The tetraquark system in a chiral quark model

The S-wave tetraquarks, with spin-parities , , and , in both isoscalar and isovector sectors, are systematically studied using a chiral quark model. The meson-meson, diquark-antidiquark, and K-type arrangements of quarks and all possible color wave functions are comprehensively considered. The four-body system is solved using the Gaussian expansion method, a highly efficient computational approach. Additonally, a complex-scaling formulation of the problem is established to disentangle bound, resonance, and scattering states. This theoretical framework has already been successfully applied in various tetra- and penta-quark systems. For the complete coupled channel and within the complex-range formulation, several narrow resonances of and systems are obtained, in each allowed -channel, within the energy regions of GeV and GeV, respectively. The predicted exotic states, which indicate a richer color structure when going towards multiquark systems beyond mesons and baryons, are expected to be confirmed in future high-energy particle and nuclear experiments.


I. INTRODUCTION
T ++ c s (2900) 2.908 ± 0.011 ± 0.020 0.136 ± 0.023 ± 0.011 I(J P ) 1(0 + ) In 2020, two charm-strange resonances, and , which are presumably tetraquark candidates, were reported by the LHCb collaboration for decays [1,2].In 2023, this collaboration announced two new resonant states, and , in a combined amplitude analysis of and decays [3,4].These observations indicated that the signals may be open-charm tetraquark candidates with minimal quark content .Without the ability to disentangle the masses and widths of these two resonances, the LHCb collaboration determined that they are GeV and GeV in both cases; moreover, their quantum numbers were determined to be .These observations have inspired numerous theoretic-T c s(2900)

I(J P
) = 1(0 + ) T c s 2.1 − 3.0 al investigations.Generally, the states can be fully identified as a molecular structures in the channel using phenomenological models such as those presented in Refs.[5−9].However, interpretations of compact configurations [10,11], threshold effects [12], and triangle anomalies [13] have also been proposed through various effective field theory approaches and quark models based on the color flux-tube.Additionally, the strong decay properties [14−16] and production mechanisms [17−20] of the exotic states have been investigated.In addition, more resonances within an energy region of GeV have been predicted theoretically [5,7,10].
In addition to the mentioned and states, dozens of exotic hadrons, whose masses are generally near the threshold of two conventional heavy flavored hadrons, have been reported experimentally within the last 20 years.Owing to the rate at which they were being discovered, they have garnered considerable theoretical interest, and experiments through various approaches have been developed by theorists to reveal the nature of the unexpected exotic states, which are overall good candidates of multiquark systems.Particularly, many extensive reviews [21−38], which explain in detail particular theoretical methods and thus capture particular interpretations of exotic hadrons, can be found in the literature.
Herein, we comprehensively investigate tetraquark systems with spin-parities of , , and in the isospin and sectors.A variational formalism based on a highly efficient numerical approach called the Gaussian expansion method (GEM) [39] is used to solve the four-body Hamiltonian, which is based on a chiral quark model that has been used to describe reasonably well various tetra-and penta-quark systems [40−49].Moreover, bound, resonant, and scattering states can be fully disentangled by solving the complex scaled Schrödinger equation formulated using the complex scaling method (CSM).Furthermore, the mesonmeson, diquark-antidiquark, and K-type arrangements of quarks, as well as their couplings with all possible color wave functions, are considered.

qq sQ
The remainder of this manuscript is organized as follows.Sec.II presents the theoretical framework and a brief description of the chiral quark model and tetraquark wave functions.Sec.III analyzes and discusses the calculated results.Finally, a summary is presented in Sec.IV.

II. THEORETICAL FRAMEWORK qq sQ qq sQ
Phenomenological models continue to be the main tools used to study the nature of multiquark candidates observed experimentally.Hence, tetraquark systems are systematically investigated using a chiral quark model.Moreover, a highly accurate computing approach for the few-body system, the GEM, and an effective CSM are adopted to investigate the bound, resonant, and scattering states of the multiquark system.The theoretical formalism employed herein is described in detail in Ref. [26], and we shall focus on the most relevant features of the model and the numerical approach for tetraquarks.

