Exotic double-charm molecular states with hidden or open strangeness and around $4.5\sim 4.7$ GeV

In this work, we investigate the interactions between the charmed-strange meson ($D_s, D_s^{*}$) in $H$-doublet and the (anti-)charmed-strange meson ($D_{s1}, D_{s2}^{*}$) in $T$-doublet, where the one boson exchange model is adopted by considering the $S$-$D$ wave mixing and the coupled-channel effects. By extracting the effective potentials for the discussed $H_s\bar{T}_s$ and $H_s{T}_s$ systems, we try to find the bound state solutions for the corresponding systems. We predict the possible hidden-charm hadronic molecular states with hidden strangeness, i.e., the $D_s^{*}\bar D_{s1}+c.c.$ states with $J^{PC}$=$0^{--}, 0^{-+}$ and the $D_s^{*}\bar D_{s2}^{*}+c.c.$ states with $J^{PC}$=$1^{--}, 1^{-+}$. Applying the same theoretical framework, we also discuss the $H_s T_s$ systems. Unfortunately, the existence of the open-charm and open-strange molecular states corresponding to the $H_s T_s$ systems can be excluded.


I. INTRODUCTION
Studying exotic hadronic states, which are very different from conventional mesons and baryons, is an intriguing research frontier full of opportunities and challenges in hadron physics. As an important part of hadron spectroscopy, exotic state can be as a good platform for deepening our understanding of non-perturbative behavior of quantum chromodynamics (QCD). Since the observation of the charmonium-like state X(3872) [1], a series of X/Y/Z/P c states have been observed in the past 17 years, which stimulated extensive discussions on exotic hadronic state assignments to them. Because of the masses of several X/Y/Z/P c states are close to the thresholds of two hadrons, it is natural to consider them as the candidates of the hadronic molecules, which is the reason why exploring hadronic molecular states has become popular. The theoretical and experimental progress on the hidden-charm multiquark states can be found by Refs. [2][3][4][5][6][7][8]. Among them, a big surprise is the observation from the LHCb Collaboration of three P c states (P c (4312), P c (4440), and P c (4457)) in 2019 [9], which provides strong evidence to support these P c states as the Σ cD ( * ) -type hidden-charm pentaquark molecules [10][11][12][13][14][15][16].
Before presenting our motivation, we firstly need to give a brief review of how these observed XYZ states were decoded as the corresponding hidden-charm molecular states. In 2003, X(3872) was discovered by Belle [1]. For solving its low mass puzzle, the DD * molecular state explanation was proposed in Ref. [17]. Later, more theoretical groups joined the discussion of whether X(3872) can be assigned as DD * molecular state [18,19], where more and more effects were added in realistic calculation of the DD * interaction [20][21][22][23][24][25]. At the same time, another special XYZ state is Z + (4430), which was observed by Belle [26]. In Refs. [27][28][29], a dynamic calculation of Z + (4430) as D * D (′) 1 molecular state was performed. Later, the observed Y(4140) [30] also stimulated a universal molecular state explanation to Y(3930) and Y(4140) [31], which is due to the similarity between Y(3930) [32] and Y(4140) [30]. Additionally, Y(4260) [33] as a DD 1 molecular state was given [34] and discussed [35,36]. Besides Y(4140) [30], the above studies on these typical charmonium-like XYZ are mainly involved in hidden-charm hadronic molecular states without strangeness. These observed XYZ states also result in several systematic theoretical calculations of the interaction between charmed and anti-charmed mesons [37][38][39][40][41]. As good candidate of hidden-charm molecular state with strangeness, Y(4274) [42] has D sDs0 (2317) configuration [43]. Obviously, these studies enlarged our knowledge of hidden-charm hadronic molecular states with mass below 4.5 GeV.
In the past years, more XYZ states with higher mass were announced by many experiment collaborations [44][45][46][47]. In 2019, the BESIII Collaboration announced a white paper on the future physics program [48], where they plan to perform a detailed scan of cross sections between 4.0 and 4.6 GeV and take more data above 4.6 GeV. These new measurements will not only result in complement the higher radial and orbital charmonium family [49,50], but also provide us a good opportunity to study exotic state assignments to the XYZ states above 4.5 GeV.
For hidden-charm molecular states with and without strangeness, which has mass below 4.5 GeV, we have abundant theoretical study. However, our knowledge of hiddencharm molecular states above 4.5 GeV is still not enough. Considering this research status and future experimental plan, we propose to explore exotic double-charm molecular states with hidden or open strangeness existing in mass range around 4.5 ∼ 4.7 GeV, which are relevant to the interactions between S -wave charmed-strange meson in H-doublet and Pwave (anti-)charmed-strange meson in T -doublet. Generally, the H sT s system corresponds to hidden-charm and hiddenstrange hadronic molecular state with the (cs)(cs) configuration while the H s T s system is involved in open-charm and open-strange hadronic molecular state with the (cs)(cs) configuration. In the following, the H sT s and H s T s systems will be main body of this work.
For obtaining the interaction information of the H sT s and H s T s systems, we apply one boson exchange (OBE) model [51,52] to deduce the effective potentials in coordinate space. With this effective potential reflecting the interaction of the H sT s and H s T s systems, we try to find bound state solutions of these discussed H sT s and H s T s systems, by which we may predict possible exotic double-charm molecular states with hidden or open strangeness around 4.5 ∼ 4.7 GeV mass range. Further suggestion of experimental search for this new type of hadronic molecular states will be given. This paper is organized as the follows. After introduction, we illustrate the deduction of the effective potentials of these discussed H sT s and H s T s systems in Sec. II. With this preparation, we present the numerical results of finding the bound state solutions in Sec. III. Finally, this work ends with a short summary in Sec. IV.

