Tetra- and penta-quark structures in the constituent quark model

With the development of high energy physics experiments, a large amount of exotic states in the hadronic sector have been observed. In order to shed some insights on the nature of the tetraquark and pentaquark candidates, a constituent quark model, along with the gaussian expansion method, has been employed systematically in the real- and complex-range investigations. We review herein the double- and full-heavy tetraquarks, but also the hidden-charm, -bottom and doubly charmed pentaquarks. Several experimentally observed exotic hadrons are well reproduced within our approach; moreover, their possible compositeness are suggested and other properties analyzed accordingly such as their decay widths and general patterns in the spectrum. Besides, we report also some predictions not yet seen by experiment within the studied tetraquark and pentaquark sectors.


Introduction
A Large amount of conventional hadrons, 3-quarks baryons and quark-antiquark mesons, could be described well in a constituent quark model [1][2][3] which was firstly proposed by M. Gell-Mann in 1964 [4]. Especially, this 'toy model' had successfully predicted the existence of the Ω baryon. However, with decades of experimental efforts in high energy physics, many exotic states have been observed by each collaboration, e.g., BABAR, Belle, BES, CDF, CLEO-c, LHCb, etc. The remarkable example should be the X(3872) which was firstly announced through the B ± → K ± π + π − J/ψ decay by the Belle Collaboration in 2003 [5] and later on, CDF [6], D0 [7] and BABAR [8] collaborations confirmed this charmoniumlike state. Meanwhile, dozens of so-called XYZ particles have also been observed worldwide in B factories, τ-charm facilities and hadron-hadron colliders. In particular, Y(3940) was discovered by the Belle collaboration through analyzing the ω J/ψ invariant mass spectrum in 2004 [9]. Besides, a charged charmoniumlike state Z + (4430) was observed in the B → Kπ ± ψ(3686) decay by the same collaboration in 2008 [10]. Y(4260) was firstly observed in the e + e − → γπ + π − J/ψ channel by the BABAR collaboration in 2005 [11] and this state was confirmed by the CLEO [12] and Belle [13] collaborations within the same decay process. Additionally, the other exotic states such as Y(4140), Y(4274), Z + (4051) and Z + (4200), etc., are all collected in the Particle Data Group [14]. Among the tetraquark sector, a fully heavy 4-body system QQQQ is quite appealing. In 2017, a benchmark measurement of the Υ(1S) pair production at √ s=8 TeV in pp collision was implemented by the CMS collaboration [15], then an excess at 18.4 GeV in the Υ + − decay Submitted to Journal Not Specified, pages 1 -92 www.mdpi.com/journal/notspecified The observations of the hidden-charm pentaquarks P + c (4380), P + c (4312), P + c (4440) and P + c (4457) bring great interest in theoretical investigations. Especially, during 2010 to 2013 which is before the announcement of P + c (4380) by the LHCb collaboration [22], several narrow hidden-charm resonances ∼4.3 GeV were predicted by means of coupled-channel unitary studies [102][103][104][105][106] and possible loosely bound hidden-charm molecular states were discussed in the one-boson-exchange model [107]. Then great deals of subsequent theoretical works devoted to the interpretation of the nature this exotic state, particularly, Σ ( * ) cD ( * ) molecular state with quantum numbers I(J P ) = 1 2 ( 3 2 − ) is preferred by the boson exchange model [108], the constituent quark model [109,110], the Bethe-Salpeter equation [111], QCD sum rules [112][113][114], etc. The spin-parity of 3 2 − is also suggested in a diquark-triquark model investigation [115]. Furthermore, some other non-resonance explanations were also proposed such as kinematic effects and triangle singularities [116][117][118]. Strong decay property of P + c (4380) is studied in a molecular configuration [119].
Furthermore, in 2019 the three newly announced pentaquarks P + c (4312), P + c (4440) and P + c (4457) by the LHCb collaborattion [24] triggered many theoretical investigations again. The main interpretation with Σ ( * ) cD ( * ) molecular configurations are provided by effective field theory [120,121], QCD sum rules [133], potential models [125][126][127][128][129][130], heavy quark spin multiplet structures [122,124] and heavy hadron chiral perturbation theory [132], etc. Moreover, the production [134,135] and decay properties [123] of these pentaquarks were also investigated. Therein, through Tables III and IV presented in our systematically study on the hidden-charm pentaquark states in 2017 [109], one could notice that the new reported states are described well if with the following assignments that P  [120][121][122][123]. Meanwhile, there are also many works devoted to the investigations on other kinds of pentaquark states. E.g.,Qqqqq bound state is unavailable in a quark model formalism [136]. However, narrow resonances in doubly heavy pentaquarks are possible in potential models [137][138][139]. Besides, triply charmed pentaquarks like Ξ cc D ( * ) molecular state is suggested within one-boson-exchange model [140] and QCD sum rules [141]. Additionally, some general reviews on the exotic states of tetraquark and pentaquark can be referenced in Refs. [142,143].
Apparently, much more investigations on the exotic hadronic states in the future experiments e.g., ATLAS, CMS and LHCb collaborations are necessary. In this review, we mainly focus on a summary of the doubly-, fully-heavy tetraquarks and hidden-, doubly-heavy pentaquarks which were all systematically studied in the framework of constituent quark model. By comparing various calculated bound and resonance states of tetraquark and pentaquark in one theoretical framework, some general or universal features on multi-quark systems are expected, and with a purpose in shedding some insights on the future investigations in hadron physics both experimentally and theoretically.
The structure of this article is organized as follows: Sec. 2 devotes to the theoretical framework where our constituent quark model and wave-functions of tetra-and penta-quark states are illustrated. Then theoretical results along with discussions on each kinds of tetraquarks and pentaquarks are presented in Sec. 3, respectively. The last section is about a summary.

Theoretical Framework
Among all the methods in dealing with the issues on hadron physics, which is located in the QCD's non-perturbative energy region, the QCD-inspired phenomenology approach, constituent quark model is still a powerful and major way applied to the baryon and meson spectra [1, 3,[144][145][146][147], hadron-hadron interaction [148][149][150][151][152] and exotic states [109,[153][154][155][156], etc. Therefore, the S-wave systems of doubly-heavy tetraquark, hidden-, doubly-heavy pentaquark in each allowed quantum numbers I(J P ) states are analyzed in this formalism. Meanwhile, the widely accepted Lattice QCD which is based on the first principle has also made prominent achievements in studing the multi-quark systems [157,158] and the hadronic interactions [159][160][161]. Hence a potential model which is according to the Lattice QCD investigations on the interaction of QQ pair is also employed in the investigation on fully-heavy tetraquark states. Particularly, the doubly-, fully-heavy tetrauqarks with spin-parity J P = 0 + , 1 + and 2 + , and in I = 0 or 1 isospin sectors; the hidden-charm, bottom and doubly charmed pentaquarks with quantum numbers J P = 1 2 − , 3 2 − and 5 2 − , and in the I = 1 2 or 3 2 isospin sectors are investigated. The wave-function of multi-quark systems are exactly constructed in the non-relativistic quantum mechanics range. Specifically, wave-functions of the color, spin, flavor and spatial degrees of freedom are considered in all possible meson-meson color-singlet, hidden-color, diquark-antidiquark and K-types channels for 4-quark systems, baryon-meson color-singlet and hidden-color channels for 5-quark systems. Besides, the couplings of these different configurations in one system are also considered. When solving the eigenvalue problem on 4-and 5-quark systems, both the real-and complex-range calculations are implemented. In particular, the bound, resonance and scattering states can be classified simultaneously in the later framework according to the so-called ABC theorem [162,163]. The crucial manipulation in the complex scaling method (CSM) is to transform the coordinates of relative motions between quarks with a complex rotation r → re iθ , and then a complex scaled Schrödinger equation [H(θ) − E(θ)] Ψ(θ) = 0 will be solved. In particular, Fig. 1 presents a schematic distribution of the complex energy of 2-body system by the CSM according to Ref. [164].

