The Genuine Resonance of Full-Charm Tetraquarks

In this work, the genuine resonance states of full-charm tetraquark systems with quantum numbers JPC = 0++, 1+−, 2++ are searched in a nonrelativistic chiral quark model with the help of the Gaussian Expansion Method. In this calculation, two structures, meson-meson and diquark– antidiquark, as well as their mixing with all possible color-spin configurations, are considered. The results show that no bound states can be formed. However, resonances are possible because of the color structure. The genuine resonances are identified by the stabilization method (real scaling method). Several resonances for the full-charm system are proposed, and some of them are reasonable candidates for the full-charm states recently reported by LHCb.

Recently, the tetraquark of the all-heavy system, such as cccc and bbbb, has received considerable attention due to the development of experiments. If the cccc or bbbb states steadily exist, they are most likely to be observed at LHC, Belle II and other facilities. Double J/ψ production becomes possible [10][11][12][13][14] and an enhancement in the differential production cross-section for J/ψ pairs between 6 GeV and 8 GeV can be observed [10,11]. The CMS collaboration measured pair production of Υ(1S) [15]. There was also a claim for the existence of a full-bottom tetraquark state bbbb [16], with a global significance of 3.6σ and a mass of around 18.4 GeV, almost 500 MeV below the threshold of ΥΥ. Very recently, the LHCb collaboration reported their newest finding that the full-charm states have been observed: there is a broad structure in the range 6.2∼6.8 GeV, a narrower structure at 6.9 GeV with a significance of about 5σ, and a structure around 7.2 GeV [17]. This discovery brought about widespread theoretical attention [18][19][20][21][22][23][24][25].
In fact, whether or not observable states of fully-heavy tetraquarks exist has been debated for more than forty years. Theoretically, various methods are applied to study the full-heavy tetraquark states. Some work has suggested that stable bound states should exist [26][27][28][29][30][31][32][33]. Iwasaki [26] first argued that the bound state of c 2c2 can exist and estimated its mass to be in the neighborhood of 6 GeV or 6.2 GeV based on a string model. Heller et al. Claimed that the dimensions cccc and bbbb are bound, and that the binding energy ranges from 0.16∼0. 22 GeV based on the potential energy arising from the MIT bag model [27]. Lloyd et al. have used a parametrized Hamiltonian to calculate the spectrum of the all-charm tetraquark state and found several close-lying bound states with two sets of parameters based on large but finite oscillator bases. For example, the lowest state with quantum number J PC = 0 ++ had a mass below the threshold of two η c (1S), 5967.2 MeV [28]. Berezhnoy et al. showed that the masses of cccc and bbbb states are under the thresholds for J = 1, 2 by taking a diquark and antidiquark as point particles and employing the hyperfine interaction between them [29]. In a moment QCD sum rule approach, Chen et al. studied the fully-heavy tetraquark states, and discovered that the masses of the bbbb tetraquarks are below the thresholds of Υ(1S)Υ(1S) and η b (1S)η b (1S), This paper is organized as follows. In Section 2, we briefly discuss the chiral quark model, the wave functions of the full-heavy tetraquark, and the Gaussian Expansion Method. In Section 3, the numerical results and discussion are presented. Some conclusions and a summary are given in Section 4.

The Chiral Quark Model
The chiral quark model has been applied both in explaining the hadron spectra and hadron-hadron interactions successfully. We can find the details for the model in [52,53]. Three parts of the Hamiltonian are included for the cccc fully-heavy system: quark rest mass, kinetic energy, and potential energy: m i is the constituent mass of i-th quark/antiquark, and µ ij is the reduced mass of two interacting quarks, with V C and V G represents the quark confinement and the one-gluon-exchange potential, respectively. The detailed forms for the two potentials are shown below [52]: λ c are the SU(3) color Gell-Mann matrices and σ are the SU(2) Pauli matrices; r 0 (µ ij ) = s 0 /µ ij and α s is an effective scale-dependent running coupling [53], All the model parameters are listed in Table 1, which are determined by fitting the light and heavy meson spectra. In Table 2, the theoretical and experimental masses for the lowlying cc are demonstrated in the chiral quark model. Because the orbital-spin interactions are not taken into account in the present calculation, the P-wave states χ cJ (J = 0, 1, 2) have the same mass.

The Wave Functions of cccc System
The wave functions of four-quark states can be constructed in two steps for both the meson-meson and diquark-antidiquark structure. First we obtain the wave functions for two-body sub-clusters, and, then, secondly, couple the wave functions of these two sub-clusters to construct the total wave functions for a four-quark state with exclusive quantum number I J PC . To save space, only the wave functions for each degree of freedom are shown below.
