Possible triply heavy tetraquark states

In the present work, the triply heavy tetraquarks states $QQ\bar{Q}\bar{q}$ with $Q=(c, b)$ and $q=(u, d, s)$ with all possible quantum numbers are systematically investigated in the framework of the chiral quark model with the resonance ground method. Two kinds of structures, including the meson-meson configuration (the color-singlet channels and the hidden-color channels) and the diquark-antidiquark configuration (the color sextet-antisextet and the color triplet-antitriplet), are considered. In the considered system, several bound states are obtained for the $cc\bar{c}\bar{q^{'}}$, $bb\bar{c}\bar{q^{'}}$ and $bc\bar{c}\bar{q}$ tetraquarks. From the present estimations, we find that the coupled channel effect is of great significance for forming the below thresholds tetraquark states, which are stable for strong decays.


I. INTRODUCTION
Searching for multiquark states has become one of the most important and interesting topics of hadron physics, and the experimental observations and theoretical investigations shall deepen our understanding of the nonperturbative QCD [1][2][3][4][5][6][7].At the early beginning of the quark model, the notion of multiquark states had been proposed [8].But there had been no progress on the experimental side for a long time.A turning point came in the year of 2003, when the Belle Collaboration reported their observation of a new charmonium-like state X(3872) in the exclusive B ± → K ± π + π − J/ψ decays [9].Since then, a growing number of new hadron states have been observed experimentally, which attract the great interest of experimentists and theorists.
Among the new hadron states observed in the recent two decades, there are some good candidates of QCD exotic states, which can be classified into different categories according to different criteria.For example, for the charmonium-like states, we can divide them into two types according to the carried charges, i.e., the neutral and charged categories.One can also classify the new hadron states by their most possible quark components into tetraquark, pentaquark states, etc.It is interesting to notice that almost all the new hadron states have at least one heavy constituent quark or antiquark component.Since the mass of the heavy quarks is much larger than that of the light quarks, one can usually discuss the properties of hadrons with heavy quark components in the heavy quark limit.In this case, the number of heavy constituent quark/antiquark can also be used to classify the new hadron states.According to this criterion, we separate the observed new hadron states into three types, which are states with one, two, and four heavy quark/antiquark components, respectively.In the following, we select some typical examples for each type and present a short review.
The observations of the fully heavy tetraquark states makes tetraquark spectroscopy abundant and systematic.However, one can find that the tetraquark states with three heavy quark/antiquarks, i.e., QQ Q q (q = u, d, s), absent experimentally.The triply heavy tetraquark states are different from the already discovered quarkonium-like states, it might in sense offer a new platform of studying the internal structure of the exotic states.On the theoretical side, in the frame of colormagnetic interactions, the triply heavy tetraquark states were systematically investigated and some exotic tetraquark states were predicted [127].The QCD sum rule estimations indicated that the triply heavy tetraquarks states, ccc q, cc b q and bc b q, with quantum numbers J P = 0 + and J P = 1 + are all heavier than the corresponding meson-meson thresholds, while the bb b q tetraquarks were expected to be stable for strong decay [129].However, the estimations in the extended chromomagnetic model [128], nonrelativistic quark model [126], extended relativized quark model [130] indicated that there was no bound triply-heavy tetraquark state.In a word, the existence of the triply heavy tetraquark states is still an open question.In the present work, we employ a nonrelativistic chiral quark model (ChQM) to estimate the mass spectra of the S -wave triply heavy tetraquark states with the possible J P quantum numbers to be 0 + , 1 + , 2 + , to further check the existence of triply heavy tetraquark states.
The work is organized as follows.In Section II and section III, the theoretical framework utilized in present estimations is presented, which includes the chiral quark model and the Resonating Group Method (RGM).Section IV is devoted to the analysis and discussion of the obtained results.In the last section, we give a short summary.

