Investigating full-heavy tetraquarks composed of cc ¯ c ¯ b and bb ¯ b ¯ c

The full-heavy tetraquarks cc ¯ c ¯ b and bb ¯ b ¯ c are systematically investigated within a quark model. The meson–meson structure, diquark–antidiquark structure and K-structure are considered in this work. There is no bound state for cc ¯ c ¯ b and bb ¯ b ¯ c systems in I J P = 00 + , 01 + and 02 + channels. However, for cc ¯ c ¯ b system, three possible resonance states with energy of 10 , 079 MeV, 10 , 081 MeV and 10 , 177 MeV are found in I J P = 00 + , 01 + and 02 + , respectively, and their decay width (cid:2) are 6.7–8.4 MeV, 1.4–7.2 MeV and 9.1–11.1 MeV. For bb ¯ b ¯ c system, there also exist three possible resonance states with energy of 16 , 474 MeV, 16 , 474 MeV and 16 , 541 MeV in I J P = 00 + , 01 + and 02 + , respectively, and the decay widths (cid:2) of them are 2.2–6.1 MeV, 2.2–6.9 MeV and 5.3–8.5 MeV. bc ¯ c ¯ c and cb ¯ b ¯ b systems will have the same results as cc ¯ c ¯ b and bb ¯ b ¯ c , respectively. These full-heavy resonance states are worthy to be searched in the future experiments.


I. INTRODUCTION
In the past few decades, many discoveries about exotic states inspired extensive interest in probing the structures of the multiquark hadrons.Gell-Man and Zweig have been proposed and invented multiquark states in the early quark models [1,2].In 2003, Belle collaboration announced the observation of exotic state X(3872) [3], and other experimental groups also found this exotic state [4][5][6], which make the tetraquark state became a heated research area.After that, plenty of exotic states were observed and investigated, for review articles, please see [7][8][9][10][11][12][13][14][15][16][17].In 2017, the CMS collaboration implemented a benchmark measurement of the Υ(1S) pair production at √ s = 8 TeV in pp collision [18].However, no evidence has been found in the Υ(1S)µ + µ − invariant mass spectrum by the LHCb collaboration [19].Therefore, the fully-bottom tetraquark needs more experiments to confirm its signal.For fully-charm tetraquark systems, J/Ψ pair production and double cc production have also been observed in experiments [20,21].The J/ΨJ/Ψ and η c (1S)η c (1S) channels are suggested to search for the doubly hidden-charm.In 2020, the LHCb collaboration reported their elementary conclusion on the observations of full-charm states, a narrower structure at 6.9 GeV with significance about 5σ, a broad structure in the range 6.2 to 6.8 GeV, and there is also a hint for a structure around 7.2 GeV [22].These experimental results greatly stimulated much interest in full-heavy tetraquark states.In 2022, based on all the proton-proton collisions collected from 2016 to 2018, the CMS collaboration has observed three structures X(6600), X(6900), and X(7300) in the invariant mass spectra of the charm quarks [23].
In 1975, Y. Iwasaki proposed that full-charm tetraquark with the mass of about 6.0 GeV or 6.2 GeV is a sharp resonance state [24].R.J. Lloyd et al. investigated cccc states and find several close-lying bound states [25].Berezhnoy et al. obtained the mass M 0 (cccc) = 6124 MeV and M 0 (bb bb ) = 18857 MeV without hyperfine splitting, which involving charmed and bottom tetraquarks, respectively [26].M. Karliner et al. discovered M (cccc) = 6192 ± 25MeV and M (bb bb ) = 18826 ± 25MeV for J P C = 0 ++ involving charmed and bottom tetraquarks, respectively [27].W. Chen et al. investigated cccc and bb bb states by a moment QCD sum rule method, and concluded that the mass of ΥΥ and η b η b are below and close to the corresponding thresholds except one current of J P C = 0 ++ , while the mass of cccc all above the thresholds [28].J. Wu et al. discussed the QQ Q Q configuration, and the result shows that the lowest J P = 1 + state of ccc b system should be less stable than that of bb bc as the cc interaction is stronger than the bb interaction [29].M. N. Anwar et al. found that the ground state bb bb tetraquark mass is predicted to be M (bb bb ) = 18.72 ± 0.02 GeV [30]. A. Esposito et al. proposed a model based on the conjecture of a short range diquark repulsion in a compact tetraquark to estimate masses and widths of tetraquarks, and found that the bb bb system with the mass of 18.8 GeV [31].G.J. Wang et al. systematically calculate the mass spectra of the Swave fully-heavy tetraquark states cccc, bb bb and bbcc in two nonrelativistic quark models, and the numerical results shows that the ground QQ Q Q tetraquark states are located above the corresponding scattering states [32].M.S. Liu et al. studied the all-heavy tetraquark systems, i.e., cccc, bb bb , bbcc/cc bb , bccc/cc bc, bc bb /bb bc, and bc bc within a potential model by including the linear confining potential, Coulomb potential, and spin-spin interactions, and the results showed that all diquark-antidiquark states are found to have masses above the corresponding the thresholds of (Q Q) − (Q Q) structure [33].Moreover, there are also many researches about the narrow structure X(6900) reported by LHCb in 2020 [34][35][36][37][38][39][40][41][42][43][44], and more papers can be found in Ref. [17] and reference there in.Some of these researches interprets X(6900) as excited state of diquark-antidiquark structure in the cccc system by various quark model [34,37,39,[41][42][43][44] and by QCD sum rules method [35,38,40], while other study suggest that the component of it contains the excited state of meson-meson structure mixing with diquark-antiquark structure [36].
In this paper, we investigate the possible resonance states of ccc b and bb bc systems in IJ P = 00 + , 01 + , and 02 + channels by using a quark model.The four-body configurations, which refers to meson-meson structure, diquark-antidiquark structure, and K-structure, as well as their couplings, are considered in the calculation.In Section.II, we give the introduction of the construction of wave functions.The numerical results and discussions are shown in Section.III.The summary is given in Section.IV.