II. EFFECTIVE POTENTIALS INVOLVED IN THE H sT s AND H s T s SYSTEMS
In this section, we deduce the effective potentials in the coordinate space for the H sT s and H s T s systems, where the OBE model is adopted in concrete calculation. Here, we need to emphasize that the OBE model was extensively applied to study these observed X/Y/Z/P c states [2,5].

A. Effective Lagrangians
When describing the interactions quantitatively at the hadronic level, we use the effective Lagrangian approach. For writing out the compact effective Lagrangians related to the charmed meson in H-doublet and the (anti-)charmed meson in T -doublet, it is convenient to introduce the super-fields H (Q) a , , and their corresponding conjugate fields [34]. According to the heavy quark limit [53], the super-fields H (Q) a and T (Q)µ a corresponding to the heavy-light meson Qq can be defined by [34] Here, P ± = (1 ± v /)/2 denotes the projection operator, and v µ = (1, 0, 0, 0) is the four velocity under the non-relativistic approximation. Besides, their conjugate fields can be expressed as (2.1) On the basis of the heavy quark symmetry, the chiral symmetry, and the hidden local symmetry [53][54][55][56], the compact effective Lagrangians depicting the interactions between the (anti-)charmed mesons and light pseudoscalar and vector mesons were constructed in Ref. [34], i.e., where the axial current A µ , the vector current V µ , and the vector meson field strength tensor F µν (ρ) are given by respectively. Here, the pseudo-Goldstone and vector meson fields can be written as ξ = exp(iP/ f π ) and ρ µ = ig V V µ / √ 2, respectively. The light pseudo-scalar meson matrix P and the light vector meson matrix V µ have the standard form, i.e., (2.5) In addition, the covariant derivatives can be written as D µ = ∂ µ + V µ and D ′ µ = ∂ µ − V µ . With the above preparation, we can expand the compact effective Lagrangians to the leading order of the pseudo-Goldstone field. The expanded effective Lagrangians for the (anti-)charmed mesons and the exchanged light mesons are collected in Appendix A. Here, the normalized relations for the pseudo-scalar charmed-strange meson D s , the vector charmed-strange meson D * s , the axial-vector charmed-strange meson D s1 , and the tensor charmed-strange meson D * s2 can be expressed as respectively. Here, the polarization vector ǫ µ m (m = 0, ±1) with spin-1 field is written as ǫ µ ± = (0, ±1, i, 0) / √ 2 and ǫ µ 0 = (0, 0, 0, −1) in the static limit, and the polarization tensor ζ µν m (m = 0, ±1, ±2) with spin-2 field is constructed as ζ µν m = m 1 ,m 2 1, m 1 ; 1, m 2 |2, m ǫ µ m 1 ǫ ν m 2 [57].