A. Hamiltonian
The four-body problem is studied using a complex scaled Schrödinger equation: where the general form of the four-body Hamiltonian for a QCD-inspired chiral quark model is expressed as where is the constituent quark mass, is the momentum of a quark, is the center-of-mass kinetic energy, and the last term is the two-body potential.By introducing an artificial parameter in the Hamiltonian, called the rotated angle θ, we find three types of complex eigenvalues, viz.bound, resonance, and scattering states can be simultaneously studied.In particular, the bound and resonance states are independent of θ, with the former always placed on the real-axis of the complex energy plane, and the latter located above the threshold line with a total decay width of .Meanwhile, the energy dots corresponding to scattering states are unstable under rotations of θ and align along the threshold line, which also changes with θ.

qq sQ
The dynamics of tetraquark systems is driven by two-body complex scaled potentials: ) . (3) In particular, dynamical chiral symmetry breaking, colorconfinement, and perturbative one-gluon exchange interactions, which are considered the most relevant features of QCD at its low energy regime, are taken into account.
Because the low-lying S-wave positive parity tetraquark states are investigated herein, only the central and spin-spin terms of the potential are considered.
A consequence of the dynamical breaking of chiral symmetry is that Goldstone boson exchange interactions appear between constituent light quarks u, d, and s.Thus, the complex scaled chiral interaction can be expressed as where Gang Yang, Jialun Ping, Jorge Segovia Chin.Phys.C 48, 073106 (2024) 073106-2 where is the Yukawa function.The physical η meson, instead of the octet one, is considered by introducing a model parameter of angle .
represents the flavor Gell-Mann matrices.Obtained from their experimental values, , , and are the masses of the Goldstone bosons.The value of is determined through the relation [50].Finally, the chiral coupling constant, , is determined from the coupling constant through Color confinement should be encoded in the non-Abelian character of QCD.On one hand, lattice-regularized QCD has demonstrated that multi-gluon exchanges produce an attractive linearly increasing potential proportional to the distance between infinite-heavy quarks [51].On the other hand, the spontaneous creation of lightquark pairs from the QCD vacuum may result, at the same scale, in a breakup of the created color flux-tube [51].We can phenomenologically describe the above two observations as where denotes the color Gell-Mann matrices, and , , and Δ are model parameters.When the rotated angle θ is , the real-range potential in Eq. ( 10) is linear at short inter-quark distances with an effective confinement strength , whereas it becomes a constant at large distances, .Beyond the chiral symmetry breaking energy scale, we also expect the dynamics to be governed by perturbative effects of QCD.In particular, the one-gluon ex-change potential, which includes the so-called Coulomb and color-magnetic interactions, is the leading order contribution: where denotes the Pauli matrices, and depends on the reduced mass of a pair.Moreover, the regularized contact term is α s (µ i j ) An effective scale-dependent strong coupling constant, , provides a consistent description of mesons and baryons from light to heavy quark sectors.The frozen coupling constant is expressed as (for instance, Ref. [52]) where , , and are model parameters.All the discussed model parameters are summarized in Table 1.They have been fixed along the last two decades by thorough studies of hadron phenomenology such as meson [53−55] and baryon [56−58] spectra, hadron decays and reactions [59−61], the coupling between con- ventional hadrons and hadron-hadron thresholds [62−64], and molecular hadron-hadron formation [65−67].Furthermore, for a later discussion, Table 2 lists theoretical and experimental (if available) masses of and states of and mesons.

B. Wave function
The S-wave tetraquark configurations are shown in Fig. 1.Figs. 1(a) and (b) show the meson-meson structures, Fig. 1(c) shows the diquarkantidiquark arrangement, and (d) to (f) show the K-type configurations.For solving a manageable -body problem for fully-coupled channels, the K-type configurations are occasionally restricted, as in our previous investigations [41,42].Furthermore, only one configuration would be sufficient for the calculation if all radial and orbital excited states are considered; however, this is inefficient.Therefore, a more economic method to use is the combination of the different mentioned structures.At the quark level, the total wave function of a tetraquark system is the internal product of color, spin, flavor, and space wave functions.First, regarding the color degree-of-freedom, the colorless wave function of a -quark system in meson-meson configuration can be obtained using either two coupled color-singlet clusters, : 8 ⊗ 8 or two coupled color-octet clusters, : (3 brrb + 3ḡrrg + 3 bgḡb + 3ḡb bg + 3rgḡr The first color state is the so-called color-singlet channel, and the second one is the named hidden-color channel.

