PHYSICAL REVIEW D 102, 094006 (2020)

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

† Fu-Lai Wang and Xiang Liu * School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China and Research Center for and CSR Physics, Lanzhou University and Institute of Modern Physics of CAS, Lanzhou 730000, China

(Received 1 September 2020; accepted 16 October 2020; published 10 November 2020)

In this work, we investigate the interactions between the charmed-strange (Ds;Ds )inH-doublet and the (anti-)charmed-strange meson (Ds1;Ds2)inT-doublet, where the one exchange model is adopted by considering the S-D wave mixing and the coupled-channel effects. By extracting the effective ¯ potentials for the discussed HsTs and HsTs systems, we try to find the solutions for the corresponding systems. We predict the possible hidden-charm hadronic molecular states with hidden ¯ PC 0−− 0−þ ¯ strangeness, i.e., the Ds Ds1 þ c:c: states with J ¼ ; and the Ds Ds2 þ c:c: states with PC −− −þ J ¼ 1 ; 1 . Applying the same theoretical framework, we also discuss the HsTs systems. Unfortunately, the existence of the open-charm and open-strange molecular states corresponding to the HsTs systems can be excluded.

DOI: 10.1103/PhysRevD.102.094006

¯ ðÞ I. INTRODUCTION strong evidence to support these Pc states as the ΣcD - – Studying exotic hadronic states, which are very type hidden-charm [10 16]. different from conventional and , is an Before presenting our motivation, we first need to give a intriguing research frontier full of opportunities and chal- brief review of how these observed XYZ states were lenges in hadron physics. As an important part of hadron decoded as the corresponding hidden-charm molecular states. In 2003, Xð3872Þ was discovered by Belle [1]. spectroscopy, exotic state can be as a good platform for ¯ deepening our understanding of nonperturbative behavior For solving its low mass puzzle, the DD molecular state explanation was proposed in Ref. [17]. Later, more theo- of quantum chromodynamics (QCD). Since the observation 3872 of the charmoniumlike state Xð3872Þ [1], a series of retical groups joined the discussion of whether Xð Þ can be assigned as DD¯ molecular state [18,19], where X=Y=Z=Pc states have been observed in the past 17 years, which stimulated extensive discussions on exotic hadronic more and more effects were added in realistic calculation of ¯ – state assignments to them. Because of the masses of several the DD interaction [20 25]. At the same time, another þ 4430 X=Y=Z=P states are close to the thresholds of two special XYZ state is Z ð Þ, which was observed by c – , it is natural to consider them as the candidates Belle [26]. In Refs. [27 29], a dynamic calculation of þ ¯ ð0Þ of the hadronic molecules, which is the reason why Z ð4430Þ as D D1 molecular state was performed. Later, exploring hadronic molecular states has become popular. the observed Yð4140Þ [30] also stimulated a universal The theoretical and experimental progress on the hidden- molecular state explanation to Yð3930Þ and Yð4140Þ [31], charm multiquark states can be found by Refs. [2–8]. which is due to the similarity between Yð3930Þ [32] and ¯ Among them, a big surprise is the observation from the Yð4140Þ [30]. Additionally, Yð4260Þ [33] as a DD1 LHCb Collaboration of three Pc states (Pcð4312Þ, molecular state was given [34] and discussed [35,36]. 4140 Pcð4440Þ, and Pcð4457Þ) in 2019 [9], which provides Besides Yð Þ [30], the above studies on these typical charmoniumlike XYZ are mainly involved in hidden-charm hadronic molecular states without strangeness. These *Corresponding author. [email protected] observed XYZ states also result in several systematic † [email protected] theoretical calculations of the interaction between charmed and anticharmed mesons [37–41]. Since the Yð4274Þ [42] Published by the American Physical Society under the terms of is observed in the J=ψϕ invariant mass spectrum and is just the Creative Commons Attribution 4.0 International license. ¯ below the threshold of the DsDs0ð2317Þ channel, the Further distribution of this work must maintain attribution to 4274 ¯ 2317 the author(s) and the published article’s title, journal citation, Yð Þ fits well to be the S-wave DsDs0ð Þ molecular and DOI. Funded by SCOAP3. state with JP ¼ 0− in Ref. [43]. Obviously, these studies

2470-0010=2020=102(9)=094006(17) 094006-1 Published by the American Physical Society FU-LAI WANG and XIANG LIU PHYS. REV. D 102, 094006 (2020) enlarged our knowledge of hidden-charm hadronic molecu- where the OBE model is adopted in concrete calculation. lar states with mass below 4.5 GeV. Here, we need to emphasize that the OBE model was In the past years, more XYZ states with higher mass were extensively applied to study these observed X=Y=Z=Pc announced by many experiment collaborations [44–47]. states [2,5]. In 2019, the BESIII Collaboration announced a white paper on the future physics program [48], where they plan A. Effective Lagrangians 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 When describing the interactions quantitatively at the measurements will not only result in complement the hadronic level, we use the effective Lagrangian approach. higher radial and orbital charmonium family [49,50],but For writing out the compact effective Lagrangians related also provide us a good opportunity to study exotic state to the charmed meson in H-doublet and the (anti-)charmed assignments to the XYZ states above 4.5 GeV. meson in T-doublet, it is convenient to introduce the super- ðQÞ ðQÞμ ðQ¯ Þ ðQ¯ Þμ For hidden-charm molecular states with and without fields Ha , Ta , Ha , Ta , and their corresponding strangeness, which has mass below 4.5 GeV, we have conjugate fields [34]. According to the heavy limit abundant theoretical study. However, our knowledge of ðQÞ ðQÞμ [53], the super-fields Ha and Ta corresponding to the hidden-charm molecular states above 4.5 GeV is still not heavy-light meson Qq¯ can be defined by [34] enough. Considering this research status and future exper- imental plan, we propose to explore exotic double-charm molecular states with hidden or open strangeness existing ðQÞ P ðQÞμγ − ðQÞγ 4 5 ∼ 4 7 Ha ¼ þðDa μ Da 5Þ; in mass range around . . GeV, which are relevant to rffiffiffi   the interactions between S-wave charmed-strange meson in ðQÞμ ðQÞμν 3 ðQÞ μν 1 ν μ μ P D γν − γ5 g − γ γ − H-doublet and P-wave (anti-)charmed-strange meson in T- Ta ¼ þ 2a 2D1aν 3 ð v Þ ; ¯ doublet. Generally, the HsTs system corresponds to hid- den-charm and hidden-strange hadronic molecular state ¯ ¯ ¯ with the ðcsÞðcsÞ configuration while the HsTs system is respectively. Meanwhile, the antimeson Qq superfields involved in open-charm and open-strange hadronic ðQ¯ Þ ðQ¯ Þμ ¯ ¯ Ha and Ta can be obtained by the charge conjugate molecular state with the ðcsÞðcsÞ configuration. In the ðQÞ ðQÞμ ¯ transformation for the superfields Ha and Ta [34], following, the HsTs and HsTs systems will be main body of this work. where its expression denotes ¯ For obtaining the interaction information of the HsTs and HsTs systems, we apply one boson exchange (OBE) ðQ¯ Þ ¯ ðQ¯ Þμ ¯ ðQ¯ Þ model [51,52] to deduce the effective potentials in coor- Ha ¼ðDa γμ − Da γ5ÞP−; rffiffiffi   dinate space. With this effective potential reflecting the 3 1 ¯ ðQ¯ Þμ ¯ ðQ¯ Þμν ¯ ðQ¯ Þ μν μ μ ν interaction of the HsTs and HsTs systems, we try to find Ta ¼ D γν − D γ5 g − ðγ −v Þγ P−: ¯ 2a 2 1aν 3 bound state solutions of these discussed HsTs and HsTs systems, by which we may predict possible exotic double- charm molecular states with hidden or open strangeness Here, P ¼ð1 vÞ==2 denotes the projection operator, and around 4.5 ∼ 4.7 GeV mass range. Further suggestion of vμ ¼ð1; 0; 0; 0Þ is the four velocity under the nonrelativ- experimental search for this new type of hadronic molecu- istic approximation. Besides, their conjugate fields can be lar states will be given. expressed as This paper is organized as the follows. After introduc- tion, we illustrate the deduction of the effective potentials ¯ of these discussed H T and H T systems in Sec. II. With ¯ ¯ s s s s ¯ γ †γ ðQÞ ðQÞμ ðQÞ ðQÞμ this preparation, we present the numerical results of finding X ¼ 0X 0;X¼ Ha ;Ta ;Ha ;Ta : ð2:1Þ the bound state solutions in Sec. III. Finally, this work ends with a short summary in Sec. IV. On the basis of the heavy quark symmetry, the chiral symmetry, and the hidden local symmetry [53–56], II. EFFECTIVE POTENTIALS INVOLVED the compact effective Lagrangians depicting the inter- IN THE H T¯ AND H T SYSTEMS s s s s actions between the (anti-)charmed mesons and light In this section, we deduce the effective potentials pseudoscalar and vector mesons were constructed in ¯ in the coordinate space for the HsTs and HsTs systems, Ref. [34], i.e.,

