The study of double-charm and double-strange tetraquarks

Over the past decades, dozens of exotic hadron states have been reported in experiments worldwide. These states provide us an ideal platform to deepen our understanding of the non-perturbative quantum chromodynamics (QCD). Although none of the exotic states is now definitely confirmed by experiment, more and more theoretical work have been done to investigate the exotic states. Searching for new exotic states is one of the most significant research topics in the hadron physics.

The hidden-charm and hidden-strange tetraquark, which is composed of $cs\bar{c}\bar{s}$, is one type of the exotic states. In 2009, the CDF Collaboration reported the $X(4140)$ in the $J/\psi\phi$ invariant mass distribution CDF . Later, the $X(4140)$ was successively observed by other experiments, such as Belle Bell , CMS CMS , D0 D0 , BABAR BABAR , and LHCb LHCb-CCSS collaborations. In 2010, a narrow resonance $X(4350)$ was observed in the $\gamma\gamma\longrightarrow J/\psi\phi$ process by the Belle collaboration Bell . In 2016, LHCb collaboration reported four exotic states $X(4140)$, $X(4274)$, $X(4500)$, and $X(4700)$ from the amplitude analysis of the $B^{+}\rightarrow J/\psi\phi K^{+}$ decays LHCb1 ; LHCb2 . In 2017, the $X(4274)$ was reported by the CDF collaboration in the process of $B^{+}\rightarrow J/\psi\phi K^{+}$ CDF2 . In 2020, the LHCb collaboration reported a new state $\chi_{c0}(3930)$, which is just below the $D_{s}^{-}D_{s}^{+}$ threshold LHCb_X3930.1 ; LHCb_X3930.2 . In 2021, two new hadron states $X(4685)$ and $X(4630)$ were reported with the help of the improved full amplitude analysis of the $B^{+}\longrightarrow J/\psi\phi K^{+}$ decay by the LHCb collaborationLHCb-CCSS-2021 .

These experimental developments aroused great interests in studying the hidden-charm and hidden-strange tetraquarks. In Ref. XJL , Liu et al. investigated the tetraquark composed of $cs\bar{c}\bar{s}$, and found that the $X(4350)$, $X(4500)$, and $X(4700)$ could be explained as the compact tetraquarks with $IJ^{P}=00^{+}$ and the $X(4274)$ was explained as a compact tetraquark with $IJ=01^{+}$. In the framework of the chiral quark model, Yang et al. found that the $X(4274)$ could be the $cs\bar{c}\bar{s}$ tetraquark with $J^{PC}=1^{++}$ and the $X(4350)$ could be assigned as a candidate of the compact tetraquark with $J^{PC}=0^{++}$. For the $X(4700)$, it is explained as the $2S$ radial excited tetraquark with $J^{PC}=0^{++}$ YFY . In Ref. CHX , the $cs\bar{c}\bar{s}$ tetraquarks were studied within the diquark-antidiquark configuration by using the QCD sum rule. They claimed that the $X(4140)$ and $X(4274)$ could be assigned as the $S-$wave $cs\bar{c}\bar{s}$ tetraquarks with opposite color structures and both the $X(4500)$ and $X(4700)$ were the $D-$wave $cs\bar{c}\bar{s}$ tetraquarks with opposite color structures too. More results and discussions are given in Refs. JWu ; WZG1 ; QFL ; PGO ; LM ; DCR1 ; ESS .