II.
THE QUARK MODEL AND WAVE FUNCTIONS OF ccc b AND bb bc SYSTEMS Quantum chromodynamics(QCD) is recognized as a fundamental theory of strong interaction and the basic theory of multiquark states.However, with nonperturbative properties of QCD in the low energy region, it is inappropriate to directly use QCD theory to solve specific problems, such as hadron-hadron interactions and multiquark states.Therefore, some researchers have proposed and developed some quark models to solve these problems in low-energy regions.
For full-heavy tetraquark systems, the potential is composed of the confinement and the one-gluonexchange.The gluonic potential have various forms, such as linear confinement [36,39,[41][42][43][44][45], square confinement [34,46], and exponential confinement [36,37].The mass spectrum of these models is all in consistent with experimental results, and they all explained X(6900) [34,36,37,39,[41][42][43][44].In other words, the forms of the confinement and the one-gluon-exchange in fullheavy tetraquark systems need more experimental data to be determined.Here we use one of these forms [36,47] to study possible resonance state in ccc b and bb bc systems.
The Hamiltonian in this model for the present study is written as: where m i and p i refer to the mass and the kinetic of i-th quark(antiquark), T CM is the kinetic energy of the center of mass in tetraquark system; V CON ij and V OGE ij are the interactions of the confinement and the one-gluonexchange between the i-th and j-th quark, respectively.The V CON ij is written as and the V OGE ij express as where r ij stands for the distance between the two quarks/antiquarks, and σ indicates the SU(2) Pauli matrices, while λ c is SU(3) color Gell-Mann matrices, respectively.α s is an effective scale-dependent running coupling.µ is the reduced mass of two quarks, and it written as µ ij = mimj mi+mj .The a c , ∆, µ 0 , Λ 0 , µ c , and r 0 are parameters listed in Table I, and these parameters are taken from Ref. [47].The mass of mesons that related to the present work are shown in Table II.According to different coupling methods, ccc b and bb bc systems have three structures are considered, i.e., meson-meson structure, diquark-antidiquark structure and K-structure, and are shown in Fig. 1, Fig. 2 and Fig. 3.The hollow circles and black discs refer to quark and antiquark, respectively.The total wave functions of each structure are constructed by four parts: orbit, spin, flavor and color wave functions.The mesonmeson structure and diquark-antidiquark structure are constructed by coupling two sub-clusters wave functions, while the K-structure is coupling a quark on the basis of quark-antiquark system and then coupling another antiquark.In the following discussions, the spin, flavor and color wave functions of meson-meson structure and Kstructure are written in order of N 1 N 2 N 3 N 4 , while that of diquark-antidiquark structure is written in order of The total function of meson-meson structure (Ψ m ), diquark-antidiquark structure (Ψ di ) and K-structure (Ψ K ) are written as, For ccc b and bb bc systems, the two quarks with labels of 1 and 3 are identical particles, while the two antiquarks with labels of 2 and 4 are not identical particles.Therefore, the antisymmetrization operator A of mesonmeson structure, diquark-antidiquark structure and Kstructure is The Φ L , χ σ , χ f and χ c refers to orbit, spin, flavor and color wave function, respectively.

