$QQ\bar{s}\bar{s}$ tetraquarks in the chiral quark model

The low-lying $S$-wave $QQ\bar{s}\bar{s}$ ($Q=c, b$) tetraquark states with $IJ^P=00^+$, $01^+$ and $02^+$ are systematically investigated in the framework of complex scaling range of chiral quark model. Every structure including meson-meson, diquark-antidiquark and K-type configurations, and all possible color channels in four-body sector are considered by means of a commonly extended variational approach, Gaussian expansion method. Several narrow and wide resonance states are obtained for $cc\bar{s}\bar{s}$ and $bb\bar{s}\bar{s}$ tetraquarks with $IJ^P=00^+$ and $02^+$. Meanwhile, narrow resonances for $cb\bar{s}\bar{s}$ tetraquarks are also found in $IJ^P=00^+$, $01^+$ and $02^+$ states. These results confirm the possibility of finding hadronic molecules with masses $\sim\,0.6\,\text{GeV}$ above the noninteracting hadron-hadron thresholds.


I. INTRODUCTION
We are witnessing in the last two decades of a big experimental effort for understanding the heavy-flavor quark sectors of both meson and baryon systems. Many experiments have been settled worldwide such as Bfactories (BaBar, Belle, and CLEO), τ -charm facilities (CLEO-c and BES) and hadron-hadron colliders (CDF, D0, LHCb, ATLAS, and CMS) providing a sustained progress in the field with new measurements of conventional and exotic heavy-flavored hadrons.
Within the baryon sector, and attending mostly to the spectrum, five excited Ω c baryons were announced three years ago by the LHCb collaboration in the Ξ + c K − mass spectrum [1] and, very recently, the same collaboration has reported additional four narrow excited states of the Ω b system in the Ξ 0 b K − mass spectrum [2]. In 2019, two excited bottom baryons, Λ 0 b (6146) and Λ 0 b (6152), were discovered in the LHCb experiment [3]. Later on, the LHCb collaboration also announced one more Λ 0 b baryon around 6070 MeV in the Λ 0 b π + π − invariant mass spectrum [4], which is consistent with the reported one of the CMS collaboration [5]. Additionally, three excited Ξ 0 c states were announced by the LHCb collaboration in the Λ + c K − mass spectrum [6]. All of these newly discovered baryons undoubtedly complement the scarce data on heavy flavor baryons in the Review of Particle Physics (RPP) of the Particle Data Group [7]. Furthermore, these experimental findings trigger a large number of theoretical investigations. The three-quark structure of the new Ω c baryons has been claimed by QCD sum rules [8] and different potential models [9][10][11]. Also, the description of the Ω b signals as P -wave conventional baryons is preferred by phenomenological quark model approach [12,13], heavy quark effective theory [14] and QCD sum rules [15]. Meanwhile, the baryon-meson molecular interpretation has been suggested for the excited Ω b baryons in Ref. [16]. The Λ 0 b (6072), Λ 0 b (6146) and Λ 0 b (6152) have been identified as radial and angular excitations within QCD sum rules [17][18][19][20] and chiral quark models [21,22]. However, theDΛ −DΣ molecular configurations have been also suggested for these states in Ref. [23].
Apart from conventional heavy flavored baryons, there are limited results on open-bottom mesons and detailed studies of the open-charm ones were not undertaken until large datasets were obtained by CLEO at discrete energy points and by the B-factory experiments using radiative returns to obtain a continuous exposure of the mass region. The picture that has emerged is complex due to the many thresholds in the region. This resembles the experimental situation found in the heavy quarkonium spectrum with the observation of more than two dozens of unconventional charmonium-and bottomonium-like states, the so-called XYZ mesons. However, still successful observations of 6 new conventional heavy quarkonium states (4 cc and 2 bb) have been made.