Chiral Quark Model
The general form of a N-body Hamiltonian can be written as where the kinetic energy of central mass T CM is subtracted during calculation and this is owing to the internal relative motions of systems are crucial. Besides the two-body potential in a chiral quark model contains the color confinement, one gluon exchange and Goldstone-Boson exchange interactions. Furthermore, only the central parts of potential listed in Eq. (2) are considered, the spin-orbit and tensor contributions are ignored at present. Firstly, color-confining should be encoded in the non-Abelian gauge feature of QCD. On one hand, multi-gluon exchanges induce a linearly rising attractive potential which is proportional to the distance between two infinite-heavy quarks has been demonstrated by the investigation of Lattice QCD [165]. On the other hand, light-quark pairs spontaneously created in the QCD vacuum may also lead to a breakup of the created color flux-tube at the same scale [165]. Accordingly, these two phenomenological features are mimicked in the expression: particularly, a c , µ c and ∆ are the chiral quark model parameters, λ c represents the Gell-Mann matrices in SU(3) color. One can see that we have a linear potential with an effective confinement strength σ = −a c µ c ( λ c i · λ c j ) if two quarks are extremely close; however, it will turns to be a constant at large distance.
Secondly, the one gluon exchange potential which includes a coulomb interaction and a color-magnetism one is given by where m i is the constituent quark mass and the Pauli matrices in spin degree of freedom are denoted by σ.
The contact term of spatial part in color-magnetism interaction has been regularized as δ( r ij ) ∼ 1 4πr 2 0 e −r ij /r 0 r ij , with r 0 (µ ij ) =r 0 /µ ij a regulator which depends on the reduced quark mass µ ij . According to Ref. [166], a parameterized scheme for the QCD strong coupling constant α s is used herein and the detail is in which α 0 , µ 0 and Λ 0 are all of the model parameters.
Finally, the central parts of chiral potentials which include the pion, kaon, η and σ exchange interactions are written as below where Y(x) = e −x /x is the Yukawa function and λ a is the SU(3) flavor matrix of Gell-Mann. By introducing an angle θ p , the physical η meson is considered, meanwhile m π , m K and m η are the experimental masses of the SU(3) Goldstone-bosons. As for the σ term which is simulated according to the ππ resonance, its value is determined by the PCAC relation m 2 σ m 2 π + 4m 2 u,d [167]. Finally, the chiral coupling constant, g ch is determined from the πNN coupling constant as following which assumes that SU(3) flavor is an exact symmetry only broken by the different mass of the strange quark. The chiral quark model parameters are listed in Table 1 and they have been fixed in advance reproducing hadron [1, 3,[144][145][146][147], hadron-hadron [148][149][150][151][152] and multiquark [109,153,154] phenomenology. In particular, with an application in the study of hidden-charm pentaquark states in Ref. [109], not only the P + c (4380) but also the later three newly observed P + c particles are all successfully interpreted.  36.976 r 0 (MeV fm) 28.170

Cornell Potential
In the fully-heavy tetraquark systems QQQQ (Q = c, b), the interplay between a pair of heavy quarkonium can be well approximated by the Cornell potential (linear confinement and Coulomb interactions) along with a spin-spin dependent interaction according to the Lattice QCD investigation [56]. Generally, this concise character can be incorporated into the following form for four-body systems, where the center-of-mass kinetic energy T CM is also subtracted without losing a generality as the case in the chiral quark model. Besides, the two-body interactions read as The three parameters α, σ and β in Eq. (13) which relate to the coulomb, confinement and spin-spin interactions are determined by Ref. [168] and their values are listed in Table 2. Additionally, Table 3 presents the theoretical and experimental masses of the S-wave QQ mesons, apparently, the deviations for each states are acceptable. Meanwhile, based on the investigations by quark model and lattice QCD [169], the quark-quark interaction V QQ is just half of V QQ . This conclusion will be employed in our study in the fully-heavy tetraquark states.  Table 3. Theoretical and experimental masses of the S-wave QQ mesons, unit in MeV.

Wave-function of Multi-quark System
There are four degrees of freedom in the quark level: color, spin, flavor and spatial. A complete antisymmetry N-quarks wave-function which fulfills the Pauli principle is written as In Eq. (14) ψ L , χ σ i S , χ f j I and χ c k stand for the spatial, spin, flavor and color wave-functions, respectively. Besides, A is the antisymmetry operator of system by considering the nature of two identical particles interchange i.e., qq in SU(3) flavor and QQ in charm or bottom sectors, etc. Fig. 2 shows six configurations in the double-heavy tetraquark states. In particular, Panel (a) is the dimeson structure, panel (b) is the diquark-antidiquark configuration and the other panels are of the K-types. Furthermore, the light flavor antiquarksq can be naturally switched withQ in the fully-heavy sectors. Accordingly, there are four exchange terms included in the antisymmetry operator for both the double-heavy and fully-heavy tetraquark states which the two quarks and antiquarks should be the same flavor and read as However, due to the asymmetry between cand b-quark, there are only two exchanges for theqcqb system, namely for the (udQ)(Qu) + (uuQ)(Qd) structure, and for the (uud)(QQ) configuration, respectively. However, for the doubly-heavy case in Fig. 4, the antisymmetry operators are and the above equations from (19) to (22) represent the results in configurations (a) to (d) of Fig. 4, respectively. In the following parts, we will introduce the wave-functions of tetraquarks and pentaquarks in each four degrees of freedom.

Color wave-function
Much richer color structures in multi-quark systems we will have than those in conventional hadrons (qq meson and qqq baryon). The wave-functions in color degree of freedom for each configurations are discussed according to the classification of tetraquark and pentaquark states, respectively.

• Tetraquark
Theoretically, the colorless wave-function of tetraquark in meson-meson configuration presented in Fig. 2(a) can be obtained through two channels, a color-singlet and a hidden-color. However, just the former channel is enough if all spatial excitation states are considered for the multi-quark systems [170,171]. Herein, a more economical approach by employing all possible color configurations along with their couplings is favored. Therefore, in the group of SU(3) color, the wave-functions of color-singlet (two color-singlet clusters coupling, 1 × 1) and hidden-color (two color-octet clusters coupling, 8 × 8) channels in meson-meson configuration of Fig. 2(a) are marked with χ c 1 and χ c 2 , respectively, (3brrb + 3ḡrrg + 3bgḡb + 3ḡbbg + 3rgḡr Meanwhile, as for the diquark-antidiquark channel shown in Fig. 2(b), the color wave-functions are χ c 3 (color triplet-antitriplet clusters coupling, 3 ×3) and χ c 4 (color sextet-antisextet clusters coupling, 6 ×6), respectively. In particular, it is symmetry for the two quarks (antiquarks) interchange in eq. (25) and antisymmetry in eq. (26).
The rest four structures from Fig. 2(c) to 2(f) are about K-types which the 4-quark wave-functions are constructed through the coupled-quarks in turn. Especially, their color bases are obtained by the following coupling coefficients according to the SU(3) color group 1 .
• Pentaquark It is the same as 4-quarks system we discussed before, hadron-hadron structures, diquark-diquark-antiquark ones and even much more color configurations are involved in the 5-quarks sector. However, due to enormous computation in exactly solving the 5-body Schrödinger equation, only baryon-meson configuration which the color singlet channels (k = 1) and the hidden color ones (k = 2, 3) along with their couplings are considered in this work. The details of color wave-functions χ c k are as below, while k = 2 and 3 is of a symmetry and antisymmetry wave-function, respectively. The sub-clusters bases are

Spin wave-function
In the 4-quark and 5-quark systems, the total spin S can take values from 0 to 2 for the former case and 1 2 to 5 2 for the later one, respectively. Their spin wave-functions for one certain configuration listed from Fig. 2 to 4 are obtained by the couplings of Clebsh-Gordon coefficients in the spin SU(2) group. We now proceed to describe them in the tetraquark and pentaquark states.

• Tetraquark
The spin wave-function χ σ i S,M S of 4-quark system is organized by two sub-clusters for the dimeson and the diquark-antidiquark structures, and couplings in an increased sequence of quark numbers for K-types. Furthermore, because no spin-orbital dependent potential is included in the model, the third component (M S ) of total spin can be taken the same value as S without losing generality. The details are written as In the above equations, the superscripts l 1 ...l 4 and m 1 ...m 6 are signs for each structures presented in Fig. 2, their specific assignments are summarized in Table 4. Meanwhile, the necessary sub-clusters bases are read as with χ σ  for S = 5/2. These expressions can be obtained easily using SU(2) algebra and considering the 3-quark and quark-antiquark sub-clusters individually. The details read as

Flavor wave-function
A similar procedure can be implemented in the iso-spin space and the total flavor wave-function of multi-quark system is introduced according to each configuration of state.

• Tetraquark
Generally, there are two kinds of 4-quark systems that we are dealing with, the doubly-and fully-heavy tetraquark states. Hence, the well defined isospin quantum number I can be taken either 0 or 1 for QQqq systems but only the iso-scalar state I = 0 will be considered for QQQQ sectors (Q = c, b and q = u, d, s). Herein, we use χ f i I,M I to represent the flavor wave-functions and the superscript i = 1, 2 and 3 stand for ccqq, bbqq and cbqq systems, respectively. The specific expressions are as following, The third component of the isospin M I is also set to be equal to the absolute value of total isospin I as the case in spin space. This is reasonable since no symmetry broken interaction on isospin included in our model. Meanwhile, it is a trivial results for the fully-heavy tetraquark states i.e., cccc and bbbb.
• Pentaquark Three 5-quark systems are studied in this work, namely, hidden-charm, -bottom and doubly charmed pentaquarks. Accordingly, isospin equals to 1 2 and 3 2 are both allowed. However, only the hidden-charm pentaquark state in I = 1 2 sector is discussed in our earliest work. The total 5-quark flavor wave-function is obtained by a coupling with the bases of two sub-clusters which are baryon and meson, respectively.
In particular, in uudQQ (Q = c, b) systems, we have two kinds of separation, one is (udQ)(Qu) + (uuQ)(Qd) and the other is (uud)(QQ) as illustrated in Fig. 3. The wave-functions are as below where the necessary bases on sub-clusters are As for the QQqqq (Q = c, b) tetraquarks shown in Fig. 4 where the complete configurations in baryon-meson sector are considered, their flavor wave-functions with I = 1/2 and 3/2 read as where the third component of isospin is still chosen to be the same as total one, and the superscript n which value is from 1 to 4 marks each four configurations in Fig. 4. The flavor wave functions for the baryon and meson clusters are