⊗ 3
The color wave functions associated to the diquarkantidiquark structure are the coupled color triplet-antitriplet clusters, : 6 ⊗ 6 and the coupled color sextet-antisextet clusters, : (2rrrr + 2ḡgḡg + 2 bb bb + rr ḡg + ḡrrg + ḡgrr + rgḡr + rr bb + brrb + bbrr Meanwhile, the possible color-singlet wave functions of three K-type structures are given by For the flavor degree-of-freedom, both iso-scalar ( ) and iso-vector ( ) channels of tetraquarks should be considered.In particular, for meson-meson and part of K-type (Fig. 1(d) and (e)) configurations, the flavor wave functions, which are denoted as , are Additionally, using similar notations and , where the superscripts and denote symmetry and antisymmetry properties between the -pair, respectively, the wave functions of diquark-antidiquark and K-type (Fig. 1(f)) structures are given by

M I
Herein, the third component of the isospin, , is fixed to zero for simplicity, and this is because there is no flavordependent interaction in the Hamiltonian that can distinguish the third component of the isospin I.
Now, let us consider the S-wave ground states with spin (S) ranging from to .Therefore, the spin wave functions, , are given by ( can be set to be equal to S without losing generality): (S , M S ) = (0, 0) for , by for , and by qq sQ (q = u, d; Q = c, b) for .The superscripts and determine the spin wave function for each configuration of the tetraquark system, their values are listed in Table 3.Furthermore, the expressions above are obtained by considering the coupling between two sub-clusters whose spin wave functions are given by trivial SU(2) algebra, and the necessary basis is expressed as ) .
The Rayleigh-Ritz variational principle, which is one of the most used tools to solve eigenvalue problems, is employed to solve the Schrödinger-like four-body system equation.Generally, within a complex-scaling theoretical framework, the spatial wave function is expressed as follows: where the internal Jacobi coordinates are defined as for the meson-meson configurations of Figs. 1 and ; and as for the diquark-antidiquark structure of Fig. 1 .The remaining K-type configurations shown in Fig. 1 to are ( take values according to panels to of Fig. 1): It is obvious now that the center-of-mass kinetic term can be completely eliminated for a non-relativistic system defined in any of the above sets of relative motion coordinates.
The basis expansion of the genuine wave function of Eq. ( 45) is a crucial aspect in the Rayleigh-Ritz variational method.By employing the GEM [39], which has proven to be efficient in solving the bound-state problem of multi-body systems, we expand all the spatial wave functions corresponding to the four relative motions with Gaussian basis functions, whose sizes are obtained using geometric progression.Hence, the form of orbital wave functions, ϕ, in Eq. ( 45) for a S-wave tetraquark system is simply expressed as Finally, the complete wave function, which fulfills the Pauli principle, is written as Table 3.Values of the superscripts and that specify the spin wave function for each configuration of tetraquark systems.
Di-meson Diquark-antidiquark where is the anti-symmetry operator of tetraquark systems, which considers the use of flavor symmetry.According to Fig. 1, it is defined as This is necessary in our theoretical framework because the complete wave function of the four-quark system is constructed from two sub-clusters: meson-meson, diquark-antidiquark, and K-type configurations.
Because the anti-symmetry operator and Gaussian basis functions are employed, the different channels, which include meson-meson in singlet-and hidden-color arrangements, diquark-antidiquark, and K-type configurations, are not orthogonal to each other.This is inevitable in our theoretical framework; however, the off-diagonal matrix elements are very small numerically and negligible compared with the diagonal ones.Accordingly, a quantitative analysis of the inter-quark distances, and a qualitative survey of dominant components, qq sQ can be conducted.In the formula above, and are the left and right generalized eigenvectors, respectively, of the complete anti-symmetric complex wave-function.Additionally, note that only the real part of these complex quantities are discussed herein to gain insights into the nature of tetraquarks.
qq sQ In the next section, which discusses the computed results on the tetraquarks, we first study the systems through a real-range analysis, viz., the rotated angle θ is equal to .In this case, when a complete coupled-channel calculation of matrix diagonalization is performed, possible resonant states are embedded in the continuum.However, we can employ the CSM, with appropriate nonzero values of θ, to disentangle bound, resonance, and scattering states in a complex energy plane.Accordingly, for solving manageable eigevalue problems, the artificial parameter of rotated angle is ranged form to .Meanwhile, with the cooperation of real-and complex-range computations, available exotic states, which are first obtained within a complex-range analysis and then can be identified among continuum states according to its mass in a real-range calculation, are further investigated by analyzing their dominant quark arrangements, sizes, and decay patterns.

III. RESULTS
The S-wave tetraquarks are systematically studied by including meson-meson, diquark-antidiquark, and K-type configurations.Therefore, the total angular momentum, J, coincides with the total spin, S, and can have values of , , and .Thus, the parity of tetraquark system is positive.Furthermore, both the iso-scalar and -vector sectors of tetraquarks are considered.qq sQ Tables 4 to 27 list the calculated results of low-lying tetraquark states.In particular, real-range computations on the lowest-lying masses of each tetraquark system in the allowed quantum numbers are presented in Tables 4,6,8,10,12,14,16,18,20,22,24,and 26.Therein, the considered meson-meson, diquark-antidiquark, and K-type configurations are listed in the first column; if available, the experimental value of the noninteracting di-meson threshold is labeled in parentheses.In the second column, each channel is assigned with an index, which indicates a particular combination of spin ( ), flavor ( ), and color ( ) wave functions, which are shown explicitly in the third column.The theoretical mass calculated in each channel is shown in the fourth column, and the coupled result for each type of configuration is presented in the last one.The last row of the table indicates the lowest-lying mass, which is obtained in a complete coupled-channel calculation within the realrange formalism.

I(J P
) qq sQ In a further step, a complete coupled-channel calculation is performed using the CSM in each tetraquark system.Figures 2 to 13 show the distribution of complex eigenenergies, and therein, the obtained resonance states are indicated inside circles.Several insights about the nature of these resonances are given by calculating their interquark sizes and dominant components; correspondingly, results are listed among Tables 5,7,9,11,13,15,17,19,21,23,25 and 27.In particular, because the flavor symmetry is considered for the tetraquark systems, four types of quark distances, which are , , , and , are calculated.Finally, a summary of our most salient results is presented in Table 28.
To explain our delicate compromise between obtaining reliable results and reasonable times of computation, let us explain our method of obtaining results when we perform a fully-coupled channel computation using the CSM.First, the bound, resonance, and scattering states are complex values distributed in a two dimensional plane, we can observe in the figures below that rotations The tetraquark system in a chiral quark model Chin.Phys.C 48, 073106 (2024) 073106-7 from to enable us to reasonably distinguish all bound, resonance, and scattering states when the imaginary part of the complex energy is lower than .Second, regardless of how clear the figure may be, we follow the behavior of each energy point with respect to the rotated angle from zero to six degrees; thus, we can distinguish wether they belong to a particular mesonmeson threshold or can be something else.In the second case, we perform some extra internal assessments before claiming that the state may be a resonance (there is no doubt when an energy point is a bound state because it is localized in the real axis and does not move with respect the angle).Third, we require a black-circle ( ), red triangle ( ), and blue square ( ) leaving the meson-meson lines and forming a cluster to attract our attention; subsequently, we make some extra assessments related with the evolution of the energy point with respect the theta parameter and its connection with the case to finally claim that it is actually a resonance.

qq sQ
Now, we describe in detail our theoretical findings for each sector of tetraquarks.
qqsc (q=u, d) A. tetraquarks 2.8 3.5 Several resonances whose masses range from to GeV are obtained in this tetraquark sector.Each isoscalar and -vector sectors with total spin and parity , , and are discussed individually below.
The sector: Four meson-meson configurations, , , and in both color-singlet and -octet channels, four diquark-antidiquark structures, along with three K-type configurations are individually calculated in Table 4.The lowest channel is the colorsinglet state of , whose theoretical mass is MeV, the other three meson-meson configurations with the same color channel are in an energy region of to GeV.No bound state is found.Additionally, the single channel calculations are performed in each exotic structure.The hidden-color channels of di-meson configurations are generally located in the GeV interval.This result also holds for the diquark-antidiquark and Ktype structures, although the lowest masses of and -type channels are GeV.well presented.However, in addition to the vast majority of scattering dots, one stable resonance pole is found and circled.The mass and width are and MeV, respectively.
Table 5 shows the compositeness of the resonance Lowest-lying tetraquark states with calculated within the real range formulation of the chiral quark model.The allowed meson-meson, diquark-antidiquark, and K-type configurations are listed in the first column; when possible, the experimental value of the non-interacting meson-meson threshold is labeled in parentheses.Each channel is assigned an index in the second column; it reflects a particular combination of spin ( ), flavor ( ), and color ( ) wave functions, which are shown explicitly in the third column.The theoretical mass obtained in each channel is shown in the fourth column, and the coupled result for each type of configuration is presented in the fifth column.When a complete coupled-channel calculation is performed, the last row of the table indicates the calculated lowest-lying mass (unit: MeV).

Channel
Index state.First, it is a loosely-bound structure with quark-antiquark distances of and fm for the and pairs, respectively.Moreover, a strong coupling exists among singlet-, hidden-color, diquark-antidiquark, and K-type channels.The golden decays for this resonance are and , which are the dominant components (21.3%) of the color-singlet channels.
The sector: 36 channels contribute to this case, and results in real-range calculations are listed in Table 6.First, the lowest mass, MeV, in a single channel computation is the theoretical threshold value of .The other di-meson channels, which include , , and , are generally located in GeV.All these states have a scattering nature.Six channels are also included in each exotic configuration.By referencing the calculated data on each channel, we find that the lowest masses in the hidden-color, diquark-antidiquark, and K-type configurations are all within GeV.Furthermore, color resonances with structures of diquarkantidiquark and -type are still obtained at approximately GeV.In a further step, the lowest coupled-channel masses qq sc I(J P ) = 0(0 + ) Fig. 2. (color online) Complete coupled-channel calculation of the tetraquark system with quantum numbers.
Di Table 5. Compositeness of the exotic resonance obtained in a complete coupled-channel calculation in the state of the tetraquark.In particular, the first column shows the resonance pole labeled by (unit: MeV); the second one shows the distance between any two quarks or quark-antiquark ( ) pair (unit: fm) and the component of resonance state (S: di-meson structure in color-singlet channel; H: di-meson structure in the hidden-color channel; : diquarkantiquark configuration; K: K-type configuration).
Resonance Structure 3006 + i6.3 r q q : 1.64 r q q : 2. , and GeV, respectively.These results indicate that the coupling effect is significantly weak in color-singlet channels, but it becomes stronger in other configurations.However, the bound state is still unavailable even in a complete coupled-channel scenario.
2.5 − 3.4 To find a possible resonance state in an excited energy region of GeV, the fully coupled-channel calculation is further performed using the CSM, and results are plotted in a complex energy plane in Fig. 3. Therein, seven meson-meson scattering states are generally presented.They are ground states of , , and and the first radial excited state of .However, three stable poles are obtained within the radial excited energy region, and their complex energies are , , and MeV, respectively. i.e.
Table 7 shows particular features of the three resonances.First, their dominant components are all of exotic color structure, , the hidden-color, diquark-antidiquark, and K-type configurations.Moreover, the coupling among these three sectors is strong.The color resonances are also confirmed by calculating their sizes, with internal quark distances of about fm.These resonances are expected to be experimentally studied in the golden decay channels.
The state: Two meson-meson channels, and , should be considered in the highest spin state.Table 8 shows that their lowest masses are simply the theoretical threshold values; hence, no bound state is found.Meanwhile, ten other channels of exotic structures are generally located in a mass region of GeV.When coupled-channel calculations are performed in each specific configuration, color resonances are located at GeV, and the scattering state of , which is the lowest-lying channel, remains at GeV.This extremely weak coupling effect also holds for the complete coupled-channel study.
Nevertheless, three narrow resonances are obtained in a complex analysis on the fully coupled-channel computation.Figure 4 shows the two scattering states of and within GeV.Moreover, three stable poles against the descending cut lines occur when the rotated angle is varied from to .
The naturalness of these narrow resonances can be inferred from Table 9.In particular, the complex energies of these resonances are , , and MeV, respectively.Compact structures are dominant when referring to their interquark sizes, which are about fm.Furthermore, strong couplings exist among the color-singlet, hidden-color, diquark-antidiquark, and K-type configurations of these states.Because the singlet-color component are of and for these resonances, they can be experimentally in-  q q : 1.14 r q q : 1.13 r c q : 1.14 r qc : 1.37 ; ; ; Di S: 9.6%; H: 13.9%; : 34.5%; K: 42.0% 3346 + i22.0 r q q : 1.54 r q q : 1.75 r c q : 1.49 vestigated in any of the mentioned two-body strong decay processes.
The sector: This case is similar to the channel, i.e., 24 channels are investigated, as shown in Table 10.First, , , , and channels in both singlet-and hidden-color configurations are calculated.The lowest-lying state is the scattering state with a theoretical threshold value of MeV.Moreover, the other three meson-meson structures in color-singlet channels are also unbound, and the four hidden-color channels are generally located at GeV.For the single channel computations of diquark-antidiquark and K-type configurations, the lowest masses are generally distributed within an energy region of GeV, except for a -type channel with mass at MeV.
We do not find bound states when partially and fully coupled-channel calculations, in the real-range approximation, are performed.The complete coupled-channels study, in the complex range formulation, delivers six continuum states of , , , , and , which are clearly presented in the GeV energy region of Fig. 5. Additionally, one stable resonance is obtained and circled in Fig. 5; its complex energy is predicted to be GeV.This state should be tentatively assigned to the experimentally observed state [3, 4]; however, its theoretical width is smaller than the experimental one.Thus, we cannot be optimistic in the assignment.Note that only two-body strong decay processes are considered, and the other channels not considered herein must contribute to its total decay width.The sector: Among the six channels of the considered tetraquark configurations, which include singlet-, hidden-color, diquark-antidiquark, and Ktype structures, the lowest-lying state is the scattering state with a theoretical threshold at MeV.Meanwhile, this unbound nature remains unchanged in coupled-channel computations.The other five di-meson configurations in the color-singlet channels, which are and , are also unbound.The masses of exotic color channels are generally located in the energy region of GeV, except for a -type channel whose calculated mass is MeV.When coupledchannel calculations are performed in each of these structures, the lowest masses of hidden-color, diquark-antidiquark, , , and channels are , , , , and MeV, respectively.Although bound states are unavailable, the mentioned excited states obtained in each exotic color configuration may be good candidates of color resonances for the tetraquark system.

− 3.15
Furthermore, Fig. 6 shows the distribution of complex energies in the fully coupled-channel calculation using the CSM.In particular, the top panel presents six scattering states, which have been discussed above.Within GeV, two stable poles are obtained, and they are indicated within circles.The lower resonance pole is qq sc I(J P ) = 1(0 + ) Fig. 5. (color online) Complete coupled-channel calculation of the tetraquark system with quantum numbers.
Table 11.Compositeness of the exotic resonance obtained in a complete coupled-channel calculation in the state of the tetraquark.The results are presented in the same manner as that in Table 5.
Resonance Structure 2770 + i1.5 r q q : 1.05 r q q : 1.80 Some insights into the nature of the narrow resonances can be found in Table 13.In particular, a compact tetraquark structure is predicted for the lower resonance, and its size is about fm.The coupling is strong among color-singlet (27%), diquark-antidiquark (14%), and K-type (56%) channels.Moreover, the dominant meson-meson decay channel is , which is expected to be confirmed in future experiments.In contrast, the higher resonance is a loosely-bound structure with a size of fm.The ratios between the components are similar to those at a lower resonance.This suggests that this state should be further studied using high energy experiments in the and decay channels.
The sector: Twelve channels are considered in the highest spin and isospin tetraquark state, The complete coupled-channel calculation using CSM is shown in Fig. 7.In particular, two scattering states of and are well presented within GeV.However, four stable poles occur above the threshold lines.(color online) Complete coupled-channel calculation of the tetraquark system with quantum numbers.In particular, the bottom panel shows enlarged parts of dense energy region from to .
Table 13.Compositeness of exotic resonances obtained in a complete coupled-channel calculation in the state of the tetraquark.The results are presented in the same manner as that in Table 5.

