Where are the hidden-charm hexaquarks?

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 XY Z states and P c states have been observed [1][2][3][4][5][6][7][8][9][10][11]. The present situation of hadronic states is far beyond the conventional quark model. The first doubly charm tetraquark T + cc with the configuration ccūd was observed by the LHCb Collaboration [12], and this newly discovered particle is explicitly an exotic state which cannot be classified into the conventional mesons.
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. * zhliu20@lzu.edu.cn † anht14@lzu.edu.cn ‡ liuzhanwei@lzu.edu.cn § xiangliu@lzu.edu.cn Very recently, BESIII collaboration measured the cross section of the process e + e − → π + π − ψ(3686) and further confirms the existence of three charmonium-like states wherein Y (4660) is closed to the threshold of Λ c -Λ c systems [46]. Before this, the structure Y (4660) has been observed in the process of e + e − → γ ISR π + π − ψ(3686) in the Belle and BarBar experiments [47][48][49]. Y (4660) was interpreted as a higher charmonium in Ref. [50] and a hexaquark state configured by the triquark-antitriquark clusters in Ref. [53]. The charmonium states can very likely be bound inside light hadronic matters, and such hadrocharmonium may explain the properties of the Y (4660) peak [51]. 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. [52].
The Λ c -Λ c structure was introduced to explain the production and decays of Y (4260) in Refs. [53][54][55]. Y (4630) was observed in process e + e − → Λ cΛc in the Belle experiments [56] and is considered as a candidate of Λ cΛc bound state [57]. Especially, heavy baryon chiral perturbation theory was applied to systemically study the Λ c -Λ c , Σ c -Σ c , and Λ b -Λ b systems [58], and the results suggest that Y (4260) and Y (4360) could be Λ c -Λ c baryonia. The two states are also suggested to be a mixture, with mixing close to maximal, of two states of hadrochamonium [59].
The masses of baryonia with the open and hidden charm, bottomness and strangeness are studied in the framework of dispersion relation technique in Refs. [60][61][62]. The heavy baryon-antibaryon molecule states are investigated within the effective field theory [63]. The hidden-charm and hidden-bottom hexaquark states were discussed within the QCD sum rules [64,65].
These work stimulate us to further study the hiddencharm 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 [66,67]. The CMI model has been successfully adopted to study the mass spectra and stabil-ity of multiquark states [68][69][70][71][72][73][74][75][76][77][78][79][80][81][82][84][85][86][87][88][89]. 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 ⊗ color ⊗ 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 where m i is the effective mass of the i-th constituent (anti) quark, and λ i and σ i are Gell-Mann and Pauli matrices, respectively. For the antiquark, λq = −λ * q and σq = σ * q . The dynamical effect of spatial wavefunctions plays an important role in the study of hadron spectrum. Chromomagnetic interaction is nonrelativistic in the Schrödinger equation in Ref. [83] wherein the authors used the spatial wave functions with harmonicoscillator expansion. The C ij is effective coupling constant between the i-th (anti) quark and j-th (anti) quark 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 heavylight system, and introduced the color interaction (the spin-independent color Coulomb-like terms in the onegluon-exchange interactions) in Refs. [84][85][86]. We also introduce a color term into our model Refs. [84,86] The nonvanishing color interaction coefficient A ij implies a change of the effective masses. We can rewrite the CMI Hamiltonian as Ref. [86] where 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 [86]. In the present work, v qq and m qq are only determined by vector mesons (q = n, s and n = u, d). We present the obtained effective coupling parameters in Table I.

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 flavorcolor-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][(q 3q4 )c] 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 cc, ss, and nn pairs. Based on these, we list all the possible flavor combinations for the hidden-charm hexaquark system in Table II. In Table II, 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 II, nncnnc has isospin I = (2, 1, 0) and nscnsc has isospin I = (1, 0). In the second line, the isospin I can be (3/2, 1/2) for nscnnc, (1, 0) for nncssc, and 1/2 for nscssc.
Next, we briefly introduce the color wavefunctions for all hexaquark systems. They can be deduced from the following direct product:  (6) where A (S) means totally symmetric (antisymmetric), and MS (MA) means that q 1 q 2 orq 3q4 is symmetric (antisymmetric). Here, the color-singlet wavefunctions for the hexaquarks are shown in Table III. In the notation |[(q 1 q 2 ) color1 c] color3 [(q 3q4 ) color2c ] color4 , the color1, color2, color3, and color4 stand for the color representations of q 1 q 2 ,q 3q4 , q 1 q 2 c, andq 3q4c , respectively.
Considering the Pauli principle, we obtain 54 types of total wavefunctions and present them in the first part of Table IV. Some wavefunctions are the eigenstates of C parity like [φ SS ⊗ χ 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 IV. We introduce notations δ A 12 , δ S 12 , δ A 34 , and δ S 34 . When the two light quarks or antiquarks are antisymmetric (symmetric) in the flavor space, δ A 12 = 0 (δ S 12 = 0), or else δ A 12 = 1 (δ S 12 = 1). The hidden-charm hexaquark states can be categorized into 6 classes, and we present them in third part of Table IV.