B. Effective potentials
For getting the effective potentials in the coordinate space, we follow the standard strategy in Ref. [58]. Firstly, we write out the scattering amplitudes M(h 1 h 2 → h 3 h 4 ) of the involved scattering processes h 1 h 2 → h 3 h 4 . For the D ′ D systems, there exist the direct channel and crossed channel [27], which are depicted in Fig. 1, where the notations D ′ and D stand for two different charmed-strange mesons. Secondly, we have with the Breit approximation [59,60] and the non-relativistic normalization. Here, is the effective potentials in the momentum space, and m i (i = h 1 , h 2 , h 3 , h 4 ) represent the masses of the initial and final states.
Thirdly, we need to get the effective potentials in the co- For compensating the off-shell effects of the exchanged light mesons and reflecting the inner structures of the interaction vertex, the form factor should be introduced in every interaction vertex [51,52]. We should indicate that the form factor also plays the role to regulate the effective potentials in the coordinate space since these effective potentials in the coordinate space have the singular delta-function terms [58].
In this work, we introduce the monopole type form factor F (q 2 , m 2 E ) = (Λ 2 − m 2 E )/(Λ 2 − q 2 ) in the OBE model [51,52]. Here, Λ, m E , and q are the cutoff parameter, the mass, and the four momentum of the exchanged light mesons, respectively.
In order to obtain the effective potentials of these focused systems, we need to construct the flavor and spinorbital wave functions of systems. For the hidden-charm and hidden-strange D ′ D molecular systems, we need distinguish the charge parity quantum numbers due to the charge conjugate transformation invariance. The flavor wave function |I, I 3 can be expressed as |0, 0 = |D ′+ D − + cD + D ′− / √ 2 [20,22,38]. Here, we want to emphasize that the coefficients c = ±1 correspond to the charge parity quantum numbers, i.e., C = −c · (−1) J 1 −J 2 −J , where the notations J, J 1 , and J 2 stand for the total angular momentum quantum numbers of the D ′ D, the charmed-strange mesons D ′ , and the charmedstrange mesons D, respectively.
And then, we may get the spin-orbital wave functions | 2S +1 L J for all of the investigated hidden-charm and hiddenstrange molecular systems, i.e., In the above expressions, the notations D 0 , D 1 , and D 2 denote the charmed-strange mesons with the total angular momentum quantum numbers J = 0, 1, and 2, respectively. The constant C e, f ab,cd is the Clebsch-Gordan coefficient, and |Y L,m L stands for the spherical harmonics function.
Through the above preparation, we can derive the effective potentials in the coordinate space for all of the investigated hidden-charm and hidden-strange systems, which are shown in Appendix B. We need to emphasize that the total effective potentials contain the direct channel and crossed channel contributions, which can be written in a general form, i.e., (2.11) where V D (r) and V C (r) are the effective potentials corresponding to the direct channel and crossed channel, respectively.
In the present work, we will also discuss the bound state properties of the S -wave H s T s systems. Their OBE effective potentials can be related to the effective potentials of the H sT s systems by the G-parity rule [61], i.e., where G i is the G-parity of some exchanged mesons. We need to emphasize that we only need to consider the direct diagram contribution to the H s T s systems since the charge conjugate transformation invariance for the H s T s systems does not exist.