II. THE CHIRAL QUARK MODEL
In the quark model, the Hamiltonian of a hadron is generally written as [131], with m i and p i are the mass and momentum of the ith quark, respectively.T CM is the center-of-mass kinetic energy, which is usually subtracted without losing generality since one mainly focuses on the internal relative motions.V(r i j ) indicates the interaction potential between the ith and jth quarks.
As for the ChQM, it is constructed based on the fact that the light current quarks are nearly massless, which lead to the chiral symmetry.However, due to the interactions of the quarks with the gluon medium, the current quarks become dressed and such dressed current quarks can be approximately described by the massive constituent quarks.In practice, the masses of the constituent quarks in the ChQM are determined by reproduce the spectrum of the conventional hadrons, and this model has been widely used to investigate the study of the spectra of mesons containing heavy quarks [132][133][134][135], the electromagnetic, weak and strong decays and reactions of mesons as well [135][136][137][138][139][140], the phenomena related to multiquark structures [141][142][143][144][145][146][147].In addition, in the ChQM, the interaction potential usually includes the Goldstone-boson exchange potentials, the perturbative one-gluon interaction, and a confinement potential.Furthermore, when one only considers the S −wave tetraquark system, the spin-orbit and tensor contributions can be ignored, thus the two body interaction potential reads, ( where V OGE (r i j ) indicates the potential resulted from one gluon exchange, and its concrete form is where σ and λ c are the Pauli matrices and SU(3) color matrix, respectively.α i j s is the QCD-inspired scale-dependent quarkgluon coupling constant, which offers a consistent description of mesons from light to heavy-quark sectors, and it can be determined by the mass splits between different mesons 1 .As for the confinement potential, the harmonic oscillator potential is adopted, which is, where a c represents the strength of the confinement potential and V 0 i j is the zero-point energies, which can be determined by the mass shift between different mesons.The Goldstone-boson exchange interactions between light quarks appear because of the dynamical breaking of chiral symmetry.For the QQ Q q with (Q = (c, b), q = (u, d, s)) systems, the π, K and η exchange interactions do not work due to the quark components.Thus in this paper, the Goldstoneboson exchange interactions are not considered.
The concrete values of these parameters are collected in Table I.In addition, the details of how to obtain these parameters can also be found in Ref. [148].The calculated mesons masses in comparison with experimental values are shown in Table II.It should be noticed that the parameters in the potentials are obtained by reproducing the mass spectra of conventional mesons, but the two-body quark-quark interaction potentials could be extended to investigate the multiquark system, where the difference between the color configurations is reflected by the product of the SU(3) color matrix λ c i • λ c j .In the present work, the triply heavy tetraquark systems are estimated by using the resonating group method [149].In this FIG.
1: Two types of configurations in QQ Q q, QQ Q′ q and QQ ′ Q q tetraquarks.For the QQ Q q system, there are two structures: the meson-meson configuration (diagram(a)) and the diquarkantidiquark configuration (diagram(b)).For the QQ Q′ q system, diagrams (c) and (d) correspond to the meson-meson and the diquarkantidiquark configurations, respectively.For the QQ ′ Q q system, diagrams (e) and (f) correspond to meson-meson configuration, while diagram (g) refers to the diquark-antidiquark configuration.method, the multiquark system can be divided into two clusters, which are frozen inside, so one only needs to consider the relative motion between the two clusters.The conventional ansatz for two-cluster (cluster A and B) wave functions is, where A is the antisymmetry operator of triply heavy tetraquarks.
For QQ Q q system, one has, TABLE III: All the possible channels for different J P quantum numbers, where [i, j, k] denotes the channels with i, j, and k to be the indices of flavor, spin, and color, respectively.
this antisymmetry operator becomes, for QQ Q′ q system, and for QQ ′ Q q system, due to the absence of any homogeneous quarks, then antisymmetry operator becomes a unit operator, which is, Moreover, [σ] = [222] gives the total color symmetry, and I, S , L and J represent flavor, spin, orbital and total angular momenta, respectively.ψ A and ψ B are the two-quark cluster wave functions, which are, where η I , S , and χ represent the flavor, spin, and internal color terms of the cluster wave functions, respectively.According to Fig. 1, we define different Jacobi coordinates for different diagrams.As for the meson-meson configuration in Fig. 1, the Jacobi coordinates are, where the subscript q/ q indicates the quark/antiquark particle, while the number indicates the quark position in Fig. 1.By interchanging r q 1 with r q 3 , one can obtain the Jacobi coordinates in Fig. 1-(f).As for the diquark-antidiquark configuration, one can also obtain the Jacobi coordinates corresponding to the diagrams in Fig. 1 by interchanging r q 3 with r q2 .
From the variational principle, after variation with respect to the relative motion wave function with H(R, R ′ ) and N(R, R ′ ) to be the Hamiltonian and normalization kernels, respectively.The eigenenergy E and the wave functions are obtained by solving the above RGM equation.In the present estimation, the function χ(R) can be expanded by gaussian bases, which is, where C i,L is the expansion coefficient, and n is the number of gaussian bases, which is determined by the stability of the results.S i is the separation of two reference centers.R is the dynamic coordinate defined in Eq. (11).After including the motion of the center of mass, i.e., With the above formula, one can rewrite the wave function in Eq. ( 5) as, where φ α (S i ) and φ β (−S i ) are the single-particle orbital wave functions with different reference centers, which are, With the reformulated ansatz as shown in Eq. ( 15), the RGM equation becomes an algebraic eigenvalue equation, which is, with N L ′ i, j and H L,L ′ i, j to be the overlap of the wave functions and the matrix elements of the Hamiltonian, respectively.By solving the generalized eigenvalue problem, we can obtain the energies of the tetraquark systems E and the corresponding expansion coefficient C j,L .Finally, the relative motion wave function between two clusters can be obtained by substituting the C j,L into Eq.( 13).
Besides the space part, we present the flavor, spin, and color parts of the wave function in Appendix-A.It is worth noting that after applying the antisymmetry operator, some wave functions may vanish, which means that some states are forbidden.For example, for the cc b q system with J P = 0 + , when considering the diquark-antidiquark structure with the spin wave function forced to choose S 1 0 , the color wave function χ c 3 would be excluded due to the constraints that the total wave function must be antisymmetric.