(1) Diquark-antidiquark structure We denote α and β as the spin-up and spin-down states of the quarks and the spin wave functions for the four-quark states takes the form of There are six spin wave functions for four-quark states in total, which are marked as the superscript σi (i = 1∼6) of χ. The subscripts of χ represent the total spin and the third projection of total spin of the system, and only one component (M S = S) is needed for a given total spin S. For the cccc system, the flavor wave function reads as follows: The marks d0 of χ represent the diquark-antidiquark structure with an isospin that equals zero. For the color freedom, we need to obtain the color singlet wave functions for the four-quark states. The detailed coupling pathways can be found in our previous work [54].
(2) Meson-meson structure For the spin freedom, the wave functions are independent of the structure of the system and they are the same as those for the diquark-antidiquark structure, Equation (5).
The orbital wave functions for the four-quark states can be obtained by coupling the orbital wave function for each relative motion of the system: where Ψ l 1 and Ψ l 2 are the orbital wave functions of the two sub-clusters, with an angular momentum of l 1 and l 2 , respectively. Ψ L r (r 1234 ) is the wave function of the relative motion between two sub-clusters with orbital angular momentum L r . The total orbital angular momentum of the four-quark state is L. "[ ]" denotes the Clebsh-Gordan coupling. As a preliminary calculation, here, we take an approximation that all angular momenta (l 1 , l 2 , L r , L) are set to zero. The Jacobi coordinates used are defined as follows: For the diquark-antidiquark structure, the two quarks cc in one cluster are numbered as 1, 2, and the two antiquarkscc in the other cluster are numbered as 3, 4; for the meson-meson structure, numbers 1, 2 denote the antiquark and quarkcc in one cluster, and numbers 3, 4 denote the antiquark and quarkcc in the other cluster. In the mixing calculations for the two structures, the marks of the quarks, antiquarks in the diquark-antidiquark structure are changed to be consistent with the numbering scheme in the meson-meson structure. In GEM, the orbital wave function is expanded by a series of Gaussians [55]: where N nl are normalization constants, (13) c n are the variational parameters, which are determined dynamically. The Gaussian size parameters are chosen according to the following geometric progression: This procedure enables optimized use of Gaussians, with as small a number of Gaussians used as possible. So, we obtain the final channel wave function for the four-quark system cccc in the diquark-antidiquark structure: where A 1 is the antisymmetrization operator, for the cccc system, In the meson-meson structure, the final channel wave function for the cccc system is written as where A 2 is the antisymmetrization operator, for the cccc system, Finally, we obtain the eigenenergies for the cccc system by solving a Schrödinger equation: Ψ I M I J M J is the wave function of the four-quark states, which is the linear combinations of the above channel wave functions, Equation (15) in the pure diquark-antidiquark structure or Equation (17) in the pure meson-meson structure, or both wave functions of Equations (15) and (17), respectively.

Results and Discussions
In this work, we estimated the masses of the tetraquark states for cccc with the quantum numbers J PC = 0 ++ , 1 +− , 2 ++ in the chiral quark model by adopting GEM. The pure meson-meson and the pure diquark-antidiquark structure, along with the dynamical mixing of these two structures are considered, respectively. In our calculations, all possible color, and spin configurations are included, and the approximation that all orbital angular momenta are set to 0 is used. In Table 3, we give the possible channels and corresponding wave functions. Table 3. The possible channels and corresponding wave functions for cccc tetraquarks with quantum numbers J PC = 0 ++ , 1 +− , 2 ++ . The channels (DA) represent the diquark-antidiquark structures and the superscripts of the channels represent the total angular momentum J.
The single-channel and channel-coupling calculations are performed in the present work. To determine whether or not any bound states exist, Table 4 gives the low-lying energies for cccc tetraquarks with quantum numbers J PC = 0 ++ , 1 +− , 2 ++ , respectively. From the table, we find that all the energies obtained are above the corresponding theoretical thresholds, so no bound states are formed in our calculations. In addition, the energies of the hidden-color states with color configurations 8 × 8,3 × 3 and 6 ×6 are higher than those with color 1 × 1.