Recently, the BESIII collaboration reported a new structure $Z_{cs}(3985)^{-}$ near the $D_{s}^{-}D^{*0}/D_{s}^{*-}D^{0}$ thresholds in the processes of $e^{+}e^{-}\rightarrow K^{+}(D_{s}^{-}D^{*0}+D_{s}^{*-}D^{0})$. The mass and width of this state are ($3982.5^{+1.8}_{-2.6}\pm 2.1$) Mev and ($12.8^{+5.3}_{-4.4}\pm 3.0$) Mev, respectively BESIII:2020qkh . From the production mode, the minimum quark component of $Z_{cs}(3985)^{-}$ is $cs\bar{c}\bar{u}$, so it is the first candidate of the charged hidden-charm tetraquark state with strangeness. Later, the LHCb collaboration reported two new states $Z_{cs}(4000)^{+}$ and $Z_{cs}(4220)^{+}$ with the quark content of $cu\bar{c}\bar{s}$ decaying to the $J/\psi K^{+}$ LHCb-CCSS-2021 . The decay widths of these two states are $\Gamma=131\pm 15\pm 26$ MeV and $\Gamma=233\pm 52^{+97}_{-73}$ MeV, respectively. Obviously, the masses of $Z_{cs}(3985)^{-}$ and $Z_{cs}(4000)^{+}$ are close, but the decay widths are largely different. These observations immediately stimulated a lot of theoretical studies of the open-strange and hidden-charm tetraquarks  JinX ; Sun:2020hjw ; Cao:2020cfx ; Du:2020vwb ; Wang:2020kej ; Yang:2020nrt ; Azizi:2020zyq ; Wang:2020htx ; Simonov:2020ozp ; Wang:2020iqt ; Chen:2020yvq ; Meng:2020ihj ; Wan:2020oxt ; Rossi:2020ezg ; Yan:2021tcp ; Wang:2020rcx Some works showed that $Z_{cs}(3985)^{-}$ and $Z_{cs}(4000)^{+}$ are the same states Ortega:2021enc ; Giron:2021sla , while some indicated that they are not the same states Yang:2020nrt ; Meng:2021rdg ; Chen:2021erj ; Shi:2021jyr .

Inspired by the study progress of the hidden-charm and hidden-strange tetraquarks, and the open-strange and hidden-charm tetraquarks, it is natural to investigate the existence of the open-charm and open-strange tetraquarks. Very recently, the Belle collaboration searched for the double-heavy tetraquark state candidates $X_{cc\bar{s}\bar{s}}$ decaying to $D^{+}_{s}D^{+}_{s}$ and $D^{*+}_{s}D^{*+}_{s}$, but no significant signals were observed Belle2021 . In Ref. Yanggang11 , the double-charm and double-strange tetraquarks $cc\bar{s}\bar{s}$ were studied within the chiral quark model, and some resonance states with $IJ^{P}=00^{+}$ and $02^{+}$ were obtained. In Ref. R. Molina , the $D_{s}^{*+}D_{s}^{*+}$ systems with $IJ^{P}=00^{+}$ and $02^{+}$ were studied in a coupled channel unitary approach, but no any bound state was found because of the strong repulsion.

As is commonly believed that QCD is the fundamental theory of the strong interaction. However, the low energy physics of QCD, such as hadron structure, hadron-hadron interactions, and the structure of multiquark systems, is much harder to calculate directly from QCD. The quark delocalization color screening model (QDCSM), which was developed in 1990s with the aim of explaining the similarities between nuclear and molecular forces QDCSM0 , is one of the effective approaches for studying the multiquark systems. Two new ingredients were introduced: quark delocalization (to enlarge the model variational space to take into account the mutual distortion or the internal excitations of nucleons in the course of their interactions, the distortion of wave functions in the existence of other nucleons is also considered in the quark-meson-coupling model QMC ) and color screening (assuming the quark-quark interaction dependent on quark states aimed to take into account the QCD effect which has not yet been included in the two-body confinement and effective one gluon exchange). This model has been applied to the study of the dibaryos Ping1 ; Huang2 , pentaquarks Pc_Huang and some tetraquark systems XJL . It is also interesting to extend this model to study the open-charm and open-strange tetraquarks. As the first step, we investigate the existence of the double-charm and double-strange tetraquarks $cc\bar{s}\bar{s}$ in this work. Different structures and the effect of the channel-coupling are considered.

The structure of this paper is as follows. Section II gives a brief description of the quark model and wave functions. Section III is devoted to the numerical results and discussions. The summary is shown in the last section.