Exotic states such as tetraquarks and pentaquarks have lastly received considerable attention within the hadron physics community. Related with the first structures, the best known is the X(3872), which was observed in 2003 as an extremely narrow peak in the B + → K + (π + π − J/ψ) channel and at exactly theD 0 D * 0 threshold [24,25], and it is suspected to be a cncn (n = u or d quark) tetraquark state whose features resemble those of a molecule, but some experimental findings forbid to discard a more compact, diquark-antidiquark, component or even some cc trace in its wave function. On the other hand, there are no doubts of the tetraquark character of the Z c 's [26,27] and Z b 's [28,29] states due to its non-zero charge. The most prominent examples of the second mentioned structures are the hiddencharm pentaquarks P + c (4312), P + c (4380), P + c (4440) and P + c (4457) reported in 2015 and 2019 by the LHCb collaboration in the Λ 0 b decay, Λ 0 b → J/ψK − p [30,31]. The discussion about the nature of these exotic sig-arXiv:2007.05190v1 [hep-ph] 10 Jul 2020 nals are carried out by various theoretical approaches. In particular, the three newly announced hidden-charm pentaquarks, P + c (4312), P + c (4440) and P + c (4457) are favored to be molecular states of Σ cD * in, for instance, effective field theories [32,33], QCD sum rules [34], phenomenological potential models [35][36][37][38][39][40], heavy quark spin symmetry formalisms [41,42] and heavy hadron chiral perturbation theory [43]. Moreover, their photoproduction [44,45] and decay properties [46] have been also discussed. As for the other types of pentaquarks, bound states of theQqqqq system are not found within a constituent quark model [47]. Using the same approach, several narrow double-heavy pentaquark states are found to be possible in the systematical investigations of Refs. [48][49][50]. Moreover, within the one-bosonexchange model, possible triple-charm molecular pentaquarks Ξ cc D ( * ) are suggested [51]. In the tetraquark sector, double-heavy tetraquarks are studied using QCD sum rules [52], quark models [53,54] and even latticeregularized QCD computations [55]. Besides, theoretical techniques such as diffusion Monte Carlo [56], Bethe-Salpeter equation [57], QCD sum rules [58,59] and effective phenomenological models [60][61][62][63][64] have recently contributed to the investigations of fully heavy tetraquarks QQQQ. Some reviews on both tetraquark and pentaquark systems can be found in Refs. [65,66].
Our QCD-inspired chiral quark model explained successfully the nature of the P + c states in Ref. [67], even before the last updated data reported by the LHCb collaboration [30]. Based on such fact, the hiddenbottom [68] and double-charm pentaquarks [50] were systematically investigated within the same theoretical framework, finding several either bound or resonance states. Reference [54] reported results on the doubleheavy tetraquarks QQqq (Q = c, b and q = u, d), its natural extension should be the QQss tetraquark sector with the hope of finding either bound or resonance states. In order to do so, we have recently established a complex scaling range formalism of the chiral quark model which allows us to determine (if exist) simultaneously scattering, resonance and bound states. We shall study herein the QQss tetraquarks in the spin-parity channels J P = 0 + , 1 + and 2 + , and in the isoscalar sector I = 0. Another relevant feature of our study is that all configurations: meson-meson, diquark-antidiquark and K-type for four-body systems are considered; moreover, every possible color channel is taken into account, too. Finally, the Rayleigh-Ritz variational method is employed in dealing with the spatial wave functions of tetraquark states, which are expanded by means of the well-known Gaussian expansion method (GEM) of Ref. [69].
The present manuscript is arranged as follows. Section II is devoted to briefly describe our theoretical approach which includes the complex-range formulation of the chiral quark model and the discussion of the QQss wave-functions. Section III is devoted to the analysis and discussion of the obtained results. The summary and some prospects are presented in Sec. IV.

II. THEORETICAL FRAMEWORK
The complex scaling method (CSM) applied to our chiral quark model has been already explained in Refs. [50,54]. The general form of the four-body complex Hamiltonian is given by where the center-of-mass kinetic energy T CM is subtracted without loss of generality since we focus on the internal relative motions of quarks inside the multi-quark system. The interplay is of two-body potential which includes color-confining, V CON , one-gluon exchange, V OGE , and Goldstone-boson exchange, V χ , respectively, (2) In this work, we focus on the low-lying positive parity QQss tetraquark states of S-wave, and hence only the central and spin-spin terms of the potentials shall be considered.