Spatial wave-function
The few-body bound state problem is solved in an exact and efficient variational method, Gaussian expansion method (GEM) [172]. In this theoretical framework, the intrinsic spatial wave-function of state is fitted by various widths (ν n ) of Gaussian bases which are taken as the geometric progression form. Eq. (121) presents a general expression of the orbital wave-function, where N nl is the normalization constants The angle part of space is trivial in the S-wave multi-quark state, therein the angular matrix element is just a constant due to Y 00 = √ 1/4π. However, as to avoid laborious Racah algebra in solving the angular excitation state, a powerful technique named infinitesimally shifted Gaussian (ISG) [172] is employed. With the spherical harmonic function absorbed into a shifted vector D, the new function is Their applications in the tetraquark and pentaquark states will be discussed individually.
• Tetraquark The 4-body system is investigated in a set of relative motion coordinates, the spatial wave function is, Particularly, the three internal Jacobi coordinates for Fig. 2(a) of meson-meson configuration are read as and the diquark-antdiquark structure of Fig. 2(b) are defined as, Moreover, the other K-type configurations from Fig. 2(c) to 2(f) are, values of the above subscripts i, j, k, l are according to the definitions of each configuration in Fig. 2.
Obviously, the center-of-mass kinetic term T CM can be completely eliminated for a nonrelativistic system in these sets of relative motion coordinates.

• Pentaquark
The spatial wave-function of 5-body system is also constructed in the relative motion coordinates, eq. (134) presents a general form.
where in a baryon-meson configuration, the four Jacobi coordinates are defined as Besides, the T CM part can also be entirely deducted in a 5-quark non-relativistic system by this set of Jacobi coordinates.

Results and Discussions
In the constituent quark model formalism, the possible low-lying bound and resonance states of doubly-, fully-heavy tetraquarks, hidden-charm, -bottom and doubly charmed pentaquarks are systematically investigated by means of the computational approach, Gaussian expansion method. The obtained results along with their corresponding discussions are organized as follows.

Doubly and Fully Heavy Tetraquarks
In this part, the S-wave QQqq and QQQQ (Q = c, b, q = u, d, s) tetraquark states with J P = 0 + , 1 + and 2 + , the isospin I = 0 or 1 are studied in the chiral quark model and Cornell potential model, respectively. We will discuss them one by one.

Double-charm tetraquarks
According to the Pauli principle, all possible couplings in spin, flavor and color degrees of freedom for the S-wave tetraquark states are considered. Table 5 presents the allowed meson-meson and diqurak-antidiquark channels for doubly charmed tetraquarks in J P = 0 + , 1 + and 2 + , I = 0 and 1 states. However, bound and resonance states are only obtained in the I(J P ) = 0(1 + ) quantum state. Their calculated masses are listed in Table 6 where two dimeson channels, D + D * 0 and D * + D * 0 , two diquark-antidiquark channels, (cc) * (ūd) and (cc)(ūd) * along with their couplings are all considered. Particularly, the first column lists the allowed channels, their related experimental threshold values (E ex th ) are also marked in the parentheses. The color-singlet (S), hidden-color (H) channels and their couplings for dimeson configurations are listed in the second column. The computed masses (M) for each channels along with their binding energy (E B ), which is obtained by calculating the difference between the theoretical threshold (E th ) and the tetraquark mass (M), E B = M − E th , are presented in the 3rd and 4th columns, respectively. Then, the re-scaled masses (M ), whose theoretical uncertainties coming from the model calculation of meson spectra are avoided, for meson-meson structures are listed in the last column, and they are obtained by comparing the experimental threshold values and binding energies, Firstly, in the single channel computation for color-singlet (S) and hidden-color cases (H) of the D ( * )+ D * 0 structures, the lowest masses are all above threshold values. However, loosely bound states of D + D * 0 and D * + D * 0 are available in a coupled-channels calculation (S+H). Then after a mass shift correction which is according to the difference between the theoretical and experimental thresholds, the re-scaled masses of these two bound states are 3876 MeV and 4017 MeV, respectively. Furthermore, these two bound state can be identified as the molecule states of D ( * )+ D * 0 due to more than 95% contributions come from the color-singlet channels.
Deeply bound diquark-antidiquark channel (cc) * (ūd) with a binding energy ∼−140 MeV is found and the theoretical mass is 3778 MeV. However, another diquark-antidiquark (cc)(ūd) * state is unbound and its mass is above the D + D * 0 and D * + D * 0 theoretical thresholds with E B = +305 MeV and +186 MeV, respectively. In a further step, we performed a complete coupled-channels calculation for the channels listed in Table 5, and the lowest-lying bound state mass is 3726 MeV. By analyzing the distance between any two quarks of ccqq system in Table 8, the nature of compact double-charm tetraquark state is clearly presented. The general size of this tetraquark state is around 0.67 fm. Meanwhile, tightly bound and compact structure of the obtained tetraquark state is also confirmed in Table 7 where each component in the coupled-channels calculation is presented and the two dominant channels are the color-singlet channel D + D * 0 (25.8%) and diquark-antidiquark (cc) * (qq) one (36.7%).
As to find possible double-charm tetraquark resonance in excitation state, the complex scaling method is employed in the complete coupled channels calculation too. Fig. 5 shows the distributions of calculated complex energies in the I(J P )= 0(1 + ) channel. Apparently, the bound stat is independent of the rotated angle which is varied from 0 • to 6 • and still locates at 3726 MeV of real-axis. The other energy points are generally aligned along the D ( * )+ D * 0 threshold lines which are scattering states. However, one possible resonance state whose mass and width are ∼4312 MeV and ∼16 MeV, respectively, is obtained in the complex plane and it is marked in a big orange circle with three calculated pole almost overlapping. This unchanged pole is far from the D + D * 0 threshold lines, therefore, it can be identified as a D * + D * 0 resonance. Table 5. All possible channels for ccqq (q = u or d) tetraquark systems. Table 6. Lowest-lying states of double-charm tetraquarks with quantum numbers I(J P ) = 0(1 + ), unit in MeV.

Channel
Mixed 3726 Table 7. Component of each channel in coupled-channels calculation with I(J P ) = 0(1 + ), the numbers 1 and 8 of superscript are for singlet-color and hidden-color channel respectively.
0.2% Table 8. The distance, in fm, between any two quarks of the found tetraquark bound-states in coupled-channels calculation (q = u, d). •

Double-bottom tetraquarks
In the bbqq (q = u, d) sector, possible B ( * )−B * 0 meson-meson channels and (bb) ( * ) (qq) ( * ) diquark-antidiquark structures in each quantum states are listed in Table 9. However, it is similar to the doubly charmed case, bound and resonance states are only found in the I(J P ) = 0(1 + ) state. Table 10 shows the calculated results, and the arrangements of each columns are similar to Table 6. Firstly, one can notice that bound states of B −B * 0 and B * −B * 0 in color-singlet channels are obtained, and the ∼−10 MeV binding energy is owing to much heavier b-flavored quarks included. Additionally, in a coupled-channels calculation with hidden-color channels included, a deeper binding energies (∼−35 MeV) are obtained for these two dimeson channels. The percentages of color-singlet channel and hidden-color on are around 80% and 20%, respectively. By considering the systematic uncertainty during calculation, the modified masses for B −B * 0 and B * −B * 0 bound state are 10569 MeV and 10613 MeV respectively.
There are two diquark-antidiquark channels under investigated, (bb) * (ūd) and (bb)(ūd) * , the calculated masses are 10261 MeV and 10787 MeV, respectively. Clearly, the former structure is a tightly bound tetraquark state with binding energy E B = −336 MeV. However, the other one is 190 MeV above the B −B * 0 theoretical threshold. Our result on this diquark-antidiquark bound state is supported by Refs. [26][27][28]30], and only ∼130 MeV lower than the calculated value in Ref. [28].
In the third step, a complete coupled-channels calculation is performed. Particularly, two bound states which masses are 10238 MeV and 10524 MeV are obtained. The first state is close to the (bb) * (ūd) channel and 23 MeV lower by the coupling effect. The second bound state is below the B −B * 0 theoretical threshold with 73 MeV. Furthermore, Table 11 presents the components of these two bound states in coupled-channels computation. There are both around 42% (bb) * (ūd) channel and ∼20% B ( * )−B * 0 in the color-singlet channels of these tetraquark states. Accordingly, they can be identified as compact bound states in an analysis of the internal structure which the distance between any two quarks are calculated in Table 12. Therein, the general size is less than 0.83 fm and the values (0.328 fm and 0.711 fm) on two bottom quarks are even small for the two obtained bound states.
In a complex range investigation on the bbqq tetraquark in I(J P ) = 0(1 + ) state, apart from the original two bound states, one narrow resonance state is also found. Fig. 6 shows the distributions of the calculated energy points in the complete coupled case and the rotated angle θ is also taken from 0 • to 6 • . In this range, the threshold lines of two meson-meson channels B −B * 0 and B * −B * 0 are well established and the two bound states is stable in the real-axis at 10238 MeV and 10524 MeV, respectively. Meanwhile, a fixed resonance pole at ∼10.8 GeV is obtained with the variation of θ. We marked it with a big orange circle in Fig. 6, besides the theoretical mass and width of this narrow resonance is 10814 MeV and 2 MeV, respectively. Because it is closer to the B * −B * 0 threshold lines, this meson-meson resonance is expected to be confirmed in the future experiment. Table 9. All possible channels for bbqq (q = u or d) tetraquark systems.  Table 11. Component of each channel in coupled-channels calculation with I(J P ) = 0(1 + ), the numbers 1 and 8 of superscript are for singlet-color and hidden-color channel respectively.