In this work, we investigate the double-charm and double-strange tetraquark system $cc\bar{s}\bar{s}$ in the framework of QDCSM. Two structures: meson-meson and diquark-antiqiquark structures, as well as the channel-coupling of the two configurations are considered. Since we are focus on the $S-$wave states, the orbital angular momentum is set to be zero. The spin quantum number of the $cc\bar{s}\bar{s}$ system can be $0,~{}1$ and $2$, so the total angular momentum can be $J=0,~{}1$ and $2$ for this system. The isospin of the $c$ or $s$ quark is zero. In this way, the quantum number of the $cc\bar{s}\bar{s}$ tetraquark system can be $IJ^{P}=00^{+},01^{+}$ and $02^{+}$. The energy of the $cc\bar{s}\bar{s}$ tetraquark systems for both the meson-meson and diquark-antidiquark structures, as well as the channel coupling of these two structures are listed in Table 1, where the $E_{sc}$ is the energy of every single channel, $E_{cc}$ shows the energy by channel coupling of one certain configuration, and $E_{mix}$ is the lowest energy of the system by coupling all channels of both two configurations.

For the $IJ^{P}=00^{+}$ system, there are four channels, which are $D_{s}^{+}D_{s}^{+}$, $D_{s}^{*+}D_{s}^{*+}$, and two diquark-antidiquark channels with the color configurations $(\bar{3}\times 3)$ and $(\bar{6}\times 6)$. For the meson-meson structure, the energies of both the $D_{s}^{+}D_{s}^{+}$ and $D_{s}^{*+}D_{s}^{*+}$ channels are above the corresponding threshold, which means that neither of them are bound state. The lowest energy is almost unchanged after the channel-coupling calculation, which means that the effect of the channel-coupling here is very small. This is mainly due to the large mass gap between the $D_{s}^{+}D_{s}^{+}$ and $D_{s}^{*+}D_{s}^{*+}$ channels. For the diquark-antidiquark structure, it is obviously that the energies of both two channels are much higher than those of the meson-meson structure. Although the energy is pushed down about $60$ MeV by coupling these two channels, it is still higher than the meson-meson structure. By coupling all channels of both two structures, the lowest energy is still above the threshold of the $D_{s}^{+}D_{s}^{+}$, which indicates that there is no bound state for the $IJ^{P}=00^{+}$ system.

For the $IJ^{P}=01^{+}$ system, the threshold of $D_{s}^{+}D_{s}^{*+}$ is $4080$ MeV and the energies of the two structures are $4086$ MeV, $4390$ MeV, respectively, both of which are higher than the threshold of $D_{s}^{+}D_{s}^{*+}$. Then, channel-coupling of two structures have been performed and the energy $E_{mix}=4083$ MeV is obtained, which is still higher than the threshold. Therefore, no any bound state is obtained for the $IJ^{P}=01^{+}$ system.

For the $IJ^{P}=02^{+}$ system, the case is similar with the one of the $IJ^{P}=01^{+}$ system. The energy of each single channel is above the threshold of the $D_{s}^{*+}D_{s}^{*+}$. The channel-coupling cannot help too much. So there is no bound state for the $IJ^{P}=02^{+}$ system, either.

Although there is no any bound state for the $IJ^{P}=00^{+},01^{+}$ and $02^{+}$ systems, some resonance states are still possible. Since the colorful subclusters diquark and antidiquark cannot fall apart directly due to the color confinement, it is possible for them to be resonance states. To find out if there is any resonance state, stabilization method, also named as a real scaling method, which has proven to be a valuable tool for estimating the energies of the metastable states of electron-atom, electron-molecule, and atom-diatom complexes real_method1 , is employed to find the genuine resonances. In this method, with the increase of the distance between two clusters, the continuum state will fall off towards its threshold, while a resonance state will tend to be stable. In this situation, the resonance line acts as an avoid-crossing structure and it will appear repeatedly with the increment of the distance between two clusters. Then the resonance line corresponds to the energy of the resonance state. This method has been successfully applied to the pentaquark systems real_method2 ; real_method3 , the fully-heavy tetraquark systems real_method4 , and the $c\bar{c}s\bar{s}$ tetraquark systems XJL . Here, we calculate the energy eigenvalues of the $cc\bar{s}\bar{s}$ tetraquark systems by taking the value of the distance ($S$) between two clusters from $4.5$ fm to $8.5$ fm to see if there is any stable state. The results of the $cc\bar{s}\bar{s}$ tetraquark systems with $IJ^{P}=00^{+},01^{+}$ and $02^{+}$ are shown in Figs.1-3, respectively.

