . The heavy baryon-antibaryon molecule states are investigated within the effective field theory Lu:2017dvm . The hidden-charm and hidden-bottom hexaquark states were discussed within the QCD sum rules Wan:2019ake ; Chen:2016ymy .

These work stimulate us to further study the hidden-charm hexaquark states. In this work we systemically investigate their mass spectra, stability, and two-body decay within the chromo-magnetic interaction (CMI) model.

The simple chromo-magnetic interaction arises from the one-gluon-exchange potential and further causes the mass splittings DeRujula:1975qlm ; Liu:2019zoy . The CMI model has been successfully adopted to study the mass spectra and stability of multiquark states Luo:2017eub ; Wu:2016gas ; Wu:2018xdi ; Chen:2016ont ; Wu:2016vtq ; Liu:2016ogz ; Wu:2017weo ; Zhou:2018pcv ; Li:2018vhp ; An:2019idk ; Cheng:2020irt ; Cheng:2019obk ; Hogaasen:2013nca ; Weng:2018mmf ; Weng:2019ynva ; Weng:2020jao ; Cheng:2020nho ; An:2020jix ; Karliner:2016zzc ; Weng:2021hje ; Zhao:2014qva . The method can catch the basic features of hadron spectra, since the mass splittings between hadrons reflect the basic symmetries of their inner structures.

This paper is organized as follows. In Sec. II, the adopted CMI model and relevant parameters are introduced. We construct the flavor $\otimes$ color $\otimes$ spin wavefunctions for the $S$ -wave hidden-charm hexaquark system in Sec. III, and study the mass spectrum and the two-body decays through the strong interaction in Sec. IV. A short summary follows in Sec. V.

II THE Hamiltonian in the CMI model

In the CMI model, the Hamiltonian has a simple form

	$\displaystyle H$	$\displaystyle=$	$\displaystyle\sum_{i}^{6}m_{i}+H_{\rm CMI},$
	$\displaystyle H_{\rm CMI}$	$\displaystyle=$	$\displaystyle-\sum_{i<j}C_{ij}\bm{\lambda}_{i}\cdot\bm{\lambda}_{j}\bm{\sigma}_{i}\cdot\bm{\sigma}_{j},$		(1)

where $m_{i}$ is the effective mass of the $i$ -th constituent (anti) quark, and $\bm{\lambda}_{i}$ and $\bm{\sigma}_{i}$ are Gell-Mann and Pauli matrices, respectively. For the antiquark, $\bm{\lambda}_{\bar{q}}=-\bm{\lambda}_{q}^{*}$ and $\bm{\sigma}_{\bar{q}}=\bm{\sigma}_{q}^{*}$ . The dynamical effect of spatial wavefunctions plays an important role in the study of hadron spectrum. Chromomagnetic interaction is nonrelativistic in the Schr $\rm\ddot{o}$ dinger equation in Ref. Godfrey:1985xj wherein the authors used the spatial wave functions with harmonic-oscillator expansion. The $C_{ij}$ is effective coupling constant between the $i$ -th (anti) quark and $j$ -th (anti) quark

C_{ij}=\frac{\pi\Braket{\alpha_{s}(r)\delta^{3}(\bm{r})}}{6m_{i}m_{j}},

(2)

which is directly related to the spatial wavefunctions and the constituent quark masses. We focus on ground states in $S$ -wave, and we simply suppose it does not change for various hexaquark systems.

Høgaasen et al. found out that the $b$ quark mass in bottomonium is much lighter than the one in the heavy-light system, and introduced the color interaction (the spin-independent color Coulomb-like terms in the one-gluon-exchange interactions) in Refs. Hogaasen:2013nca ; Karliner:2016zzc ; Weng:2018mmf . We also introduce a color term into our model Refs. Hogaasen:2013nca ; Weng:2018mmf

H_{\text{C}}=-\sum_{i<j}A_{ij}\bm{\lambda}_{i}\cdot\bm{\lambda}_{j}.

(3)

The nonvanishing color interaction coefficient $A_{ij}$ implies a change of the effective masses. We can rewrite the CMI Hamiltonian as Ref. Weng:2018mmf

\displaystyle H=-\frac{3}{4}\sum_{i<j}m_{ij}\bm{\lambda}_{i}\cdot\bm{\lambda}_{j}-\sum_{i<j}v_{ij}\bm{\lambda}_{i}\cdot\bm{\lambda}_{j}\bm{\sigma}_{i}\cdot\bm{\sigma}_{j},

(4)

where

\displaystyle m_{ij}=\frac{1}{4}\left(m_{i}+m_{j}\right)+\frac{4}{3}A_{ij}.

(5)

To estimate the mass spectra of the hidden-charm hexaquark states, we extract the effective coupling parameters $m_{ij}$ and $v_{ij}$ from the conventional hadron masses Weng:2018mmf . In the present work, $v_{q\bar{q}}$ and $m_{q\bar{q}}$ are only determined by vector mesons ( $q=n,s$ and $n=u,d$ ). We present the obtained effective coupling parameters in Table 1.

Table 1: The effective coupling parameters in units of MeV.

$m_{nn}$	$m_{ns}$	$m_{ss}$	$m_{nc}$	$m_{n\bar{n}}$	$m_{n\bar{s}}$	$m_{s\bar{s}}$	$m_{n\bar{c}}$	$m_{c\bar{c}}$
182.2	226.7	262.3	520.0	166.49	204.2	241.1	493.3	767.1
$v_{nn}$	$v_{ns}$	$v_{ss}$	$v_{nc}$	$v_{n\bar{n}}$	$v_{n\bar{s}}$	$v_{s\bar{s}}$	$v_{n\bar{c}}$	$v_{c\bar{c}}$
19.1	13.3	12.2	3.9	20.5	14.2	10.3	6.6	5.3

III The wavefunctions

In order to calculate the CMI Hamiltonian, we need to exhaust all the possible spin and color wavefunctions of hexaquark states and combine them with the corresponding flavor wavefunctions. The constructed flavor-color-spin wavefunctions should be fully antisymmetric when exchanging identical quarks because of Pauli principle. The wavefunctions do not change with different sets of basis, and we use the $|[(q_{1}q_{2})c][(\bar{q}_{3}\bar{q}_{4})\bar{c}]\rangle$ basis to construct the hidden-charm hexaquarks wavefunctions.

Firstly, we discuss the flavor wavefunctions. The mass hierarchy for $c$ , $s$ and $ud$ quarks is obvious and we neglect the mixing effect among the $c\bar{c}$ , $s\bar{s}$ , and $n\bar{n}$ pairs. Based on these, we list all the possible flavor combinations for the hidden-charm hexaquark system in Table 2.

In Table 2, the three subsystems of the first line are pure neutral particles and $C$ parity is “good” quantum number. For the six subsystems of the second line, every subsystem has a charge conjugation anti-partner, thus they have the same mass spectra, and we only need to discuss one of two relevant subsystems. In the first line of Table 2, $nnc\bar{n}\bar{n}\bar{c}$ has isospin $I=(2,1,0)$ and $nsc\bar{n}\bar{s}\bar{c}$ has isospin $I=(1,0)$ . In the second line, the isospin $I$ can be $(3/2,1/2)$ for $nsc\bar{n}\bar{n}\bar{c}$ , $(1,0)$ for $nnc\bar{s}\bar{s}\bar{c}$ , and 1/2 for $nsc\bar{s}\bar{s}\bar{c}$ .

Table 2: All possible flavor combinations for the hidden-charm hexaquark system.

System	Flavor combinations
$qqc\bar{q}\bar{q}\bar{c}$	$nnc\bar{n}\bar{n}\bar{c}$	$ssc\bar{s}\bar{s}\bar{c}$	$nsc\bar{n}\bar{s}\bar{c}$
$qqc\bar{q}\bar{q}\bar{c}$	$nsc\bar{n}\bar{n}\bar{c}$ ( $nnc\bar{n}\bar{s}\bar{c}$ )	$nnc\bar{s}\bar{s}\bar{c}$ ( $ssc\bar{n}\bar{n}\bar{c}$ )	$nsc\bar{s}\bar{s}\bar{c}$ ( $ssc\bar{n}\bar{s}\bar{c}$ )

Next, we briefly introduce the color wavefunctions for all hexaquark systems. They can be deduced from the following direct product:

$\begin{split}&([3]\otimes[3]\otimes[3])\otimes([\bar{3}]\otimes[\bar{3}]\otimes[\bar{3}])\\ =&([1_{\rm A}]\oplus[8_{\rm MA}]\oplus[8_{\rm MS}]\oplus[10_{\rm S}])\otimes([1_{\rm A}]\oplus[8_{\rm MA}]\oplus[8_{\rm MS}]\oplus[\bar{10}_{S}])\\ \rightarrow&([1_{\rm A}]\otimes[1_{\rm A}])\oplus([8_{\rm MA}]\otimes[8_{\rm MA}])\oplus([8_{\rm MS}]\otimes[8_{\rm MA}])\oplus\\ &([8_{\rm MA}]\otimes[8_{\rm MS}])\oplus([8_{\rm MS}]\otimes[8_{\rm MS}])\oplus([10_{\rm S}]\otimes[\bar{10}_{\rm S}]),\end{split}$

(6)

where A (S) means totally symmetric (antisymmetric), and MS (MA) means that $q_{1}q_{2}$ or $\bar{q}_{3}\bar{q}_{4}$ is symmetric (antisymmetric). Here, the color-singlet wavefunctions for the hexaquarks are shown in Table 3. In the notation $|[(q_{1}q_{2})^{\rm color1}c]^{\rm color3}[(\bar{q}_{3}\bar{q}_{4})^{\rm color2}\bar{c}]^{\rm color4}\rangle$ , the color1, color2, color3, and color4 stand for the color representations of $q_{1}q_{2}$ , $\bar{q}_{3}\bar{q}_{4}$ , $q_{1}q_{2}c$ , and $\bar{q}_{3}\bar{q}_{4}\bar{c}$ , respectively.

Table 3: All possible color and spin wavefunctions for the hidden-charm hexaquark system.