By transforming the coordinates of relative motions between quarks as r ij → r ij e iθ , the complex scaled Schrödinger equation is solved, giving eigenenergies that can be classified into three kinds of poles: bound, resonance and scattering ones, in a complex energy plane according to the so-called ABC theorem [70,71]. In particular, the resonance pole is independent of the rotated angle θ, i.e. it is fixed above the continuum cut line with a resonance's width Γ = −2 Im(E). The scattering state is just aligned along the cut line with a 2θ rotated angle, whereas a bound state is always located on the real axis below its corresponding threshold.
The two-body potentials in Eq. (2) mimic the most important features of QCD at low and intermediate energies. Firstly, color confinement should be encoded in the non-Abelian character of QCD. It has been demonstrated by lattice-QCD that multi-gluon exchanges produce an attractive linearly rising potential proportional to the distance between infinite-heavy quarks [72]. However, the spontaneous creation of light-quark pairs from the QCD vacuum may give rise at the same scale to a breakup of the created color flux-tube [72]. Therefore, the following expression when θ = 0 • is used for the confinement potential: where a c , µ c and ∆ are model parameters, and the SU (3) color Gell-Mann matrices are denoted as λ c . One can see in Eq. (4) that the potential is linear at short interquark distances with an effective confinement strength σ = −a c µ c ( λ c i · λ c j ), while V CON becomes constant (∆ − a c )( λ c i · λ c j ) at large distances. Secondly, the QCD's asymptotic freedom is expressed phenomenologically by the Fermi-Breit reduction of the one-gluon exchange interaction which, in the case of hadron systems with ≥ 3 quarks, consists on a Coulomb term supplemented by a chromomagnetic contact interaction given by where m i and σ are the quark mass and the Pauli matrices, respectively. The contact term of the central potential in complex range has been regularized as The QCD-inspired effective scale-dependent strong coupling constant, α s , offers a consistent description of mesons and baryons from light to heavy quark sectors in wide energy range, and we use the frozen coupling constant of, for instance, Ref. [73] in which α 0 , µ 0 and Λ 0 are parameters of the model. Thirdly, the Goldstone-boson exchange interactions between light quarks, and constituent quark masses, appear because the breaking of chiral symmetry in a dynamical way. Therefore, the following two terms of the chiral potential must be taken into account between the (ss)-pair for QQss tetraquarks: where Y (x) = e −x /x is the standard Yukawa function. The pion-and kaon-exchange interactions do not appear because no up-and down-quarks are considered herein. Furthermore, the physical η meson is taken into account by introducing the angle θ p . The λ a are the SU(3) flavor Gell-Mann matrices. Taken from their experimental values, m π , m K and m η are the masses of the SU(3) Goldstone bosons. The value of m σ is determined through the PCAC relation m 2 σ m 2 π + 4m 2 u,d [74]. Finally, the chiral coupling constant, g ch , is determined from the πN N coupling constant through which assumes that flavor SU(3) is an exact symmetry only broken by the different mass of the strange quark. The model parameters, which are listed in Table I, have been fixed in advance reproducing hadron [75][76][77][78][79][80][81][82][83][84], hadron-hadron [85][86][87][88][89] and multiquark [11,67,68,90] phenomenology. Additionally, in order to help on our analysis of the QQss tetraquarks in the following section, Table II lists the theoretical and experimental masses of the ground state and its first radial excitation (if available) for the D ( * )+ s andB ( * ) s mesons. Besides, their mean-square radii are collected in Table II. Figure 1 shows six kinds of configurations for doubleheavy tetraquarks QQss (Q = c, b). In particular, Fig. 1(a) is the meson-meson (MM) structure, Fig. 1(b) is the diquark-antidiquark (DA) one, and the other K-type configurations are from panels (c) to (f). All of them, and their couplings, are considered in our investigation. However, for the purpose of solving a manageable 4-body problem, the K-type configurations are restricted to the case in which the two heavy quarks of QQss tetraquarks are identical. It is important to note herein that just one configuration would be enough for the calculation, if all radial and orbital excited states were taken into account; however, this is obviously much less efficient and thus an economic way is to combine the different configurations in the ground state to perform the calculation.  