A. Gaussian expansion method(GEM)
This work focuses on radial excitation state of these tetraquark systems.The GEM has been successfully used in the calculation of few-body systems due to it just demands a small number of Gaussians to stabilize the system [49].Therefore, this work calculates the orbit wave function by GEM.In GEM, the radial part of orbit wave function of meson-meson structure, diquark-antidiquark structure and K-structure are all expanded by Gaussians: N n and c n means normalization constants and the variational parameters, respectively.n max is the Gaussian number, while r max and r min are parameters of the Gaussian size ν n , and the formulas of ν n are chosen according to the following geometric progression: For ccc b systems, the internal parameters clusters are chosen as r min = 0.01 (fm), r max = 2 (fm), n max = 12, and the parameters between the clusters are chosen as r min = 0.01 (fm), r max = 6 (fm), n max = 16 .For bb bc systems, the internal parameters clusters are chosen as r min = 0.08 (fm), r max = 2 (fm), n max = 15, and the parameters between the clusters are chosen as r min = 0.01 (fm), r max = 6 (fm), n max = 16.To test the stability, a second calculation was performed, which takes r min = 0.01 (fm), r max = 2 (fm), n max = 15 for internal clusters and r min = 0.01 (fm), r max = 6 (fm), n max = 20 between the clusters.The result of the energy for ground states and excited states are the same as the first calculation.Hence, the energy calculated by the Gauss number used for the first time is enough to stabilize the present study.

B. Spin wave function
The spin wave functions of meson-meson structure, diquark-antidiquark structure and K-structure are denoted by the subscript "χ σi S,Sz ", the S and S z refers to spin angular momentum and its third component, and i means the number of spin wave functions.α and β represent spin wave functions of spin-up (1 0) and spin-down (0 1) wave functions.Tables III and IV lists the spin wave function of quark, meson and baryon, and Tables V and VI lists all spin wave function of meson-meson structure, diquark-antidiquark structure and K-structure.

Meson Spin wave function
The spin wave function of meson-meson structure and diquark-antidiquark structure.

Number Spin wave function
TABLE VI.The spin wave function of K-structure.

C. Flavor wave function
The total flavor wave functions are where χ f1 m and χ f2 di refers to flavor of meson-meson structure and diquark-antidiquark structure, while χ f3 K is same as χ f1 m and represents flavor of K-structure.

D. Color wave function
The color wave function of meson-meson structure, diquark-antidiquark structure and K-structure have six channels, and they were written in Table VII.
Classify all color wave functions to 1 1 and 8 8 have physical sense.We can separate color configurations 1 1 to infinitely far, because the expectation value of the operator λ c i • λ c j is zero, when i and j belongs to different cluster.However, if we try to separate color octet-octet 8 8 structure, the operator λ c i • λ c j provide attractive force, when i and j belongs to different cluster.As confinement potential existent in infinite far, the energy of 8 8 goes to an infinite value due to the color confinement.In this sense, the physical channel with the color configurations 3 3, 6 6 and 8 8 are "compact state".
Because |ψ c , where Ψ i is a total wave function of a physical channel expressed in Eqs. ( 5)- (7).So, the component of physical channel with 1 1 is calculated by: and the component of physical channel with 8 8 is: One can easily show that P 1 1 + P 8 8 ≡ 1.

III. RESULTS AND DISCUSSIONS
We investigate full-heavy tetraquarks ccc b and bb bc systems in three kind of quark structures, i.e., meson-meson structure, diquark-antidiquark structure and Kstructure, and construct all possible physical channels according to the quantum numbers IJ P = 00 + , 01 + , and 02 + .The physical channel and the mass are shown in Tables VIII and IX.The first column of these tables shows the wave function of orbit, spin, flavor and color for each channel.We denote the total function by the subscript [i, j, k] type , "i, j, k" indicates spin, flavor and color, and "type" denotes mesonmeson structure, diquark-antidiquark structure and Kstructure, i.e., "m", "di" and "K", respectively.The second columns "channels" enumerates the subscript [χ in Kstructure, respectively.The columns headed with E th means the theoretical thresholds for two meson systems, which is the energy when separate two color singlet mesons infinitely far from each other.
For a given IJ P , the E sc refers to the energy of each single-channel, and the E cc denotes the lowest energies of the coupling of all channels, respectively [39].Which are written as: Where c i is the eigenvector of corresponding energy.
There are no bound states in Tables VIII and IX.However, some resonance state could exist in the higher energy region.The ground and excited states of "compact state", i.e., color configurations 3 3, 6 6 and 8 8, could be the candidate for resonance state.To assess the stability of these resonance states, we use the real-scaling method(RSM) [50].The RSM is a way to find the possible resonance state.In this method, the Gaussian size parameters r n between two color-singlet sub-clusters are scaled by multiplying a factor α, i.e., r → αr.The resonance state might exist if the avoid-crossing structure appears repeatedly with the increasing α.The repeated avoid-crossing structures are caused by the interaction between the interaction of scattering state with higher energy and resonance state at larger distances.Then, the decay widths could be calculated by the following formula [50].
TABLE VIII.The energies (in MeV) of all structures for tetraquarks ccc b.
Here V (α c ) stands for half of the minimal energy difference between resonance state and scattering state, while S r and S c represent the slope of the resonance state and the scattering state, respectively.The avoid-crossing structures could be caused by two ways: (i) the interaction between the scattering and resonance states with the increasing α, and these avoidcrossing structures would be possible resonance states, (ii) the interaction between two scattering states with different decay rates, which means its dominated component of the structure are scattering states, so these avoidcrossing structures couldn't regard as resonance states.
To determine a state is a true resonance or not is whether it have resonance mechanism.For present study, because there is no bound state of meson-meson structure, the resonance mechanism is whether the avoid-    From Figs. 4-9 we found that, lots of avoid-crossing structures appears in IJ P = 00 + , 01 + and 02 + channels.However, only six states have resonance mechanism, i.e., they correspond to the excited states of their "compact state".We found 3 resonant states in ccc b channel, i.e., R(10079) in IJ P = 00 + , R(10081) in IJ P = 01 + and R(10177) in IJ P = 02 + , and 3 resonant states in  There are many avoid-crossing structures on the resonant line, so we calculate the maximum decay width from a single avoid-crossing structure as the minimum total decay width of the system, and sum over all decay widths as a maximal total decay width.The 1S of "compact states" in ccc b and bb bc could not form a resonance through RSM.The reason would be that these 1S states are strongly coupled with scattering state and decay to threshold quickly.The 2S of "compact states" of IJ P = 02 + in ccc b and bb bc systems are very close to the threshold of J/Ψ(2S)B * c (1S) and Υ(1S)B * c (2S), so they couple to threshold very strong, and hard to form a resonance line.

IV. SUMMARY
We investigate the full-heavy tetraquarks ccc b and bb bc in meson-meson structure, diquark-antidiquark structure and K-structure within the framework of the quark model, and consider the combination of all possible color, flavor, spin configurations.In both of ccc b and bb bc systems, we couple all channels in quantum numbers of IJ P = 00 + , 01 + and 02 + , and found that there is no bound state exists.However, through RSM, we found three possible resonance states R(10079), R(10081) and R(10177) in ccc b system, and three possible resonance states R(16474), R(16474) and R(16541) in bb bc system.The decay width of R(10079), R(10081) and R(10177) are 6.For the present study, the annihilation interaction is inversely proportional to the masses of the interacting quarks, and the one-gluon-annihilation process must be very weak for heavy quarks [51].Therefore, the rearrangement decay is the mainly decay be-haviors if the energy below its reference thresholds of D + D − B c (10014MeV), D + B η c (10131MeV) for ccc b system and B + B − B c (16834MeV), D − B − η b (16548MeV) for bb bc system.In this means, R(10079), R(10081) and R(10177) could be found in D + D − B c (10014MeV) channel.The full consideration of 4-6 mixing could be calculated in the future works.
[39] also studied similar system.To evaluate the parameter dependence of each model, we calculate the error m err by: Here m i refers to the energy of meson with each parameter has changed ±1%, while m represents the average energy value of each parameter(except m c and m b ) changed by 1%, and "N" means the number of changed parameters from three models.The value of "N" in present study, Ref. [33] and Ref. [39] is 14, 14 and 12, respectively.[33,39], we found that three model can explain experiment with a parameter error about 3 − 5 MeV.
The main difference of present work and Refs.[33,39] are: 1. M.S Liu et al. [33] have studied the meson-meson structure and diquark-antidiquark structure in ccc b and bb bc systems by a quark model.However, there are two main differences between our study and theirs: (1) they use different potential of Here α ij and σ ij are parameter related to the types of two quarks, and b is the strength of confinement.(2) Their results reveals that the 1S of diquark-antidiquark structure with sextetantisextet and triplet-antitriplet from single channel calculation would be possible resonance state, but in present study 1S of diquark-antidiquark structure form a resonance through RSM.The reason would be that these of diquark-antidiquark structure are strongly coupled with scattering state and decay to threshold quickly.
2. G. Yang et al. [39] employs a potential model inspired by the Lattice-QCD investigation of Ref. [52] to study ccc b and bb bc systems, and the Kstructure are also considered in their calculation.In their model, the gluonic potential is ], where the model parameters α, β, γ, and σ can be determined via a calculation of the mass spectrum of the S-wave Q Q mesons [39], while the V CON ij and V OGE ij in our model are described as Eq. ( 2) and Eq.(3).According to their results, they found 7 possible resonances in ccc b system and 3 possible resonances in bb bc system [39].According to the data in Table XI and compared with their conclusions, the difference between our study and theirs mainly reflected in the following: (1) Firstly, they employ a linear confinement potential, while we use a confinement potential with screen effects, and the magnitude of attractive force comes from linear confinement are more than ours in medium and long range.Because the size of excited meson is larger than ground state meson, there model have more deviation between theoretical prediction and experiment result in excited state.(2) In addition, their conclusions points that all the solutions are possible resonance states, but does not analyze the resonance mechanism.We found that the excited states of "compact states" could be candidate of resonance state, and found the corresponding resonant line by RSM.
Since current experimental data are not sufficient to determine which form of confinement potential is better, we expected that more experimental observation data in the future could help.To evaluate the stability of possible resonance states, we calculate the energy E of R(10081) and the error E(1%) calculated by change ±1% of each parameter(except m c and m b ), which is list in Table XII.According to Tables XI and XII we conclude that the result of present model has little effects on the changing of parameters.
In addition, all six possible resonance states are below P −wave meson composed of cc and b b.In higher energy region, P −wave meson might play a significant role, which demands careful consideration.We leave it in the future works.

TABLE II .
[48]masses of the mesons obtained from present quark model related to this work.Experimental values are taken from the Particle Data Group (PDG)[48].

TABLE III .
The spin wave function of meson.

TABLE IV .
The spin wave function of quark and baryon.

TABLE VII .
The color wave function of meson-meson structure, diquark-antidiquark structure and K-structure.

TABLE IX .
The energies (in MeV) of the meson-meson structure for tetraquarks bb bc.

TABLE X .
The max decay width of single avoid-crossing and the total decay width Γ total (MeV) of the resonance states in ccc b and bb bc systems.The last column means the energy level of the "compact states".System IJ P Resonance state Γ Γ total State R(16474) in IJ P = 00 + , R(16474) in IJ P = 01 + and R(16541) in IJ P = 02 + .Each of these resonance states has more than 30% of compact states.Table X lists the decay width of the resonances states of ccc b and bb bc systems.

TABLE XI .
[33,39]ical and experimental mass of twelve mesons of present work and Refs.[33,39] Table XI lists theoretical and experimental mass of twelve mesons of present work and Refs.

TABLE XII .
The energy E of R(10081) and the error E(1%) calculated by change ±1% of each parameter(except mc and m b ).