Color wavefunctions $\phi_{1}^{\rm AA}=|[(q_{1}q_{2})^{\bar{3}}c]^{1}[(\bar{q}_{3}\bar{q}_{4})^{3}\bar{c}]^{1}\rangle$ $\phi_{2}^{\rm MAMA}=|[(q_{1}q_{2})^{\bar{3}}c]^{8}[(\bar{q}_{3}\bar{q}_{4})^{3}\bar{c}]^{8}\rangle$ $\phi_{3}^{\rm MSMA}=|[(q_{1}q_{2})^{6}c]^{8}[(\bar{q}_{3}\bar{q}_{4})^{3}\bar{c}]^{8}\rangle$ $\phi_{4}^{\rm MAMS}=|[(q_{1}q_{2})^{\bar{3}}c]^{8}[(\bar{q}_{3}\bar{q}_{4})^{\bar{6}}\bar{c}]^{8}\rangle$ $\phi_{5}^{\rm MSMS}=|[(q_{1}q_{2})^{6}c]^{8}[(\bar{q}_{3}\bar{q}_{4})^{\bar{6}}\bar{c}]^{8}\rangle$ $\phi_{6}^{\rm SS}=|[(q_{1}q_{2})^{6}c]^{10}[(\bar{q}_{3}\bar{q}_{4})^{\bar{6}}\bar{c}]^{\bar{10}}\rangle$ Spin wavefunctions Spin=0: $\chi_{1}^{\rm MSMS}=|[(q_{1}q_{2})_{1}c]_{\frac{1}{2}}[(\bar{q}_{3}\bar{q}_{4})_{1}\bar{c}]_{\frac{1}{2}}\rangle_{0}$ $\chi_{2}^{\rm SS}=|[(q_{1}q_{2})_{1}c]_{\frac{3}{2}}[(\bar{q}_{3}\bar{q}_{4})_{1}\bar{c}]_{\frac{3}{2}}\rangle_{0}$ $\chi_{3}^{\rm MSA}=|[(q_{1}q_{2})_{1}c]_{\frac{1}{2}}[(\bar{q}_{3}\bar{q}_{4})_{0}\bar{c}]_{\frac{1}{2}}\rangle_{0}$ $\chi_{4}^{\rm AMS}=|[(q_{1}q_{2})_{0}c]_{\frac{1}{2}}[(\bar{q}_{3}\bar{q}_{4})_{1}\bar{c}]_{\frac{1}{2}}\rangle_{0}$ $\chi_{5}^{\rm AA}=|[(q_{1}q_{2})_{0}c]_{\frac{1}{2}}[(\bar{q}_{3}\bar{q}_{4})_{0}\bar{c}]_{\frac{1}{2}}\rangle_{0}$ Spin=1: $\chi_{6}^{\rm MSMS}=|[(q_{1}q_{2})_{1}c]_{\frac{1}{2}}[(\bar{q}_{3}\bar{q}_{4})_{1}\bar{c}]_{\frac{1}{2}}\rangle_{1}$ $\chi_{7}^{\rm SS}=|[(q_{1}q_{2})_{1}c]_{\frac{3}{2}}[(\bar{q}_{3}\bar{q}_{4})_{1}\bar{c}]_{\frac{3}{2}}\rangle_{1}$ $\chi_{8}^{\rm MSS}=|[(q_{1}q_{2})_{1}c]_{\frac{3}{2}}[(\bar{q}_{3}\bar{q}_{4})_{1}\bar{c}]_{\frac{1}{2}}\rangle_{1}$ $\chi_{9}^{\rm SMS}=|[(q_{1}q_{2})_{1}c]_{\frac{1}{2}}[(\bar{q}_{3}\bar{q}_{4})_{1}\bar{c}]_{\frac{3}{2}}\rangle_{1}$ $\chi_{10}^{\rm MSA}=|[(q_{1}q_{2})_{1}c]_{\frac{1}{2}}[(\bar{q}_{3}\bar{q}_{4})_{0}\bar{c}]_{\frac{1}{2}}\rangle_{1}$ $\chi_{11}^{\rm SA}=|[(q_{1}q_{2})_{1}c]_{\frac{3}{2}}[(\bar{q}_{3}\bar{q}_{4})_{0}\bar{c}]_{\frac{1}{2}}\rangle_{1}$ $\chi_{12}^{\rm AMS}=|[(q_{1}q_{2})_{0}c]_{\frac{1}{2}}[(\bar{q}_{3}\bar{q}_{4})_{1}\bar{c}]_{\frac{1}{2}}\rangle_{1}$ $\chi_{13}^{\rm AS}=|[(q_{1}q_{2})_{0}c]_{\frac{1}{2}}[(\bar{q}_{3}\bar{q}_{4})_{1}\bar{c}]_{\frac{3}{2}}\rangle_{1}$ $\chi_{14}^{\rm AA}=|[(q_{1}q_{2})_{0}c]_{\frac{1}{2}}[(\bar{q}_{3}\bar{q}_{4})_{0}\bar{c}]_{\frac{1}{2}}\rangle_{1}$ Spin=2: $\chi_{15}^{\rm SS}=|[(q_{1}q_{2})_{1}c]_{\frac{3}{2}}[(\bar{q}_{3}\bar{q}_{4})_{1}\bar{c}]_{\frac{3}{2}}\rangle_{2}$ $\chi_{16}^{\rm SMS}=|[(q_{1}q_{2})_{1}c]_{\frac{3}{2}}[(\bar{q}_{3}\bar{q}_{4})_{1}\bar{c}]_{\frac{1}{2}}\rangle_{2}$ $\chi_{17}^{\rm MSS}=|[(q_{1}q_{2})_{1}c]_{\frac{1}{2}}[(\bar{q}_{3}\bar{q}_{4})_{1}\bar{c}]_{\frac{3}{2}}\rangle_{2}$ $\chi_{18}^{\rm SA}=|[(q_{1}q_{2})_{1}c]_{\frac{3}{2}}[(\bar{q}_{3}\bar{q}_{4})_{0}\bar{c}]_{\frac{1}{2}}\rangle_{2}$ $\chi_{19}^{\rm AS}=|[(q_{1}q_{2})_{0}c]_{\frac{1}{2}}[(\bar{q}_{3}\bar{q}_{4})_{1}\bar{c}]_{\frac{3}{2}}\rangle_{2}$ Spin=3: $\chi_{20}^{\rm SS}=|[(q_{1}q_{2})_{1}c]_{\frac{3}{2}}[(\bar{q}_{3}\bar{q}_{4})_{1}\bar{c}]_{\frac{3}{2}}\rangle_{3}$

Lastly, the spin wavefunctions for the hidden-charm hexaquark states are also shown in Table 3. In the notation $|[(q_{1}q_{2})_{\rm spin1}c]_{\rm spin3}[(\bar{q}_{3}\bar{q}_{4})_{\rm spin2}\bar{c}]_{\rm spin4}\rangle_{\rm spin5}$ , the spin1, spin2, spin3, spin4, and spin5 represent the spins of $q_{1}q_{2}$ , $\bar{q}_{3}\bar{q}_{4}$ , $q_{1}q_{2}c$ , $\bar{q}_{3}\bar{q}_{4}\bar{c}$ , and the total spin, respectively.

Considering the Pauli principle, we obtain 54 types of total wavefunctions and present them in the first part of Table 4. Some wavefunctions are the eigenstates of $C$ parity like $[\phi^{\rm SS}\otimes\chi^{\rm SS}]$ , but others are not. For the neutral states, we need do linear superposition to construct eigen wavefunctions of $C$ parity, and present them in the second part of Table 4. We introduce notations $\delta_{12}^{A}$ , $\delta_{12}^{S}$ , $\delta_{34}^{A}$ , and $\delta_{34}^{S}$ . When the two light quarks or antiquarks are antisymmetric (symmetric) in the flavor space, $\delta_{12}^{A}=0$ ( $\delta_{12}^{S}=0$ ), or else $\delta_{12}^{A}=1$ ( $\delta_{12}^{S}=1$ ). The hidden-charm hexaquark states can be categorized into 6 classes, and we present them in third part of Table 4.

Table 4: All possible types of total wavefunctions and different classes of the hidden-charm hexaquark system