IV. RESULTS AND DISCUSSIONS
In the present calculation, the triply heavy tetraquark systems are evaluated by taking into account the meson-meson and diquark-antidiquark configurations in the ChQM, which have been shown in Fig 1 .To exhaust all possible configurations of the QQ Q q systems, we divide them into three classes, which are, the QQ Q q system including ccc q and bb bq, the QQ Q′ q system including cc b q and bbcq, and the QQ ′ Q q system including cb b q and cbcq.Moreover, in the present work, only the S −wave triply-heavy tetraquark states are evaluated, which indicates that the total orbital angular momenta L is equal to zero.Then, the total angular momentum, J, coincides with the total spin, S , and can take values of 0, 1, and 2, then the possible J P quantum numbers of the tetraquark states could be 0 + , 1 + , and 2 + .All the possible channels would be considered through the symmetry of the wave functions and all the allowed channels are listed in Table III.From Table III, one can find that in the ChQM the color singletsinglet (1 c × 1 c ) and the color octet-octet (8 c × 8 c ) structure have been taken into account for the meson-meson configuration.Moreover, for the diquark-antidiquark configuration, both antitriplet-triplet ( 3c × 3 c ) and sextet-antisextet (6 c × 6c ) color structures have also been considered.
Our estimations of the eigenenergies of the triply tetraquark states are presented in Tables IV-XI.In these tables, all the allowed meson-meson and diquark-antidiquark configurations are listed.In the meson-meson channels, (M 1 M 2 ) 1 and (M 1 M 2 ) 8 indicate the color singlet-singlet (1 c × 1 c ) and the color octet-octet (8 c × 8 c ) structures, respectively.E th is the experimental value of the thresholds for the physical channels.In the present work, the single-channel and channel-coupling calculations are all considered, and E sc , E CC1 , E CC2 and E CC are the estimated values of the eigenenergies of every single channel, the coupled channel for the meson-meson configurations, the coupled channel for the diquark-antidiquark configurations, and the one estimated by simultaneously considering the meson-meson and diquark-antidiquark configurations, respectively.P indicates the percentages of each channel for the lowest-lying eigenenergies E CC .
A. The QQ Q q systems Our estimations for the ccc q tetraquark system are presented in Table IV.For the case of J P = 0 + , one can find there are four channels in the meson-meson configurations, and two channels in the diquark-antidiquark configurations.For the cccn, n = {u, d} tetraquark states, the lowest threshold of the physical channel is 4849 MeV, which is the threshold of η c D. Form the table, one can find the eigenenergies of every single channel in both the meson-meson and diquarkantidiquark configurations are all above the lowest threshold of the allowed physics channel, which indicates that all these tetraquark states can decay into η c D. When one couple all the channels in a certain configuration, one can find the estimated eigenenergies are 4851 and 5415 MeV for the mesonmeson and diquark-antidiquark configurations, respectively, which is still a bit higher than the threshold of η c D. After considering both the meson-meson and diquark-antidiquark configurations simultaneously, we find the eigenenergy of cccn tetraquark state is about 4851 MeV, which is about 2 MeV above the threshold of η c D. As for ccc s tetraquark states with J P = 0 + , the lowest threshold of the physical channel is 4952 MeV, which is the threshold of η c D + s .As for ccc s tetraquark state with J P = 0 + , the single channel estimations show that all the tetraquark states are heavier than η c D + s .The eigenenergies of the coupled-channel estimations in mesonmeson and diquark-antidiquark configurations are 4954 MeV and 5487 MeV, respectively, which are all above the threshold of η c D + s .Moreover, the full coupled-channel estimations, i.e., considering the meson-meson and diquark-antidiquark configurations simultaneously, indicate the eigenenergy of the ccc s tetraquark state is 4594 MeV, which indicates that in this case, the effects of channel coupling is rather weak.It is worth noting that in the single-channel estimation the eigenen- ergy for the lowest physical meson-meson channel is several hundred MeV below the ones of other channels, thus, in the coupled-channel estimations, the mixings between different channels are expected to be small due to the large eigenenergy splittings.
As for the cccq tetraquark system with J P = 1 + , there are nine channels in this case, which include three color singlet channels and three hidden color channels in the meson-meson configuration, while there are three channels in the diquarkantidiquark configuration.The lowest physical meson-meson threshold is the one of J/ψD, which is 4962 MeV.In the single channel estimations, no bound state is found.The eigenenergies estimated in the coupled-channel estimations of the meson-meson and diquark-antidiquark configurations are 4964 and 5363 MeV, respectively, which are all above the threshold of J/ψD.By considering both the meson-meson and diquark-antidiquark configurations simultaneously, the eigenenergy of the tetraquark state with J P = 1 + is estimated to be 4963 MeV, and the effect of the channel coupling is rather weak, which is similar to the case of J P = 0 + .As for the ccc s tetraquark system, the lowest physical threshold is the one of J/ψD + s , which is 5065 MeV.Similar to the case of cccn system, the eigenenergies obtained in the single channel are all above the threshold of J/ψD + s .In addition, when we consider the channel coupling in the meson-meson and diquarkantidiquark configurations individually, the eigenenergies of the tetraquark state are estimated to be 5067 and 5442 MeV.After considering the meson-meson and diquark-antidiquark configurations simultaneously, we obtain the eigenenergy of ccc s tetraquark state with J P = 1 + is 5066 MeV, which is still a bit higher than the threshold of J/ψD + s .