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ðQÞ ¯ ðQÞ ¯ ðQ¯ Þ ðQ¯ Þ ðQÞμ ¯ ðQÞ ¯ ðQ¯ Þμ ðQ¯ Þ L ¼ ighH =A γ5Ha iþighHa =A γ5H iþikhT =A γ5Taμ iþikhTa =A γ5T μ i b ba ab b b ba  ab b ¯ ¯ ðQÞμ h1 h2 ¯ ðQÞ ¯ ðQÞ h1 ⃖0 h2 ⃖0 ðQÞμ i T Dμ=A DAμ γ5H : : i H =ADμ AμD γ5T : : þ½h b Λ þ Λ a iþH c þ½h a Λ þ Λ b iþH c χ χ ba χ χ ab ¯ ¯ ðQÞ β μ V − ρ λσμν ρ ¯ ðQÞ − ¯ ðQÞ β μ V − ρ − λσμν ρ ðQÞ þhiHb ð v ð μ μÞþ Fμνð ÞÞbaHa i hiHa ð v ð μ μÞ Fμνð ÞÞabHb i ¯ ¯ ðQÞ β00 μ V − ρ λ00σμν ρ ¯ ðQÞλ − ¯ ðQÞ β00 μ V − ρ − λ00σμν ρ ðQÞλ þhiTbλ ð v ð μ μÞþ Fμνð ÞÞbaTa i hiTaλ ð v ð μ μÞ Fμνð ÞÞabTb i ¯ ¯ ðQÞμ ζ V − ρ μ γν ρ ¯ ðQÞ − ¯ ðQÞ ζ V − ρ − μ γν ρ ðQÞμ þ½hTb ði 1ð μ μÞþ 1 Fμνð ÞÞbaHa iþH:c: ½hHa ði 1ð μ μÞ 1 Fμνð ÞÞabTb iþH:c:;

− pffiffiffiffiffiffiffiffi where the axial current Aμ, the vector current Vμ, and the 0 ¯ 0 h jDsjcsð Þi ¼ mDs ; field strength tensor FμνðρÞ are given by μ − μpffiffiffiffiffiffiffiffi h0jDs jcs¯ð1 Þi ¼ ϵ mD ; ffiffiffiffiffiffiffiffiffis 0 μ ¯ 1þ ϵμp 1 h jDs1jcsð Þi ¼ mD 1 ; A ξ†∂ ξ − ξ∂ ξ† pffiffiffiffiffiffiffiffiffis μ ¼ ð μ μ Þμ; ð2:2Þ μν þ μν 2 h0jD 2 jcs¯ð2 Þi ¼ ζ m ; s Ds2 ϵμ 0 1 1 respectively. Here, the polarization vector mðm ¼ ffiffiffi; Þ V ξ†∂ ξ ξ∂ ξ† μ p μ ¼ ð μ þ μ Þμ; ð2:3Þ with spin-1 field is written as ϵ ¼ð0; 1;i;0Þ= 2 and 2 μ ϵ0 ¼ð0; 0; 0; −1Þ in the static limit, and the polarization ζμν 0 1 2 tensor mPðm ¼ ; ; Þ with spin-2 field is constructed ρ ∂ ρ − ∂ ρ ρ ρ ζμν 1 ;1 2 ϵμ ϵν Fμνð Þ¼ μ ν ν μ þ½ μ; ν; ð2:4Þ as m ¼ m1;m2 h ;m1 ;m2j ;mi m1 m2 [57]. respectively. Here, the pseudo-Goldstone and vector B. Effective potentials meson fields can be written as ξ exp iP=fπ and pffiffiffi ¼ ð Þ For getting the effective potentials in the coordinate ρ V 2 μ ¼ igV μ= , respectively. The light pseudoscalar space, we follow the standard strategy in Ref. [58]. First, P V meson matrix and the light vector meson matrix μ we write out the scattering amplitudes Mðh1h2 → h3h4Þ have the standard form, i.e., of the involved scattering processes h1h2 → h3h4. For the D0D¯ systems, there exist the direct channel and crossed 0 1 π0 η channel [27], which are depicted in Fig. 1, where the pffiffi þ pffiffi πþ Kþ B 2 6 C notations D0 and D stand for two different charmed-strange B − π0 η 0 C π − pffiffi pffiffi mesons. In Fig. 1, according to requirement of the spin- P ¼ B 2 þ 6 K C; @ qffiffi A parity conservation, we may determine the exchanged K− K¯ 0 − 2η for certain hadron-hadron system in the frame- 3 ¯ 0 1 work of the OBE model. Here, we take the DsDs1 system ρ0 ω pffiffi pffiffi ρþ Kþ as an example to illustrate it. Since the D D η and B 2 þ 2 C s s B C D D 1η vertexes are strictly forbidden by the spin-parity − ρ0 ω 0 s s V μ ¼ B ρ − pffiffi pffiffi C : ð2:5Þ @ 2 þ 2 K A − ¯ 0 ϕ K K μ

In addition, the covariant derivatives can be written as Dμ ¼ 0 ∂μ þ Vμ and Dμ ¼ ∂μ − Vμ. 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 pseudoscalar charmed-strange meson Ds, the vector FIG. 1. The direct channel and crossed channel Feynman ¯ charmed-strange meson Ds , the axial-vector charmed- diagrams for the D0D systems. The thick (thin) lines stand for 0 strange meson Ds1, and the tensor charmed-strange meson the charmed-strange mesons D ðDÞ, while the dashed lines Ds2 can be expressed as represent the exchanged light mesons.