Four fundamental degrees of freedom at the quark level: color, spin, flavor and space are generally accepted by QCD theory and the multiquark system's wave function is an internal product of color, spin, flavor and space terms. Firstly, concerning the color degree-of-freedom, plenty of color structures in multiquark system will be available with respect those of conventional hadrons (qq mesons and qqq baryons). The colorless wave function of a 4-quark system in di-meson configuration, i.e. as illustrated in Fig. 1(a), can be obtained by either a colorsinglet or a hidden-color channel or both. However, this is not the unique way for the authors of Refs. [91,92], who assert that it is enough to consider the color singlet channel when all possible excited states of a system are included. 1 The SU (3) color wave functions of a colorsinglet (two coupled color-singlet clusters, 1 c ⊗ 1 c ) and hidden-color (two coupled color-octet clusters, 8 c ⊗ 8 c ) channels are given by, respectively, (3brrb + 3ḡrrg + 3bgḡb + 3ḡbbg + 3rgḡr In addition, the color wave functions of the diquarkantidiquark structure shown in Fig. 1(b) are χ c 3 (color triplet-antitriplet clusters, 3 c ⊗3 c ) and χ c 4 (color sextetantisextet clusters, 6 c ⊗6 c ), respectively: (2rrrr + 2ḡgḡg + 2bbbb +rrḡg +ḡrrg +ḡgrr +rgḡr +rrbb +brrb +bbrr +rbbr +ḡgbb +bgḡb +bbḡg +ḡbbg) .
Meanwhile, the colorless wave functions of the K-type structures shown in Fig. 1(c) to 1(f) are obtained by following standard coupling algebra within the SU (3) color group: 2 • K 1 -type of Fig. 1 [211], [11] 10 ; 1 After a comparison, a more economical way of computing through considering all the possible color structures and their coupling is preferred. 2 The group chain of K-type is obtained in sequence of quark number. Moreover, each quark and antiquark is represented, respectively, with [1] and [11] in the group theory.
• K 4 -type of Fig. 1 [221], [1] 12 . These group chains will generate the following K-type color wave functions whose subscripts correspond to those numbers above: (rbbr +ḡbbg +bgḡb +ḡrrg +brrb) 6 (rbbr +rrbb +ḡbbg +ḡgbb +rgḡr +rrḡg +bbḡg +bgḡb +ḡgrr +ḡrrg +bbrr +brrb) As for the flavor degree-of-freedom, since the quark content of the tetraquark systems considered herein are two heavy quarks, (Q = c, b), and two strange antiquarks, s, only the isoscalar sector, I = 0, will be discussed. The flavor wave functions denoted as χ f i I,M I , with the superscript i = 1, 2 and 3 referring to ccss, bbss and cbss systems, can be written as where, in this case, the third component of the isospin M I is equal to the value of total one I. The total spin S of tetraquark states ranges from 0 to 2. All of them shall be considered and, since there is not any spin-orbit potential, the third component (M S ) can be set to be equal to the total one without loss of generality. Therefore, our spin wave functions χ σi S,M S are given by The superscripts l 1 , . . . , l 4 and m 1 , . . . , m 6 are numbering the spin wave function for each configuration of tetraquark states, their specific values are shown in Table III. Furthermore, these expressions are obtained by considering the coupling of two sub-cluster spin wave functions with SU(2) algebra, and the necessary bases read as Among the different methods to solve the Schrödingerlike 4-body bound state equation, we use the Rayleigh-Ritz variational principle which is one of the most extended tools to solve eigenvalue problems because its simplicity and flexibility. Meanwhile, the choice of basis to expand the wave function solution is of great importance. Within hte CRM, the spatial wave function can be written as follows where the internal Jacobi coordinates for the mesonmeson configuration ( Fig. 1(a)) are defined as and for the diquark-antdiquark one ( Fig. 1(b)) are The Jacobi coordinates for the remaining K-type configurations shown in Fig. 1, panels (c) to (f), are (i, j, k, l are according to the definitions of each configuration in Fig. 1): Obviously, the center-of-mass kinetic term T CM can be completely eliminated for a non-relativistic system when using these sets of coordinates. A very efficient method to solve the bound-state problem of a few-body system is the Gaussian expansion method (GEM) [69], which has been successfully applied by us in other multiquark systems [50,54,67,68]. The Gaussian basis in each relative coordinate is taken with geometric progression in the size parameter. 