For the case of cccn tetraquark states with J P = 2 + , there are two channels in the meson-meson configuration and only one channel in the diquark-antidiquark channel.The physi- cal meson-meson threshold is 5104 MeV.Our single channel estimations indicate that the eigenenergies are all above the threshold of J/ψD * , and after considering the channel coupling in the meson-meson configuration, the eigenenergy is estimated to be 5106 MeV, which is still above the threshold of J/ψD * .When we include the meson-meson and diquarkantidiquark configuration simultaneously, the eigenenergy is estimated to be 5095 MeV, which is about 9 MeV below the threshold of J/ψD * and then this tetraquark state can not decay into J/ψD * .Moreover, our estimations indicate in this states the dominant component is J/ψD * , which is about 75%, while the fractions of the hidden color channel, (J/ψD * ) 8 , and the diquark-antidiquark channel, (cc)(cn), are about 11% and 13%, respectively, which indicate the effect of coupled channel plays an important role in the existence of below threshold cccn tetraquark state.Different from the cccn system, our estimations find there are no below threshold ccc s tetraquark state In a very similar way, we can estimate the bb b q tetraquark system, and our results are listed in Table V.Our estimations indicate that there are no below threshold bb b q tetraquark states.However, within the framework of QCD sum rules, the bb b q tetraquark states with J P = 0 + and J P = 1 + may be stable due to obtaining the masses below the threshold η b B and η b B * [129], which is different from our conclusions.It is interesting to notice that for the cccn system, we find one below threshold tetraquark state with J P = 2 + , while the mass of the corresponding state in bb bn sector is above the threshold of Υ B * .To find which interaction plays the dominant role in forming a below threshold cccn tetraquark state with J P = 2 + and further check the influence of the coupled channel effect, we list the contribution of each term in the system hamiltonian in Table VI.As we have discussed in the above section, the potential resulting from the Goldstone-boson ex- The average values of each operator in the Hamiltonian of the cccn and bb b n tetraquark system in unit of MeV.E M(J/ψD * ) and E M(ΥB * ) stand for the sum of the theoretical thresholds of J/ψD * and ΥB * channel, where the distance between two mesons are very large and the interactions between them are ignored.change disappeared due to the quark components of the triply heavy tetraquark system.For the cccn tetraquark system with J P = 2 + , E M(J/ψD * ) refers to the sum of the theoretical threshold of J/ψD * , which indicates the interactions between J/ψ and D * to be zero and the system wave function is the product of the ones of J/ψ and D * .In this case, the average value of the kinetic operator is 1800.1 MeV, and the ones of confinement and OGE terms are -1812.8MeV and -380.5 MeV, respectively, one can obtain the threshold of J/ψD * by summing over the average values of different terms and the masses of the constituent quarks.In a similar way, one can obtain the average value of the operators in the single E (J/ψD * ) 1 channel estimation, the coupled channel estimations of meson-meson configuration (E cc1 ) and diquark-antidiquark configuration E cc2 , and the coupled channel estimation of both meson-meson and diquark-antidiquark configurations E cc .For simplify, we can define the ∆E as the difference of the average values of operators between single/coupled channel cases and E M(J/ψD * ) .If the sum of ∆E for all the operators is negative, the tetraquark states are below the threshold of J/ψD * .From the table, one can find the sum of ∆E for a single channel, coupled channel of each configuration is positive, while the coupled channel of both configurations is negative, which indicates the cccn tetraquark state with J P = 2 + is a below threshold state and the coupled channel effects between different configurations are essential in forming a below threshold tetraquark state.From the table, this result is mainly due to the strong attraction of the interaction of OGE term under the coupling of all configurations.As for bb bn tetraquark state with J P = 2 + , one can find that all the ∆E is positive, which indicates the tetraquark state is above the threshold of ΥB * .
B. The QQ Q′ q system In Table VII, we present our estimations of the eigenenergies of the cc b q system with J P = 0 + , 1 + and 2 + , respectively.For the case of cc bn tetraquark with J P = 0 + , we find there are four meson-meson channels and two diquarkantidiquark channels.The lowest physical threshold of cc bn is the one of B + c D, which is 8140 MeV.The eigenenergies obtained from the single channel, coupled channel in each configuration, and the full coupled channel estimations are all above the threshold of B + c D. From the full coupled channel estimations one can find the dominant component of cc bn tetraquark state with J P = 0 + is B + c D. As for the cc bn tetraquark states with J P = 1 + , there are six meson-meson and three diquark-antidiquark channels, respectively.The lowest physical threshold is the one of B + c D * , which is 8282 MeV.Similar to the case of 0 + , the eigenenergies obtained from the single channel, coupled channel in each configuration, and the full coupled channel estimations are all above the threshold of B + c D * .Similarly, there are two meson-meson and one diquark-antidiquark channels in the cc bn tetraquark system with J P = 2 + , and our estimations also indicate that there is no below threshold cc bn tetraquark state with J P = 2 + .Similarly, we can analyze the cc b s tetraquark system, and we find all the eigenenergies of the cc b s tetraquark are above the lowest thresholds of the corresponding physical channels.