094006-3 FU-LAI WANG and XIANG LIU PHYS. REV. D 102, 094006 (2020) X μ ϕ ¯ S;mS J;M ν conservation, there only exists the exchange contribution jD1D1i¼ C 0 C ϵmϵ 0 jY i; ð2:9Þ 1m;1m SmS;LmL m L;mL ¯ 0 to the direct and crossed channels for the DsDs1 system. m;m ;mS;mL Second, we have X ¯ S;mS J;M λ μν jD1D2i¼ C 0 C ϵ ζ 0 jY i: ð2:10Þ 1m;2m SmS;LmL m m L;mL 0 → Mðh1h2 → h3h4Þ m;m ;mS;mL Vh1h2 h3h4 − pffiffiffiffiffiffiffiffiffiffiffiffiffiffi E ðqÞ¼ Q ð2:6Þ 2mi i In the above expressions, the notations D0, D1, and D2 denote the charmed-strange mesons with the total angular with the Breit approximation [59,60] and the nonrelativistic momentum quantum numbers J ¼ 0, 1, and 2, respectively. h1h2→h3h4 V e;f normalization. Here, E ðqÞ is the effective poten- The constant Cab;cd is the Clebsch-Gordan coefficient, and tials in the momentum space, and m i h1;h2;h3;h4 ið ¼ Þ jY i stands for the spherical harmonics function. represent the masses of the initial and final states. L;mL Through the above preparation, we can derive the Thirdly, we need to get the effective potentials in the effective potentials in the coordinate space for all of the Vh1h2→h3h4 coordinate space E ðrÞ by performing the Fourier investigated hidden-charm and hidden-strange systems, Vh1h2→h3h4 transformation to E ðqÞ, i.e., which are shown in Appendix B. We need to emphasize that the total effective potentials contain the direct channel Z 3 → d q → and crossed channel contributions, which can be written in Vh1h2 h3h4 iq·rVh1h2 h3h4 F 2 2 2 E ðrÞ¼ 3 e E ðqÞ ðq ;mEÞ: a general form, i.e., ð2πÞ

VTotalðrÞ¼VDðrÞþcVCðrÞ; ð2:11Þ For compensating the off-shell effects of the exchanged light mesons and reflecting the inner structures of the where VDðrÞ and VCðrÞ are the effective potentials corre- interaction vertex, the form factor should be introduced in sponding to the direct channel and crossed channel, every interaction vertex [51,52]. We should indicate that the respectively. form factor also plays the role to regulate the effective In the present work, we will also discuss the bound state potentials in the coordinate space since these effective properties of the S-wave HsTs systems. Their OBE potentials in the coordinate space have the singular delta- effective potentials can be related to the effective potentials function terms [58]. In this work, we introduce the of the H T¯ systems by the G-parity rule [61], i.e., F 2 2 Λ2 − 2 s s monopole type form factor ðq ;mEÞ¼ð mEÞ= 2 2 X Λ − Λ → H T¯ →H T¯ ð q Þ in the OBE model [51,52]. Here, , mE, and HsTs HsTs r s s s s r V ð Þ¼ GiVi ð Þ; ð2:12Þ q are the cutoff parameter, the mass, and the four momen- i tum of the exchanged light mesons, respectively. In order to obtain the effective potentials of these focused where Gi is the G-parity of some exchanged mesons. We systems, we need to construct the flavor and spin-orbital need to emphasize that we only need to consider the direct wave functions of systems. For the hidden-charm and diagram contribution to the HsTs systems since the charge 0 ¯ hidden-strange D D molecular systems, we need distin- conjugate transformation invariance for the HsTs systems guish the charge parity quantum numbers due to the does not exist. charge conjugate transformation invariance. The flavor wave function I;I3 can be expressed as 0; 0 j piffiffiffi j i¼ III. NUMERICAL RESULTS jD0þD− þ cDþD0−i= 2 [20,22,38]. Here, we want to For describing the interactions quantitatively, we need emphasize that there exists a relation of c and the C parity, J1−J2−J the values of the coupling constants. The pionic coupling i.e., C ¼ −c · ð−1Þ , where the notations J, J1, and J2 constant g can be determined by reproducing the stand for the total angular momentum quantum numbers of experimental width of the process Dþ → D0πþ [62,63]. the D0D¯ , the charmed-strange mesons D0, and the charmed- D With the available experimental information, the authors strange mesons , respectively. 0 extracted the coupling constant h ¼ðh1 þ h2Þ=Λχ [55]. And then, we may get the spin-orbital wave functions 2Sþ1 According to the vector meson dominance mechanism [63], j LJi for all of the investigated hidden-charm and the coupling constants β, ζ1, μ1, and g can be obtained hidden-strange molecular systems, i.e., V [63,64]. Among them, the coupling constants ζ1 and μ1 are X ¯ J;M μ consistent with the numerical results in Refs. [65,66]. The jD0D1i¼ C1 ϵmjY i; ð2:7Þ m;LmL L;mL coupling constant λ may be fixed by comparing the form m;mL factor obtained from the lattice QCD with this calculated X via the light cone sum rule [63]. In addition, the coupling ¯ J;M μν jD0D2i¼ C ζm jY i; ð2:8Þ 2m;LmL L;mL constants with the charmed meson in T-doublet can be m;mL estimated with the in Refs. [58,67,68].

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TABLE I. The summary of the coupling constants and the TABLE II. The relevant quantum numbers JP and possible 2Sþ1 ¯ hadron masses adopted in our calculations. Unit of the coupling channels j LJi involved in the S-wave HsTs systems. Here, 0 00 −1 constants h , λ, λ , and μ1 is GeV , and the coupling constant fπ “...” means that the S-wave components for the corresponding is given in units of GeV. channels do not exist.

0 00 g; k h fπ β; β P ¯ ¯ ¯ ¯ ð Þjj ð Þ J DsDs1 DsDs2 Ds Ds1 Ds Ds2 −0 90 0 90 − 1 5 (0.59,0.59) 0.55 0.132 ð . ; . Þ 0 ...... j S0i=j D0i ... λ λ00 ζ μ − 3 3 3 3;5 3 3;5;7 Coupling constants ð ; Þj1j 1 gV 1 j S1i=j D1i ... j S1i=j D1ijS1i=j D1i ð−0.56; 0.56Þ 0.20 0 5.83 − 5 5 5 1;3;5 5 3;5;7 2 ... j S2i=j D2ijS2i=j D2ijS2i=j D2i − 7 3;5;7 3 ...... j S3i=j D3i ηϕDs Ds 547.86 1019.46 1968.34 2112.20 Hadron masses (MeV) Ds1 Ds2 may be regarded as an ideal molecular state [51,52,58].In 2535.11 2569.10 this work, we also consider the S-D wave mixing effect to the ¯ S-wave HsTs systems. The relevant spin-orbit wave func- 2Sþ1 Meanwhile, the corresponding phase factors between these tions j LJi are summarized in Table II, where S, L, and J coupling constants related to the effective potentials in the denote the spin, angular momentum, and total angular direct channels are fixed with the quark model [69].In momentum quantum numbers, respectively. addition, we also need the parameters of the hadron masses By solving the Schrödinger equation [70]. The values of the coupling constants and the hadron   1 l l 1 masses are listed in Table I. Thus, the cutoff Λ in the form − ∇2 − ð þ Þ ψ ψ ψ 2 ðrÞþVðrÞ ðrÞ¼E ðrÞ; ð3:1Þ factor is only one free parameter in our numerical analysis, 2μ r we attempt to find the loosely bound solutions by varying the cutoff parameters Λ from 1 to 5 GeV in the following.1 we can find bound state solutions of these discussed 2 1 ∂ 2 ∂ m1m2 systems. Here, ∇ ¼ 2 r and μ ¼ as the In this work, we take cutoff range around 1 GeV to discuss r ∂r ∂r m1þm2 possible hadron molecular states with hidden-charm and reduced mass for the investigated system. The bound state hidden-strange. solutions include the E and the radial wave function ψðrÞ. In addition, we can further calculate the A. The hidden-charm and hidden-strange root-mean-square radius rRMS and the probability of molecular systems the individual channel Pi. In the following, we present the numerical results for single channel and coupled- The hadronic molecular state is a loosely bound state, channel cases, respectively. 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 1. The single channel analysis information to identify the hadronic molecular candidates. In our numerical analysis, we first give the results Besides, it is important to note that the S-wave bound states without considering the S-D wave mixing effect. After should first appear since there exist the repulsive centrifu- that, we further take into account the S-D wave mixing 2 gal potential lðl þ 1Þ=2μr for the higher partial wave effect, and repeat the numerical analysis. For the S-wave states (μ and l respectively are the reduced mass and the ¯ angular momentum quantum number for the investigated TABLE III. Bound state solutions for the S-wave DsDs1 ¯ Λ system). Thus, we are mainly interested in the S-wave HsTs system. The cutoff , the binding energy E, and the root- systems in this work. mean-square radius rRMS are in units of GeV, MeV, and fm, In fact, the S-D wave mixing effect may play an important respectively. Here, we label the major probability for the role to modify the bound properties of the deuteron, which 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 1For the Yð4274Þ [42], the spin-parity quantum number JPC ¼ with the S-D wave mixing effect. 1þþ was measured by the LHCb Collaboration later [44], which PC PC Λ Λ 3S 3D is not consistent with the J quantum number corresponding to J ErRMS ErRMS Pð 1= 1Þ ¯ 2317 the assignment of the S-wave DsDs0ð Þ molecular state even. −0 30 −0 30 100 00 0 Considering this situation, we did not adopt the observed 4.56 . 4.73 4.56 . 4.73 . =oð Þ 1−− −0 67 −0 67 100 00 0 Yð4274Þ to fix the Lambda cutoff in this work. As is well 4.78 . 3.63 4.78 . 3.63 . =oð Þ known, the deuteron is a loosely bound state composed of a 5.00 −1.15 2.89 5.00 −1.15 2.89 100.00=oð0Þ and a , and may be regarded as an ideal molecular 3.85 −0.31 4.72 3.85 −0.31 4.72 100.00=oð0Þ state. By reproducing the binding energy and root-mean-square 1−þ 4.43 −2.26 2.12 4.43 −2.26 2.12 100.00=oð0Þ radius of deuteron with the OBE potential model, the Λ cutoff 5.00 −5.07 1.46 5.00 −5.07 1.46 100.00=oð0Þ should be around 1 GeV [51,52,58].