•
Charm-bottom tetraquarks Table 13 lists the allowed channels of cbqq tetraquark with J P = 0 + , 1 + and 2 + , I = 0 and 1, respectively. However, some bound and resonance states are only found in the iso-scalar sector, besides the calculated results on meson-meson configurations are supported by the investigation of Ref. [173]. We will discuss these tetraquark states according to I(J P ) quantum numbers respectively. Meanwhile, the arrangements of each columns in Tables 14, 15 and 16 are still the same as those in Table 6. Table 13. All possible channels for cbqq (q = u or d) tetraquark systems. For a brief purpose, only the D ( * )0 B ( * )0 structures are listed and the corresponding D ( * )+B( * )− ones are absent in I = 0. However, all these configurations are still employed in constructing the wavefunctions of 4-quark systems. Table 14 summarizes the calculated results of each meson-meson, diquark-antiquark channels along with their couplings. Weakly bound states of D 0B0 and D * 0B * 0 in color-singlet channels are obtained firstly, the binding energies are −4 MeV and −9 MeV, respectively. Then in a coupled-channels computation which the hidden-color channels are included, bound state in the D * 0B * 0 channel is further pushed with E B = −39 MeV, however this coupling effect is quite weak in the D 0B0 channel. These features on binding energies are confirmed by the investigation of each components shown in Table 14, where the proportions of color-singlet channels in the D 0B0 and D * 0B * 0 are 96.4% and 87.8%, respectively.

I=0 I=1
In the diquark-antidiquark sector, by comparing with the D 0B0 theoretical threshold value, one tightly bound state (cb)(ūd) with E B = −148 MeV and one excited state (cb) * (ūd) * with E B = +306 MeV are obtained. Furthermore, the lowest bound state is found at 6980 MeV in the fully coupled-channels investigation. Obviously, this should be a compact charm-bottom tetraquark state, since its nature can be confirmed by analyzing the component and inner structure presented in Table 17 and 18, respectively. In particular, size of the tetraquark in 0(0 + ) state is less than 0.66 fm and almost 50% is the (cb)(ūd) channel, the sub-dominant components are 26.4% (D 0B0 ) 1 and 21.5% (D * 0B * 0 ) 1 channels.
The fully coupled-channels calculation is performed in a complex-range where the rotated angle θ is taken from 0 • to 6 • , and then, the nature of bound tetraquark state is clearly shown in Fig. 7, where the calculated dots are always fixed in the real-axis and at 6980 MeV. The other energy points which are generally aligned along the corresponding D 0B0 and D * 0B * 0 threshold lines are of scattering states. However, we also find a narrow resonance state which mass is around 7.7 GeV and width is ∼12 MeV. Although this resonance pole, marked with orange circle, is both above D 0B0 and D * 0B * 0 thresholds, we can still identify it as a D * 0B * 0 molecule resonance which is farther away from D 0B0 lines. Table 14. Lowest-lying states of charm-bottom tetraquarks with quantum numbers I(J P ) = 0(0 + ), unit in MeV.

Channel
Color   Table 15. Similar to the above discussed doubly heavy tetraquarks, four conclusions can be drawn in a real-range calculation. (I) Loosely bound states are obtained in the color-singlet channels of D 0B * 0 , D * 0B0 and D * 0B * 0 , their weak binding energies are E B = −3 MeV, −2 MeV and −2 MeV, respectively. (II) The coupling between singlet-and hidden-color channels in meson-meson configuration are weak with the majority component ( more than 90%) is the former channel. (III) One tightly bound diquark-antidiquark channel (cb) * (ūd) is found and the theoretical mass is 7039 MeV, the other two diquark-antidiquark channels' masses are above 7.5 GeV. (IV) In a fully coupled-channels calculation, the lowest mass of bound state reduced to 6997 MeV.
In order to have a better insight into the nature of the obtained bound state in the complete coupled-channels calculation, we may also focus on the results on the components and structures of the tetraquark bound state. As shown in Table 17 and 18, the dominant contribution 46.4% is from (cb) * (ūd) channel and other three sub-dominant channels are the color-singlet channels of D 0B * 0 , D * 0B0 and D * 0B * 0 , their contributions are 20.2%, 11.6% and 16.8%, respectively. This strong coupling effect leads to a compact structure which size is less than 0.67 fm again.
With the complete coupled-channels computation extended to a complex-range which θ is chosen still from 0 • to 6 • , the bound state is confirmed again. Moreover, one more resonance state is found. In Fig. 8 one can notice that apart from the most scattering points which are the D 0B * 0 , D * 0B0 and D * 0B * 0 channels, one bound state at 6997 MeV of real-axis and one narrow resonance state with mass and width is 7327 MeV and 2.4 MeV are obtained. Due to the resonance pole is located in the region between D * 0B0 and D * 0B * 0 thresholds, it can be identified as the D * 0B0 resonance state according to the definition in CSM. Table 15. Lowest-lying states of charm-bottom tetraquarks with quantum numbers I(J P ) = 0(1 + ), unit in MeV.

Channel
Color For the highest spin state, only one meson-meson D * 0B * 0 channel and one diquark-antidiquark (cb) * (ūd) * . Firstly, in the single channel calculations, only the D * 0B * 0 color-singlet channel is loosely bound with E B = −2 MeV. Furthermore, this fact is not changed by the coupling with a hidden-color channel and only 1 MeV decreased in the complete coupled-channels case, the lowest mass is 7333 MeV. Hence, a molecular-type structure is possible for the obtained bound state and it has been supported by calculating the quark distances which is already beyond 1.6 fm from Table 18 and ∼99% contributions come from color-singlet channel of D * 0B * 0 .
In contrast to the previous tetraquark states, no resonance is found in 02 + state. It is clearly shown in Fig. 9 that all of the calculated poles are aligned along the D * 0B * 0 threshold lines except the weakly bound state at 7333 MeV. Table 16. Lowest-lying states of charm-bottom tetraquarks with quantum numbers I(J P ) = 0(2 + ), unit in MeV.

Channel
Color Complex energies of charm-bottom tetraquarks with I(J P ) = 0(2 + ) in the coupled channels calculation, θ varying from 0 • to 6 • . Table 17. Component of each channel in coupled-channels calculation, the numbers 1 and 8 of superscript are for singlet-color and hidden-color channel respectively, (q = u, d).

QQss Tetraquarks
As for a natural extension of the work on QQqq (Q = c, b and q = u, d) systems, the double-heavy tetraquark state in strange quark sector is investigated herein. In addition, for this 4-quark system, a complete set of configurations including meson-meson, diquark-antidiquark and K-type structures (Fig. 2) is included. As for a clarify purpose, masses and mean square radii of the Qs mesons are listed in Table 19. These results will be useful in identifying possible QQss bound or resonance states. Furthermore, Tables ranging from 20 to 28 summarized our theoretical findings. Particularly, in those tables, the first column shows the allowed channels and, in the parenthesis, the noninteracting meson-meson threshold value of experiment. Color-singlet (S), hidden-color (H) along with other configurations are indexed in the second column, respectively, the third column lists the necessary bases in spin, flavor and color degrees of freedom, the fourth and fifth columns refer to the theoretical mass of each channels and their couplings.