As for the bbcq tetraquark system, the estimated eigenenergies are listed in Table VIII.For the bbcn tetraquark states with J P = 0 + , we find the lowest threshold of physical channel is the one of B − c B, which is 11554 MeV.The eigenenergies obtained from the single channel estimations and coupled channel estimations in each configuration are above the threshold of B − c B. While considering the coupled channel effects of meson-meson and diquark-antidiquark configurations simultaneously, we find the eigenenergies of bbcn tetraquark with J P = 0 + is 11552 MeV, which is about 2 MeV below the threshold of B − c B. In this tetraquark state, the dominant component is B − c B and its percentage is about 94.25.As for the bbcn tetraquark states with J P = 1 + , the lowest physical channel is B − c B * with the threshold to be 11579 MeV.We find that the eigenenergies obtained from single channel estimations and coupled channel estimations in each configuration are all above the threshold of B − c B * , while the  From our estimations, we find there is no below threshold QQ Q′ s tetraquark state.But for QQ Q′ n tetraquark system, we find the eigenenergies of all the S -wave ground bbcn tetraquark states with different J P quantum numbers are below the lowest threshold of the corresponding physical channels, which is much different with cc bn case.To further compare the spectrum of cc bn and bbcn, we list the average values of each operator in the Hamiltonian of the tetraquark systems in Table IX.It is interesting to notice that in the full coupled channel estimation all the eigenenergies of the bbcn tetraquark states are below the corresponding lowest physical threshold, while the eigenenergies of the cc bn are all above the corresponding lowest physical threshold.By comparing the average values of the operators in the Hamiltonian of the cc bn and bbcn tetraquark system, one finds the dominant difference is the average values of V OGE , especially in the case of coupled channel estimations in the diquark-antidiquark configurations.The average values of V OGE are negative, which indicates that the OGE potential is attractive.However, for the cc bn tetraquark states with J P = 0 + and 2 + , the attractions become weak when we consider coupled channel effects in each configuration, and for J P = 2 + case, the attraction becomes stronger in the diquark-antidiquark coupled channel estimations.For the bbcn tetraquark states, we find that the attractions become much stronger in the diquark-antidiquark coupled channel estimations, although the attractions caused by the confinement potential become weak and the eigenenergies obtained in the diquark-antidiquark coupled channel estimations are still above the corresponding lowest physical threshold.But when we consider the coupled channel effects in both configurations, the eigenenergies of the bbcn are below the corresponding lowest threshold of the physical channels.
C. The QQ ′ Q q system Similar to the cases of QQ Q′ q and QQ ′ Q q tetraquark states, we can estimate the eigenenergies of QQ ′ Q q tetraquark states.Our estimations of the eigenenergies of the cbcq and bc b q tetraquark states are collected in Table X and XI.From Table X, one can find the eigenenergies of bccn tetraquark state with J P = 0 + obtained in the single channel estimations, the coupled channel estimations in each configuration and the full coupled channel estimations are all above the threshold of DB − c , which is 8140 MeV.Similarly, we also find that the eigenenergies of the bcc s tetraquark states with J P = 0 + are all above the threshold of D + s B − c .As for bccn tetraquark state with J P = 1 + , we find that the eigenenergies obtained in the single channel estimations and the coupled channel estimations in the meson-meson and diquark-antidiquark configurations are all above the threshold of DB * − c , however, when considering the coupled channel effects in both meson-meson and diquarkantidiquark configurations, one obtains the eigenenergy to be 8159 MeV, which is 6 MeV below the threshold of DB * − c .In this tetraquark state, the dominant component is DB * − c with a percentage to be 91.57.As for bcc s tetraquark state with J P = 1 + , we find that the eigenenergies obtained in the single channel estimations, the coupled channel estimations in each configuration and the full coupled channel estimations are all above the threshold of D + s B * − c .As for the case of J p = 2 + , the eigenenergies of bccn and bcc s obtained in the full coupled channel estimations are 8273 MeV and 8410 MeV, which are below the threshold of D * B * − c and D * + s B * − c , respectively.In the bc bn tetraquark state with J P = 2 + , the dominant components are (D * B * − c ) 1 , (J/ψB * ) 8 and (bc)(cn) with [i, j, k] = [7, 6, 4], the corresponding percentages of these components are 72.10,11.04 and 10.77, respectively.As for bc b s tetraquark state with J P = 2 + , the dominant component is (D * + s B * − c ) 1 with a percentage to be 95.29.
As for the bc b q tetraquark system, the eigenenergies estimated in the ChQM are collected in Table XI.From the table, one can find that the eigenenergies obtained in the single channel estimations, coupled channel estimations in each ∼ 0% [7,3,3] (bc)(cn) 8828 ∼ 0% (bc)(c s) ∼ 0% [7,3,4] (bc)(cn) 8640 ∼ 0% (bc)(c s) ∼ 0% [7,4,3] (bc)(cn) 8858 ∼ 0% (bc)(c s) ∼ 0% [7,4,4] (bc)(cn) 8582 1.17% (bc)(c s) ∼ 0% [7,5,3] (bc)(cn) 8803 ∼ 0% (bc)(c s) ∼ 0% [7,5,4] ( configuration and the full coupled channel estimations are all above the corresponding lowest physical threshold, which is different with the bccq tetraquark states, where one find three below threshold tetraquark states.To further analyze the role of the coupled channel effects, we estimate the average values of the operators in the Hamiltonian of Q ′ Q Q q system, which are collected in Tables XII and XIII.From the tables, one can find the average values of kinetic terms increase when we include the interaction between mesons and coupled-channel effects.In the full coupled-channel estimations, we find the attraction from confinement potential becomes stronger for bccn tetraquark states with J P = 1 + and J p = 2 + , but the attraction from the OGE potential becomes weak for the bccn tetraquark states with J P = 1 + , while this attraction becomes strong for the bccn tetraquark states with J P = 2 + .As for bcc s tetraquark states, the full coupled-channel estimations indicate the average values of H T , V Con and V OGE are close to those of E M(ΥD) , and the sum of these terms are positive.
As for the bcc s tetraquark state with J P = 2 + , the estimations indicate the confinement potential becomes strong in the full coupled-channel estimation.