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¯ ¯ PC 1−− TABLE IV. Bound state solutions for the S-wave DsDs2 bound states for the S-wave DsDs1 states with J ¼ system. Conventions are the same as Table III. Here, “×” and 1−þ. Thus, these states as the candidates of the indicates no binding energy when scanning the Λ range 1–5 GeV. hadronic molecular states are no priority. Besides, we also ¯ PC 5 5 notice that the D D 1 system without and with the S-D J Λ Er Λ Er Pð S2= D2Þ s s RMS RMS wave mixing effect have the same bound state properties in 2−− −0 22 99 87 0 13 × × × 4.70 . 5.00 . = . our calculation, which is not surprising since the contri- −2 87 99 53 0 47 × × × 4.76 . 1.81 . = . bution of the tensor forces from the S-D wave mixing effect × × × 4.82 −9.76 1.00 99.07=0.93 ¯ for the DsDs1 interactions disappears. 2−þ −0 31 −0 34 99 99 0 01 −− 2.44 . 4.69 2.41 . 4.58 . = . For the S-wave D D¯ state with JPC 2 , there do not −2 95 −3 35 99 97 0 03 s s2 ¼ 2.52 . 1.87 2.49 . 1.76 . = . exist the loosely bound state solutions if we only consider 2.60 −10.16 1.06 2.56 −10.26 1.05 99.93=0.07 the S-wave component until we increase the cutoff param- Λ ¯ eter to be around 5 GeV. There exist weakly bound state HsTs systems, the relevant numerical results are collected solutions when the value of the cutoff parameter is taken – in Tables III VI within the OBE model, which include the around 4.70 GeV if adding the contributions of the S-D cutoff parameter Λ, the binding energy E, the root-mean- ¯ wave mixing effect. For the S-wave DsDs2 state with square radius r , and the probability of the individual − RMS JPC ¼ 2 þ, we can find the loosely bound state solutions channel P . i when the cutoff parameter Λ larger than 2.40 GeV, even if Since the D D η and D D 1η vertexes are forbidden by s s s s the S-D wave mixing effect is included in our calculation. the spin-parity conservation, there only exist the ϕ According to our quantitative analysis, it is obvious that the exchange contribution to the direct and crossed channels ¯ corresponding cutoff parameters are far away from the for the DsDs1 system. By performing numerical calcula- ¯ tions, we can find that there exist the loosely bound state usual value around 1 GeV for the S-wave DsDs2 bound PC −− −þ ¯ PC −− states with J ¼ 2 and 2 [51,52,58]. Here, the large solutions for the S-wave DsDs1 states with J ¼ 1 and 1−þ when the cutoff parameters Λ are larger than 4.56 GeV cutoff parameter means that the attractive forces are not and 3.85 GeV, respectively. However, such cutoff param- strong enough to form these loosely bound states. Thus, we eters are unusual and deviate from the reasonable range conclude that our numerical results disfavor the existence around 1.00 GeV [51,52,58], which reflects that the ϕ of the hadronic molecular state candidates for the S-wave ¯ PC 2−− 2−þ exchange interaction is not strong enough to generate the DsDs2 states with J ¼ and .

¯ TABLE V. Bound state solutions for the S-wave Ds Ds1 system. Conventions are the same as Table III.

PC 1 5 J Λ ErRMS Λ ErRMS Pð S0= D0Þ 0−− 1.68 −0.41 4.14 1.68 −0.42 4.09 100.00=oð0Þ 1.72 −4.59 1.38 1.72 −4.63 1.38 100.00=oð0Þ 1.75 −10.27 0.95 1.75 −10.32 0.95 100.00=oð0Þ 0−þ 1.55 −0.22 4.98 1.55 −0.35 4.48 99.95=0.05 1.59 −3.91 1.57 1.59 −4.28 1.51 99.89=0.11 1.62 −9.75 1.03 1.62 −10.26 1.01 99.87=0.13

PC Λ Λ 3 3 5 J ErRMS ErRMS Pð S1= D1= D1Þ 1−þ 1.83 −0.30 4.63 1.82 −0.23 4.95 99.96=0.04=oð0Þ 1.89 −3.84 1.59 1.88 −3.54 1.66 99.89=0.11=oð0Þ 1.94 −10.06 1.03 1.93 −9.53 1.05 99.86=0.13=0.01 1−− 2.00 −0.32 4.51 1.99 −0.23 4.90 100.00=oð0Þ=oð0Þ 2.07 −4.20 1.47 2.06 −4.01 1.51 99.99=oð0Þ=0.01 2.13 −10.82 0.95 2.12 −10.71 0.95 99.97=0.01=0.02

PC Λ Λ 5 1 3 5 J ErRMS ErRMS Pð S2= D2= D2= D2Þ 2−− 3.313 −0.33 4.46 3.119 −0.29 4.67 99.55=0.40=oð0Þ=0.05 3.316 −5.60 1.23 3.125 −3.81 1.55 98.28=1.59=oð0Þ=0.13 3.318 −13.00 0.78 3.130 −10.75 0.92 97.03=2.81=oð0Þ=0.16 2−þ 2.96 −0.29 4.84 2.86 −0.31 4.76 99.97=0.02=oð0Þ=0.01 3.20 −3.27 1.87 3.04 −3.07 1.92 99.85=0.09=oð0Þ=0.06 3.43 −10.00 1.17 3.22 −10.20 1.15 99.59=0.21=oð0Þ=0.20

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¯ TABLE VI. Bound state solutions for the S-wave Ds Ds2 system. Conventions are the same as Table III. PC Λ Λ 3S 3D 5D 7D J ErRMS ErRMS Pð 1= 1= 1= 1Þ 1−− 1.70 −0.60 3.65 1.69 −0.50 3.90 99.97=0.02=oð0Þ=0.01 1.74 −4.15 1.49 1.73 −4.01 1.51 99.94=0.05=oð0Þ=0.01 1.78 −10.68 0.96 1.77 −10.69 0.96 99.91=0.08=oð0Þ=0.01 1−þ 1.61 −0.56 3.76 1.60 −0.28 4.67 99.99=oð0Þ=oð0Þ=0.01 1.65 −5.12 1.36 1.64 −4.28 1.48 99.96=0.01=oð0Þ=0.03 1.68 −11.56 0.93 1.67 −10.37 0.98 99.94=0.02=oð0Þ=0.04