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 hiddencharm hexaquark states.
From the eigenvalues, we present the mass spectra in Fig. 1 (for nncnnc, sscssc, and nncssc), Fig. 2 (for nncnsc and nscnsc), and Fig. 3 (for nscssc). 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).
The partial width of the two body L-wave "OZIsuperallowed decay" mode reads [87][88][89] where α 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, γ 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, Σ c (Ξ * c ) and Σ * c (Ξ c ) have the same spatial wavefunction. Based on these, we assume the γ i relationships for different hidden-charm hexaquark states presented in Table V. We find that the (k/m) 2 is of 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 VI, we calculate the values of k ·|c i | 2 for each decay process and present them in Table VII. The blank area in Tables VI and VII means that the hexaquark state is forbidden to decay through this channel because of the quantum number conservation. According to the γ i relationships in Table V and the values of k · |c i | 2 in Table  TABLE IV. 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 All possible types of total wavefunctions for pure neutral hexaquark system Different classes of the hidden-charm hexaquark system VII, we can roughly estimate the relative decay widths for different two-body decay processes of a hexaquark state.
Firstly, we discuss the nncnnc subsystem based on Fig.  1 (a). They have the same mass range as the excited states of cc. The nncnnc subsystem can be divided into three situations: (nn) I=1 c(nn) I=1c , (nn) I=0 c(nn) I=1c , and (nn) I=0 c(nn) I=0c .
As for (nn) I=1 c(nn) I=0c states, they are antiparticles of the (nn) I=0 c(nn) I=1c states, thus they have the same mass spectra. We find no relative "stable" states for the nncnnc 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(nn) I=1c (nn) I=0 c(nn) I=1c , and (nn) I=0 c(nn) I=0c . For example, both (nn) I=1 c(nn) I=1c and (nn) I=0 c(nn) I=1c 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(nn) I=1c states are usually larger than those of (nn) I=0 c(nn) I=1c states which are generally larger than those of (nn) I=0 c(nn) I=0c states. In the conventional baryon sectors, the I = 1 one is usually heavier than the I = 0 one, for example, see [Σ(1189)(I = 1) vs Λ(1116)(I = 0)] and [Σ c (2455)(I = 1) vs Λ c (2286)(I = 0)]. In our work, the wave functions of hexaquark states   can be regarded as "baryon ⊗ 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. [69,86,87]. The total isospin of (nn) I=1 c(nn) I=1c states can be I = 2, 1, and 0. Note that the symmetry property of (nn) I=1 c(nn) I=1c is determined from I nn and Inn. Thus, the (nn) I=1 c(nn) I=1c states are degenerate for the total isospin of I = 2, 1, and 0 in the CMI model.
There are some nncnnc neutral states with exotic quantum numbers J P C = 0 −− , 1 −+ , and 3 −+ which the traditional mesons (qq) cannot have. These exotic quantum number can help identify hidden-charm hexaquark states.
The notation H n 2n2 (5036, 2 − , 3 −− ) is for a hexaquark state nncnnc with mass around 5036 MeV and I G (J P C ) = 2 − (3 −− ). According to the Table VI, the overlap between H n 2n2 (5036, 2 − , 3 −− ) and Σ * cΣ * 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 2n2 (5060, 2 + , 2 −+ ), H n 2n2 (5066, 2 + , 0 −+ ), and others have similar situations. These kinds of hexaquarks deserve a more careful study.
Meanwhile, all of these have many different rearrangement decay channels according to Fig. 1 (b) and thus their widths are relative broad.

C. The nncssc subsystem
According to Fig. 1 (c), we discuss the mass spectra and decay behaviour of nncssc subsystem. For the I = 1 nncssc states, they are explicitly exotic states. There are still no relative stable states in nncssc subsystem.
There are four different decay channels for the I = 1 states: Σ * We discuss the mass spectra and decay behaviors of nscnnc subsystem based on Fig. 2 (a). The nncnsc states are antiparticles of the nscnnc states, and thus they have the same mass spectra. The nscnnc subsystem can be divided into two situations: nsc(nn) I=1c and nsc(nn) I=0c . For the nsc(nn) I=1c 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(nn) I=1c subsystem. The nscnnc states with I = 3/2 are explicitly exotic and thus easily identifiable as candidates for the hidden-charm hexaquark state.
From Table VII, there are 6 and 3 possible baryon-antibaryon channels for the nsc(nn) I=1c and nsc(nn) I=0c subsystems, respectively.
E. The nscnsc subsystem Here, we discuss the nscnsc 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 nscnsc subsystem, the values of δ A 12 , δ S 12 , δ A 34 , and δ S 34 from Table IV are all 1 and the obtained mass spectra is more complicated than other subsystems. There are no relative stable states for the nscnsc subsystem. Similar to the nncnnc states, the mass spectra of nscnsc 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 P C = 0 −− .

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 hiddencharm hexaquark states in the framework of the CMI model.  Firstly, we introduce the CMI model and extract the corresponding coupling constants from traditional hadrons. Next, we construct the flavor ⊗ color ⊗ 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 · |c i |. Lastly, we specifically discuss the stability, the possible quark rearrangement decay channels, and the relative decay width ratios.
For nncnnc, sscssc, and nscnsc subsystems, they are pure neutral particles (except (nn) I=0 c(nn) I=1c subsys-tem), 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 P C = 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 VIII. We find ten relative stable states, which are below all allowed rearrangement decay channels. These states belong to the nncnnc subsystem, nscnnc subsystem and nscnsc subsystem respectively. We think the H nsn 2 (3578, 1/2, 0 − ) and H nsn 2 (3670, 1/2, 1 − ) states are better stable candidates which could be first searched for in experiments.   In order to check the uncertainty of our framework, we also determine the v qq and m qq 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 qq and m qq are not the same as those obtained with the vector mesons. For example, v nn and m nn 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(nn) I=0c and sscssc systems with I G (J P C ) = 0 − (1 −− ) in Table IX. 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 qq becomes larger while the new m qq becomes smaller. Secondly, the mass of the (nn) I=0 c(nn) I=0c ground state with I G (J P C ) = 0 − (1 −− ) changes about 399 MeV while that for the sscssc case only varies about 166 MeV. That is, the uncertainty reduces when the number of n/n quark in hexaquark states decreases, which is because the mass difference between π and ρ 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.