All possible types of total wavefunctions for hexaquark system without $C$ parity $[\phi^{\rm AA}\otimes\chi^{\rm SS}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $[\phi^{\rm MAMA}\otimes\chi^{\rm SS}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $[\phi^{\rm MSMA}\otimes\chi^{\rm SS}]\delta_{12}^{\rm S}\delta_{34}^{\rm A}$ $[\phi^{\rm SS}\otimes\chi^{\rm SS}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $[\phi^{\rm MSMA}\otimes\chi^{\rm SS}]\delta_{12}^{\rm S}\delta_{34}^{\rm A}$ $[\phi^{\rm MSMS}\otimes\chi^{\rm SS}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $[\phi^{\rm AA}\otimes\chi^{\rm SA}]\delta_{12}^{\rm A}\delta_{34}^{\rm S}$ $[\phi^{\rm MAMA}\otimes\chi^{\rm SA}]\delta_{12}^{\rm A}\delta_{34}^{\rm S}$ $[\phi^{\rm MSMA}\otimes\chi^{\rm SA}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $[\phi^{\rm SS}\otimes\chi^{\rm SA}]\delta_{12}^{\rm S}\delta_{34}^{\rm A}$ $[\phi^{\rm MSMA}\otimes\chi^{\rm SA}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $[\phi^{\rm MSMS}\otimes\chi^{\rm SA}]\delta_{12}^{\rm S}\delta_{34}^{\rm A}$ $[\phi^{\rm AA}\otimes\chi^{\rm AS}]\delta_{12}^{\rm S}\delta_{34}^{\rm A}$ $[\phi^{\rm MAMA}\otimes\chi^{\rm AS}]\delta_{12}^{\rm S}\delta_{34}^{\rm A}$ $[\phi^{\rm MSMA}\otimes\chi^{\rm AS}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $[\phi^{\rm SS}\otimes\chi^{\rm AS}]\delta_{12}^{\rm A}\delta_{34}^{\rm S}$ $[\phi^{\rm MSMA}\otimes\chi^{\rm AS}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $[\phi^{\rm MSMS}\otimes\chi^{\rm AS}]\delta_{12}^{\rm A}\delta_{34}^{\rm S}$ $[\phi^{\rm AA}\otimes\chi^{\rm AA}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $[\phi^{\rm MAMA}\otimes\chi^{\rm AA}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $[\phi^{\rm MSMA}\otimes\chi^{\rm AA}]\delta_{12}^{\rm A}\delta_{34}^{\rm S}$ $[\phi^{\rm SS}\otimes\chi^{\rm AA}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $[\phi^{\rm MSMA}\otimes\chi^{\rm AA}]\delta_{12}^{\rm A}\delta_{34}^{\rm S}$ $[\phi^{\rm MSMS}\otimes\chi^{\rm AA}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $[\phi^{\rm AA}\otimes\chi^{\rm SMS}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $[\phi^{\rm MAMA}\otimes\chi^{\rm SMS}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $[\phi^{\rm MSMA}\otimes\chi^{\rm SMS}]\delta_{12}^{\rm S}\delta_{34}^{\rm A}$ $[\phi^{\rm SS}\otimes\chi^{\rm SMS}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $[\phi^{\rm MSMA}\otimes\chi^{\rm SMS}]\delta_{12}^{\rm S}\delta_{34}^{\rm A}$ $[\phi^{\rm MSMS}\otimes\chi^{\rm SMS}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $[\phi^{\rm AA}\otimes\chi^{\rm MSS}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $[\phi^{\rm MAMA}\otimes\chi^{\rm MSS}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $[\phi^{\rm MSMA}\otimes\chi^{\rm MSS}]\delta_{12}^{\rm S}\delta_{34}^{\rm A}$ $[\phi^{\rm SS}\otimes\chi^{\rm MSS}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $[\phi^{\rm MSMA}\otimes\chi^{\rm MSS}]\delta_{12}^{\rm S}\delta_{34}^{\rm A}$ $[\phi^{\rm MSMS}\otimes\chi^{\rm MSS}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $[\phi^{\rm AA}\otimes\chi^{\rm MSA}]\delta_{12}^{\rm A}\delta_{34}^{\rm S}$ $[\phi^{\rm MAMA}\otimes\chi^{\rm MSA}]\delta_{12}^{\rm A}\delta_{34}^{\rm S}$ $[\phi^{\rm MSMA}\otimes\chi^{\rm MSA}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $[\phi^{\rm SS}\otimes\chi^{\rm MSA}]\delta_{12}^{\rm S}\delta_{34}^{\rm A}$ $[\phi^{\rm MSMA}\otimes\chi^{\rm MSA}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $[\phi^{\rm MSMS}\otimes\chi^{\rm MSA}]\delta_{12}^{\rm S}\delta_{34}^{\rm A}$ $[\phi^{\rm AA}\otimes\chi^{\rm AMS}]\delta_{12}^{\rm S}\delta_{34}^{\rm A}$ $[\phi^{\rm MAMA}\otimes\chi^{\rm AMS}]\delta_{12}^{\rm S}\delta_{34}^{\rm A}$ $[\phi^{\rm MSMA}\otimes\chi^{\rm AMS}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $[\phi^{\rm SS}\otimes\chi^{\rm AMS}]\delta_{12}^{\rm A}\delta_{34}^{\rm S}$ $[\phi^{\rm MSMA}\otimes\chi^{\rm AMS}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $[\phi^{\rm MSMS}\otimes\chi^{\rm AMS}]\delta_{12}^{\rm A}\delta_{34}^{\rm S}$ $[\phi^{\rm AA}\otimes\chi^{\rm MSMS}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $[\phi^{\rm MAMA}\otimes\chi^{\rm MSMS}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $[\phi^{\rm MSMA}\otimes\chi^{\rm MSMS}]\delta_{12}^{\rm S}\delta_{34}^{\rm A}$ $[\phi^{\rm SS}\otimes\chi^{\rm MSMS}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $[\phi^{\rm MSMA}\otimes\chi^{\rm MSMS}]\delta_{12}^{\rm S}\delta_{34}^{\rm A}$ $[\phi^{\rm MSMS}\otimes\chi^{\rm MSMS}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ All possible types of total wavefunctions for pure neutral hexaquark system $[\phi^{\rm SS}\otimes\chi^{\rm SS}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $[\phi^{\rm SS}\otimes\chi^{\rm MSMS}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $[\phi^{\rm SS}\otimes\chi^{\rm AA}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $[\phi^{\rm MSMS}\otimes\chi^{\rm SS}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $[\phi^{\rm MSMS}\otimes\chi^{\rm AA}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $[\phi^{\rm MSMS}\otimes\chi^{\rm MSMS}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $[\phi^{\rm AA}\otimes\chi^{\rm SS}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $[\phi^{\rm AA}\otimes\chi^{\rm MSMS}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $[\phi^{\rm AA}\otimes\chi^{\rm AA}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $[\phi^{\rm MAMA}\otimes\chi^{\rm SS}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $[\phi^{\rm MAMA}\otimes\chi^{\rm AA}]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $[\phi^{\rm MAMA}\otimes\chi^{\rm MSMS}]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $\frac{1}{\sqrt{2}}[(\phi^{\rm MAMS}\pm\phi^{\rm MSMA})\otimes\chi^{\rm SS}]\delta_{12}^{\rm S}\delta_{12}^{\rm A}\delta_{34}^{\rm S}\delta_{34}^{\rm A}$ $\frac{1}{\sqrt{2}}[(\phi^{\rm MAMS}\pm\phi^{\rm MSMA})\otimes\chi^{\rm MSMS}]\delta_{12}^{\rm S}\delta_{12}^{\rm A}\delta_{34}^{\rm S}\delta_{34}^{\rm A}$ $\frac{1}{\sqrt{2}}[(\phi^{\rm MAMS}\pm\phi^{\rm MSMA})\otimes\chi^{\rm AA}]\delta_{12}^{\rm S}\delta_{12}^{\rm A}\delta_{34}^{\rm S}\delta_{34}^{\rm A}$ $\frac{1}{\sqrt{2}}[\phi^{\rm SS}\otimes(\chi^{\rm SA}\pm\chi^{\rm AS})]\delta_{12}^{\rm S}\delta_{12}^{\rm A}\delta_{34}^{\rm S}\delta_{34}^{\rm A}$ $\frac{1}{\sqrt{2}}[\phi^{\rm AA}\otimes(\chi^{\rm SA}\pm\chi^{\rm AS})]\delta_{12}^{\rm S}\delta_{12}^{\rm A}\delta_{34}^{\rm S}\delta_{34}^{\rm A}$ $\frac{1}{\sqrt{2}}[\phi^{\rm MSMS}\otimes(\chi^{\rm SA}\pm\chi^{\rm AS})]\delta_{12}^{\rm S}\delta_{12}^{\rm A}\delta_{34}^{\rm S}\delta_{34}^{\rm A}$ $\frac{1}{\sqrt{2}}[\phi^{\rm SS}\otimes(\chi^{\rm MSA}\pm\chi^{\rm AMS})]\delta_{12}^{\rm S}\delta_{12}^{\rm A}\delta_{34}^{\rm S}\delta_{34}^{\rm A}$ $\frac{1}{\sqrt{2}}[\phi^{\rm AA}\otimes(\chi^{\rm MSA}\pm\chi^{\rm AMS})]\delta_{12}^{\rm S}\delta_{12}^{\rm A}\delta_{34}^{\rm S}\delta_{34}^{\rm A}$ $\frac{1}{\sqrt{2}}[\phi^{\rm MSMS}\otimes(\chi^{\rm MSA}\pm\chi^{\rm AMS})]\delta_{12}^{\rm S}\delta_{12}^{\rm A}\delta_{34}^{\rm S}\delta_{34}^{\rm A}$ $\frac{1}{\sqrt{2}}[\phi^{\rm SS}\otimes(\chi^{\rm SMS}\pm\chi^{\rm MSS})]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $\frac{1}{\sqrt{2}}[\phi^{\rm AA}\otimes(\chi^{\rm SMS}\pm\chi^{\rm SMS})]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $\frac{1}{\sqrt{2}}[\phi^{\rm MSMS}\otimes(\chi^{\rm SMS}\pm\chi^{\rm SMS})]\delta_{12}^{\rm S}\delta_{34}^{\rm S}$ $\frac{1}{\sqrt{2}}[\phi^{\rm MAMA}\otimes(\chi^{\rm MSA}\pm\chi^{\rm AMS})]\delta_{12}^{\rm S}\delta_{12}^{\rm A}\delta_{34}^{\rm S}\delta_{34}^{\rm A}$ $\frac{1}{\sqrt{2}}[\phi^{\rm MAMA}\otimes(\chi^{\rm SA}\pm\chi^{\rm AS})]\delta_{12}^{\rm S}\delta_{12}^{\rm A}\delta_{34}^{\rm S}\delta_{34}^{\rm A}$ $\frac{1}{\sqrt{2}}[\phi^{\rm MAMA}\otimes(\chi^{\rm SMS}\pm\chi^{\rm SMS})]\delta_{12}^{\rm A}\delta_{34}^{\rm A}$ $\frac{1}{\sqrt{2}}[(\phi^{\rm MAMS}\otimes\chi^{\rm SA})\pm(\phi^{\rm MSMA}\otimes\chi^{\rm AS})]\delta_{12}^{\rm S}\delta_{12}^{\rm A}\delta_{34}^{\rm S}\delta_{34}^{\rm A}$ $\frac{1}{\sqrt{2}}[(\phi^{\rm MAMS}\otimes\chi^{\rm MSA})\pm(\phi^{\rm MSMA}\otimes\chi^{\rm AMS})]\delta_{12}^{\rm S}\delta_{12}^{\rm A}\delta_{34}^{\rm S}\delta_{34}^{\rm A}$ $\frac{1}{\sqrt{2}}[(\phi^{\rm MAMS}\otimes\chi^{\rm AS})\pm(\phi^{\rm MSMA}\otimes\chi^{\rm SA})]\delta_{12}^{\rm S}\delta_{12}^{\rm A}\delta_{34}^{\rm S}\delta_{34}^{\rm A}$ $\frac{1}{\sqrt{2}}[(\phi^{\rm MAMS}\otimes\chi^{\rm MSA})\pm(\phi^{\rm MSMA}\otimes\chi^{\rm AMS})]\delta_{12}^{\rm S}\delta_{12}^{\rm A}\delta_{34}^{\rm S}\delta_{34}^{\rm A}$ Different classes of the hidden-charm hexaquark system $\delta_{12}^{\rm A}=1$ , $\delta_{34}^{\rm A}=1$ , $\delta_{12}^{\rm S}=0$ , $\delta_{34}^{\rm S}=0$ : $(nn)^{I=1}c(\bar{n}\bar{n})^{I=1}\bar{c}$ , $\delta_{12}^{\rm A}=0$ , $\delta_{34}^{\rm A}=1$ , $\delta_{12}^{\rm S}=1$ , $\delta_{34}^{\rm S}=0$ : $(nn)^{I=0}c(\bar{n}\bar{n})^{I=1}\bar{c}$ , $(nn)^{I=1}c\bar{s}\bar{s}\bar{c}$ , $ssc\bar{s}\bar{s}\bar{c}$ $(nn)^{I=0}c\bar{s}\bar{s}\bar{c}$ $\delta_{12}^{\rm A}=0$ , $\delta_{34}^{\rm A}=0$ , $\delta_{12}^{\rm S}=1$ , $\delta_{34}^{\rm S}=1$ : $(nn)^{I=0}c(\bar{n}\bar{n})^{I=0}\bar{c}$ $\delta_{12}^{\rm A}=1$ , $\delta_{34}^{\rm A}=1$ , $\delta_{12}^{\rm S}=1$ , $\delta_{34}^{\rm S}=0$ : $nsc(\bar{n}\bar{n})^{I=1}\bar{c}$ , $nsc\bar{s}\bar{s}\bar{c}$ $\delta_{12}^{\rm A}=1$ , $\delta_{34}^{\rm A}=0$ , $\delta_{12}^{\rm S}=1$ , $\delta_{34}^{\rm S}=1$ : $nsc(\bar{n}\bar{n})^{I=0}\bar{c}$ $\delta_{12}^{\rm A}=1$ , $\delta_{34}^{\rm A}=1$ , $\delta_{12}^{\rm S}=1$ , $\delta_{34}^{\rm S}=1$ : $nsc\bar{n}\bar{s}\bar{c}$

IV Numerical results and discussion

Sandwiching the CMI Hamiltonian between the two wavefunctions with the same quantum number, we obtain the Hamiltonian matrices. Based on the corresponding eigenvalues and eigenvectors, we discuss the mass gaps, stabilities, and strong decay behaviors of all the hidden-charm hexaquark states.

From the eigenvalues, we present the mass spectra in Fig. 1 (for $nnc\bar{n}\bar{n}\bar{c}$ , $ssc\bar{s}\bar{s}\bar{c}$ , and $nnc\bar{s}\bar{s}\bar{c}$ ), Fig. 2 (for $nnc\bar{n}\bar{s}\bar{c}$ and $nsc\bar{n}\bar{s}\bar{c}$ ), and Fig. 3 (for $nsc\bar{s}\bar{s}\bar{c}$ ). Moreover, we also plot the corresponding thresholds which they can decay to through quark rearrangements. In convenience, we label the spin (isospin) of the rearrangement decay channel in the superscript (subscript).

Refer to caption — Figure 1: Relative positions (units: MeV) for three kinds of hexaquark states. In the $nnc\bar{n}\bar{n}\bar{c}$ subsystem, the black lines show the $(nn)^{I=1}c(\bar{n}\bar{n})^{I=1}\bar{c}$ hexaquark states, the red lines show the $(nn)^{I=0}c(\bar{n}\bar{n})^{I=1}\bar{c}$ hexaquark states, and the orange lines show the $(nn)^{I=0}c(\bar{n}\bar{n})^{I=0}\bar{c}$ hexaquark states. The solid and dashed short lines are to differentiate the positive and negative $C$ parity and it’s the same as $ssc\bar{s}\bar{s}\bar{c}$ subsystem. In the $nnc\bar{s}\bar{s}\bar{c}$ subsystem, the black (red) lines represent the $nnc\bar{s}\bar{s}\bar{c}$ hexaquark states with $I=1(0)$ . The dotted lines denote various baryon-antibaryon or meson-meson-meson thresholds. Some meson-meson-meson thresholds have specific $C$ parity, we label thresholds which have positive (negative) $C$ parity with blue (green). When the spin (isospin) of an initial hexaquark state is equal to the number in the superscript (subscript) of a baryon-antibaryon (meson-meson-meson) state, it can decay into these state through $S$ -wave. Moreover, the stable states are marked with “ ${\dagger}$ ”.

In addition to the mass spectra, we discuss the two body strong decay based on the obtained eigenvectors. According to Table 3 and Table 4, we can find that there are overlaps between hexaquark states and particular baryon-antibaryon states. In the $qqc\otimes\bar{q}\bar{q}\bar{c}$ configuration, the color wavefunction of the hexaquark states falls into three categories: $|(qqc)^{1_{c}}(\bar{q}\bar{q}\bar{c})^{1_{c}}\rangle$ , $|(qqc)^{8_{c}}(\bar{q}\bar{q}\bar{c})^{8_{c}}\rangle$ , and $|(qqc)^{10_{c}}(\bar{q}\bar{q}\bar{c})^{\bar{10}_{c}}\rangle$ . The $|(qqc)^{1_{c}}(\bar{q}\bar{q}\bar{c})^{1_{c}}\rangle$ can easily dissociate into an S-wave baryon and S-wave antibaryon (the “OZI-superallowed” decay mode). In contrast, the $|(qqc)^{8_{c}}(\bar{q}\bar{q}\bar{c})^{8_{c}}\rangle$ and $|(qqc)^{10_{c}}(\bar{q}\bar{q}\bar{c})^{\bar{10}_{c}}\rangle$ fall apart through the gluon exchange. For simplicity, we only focus on the “OZI-superallowed” decay mode.