PC Λ Λ 5 3 5 7 J ErRMS ErRMS Pð S2= D2= D2= D2Þ 2−þ 2.67 −0.28 4.69 2.63 −0.26 4.77 99.99=oð0Þ=0.01=oð0Þ 2.86 −3.58 1.60 2.80 −3.68 1.58 99.96=oð0Þ=0.04=oð0Þ 3.05 −10.08 0.98 2.96 −10.43 0.96 99.91=oð0Þ=0.09=oð0Þ 2−− 1.91 −0.47 4.06 1.90 −0.26 4.82 99.99=oð0Þ=0.01=oð0Þ 1.96 −3.57 1.65 1.96 −3.81 1.60 99.97=oð0Þ=0.03=oð0Þ 2.01 −10.20 1.01 2.01 −10.62 1.00 99.96=oð0Þ=0.04=oð0Þ

PC 7 3 5 7 J Λ ErRMS Λ ErRMS Pð S3= D3= D3= D3Þ 3−− 4.18 −0.28 4.87 3.87 −0.29 4.80 99.96=0.03=oð0Þ=0.01 4.59 −2.56 2.07 4.04 −1.97 2.32 99.83=0.13=oð0Þ=0.04 5.00 −6.36 1.40 4.20 −9.89 1.15 98.87=0.97=oð0Þ=0.16

In this work, we need to focus on the q4 correction (1) When tuning the cutoff parameters Λ from 1 GeV to terms in the effective potentials, the expressions of these 5 GeV, we can obtain the loosely bound solutions for ¯ PC −− −þ −− −þ correction terms are a little tricky (see Eq. (B5) of the the Ds Ds1 states with J ¼ 0 ; 0 ; 1 ; 1 ; 2−− 2−þ ¯ PC 1−− Appendix B for more details). Through our numerical ; and the DsDs2 states with J ¼ ; ¯ −þ −− −þ −− ¯ analysis for the DsDs2 system, we can obtain an inequality 1 ; 2 ; 2 ; 3 , where the S-wave Ds Ds1 and Λ PC 2−− Λ PC 2−þ ¯ ðJ ¼ Þ > ðJ ¼ Þ when taking the same Ds Ds2 states with lower spin can be bound more binding energy. This difference is caused by the q4 tightly; correction terms in the effective potentials. Usually, a (2) If strictly considering this criterion of the cutoff value loosely bound state with smaller cutoff parameter corre- Λ around 1 GeV [51,52,58], our results disfavor the sponds to the more attractive interaction, which means that existence of the hidden-charm and hidden-strange the q4 correction terms in the effective potentials are molecular candidates for the S-wave ¯ ðÞ ¯ ¯ favorable for forming the DsDs2 molecular state with Ds Ds1ðDs2Þ states; JPC ¼ 2−þ. In contrast, these correction terms are unfav- (3) If the cutoff Λ smaller than 1.70 GeV is a reasonable ¯ PC 2−− 2 ¯ orable for forming the DsDs2 state with J ¼ . Based input parameter, our results suggest that the Ds Ds1 4 PC 0−− 0−þ ¯ on the analysis mentioned above, it is clear that the q states with J ¼ ; and the DsDs2 states correction terms in the interactions play an important with JPC ¼ 1−−; 1−þ are good candidates of the role to modify the behavior of the loosely bound state in hidden-charm and hidden-strange molecular states. ¯ some cases. Here, we need to emphasize that the DsDs1 state ¯ ¯ PC 0−− ¯ PC Besides the S-wave DsDs1ðDs2Þ systems, we also with J ¼ and the Ds Ds2 state with J ¼ −þ ¯ investigate the bound state properties of the S-wave 1 are very different from the DsDs1 state with ¯ ¯ − −− D D 1 D systems in the current work. For the PC 0 þ ¯ PC 1 s s ð s2Þ J ¼ and the DsDs2 state with J ¼ , ¯ ¯ η ϕ Ds Ds1ðDs2Þ systems, we notice that the and exchange since they have the exotic spin-parity quantum contributions to the direct and crossed diagrams are also allowed, and their interactions are simultaneously associ- ated with the total angular momentum J and the charge 2In this work, we take a Λ range from 1 GeV to 5 GeV when parity C. finding the bound state solution of these discussed systems. ¯ According to the experience of studying deuteron, the cutoff From the numerical results for the S-wave Ds Ds1 and ¯ value should not be far away from 1 GeV. Thus, in this work, we Ds Ds2 systems with and without considering the S-D wave take such so-called criteria to make discussion of possible mixing effect, we can find several interesting results: molecular states.

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numbers apparently different from the conventional We may predict the existence of four possible hidden- hadrons. charm and hidden-strange molecular states, which are the ¯ PC 0−− 0−þ ¯ (4) The S-D wave mixing effect plays a minor role in Ds Ds1 states with J ¼ ; and the Ds Ds2 states ¯ ¯ PC −− −þ generating the S-wave DsDs1 and DsDs2 bound with J ¼ 1 ; 1 . These predictions can be accessible ¯ states in many cases. at future experiment. Moreover, the Ds Ds1 state with PC 0−− ¯ PC 1−þ J ¼ and the Ds Ds2 state with J ¼ have 2. The coupled-channel analysis the typical exotic spin-parity quantum numbers, which ¯ can be distinguished with conventional mesons. In this subsection, we discuss the HsTs systems with J ¼ 1, 2 by considering the coupled-channel effect. In the B. The open-charm and open-strange coupled-channel approach, we need to emphasize that the molecular systems binding energy E is determined by the lowest mass threshold among various involved channels [71]. The In the above subsection, we mainly discussed the bound ¯ state properties of the S-wave H T¯ systems, which relevant numerical results for the HsTs coupled systems s s with JPC ¼ 1−−; 1−þ; 2−−; 2−þ for the different lowest stimulates our interest to investigate the behavior of the mass thresholds are given in Table VII. open-charm and open-strange molecular systems com- By comparing the numerical results of the single channel posed by a charmed-strange meson in H-doublet and a and coupled-channel cases, it is obvious that the bound charmed-strange meson in T-doublet, which have typical state properties will change accordingly after considering exotic state configurations totally different from the con- the coupled-channel effect, i.e., the cutoff parameters in the ventional hadrons. coupled-channel analysis are smaller than that in the single channel analysis with the same binding energy in many 1. The single channel analysis ¯ PC −− cases, especially for the DsDs1 state with J ¼ 1 . In the following, we try to search for the bound state However, our result indicates that the coupled-channel solutions for the S-wave H T systems by solving the ¯ s s effect to the S-wave HsTs systems is not obvious. coupled-channel Schrödinger equation. When considering

¯ TABLE VII. Bound state solutions for the S-wave HsTs coupled systems. Conventions are the same as Table III. Here, “Threshold” 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.