Meson nL
The. Exp.
There is no bound state in the doubly charmed ccss system, but resonances are found in the I(J P ) = 0(0 + ) and 0(2 + ) quantum states. Obviously, these results are different from the ccqq tetraquark states. We will discuss them in the following parts. Table 20 presents all of the possible channels in ccss system with I(J P ) = 0(0 + ) state. It is clearly to notice that the meson-meson channels of D + s D + s and D * + s D * + s both in color-singlet and hidden-color states are unbound. Moreover, the coupled results in these two color configurations are still unchanged with the obtained masses are 3978 MeV and 4377 MeV, respectively. As for the two diquark-antidiquark channels, masses of (cc)(ss) and (cc) * (ss) * are both around 4.4 GeV which is above the D ( * )+ s D ( * )+ s threshold values. Their coupled-mass 4379 MeV is quite close to the value of hidden-color channels. However, although there is a strong coupling between them, it is still not enough to have a bound state. Additionally, bound state is still unavailable in the K-type configurations, mass of the four K-type channels are in the region from 4.2 GeV to 4.8 GeV and there is a degeneration at 4.4 GeV for (cc) ( * ) (ss) ( * ) , K 3 and K 4 channels. Finally, the lowest mass (3978 MeV) in the complete coupled-channels calculation is the same as that in the color-singlet channels. Therefore, no bound state is found in ccss tetraquark with I(J P ) = 0(0 + ) state.
Apart from the above real-range study, Fig. 10 shows the fully coupled-channels computation in the complex scaling method. Particularly, in the mass region from 3.9  All of the above channels: 2. I(J P ) = 0(1 + ) state Two dimeson channels, D + s D * + s and D * + s D * + s , one diquark-antidiquark channels, (cc) * (ss) * and four-K-types configurations are studied in Table 21. Firstly, each of the channels along with the couplings in one certain configuration are all unbound. Specifically, the couplings are quite weak in both the color-singlet and hidden-color channels of meson-meson configurations. However, several to hundreds of MeV decreased are obtained in the coupling computations of K-types and their coupled-masses are ∼4.4 GeV. Then in a complete coupled-channels investigation, the lowest state is still unbound with mass equals to the theoretical threshold of D + s D * + s , 4105 MeV. Meanwhile, in comparison with the results of ccqq tetraquarks in Table 6, one can find that character of the D + s D * + s state is opposite to D + D * 0 case which has a binding energy ∼200 MeV. Fig. 11 shows the distributions of complex energies of the D + s D * + s and D * + s D * + s scattering states in the complete coupled-channels calculation. In 4.1 GeV to 5.0 GeV energy gap, the ground states and first radial excitation ones are clearly shown. In spite of three slowly descended poles between 4.55 GeV and 4.70 GeV, the nature of resonance state is in contrast to them and neither bound state nor resonance one can be obtained in this quantum state. All of the above channels: Nevertheless, three resonance states are found in the complex-range calculation which all of the channels listed in Table 22 are considered. In Fig. 12 we can see it clearly that apart from the continuum states of D * + s D * + s in the ground and first radial excitation states, three almost fixed poles are obtained at ∼4.8 GeV. In particular, the three orange circles marked the obtained resonance states of D * + s D * + s molecule state, their masses and widths are (4821 MeV, 5.58 MeV), (4846 MeV, 10.68 MeV) and (4775 MeV, 23.26 MeV), respectively. These poles are also around 0.6 GeV above two non-interacting D * + s mesons threshold, and ∼0.1 GeV below its first radial excitation state.  • bbss tetraquarks In this part, we study the I(J P ) = 0(0 + ), 0(1 + ) and 0(2 + ) states for bbss tetraquarks. It is similar to the ccss systems that only narrow resonances are found in 0(0 + ) and 0(2 + ) states. Let us proceed to discuss them in detail.
1. I(J P ) = 0(0 + ) state In this quantum state as shown in Table 23, there are two meson-meson, diquark-antidiquark and 12 K-type channels under investigated. Theoretical masses of each single channels locate in the region from 10.71 GeV to 11.45 GeV, besides, the coupled-mass ∼10.9 GeV inB 0 sB 0 s andB * sB * s channels is comparable with those of K 1 , K 2 , K 3 and K 4 channels. Although the color-singlet channels mass is the lowest, on bound state is found in the coupled-channels calculation which includes the fully channels case.
Additionally, Fig. 13 shows the results of complete coupled-channels by complex scaling method. The states of ground and first radial excitation forB 0  All of the above channels: 2. I(J P ) = 0(1 + ) state In contrast the obtained deeply bound and narrow resonance states of bbqq tetraquarks in 01 + state, bound state is forbidden as the ccss system. Firstly, in Table 24, masses of the meson-meson channels ofB 0 sB * s andB * sB * s in color-singlet state are 10755 MeV and 10800 MeV, respectively. These results are not changed when their hidden-color channels included. As for the other exotic structures, i.e., diquark-antidiquark bbss, and K-type channels, the theoretical masses are all ∼10.9 GeV. Besides, these excited states do not help in forming a bound state in the complete coupled-channels calculation which the lowest mass is still 10755 MeV. The above conclusion is clear shown in Fig. 14 which is the results in complex-range study. In the mass region from 10.7 GeV to 11.3 GeV, only two scattering state ofB 0 sB * s andB * sB * s channels are obtained, the other bound or resonance state is unavailable in the present theoretical framework. Specifically, one may notice a gradually varied dots at 11.15 GeV, however, these unstable poles still can not be identified as a regular resonance state.  3. I(J P ) = 0(2 + ) state OneB * sB * s meson-meson channel is studied in Table 25, but the calculated mass 10.8 GeV in color-singlet channel is just the theoretical threshold value. The other hidden-color, diqaurk-antidiquark and K-type structures are all above 10.9 GeV in the coupled-channels calculation of each configurations except the 10.87 MeV for K 1 and K 2 channels. Meanwhile, it is the same as the several cases before that the lowest energy level is still unbound in the fully coupled-channels computation and the coupled mass is 10.8 GeV.
Nevertheless, when the investigation extended to the complex-range, new exotic states are obtained. In Fig. 15, three narrow resonance states at ∼11.35 GeV are marked with orange circles which are near the real-axis. With a angle θ varied less than 6 • , masses and widths of these three fixed poles are (11.33 GeV, 1.48 MeV), (11.36 GeV, 4.18 MeV) and (11.41 GeV, 2.52 MeV), respectively. Obviously, they are also around 0.6 GeV above theB * sB * s threshold, therefore the nature of twoB * s mesons molecule states can be drawn herein. All of the above channels:

• cbss tetraquarks
Bound state is also not found in this sector, however, some narrow resonance states in I(J P ) = 0(0 + ), 0(1 + ) and 0(2 + ) are obtained. The following parts are devoted to the discussions on them.
1. I(J P ) = 0(0 + ) state As shown in Table 26, two meson-meson syructures, D + sB 0 s and D * + sB * s , two diquark-antidiquark channels, (cb)(ss) and (cb) * (ss) * , 14 K-type ones are considered in this quantum state. Firstly, the calculated masses of these channels are located in the energy region from 7.34 GeV to 8.67 GeV and no bound state is found. Then this result remains in the coupled-channels computations for the dimeson, diquark-antidiquark and K-type configurations. In particular, the coupling in D + sB 0 s and D * + sB * s channels is extremely weak. However, dozens to hundreds MeV decreased in the other configurations couplings, and they are all above 7.6 GeV. Finally, in the real-range fully coupled-channels calculation, the lowest energy level is still at 7344 MeV.
In a further investigation which the CSM is employed, one can find that two resonance states are marked in Fig. 16. Around 0.5 GeV higher than the D * + sB * s threshold value, two narrow resonance poles with masses and widths equal to (7.92 GeV, 1.02 MeV) and (7.99 GeV, 3.22 MeV), respectively are stable against the variation of rotated angle θ. Herein, we can identify them as the molecule states of D * + sB * s .    All of the above channels:

I(J P ) = 0(2 + ) state
There is both one channel for the dimeson D * + sB * s and diquark-antidiquark (cb) * (ss) * configuration. Besides, 7 K-type channels are listed in Table 28. First of all, it is similar to the other QQss state, bound state is impossible neither in the single channel calculation nor the coupled-channels case. Theoretical threshold of the lowest state D * + sB * s is 7516 MeV and the other excited states in coupled-channels are all above 7.72 GeV, in particular, there is a degeneration between K 3 and K 4 channels which coupled masses are 7697 MeV.
Unlike the results of cbqq tetraquarks in 02 + state, two resonance states are obtained in the complex analysis of the complete coupled-channels of cbss tetraquarks. In Fig. 18, two stable resonance poles circled with orange are ∼0.6 GeV above the D * + sB * s threshold value and ∼0.1 GeV below its first radial excitation state. Hence, they can be identified as the D * +

QQQQ Tetraquarks
As the previous discussion on the fully heavy tetraquark states, recent experimental progress on the cccc system by the LHCb collaboration [21] triggers a revived interest in the QQQQ system. Herein, a potential model which is based on the well investigated phenomenon of heavy quark pair by Lattice QCD are employed and two sectors, fully charm and fully bottom tetraquarks, will be discussed in the following parts. •