For the $cc\bar{s}\bar{s}$ system with $IJ^{P}=00^{+}$ in Fig. 1, it is clearly that the first two horizontal lines locate at the corresponding physical threshold of two channels $D_{s}^{+}D_{s}^{+}$ and $D_{s}^{*+}D_{s}^{*+}$. We mark them with red lines. Another two lines around $4256$ MeV and $4308$ MeV are stable with the variation of the distance between two clusters, so both of them are on behalf of resonance states. We mark them with blue lines. Besides, we also calculate the component of each channel for these two resonance states. For the resonance state with the mass of $4256$ MeV, the proportion of the channels with the meson-meson structure is about $90\%$, while the one of the diquark-antidiqaurk structure is about $10\%$. It indicates that this state is more likely to be the molecular structure. For the resonance state with the mass of $4308$ MeV, the proportion of the channels with the meson-meson structure is about $40\%$, while the one of the diquark-antidiqaurk structure is about $60\%$. It means that this resonance state is more inclined to be the compact structure rather than the molecular structure.

Moreover, the decay width of this resonance state can be calculated by the formula real_method1 :

where the $V(S)$ is the minimal energy difference between the resonance state and the scattering state, $k_{r}$ and $k_{c}$ stand for the slope of the resonance state and the scattering state, respectively. Then, we obtain the width of these two resonance state is about $60$ MeV and $19$ MeV, respectively.

The results of the $cc\bar{s}\bar{s}$ systems with $IJ^{P}=01^{+}$ and $02^{+}$ are shown in Fig. 2 and Fig. 3, respectively. The first horizontal line in two figures represents the threshold of the $D_{s}^{+}D_{s}^{*+}$ and $D_{s}^{*+}D_{s}^{*+}$, respectively. It is obvious that with the increase of the distance between two clusters, the energy of the continuum state falls off towards its threshold. So there is no any resonance state for the $cc\bar{s}\bar{s}$ systems with $IJ^{P}=01^{+}$ and $02^{+}$.

In this work, we systematically investigate the low-lying double-charm and double-strange tetraquark systems in the framework of the QDCSM. Two structures, meson-meson and diquark-antidiquark, as well as the coupling of these two configurations are considered. The dynamical bound-state calculation is carried out to search for any bound state in the $cc\bar{s}\bar{s}$ systems. Besides, both the single channel and the channel coupling calculation are performed to investigate the effect of the channel coupling. Meanwhile, a stabilization calculation is carried out to find any resonance state.

The bound-state calculation shows that there is no any bound state for the $cc\bar{s}\bar{s}$ system in QDCSM. However, two resonance states with $IJ^{P}=00^{+}$ are obtained. One is more possible to be a molecular resonance state with a mass and width around $4256$ MeV and $60$ MeV, respectively; another one is more likely to be a compact resonance state with a mass and width around $4308$ MeV and $19$ MeV, respectively. Our results show that the coupling calculation is indispensable to explore the resonance states. In the work of the chiral quark model Yanggang11 , there was no any bound state for the $cc\bar{s}\bar{s}$ system. However, several resonance states were obtained for $cc\bar{s}\bar{s}$ tetraquarks, which were one $IJ^{P}=00^{+}$ state with the resonance mass around $4.9$ GeV and three $IJ^{P}=02^{+}$ states with the resonance mass around $4.8$ GeV. Besides, the work of Ref. R. Molina also studied the state with $C=2$, $S=2$, $I=0$, and $J=0,~{}2$, and no any state was obtained in this sector. However, only the $D^{*}_{s}D^{*}_{s}$ channel was studied there. The coupling with other channels is worthy of consideration to find some resonance states.

Besides, although no significant signals were observed in present experiment at the Belle collaboration Belle2021 , there is still some structures around $4.3$ GeV in the distributions of $M_{D^{+}_{s}D^{+}_{s}}$ and $M_{D^{*+}_{s}D^{*+}_{s}}$. We suggest that the experiment check with a larger amount of data in the future.