PC Λ ¯ ¯ ¯ ¯ J Threshold ErRMS PðDsDs1=DsDs2=Ds Ds1=Ds Ds2Þ 1−− ¯ −0 32 91 19 0 8 18 DsDs1 1.92 . 4.42 . = =oð Þ= . 1.93 −3.38 1.47 72.37= =oð0Þ=27.63 1.94 −9.24 0.83 57.50= =oð0Þ=42.50 ¯ −2 17 0 100 00 Ds Ds1 1.86 . 0.56 = =oð Þ= . 1.87 −6.01 0.54 = =oð0Þ=100.00 1.88 −10.06 0.52 = =oð0Þ=100.00 1−þ ¯ −0 50 69 27 0 30 73 DsDs1 1.87 . 3.36 . = =oð Þ= . 1.88 −8.88 0.61 25.86= =oð0Þ=74.14 1.89 −20.56 0.39 16.13= =oð0Þ=83.87 ¯ −0 43 0 100 00 Ds Ds1 1.74 . 0.58 = =oð Þ= . 1.75 −5.27 0.55 = =oð0Þ=100.00 1.76 −10.46 0.52 = =oð0Þ=100.00 2−− ¯ −3 24 0 0 100 00 DsDs2 2.27 . 0.28 =oð Þ=oð Þ= . 2.28 −13.95 0.27 =oð0Þ=oð0Þ=100.00 2.29 −25.24 0.26 =oð0Þ=oð0Þ=100.00 ¯ −3 89 0 100 00 Ds Ds1 2.11 . 0.54 = =oð Þ= . 2.12 −8.00 0.51 = =oð0Þ=100.00 2.13 −12.41 0.49 = =oð0Þ=100.00 2−þ ¯ −0 33 99 94 0 06 0 DsDs2 2.40 . 4.61 = . = . =oð Þ 2.49 −3.47 1.72 =99.81=0.19=oð0Þ 2.57 −10.76 1.01 =99.64=0.36=oð0Þ ¯ −0 31 100 00 0 Ds Ds1 2.86 . 4.74 = = . =oð Þ 3.04 −3.07 1.90 = =100.00=oð0Þ 3.22 −10.18 1.13 = =100.00=oð0Þ

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TABLE VIII. Bound state solutions for the S-wave HsTs the corresponding cutoff is deviated from the usual value systems. Conventions are the same as Table III. around 1 GeV for these investigated open-charm and open-

P − strange systems [51,52,58]. It means that S-wave bound Ds Ds1ðJ ¼ 2 Þ states for the HsTs systems do not exist even the coupled- Λ Λ 5S 1D 3D 5D ErRMS ErRMS Pð 2= 2= 2= 2Þ channel effect is included. 4.69 −1.73 1.90 3.32 −0.64 3.49 97.11=0.13=oð0Þ=2.76 In short summary, since the attractive interactions 4.70 −5.74 0.98 3.36 −4.90 1.33 93.64=0.27=oð0Þ=6.09 between a charmed-strange meson in H-doublet and a 4.71 −10.84 0.70 3.39 −10.27 0.93 91.95=0.33=oð0Þ=7.72 charmed-strange meson in T-doublet are not strong enough to make the hadronic component bound together when P 2− taking reasonable input parameters. Thus, we conclude that Ds Ds2ðJ ¼ Þ the quantitative analysis does not support the existence of Λ Er Λ Er 5S 3D 5D 7D RMS RMS Pð 2= 2= 2= 2Þ the open-charm and open-strange tetraquark molecular × × × 4.66 −0.25 4.85 98.21=0.49=0.98=0.33 ðÞ candidates with the S-wave Ds D 1ðD Þ systems, such × × × 4.76 −3.52 1.62 94.49=1.44=3.06=1.02 s s2 qualitative conclusion can be further tested in future × × × 4.86 −10.32 0.99 91.66=2.09=4.69=1.56 experiments and other theoretical approaches. P 3− Ds Ds2ðJ ¼ Þ 7 3 5 7 IV. SUMMARY Λ ErRMS Λ ErRMS Pð S3= D3= D3= D3Þ Exploring the exotic hadronic states has become an 4.20 −2.59 1.53 3.14 −0.65 3.46 96.94=0.09=0.31=3.86 intriguing research issue full of challenges and opportu- 4.21 −6.99 0.90 3.17 −3.89 1.48 93.83=0.13=0.44=5.60 4.22 −12.40 0.66 3.20 −9.38 0.97 91.80=0.16=0.58=7.47 nities, which has been inspired by the abundant observa- tions of a series of X=Y=Z=Pc states since 2003 [2,5]. Among different exotic hadronic configurations, hadronic molecular state has aroused heated discussion. By the the S-D wave mixing effect and scanning the cutoff efforts from both theorist and experimentalist, the proper- Λ parameters from 1 GeV to 5 GeV, we list these typical ties of hidden-charm molecular states with masses below Λ values of the cutoff parameter , the binding energy E, the 4.5 GeV, which are from the interaction between charmed root-mean-square radius rRMS, and the probability of the meson and anti-charmed meson, become more and more individual channel Pi for the S-wave HsTs systems in clearer. Table VIII. According to the theoretical results of studying Recently, the observation of XYZ charmoniumlike states Λ the deuteron via the OBE model, the cutoff parameter in experiment collaborations [44–47] and the announced should be around 1 GeV [51,52,58]. If taking this criterion, white paper on the future physics program by the BESIII our results disfavor the existence of the open-charm and Collaboration [48] show that hunting new XYZ charmo- ðÞ open-strange molecular states for the S-wave Ds Ds1ðDs2Þ niumlike states with mass above 4.5 GeV becomes pos- systems since the cutoff parameter Λ is obviously far away sible. Considering the close relation of XYZ states with from 1 GeV. hidden-charm hadronic molecular states, we propose to perform theoretical study of hidden-charm hadronic molecular states composed of the charmed-strange meson 2. The coupled-channel analysis in H-doublet and the anticharmed-strange meson in T- ¯ Similar to the S-wave HsTs systems, we also present the doublet. The discussed hidden-charm and hidden-strange bound state properties for the S-wave HsTs coupled- hadronic molecular states just exist in the mass range channel systems in Table IX. Nevertheless, it is clear that around 4.5 ∼ 4.7 GeV. We predict the existence of the coupled-channel effect plays a positive but minor effect ¯ PC −− −þ the DsDs1 molecular states with J ¼ 0 ; 0 and the ðÞ ¯ PC 1−− 1−þ to generate these S-wave Ds Ds1ðDs2Þ bound states, where Ds Ds2 molecular states with J ¼ ; . Here, the

TABLE IX. Bound state solutions for the S-wave HsTs coupled-channel systems. Conventions are the same as Table VII.

P Λ J Threshold ErRMS PðDsDs1=DsDs2=Ds Ds1=Ds Ds2Þ 2− −1 75 60 58 34 02 5 40 DsDs2 2.92 . 1.62 = . = . = . 2.93 −7.86 0.67 =47.20=45.48=7.32 2.94 −15.15 0.47 =42.22=49.65=8.13 Ds Ds1 3.25 −0.30 4.51 = =99.88=0.12 3.29 −4.45 1.36 = =99.58=0.42 3.32 −10.19 0.91 = =99.37=0.63

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¯ PC 0−− ¯ Ds Ds1 molecular state with J ¼ and the Ds Ds2 the predicted vector hidden-charm and hidden-strange PC 1−þ molecular state by the accumulated data from eþe− molecular state with J ¼ have the exotic spin-parity pffiffiffi quantum numbers, which can be totally distinguished with collision with s > 4.5 GeV. And, the remaining three the conventional meson states. molecular states with JPC ¼ 0−−; 0−þ; 1−þ predicted in this We also extend our theoretical framework to study open- work can be accessible at the LHCb and BelleII by charm and open-strange molecular systems, which have decays. Additionally, theoretical study of this new type of components, a charmed-strange meson in H-doublet and a hadronic molecular state by other approaches is also charmed-strange meson in T-doublet. However, our cal- encouraged. We believe that these investigations will make culation does not support the existence of such hadronic our knowledge of exotic hadronic molecular states become molecular states. more abundant. The possible hidden-charm hadronic molecular states with hidden strangeness can be searched in their possible ACKNOWLEDGMENTS two-body strong decay channels, where they can decay into This work is supported by the China National Funds for a charmonium plus a ϕ meson, and a charmed-strange Distinguished Young Scientists under Grant No. 11825503 meson plus an anticharmed-strange meson if kinetically and by the National Program for Support of Top-notch allowed, i.e., ηcð1SÞϕ, J=ψϕ, χcJð1PÞϕðJ ¼ 0; 1; 2Þ, ¯ ¯ ¯ Young Professionals. DsDs, DsDs, Ds Ds , and so on. These decay modes can provide valuable information when searching for possible hidden-charm hadronic molecular states with hidden APPENDIX A: THE EFFECTIVE LAGRANGIANS strangeness experimentally. The expanded effective Lagrangians for depicting the Considering the experimental potential, we strongly interactions of the (anti-)charmed mesons with the light suggest that the BESIII Collaboration should focus on mesons are expressed as