3 Therefore, the form of the orbital wave functions, φ's, in Eq. (41) is As one can see, the Jacobi coordinates are all transformed with a common scaling angle θ in the complex scaling method. In this way, both bound states and resonances can be described simultaneously within one scheme. Moreover, only S-wave state of double-heavy tetraquarks are investigated in this work and thus no laborious Racah algebra is needed during matrix elements calculation. Finally, in order to fulfill the Pauli principle, the complete wave-function is written as where A is the antisymmetry operator of QQss tetraquarks when considering interchange between identical particles (ss, cc and bb). This is necessary because the complete wave function of the 4-quark system is constructed from two sub-clusters: meson-meson, diquarkantidiquark and K-type structures. In particular, when the two heavy quarks are of the same flavor (QQ = cc or bb), the operator A with the quark arrangementssQsQ is defined as However, due to the fact that c-and b-quarks are distinguishable particles, the operator A consists only on two terms for thescsb system, and read as

III. RESULTS
The low-lying S-wave states of QQss (Q = c, b) tetraquarks are systematically investigated herein. The parity for different QQss tetraquarks is positive under our assumption that the angular momenta l 1 , l 2 , l 3 , which appear in Eq. (41), are all 0. Accordingly, the total angular momentum, J, coincides with the total spin, S, and can take values 0, 1 and 2. Note, too, the value of isospin can only be 0 for the QQss system. For ccss, bbss and cbss systems, all possible meson-meson, diquark-antidiquark and K-type channels for each IJ P quantum numbers are listed in Table IV, V, VI, VII,  TABLE IV. All possible channels for IJ P = 00 + ccss and bbss tetraquark systems. The second column shows the necessary basis combination in spin (χ σ i J ), flavor (χ f j I ) and color (χ c k ) degrees of freedom. Particularly, the flavor indices (j) 1 and 2 are of ccss and bbss, respectively. The superscript 1 and 8 stands for the color-singlet and hidden-color configurations of physical channels.
Index VIII and IX, respectively. The second column shows the necessary basis combination in spin (χ σi J ), flavor (χ fj I ), and color (χ c k ) degrees-of-freedom. The physical channels with color-singlet (labeled with the superindex 1), hidden-color (labeled with the superindex 8), diquarkantidiquark (labeled with (QQ)(ss)) and K-type (labeled from K 1 to K 4 ) configurations are listed in the third column.
Tables ranging from X to XIX summarize our calculated results (mass and width) of the lowest-lying QQss tetraquark states and possible resonances. In particular, results of ccss tetraquarks with I(J P ) = 0(0 + ), 0(1 + ) and 0(2 + ) are listed in Tables X, XI and XII; those of bbss tetraquarks are shown in Tables XIII, XIV and XV; and  Tables XVI, XVII and XVIII collect the cbss cases. In these tables, the first column lists the physical channel of meson-meson, diquark-antidiquark and K-type (if it fulfills Pauli principle), and the experimental value of the noninteracting meson-meson threshold is also indicated in parenthesis; the second column signals the discussed channel, e.g. color-singlet (S), hidden-color (H), etc.; the third column shows the theoretical mass (M ) of each single channel; and the fourth column shows a coupled calculation result for one certain configuration. Moreover, the complete coupled channels results for each quantum TABLE V. All possible channels for IJ P = 01 + ccss and bbss tetraquark systems. The second column shows the necessary basis combination in spin (χ σ i J ), flavor (χ f j I ) and color (χ c k ) degrees of freedom. Particularly, the flavor indices (j) 1 and 2 are of ccss and bbss, respectively. The superscript 1 and 8 stands for the color-singlet and hidden-color configurations of physical channels.