III. NUMERICAL RESULTS
For describing the interactions quantitatively, we need the values of the coupling constants. The pionic coupling constant g can be determined by reproducing the experimental width of the process D * + → D 0 π + [62,63]. With the available experimental information, the authors extracted the coupling constant h ′ = (h 1 + h 2 )/Λ χ [55]. According to the vector meson dominance mechanism [63], the coupling constants β, ζ 1 , µ 1 , and g V can be obtained [63,64]. Among them, the coupling constants ζ 1 and µ 1 are consistent with the numerical results in Refs. [65,66]. The coupling constant λ may be fixed by comparing the form factor obtained from the lattice QCD with this calculated via the light cone sum rule [63]. In addition, the coupling constants with the charmed meson in T -doublet can be estimated with the quark model in Refs. [58,67,68]. Meanwhile, the corresponding phase factors between these coupling constants related to the effective potentials in the direct channels are fixed with the quark model [69]. In addition, we also need the parameters of the hadron masses [70]. The values of the coupling constants and the hadron masses are listed in Table I. Thus, the cutoff Λ in the form factor is only one free parameter in our numerical analysis, we attempt to find the loosely bound solutions by varying the cutoff parameters Λ from 1 to 5 GeV in the following. As commonly believed, a loosely bound state with the cutoff parameter around 1 GeV can be regarded as the hadronic molecular candidate according to the theoretical results of the deuteron [51,52,58], which provides us a suitable benchmark.
Hadron masses A. The hidden-charm and hidden-strange molecular systems The hadronic molecular state is a loosely bound state, where the binding energy should be tens of MeV, and the typical size should be larger than the size of all the included hadrons [71]. The above criteria may provide us the critical information to identify the hadronic molecular candidates. Besides, it is important to note that the S -wave bound states should firstly appear since there exist the repulsive centrifugal potential ℓ(ℓ + 1)/2µr 2 for the higher partial wave states (µ and ℓ respectively are the reduced mass and the angular momentum quantum number for the investigated system). Thus, we are mainly interested in the S -wave H sT s systems in this work.
In fact, the S -D wave mixing effect may play an important role to modify the bound properties of the deuteron, which may be regarded as an ideal molecular state [51,52,58]. In this work, we also consider the S -D wave mixing effect to the S -wave H sT s systems. The relevant spin-orbit wave functions | 2S +1 L J are summarized in Table II, where S , L, and J denote the spin, angular momentum, and total angular momentum quantum numbers, respectively. The relevant quantum numbers J P and possible channels | 2S +1 L J involved in the S -wave H sT s systems. Here, "..." means that the S -wave components for the corresponding channels do not exist.
By solving the Schrödinger equation we can find bound state solutions of these discussed systems.
Here, ∇ 2 = 1 r 2 ∂ ∂r r 2 ∂ ∂r and µ = m 1 m 2 m 1 +m 2 as the reduced mass for the investigated system. The bound state solutions include the binding energy E and the radial wave function ψ(r). In addition, we can further calculate the root-mean-square radius r RMS and the probability of the individual channel P i . In the following, we present the numerical results for single channel and coupled-channel cases, respectively.