2ig μ 2g μ μ† 2ig μ† L − αε λ†∂νP − † ∂ P αε ¯ ¯ λ∂νP HHP ¼ v αμνλDb Da ba ðDb Da þ DbDa Þ μ ba þ v αμνλDa Db ab fπ fπ fπ 2 g ¯ μ† ¯ ¯ † ¯ μ ∂ P þ ðDa Db þ DaDb Þ μ ab; ðA1Þ fπ pffiffiffi pffiffiffi pffiffiffi L − 2β † V 2β μ† V − 2 2 λ μ ν† ∂ V − ∂ V HHV ¼ gVDbDav · ba þ gVDbμDa v · ba i gV Db Da ð μ ν ν μÞba pffiffiffi pffiffiffi pffiffiffi − 2 2λ λε μ† μ † ∂αV β 2β ¯ ¯ † V − 2β ¯ ¯ μ† V gVv λμαβðDbDa þ Db DaÞ ba þ gVDaDbv · ab gVDaμDb v · ab pffiffiffi pffiffiffi − 2 2 λ ¯ μ† ¯ ν ∂ V − ∂ V − 2 2λ λε ¯ μ† ¯ ¯ † ¯ μ ∂αV β i gVDa Db ð μ ν ν μÞab gVv λμαβðDa Db þ DaDb Þ ab; ðA2Þ rffiffiffi 5ik 2ik 2 k μλ μλ† L − ϵμνρτ † ∂ P ϵμνρτ α† ∂ P − † ∂ P TTP ¼ vτD1bμD1aν ρ ba þ vνD2bρD2aατ μ ba ðD2b D1aμ þ D1bμD2a Þ λ ba 3fπ fπ 3 fπ rffiffiffi 5ik 2ik 2 k μλ μλ† − εμνρτ ¯ † ¯ ∂ P εμνρτ ¯ α† ¯ ∂ P ¯ ¯ † ¯ ¯ ∂ P vνD1aρD1bτ μ ab þ vνD2aρD2bατ μ ab þ ðD2a D1bμ þ D1aμD2b Þ λ ab; ðA3Þ 3fπ fπ 3 fπ pffiffiffi pffiffiffi 5 2 λ00 pffiffiffi 00 μ† i gV ν† μ ν μ† 00 λν † L V − 2β g v V D1 μD D D − D D ∂μV ν 2β g v V D D TT ¼ Vð · baÞ b 1a þ 3 ð 1b 1a 1b 1aÞ ba þ Vð · baÞ 2b 2aλν pffiffiffi β00 00 λν μ† λν† μ i gV λαρτ † † þ 2 2iλ g × ðD D − D D Þ∂μV ν þ pffiffiffi ϵ vρðv · V ÞðD D − D1 αD Þ V 2b 2aλ 2a 2bλ ba 3 ba 1bα 2aλτ b 2aλτ 2λ00 gV μλντ α† α † λαρν † μ †μ þ pffiffiffi ½3ϵ vλðD D þ D D Þ∂μV ν þ 2ϵ vρðD D þ D1 αD Þ × ð∂μV ν − ∂νV μÞ 3 1a 2bατ 1b 2aατ ba 1bα 2aλ b 2aλ ba pffiffiffi pffiffiffi 5 2 λ00 pffiffiffi 00 ¯ ¯ μ† i gV ¯ ν ¯ μ† ¯ ν† ¯ μ 00 ¯ λν ¯ † 2β g v V D1 μD D D − D D ∂μV ν − 2β g v V D D þ Vð · abÞ a 1b þ 3 ð 1a 1b 1a 1bÞ ab Vð · abÞ 2a 2bλν pffiffiffi β00 00 ¯ λν† ¯ μ ¯ λν ¯ μ† i gV λαρτ ¯ † ¯ ¯ ¯ † þ 2 2iλ g ðD D − D D Þ∂μV ν þ pffiffiffi ε vρðv · V ÞðD D − D1 αD Þ V 2a 2bλ 2a 2bλ ab 3 ba 1aα 2bλτ a 2bλτ 2λ00 gV μλντ ¯ α† ¯ ¯ α ¯ † λαρν ¯ † ¯ μ ¯ ¯ †μ þ pffiffiffi ½3ε vλðD D þ D D Þ∂μV νþ2ε vρðD D þ D1 αD Þ × ð∂μV ν − ∂νV μÞ ; ðA4Þ 3 1a 2bατ 1a 2bατ ab 1aα 2bλ a 2bλ ab

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2 0 L − h μν† μν † ∂ ∂ P HTP ¼ ðDbD2a þ D2b DaÞ μ ν ba fπ pffiffiffi 0 2h μ μ† − ffiffiffi 3 ν† ν ∂ ∂ P λ† λ † ∂ ∂ν − μ ν∂ ∂ P p ½ ðDb D1a þ D1bDa Þ μ ν baþðDλbD1a þ D1bDλaÞð ν v v μ νÞ ba 3fπ 0 0 2ih μ† μ 2h μν† μν − ελνρτ − † ∂ ∂ P ¯ ¯ ¯ ¯ † ∂ ∂ P vλðDbτD2aρ D2bρDaτÞ μ ν ba þ ðDaD2b þ D2a DbÞ μ ν ab fπ fπ pffiffiffi 0 2h μ μ† ffiffiffi 3 ¯ ¯ ν† ¯ ν ¯ ∂ ∂ P ¯ ¯ λ† ¯ λ ¯ † ∂ ∂ν − μ ν∂ ∂ P þ p ½ ðDa D1b þ D1aDb Þ μ ν abþðDλaD1b þ D1aDλbÞð ν v v μ νÞ ab 3fπ 0 2ih μ† μ − ελνρτ ¯ ¯ − ¯ † ¯ ∂ ∂ P vλðD2aτ Dbν DaνD2bτÞ μ ρ ab; ðA5Þ fπ

2 † † μ 2i ν† † μ L V ¼ − pffiffiffi ζ1g ðD D þ D1 μD ÞV þ pffiffiffi μ1g ½ðD D − D1 νD Þv ð∂μV ν − ∂νV μÞ HT 3 V b 1aμ b a ba 3 V b 1a b a ba μ i μνρτ † † 1gV αβρτ μ † † þ pffiffiffi ζ1g ε vνðD D τ − D D1 μÞV ρ þ pffiffiffi ½ε vβv ðD D τ þ D D1 αÞ∂μV ρ 3 V 1bμ a bτ a ba 3 1bα a bτ a ba pffiffiffi −εμβλτ ρ † † ∂ V 2ζ † μ μ† V ν vλv ðD1bβDaτ þ DbτD1aβÞ μ baρþ 1gVðD2bμνDa þ Db D2aμνÞ ba pffiffiffi 2 μ † λ − λ† λ† − λ † ∂ V ν þ i 1gV½vνðD2bλμDa Db D2aλμÞþvμðD2bλνDa Db D2aλνÞ μ ba 2 2 ¯ ¯ † ¯ ¯ † μ i ¯ ¯ ν† ¯ ¯ † μ þ pffiffiffi ζ1g ðD D þ D1 μD ÞV − pffiffiffi μ1g ½ðD D − D1 νD Þv ð∂μV ν − ∂νV μÞ 3 V a 1bμ a b ab 3 V a 1b a b ab μ i μνρτ ¯ † ¯ ¯ † ¯ 1gV αβρτ μ ¯ † ¯ ¯ † ¯ þ pffiffiffi ζ1g ε vμðD D − D τD1 ρÞV ν þ pffiffiffi ½ε vαv ðD D þ D ρD1 βÞ∂μV τ 3 V 1aρ bτ a b ab 3 1aβ bρ a b ab pffiffiffi εμβλρ τ ¯ † ¯ ¯ † ¯ ∂ V − 2ζ ¯ † ¯ μ ¯ μ† ¯ V ν þ vβv ðD1aλDbρ þ DaρD1bλÞ μ abτ 1gVðD2aμνDb þ Da D2bμνÞ ab pffiffiffi − 2 μ ¯ † ¯ λ − ¯ λ† ¯ ¯ ¯ λ† − ¯ λ ¯ † ∂ V ν i 1gV½vνðD2aλμDb Da D2bλμÞþvμðD2aλνDb Da D2bλνÞ μ ab: ðA6Þ

APPENDIX B: THE DETAILS OF THESE OBTAINED EFFECTIVE POTENTIALS Before presenting the effective potentials, we first focus on several typical Fourier transforms, i.e.,     1 Λ2 − m2 2 F ¼ YðΛ;m;rÞ; ðB1Þ q2 þ m2 Λ2 þ q2     q2 Λ2 − m2 2 F ¼ −ZYðΛ;m;rÞ; ðB2Þ q2 þ m2 Λ2 þ q2     ða · qÞðb · qÞ Λ2 − m2 2 1 1 F ¼ − ða · bÞZYðΛ;m;rÞ − Tða; bÞT YðΛ;m;rÞ; ðB3Þ q2 þ m2 Λ2 þ q2 3 3     ða × qÞ · ðb × qÞ Λ2 − m2 2 2 1 F ¼ − ða · bÞZYðΛ;m;rÞþ Tða; bÞT YðΛ;m;rÞ; ðB4Þ q2 þ m2 Λ2 þ q2 3 3     ða · qÞðb · qÞðc · qÞðd · qÞ Λ2 − m2 2 1 F ¼ ½ða · bÞðc · dÞþða · cÞðb · dÞþða · dÞðb · cÞZZYðΛ;m;rÞ q2 þ m2 Λ2 þ q2 27 1 TT Λ þ 27 ½Tða; bÞTðc; dÞþTða; cÞTðb; dÞþTða; dÞTðb; cÞ Yð ;m;rÞ 1 þ 54 ½ða · bÞTðc; dÞþða · cÞTðb; dÞþða · dÞTðb; cÞþðc · dÞTða; bÞ þðb · dÞTða; cÞþðb · cÞTða; dÞfT ; ZgYðΛ;m;rÞ: ðB5Þ

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Z 1 ∂ 2 ∂ T ∂ 1 ∂ T Z TZ ZT In the above expressions, the operators are defined as ¼ r2 ∂r r ∂r, ¼ r ∂r r ∂r, f ; g¼ þ , and Tða; bÞ¼3ðrˆ · aÞðrˆ · bÞ − a · b. Through the above preparation, we can write out the effective potentials in the coordinate space for all of the investigated systems, which include ¯ ¯ (i) DsDs1 → DsDs1 process:

A A0 V C 1 þ 1 D ¼ 2 2 Yϕ; ðB6Þ

A A0 V E 2 þ 2 C ¼ 3 2 Yϕ0: ðB7Þ

¯ → ¯ (ii) DsDs2 DsDs2 process:

A A0 V C 3 þ 3 D ¼ 2 2 Yϕ; ðB8Þ

 2 A A0 A A0 A A0 V B 4 þ 4 ZZ 5 þ 5 TT 6 þ 6 T Z C ¼ 3 2 þ 2 þ 2 f ; g Yη1: ðB9Þ

¯ ¯ (iii) DsDs1 → DsDs1 process:  5 5 V A A Z A T C A D A T − 2A Z D ¼ 27 ½ 8 þ 9 Yη þ 2 7 þ 9 ð 9 8 Þ Yϕ; ðB10Þ

V B A ZZ A TT A T Z E A C ¼ 9 ½ 10 þ 11 þ 12f ; gYη2 þ 12 8Yϕ2: ðB11Þ

¯ → ¯ (iv) DsDs2 DsDs2 process:  2 A A0 A A0 V A 14 þ 14 Z 15 þ 15 T D ¼ 9 2 þ 2 Yη    2 A A0 A A0 A A0 D 15 þ 15 T − 2 14 þ 14 Z C 13 þ 13 þ 3 2 2 þ 2 2 Yϕ; ðB12Þ

 2 A A0 A A0 A A0 A A0 V B 16 þ 16 ZZ 17 þ 17 TT 18 þ 18 T Z E 19 þ 19 C ¼ 3 2 þ 2 þ 2 f ; g Yη3 þ 2 2 Yϕ3: ðB13Þ

2 02 2 ββ00 2 λλ00 2 ζ2 2 Λ Here, A ¼ gk=fπ, B ¼ h =fπ, C ¼ gV, D ¼ gV, and E ¼ 1gV. In the above expressions, the function Yð i;mi;rÞ reads as

− −Λ 2 2 e mir − e ir Λ − m i i −Λir Yi ≡ YðΛi;mi;rÞ¼ − e ðB14Þ 4πr 8πΛi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 with m ¼ m − q and Λ ¼ Λ − q . The variables q are defined as q0 ¼ m − m , q1 ¼ m − m , i i i i i Ds1 Ds Ds2 Ds q2 ¼ m − m , and q3 ¼ m − m . In addition, we introduce several operators, which include Ds1 Ds Ds2 Ds

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† 0 † † 0 † A1 ¼ ϵ · ϵ ; A ¼ ϵ · ϵ ; A2 ¼ ϵ · ϵ ; A ¼ ϵ · ϵ ; X4 2 1 3 1 X3 2 2 4 1 A ϵ† ϵ ϵ† ϵ A0 ϵ† ϵ ϵ† ϵ 3 ¼ ð 4m · 2aÞð 4n · 2bÞ; 3 ¼ ð 3m · 1aÞð 3n · 1bÞ; 2 X 2 X A ϵ† ϵ ϵ† ϵ ; A0 ϵ† ϵ ϵ† ϵ ; 4 ¼ 27 ð 3m · 2aÞð 3n · 2bÞ 4 ¼ 27 ð 4m · 1aÞð 4n · 1bÞ X X 1 † † 2 † † A5 T ϵ ; ϵ T ϵ ; ϵ T ϵ ; ϵ T ϵ ; ϵ ; ¼ 27 ð 3m 3nÞ ð 2a 2bÞþ27 ð 3m 2aÞ ð 3n 2bÞ 1 X 2 X A0 T ϵ† ; ϵ† T ϵ ; ϵ T ϵ† ; ϵ T ϵ† ; ϵ ; 5 ¼ 27 ð 4m 4nÞ ð 1a 1bÞþ27 ð 4m 1aÞ ð 4n 1bÞ 2 X 2 X † † 0 † † A6 ϵ ϵ T ϵ ; ϵ ; A ϵ ϵ T ϵ ; ϵ ; ¼ 27 ð 3m · 2aÞ ð 3n 2bÞ 6 ¼ 27 ð 4m · 1aÞ ð 4n 1bÞ A ϵ† ