Study on $Z_{cs}$ and excited $B_s^0$ states in the chiral quark model

Stimulated by the newly observed charged hidden-charm state $Z_{cs}(3985)^-$ by BESIII Collaboration, $Z_{cs}(4000)^+$, $Z_{cs}(4220)^+$ and the excited $B_s^0$ states by LHCb Collaboration, a full calculation including masses and decay widths is emerged in the chiral quark model. For $Z_{cs}$ states, we assign quantum numbers $I(J^P)=\frac{1}{2}(1^+)$ and quark composition $c\bar{c}s\bar{u}$ according to the experiment. For $B_s^0$ states, systematically investigations are performed with $I(J^P)=0(0^+), 0(1^+), 0(2^+)$ both in 2-body $b\bar{s}$ and 4-body $b\bar{s}q\bar{q}~(q=u~\rm{or}~d)$ systems. Each tetraquark calculation takes all structures including meson-meson, diquark-antidiquark and all possible color configurations into account. Among the numerical techniques to solve the 2-body and 4-body Schr{\"o}dinger equation, the spatial wave functions are expanded in series of Gaussian basis functions for high precision, which is the way Gaussian expansion method so called. Our results indicate that the low-lying states of 4-quark system are all higher than the corresponding thresholds either for $c\bar{c}s\bar{u}$ or for $b\bar{s}q\bar{q}$ systems. With the help of the real scaling method, we found two resonance states with masses of 4023 MeV and 4042 MeV for $c\bar{c}s\bar{u}$ system. The state $c\bar{c}s\bar{u}(4042)$ has a consistent mass and decay width with the recent observed state $Z_{cs}(3985)^-$. For $b\bar{s}q\bar{q}$ system with $J=0$, some resonance states are also found. The newly observed excited $B_s^0$ states can be accommodated in the chiral quark model as $2S$ or $1D$ states, and the mixing with four-quark states are also needed to be considered.


I. INTRODUCTION
Recently, for the first time, the BESIII Collaboration has reported a structure Z cs (3985) − in the K + recoilmass spectrum near the D − s D * 0 and D * − s D 0 mass thresholds in the process of e + e − → K + (D − s D * 0 + D * − s D 0 ) at the center-of-mass energy √ s = 4.681 GeV, with the mass and narrow decay width [1], M Zcs = (3982 +1.8 −2.6 ± 2.1) MeV, Γ Zcs = (12.8 +5. 3 −4.4 ± 3.0) MeV, (1) and the significance was estimated to be 5.3σ. Soon afterwards, a significant state Z cs (4000) + , with a mass of 4003±6 +4 −14 MeV, a width of 131±15±26 MeV, and spinparity J P = 1 + , was observed by LHCb Collaboration, including another exotic state Z cs (4220) + [2]. The discovery of the charged heavy quarkonium-like structures with strangeness could shed light on the properties of the charged exotic Z states reported before [3].
(3) * xychen@jit.edu.cn † 181001003@njnu.edu.cn ‡ cy1208@nuaa.edu.cn For Z cs , it is classified into the exotic state as the strange partner of Z c (3900) and has intensively attracted more attentions and investigations theoretically within a very short time [5][6][7][8][9][10][11][12][13][14]. These explanations basically cover various exotic hadron configurations. One feature of the Z cs is that its mass is on the verge of theD s D * orD * s D threshold, so a molecular resonance is suggested. For example, Lu Meng et al. obtained the mass and width of Z cs in good agreement with the experimental results by considering the coupled-channel effect and strongly supported the Z cs states as the U/V -spin partner states of the charged Z c (3900) [5]. In chiral effective field theory up to the next-to-leading order, Z cs also was regarded as the partner of the Z c (3900) in the SU(3) flavor symmetry and theD s D * /D * s D molecular resonance [6]. In the QCD sum rule, Z cs can be well defined as a diquarkantidiquark candidate with quark contentccus [8].
On the contrary, J. Ferretti pointed out that the meson-meson molecular model could not be used to describe heavy-light tetraquarks with non-null strangeness content, and in the case of cscn (n = u or d) configurations, the compact tetraquark ground-state is about 200 MeV below the lowest energy hadro-charmonium state, η c K [13]. Another explanation is that Z cs can be naturally regarded as a reflection structure from a charmedstrange meson D * s2 (2573) by Lanzhou group [14]. By adopting a one-boson-exchange model and considering the coupled channel effect, Ref. [10] excluded Z cs as a D * 0 D − s /D 0 D * − s /D * 0 D * − s resonance. As more and more exotic states named as XY Z have been observed in different experiments, their structures are still inexplicable and controversial theoretically. Investigating for charged charmonium-like states can extend our knowledge of hadrons and our understanding of the nature of strong interaction. So we believe that the study of the newly observed Z cs states in the chiral quark model can provide some useful information on exotic hadrons. In present work, for Z cs , the quantum number is assigned as I(J P ) = 1 2 (1 + ) and the quark composition is ccsū. Now let's turn to the newly observed excited B 0 s states. In the past few years, many experiment collaborations such as CDF, D0, and LHCb have made contributions to find the radial and orbital excitations of the bottom and bottom-strange meson families. More and more higher excitations emerged in experiments [15][16][17][18]. Therewith many theoretical studies of the bottom and bottomstrange mesons follow close on another [19][20][21][22][23][24]. The mass spectrum and strong decay patterns are studied most in conventional 2-body quark-antiquark system, which can describe the ground states very well, but has poor understanding for higher excitations of bottom and bottomstrange mesons. Studying the B and B s mesons will help us not only understand of excited mesons, but also put the discovered excited charm and chram-strange mesons into the larger context, since the present situation of experimental exploration of bottom and bottom-strange states is very similar to that of D and D s states in 2003 [25][26][27][28][29][30][31]. With newly observation of the excited B 0 s states by LHCb Collaboration [4], now it is a good time to carry out a comprehensive theoretical study on higher bottom-strange mesons. In this work, all possible quantum numbers with I(J P ) = 0(0 + ), 0(1 + ), 0(2 + ) are studied for B 0 s states. Considering the possible limitation of quark-antidiquark system in conventional quark model in describing the higher excitations and the possible production of quark-antiquark pair in the vacuum, we obtain the masses of B 0 s states in 2-body quark-antidiquark system and 4-body bsqq (q = u or d) system, respectively. bsss is not included here because of its high energy.
Each tetraquark calculation takes into account the mixing of structures, such as meson-meson and diquarkantidiquark structure, along with all possible color, spin configurations. In the meantime, in order to find possible stable resonance states, high precision computing method Gaussian expansion method (GEM) [32] and an useful stabilization real scaling method are both employed [33,34] in our calculations.
The paper is arranged as follows. Theoretical framework including the chiral quark model, the wave functions of Z cs and B 0 s , along with GEM are introduced in Section II. In Section III, the numerical results and discussion are presented. A short summary is given in Section IV.

II. THEORETICAL FRAMEWORK
Chiral quark model: the review of the chiral quark model and GEM has been introduced in Refs. [35][36][37], and here they will be introduced briefly and we mainly focus on the relevant features of Z cs and B 0 s states. The Hamiltonian of the chiral quark model can be writ- The potential energy: V C,G,χ,σ ij represents the confinement, one-gluon-exchange, Goldston boson exchange and σ exchange, respectively. The detailed forms can be referred to Eq. (13) in Ref. [35], which are omit here for space saving. All the model parameters are determined by fitting the meson spectrum, from light to heavy; and the resulting values are listed in Table I.
It is to be noted that, only V χ=η of Goldston boson exchange plays a role between u and s quark, and for u andū interacting quark-pair, not only V χ=π,η , but also V σ works; for other quark pairs such as, (Q, u), (Q, s), is considered without Goldstone bosons and σ exchange.
Wave functions: There are three quark configurations for Z cs and bsqq system , two meson-meosn structures and one diquark-antidiquark structure, which are shown in Fig.1. For spin part, the wave functions for 2-body system are then total six wave functions of 4-body system are obtained easily, which are shown in Table II(the first column). The subscripts SM S of χ represents the total spin  Table II), χ c1 color singlet-singlet (1 ⊗ 1) and χ c2 color octet-octet (8 ⊗ 8) for meson-meson structure and χ c3 color antitriplet-triplet (3 ⊗ 3) and χ c4 sextet-antisextet (6 ⊗6) for diquark-antidiquark structure.
So we can get all allowed spin ⊗ flavor ⊗ color channels of Z cs and bsqq system by taking meson-meson structures, diquark-antidiquark structure, along with all kinds of color spin configurations into account, which are shown in Table III. Next, let's discuss the orbital wave functions for 4-body system. They can be obtained by coupling the orbital wave function for each relative motion of the system, where l 1 and l 2 is the angular momentum of two subclusters, respectively. Ψ Lr (r 1234 ) is the wave function of the relative motion between two sub-clusters with orbital angular momentum L r . L is the total orbital angular momentum of four-quark states. Because of the positive parity (P = (−1) l1+l2+Lr =+) for Z cs and bsqq, it is natural to assume that all the orbital angular momenta are zeros. With the help of Gaussian expansion method (GEM), the spatial wave functions are expanded in series of Gaussian basis functions.
where N nl are normalization constants, c n are the variational parameters, which are determined dynamically. The Gaussian size parameters are chosen according to the following geometric progression This procedure enables optimization of the expansion using just a small numbers of Gaussians. Finally, the complete channel wave function Ψ MI MJ IJ for four-quark system is obtained by coupling the orbital and spin, flavor, color wave functions get in Table III. At last, the eigenvalues of four-quark system are obtained by solving the Schrödinger equation To obtain stable results in our work, the Gaussian width and Gaussian number of each inner cluster takes, r 1 = 0.1 fm, r n = 2 fm, n = 12. For the relative motion between two sub-clusters r 1 = 0.1 fm, r n = 6 fm, n = 7.

III. CALCULATIONS AND ANALYSIS
In the present work, we calculated the mass spectrum of newly observed Z cs and excited B 0 s states in the chiral quark model. For the excited B 0 s states, we firstly treat them as ordinary quark-antiquark states. Using the model parameters given in Table I, Table IV, we can see that the 2S and 1D states have masses between 6000 MeV and 6200 MeV, so it is possible that the newly observed excited B 0 s are 2S or 1D states of bs. However, for the excitation energy as high as 700 MeV, the excitation of the light quark-antiquark pair from vaccuum is highly favored. So considering the excited B 0 s states as four-quark bsqq (q = u or d) system is also necessary. In the following the four-quark bsqq (q = u or d) system with quantum numbers I(J P ) = 0(0 + ), 0(1 + ), 0(2 + ) is investigated.

system
Zcs bsqq channel (spin · flavor · color) For Z cs , the minimal quark component should be ccsū rather than a pure cc since it is observed as a charged particle with strangeness. Both of Z cs and bsqq states have two kinds of meson-meson structures and one diquarkantidiquark structure, which are shown in Fig. 1. Along with all possible color and spin configurations, we take all kinds of structures into account. Table V gives the masses of some relevant quark-antiquark mesons in present work in the chiral quark model. From the table, we can see that the chiral quark model is very successful in describing the meson spectra. And the mass spectra of Z cs and bsqq system are demonstrated in Table VI. From the table, both for Z cs and bsqq system, we can easily found that the low-lying energies in diquarkantidiquark are all much larger than those in mesonmeson structures. All of them are higher than the lowest  there may exist resonances even with the higher energies. Using the stabilization method (real scaling method), we try to find possible resonance for ccsū and bsqq system. To realize the real scaling method here, we multiply the Gaussian size parameters r n in Eq. (9) by a factor α, r n → αr n only for the meson-meson structure with color singlet-singlet configuration. Then we can locate the resonances of ccsū and bsqq system with respect to the scaling factor α, which takes the values from 1.0 to 3.5. With the variation of α, the scattering states will level off to corresponding thresholds, but a resonance will appear as a avoid-crossing structure, which is illustrated in Fig. 2 [33]. The above line represents a scattering state, and it will fall down to the threshold. The down line is the resonance state, which try to keep stable. The resonance state will interact with the scattering state, which can bring about a avoid-crossing point in Fig. 2. With the increasing of the scaling factor α, if we can observe repeated avoid-crossing point, it will be a resonance [33].   Table III). It is the same with bsqq. The last column gives the theoretical lowest thresholds (unit: MeV).
Ecc the lowest thresholds   To make it clear for the reader, we illustrated the stabilization plots of the energies from 3600 MeV to 4100 MeV for Z cs states, respectively in Fig. 3, Fig. 4, Fig.  5. In Fig. 3, we see the first green horizontal line, which represents the lowest threshold J/ψK(1 ⊗ 0 → 1). In the energy region 3900 MeV to 4000 MeV (Fig. 4), there are two thresholds D * 0 D − s (1 ⊗ 0 → 1) and D 0 D * s (0 ⊗ 1 → 1). In higher energy range 4000 MeV to 4100 MeV in Fig. 5, two thresholds η c K * (0 ⊗ 1 → 1) and D from the experimental values of Z cs (3985) − observed by BESIII and Z cs (4000) + observed by LHCb.
For bsqq system, we show the results with all possible quantum numbers I(J P ) = 0(0 + ), 0(1 + ) and 0(2 + ) in Figs. (6)(7)(8)(9)(10). Figs. (6)-(8) represent the bsqq system for 0(0 + ). In the energy range 5700 MeV to 6000 MeV (Fig. 6), there is one threshold BK, and no resonance is found. Above 6000 MeV in Figs. 7 and 8, we find several resonances, such as 6050 MeV, 6078 MeV, 6140 MeV, 6155 MeV and 6241 MeV. From Fig. 7, we can found that resonance states with energies 6140 MeV, 6155 MeV have the same resonant line. To identify which state is the real resonance state, we calculate the proportions of total 12 channels for these two eigen-states. And we find that at the avoid-crossing point for state with 6140 MeV, the color singlet channels B 0 s η and B * s ω play a major role. But for state with 6155 MeV, the hidden-color channels occupy an important role. So we abandon the state with energy 6140 MeV and the state with energy 6155 MeV is the real resonant state in our calculation. For 0(1 + ) and 0(2 + ) states in Fig. 9 and Fig. 10, we cannot find any resonance states in our work.
Besides, we calculated the decay widths of these reso-  nance states using the formula taken from reference [33], where, V (α) is the difference between the two energies at the avoid-crossing point with the same value α. S r and S c are the slopes of scattering line and resonance line, respectively. For each resonance, we get the decay width at the first and the second avoid-crossing point, and we finally give the average decay width of these two values.
The results are shown in    [4]. Combining with the results of bs system, it is possible that the newly observed excited B 0 s states are mixing states of bs and bsqq (q = u, d). The unquenched quark model should be invoked to study the highly excited mesons.

IV. SUMMARY
Motivated by the recent experimental information from BESIII and LHCb Collaboration, we calculated the mass spectrum of the Z cs with I(J P ) = 1 2 (1 + ) and B 0 s states with I(J P ) = 0(0 + ), 0(1 + ), 0(2 + ) in the framework of the chiral quark model using the Gaussian expansion method. Meson-meson and diquark antidiquark structures, and the coupling of them are considered.
For Z cs state with quark component ccsū, we found that the low-lying eigenvalues are all higher than the corresponding thresholds in either structure, leaving no space for a bound state. But we found two resonances with mass 4023 MeV and 4042 MeV for ccsū system with the help of the real scaling method, and the decay width is 3.1 MeV and 13.7 MeV, respectively. The state ccsū(4042) has a consistent mass with the recent observed state Z cs (3985) − and Z cs (4000) + , but the decay width is close to the experimental value of Z cs (3985) − and far narrower than the experimental value Z cs (4000) + .
To find the excited B 0 s state observed by LHCb Collaboration, we give the mass spectrum both in 2-body bs system and 4-body bsqq(q = u or d) system by considering the possible production of quark-antiquark pair in the vacuum. For quark-antiquark system, the 2S and 1D states have masses close to the newly observed B 0 s , so the chiral quark model can accommodate these excited B 0 s states. For four-quark system, no bound state is found. However several resonances are emerged. They have energies, 6050 MeV, 6078 MeV, 6155 MeV, and 6241 MeV. The decay width are all relatively narrow, with 7.8 MeV, 44.1 MeV, 8.7 MeV and 4.1 MeV, respectively. Comparing with the experimental data, we found that it is also possible to interpret the observed B 0 s states as fourquark states. Therefore the better way to investigate the highly excited states is to invoke the unquenched quark model [35], which is our future work.
These possible resonant states should be tested in more precise experimental data in the future and we need more experimental studies on the dominant decay channels of Z cs and B s to figure out their inner configurations.