Fully-charm tetraquarks
All of the three spin-parity channels, J P = 0 + , 1 + and 2 + are studied in the cccc tetraquark sector and no bound state is obtained. However, some resonance states are possible and we will introduce them individually. Particularly, in Tables 29, 32  1. I(J P ) = 0(0 + ) state Two meson-meson channels, η c η c , J/ψJ/ψ and two diquark-antidiquark structures, (cc)(cc), (cc) * (cc) * are investigated herein. Table 29 lists the calculated masses for them along with the coupling. Clearly, the lowest energy level is 6469 MeV of (cc) * (cc) * channel and the other diquark-antidiquark structure of (cc)(cc) is at 6683 MeV. Besides, the calculated masses of color-singlet and hidden-color channels in the dimeson configurations, η c η c and J/ψJ/ψ, are degenerated since there is no color-dependent interaction in the Cornell-like model. It is 6536 MeV in the η c η c channel and 6657 MeV in the di-J/ψ one. Herein, the fully coupled-channels calculation is important for different color configurations are not orthogonal. The bottom of Table 29 presents two resonances' masses which are obtained in such computation. In particular, they are 6423 and 6650 MeV, respectively, and the first resonance state is around 50 MeV lower than the mass of (cc) * (cc) * channel.
In order to have better identifications of the nature of these two resonance states in the fully coupled-channels calculation. The components and their internal structures are analyzed in Table 30 and 31, respectively. Firstly, the couplings are strong for both of these two resonance states. Besides, the meson-meson configuration dominates them. From Table 30 one can find that the components of η c η c and J/ψJ/ψ, which are the sum of their corresponding singlet-and hidden-color channels, are (49%, 45%) for the first resonance state and (31%, 48%) for another one. Furthermore, they are both of a compact tetraquark configurations with sizes ∼0.34 fm.
We also extend the investigation on the complete coupled-channels from a real-range to a complex one, the results are shown in Fig. 19. In the 6 − 10 GeV energy region, there are two fixed poles in the real-axis when the rotated angle θ varied from 0 • to 6 • . Actually, masses of these two stable poles are just 6423 MeV and 6650 MeV. This fact confirms the two previously obtained resonance states in the fully coupled-channels calculation of real-range. Moreover, the other energy points are unstable and always descend more or less along with the change of θ. Table 29. Possible resonance states of fully-charm tetraquarks with quantum numbers I(J P ) = 0(0 + ), unit in MeV.

Channel
Index   Table 31. The distances, between any two quarks of the found fully-charm resonance states with I(J P ) = 0(0 + ) in coupled-channels calculation, unit in fm. 2. I(J P ) = 0(1 + ) state In this sector we still do not find any bound state of cccc system, however, resonance states with masses around 6.6 GeV are available. In particular, three almost degenerate states with masses ∼6.67 GeV are presented in Table 32, they are the color-singlet and hidden-color channel of η c J/ψ and (cc) * (cc) * diquark-antidiquark channel, respectively. However, no stable resonance state can be found in the di-J/ψ channel. Then in a fully coupled-channels case, which both the meson-meson and diquark-antidiquark structures are considered, lower mass at 6627 MeV is obtained for the I(J P ) = 0(1 + ) state. Herein, the coupling between color-singlet and hidden-color channels of η c J/ψ is also strong that the contributions are about 56% and 35% for them, respectively. Only less than 10% is from the (cc) * (cc) * channel. From Table 33 we can conclude that it is a compact tetraquark state, which size is ∼0.35 fm, in the I(J P ) = 0(1 + ) state.
Additionally, the complete coupled-channels investigation is also performed in the complex-range framework where a rotated angle is varied from 0 • to 6 • . The calculated results of complex energies from 6.5 GeV to 10.0 GeV are presented in Fig. 20. Obviously, there is a fixed pole in the real-axis and circled with orange. It can be identified as a η c J/ψ resonance with mass at 6627 MeV. However, the other poles should be corresponded to the scattering states of η c J/ψ in higher energy region.  Table 33. The distances, between any two quarks of the found fully-charm resonance states with I(J P ) = 0(1 + ) in coupled-channels calculation, unit in fm. 3. I(J P ) = 0(2 + ) state Similar to the 0(1 + ) quantum state, two meson-meson J/ψJ/ψ channels and one diquark-antidiquark (cc) * (cc) * channel are considered in the highest spin state of tetraquark. In particular, masses of them are all around 7.03 GeV and the (cc) * (cc) * channel is the lowest one with mass at 7026 MeV. Then in their coupling calculation, the mixed mass is 7014 MeV shown in Table 34. Meanwhile, the percentages of [J/ψJ/ψ] 1 , [J/ψJ/ψ] 8 and (cc) * (cc) * are around 53%, 33% and 14%, respectively. This strong coupling fact also leads to a compact tetraquark configuration with size around 0.38 fm listed in Table 35.
By employing the complex scaling method in our model investigation of the complete coupled-channels, the above conclusions are confirmed too. Particularly, one stable resonance pole against the two-body strong decay is obtained at 7014 MeV of Fig. 21. Obviously, this resonance mass is quite close to the reported new structure at 6.9 GeV by the LHCb collaboration [21]. Hence it can be identified as a compact fully charmed tetraquark in 0(2 + ) state. However, the other complex energy points always move along with the variation of angle θ. Table 34. Possible resonance states of fully-charm tetraquarks with quantum numbers I(J P ) = 0(2 + ), unit in MeV.

Channel
Index  Table 35. The distances, between any two quarks of the found fully-charm resonance states with I(J P ) = 0(2 + ) in coupled-channels calculation, unit in fm. •

Fully-bottom tetraquarks
Bound states and resonances in J P =0 + , 1 + and 2 + , I = 0 are found in the bbbb tetraquark systems. Furthermore, they are more compact than the cccc resonances. The calculated masses are listed in Tables 36, 39 and 41. Particularly, the first column shows the allowed channels and, in the parenthesis, the noninteracting meson-meson experimental threshold values. Meson-meson and diquark-antidiquark channels are indexed in the second column, respectively. The necessary bases in spin, flavor and color degrees of freedom are listed in the third column. The fourth column refers to the theoretical mass of each channels and their couplings, binding energies of the dimeson channels are listed in the last column. The details are as follows.
1. I(J P ) = 0(0 + ) state As shown in Table 36, four meson-meson configurations, which include the singlet-and hidden-color channels of η b η b and ΥΥ, along with two diquark-antidiquark ones, (bb)(bb) and (bb) * (bb) * , are calculated in the 0(0 + ) state of fully bottom tetraquark. Firstly, two low-lying stable states are found in each single channel computations. Masses of the first energy level of them are ∼18.0 GeV, and the second one is around 19.0 GeV. Obviously, the lowest-lying state of each channel is deeply bound with E b more than −800 MeV and the higeher one is a resonance state which is ∼150 MeV above the threshold. Additionally, masses of three stable states in the complete coupled-channels case are also listed in Table 36, in particular, they are 17.92 GeV, 18.01 GeV and 19.28 GeV, respectively. Generally, they are still located at 18.0 GeV and 19.0 GeV.
Natures of these three exotic states can be indicated in Table 37 and 38. In particular, the percentages of each meson-meson and diquark-antidiquark channels of bbbb tetraquark states are listed in Table 37. Therein, the diquark-antidiquark channels are less than 11% for all of these three states. Nevertheless, the couplings between color-singlet and hidden-color channels of η b η b and ΥΥ are very strong. Besides, a compact fully-bottom tetraquark configuration is shown in Table 38, where the sizes of two bound states are around 0.16 fm and 0.29 fm for the resonance one. Apparently, the conclusions are consistent with the deeply binding energies obtained before.
Additionally, we also investigate the bbbb system in a complex-range. With a complex scaling method employed in the fully coupled-channels calculation, the three fully-bottom tetraquark states, which are obtained in the real-range, are all well presented in Fig. 22 again. Therein, apart from the scattering points, which always descend with the variation of angle θ, the three poles in the real-axis and circled orange are stable. Hence, the bound and resonance states in bbbb sector are possible. However, as the statement on the fully-heavy tetraquarks in Sec. I, since no color-dependent interaction is considered in our present model, the obtained exotic states, especially the bound states of bbbb are quite negotiable. Much more efforts both theoretical and experimental are deserved [174]. Table 36. Possible bound and resonance states of fully-bottom tetraquarks with quantum numbers I(J P ) = 0(0 + ), unit in MeV.

Channel
Index  Table 37. Component of each channel in the coupled-channels calculation of fully-bottom bound and resonance states with I(J P ) = 0(0 + ). 3.5% 0.0% Table 38. The distances, between any two quarks of the found fully-bottom bound and resonance states with I(J P ) = 0(0 + ) in coupled-channels calculation, unit in fm.

I(J P ) = 0(1 + ) state
It is similar to the cccc system in 0(1 + ) state, di-Υ is a scattering state under investigation. Hence Table 39 Table 39 where the η b Υ meson-meson channel dominates the first two exotic states and a strong coupling between dimeson and diquark-antidiquark configurations is obtained for the third resonance states.
In Table 40 we can notice that sizes of the three bbbb tetraquark states are quite comparable with those in 0(0 + ) case. It is ∼0.16 fm and 0.27 fm for the bound and resonance states, respectively. Meanwhile, Fig. 23 shows the distribution of complex energies in the complete coupled-channels calculation of bbbb system. The bound state, which mass is 18.01 GeV, and the two resonances, which masses at 19.34 GeV and 19.63 GeV, respectively, are stable in the real-axis and independent of the variation of rotated angle θ. Table 39. Possible bound and resonance states of fully-bottom tetraquarks with quantum numbers I(J P ) = 0(1 + ), unit in MeV.