The partial width of the two body $L$ -wave “OZI-superallowed decay” mode reads Weng:2019ynva ; Weng:2020jao ; Weng:2021hje

\Gamma_{i}=\gamma_{i}\alpha\frac{k^{2L+1}}{m^{2L}}{\cdot}|c_{i}|^{2},

(7)

where $\alpha$ is an effective coupling constant, $m$ is the initial state mass, $k$ is the spatial momentum of the final state in the center-of-mass frame, and $c_{i}$ is overlap between the hexaquark states and the final baryon-antibaryon states. Generally, $\gamma_{i}$ depends on the spatial wavefunctions of the initial hexaquark and final baryon-antibaryon, which are different for each decay process. In the heavy quark limit, $\Sigma_{c}$ ( $\Xi^{*}_{c}$ ) and $\Sigma^{*}_{c}$ ( $\Xi^{\prime}_{c}$ ) have the same spatial wavefunction. Based on these, we assume the $\gamma_{i}$ relationships for different hidden-charm hexaquark states presented in Table 5. We find that the $(k/m)^{2}$ is of $\mathcal{O}(10^{-2})$ or even smaller, which means that the large partial wave decays are all suppressed. Thus we only need to consider the $S$ -wave two body decay modes. Employing the eigenvectors in Table 6, we calculate the values of $k\cdot|c_{i}|^{2}$ for each decay process and present them in Table 7. The blank area in Tables 6 and 7 means that the hexaquark state is forbidden to decay through this channel because of the quantum number conservation. According to the $\gamma_{i}$ relationships in Table 5 and the values of $k\cdot|c_{i}|^{2}$ in Table 7, we can roughly estimate the relative decay widths for different two-body decay processes of a hexaquark state.

Table 5: The

\gamma_{i}

relationships for different hidden-charm hexaquark subsystems.

Subsystem	$\gamma_{i}$
$nnc\bar{n}\bar{n}\bar{c}$	$\gamma_{\Sigma_{c}^{}\bar{\Sigma}_{c}^{}}=\gamma_{\Sigma_{c}^{}\bar{\Sigma}_{c}}=\gamma_{\Sigma_{c}\bar{\Sigma}^{}_{c}}=\gamma_{\Sigma_{c}\bar{\Sigma}_{c}}$ $\gamma_{\Sigma_{c}^{*}\bar{\Lambda}_{c}}=\gamma_{\Sigma_{c}\bar{\Lambda}_{c}}$
$nsc\bar{n}\bar{s}\bar{c}$	$\gamma_{\Xi_{c}^{}\bar{\Xi}^{}_{c}}=\gamma_{\Xi_{c}^{\prime}\bar{\Xi}_{c}^{}}=\gamma_{\Xi_{c}^{\prime}\bar{\Xi}_{c}^{\prime}}=\gamma_{\Xi_{c}\bar{\Xi}_{c}^{}}\approx\gamma_{\Xi_{c}\bar{\Xi}_{c}^{\prime}}=\gamma_{\Xi_{c}\bar{\Xi}_{c}}$
$nsc\bar{n}\bar{n}\bar{c}$	$\gamma_{\Xi_{c}^{}\bar{\Sigma}_{c}^{}}=\gamma_{\Xi_{c}^{}\bar{\Sigma}_{c}}=\gamma_{\Xi_{c}^{\prime}\bar{\Sigma}_{c}^{}}=\gamma_{\Xi_{c}^{\prime}\bar{\Sigma}_{c}}=\gamma_{\Xi_{c}\bar{\Sigma}_{c}^{*}}=\gamma_{\Xi_{c}\bar{\Sigma}_{c}}$
$nnc\bar{s}\bar{s}\bar{c}$	$\gamma_{\Sigma_{c}^{}\bar{\Omega}_{c}^{}}=\gamma_{\Sigma_{c}\bar{\Omega}_{c}^{}}=\gamma_{\Sigma_{c}^{}\bar{\Omega}_{c}}=\gamma_{\Sigma_{c}\bar{\Omega}_{c}}$ $\gamma_{\Lambda_{c}\bar{\Omega}_{c}^{*}}=\gamma_{\Lambda_{c}\bar{\Omega}_{c}}$
$nsc\bar{s}\bar{s}\bar{c}$	$\gamma_{\Xi_{c}^{}\bar{\Omega}_{c}^{}}=\gamma_{\Xi_{c}^{}\bar{\Omega}_{c}}=\gamma_{\Xi_{c}^{{}^{\prime}}\bar{\Omega}_{c}^{}}=\gamma_{\Xi_{c}^{{}^{\prime}}\bar{\Omega}_{c}}\approx\gamma_{\Xi_{c}\bar{\Omega}_{c}^{*}}=\gamma_{\Xi_{c}\bar{\Omega}_{c}}$
$ssc\bar{s}\bar{s}\bar{c}$	$\gamma_{\Omega_{c}^{}\bar{\Omega}_{c}^{}}=\gamma_{\Omega_{c}^{}\bar{\Omega}_{c}}=\gamma_{\Omega_{c}^{}\bar{\Omega}_{c}}=\gamma_{\Omega_{c}\bar{\Omega}_{c}}$

Table 6: The values of eigenvectors for the

nnc\bar{n}\bar{n}\bar{c}

nsc\bar{n}\bar{n}\bar{c}

nnc\bar{s}\bar{s}\bar{c}

nsc\bar{n}\bar{s}\bar{c}

nsc\bar{s}\bar{s}\bar{c}

, and

ssc\bar{s}\bar{s}\bar{c}

hexaquark subsystems. The masses are all in units of MeV.