B. tetraquarks
Three spin-parity states, , , and , with isospins of and , are investigated for the tetraquark system.Several narrow resonances are obtained in each quantum number.The details of the calculation and the related discussion are presented in the following.
The sector: Table 16 shows that 24 channels are investigated in this case.First, concerning the four color-singlet channels, which include , , , and , the lowest mass is MeV.This is the theoretical threshold value of , and the other channels are also unbound.Furthermore, the lowest-lying channels of hidden-color, diquark-antidiquark, and K-type configurations are generally located in a mass region of GeV.When coupled-channel computations are performed for each type of structure, the scattering nature of channel in di-meson structure is still obtained.Moreover, possible color resonances in and diquarkantidiquark structures are obtained at and GeV, respectively.The lowest masses of other three configurations are GeV. Finally, in a real-range computation, which is performed by including all of the above channels, the lowest mass of the tetraquark system remains at the theoretical threshold, MeV.
5.7 − 6.7 In the next step, the complete coupled-channel case is studied in a complex-range formulation.Figure 8 shows the distribution of complex energies within GeV.In particular, five scattering states, which are the ground states of , , , and , and the first radial excitation of , are well presented.Nevertheless, Fig. 7. (color online) Complete coupled-channel calculation of the tetraquark system with quantum numbers.
Table 15.Compositeness of exotic resonances obtained in a complete coupled-channel calculation in the state of tetraquark.The results are presented in the same manner as that in Table 5.
Resonance Structure 3031 + i0.7 r q q : 1.02 r q q : 1.07 r c q : 0.94 r qc : 1.00 ; ; ; Di S: 16.4%; H: 14.1%; : 27.0%; K: 42.5% 3105 + i3.7 r q q : 1.26 r q q : 1.33 r c q : 0.99 r qc : 1.26 ; ; ; Di S: 4.0%; H: 21.6%; : 28.0%; K: 46.4% 3373 + i4.2 r q q : 1.51 r q q : 1.61 r c q : 1.45 r qc : 1.55 ; ; ; Di S: 8.0%; H: 10.6%; : 26.7%; K: 54.7% 3455 + i12.9 r q q : 1.68 r q q : 1.54 r c q : 1.30 five resonances are also found and circled in Fig. 8; their complex energies are , , , , and MeV, respectively. 1.9 The details of the properties of such resonances are listed in Table 17.First, strong couplings exist among the singlet-, hidden-color, diquark-antidiquark, and K-type channels.Second, their sizes are less than fm.In particular, the two lower resonances have sizes within fm, and the other three are extended to approximately fm.Third, for the lowest resonance at GeV, the golden decay channel is the channel; the and channels are dominant meson-meson components for the two resonances at GeV, whereas the remaining two resonances at GeV can be confirmed in and channels. s The sector: Table 18 summarizes our results in the real-range formalism.First, the , , and channels in both color-singlet and -octet cases are considered.The lowest mass is GeV, which is the theoretical threshold value.Moreover, no bound states in the other meson-meson channels are found, and the six hidden-color channels are generally located within the energy range of GeV.This energy region is also shared by the diquark-antidiquark and three K-type channels.In partially coupled-channel calculations, the coupling effect is weak in singlet-and hiddencolor channels, their lowest masses are and GeV, respectively.Therefore, the bound state is still unavailable.Although strong couplings exist in diquark-antidiquark and three K-type channels, and the mass shift for the lowest-lying channel is MeV, the bound state is again not obtained.This result also holds for the fully coupled-channel calculation.
In a further complex analysis of the complete coupled-channel, five resonances are obtained, and they are indicated in Fig. 9.We can observe, in addition to the seven scattering states of , , and in the energy region of GeV, the five stable poles circled, with complex energies given by , , Table 19 lists the calculated properties of the resonances to elucidate their nature.First, the strong coupling effects of different tetraquark configurations are reflected.
Meanwhile, these resonances have sizes of about fm, except for the one at GeV, whose size is less than fm.Finally, the lowest resonance at GeV is suggested to be experimentally studied in the decay channel.The and are dominant two-body strong decay channels for the and GeV resonances, whereas the is the golden channel for the remaining two resonances at GeV.
The sector: and in both color-singlet and hidden-color channels, along with two diquark-antidiquark, and six K-type channels are considered in this case.First, the bound state is not found in single-and coupled-channel computations.The lowestlying channel is the scattering state.Additionally, other channels with exotic color structures are located in GeV.When a coupled-channel calculation is performed in each specific configuration, their lowest mass is GeV.
6239 + i0.8 6314 + i3.5 6619 + i4 6664 + i6.9 1.4 Furthermore, Fig. 10 presents results in the fully coupled-channel case using the CSM.Therein, scattering states of , , and are fully shown; additionally, four narrow resonances are also found.Table 21 presents their resonance parameters: , , , and MeV, respectively.Moreover, the size of these four resonances is approximately fm; the proportions of singlet-, hidden-color, diquark-antidiquark, and K-type channels are comparable.They can be further confirmed experimentally in qq sb I(J P ) = 0(0 + ) Fig. 8. (color online) Complete coupled-channel calculation of the tetraquark system with quantum numbers.
Table 17.Compositeness of exotic resonances obtained in a complete coupled-channel calculation in the state of the tetraquark.The results are presented in the same manner as that in Table 5.

IV. SUMMARY
qq sQ (q = u, d; Q = c, b) The S-wave tetraquarks with spin-parities of , , and and isospins of and are systematically investigated under a chiral quark model formalism.Furthermore, the color-singlet, -octet meson-meson configurations, diquark-antidiquark arrangements with their allowed color triplet-antitriplet and sextet-antisextet channels, and K-type configurations are considered.The four-body bound and resonant states are determined using a highly efficient numerical approach: the Gaussian expansion method (GEM) supplemented with a complex-scaling analysis (CSM).Three types of computations are generally presented: single, partiallycoupled, and fully-coupled channels.
qq sc qq sb I(J P ) Table 28 summarizes our theoretical findings for the and tetraquark systems.The first column shows the quantum numbers , the second one expresses the dominant configurations, and the third one lists the complex eigenenergies.

Fig. 1 .
Fig. 1. (color online) Six types of configurations are considered for the tetraquarks.Panels and are meson-meson structures, panel is diquark-antidiquark arrangement, and the K-type configurations are shown in panels to .

Fig. 6 .
Fig. 6. (color online) Complete coupled-channel calculation of the tetraquark system with quantum numbers.In particular, the bottom panel shows enlarged parts of dense energy region from to .

Fig. 9 .
Fig. 9. (color online) Complete coupled-channel calculation of the tetraquark system with quantum numbers.

Fig. 12 .
Fig. 12. (color online) Top panel: Complete coupled-channel calculation of the tetraquark system with quantum numbers.In particular, the bottom panel shows enlarged parts of dense energy region from to .

Fig. 13 .
Fig. 13.(color online) Complete coupled-channel calculation of the tetraquark system with quantum numbers.

Table 6 .
Lowest-lying tetraquark states with calculated within the real range formulation of the chiral quark model.The results are presented in the same manner as that in Table4(unit: MeV).

Table 7 .
Compositeness of exotic resonances obtained in a complete coupled-channel calculation in the state of the tetraquark.The results are presented in the same manner as that in Table5.

Table 8 .
Lowest-lying tetraquark states with calculated within the real range formulation of the chiral quark model.The results are presented in the same manner as that in Table4(unit: MeV).

Table 9 .
Compositeness of exotic resonances obtained in a complete coupled-channel calculation in the state of the tetraquark.The results are presented in the same manner as that in Table5.

Table 10 .
Lowest-lying tetraquark states with calculated within the real range formulation of the chiral quark model.The results are presented in the same manner as that in Table4(unit: MeV).

Table 12 .
Lowest-lying tetraquark states with calculated within the real range formulation of the chiral quark model.The results are presented in the same manner as that in Table4(unit: MeV).

Table 14 .
Lowest-lying tetraquark states with calculated within the real range formulation of the chiral quark model.The results are presented in the same manner as that in Table4(unit: MeV).

Table 16 .
Lowest-lying tetraquark states with calculated within the real range formulation of the chiral quark model.The results are presented in the same manner as that in Table4(unit: MeV).

Table 18 .
Lowest-lying tetraquark states with calculated within the real range formulation of the chiral quark model.The results are presented in the same manner as that in Table4(unit: MeV).

Table 19 .
Compositeness of exotic resonances obtained in a complete coupled-channel calculation in the state of the tetraquark.The results are presented in the same manner as that in Table5.

Table 23 .
First, the complex energies of the two reson-

Table 20 .
Lowest-lying tetraquark states with calculated within the real range formulation of the chiral quark model.The results are presented in the same manner as that in Table4(unit: MeV).

Table 21 .
Compositeness of exotic resonances obtained in a complete coupled-channel calculation in the state of the tetraquark.The results are presented in the same manner as that in Table5.

Table 22 .
Lowest-lying tetraquark states with calculated within the real range formulation of the chiral quark model.The results are presented in the same manner as that in Table4(unit: MeV).

Table 23 .
Compositeness of exotic resonances obtained in a complete coupled-channel calculation in the state of the tetraquark.The results are presented in the same manner as that in Table5.

Table 24 .
Lowest-lying tetraquark states with calculated within the real range formulation of the chiral quark model.The results are presented in the same manner as that in Table4(unit: MeV).

Table 25 .
Compositeness of exotic resonances obtained in a complete coupled-channel calculation in the state of the tetraquark.The results are presented in the same manner as that in Table5.

Table 26 .
Lowest-lying tetraquark states with calculated within the real range formulation of the chiral quark model.The results are presented in the same manner as that in Table4(unit: MeV).Figure13shows the distribution of complex energies in the complete coupled-channel computation using the CSM.Within the GeV energy region, the and scattering states are clearly presented.Meanwhile, four stable resonance poles are also obtained, and they are circled in the complex plane.Table27lists the properties of these resonances.In particular, their complex energies are , , * B * *

Table 27 .
Compositeness of exotic resonances obtained in a complete coupled-channel calculation in the state of the tetraquark.The results are presented in the same manner as that in Table5.

Table 28 .
Summary of resonance structures found in tetraquark systems.The first column shows the isospin, total spin, and parity of each singularity.The second column refers to the dominant configuration components; H: hidden color, : diquark-antidiquark, and K: K-type.Theoretical resonances are presented with the notation in the last column (unit: MeV).