Although there is no bound state for the fully-heavy tetraquark system, resonance states with energies higher than the corresponding thresholds are possible because of the color structures of the system. In the present work, we employ the dedicated real scaling (stabilization) method to determine the genuine resonances. Because the calculation is carried out in a finite space, all the energies obtained are discrete. The bound state has a stable energy in the channel coupling calculation, while the resonance has a quasi-stable energy with increasing space after coupling to the scattering states. The real scaling method has been often used for analyzing electron-atom and electron-molecule scattering [46]. In the present approach, the real scaling method is realized by scaling the Gaussian size parameters r n in Equation (14) just for the meson-meson structure with the 1 × 1 color configuration, i.e., r n → αr n , where α takes values between 0.9 and 1.7. (a) J PC = 0 ++ : There are six channels in this case. The stabilization plots for some channels are given. The first channel corresponds to the (η c η c ) 0 configuration. Figure 1 shows the behavior of the energy spectrum of the first channel under the scaling of space. In this figure, most energies for the states decrease with increasing α, and they are scattering states. However, there are several horizontal lines, which correspond to the thresholds, The second channel is the hidden-color channel (η 8 c η 8 c ) 0 . η 8 c is the color octet state of cc with spin 0. To check the α dependence of the energies of the hidden color channel, we also changed r n to αr n for the relative motion between two sub-clusters in the hidden color channel. Figure 2 gives a stabilization plot of the second channel, and all the energies are stable against the scaling of space. Figure 3 shows the stabilization plot for the third channel, (J/ψJ/ψ) 0 configuration. We observe similar behavior as that found for the first channel (η c η c ) 0 and some thresholds appear. The remaining three channels (J/ψ 8  show similar behavior as observed for the second channel (η 8 c η 8 c ) 0 , which is omitted here to save space. From these three figures, we can see that the energies of the hidden-color channels are independent with the scaling of space, but for the 1 × 1 color configuration. To show the coupling effect, the following calculations were carried out. We first studied the coupling effects between hidden-color channels. In Table 5, the energies for the single hidden-color channel and hidden-color channels coupling are shown. To save space, only the first ten energies are listed. From the table, we can see that the coupling has important effects on the energy spectrum because all the energies of the hidden-color channels lie in the same energy range. For example, hidden-color meson-meson structure has the lowest energy of 6328 MeV, while the diquark-antidiquark structure has a lowest energy of 6344 MeV, and the coupling between structures reduces this energy to 6253 MeV. The coupling results for all hidden-channels are stable against the scaling of space, which is shown in Figure 4. The energy of the lowest state is 6253 MeV. There is another state close to it with an energy of 6316 MeV. At around 6500 MeV, there are two states, and, in the range of 6700-7000 MeV, 8 resonance states appear. Which resonance states can be observed among these states? A calculation for the coupling of the hidden-color channels to the scattering states is required.    below 7100 MeV are missing. Only one resonance with an energy of 7138 MeV survives the coupling to the scattering states. Because there is no scattering state in the diquark-antidiquark structure, we also perform a channel coupling calculation involving two diquark-antidiquark structures, (DA) 0 1 +(DA) 0 2 , and the two scattering states, (η c η c ) 0 +(J/ψJ/ψ) 0 . The results are displayed in Figure 6. Unlike the channel coupling calculations between the hidden-color channels (η 8 c η 8 c ) 0 +(J/ψ 8 J/ψ 8 ) 0 and the two scattering states (η c η c ) 0 +(J/ψJ/ψ) 0 , several resonance states appear in the spectral region of 6700-7200 MeV. So, to identify the resonance states, full channel-coupling calculations are needed. From these calculations, we also find that the low-lying resonance states in the hidden-color channels are all missing.
To understand the reason behind the missing low-lying resonance states, we carried out the following calculations: Figure 7 illustrates the results obtained for the coupling between all hidden-color channels and the color-singlet channel (η c η c ) 0 . The lowest state with energy of 6253 MeV disappears, which means that the state strongly couples to (η c η c ) 0 . The resonance states around 6500 MeV are also missing in the coupling. Figure 8 gives the results for the coupling between all hidden-color channels and the color-singlet channel (J/ψJ/ψ) 0 . The second lowest state with energy of 6316 MeV disappears, which denotes that this state couples strongly to (J/ψJ/ψ) 0 . So, the resonance states below 6714 MeV are all missing after coupling to the scattering states. To identify the genuine resonances, the full channel-coupling calculations are performed and the results are shown in Figure 9. The first resonance state R(6763) appears at an energy of 6763 MeV. There are two other resonances below 7000 MeV, with energies of 6849 MeV and 6884 MeV, which can be good candidates for the narrow structure around 6.9 GeV reported by the LHCb collaboration. The broad structure in the range 6.2∼6.8 GeV is due to the effect of the mixture of scattering channels, J/ψJ/ψ, η c (1S)η c (2S) and J/ψψ(2S) opening and the resonance R(6763). As for the structure around 7.2 GeV, there are too many resonances here and our present calculations cannot give a clear picture. More scattering states with non-zero orbital angular momentum are required to be added.