The single channel analysis
In our numerical analysis, we firstly give the results without considering the S -D wave mixing effect. After that, we further take into account the S -D wave mixing effect, and repeat the numerical analysis. For the S -wave H sT s systems, the relevant numerical results are collected in Tables III-VI within the OBE model, which include the cutoff parameter Λ, the binding energy E, the root-mean-square radius r RMS , and the probability of the individual channel P i .
Since the D s D s η and D s D s1 η vertexes are forbidden by the spin-parity conservation, there only exist the φ exchange con-tribution to the direct and crossed channels for the D sDs1 system. By performing numerical calculations, we can find that there exist the loosely bound state solutions for the S -wave D sDs1 states with J PC = 1 −− and 1 −+ when the cutoff parameters Λ are larger than 4.56 GeV and 3.85 GeV, respectively. However, such cutoff parameters are unusual and deviate from the reasonable range around 1.00 GeV [51,52,58], which reflects that the φ exchange interaction is not strong enough to generate the bound states for the S -wave D sDs1 states with J PC = 1 −− and 1 −+ . Thus, these states as the candidates of the hadronic molecular states are no priority. Besides, we also notice that the D sDs1 system without and with the S -D wave mixing effect have the same bound state properties in our calculation, which is not surprising since the contribution of the tensor forces from the S -D wave mixing effect for the D sDs1 interactions disappears. III: Bound state solutions for the S -wave D sDs1 system. The cutoff Λ, the binding energy E, and the root-mean-square radius r RMS are in units of GeV, MeV, and fm, respectively. Here, we label the major probability for the corresponding channels in a bold manner, and the second column shows the numerical results without considering the S -D wave mixing effect while the last column shows the relevant results with the S -D wave mixing effect. For the S -wave D sD * s2 state with J PC = 2 −− , there do not exist the loosely bound state solutions if we only consider the S -wave component until we increase the cutoff parameter Λ to be around 5 GeV. There exist weakly bound state solutions when the value of the cutoff parameter is taken around 4.70 GeV if adding the contributions of the S -D wave mixing effect. For the S -wave D sD * s2 state with J PC = 2 −+ , we can find the loosely bound state solutions when the cutoff parameter Λ larger than 2.40 GeV, even if the S -D wave mixing effect is included in our calculation. According to our quantitative analysis, it is obvious that the corresponding cutoff parameters are far away from the usual value around 1 GeV for the S -wave D sD * s2 bound states with J PC = 2 −− and 2 −+ [51,52,58]. Here, the large cutoff parameter means that the attractive forces are not strong enough to form these loosely bound states. Thus, we conclude that our numerical results disfavor the existence of the hadronic molecular state candidates for the S -wave D sD * s2 states with J PC = 2 −− and 2 −+ .
In this work, we need to focus on the q 4 correction terms in the effective potentials, the expressions of these correction terms are a little tricky (see Eq.(2.5) of the Appendix B for more details). Through our numerical analysis for the  Table III. Here, "×" indicates no binding energy when scanning the Λ range 1-5 GeV.
s2 system, we can obtain an inequality Λ J PC = 2 −− > Λ J PC = 2 −+ when taking the same binding energy. This difference is caused by the q 4 correction terms in the effective potentials. Usually, a loosely bound state with smaller cutoff parameter corresponds to the more attractive interaction, which means that the q 4 correction terms in the effective potentials are favorable for forming the D sD * s2 molecular state with J PC = 2 −+ . In contrast, these correction terms are unfavorable for forming the D sD * s2 state with J PC = 2 −− . Based on the analysis mentioned above, it is clear that the q 4 correction terms in the interactions play an important role to modify the behavior of the loosely bound state in some cases.
Besides the S -wave D sDs1 (D * s2 ) systems, we also investigate the bound state properties of the S -wave D * sDs1 (D * s2 ) systems in the current work. For the D * sD s1 (D * s2 ) systems, we notice that the η and φ exchange contributions to the direct and crossed diagrams are also allowed, and their interactions are simultaneously associated with the total angular momentum J and the charge parity C.   Table III.

The coupled-channel analysis
In this subsection, we discuss the H sT s systems with J = 1, 2 by considering the coupled-channel effect. In the coupledchannel approach, we need to emphasize that the binding energy E is determined by the lowest mass threshold among various involved channels [71]. The relevant numerical results for the H sT s coupled systems with J PC = 1 −− , 1 −+ , 2 −− , 2 −+ for the different lowest mass thresholds are given in Table VII.
By comparing the numerical results of the single channel and coupled-channel cases, it is obvious that the bound state properties will change accordingly after considering the coupled-channel effect, i.e., the cutoff parameters in the  Table III.     Table III. Here, "Th." represents the lowest mass threshold, and "· · · " means that the S -wave components for the corresponding channels do not exist or the corresponding channels below the threshold considered. 3.04 −3.07 1.90 · · · /· · · /100.00/o(0) 3.22 −10.18 1.13 · · · /· · · /100.00/o(0)

The single channel analysis
In the following, we try to search for the bound state solutions for the S -wave H s T s systems by solving the coupledchannel Schrödinger equation. When considering the S -D wave mixing effect and scanning the cutoff parameters Λ from 1 GeV to 5 GeV, we list these typical values of the cutoff parameter Λ, the binding energy E, the root-mean-square radius r RMS , and the probability of the individual channel P i for the S -wave H s T s systems in Table VIII Table III.