Channel
Index  Table 40. The distances, between any two quarks of the found fully-bottom bound and resonance states with I(J P ) = 0(1 + ) in coupled-channels calculation, unit in fm. 3. I(J P ) = 0(2 + ) state In this highest spin state, one meson-meson configuration Υ(1S)Υ(1S), which include a color-singlet and a hidden-color channel, and one diquark-antidiquark configuration (bb) * (bb) * are considered. Firstly, bound states at ∼18.24 GeV are obtained in each single channels studies, and the binding energy ∼−690 MeV is a littler shallower than the other two quantum states. Furthermore, there are also three resonances found at ∼19.21 GeV. Then in a complete coupled-channels investigation, masses of the bound and resonance states are 18.19 GeV, 19.45 GeV and 19.71 GeV, respectively. Then by comparing the components of these exotic states in 1 + and 2 + states, similar features can also be drawn herein. Furthermore, these compact configurations are also confirmed in analyzing the internal structure of bbbb system in Table 42, in particular, the mean square radii are around 0.17, 0.30 and 0.27 fm for the one bound and two resonance states, respectively.
Additionally, the results obtained in a real-range investigation is supported by a complex-range one of Fig. 24. Therein, three outstanding poles in the real-axis are just the bound and resonance states found in our study. In the varied region from 0 • to 6 • of angle θ, these three poles are fixed. However, the other energy points are obviously unstable. Table 41. Possible bound and resonance states of fully-bottom tetraquarks with quantum numbers I(J P ) = 0(2 + ), unit in MeV.

Channel
Index  Table 42. The distances, between any two quarks of the found fully-bottom bound and resonance states with I(J P ) = 0(2 + ) in coupled-channels calculation, unit in fm.

Hidden and Open Heavy Flavor Pentaquarks
In this part three types of 5-quarks systems will be introduced according to the investigations by chiral quark model. Particularly, hidden charm qqqcc, hidden bottom qqqbb and doubly charmed ccqq pentaquarks in spin-parity J P = 1

Hidden Charm Pentaquarks
In the qqqcc (q = u, d) sector, several pentaquark states reported by the LHCb collaboration can be well explained in our work. In particular, the P + c (4380) can be identified as the Σ * cD molecule state with I(J P ) = − Σ cD * , respectively. Let us discuss them in the following parts.
Seven baryon-meson channels, Nη c , N J/ψ, Λ cD , Λ cD * , Σ cD , Σ cD * and Σ * cD * are studied in the lowest spin state. Table 43 shows the calculated results. Particularly, the first column lists the necessary bases in spin, flavor and color degrees of freedom. Mass of 5-quark system is in the second one. The theoretical and experimental thresholds of each channels are in the third and fifth column, respectively. In the fourth column, the binding energy of state is presented. A modified mass E which is obtained by the summation of binding energy E B and experimental threshold value E Exp th is in the last column.
Obviously, it is a scattering nature of the Nη c , N J/ψ, Λ cD and Λ cD * channels which E B is 0.
However, resonance states are available in the Σ cD , Σ cD * and Σ * cD * channels. Particularly, binding energies E B = −4 MeV, −2 MeV and −3 MeV is found for the color-singlet channels of these three states, respectively. In additional, deeper binding energies are obtained through the couplings with hidden-color channels for the Σ cD , Σ cD * and Σ * cD * , the new E B is −8 MeV, −41 MeV and −105 MeV, respectively. The main contributions are color-singlet channels for the first two states (91% in Σ cD and 67% in Σ cD * ), but ∼80% component is the hidden-color channel for Σ * cD * state. After a mass shift by considering the systematical error, the rescaled masses E for the three resonance states are 4312 MeV, 4421 MeV and 4422 MeV, respectively. Accordingly, the first one is nicely consistent with P + c (4312) and can be identified as the Σ cD state. The nature of Σ c baryon andD meson molecular state is confirmed in Table 44, in which the distances between any two quarks are shown and the obtained 2.1 fm for the cc pair in hidden charm pentaquark state is quite comparable with the size of deuteron. Besides, there is a degeneration between the coupled masses of Σ cD * and Σ * cD * states, however, the P + c (4440) is preferred to be explained as a Σ cD * state since its threshold value and the mass of color-singlet channel are more reasonable. The molecular nature for this pentaquark state can also be guessed from Table 44. Meanwhile, a complete coupled-channels mass which includes all baryon-meson structures is listed in the bottom of Table 43, clearly, 3745 MeV is just the theoretical threshold value of Nη c . Therefore, no bound state is found in the fully coupled-channels case. Table 43. The lowest eigen-energies of the udccu system with J P = 1 2 − (unit: MeV). The percentages of color-singlet (S) and hidden-color (H) channels are also given.

Exp th
Although the lowest mass remains at 3841 MeV in a fully coupled channels computation, the obtained Σ cD * resonance with mass equals to 4459 MeV is a robust support for the P + c (4457) which is also a new hidden charm pentaquark state reported by the LHCb collaboration. This exotic state can be identified as a Σ c baryon andD * meson molecular state whose theoretical size is around 2.3 fm according to the results in Table 44.  Firstly, there is -3 MeV binding energy in the color-singlet channel calculation and after a mass shift with E = E Exp th + E B , 4524 MeV resonance mass is obtained. However, the hidden-color channel mass is higher and at 5002 MeV. A strong coupling between this two channels leads to a deeper binding energy which is -89 MeV, and the modified mass in coupled-channels is 4438 MeV. The components of this resonance state is comparable with 66% for the color-singlet channel of Σ * cD * and 34% of its hidden-color one. Hence it is also a good candidate for the hidden charm pentaquark in high spin state.     Table 46. Several resonance states are found in these quantum states except the 3 2 ( 1 2 − ) case.
Particularly, from Table 47 to 51, the first column lists the channels on our obtained bound states, their experimental values of the noninteracting baryon-meson threshold (E ex th ) are also shown in parentheses. The second column labels the color-singlet (S), hidden-color (H) and coupled-channels (S+H) computation. Then, the calculated theoretical mass (M) and binding energy (E B ) of pentaquark state is presented in the third and fourth columns, respectively. A re-scaled mass (M ), which is obtained by attending to the experimental baryon-meson threshold, M = E ex th + E B , and with a purpose of removing the theoretical uncertainty, is listed in the last column. The percentages of color-singlet (S) and hidden-color (H) channels are also given when the coupled-channels calculation is performed. Hence, we will discuss them according to the I(J P ) state, respectively. configurations and they are listed in Table 47. Particularly, there are around −20 MeV binding energy for the color-singlet channels of Σ bB , Σ bB * and Σ * bB * . Their modified masses M are 11.07 GeV, 11.11 GeV and 11.13 GeV, respectively. As for their hidden-color channels, only the Σ * bB * channel with E B = −232 MeV is found, the other two are at least 120 MeV above their corresponding theoretical threshold values.
In additional, within a calculation which the coupling on color-singlet and hidden-color channels are considered, the Σ bB * and Σ * bB * resonances with much deeper binding energies (more than −90 MeV) are obtained, their coupled masses are 11.04 GeV and 10.86 GeV, respectively. Therefore, compact configuration of hidden bottom pentaquark state is favored herein. However, the coupling in Σ bB resonance is quite weak with only 2 MeV binding energy decreased and the rescaled mass is still ∼11.07 GeV, this can be identified as a molecule state. Moreover, these results are confirmed by two facts: (i) the color-singlet channel is almost 99% in Σ bB , 58% in Σ bB * and only 16% in Σ * bB * , (ii) in Table 52, the calculated size for Σ bB resonance state is around 1 fm. However, ∼ 0.7 fm for the other two resonance states and especially, the distance between a bb pair in Σ bB * and Σ * bB * states is only ∼0.3 fm.
We only find resonances in the Σ bB * , Σ * bB and Σ * bB * states. Table 48 presents the predicted mass, binding energy and components of these hidden bottom pentaquarks. Firstly, in the color-singlet channels calculation, the binding energies are −12 MeV, −15 MeV and −15 MeV for the Σ bB * , Σ * bB and Σ * bB * channels, respectively.
However, when the hidden-color channels are incorporated into computation, a strong coupling effect is obtained in the later two states and the binding energies are −67 MeV and −195 MeV, respectively. In contrast, the coupling is very weak and only 2 MeV decreased when hidden-color channel included in Σ bB * state. The modified masses are 11.12 GeV, 11.04 GeV and 10.96 GeV for the Σ bB * , Σ * bB and Σ * bB * states in the coupled-channels study, respectively. Furthermore, Table 48 also presents the percentages of singlet-and hidden-channels of the three resonance states. In particular, it is almost the complete color-singlet component in the Σ bB * state. But the main part (78%) is the hidden-color channel in Σ * bB * state, and also a considerable percentage (45%) in the Σ * bB . These results are consistent with the analysis on the internal structures of Σ ( * ) bB ( * ) resonance states. In Table 52 one can notice that a compact configuration within 0.7 fm for both the Σ * bB and Σ * bB * states confirms the previous facts of deep binding energy and strong coupling. However, it is a Σ bB * molecule state which size is around 1.1 fm in the coupled-channels calculation.
Only one Σ * bB * configuration need to be considered in the highest spin state and Table 49 shows the calculated mass of this resonance state. Clearly, there are −12 MeV binding energy in the color-singlet channel. Besides, this result is comparable with the colorless channel of Σ * bB * in the other two spin-parity states.
In a further step with hidden-color channel included, only a weak coupling with E B = −13 MeV is obtained and the rescaled mass which is obtained as the previous procedure is 11.14 GeV. Accordingly, this resonance can be identified as the Σ * bB * molecule state with 99.6% color-singlet component and ∼1.1 fm size shown in Table 52.
In this spin-parity channel, neither bound nor resonance state is found in the ∆η b and ∆Υ configurations, however, resonance states are possible in the Σ  Table 50.
Firstly, only scattering states are found in the Σ bB * , Σ * bB and Σ * bB * channels when only color-singlet configurations are included. However, hidden-color structures help in obtaining resonance states with deep binding energies. Namely, coupled-masses of these three resonance states are all below the corresponding theoretical thresholds about 110 MeV. The modified masses are 11.02 GeV, 10.99 GeV and 11.05 GeV for the Σ bB * , Σ * bB and Σ * bB * states, respectively. Furthermore, their calculated sizes listed in Table 52 are ∼0.8 fm which is a compact 5-quarks configuration. This feature is supported by the fact that the coupling between singlet-and hidden-channels is strong and the percentage of the later configuration is quite considerable, 35% in Σ bB * and both more than 81% in Σ * bB and Σ * bB * .
There are two baryon-meson channels devote to the highest spin and isospin state, ∆Υ and Σ * bB * . Firstly, in the single channel calculation of color-singlet and hidden-color configuration, there are more than −100 MeV binding energy obtained and especially E B = −179 MeV for the later structure. Additionally, a deeper binding energy which is −222 MeV is obtained in their coupled-channels computation and the modified mass is 10.93 GeV. Apparently, this can be identified a color resonance which about 80% component is the hidden-color channel of Σ * bB * . Moreover, the size ∼0.8 fm in Table 52 confirms a compact structure too.

Doubly Charmed Pentaquarks
Along with the hidden-charm pentaquark states announced experimentally, an open-flavor 5-quark system is also quite charming. In this part, the ccqqq (q = u, d) pentaquarks with spin-parity J P = 1  Table 53 and 54 listed the allowed channels in each I(J P ) states. However, due to the fact that a large amount of computational effort is needed for exactly solving the 5-body Schrödinger equation, we still perform the study in the baryon-meson sector in which both color-singlet and hidden-color channels are considered. Nevertheless, as shown in Fig. 4, there are four configurations in the 3+2 clusters. In particular, each five quarks are fixed and we only need to consider the different coupling sequences.
Moreover, the complex scaling method is also applied in this sector in order to have a better classification of bound and resonance states in the multiquark systems. Table 55 summarized our results on the obtained ccqqq pentaquark states in advance, particularly, the second column shows the channels of bound or resonance states with their theoretical masses marked in the bracket, and the last column lists the binding energy or resonance width for these exotic states. The details are going to be discussed according to the I(J P ) quantum states.  (Σ * c D * ) 8 Table 55. Possible bound and resonance states of doubly charm pentaquarks. The last column listed the binding energy or resonance width of each states.

Quantum state E B /Γ (in MeV)
Bound state Resonance state There are eleven baryon-meson channels under investigation, Ξ cc η, Ξ cc ω, Ξ cc π, Ξ cc ρ, Ξ * cc ω, Ξ * cc ρ, Λ c D, Λ c D * , Σ c D, Σ c D * and Σ * c D * . The lowest channel is Ξ cc π which experimental threshold value is 3657 MeV, but it is unbound in our study. The scattering nature is also found in the other channels no matter in the color-singlet, hidden-color structures calculation or the coupling of them. However, a possible resonance state is obtained in the Λ c D * channel which binding energy E B = −2 MeV in the color-singlet computation. The coupling with hidden-color channel is extremely weak and the coupled-mass shown in Table 57 is 4291 MeV.
In the complete coupled-channels calculation which the CSM is employed and the rotated angle θ is varied from 0 • to 6 • , we find that the previous obtained resonance state of Λ c D * at 4291 MeV in the single channel calculation is pushed above its threshold. The scattering nature is clear shown in Fig. 25 that the calculated poles are alway moving along with the Λ c D * threshold lines. Besides, this feature is also reflected by the other baryon-meson channels e.g., Ξ cc π, Λ c D, Λ c D * , etc.
Since there is a dense distribution in the mass region from 4.35 to 4.46 GeV, an enlarged part is present in the bottom panel of Fig. 25. Therein, the small separations among Ξ cc η, Ξ cc ω, Σ c D, Ξ * cc ω and Ξ cc ρ channels can be distinguished clearly. Apart from the most continuum states, one resonance, whose mass and width are ∼4416 MeV and ∼4.8 MeV, is obtained and has been circled with green in the bottom panel. We can identify it as the Σ c D molecular resonance state for this pole is above the threshold lines, a re-scaled mass, which is according to the systematic error between theoretical and experimental threshold values, equals to 4356 MeV. Meanwhile, several properties of the Σ c D(4356) remind the nature of P + c (4312), i.e., the I(J P ) quantum numbers are 1 2 1 2 − for both the P + c (4312) and the Σ c D(4356) resonances in our calculations, their mass and width are also quite comparable.     Tables 58 and 59 present the calculated mass of each baryon-meson channels in the color-singlet, hidden-color and their coupling cases. In particular, they are all scattering states listed in Table 58 and one resonance state of Σ c D * in Table 59. Obviously, all the color-singlet channels of these Ξ ( * ) cc ω,  • I(J P ) = 1 2 ( 5 2 − ) state Three baryon-meson channels Ξ * cc ω, Ξ * cc ρ and Σ * c D * contribute to the highest spin state and the calculated results on their masses in the color-singlet channel, hidden-color one and their couplings in real-range are listed in Table 60. Meanwhile, Fig. 27 shows the distributions of the complex energies in the fully coupled-channels calculation of complex-range. Obviously, no bound state is obtained neither in the single channel nor the coupled cases and resonances are also not found. Hence, it is the scattering nature of Ξ * cc ω, Ξ * cc ρ and Σ * c D * in the 1 2 ( 5 2 − ) state.   Tables 61 and 62 one can find that resonance only obtained in the Σ * c D * channel with color-singlet state and the binding energy is −3 MeV. In a subsequent calculation which its hidden-color channel included, only 1 MeV increased for the E B . The modified mass M of Σ * c D * resonance is 4523 MeV. Apart from the real scaling calculation, Fig. 28 presents the results of fully coupled-channels calculation in the complex-range. Particularly, the calculated energy points of the six channels Ξ cc π, Ξ ( * ) cc ρ and Σ ( * ) c D ( * ) distribute in the mass region 3.8 − 4.8 GeV of the top panel. When the θ varies from 0 • to 6 • , they generally present the scattering nature which the poles always moving along their threshold lines. However, much plenty of energy poles located in the 4.45 − 4.62 GeV region, so the middle panel of Fig. 28 shows an enlarged result on this part. Clearly, one resonance pole circled green is obtained herein and the theoretical mass and width is 4491 MeV and 2.6 MeV, respectively. It is both above the Σ c D and Ξ cc ρ threshold lines, however, there is much more systematic error of the later channel than Σ c D. Hence this resonance is preferred to be identified as the Σ c baryon and D meson molecular state. After a mass shift according to the experimental threshold value, the resonance mass is 4431 MeV.
Furthermore, there seems to be more structures between the threshold of Ξ cc ρ and Σ c D * , so a further enlarged part from 4.50 GeV to 4.53 GeV is shown in the bottom panel. Apparently, another Σ c D resonance state is obtained, the mass and width is 4506 MeV and 2.2 MeV, respectively. Then through a mass shift with respect to its threshold value, the rescaled mass is 4446 MeV.    Tables 63 and 64   There are only two channels under investigation in the highest spin and isospin pentaquark state, Ξ * cc ρ and Σ * c D * . In Table 65 one can see that their lowest theoretical mass in the real-range is 4488 MeV and 4551 MeV, respectively. Hence, no bound state is obtained in this case. Fig. 30 shows the distributions of complex energies of these two channels. Apart from the scattering states, one stable resonance pole is found in the green circle. It can be identified as the Ξ * cc ρ resonance with the modified mass and width is 4461 MeV and 3.0 MeV, respectively.  respectively. Meanwhile, the recently reported new structure at 6.9 GeV in the di-J/ψ invariant mass spectrum by the LHCb collaboration can be regarded as a compact fully charmed tetraquark in 02 + state, and another broad structure around 6.2∼6.8 GeV is also supported by us with several compact cccc resonances in 00 + and 01 + states. Furthermore, many other exotic states predicted in Table 66 should be expected to be confirmed in the future experiments, e.g., the LHCb, ATLAS, CMS, BESIII, Belle II, JLAB, PANDA, EIC, etc.