(b) J PC = 1 +− : There are five channels, four with a meson-meson structure and one with a diquark-antidiquark structure. The possible resonances, the results for all hiddencolor channels coupling are given in Figure 10. After coupling to the scattering channels, the remaining resonances can be read from Figure 11. From Figure 10, one can see that the energy of the lowest state is 6346 MeV in the hidden-color channels, but this state disappears after coupling the open channel, η c (1S)J/ψ. However, the second lowest state survives the coupling (see Figure 11), so we obtain a resonance state with an energy of 6383 MeV. There are several resonance states around 6. (c) J PC = 2 ++ : There are three channels, two with a meson-meson structure and one with a diquark-antidiquark structure. The results for the two hidden-color channels coupling are given in Figure 12 Table 6 collects all possible resonance states with quantum numbers J PC = 0 ++ , 1 +− , 2 ++ . The lowest observable resonance is the state with energy of 6383 MeV with J PC = 1 +− , and, below 7.0 GeV, there are nine states, which are separated into three groups. The first group has one state with an energy of 6383 MeV; the second group has three states with energies in the range of 6.75∼6.8 GeV, and the final group has five states with energies around 6.85∼6.9 GeV. The broad structure in the region 6.2∼6.8 GeV observed by the LHCb collaboration can be explained by the mixed effects of the resonance states in the first and second groups, and the thresholds for double J/ψ and η c (1S)η c (2S). The narrow structure around 6.9 GeV can be well explained by the resonances R(6849) and R(6884) with J PC = 0 ++ and/or R(6855) with J PC = 2 ++ . As for the resonances above 7.0 GeV, we obtain nine states below 7.3 GeV with double J/ψ as the final states (J PC = 0 ++ , 2 ++ ). Here, we have mentioned that not all the states can be observed, because we have not taken into account angular excited scattering states, such as χ cJ χ cJ . In order to identify the structures of these possible resonances, we calculated the distance between c andc quark, denoted as R cc , as well as the distance between c and c quark, denoted as R cc for the resonance states, respectively, which are shown in Table 7. From this table, we can see that for all the states with J PC = 0 ++ and 2 ++ , and some states with J PC = 1 +− , R cc and R cc are all in the range of 0.6∼1.0 fm, which means that the states are in the diquark-antidiquark structure. For the other states with J PC = 1 +− , R cc and R cc are larger than 1 fm, which means that the states are very likely to be molecular. The large R cc is due to the antisymmetrization and it gives the average distance between c and twoc. The distance between c andc in one sub-cluster can be extracted from R cc and R cc , which is round 0.6 fm; this is consistent with the results obtained for charmonium.

Summary
In this work, we studied the mass spectra of the fully-charm cccc system with quantum numbers J PC = 0 ++ , 1 +− , 2 ++ in the chiral quark model with the help of GEM. The dynamical mixing of the meson-meson structure and the diquark-antidiquark structure, along with all possible color, spin configurations were taken into account. The predicted masses for the lowest-lying cccc states are all above the corresponding two meson decay thresholds, leaving no space for bound states. By adopting the real scaling method, we identify the genuine resonance states for the J PC = 0 ++ , 1 +− , 2 ++ cccc system. The lowest observable resonance state has quantum numbers J PC = 1 +− with energy 6.38 GeV, and there are several states with quantum numbers J PC = 0 ++ , 2 ++ with energies in the region of 6.85∼6.90 GeV, which can be used to explain the narrow structure around 6.9 GeV in the double J/ψ invariant mass spectrum reported by the LHCb collaboration. There are many resonance states above 7.0 GeV" which are worth further study by including the angular excited scattering channels.
In general, the QQQQ (Q = b, c) resonance states mainly decay into two QQ meson final states by spontaneous dissociation. For the full-charm cccc tetraquarks, they can decay via the spontaneous dissociation mechanism since they lie above the two-charmonium thresholds. The doubly hidden-charm tetraquarks can be clearly differentiated in experiments because of their much heavier energies than conventional charmonium mesons cc. However, it is more difficult to produce cccc states because two heavy quark pairs need to be created in vacuum. However, the running of the LHC provides a chance for the observation of double heavy quarkonium. For example, the recent observations of the J/ψJ/ψ [10][11][12][13][14], J/ψΥ(1S) [56] and Υ(1S)Υ(1S) [15] events, shine a light for the production of cccc and bbbb tetraquarks. With the accumulation of experimental data and the advancement of theoretical work, the nature of the fully-heavy tetraquark states will become more clear.