The coupled-channel analysis
Similar to the S -wave H sT s systems, we also present the bound state properties for the S -wave H s T s coupled-channel systems in Table IX. Nevertheless, it is clear that the coupledchannel effect plays a positive but minor effect to generate these S -wave D ( * ) s D s1 (D * s2 ) bound states, where the corresponding cutoff is deviated from the usual value around 1 GeV for these investigated open-charm and open-strange systems [51,52,58]. It means that S -wave bound states for the H s T s systems do not exist even the coupled-channel effect is included.  In short summary, since the attractive interactions between a charmed-strange meson in H-doublet and a charmed-strange meson in T -doublet are not strong enough to make the hadronic component bound together when taking reasonable input parameters. Thus, we conclude that the quantitative analysis does not support the existence of the open-charm and open-strange tetraquark molecular candidates with the Swave D ( * ) s D s1 (D * s2 ) systems, such qualitative conclusion can be further tested in future experiment and other theoretical approaches.

IV. SUMMARY
Exploring the exotic hadronic states has become an intriguing research issue full of challenges and opportunities, which has been inspired by the abundant observations of a series of X/Y/Z/P c states since 2003 [2,5]. Among different exotic hadronic configurations, hadronic molecular state has aroused heated discussion. By the efforts from both theorist and experimentalist, the properties of hidden-charm molecular states with masses below 4.5 GeV, which are from the interaction between charmed meson and anti-charmed meson, become more and more clearer.
Recently, the observation of XYZ charmonium-like states in experiment collaborations [44][45][46][47] and the announced white paper on the future physics program by the BESIII Collaboration [48] show that hunting new XYZ charmonium-like states with mass above 4.5 GeV becomes possible. Considering the close relation of XYZ states with hidden-charm hadronic molecular states, we propose to perform theoretical study of hidden-charm hadronic molecular states composed of the charmed-strange meson in H-doublet and the anticharmed-strange meson in T -doublet. The discussed hiddencharm and hidden-strange hadronic molecular states just exist in the mass range around 4.5 ∼ 4.7 GeV. We predict the existence of the D * sD s1 molecular states with J PC =0 −− , 0 −+ and the D * We also extend our theoretical framework to study opencharm and open-strange molecular systems, which have components, a charmed-strange meson in H-doublet and a charmed-strange meson in T -doublet. However, our calculation does not support the existence of such hadronic molecular states.
Considering the experimental potential, we strongly suggest that the BESIII Collaboration should focus on the predicted vector hidden-charm and hidden-strange molecular state by the accumulated data from e + e − collision with √ s > 4.5 GeV. And, the remaining three molecular states with J PC = 0 −− , 0 −+ , 1 −+ predicted in this work can be accessible at the LHCb and BelleII by B meson decays. Additionally, theoretical study of this new type of hadronic molecular state by other approaches is also encouraged. We believe that these investigations will make our knowledge of exotic hadronic molecular states become more abundant. 3) Appendix B: The details of these obtained effective potentials Before presenting the effective potentials, we firstly focus on several typical Fourier transforms, i.e., In the above expressions, the operators are defined as Z = Through the above preparation, we can write out the effective potentials in the coordinate space for all of the investigated systems, which include • D sDs1 → D sDs1 process: • D sD * s2 → D sD * s2 process: (2.9) • D * sD s1 → D * sD s1 process: s2 process: (2.13) In the above expressions, the function Y(Λ i , m i , r) reads as In addition, we introduce several operators, which include , Here, we define = m,n,a,b C 2,m+n 1m,1n C 2,a+b 1a,1b . For these operators A (′) k , they should be sandwiched by the spin-orbital wave functions | 2S +1 L J , we present the relevant operator matrix el- Table X.