Index  Figures 2 to 10 depict the distribution of complex energies of the QQss tetraquarks in the complete coupledchannels calculation. The x-axis is the real part of the complex energy E, which stands for the mass of tetraquark states, and the y-axis is the imaginary part of E, which is related to the width through Γ = −2 Im(E). In the figures, some orange circles appear surrounding resonance candidates. They are usually ∼ 0.6 GeV above their respective non-interacting meson-meson thresholds and ∼ 0.2 GeV around their first radial excitation states; moreover, looking at the details, we shall conclude that most of these observed resonances can be identified with a hadronic molecular nature. Now let us proceed to describe in detail our theoretical findings for each sector of QQss tetraquarks.  We find only resonances in this sector with quantum numbers I(J P ) = 0(0 + ) and 0(2 + ). This result is opposite to the one found in our previous study of ccqq tetraquarks [54] and it is related with the ratio between light and heavy quarks that compose the tetraquark system. We shall proceed to discuss below the J = 0, 1 and 2 channels individually.
The I(J P ) = 0(0 + ) state: Two possible mesonmeson channels, D + s D + s and D * + s D * + s , two diquarkantidiquark channels, (cc)(ss) and (cc) * (ss) * , along with K-type configurations, are studied first in real-range calculation and our results are shown in Table X. The lowest energy level, (D + s D + s ) 1 , is unbounded and its theoretical mass just equals to the threshold value of two non-interacting D + s mesons. This fact is also found in the (D * + s D * + s ) 1 channel whose theoretical mass is 4232 MeV. As for the other exotic configurations, the obtained masses are all higher than the two di-meson channels. In particular, masses of the hidden-color channels are about 4.6 GeV, diquark-antidiquark channels are lower ∼4.4 GeV, and the other four K-type configurations are located in the mass interval of 4.2 to 4.8 GeV. Note, too, there is a degeneration between (cc) ( * ) (ss) ( * ) , K 3 and K 4 channels around 4.4 GeV.
In a further step, we have performed a coupledchannels calculation on certain configurations, and still no bound states are found. The coupling is quite weak for the color-singlet channels D + s D + s and D * + s D * + s . Hiddencolor, diquark-antidiquark and K-type structures do not shed any different with respect the color-singlet channel, coupled energies range from 4.2 to 4.4 GeV. These results confirm the nature of scattering states for D + s D + s and D * + s D * + s . Moreover, in a complete coupled-channels study, the lowest energy of 3978 MeV for D + s D + s state is remained. The real-scaling results are consistent with those of ccqq tetraquarks; however, it is invalid for resonances. Figure 2 shows the distributions of ccss tetraquarks' complex energies in the complete coupled-channels calculation. In the energy gap from 3.9 GeV to 5.0 GeV, most of poles are aligned along the threshold lines. Namely, with the rotated angle θ varied from 0 • to 6 • , the D  The lowest-lying eigen-energies of ccss tetraquarks with IJ P = 00 + in the real range calculation. 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 and fourth columns refer to the theoretical mass of each channels and their couplings. (unit: MeV) threshold, one resonance pole is found. In the yellow circle of Fig. 2, the three calculated dots with black, red and blue are almost overlapped. This resonance pole has mass and width 4902 MeV and 3.54 MeV, respectively, and it could be identified as a resonance of the D + s D + s molecular system. The I(J P ) = 0(1 + ) state: There are two mesonmeson channels, D + s D * + s and D * + s D * + s , one diquarkantidiquark channel, (cc) * (ss) * , but more K-type channels (18 channels) are allowed due to a much richer combination of color, spin and flavor wave functions which fulfills the Pauli Principle. Table XI lists the calculated masses of these channels and also their couplings.