$ssc\bar{s}\bar{s}\bar{c}$ $(I=0)$ $(nn)^{I=1}c\bar{s}\bar{s}\bar{c}$ $(I=1)$ $(nn)^{I=1}c(\bar{n}\bar{n})^{I=1}\bar{c}$ $(I=2,1,0)$ $(nn)^{I=0}c(\bar{n}\bar{n})^{I=1}\bar{c}$ $(I=1)$ $J^{PC}$ Mass ${\Omega_{c}^{*}\bar{\Omega}_{c}^{*}}$ $\Omega_{c}^{*}\bar{\Omega}_{c}$ $\Omega_{c}\bar{\Omega}_{c}^{*}$ ${\Omega_{c}\bar{\Omega}_{c}}$ $J^{P}$ Mass ${\Sigma_{c}^{*}\bar{\Omega}_{c}^{*}}$ ${\Sigma_{c}\bar{\Omega}_{c}^{*}}$ ${\Sigma_{c}^{*}\bar{\Omega}_{c}}$ ${\Sigma_{c}\bar{\Omega}_{c}}$ $J^{PC}$ Mass $\Sigma^{*}_{c}\bar{\Sigma}^{*}_{c}$ $\Sigma^{*}_{c}\bar{\Sigma}_{c}$ $\Sigma_{c}\bar{\Sigma}^{*}_{c}$ $\Sigma_{c}\bar{\Sigma}_{c}$ $J^{P}$ Mass $\Lambda_{c}\bar{\Sigma}^{*}_{c}$ $\Lambda_{c}\bar{\Sigma}_{c}$ $3^{--}$ 5534 -0.993 $3^{-}$ 5285 -0.991 $3^{--}$ 5036 -0.989 $2^{-}$ 4939 0.259 $2^{--}$ 5490 0.665 -0.665 $2^{-}$ 5319 -0.884 -0.306 -0.128 $2^{--}$ 5025 0.909 4895 -0.764 $2^{-+}$ 5539 -0.983 -0.050 -0.050 5255 -0.288 0.358 0.720 $2^{-+}$ 5060 -0.924 -0.161 -0.161 $1^{-}$ 5013 0.221 -0.015 5475 0.091 -0.664 -0.664 5233 -0.259 0.825 - 0.441 5008 0.296 -0.484 -0.484 4922 0.456 0.255 $1^{--}$ 5569 0.918 0.135 -0.135 0.133 $1^{-}$ 5415 0.688 0.3 0.275 0.188 $1^{--}$ 5143 0.672 0.327 -0.327 0.338 4876 -0.634 -0.300 5522 -0.225 0.593 -0.593 0.248 5312 0.417 -0.116 -0.685 -0.213 5130 -0.538 0.416 -0.416 0.161 4852 -0.354 0.740 5429 0.087 0.243 -0.243 -0.882 5269 -0.307 0.805 -0.191 -0.169 4954 -0.217 -0.415 0.415 0.825 4792 0.128 -0.264 $1^{-+}$ 5465 0.699 0.699 5196 -0.071 0.123 -0.439 0.774 $1^{-+}$ 5006 0.650 0.650 4686 0.188 -0.031 $0^{-+}$ 5622 -0.806 -0.235 5157 0.058 -0.173 -0.145 -0.284 $0^{-+}$ 5217 -0.704 -0.369 4605 -0.191 0.097 5495 0.447 -0.641 5150 0.308 0.244 0.067 -0.227 5066 0.531 -0.599 $0^{-}$ 4933 0.118 5435 -0.017 0.365 $0^{-}$ 5366 -0.899 -0.169 4958 0.055 0.213 4842 -0.599 5242 0.191 -0.827 4797 0.674 $nsc\bar{n}\bar{s}\bar{c}$ $(I=1,0)$ $nsc(\bar{n}\bar{n})^{I=1}\bar{c}$ $(I=3/2,1/2)$ $(nn)^{I=0}c(\bar{n}\bar{n})^{I=0}\bar{c}$ $(I=0)$ $J^{PC}$ Mass $\Xi^{*}_{c}\bar{\Xi}^{*}_{c}$ $\Xi^{*}_{c}\bar{\Xi}^{\prime}_{c}$ $\Xi^{\prime}_{c}\bar{\Xi}^{*}_{c}$ $\Xi^{*}_{c}\bar{\Xi}_{c}$ $\Xi_{c}\bar{\Xi}^{*}_{c}$ $\Xi^{\prime}_{c}\bar{\Xi}_{c}$ $\Xi_{c}\bar{\Xi}^{\prime}_{c}$ $\Xi^{\prime}_{c}\bar{\Xi}^{\prime}_{c}$ $\Xi_{c}\bar{\Xi}_{c}$ $J^{P}$ Mass $\Xi^{*}_{c}\bar{\Sigma}^{*}_{c}$ $\Xi^{*}_{c}\bar{\Sigma}_{c}$ $\Xi^{\prime}_{c}\bar{\Sigma}^{*}_{c}$ $\Xi^{\prime}_{c}\bar{\Sigma}_{c}$ $\Xi_{c}\bar{\Sigma}^{*}_{c}$ $\Xi_{c}\bar{\Sigma}_{c}$ $J^{PC}$ Mass $\Lambda_{c}\bar{\Lambda}_{c}$ $3^{--}$ 5292 -0.991 $3^{-}$ 5164 0.989 $1^{--}$ 4940 -0.319 $2^{-+}$ 5329 -0.883 -0.148 -0.148 -0.132 -0.132 $2^{-}$ 5250 0.749 0.187 0.377 0.252 4816 -0.019 5240 0.264 -0.645 -0.645 -0.095 -0.095 5134 -0.482 0.709 0.354 0.145 4584 -0.818 5216 -0.303 -0.013 -0.013 0.291 0.291 5116 -0.276 -0.554 0.705 -0.005 $0^{-+}$ 4767 0.562 5168 -0.064 -0.168 -0.168 0.535 0.535 5071 0.203 0.140 -0.013 -0.707 4649 -0.619 24 5125 -0.030 0.081 0.081 -0.006 -0.006 5051 -0.014 0.118 0.350 -0.345 $nsc(\bar{n}\bar{n}\bar{c})^{I=0}$ $(I=1/2)$ $2^{--}$ 5295 0.603 -0.603 0.131 -0.131 5017 0.080 0.180 0.156 -0.293 $J^{P}$ Mass $\Xi_{c}^{*}\bar{\Lambda}_{c}$ $\Xi_{c}^{\prime}\bar{\Lambda}_{c}$ $\Xi_{c}\bar{\Lambda}_{c}$ 24 5200 -0.026 0.026 -0.443 0.443 $1^{-}$ 5419 0.509 0.341 0.303 0.231 0.259 0.225 $2^{-}$ 5192 -0.061 5155 0.218 -0.218 -0.407 0.407 5235 -0.523 0.549 -0.036 0.067 0.163 0.104 5015 0.746 $1^{-+}$ 5326 0.347 0.347 0.194 0.194 0.219 0.219 5185 -0.337 -0.092 0.652 0.009 -0.162 -0.039 4971 -0.359 5224 -0.578 -0.578 0.069 0.069 0.108 0.108 5120 -0.18 -0.042 -0.302 0.366 0.206 -0.011 $1^{-}$ 5447 0.037 -0.023 -0.010 5134 0.066 0.066 -0.620 -0.620 0.132 0.132 5090 -0.201 -0.555 0.218 0.154 0.578 0.247 5028 0.409 -0.060 -0.009 5110 0.025 0.025 -0.157 -0.157 -0.291 -0.291 5056 0.016 0.172 0.119 -0.591 0.389 -0.102 5005 0.659 0.443 0.164 5066 0.101 0.101 -0.003 -0.003 -0.440 -0.440 5041 0.187 -0.172 -0.333 -0.129 0.246 -0.126 4970 -0.143 0.345 0.066 $1^{--}$ 5421 0.433 0.346 -0.346 0.128 -0.128 0.107 -0.107 0.317 0.142 5036 -0.175 0.022 0.033 -0.189 0.299 -0.25 4943 -0.426 0.634 0.025 5398 0.558 -0.053 0.053 0.066 -0.066 0.082 -0.082 0.066 0.025 5008 0.113 0.019 0.192 0.313 0.138 -0.734 4913 0.028 -0.033 0.078 5288 -0.537 0.388 -0.388 0.088 -0.088 0.095 -0.095 0.173 0.014 4991 0.101 0.002 -0.008 -0.302 0.014 -0.052 4896 0.171 0.215 0.432 5206 0.240 0.361 -0.361 -0.131 0.131 -0.092 0.092 -0.734 -0.079 4970 0.160 -0.163 0.070 0.021 -0.108 -0.015 4870 0.097 0.009 -0.588 5153 0.118 0.090 -0.090 -0.549 0.549 -0.141 0.141 0.394 -0.131 4946 -0.168 0.165 -0.211 0.204 0.136 -0.135 4822 0.017 -0.014 -0.279 5114 -0.065 -0.057 0.057 -0.265 0.265 0.488 -0.488 -0.158 0.129 $0^{-}$ 5285 0.845 0.256 0.093 4809 -0.226 0.017 -0.272 5073 0.182 -0.173 0.173 0.127 -0.127 0.063 -0.063 -0.059 -0.163 5151 0.333 -0.723 -0.049 4770 0.058 0.127 0.107 5050 -0.167 0.008 -0.008 0.109 -0.109 0.023 -0.023 -0.092 -0.259 5071 -0.003 0.287 0.244 $0^{-}$ 5027 0.047 -0.124 5041 0.008 0.004 -0.004 -0.001 0.001 0.008 -0.008 0.007 -0.054 5024 0.057 0.064 -0.719 4996 -0.544 -0.224 5027 0.093 0.067 -0.067 0.111 -0.111 0.083 -0.083 0.109 -0.746 5015 0.018 -0.272 0.167 4914 -0.705 0.246 5021 -0.009 0.028 -0.028 -0.006 0.006 -0.082 0.082 0.065 -0.038 4962 0.033 0.171 -0.139 4894 0.087 0.639 19 5003 0.001 -0.019 0.019 -0.124 0.124 0.296 -0.296 -0.016 -0.291 $nsc\bar{s}\bar{s}\bar{c}$ $(I=1/2)$ 4815 0.145 0.296 24 4987 -0.059 -0.030 0.030 0.046 -0.046 0.127 -0.127 -0.042 0.148 $J^{P}$ Mass ${\Xi_{c}^{*}\bar{\Omega}_{c}^{*}}$ ${\Xi_{c}^{\prime}\bar{\Omega}_{c}^{*}}$ ${\Xi_{c}^{*}\bar{\Omega}_{c}}$ ${\Xi_{c}^{\prime}\bar{\Omega}_{c}}$ ${\Xi_{c}\bar{\Omega}_{c}^{*}}$ ${\Xi_{c}\bar{\Omega}_{c}}$ $(nnc)^{I=0}\bar{s}\bar{s}\bar{c}$ $(I=0)$ 19 4978 0.047 0.028 -0.028 -0.002 0.002 -0.227 0.227 0.029 -0.049 $3^{-}$ 5413 0.991 $J^{P}$ Mass $\Lambda_{c}\bar{\Omega}_{c}^{*}$ $\Lambda_{c}\bar{\Omega}_{c}$ 24 $0^{-+}$ 5407 -0.854 -0.094 -0.094 -0.233 -0.112 $1^{-}$ 5590 -0.556 -0.295 -0.326 -0.239 -0.213 -0.199 $2^{-}$ 5135 0.412 5300 -0.064 -0.052 -0.052 0.342 0.027 5432 0.642 -0.16 -0.538 -0.162 -0.109 -0.103 5128 -0.661 5235 0.323 -0.165 -0.165 -0.732 -0.137 5390 0.192 -0.776 0.326 0.149 0.015 0.033 $1^{-}$ 5189 -0.337 -0.068 5131 0.094 -0.541 -0.541 0.309 -0.177 5332 0.178 0.102 0.502 -0.560 -0.408 -0.181 5138 -0.478 -0.225 5116 0.018 0.030 0.030 -0.016 -0.264 5307 -0.003 -0.202 0.184 -0.474 0.454 -0.041 5088 0.677 -0.0003 5047 -0.071 -0.202 -0.202 -0.175 0.733 5274 0.015 0.068 -0.176 -0.386 0.472 0.205 5071 0.139 -0.819 5009 0.019 -0.216 -0.216 0.092 -0.042 5245 -0.001 0.106 0.068 -0.138 -0.297 0.634 4984 0.006 -0.195 4976 0.041 -0.052 -0.052 -0.035 0.167 5244 0.199 -0.119 -0.113 -0.125 -0.112 0.129 $0^{-}$ 5130 0.205 $0^{--}$ 5137 0.280 -0.280 5227 -0.166 -0.298 -0.12 -0.064 -0.166 0.443 5040 0.818 5117 0.382 -0.382 5208 -0.093 0.039 -0.012 -0.091 0.260 0.182 5007 0.359 24 5084 -0.441 0.441 5183 -0.141 -0.151 0.186 0.148 0.116 -0.021 $J^{P}$ Mass $\Xi_{c}^{\prime}\bar{\Omega}_{c}^{*}$ $\Xi_{c}^{\prime}\bar{\Omega}_{c}$ $\Xi_{c}\bar{\Omega}_{c}$ 24 $J^{P}$ Mass $\Xi_{c}^{*}\bar{\Omega}_{c}^{*}$ $\Xi_{c}^{\prime}\bar{\Omega}_{c}^{*}$ $\Xi_{c}^{*}\bar{\Omega}_{c}$ $\Xi_{c}^{*}\bar{\Omega}_{c}$ Mass $\Xi_{c}^{*}\bar{\Omega}_{c}^{*}$ $\Xi_{c}^{\prime}\bar{\Omega}_{c}^{*}$ $\Xi_{c}^{*}\bar{\Omega}_{c}$ $\Xi_{c}^{*}\bar{\Omega}_{c}$ $0^{-}$ 5492 0.899 0.171 0.072 $0^{-}$ 5211 -0.054 -0.069 -0.071 $2^{-}$ 5456 0.843 0.314 0.155 0.187 5295 -0.16 -0.168 -0.126 0.855 5354 -0.216 0.829 0.071 5176 -0.035 0.239 -0.333 5371 -0.437 0.629 0.551 0.147 5251 0.057 -0.134 -0.214 0.123 5267 0.023 -0.199 -0.411 5356 0.113 -0.611 0.735 0.0003 5229 0.085 0.162 -0.710

Table 7: The values of

k\cdot|c_{i}|^{2}

for the

nnc\bar{n}\bar{n}\bar{c}

nsc\bar{n}\bar{n}\bar{c}

nnc\bar{s}\bar{s}\bar{c}

nsc\bar{n}\bar{s}\bar{c}

nsc\bar{s}\bar{s}\bar{c}

, and

ssc\bar{s}\bar{s}\bar{c}

hexaquark subsystems. The masses are all in units of MeV. The

\times

means that the decay channel is kinetically forbidden.