V. SUMMARY
To summarize, inspired by the recent observation of fully heavy tetraquark states, we perform a systematic estimation of the triply tetraquark states in a chiral quark model, where the coupled channel effects of meson-meson configuration and diquark-antidiquark configurations are included.The eigenenergies of the S -wave ground states have been estimated.After including the coupled channel effects of both configurations, We notice that the eigenenergies of some tetraquark states are below the corresponding lowest threshold of the physical channel, which indicates that these tetraquark states cannot fall apart directly and thus are stable for strong decay.In Table XIV, we collect all the stable tetraquark states estimated in the present work.For comparison, we also list the corresponding lowest thresholds of the physical channel.
Moreover, comparing with the results in Refs.[127][128][129][130], we find that the masses of the diquark-antidiquark configurations are several hundred MeV higher than those of the color-magnetic interaction model [127,128] and QCD sum rules [129], while the masses under an extended relativized quark model [130] are generally consistent with present estimations of the diquark-diquark configurations.Although there are discrepancies in the estimated masses due to different input parameters and different interactions in different models, the conclusions are basically the same for the triply heavy tetraquark system, i.e., no stable states are found in the diquark-antidiquark configurations except for the estimation of QCD sum rules [129].But when we consider the coupledchannel effects of diquark-antidiquark and meson-meson configurations simultaneously, we find there exist several stable tetraquark states which are below the corresponding lowest physical threshold, which may accessible for experiments in LHCb.For the meson-meson configurations, the color wave functions of a q q cluster are, where the subscript [111] and [21] stand for color-singlet (1 c ) and color-octet (8 c ), respectively.Then the color-singlet tetraquark SU(3) color wave functions can be constructed by two color-singlet clusters, i.e.,1 c ⊗ 1 c ) and by two color-octet clusters, i.e., 8 c ⊗ 8 c ), which are, For the diquark-antidiquark configuration, the color wave    The color-singlet wave functions of the diquark-antidiquark configuration can be the product of color sextet and antisextet clusters (6 c ⊗ 6c ) or the product of color-triplet and antitriplet cluster (6 c ⊗ 6c ), which read, C [2]  [22] + C For the flavor degree of freedom, the quark content of the investigated 4-quark system is QQ Q q, Q = {c, b}, q = {u, d, s}, the isospin could be 1/2 and 0. Here, we adopt F i m and F i d to denote the flavor wave functions of the tetraquark system in the meson-meson and diquark-antidiquark configurations, respectively.In the present work, the flavor wave function of the QQ Q q system can be categorized into three types, which are QQ Q q, QQ Q′ q and QQ ′ Q q, respectively.
For the QQ Q q system, the flavor wave functions can be, and for the QQ Q′ q system, the flavor wave functions can be read as, While the flavor wave functions for the QQ ′ Q q system read,