Firstly, the situation is similar to the IJ P = 00 + case, i.e no bound state is found in the real-range calculation. Secondly, the couplings are extremely weak for both color-singlet and hidden-color channels. In contrast, one finds binding energies for the K-type structures which go from several to hundreds of MeV. The coupled-channels The lowest-lying eigen-energies of ccss tetraquarks with IJ P = 01 + in the real range calculation. 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 and fourth columns refer to the theoretical mass of each channels and their couplings. (unit: MeV) results of these K-type configurations, along with hiddencolor and diquark-antidiquark ones, are around 4.4 GeV. Then, after mixing all of the channels listed in Table XI, the nature of the lowest energy level D + s D * + s is still unchanged, it is a scattering one. Additionally, comparing the results in Table V for ccqq tetraquarks of our previous work [54], one notices that the deeply bound state with ∼200 MeV binding energy for D + D * 0 is invalid in the D + s D * + s sector. Our results using the complex scaling method applied to the fully coupled-channels calculation are shown in Fig. 3  The lowest-lying eigen-energies of ccss tetraquarks with IJ P = 02 + in the real range calculation. 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 and fourth columns refer to the theoretical mass of each channels and their couplings. (unit: MeV) All of the above channels: 4232 IG. 3. Complex energies of ccss tetraquarks with IJ P = 01 + in the coupled channels calculation, θ varying from 0 • to 6 • . antidiquark structure and six K-type channels contribute to the highest spin channel (see Table XII). In analogy with the two previous cases, no bound state is obtained neither in each single channel calculation nor in the coupled-channels cases. The mixed results of K 1 and K 2 types are both around 4.35 GeV, which is lower than those of the other exotic configurations (∼ 4.45 GeV); however, these do not help in realizing a bound state of D * + s D * + s . Nevertheless, thrilling results are found in the complete coupled-channels study by CSM. Figure 4  , respectively. In the excited energy region which is located about 0.5 GeV higher than the D * + s D * + s threshold but 0.1 GeV below its first radial excitation, it is reasonable to find resonances whose nature is of hadronic molecules, and this conclusion has been confirmed by us in study the other multiquark systems [50,54,67].

B. The bbss tetraquarks
We proceed here to analyze the bbss tetraquark system with quantum numbers I(J P ) = 0(0 + ), 0(1 + ) and 0(2 + ). The situation is similar to the ccss case, with only narrow resonances found in the I(J P ) = 0(0 + ) and 0(2 + ) channels; meanwhile, this result is also in contrast with the one obtained for bbqq tetraquarks [54]. The details are as following.
The I(J P ) = 0(0 + ) state: Table XIII    The lowest-lying eigen-energies of bbss tetraquarks with IJ P = 01 + in the real range calculation. 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 and fourth columns refer to the theoretical mass of each channels and their couplings. IG. 5. Complex energies of bbss tetraquarks with IJ P = 00 + in the coupled channels calculation, θ varying from 0 • to 6 • . and their first radial excitation states. There are two orange circles which surround the resonance poles whose masses and widths are (11.31 GeV, 1. IG. 6. Complex energies of bbss tetraquarks with IJ P = 01 + in the coupled channels calculation, θ varying from 0 • to 6 • . conclude that, with much heavier constituent quark components included, more narrow molecular resonances will be found around 0.2 GeV interval near the first radial excitation states. The I(J P ) = 0(1 + ) state: The results listed in Table XIV highlight that tightly bound and narrow resonance states obtained in bbqq tetraquarks [54] are not found in this case. Firstly, the lowest channelB 0 sB * s is of scattering nature both in single channel calculation and coupled-channels one. Secondly, the mass of diquark-antidiquark configuration is higher than mesonmeson channels and its value of 10.91 GeV is very close to the hidden-color channels, 10.95 GeV. Furthermore, the other K-type configurations produce masses slightly lower (∼ 10.90 GeV ) than the former case. Figure 6 shows that the scattering nature ofB 0 sB * s and B * sB * s states is even clearer when the CSM is employed. More specifically, in the mass interval from 10.7 to 11.3 GeV, the calculated complex energies always move along with the varied angle θ. There is no fixed pole in the energy region which is around 0.6 GeV above theB 0 sB * s threshold. This fact is consistent with the ccss results discussed above.