$ssc\bar{s}\bar{s}\bar{c}$ $(I=0)$ $(nn)^{I=1}c\bar{s}\bar{s}\bar{c}$ $(I=1)$ $(nn)^{I=1}c(\bar{n}\bar{n})^{I=1}\bar{c}$ $(I=2,1,0)$ $(nn)^{I=0}c(\bar{n}\bar{n})^{I=1}\bar{c}$ $(I=1)$ $J^{PC}$ Mass ${\Omega_{c}^{*}\bar{\Omega}_{c}^{*}}$ $\Omega_{c}^{*}\bar{\Omega}_{c}$ $\Omega_{c}\bar{\Omega}_{c}^{*}$ ${\Omega_{c}\bar{\Omega}_{c}}$ $J^{P}$ Mass ${\Sigma_{c}^{*}\bar{\Omega}_{c}^{*}}$ ${\Sigma_{c}\bar{\Omega}_{c}^{*}}$ ${\Sigma_{c}^{*}\bar{\Omega}_{c}}$ ${\Sigma_{c}\bar{\Omega}_{c}}$ $J^{PC}$ Mass $\Sigma^{*}_{c}\bar{\Sigma}^{*}_{c}$ $\Sigma^{*}_{c}\bar{\Sigma}_{c}$ $\Sigma_{c}\bar{\Sigma}^{*}_{c}$ $\Sigma_{c}\bar{\Sigma}_{c}$ $J^{P}$ Mass $\Lambda_{c}\bar{\Sigma}^{*}_{c}$ $\Lambda_{c}\bar{\Sigma}_{c}$ $3^{--}$ 5534 69 $3^{-}$ 5285 47 $3^{--}$ 5036 552 $2^{-}$ 4939 38 $2^{--}$ 5490 124 124 $2^{-}$ 5319 238 48 9 $2^{--}$ 5025 301 4895 272 $2^{-+}$ 5539 136 1 1 5255 $\times$ 38.5 170 $2^{-+}$ 5060 206 12 12 $1^{-}$ 5013 35 0.2 5475 $\times$ 88 88 5233 $\times$ 126 44 5008 $\times$ 70 70 4922 110 59 $1^{--}$ 5569 271 10 10 12 $1^{-}$ 5415 281 65 56 30 $1^{--}$ 5143 235 71 71 88 4876 167 76 5522 $\times$ 143 143 37 5312 47 7 239 30 5130 141 109 109 19 4852 42 444 5429 $\times$ $\times$ $\times$ 252 5269 $\times$ 233 14 16 4954 $\times$ $\times$ $\times$ 228 4792 $\times$ 50 $1^{-+}$ 5465 53 53 5196 $\times$ $\times$ 209 $1^{-+}$ 5006 123 123 4686 $\times$ 0.5 $0^{-+}$ 5622 325 44 5157 $\times$ $\times$ 11 $0^{-+}$ 5217 337 121 4605 $\times$ 3 5495 $\times$ 219 5150 $\times$ $\times$ 2 5066 77 226 $0^{-}$ 4933 10 5435 $\times$ 46 $0^{-}$ 5366 377 21 4958 $\times$ 16 4842 178 5242 $\times$ 336 4797 166 $nsc\bar{n}\bar{s}\bar{c}$ $(I=1,0)$ $nsc(\bar{n}\bar{n})^{I=1}\bar{c}$ $(I=3/2,1/2)$ $(nn)^{I=0}c(\bar{n}\bar{n})^{I=0}\bar{c}$ $(I=0)$ $J^{PC}$ Mass $\Xi^{*}_{c}\bar{\Xi}^{*}_{c}$ $\Xi^{*}_{c}\bar{\Xi}^{\prime}_{c}$ $\Xi^{\prime}_{c}\bar{\Xi}^{*}_{c}$ $\Xi^{*}_{c}\bar{\Xi}_{c}$ $\Xi_{c}\bar{\Xi}^{*}_{c}$ $\Xi^{\prime}_{c}\bar{\Xi}_{c}$ $\Xi_{c}\bar{\Xi}^{\prime}_{c}$ $\Xi^{\prime}_{c}\bar{\Xi}^{\prime}_{c}$ $\Xi_{c}\bar{\Xi}_{c}$ $J^{P}$ Mass $\Xi^{*}_{c}\bar{\Sigma}^{*}_{c}$ $\Xi^{*}_{c}\bar{\Sigma}_{c}$ $\Xi^{\prime}_{c}\bar{\Sigma}^{*}_{c}$ $\Xi^{\prime}_{c}\bar{\Sigma}_{c}$ $\Xi_{c}\bar{\Sigma}^{*}_{c}$ $\Xi_{c}\bar{\Sigma}_{c}$ $J^{PC}$ Mass $\Lambda_{c}\bar{\Lambda}_{c}$ $3^{--}$ 5292 51.2 $3^{-}$ 5164 28.4 $1^{--}$ 4940 95 $2^{-+}$ 5329 247 12 12 13 13 $2^{-}$ 5250 265 22 90 52 4816 0.3 5240 $\times$ 88 88 5 5 5134 $\times$ 149 39 13 4584 108 5216 $\times$ $\times$ $\times$ 44 44 5116 $\times$ 64 114 0.01 $0^{-+}$ 4767 213 5168 $\times$ $\times$ $\times$ 107 107 5071 $\times$ $\times$ $\times$ 231 4649 160 24 5125 $\times$ $\times$ $\times$ 0 0 5051 $\times$ $\times$ $\times$ 48 $nsc(\bar{n}\bar{n})^{I=0}\bar{c}$ $(I=1/2)$ $2^{--}$ 5295 317 318 12 12 5017 $\times$ $\times$ $\times$ 24 $J^{P}$ Mass $\Xi_{c}^{*}\bar{\Lambda}_{c}$ $\Xi_{c}^{\prime}\bar{\Lambda}_{c}$ $\Xi_{c}\bar{\Lambda}_{c}$ 24 5200 $\times$ $\times$ 93 93 $1^{-}$ 5419 213 106 85 57 68 57 $2^{-}$ 5192 3 5155 $\times$ $\times$ 54 54 5235 118 179 1 4 19 10 5015 252 $1^{-+}$ 5326 63 63 28 28 41 41 5185 27 4 204 0.1 16 1 4971 40 5224 17 17 3 3 8 8 5120 $\times$ 0.4 23 78 20 0.1 $1^{-}$ 5447 2 1 0.1 5134 $\times$ $\times$ 88 88 9 9 5090 $\times$ $\times$ $\times$ 12 128 40 5028 82 2 0.1 5110 $\times$ $\times$ $\times$ $\times$ 35 35 5056 $\times$ $\times$ $\times$ 146 38 6 5005 185 116 21 5066 $\times$ $\times$ $\times$ $\times$ 44 44 5041 $\times$ $\times$ $\times$ 6 9 9 4970 6 61 3 $1^{--}$ 5421 111 88 88 15 15 12 12 84 23 5036 $\times$ $\times$ $\times$ 13 9 33 4943 30 177 0.4 5398 167 2 2 4 4 7 7 4 1 5008 $\times$ $\times$ $\times$ 23 $\times$ 249 4913 $\times$ 0.4 4 5288 $\times$ 63 63 6 6 7 7 18 0.2 4991 $\times$ $\times$ $\times$ 9 $\times$ 1 4896 $\times$ 13 109 5206 $\times$ $\times$ $\times$ 9 9 6 6 196 5 4970 $\times$ $\times$ $\times$ $\times$ $\times$ 0.1 4870 $\times$ 0.01 182 5153 $\times$ $\times$ $\times$ 96 96 11 11 $\times$ 13 4946 $\times$ $\times$ $\times$ $\times$ $\times$ 4 4822 $\times$ $\times$ 31 5114 $\times$ $\times$ $\times$ 4 4 100 100 $\times$ 11 $0^{-}$ 5285 402 53 8 4809 $\times$ $\times$ 27 5073 $\times$ $\times$ $\times$ $\times$ $\times$ 1 1 $\times$ 16 5151 $\times$ 288 2 4770 $\times$ $\times$ 2 5050 $\times$ $\times$ $\times$ $\times$ $\times$ 0 0 $\times$ 36 5071 $\times$ 26 36 $0^{-}$ 5027 1 13 5041 $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ 2 5024 $\times$ $\times$ 261 4996 169 38 5027 $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ 265 5015 $\times$ $\times$ 13 4914 173 38 5021 $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ 1 4962 $\times$ $\times$ 6 4894 2 237 19 5003 $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ 35 $nsc\bar{s}\bar{s}\bar{c}$ $(I=1/2)$ 4815 $\times$ 33 24 4987 $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ 8 $J^{P}$ Mass ${\Xi_{c}^{*}\bar{\Omega}_{c}^{*}}$ ${\Xi_{c}^{\prime}\bar{\Omega}_{c}^{*}}$ ${\Xi_{c}^{*}\bar{\Omega}_{c}}$ ${\Xi_{c}^{\prime}\bar{\Omega}_{c}}$ ${\Xi_{c}\bar{\Omega}_{c}^{*}}$ ${\Xi_{c}\bar{\Omega}_{c}}$ $(nn)^{I=0}c\bar{s}\bar{s}\bar{c}$ $(I=0)$ 19 4978 $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ 1 $3^{-}$ 5413 68 $J^{P}$ Mass $\Lambda_{c}\bar{\Omega}_{c}^{*}$ $\Lambda_{c}\bar{\Omega}_{c}$ 24 $0^{-+}$ 5407 407 9 9 44 14 $1^{-}$ 5590 216 71 88 53 45 43 $2^{-}$ 5135 78 5300 1 2 2 72 1 5432 98 13 144 17 9 9 5128 191 5235 $\times$ 19 19 245 16 5390 $\times$ 212 39 13 0.2 1 $1^{-}$ 5189 67 3 5131 $\times$ 136 136 $\times$ 22 5332 $\times$ $\times$ $\times$ 124 84 22 5138 106 32 5116 $\times$ 0.5 0.5 $\times$ 47 5307 $\times$ $\times$ $\times$ 68 91 1 5088 136 0.00005 5047 $\times$ 0.5 0.5 $\times$ 283 5274 $\times$ $\times$ $\times$ 9 72 23 5071 4 318 5009 $\times$ 3 3 $\times$ 1 5245 $\times$ $\times$ $\times$ $\times$ 15 185 4984 $\times$ 3 4976 $\times$ $\times$ $\times$ $\times$ 9 5244 $\times$ $\times$ $\times$ $\times$ 2 8 $0^{-}$ 5130 $\times$ 26 $0^{--}$ 5137 38 38 5227 $\times$ $\times$ $\times$ $\times$ $\times$ 80 5040 $\times$ 256 5117 62 62 5208 $\times$ $\times$ $\times$ $\times$ $\times$ 11 5007 $\times$ 33 24 5084 62 62 5183 $\times$ $\times$ $\times$ $\times$ $\times$ 0.1 $J^{P}$ Mass $\Xi_{c}^{\prime}\bar{\Omega}_{c}^{*}$ $\Xi_{c}^{\prime}\bar{\Omega}_{c}$ $\Xi_{c}\bar{\Omega}_{c}$ 24 $J^{P}$ Mass $\Xi_{c}^{*}\bar{\Omega}_{c}^{*}$ $\Xi_{c}^{\prime}\bar{\Omega}_{c}^{*}$ $\Xi_{c}^{*}\bar{\Omega}_{c}$ $\Xi_{c}^{*}\bar{\Omega}_{c}$ Mass $\Xi_{c}^{*}\bar{\Omega}_{c}^{*}$ $\Xi_{c}^{\prime}\bar{\Omega}_{c}^{*}$ $\Xi_{c}^{*}\bar{\Omega}_{c}$ $\Xi_{c}^{*}\bar{\Omega}_{c}$ $0^{-}$ 5492 1 0.2 0.1 $0^{-}$ 5211 $\times$ $\times$ 2 $2^{-}$ 5456 249 55 13 27 5295 $\times$ $\times$ $\times$ 294 5354 $\times$ 319 4 5176 $\times$ $\times$ 21 5371 $\times$ 108 86 13 5251 $\times$ $\times$ $\times$ 3 5267 $\times$ $\times$ 88 5356 $\times$ 70 111 0 5229 $\times$ $\times$ 209

IV.1 The $nnc\bar{n}\bar{n}\bar{c}$ subsystem

Firstly, we discuss the $nnc\bar{n}\bar{n}\bar{c}$ subsystem based on Fig. 1 (a). They have the same mass range as the excited states of $c\bar{c}$ . The $nnc\bar{n}\bar{n}\bar{c}$ subsystem can be divided into three situations: $(nn)^{I=1}c(\bar{n}\bar{n})^{I=1}\bar{c}$ , $(nn)^{I=0}c(\bar{n}\bar{n})^{I=1}\bar{c}$ , and $(nn)^{I=0}c(\bar{n}\bar{n})^{I=0}\bar{c}$ .

As for $(nn)^{I=1}c(\bar{n}\bar{n})^{I=0}\bar{c}$ states, they are antiparticles of the $(nn)^{I=0}c(\bar{n}\bar{n})^{I=1}\bar{c}$ states, thus they have the same mass spectra. We find no relative “stable” states for the $nnc\bar{n}\bar{n}\bar{c}$ system, that is, all of them can decay in $S$ -wave through strong interaction.

There are some hexaquark states which have the same quantum numbers among $(nn)^{I=1}c(\bar{n}\bar{n})^{I=1}\bar{c}$ $(nn)^{I=0}c(\bar{n}\bar{n})^{I=1}\bar{c}$ , and $(nn)^{I=0}c(\bar{n}\bar{n})^{I=0}\bar{c}$ . For example, both $(nn)^{I=1}c(\bar{n}\bar{n})^{I=1}\bar{c}$ and $(nn)^{I=0}c(\bar{n}\bar{n})^{I=1}\bar{c}$ have some states with the total isospin ${I=1}$ . The mass spectrum of these states should have been mixed, but all of transition matrix elements of CMI Hamiltonian are zero and thus they cannot be mixed under the CMI model. According to Fig. 1 (a), the masses of $(nn)^{I=1}c(\bar{n}\bar{n})^{I=1}\bar{c}$ states are usually larger than those of $(nn)^{I=0}c(\bar{n}\bar{n})^{I=1}\bar{c}$ states which are generally larger than those of $(nn)^{I=0}c(\bar{n}\bar{n})^{I=0}\bar{c}$ states. In the conventional baryon sectors, the $I=1$ one is usually heavier than the $I=0$ one, for example, see [ $\Sigma(1189)(I=1)$ vs $\Lambda(1116)(I=0)$ ] and [ $\Sigma_{c}(2455)(I=1)$ vs $\Lambda_{c}(2286)(I=0)$ ]. In our work, the wave functions of hexaquark states can be regarded as “baryon $\otimes$ antibaryon” configuration. These two factors may result into that the hexaquark with larger isospin is heavier than that with smaller isospin. The similar results can be found in Refs. Chen:2016ont ; Weng:2018mmf ; Weng:2019ynva .

The total isospin of $(nn)^{I=1}c(\bar{n}\bar{n})^{I=1}\bar{c}$ states can be $I=2$ , 1, and 0. Note that the symmetry property of $(nn)^{I=1}c(\bar{n}\bar{n})^{I=1}\bar{c}$ is determined from $I_{nn}$ and $I_{\bar{n}\bar{n}}$ . Thus, the $(nn)^{I=1}c(\bar{n}\bar{n})^{I=1}\bar{c}$ states are degenerate for the total isospin of $I=2$ , $1$ , and $0$ in the CMI model.

There are some $nnc\bar{n}\bar{n}\bar{c}$ neutral states with exotic quantum numbers $J^{PC}=0^{--}$ , $1^{-+}$ , and $3^{-+}$ which the traditional mesons ( $q\bar{q}$ ) cannot have. These exotic quantum number can help identify hidden-charm hexaquark states.

The notation $H_{n^{2}\bar{n}^{2}}(5036,2^{-},3^{--})$ is for a hexaquark state $nnc\bar{n}\bar{n}\bar{c}$ with mass around 5036 MeV and $I^{G}(J^{PC})=2^{-}(3^{--})$ . According to the Table 6, the overlap between $H_{n^{2}\bar{n}^{2}}(5036,2^{-},3^{--})$ and $\Sigma^{*}_{c}\bar{\Sigma}^{*}_{c}$ states is nearly 1, and thus the hexaquark is mainly made of a baryon and an antibaryon. It may behave like the ordinary scattering state if the inner interaction is not strong, but could also be a resonance or bound state dynamically generated by the baryon and antibaryon. The $H_{n^{2}\bar{n}^{2}}(5060,2^{+},2^{-+})$ , $H_{n^{2}\bar{n}^{2}}(5066,2^{+},0^{-+})$ , and others have similar situations. These kinds of hexaquarks deserve a more careful study.