cB
* and its percentage is about 58.26, while the (B − c B * )1 and (B − c B * )8 channels in the meson-meson configuration and (bb)(cn) channel with [i, j, k] =[4,4,4] in the diquark-antidiquark configuration are also important with the percentage to be 15.71, 6.16 and 16.57, respectively.For the J P = 2 + case, there is only one physical channel for bbcn tetraquark state, which is B * − c B * with the threshold to be 11625 MeV.Similar to the case of 0 + and 2 + , the eigenenergies obtained from the single channel estimations and the coupled channel estimations in each configuration are all above the threshold of B * − c B * .When we consider both meson-meson and diquarkantidiquark configurations simultaneously, the eigenenergy of bbcn tetraquark state with J P = 2 + is estimated to be 11613 MeV, which is about 12 MeV below the threshold of B * − c B * and the percentage of different channels are 68.00,9.79 and 22.21 for (B * − c B * ) 1 , (B * − c B * )8 channels and (bb)(cn) channel with [i, j, k] =[4,6,4], respectively.Different from the bbcn tetraquark system, our estimations indicate the eigenenergies of bbc s tetraquark states with different J P quantum numbers are all above the lowest threshold of the corresponding physical channels.
Appendix A: The wave function of the triply heavy tetraquark a.The color wave function