The I(J P ) = 0(2 + ) state: For the highest spin channel of bbss tetraquarks, Table XV summarizes our theoretical findings in real-range method. Among our results, the following are of particular interest: (i) only one di-meson channelB * sB * s exists and it is unbounded if we consider either the single channel or multi-channels coupling calculation, and (ii) the other exotic configurations which include hidden-color, diquark-antidiquark and K-type are all excited states with masses on 10.9 GeV.
In a further step, in which the complex analysis is adopted, three resonances are obtained. It is quite obvious in Fig. 7   IG. 8. Complex energies of cbss tetraquarks with IJ P = 00 + in the coupled channels calculation, θ varying from 0 • to 6 • .
The I(J P ) = 0(0 + ) channel: There are two meson-meson channels, D + sB 0 s and D * + sB * s , two diquarkantidiquark structures, (cb)(ss) and (cb) * (ss) * , and 14 K-type channels (see TableXVI). The single channel calculation produces masses which ranges from 7.34 to 8.67 The lowest-lying eigen-energies of cbss tetraquarks with IJ P = 00 + in the real range calculation. 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 and fourth columns refer to the theoretical mass of each channels and their couplings.  GeV, and all states are scattering ones. The coupledchannels study for each kind of structure reveals weak couplings in di-meson configuration of color-singlet channels and stronger ones for the other structures, with masses above 7.6 GeV.
If we now rotate the angle θ from 0 • to 6 • in a fully coupled-channels calculation, Fig. 8 shows the distribution of complex energy points of D +  The lowest-lying eigen-energies of cbss tetraquarks with IJ P = 01 + in the real range calculation. 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 and fourth columns refer to the theoretical mass of each channels and their couplings.  s , the diquark-antidiquark channels (cb)(ss) * , (cb) * (ss) and (cb) * (ss) * , and the remaining 21 channels are of K-type configurations. In our first kind of calculation, the single channel masses are located in the energy interval 7.39 to 8.23 GeV. Particularly, the color-singlet channels of di-meson configurations present   IG. 10. Complex energies of cbss tetraquarks with IJ P = 02 + in the coupled channels calculation, θ varying from 0 • to 6 • . XVIII. The lowest-lying eigen-energies of cbss tetraquarks with IJ P = 02 + in the real range calculation. 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 and fourth columns refer to the theoretical mass of each channels and their couplings. The I(J P ) = 0(2 + ) channel: Only one mesonmeson channel, D * + sB * s , one diquark-antidiquark channel, (cb) * (ss) * , and 7 K-type configurations must be considered in this case. Their calculated masses are listed in Table XVIII. As all other cases discussed above, no bound states are found neither in the single channel computation nor in the coupled-channels case. The lowest scattering state of D * + sB * s is located at 7.52 GeV and all other excited states, in coupled-channels calculation, are below 7.72 GeV.
In contrast to the cbqq tetraquarks [54], two cbss resonances are found in the complete coupled-channels calculation when complex range method is used. Figure 10 shows an orange circle, which is near the threshold lines (1S)D * + s (2S)B * s and (2S)D * + s (1S)B * s , surrounding two fixed resonance poles. The calculated masses and widths are (8.05 GeV, 1.42 MeV) and (8.10 GeV, 2.90 MeV), respectively. Apparently, these two narrow D * + sB * s resonances are ∼ 0.6 GeV higher than their threshold and this is just similar to our previous results.

IV. EPILOGUE
The QQss tetraquarks with spin-parity J P = 0 + , 1 + and 2 + , and in the isoscalar sector I = 0 have been systemically investigated. This is a natural extension of our previous work on double-heavy tetraquarks QQqq