The $nnc\bar{n}\bar{n}\bar{c}$ subsystem has one rearrangement decay mode: $nnc-\bar{n}\bar{n}\bar{c}$ . The $(nn)^{I=0}c(\bar{n}\bar{n})^{I=1}\bar{c}$ hexaquark states can decay to $\Lambda_{c}\bar{\Sigma}_{c}^{*}$ and $\Lambda_{c}\bar{\Sigma}_{c}$ , but the $J^{P}=2^{-}$ ( $0^{-}$ ) states can only decay into $\Lambda_{c}\bar{\Sigma}_{c}^{*}$ ( $\Lambda_{c}\bar{\Sigma}_{c}$ ) due to the angular momentum conservation. The $(nn)^{I=0}c(\bar{n}\bar{n})^{I=0}\bar{c}$ hexaquarks have only one decay channel $\Lambda_{c}\bar{\Lambda}_{c}$ while the $(nn)^{I=1}c(\bar{n}\bar{n})^{I=1}\bar{c}$ hexaquark states decay to $\Sigma_{c}^{(*)}\bar{\Sigma}_{c}^{(*)}$ in the “OZI-superallowed” decay mode.

One can extract the decay width information from Table 7. The $H_{n^{2}\bar{n}^{2}}(5143,2^{-},1^{--})$ and $H_{n^{2}\bar{n}^{2}}(5130,2^{-},1^{--})$ decay to all possible channel, but their partial width ratios are different. For the $H_{n^{2}\bar{n}^{2}}(5143,2^{-},1^{--})$ ,

\Gamma_{\Sigma_{c}^{*}\bar{\Sigma}_{c}^{*}}:\Gamma_{(\Sigma_{c}^{*}\bar{\Sigma}_{c})^{-}}:\Gamma_{\Sigma_{c}\bar{\Sigma}_{c}}=2.7:1.6:1,

(8)

and for $H_{n^{2}\bar{n}^{2}}(5130,2^{-},1^{--})$ ·

\Gamma_{\Sigma_{c}^{*}\bar{\Sigma}_{c}^{*}}:\Gamma_{(\Sigma_{c}^{*}\bar{\Sigma}_{c})^{-}}:\Gamma_{\Sigma_{c}\bar{\Sigma}_{c}}=7.4:11.5:1,

(9)

where $(\Sigma_{c}^{*}\bar{\Sigma}_{c})^{-}$ is short for the $(\Sigma_{c}^{*}\bar{\Sigma}_{c}-\Sigma_{c}\bar{\Sigma}_{c}^{*})/\sqrt{2}$ mode with $C=-1$ .

IV.2 The $ssc\bar{s}\bar{s}\bar{c}$ subsystem

The $ssc\bar{s}\bar{s}\bar{c}$ states can be considered as pure neutral particles. Some of them have normal quantum numbers $J^{PC}=0^{-+}$ , $1^{--}$ , $2^{-+}$ , $2^{--}$ , and $3^{--}$ , but others carry exotic quantum numbers $J^{PC}=0^{--}$ , $1^{-+}$ , and $3^{-+}$ . Meanwhile, all of these have many different rearrangement decay channels according to Fig. 1 (b) and thus their widths are relative broad.

For the two-body strong decay behaviors of the $ssc\bar{s}\bar{s}\bar{c}$ subsystem, the heaviest $H_{s^{2}\bar{s}^{2}}(5651,0^{+},0^{-+})$ has two decay modes,

\Gamma_{\Omega_{c}^{*}\bar{\Omega}_{c}^{*}}:\Gamma_{\Omega_{c}\bar{\Omega}_{c}}=7.4:1,

(10)

and its dominant decay mode is $\Omega_{c}^{*}\bar{\Omega}_{c}^{*}$ . The $H_{s^{2}\bar{s}^{2}}(5569,\\ 0^{-},1^{--})$ state can decay through all possible baryon-antibaryon channels, and

\Gamma_{\Omega_{c}^{*}\bar{\Omega}_{c}^{*}}:\Gamma_{(\Omega_{c}^{*}\bar{\Omega}_{c})^{-}}:\Gamma_{\Omega_{c}\bar{\Omega}_{c}}=22.6:1.7:1,

(11)

where the $(\Omega_{c}^{*}\bar{\Omega}_{c})^{-}$ means $(\Omega_{c}^{*}\bar{\Omega}_{c}-\Omega_{c}\bar{\Omega}_{c}^{*})/\sqrt{2}$ which is antisymmetric under the $C$ -parity transformation. Moreover, for the states $H_{s^{2}\bar{s}^{2}}(5429,0^{-},1^{--})$ and $H_{s^{2}\bar{s}^{2}}(5435,0^{+},0^{-+})$ , their masses are similar and they can only decay through ${\Omega_{c}\bar{\Omega}_{c}}$ mode. $H_{s^{2}\bar{s}^{2}}(5495,0^{+},0^{-+})$ and $H_{s^{2}\bar{s}^{2}}(5490,0^{+},2^{--})$ have similar masses but the former can decay into ${\Omega_{c}\bar{\Omega}_{c}}$ while the latter can decay through $(\Omega_{c}^{*}\bar{\Omega}_{c}-\Omega_{c}\bar{\Omega}_{c}^{*})/\sqrt{2}$ in $S$ -wave.

The rest of $ssc\bar{s}\bar{s}\bar{c}$ hexaquark states are below the baryon-antibaryon decay channels. Therefore, their mainly rearrangement decay channels should be meson-meson-meson decay channels.

IV.3 The $nnc\bar{s}\bar{s}\bar{c}$ subsystem

According to Fig. 1 (c), we discuss the mass spectra and decay behaviour of $nnc\bar{s}\bar{s}\bar{c}$ subsystem. For the $I=1$ $nnc\bar{s}\bar{s}\bar{c}$ states, they are explicitly exotic states. There are still no relative stable states in $nnc\bar{s}\bar{s}\bar{c}$ subsystem.

For the $I=0$ states, they have only two channels: ${\Lambda_{c}\bar{\Omega}_{c}^{*}}$ and ${\Lambda_{c}\bar{\Omega}_{c}}$ . The two states $H_{n^{2}\bar{s}^{2}}(5128,0,2^{-})$ and $H_{n^{2}\bar{s}^{2}}(5130,0,0^{-})$ can be distinguished by their respective decay modes. The $H_{n^{2}\bar{s}^{2}}(5128,0,2^{-})$ can dissociate into $\Lambda_{c}\bar{\Omega}_{c}^{*}$ while $H_{n^{2}\bar{s}^{2}}(5130,0,0^{-})$ can decay into $\Lambda_{c}\bar{\Omega}_{c}$ in $S$ -wave.

There are four different decay channels for the $I=1$ states: ${\Sigma_{c}^{*}\bar{\Omega}_{c}^{*}}$ , ${\Sigma_{c}\bar{\Omega}_{c}^{*}}$ , ${\Sigma_{c}^{*}\bar{\Omega}_{c}}$ , and ${\Sigma_{c}\bar{\Omega}_{c}}$ . From Table 7, for $H_{n^{2}\bar{s}^{2}}(5415,1,1^{-})$ state,

\Gamma_{\Sigma_{c}^{*}\bar{\Omega}_{c}^{*}}:\Gamma_{\Sigma_{c}\bar{\Omega}_{c}^{*}}:\Gamma_{\Sigma_{c}^{*}\bar{\Omega}_{c}}:\Gamma_{\Sigma_{c}\bar{\Omega}_{c}}=9.4:2.2:1.9:1.

(12)

and for the $H_{n^{2}\bar{s}^{2}}(5312,1,1^{-})$ state

\Gamma_{\Sigma_{c}^{*}\bar{\Omega}_{c}^{*}}:\Gamma_{\Sigma_{c}\bar{\Omega}_{c}^{*}}:\Gamma_{\Sigma_{c}^{*}\bar{\Omega}_{c}}:\Gamma_{\Sigma_{c}\bar{\Omega}_{c}}=1.6:0.2:8:1.

(13)

IV.4 The $nsc\bar{n}\bar{n}\bar{c}$ subsystem

We discuss the mass spectra and decay behaviors of $nsc\bar{n}\bar{n}\bar{c}$ subsystem based on Fig. 2 (a). The $nnc\bar{n}\bar{s}\bar{c}$ states are antiparticles of the $nsc\bar{n}\bar{n}\bar{c}$ states, and thus they have the same mass spectra.

The $nsc\bar{n}\bar{n}\bar{c}$ subsystem can be divided into two situations: $nsc(\bar{n}\bar{n})^{I=1}\bar{c}$ and $nsc(\bar{n}\bar{n})^{I=0}\bar{c}$ . For the $nsc(\bar{n}\bar{n})^{I=1}\bar{c}$ states, the mass spectra are identical for total isospin of $I=3/2$ and $1/2$ in CMI model similar to $(nn)^{I=1}c(\bar{n}\bar{n})^{I=1}\bar{c}$ subsystem. The $nsc\bar{n}\bar{n}\bar{c}$ states with $I=3/2$ are explicitly exotic and thus easily identifiable as candidates for the hidden-charm hexaquark state.

From Fig. 2 (a), we find the lowest $0^{-}$ , $1^{-}$ , $2^{-}$ , and $3^{-}$ states are relatively stable states, and especially the $H_{ns\bar{n}^{2}}(3578,1/2,0^{-})$ is below all the thresholds for rearrangement decay channels. Other three states can still decay via $D$ -wave strong interaction. For example, the $H_{ns\bar{n}^{2}}(4682,{1}/{2},3^{-})$ can decay into $KD^{*}\bar{D}$ final states via $D$ -wave.

From Table 7, there are 6 and 3 possible baryon-antibaryon channels for the $nsc(\bar{n}\bar{n})^{I=1}\bar{c}$ and $nsc(\bar{n}\bar{n})^{I=0}\bar{c}$ subsystems, respectively. The $H_{ns\bar{n}^{2}}(5071,3/2,2^{-})$ and $H_{ns\bar{n}^{2}}(5071,3/2,0^{-})$ are accidentally degenerate, but the $J^{P}=2^{-}$ state can decay into $\Xi_{c}^{*}\bar{\Sigma}_{c}^{*}$ , $\Xi_{c}^{*}\bar{\Sigma}_{c}$ , $\Xi_{c}^{\prime}\bar{\Sigma}_{c}^{*}$ , and $\Xi_{c}^{\prime}\bar{\Sigma}_{c}$ while the $J^{P}=0^{-}$ state can only decay through $\Xi_{c}\bar{\Sigma}_{c}^{*}$ , and $\Xi_{c}\bar{\Sigma}_{c}$ channels. Similarly, $H_{ns\bar{n}^{2}}(4971,1/2,2^{-})$ can only decay through ${\Xi_{c}^{*}\bar{\Lambda}_{c}}$ mode, but $H_{ns\bar{n}^{2}}(4970,1/2,1^{-})$ can only decay into ${\Xi_{c}^{\prime}\bar{\Lambda}_{c}}$ , ${\Xi_{c}^{\prime}\bar{\Lambda}_{c}}$ and ${\Xi_{c}\bar{\Lambda}_{c}}$ modes. We can distinguish $H_{ns\bar{n}^{2}}(4913,1/2,1^{-})$ and $H_{ns\bar{n}^{2}}(4914,1/2,0^{-})$ by partial decay width ratios since for the former,

\Gamma_{\Xi_{c}^{\prime}\bar{\Lambda}_{c}}:\Gamma_{\Xi_{c}\bar{\Lambda}_{c}}=0.1:1.

(14)

and for the latter,

\Gamma_{\Xi_{c}^{\prime}\bar{\Lambda}_{c}}:\Gamma_{\Xi_{c}\bar{\Lambda}_{c}}=4.6:1.