TABLE I :
The concrete values of the model parameters, which are determined by reproducing the masses of mesons listed in TableII.

TABLE II :
[150]asses (in units of MeV) of the mesons.The measured values of the masses[150]are also presented for comparison.

TABLE IV :
The lowest-lying eigenenergies of the cccn n = {u, d} and ccc s tetraquarks in the ChQM.

TABLE V :
The lowest-lying eigenenergies of the bb b n n = {u, d} and bb b s tetraquarks in the ChQM.

TABLE VII :
The lowest-lying eigenenergies of the cc bn n = {u, d} and cc b s tetraquarks in the ChQM.

TABLE VIII :
The lowest-lying eigenenergies of the bbcn n = {u, d} and bbc s tetraquarks in the ChQM.

TABLE IX :
The same as Table VI but for cc bn and bbc n tetraquark states with J P = 0 + , 1 + and 2 + .

TABLE X :
The lowest-lying eigenenergies of the bccn n = {u, d} and bcc s tetraquarks in the ChQM.

TABLE XI :
The lowest-lying eigenenergies of the bc bn n = {u, d} and bc b s tetraquarks in the ChQM.

TABLE XII :
Contributions of each term in Hamiltonian to the energy of the bccn tetraquark and bc bn tetraquark in ChQM.E M("channel") stands for the sum of the theoretical thresholds of the lowest physical channel.(unit: MeV).

TABLE XIII :
Contributions of each term in Hamiltonian to the energy of the bcc s and bc b s tetraquark in ChQM.E M("channel") stands for the sum of the theoretical thresholds of the lowest physical channel.(unit: MeV).

TABLE XIV :
Possible bound state with the different quantum number in ChQM.(unit: MeV).