(15)

IV.5 The $nsc\bar{n}\bar{s}\bar{c}$ subsystem

Here, we discuss the $nsc\bar{n}\bar{s}\bar{c}$ subsystem based on Fig. 2 (b). The subsystem is also a pure neutral subsystem, thus $C$ parity and $G$ parity are good quantum numbers. Since there is no constraint from the Pauli principle for $nsc\bar{n}\bar{s}\bar{c}$ subsystem, the values of $\delta^{A}_{12}$ , $\delta^{S}_{12}$ , $\delta^{A}_{34}$ , and $\delta^{S}_{34}$ from Table 4 are all 1 and the obtained mass spectra is more complicated than other subsystems. There are no relative stable states for the $nsc\bar{n}\bar{s}\bar{c}$ subsystem.

Similar to the $nnc\bar{n}\bar{n}\bar{c}$ states, the mass spectra of $nsc\bar{n}\bar{s}\bar{c}$ states are identical for total isospin of $I=1$ and $0$ in CMI model. From Fig. 2 (b), we find nine good exotic states candidates for quantum numbers $J^{PC}=0^{--}$ .

The mass of $H_{ns\bar{n}\bar{s}}(5117,1^{-}(0^{+}),0^{--})$ is very closed to $H_{ns\bar{n}\bar{s}}(5114,1^{-}(0^{+}),1^{--})$ , and they all can decay into $(\Xi_{c}^{\prime}\bar{\Xi}_{c}-\Xi_{c}\bar{\Xi}_{c}^{\prime})/\sqrt{2}$ . However, the $H_{ns\bar{n}\bar{s}}(5114,1^{+}(0^{-}),1^{--})$ has others decay channels. From Table 7, we obtain for $H_{ns\bar{n}\bar{s}}(5114,1^{+}(0^{-}),1^{--})$

\displaystyle\Gamma_{(\Xi_{c}^{*}\bar{\Xi}_{c})^{-}}:\Gamma_{(\Xi_{c}^{\prime}\bar{\Xi}_{c})^{-}}:\Gamma_{\Xi_{c}\bar{\Xi}_{c}}=0.6:18.1:1,

(16)

where $(\Xi_{c}^{*}\bar{\Xi}_{c})^{-}$ and $(\Xi_{c}^{\prime}\bar{\Xi}_{c})^{-}$ represent $(\Xi_{c}^{*}\bar{\Xi}_{c}-\Xi_{c}\bar{\Xi}_{c}^{*})/\sqrt{2}$ and $(\Xi_{c}^{\prime}\bar{\Xi}_{c}-\Xi_{c}\bar{\Xi}_{c}^{\prime})/\sqrt{2}$ respectively.

IV.6 The $nsc\bar{s}\bar{s}\bar{c}$ subsystem

Lastly, we discuss the mass spectra and decay behaviour of $nsc\bar{s}\bar{s}\bar{c}$ subsystem based on the Fig. 3 (a). The $ssc\bar{n}\bar{s}\bar{c}$ states are antiparticles of the $nsc\bar{s}\bar{s}\bar{c}$ states, and thus they have the same mass spectra. The restrictions from Pauli principle for the $nsc\bar{s}\bar{s}\bar{c}$ states are the same as the $nsc(\bar{n}\bar{n})^{I=1}\bar{c}$ states, and therefore the numbers of their states are equal.

From Fig. 3 (a), we easily find that there are no relative stable states. The $nsc\bar{s}\bar{s}\bar{c}$ states are higher than many different rearrangement decay channels. Therefore, they would have a relative wide width. In conclusion, we do not suggest that the experimentalists foremost find these states.

The reference baryon-antibaryon systems for the $nsc\bar{s}\bar{s}\bar{c}$ states are the $\Xi^{*}_{c}\bar{\Omega}^{*}_{c}$ , $\Xi^{\prime}_{c}\bar{\Omega}^{*}_{c}$ , $\Xi_{c}\bar{\Omega}^{*}_{c}$ , $\Xi^{*}_{c}\bar{\Omega}_{c}$ , $\Xi^{\prime}_{c}\bar{\Omega}_{c}$ , and $\Xi_{c}\bar{\Omega}_{c}$ . The mass of $H_{ns\bar{s}^{2}}(5356,1/2,2^{-})$ is close to that of $H_{ns\bar{s}^{2}}(5354,1/2,0^{-})$ . For $H_{ns\bar{s}^{2}}(5356,1/2,2^{-})$ state, it can decay through ${\Xi_{c}^{\prime}\bar{\Omega}_{c}^{*}}$ , ${\Xi_{c}^{*}\bar{\Omega}_{c}}$ , and ${\Xi_{c}\bar{\Omega}_{c}}$ channels. But $H_{ns\bar{s}^{2}}(5354,1/2,0^{-})$ can decay into ${\Xi_{c}^{\prime}\bar{\Omega}_{c}}$ and ${\Xi_{c}\bar{\Omega}_{c}}$ channels. Then we consider $H_{ns\bar{s}^{2}}(5245,1/2,2^{-})$ and $H_{ns\bar{s}^{2}}(5244,1/2,2^{-})$ , and both of them can decay through ${\Xi_{c}\bar{\Omega}_{c}^{*}}$ and ${\Xi_{c}\bar{\Omega}_{c}}$ channels in $S$ -wave. From Table 7, for $H_{ns\bar{s}^{2}}(5245,1/2,2^{-})$ ,

\displaystyle\Gamma_{\Xi_{c}\bar{\Omega}_{c}^{*}}:\Gamma_{\Xi_{c}\bar{\Omega}_{c}}=0.1:1,

(17)

and for $H_{ns\bar{s}^{2}}(5244,1/2,2^{-})$ ,

\displaystyle\Gamma_{\Xi_{c}\bar{\Omega}_{c}^{*}}:\Gamma_{\Xi_{c}\bar{\Omega}_{c}}=0.3:1.

(18)

V SUMMARY

Up to now, more and more hidden-charm tetraquark states and pentaquark states have been discovered and confirmed by different experiments. These give us a significant confidence to the existence of hidden-charm hexaquark states. Thus, we studied systemically the mass spectra, stability, and strong decay behaviors of hidden-charm hexaquark states in the framework of the CMI model.

Table 8: The relatively stable states of hidden-charm hexaquark system in CMI model. The masses are all in units of MeV.

States	$I^{G}(J^{PC})$	Masses	States	$I^{(G)}(J^{P(C)})$	Masses
$nsc\bar{n}\bar{s}\bar{c}$	$0^{+}(0^{-+})$	3815	$nsc\bar{n}\bar{n}\bar{c}$	$1/2(0^{-})$	3578
	$0^{-}(1^{--})$	4005		$1/2(1^{-})$	3670
	$0^{-}(2^{--})$	4443		$1/2(2^{-})$	4090
	$1^{+}(2^{--})$	4443		$1/2(3^{-})$	4682
	$0^{-}(3^{--})$	4794	$nnc\bar{n}\bar{n}\bar{c}$	$0^{+}(1^{-+})$	3887
	$1^{+}(3^{--})$	4794	$nnc\bar{n}\bar{n}\bar{c}$	$0^{-}(3^{--})$	4576

Firstly, we introduce the CMI model and extract the corresponding coupling constants from traditional hadrons. Next, we construct the flavor $\otimes$ color $\otimes$ spin wavefunctions based on the SU(3) and SU(2) symmetry. Meanwhile, we require the wavefunction to obey Pauli Principle. After that, we systemically calculate the mass spectra, corresponding overlap, and the values of $k\cdot|c_{i}|$ . Lastly, we specifically discuss the stability, the possible quark rearrangement decay channels, and the relative decay width ratios.

For $nnc\bar{n}\bar{n}\bar{c}$ , $ssc\bar{s}\bar{s}\bar{c}$ , and $nsc\bar{n}\bar{s}\bar{c}$ subsystems, they are pure neutral particles (except $(nn)^{I=0}c(\bar{n}\bar{n})^{I=1}\bar{c}$ subsystem), and $C$ parity and $G$ parity both are good quantum numbers. According to the mass spectra, we find that the lower isospin quantum number, the more compact hexaquark states. Here, the $J^{PC}=0^{--},1^{-+},3^{-+}$ states are good exotic states candidates, and especially the $0^{--}$ states which even the $S$ -wave tetraquark states cannot carry.

We list some possible stable hexaquark states in Table 8. We find ten relative stable states, which are below all allowed rearrangement decay channels. These states belong to the $nnc\bar{n}\bar{n}\bar{c}$ subsystem, $nsc\bar{n}\bar{n}\bar{c}$ subsystem and $nsc\bar{n}\bar{s}\bar{c}$ subsystem respectively. We think the $H_{ns{\bar{n}}^{2}}(3578,1/2,0^{-})$ and $H_{ns{\bar{n}}^{2}}(3670,1/2,1^{-})$ states are better stable candidates which could be first searched for in experiments.

Table 9: The comparison of the masses for the

(nn)^{I=0}c(\bar{n}\bar{n})^{I=0}\bar{c}

and

ssc\bar{s}\bar{s}\bar{c}

systems with

I^{G}(J^{PC})=0^{-}(1^{--})

in two scenarios. All masses are in units of MeV.

$nnc\bar{n}\bar{n}\bar{c}$		$ssc\bar{s}\bar{s}\bar{c}$
Scen.1	Scen.2	Scen.1	Scen.2
3600	3201	4836	4670
3736	3443	4940	4871
4197	3960	5100	5068
4234	4053	5175	5152
4325	4260	5275	5258
4432	4437	5287	5274
4548	4520	5329	5298
4584	4588	5429	5434
4816	4835	5522	5556
4940	5003	5569	5592

In order to check the uncertainty of our framework, we also determine the $v_{q\bar{q}}$ and $m_{q\bar{q}}$ with the masses of pseudoscalar mesons. Since the spontaneously breaking of vacuum symmetry strongly affects the properties of these pseudoscalar mesons, the parameters of $v_{q\bar{q}}$ and $m_{q\bar{q}}$ are not the same as those obtained with the vector mesons. For example, $v_{n\bar{n}}$ and $m_{n\bar{n}}$ become 29.87 MeV and 153.99 MeV in the new scenario, respectively. However, the difference between the hexquark masses of the two scenarios can be roughly used to estimate the uncertainly of our approach. We give the comparison of the $(nn)^{I=0}c(\bar{n}\bar{n})^{I=0}\bar{c}$ and $ssc\bar{s}\bar{s}\bar{c}$ systems with $I^{G}(J^{PC})=0^{-}(1^{--})$ in Table 9. Scen.1 (Scen.2) denotes the results calculated by using the parameters obtained with the vector (pseudoscalar) mesons. Firstly, one notices that the ground states differ largest from the table. The heavier the state is, the smaller the difference between the two scenarios is. These may be resulted from that the new $v_{q\bar{q}}$ becomes larger while the new $m_{q\bar{q}}$ becomes smaller. Secondly, the mass of the $(nn)^{I=0}c(\bar{n}\bar{n})^{I=0}\bar{c}$ ground state with $I^{G}(J^{PC})=0^{-}(1^{--})$ changes about 399 MeV while that for the $ssc\bar{s}\bar{s}\bar{c}$ case only varies about 166 MeV. That is, the uncertainty reduces when the number of $n/\bar{n}$ quark in hexaquark states decreases, which is because the mass difference between $\pi$ and $\rho$ mesons is much larger than those between $K$ and $K^{*}$ mesons.

In summary, we give a preliminary study about the mass spectra of hidden-charm hexaquark states. In addition to the CMI model, other non-perturbative QCD methods can also help us to understand more properties of the hexaquark states in detail such as QCD sum rule, effective fields theories and lattice QCD simulations. We hope that our study may inspire theorists and experimentalists to pay attention to these hidden-charm hexaquark states.

VI Acknowledgments

This work is supported by the China National Funds for Distinguished Young Scientists under Grant No. 11825503, National Key Research and Development Program of China under Contract No. 2020YFA0406400, the 111 Project under Grant No. B20063, and the National Natural Science Foundation of China under Grant No. 12047501. This project is also supported by the National Natural Science Foundation of China under Grants No. 12175091, and 11965016, CAS Interdisciplinary Innovation Team, and the Fundamental Research Funds for the Central Universities under Grants No. lzujbky-2021-sp24.

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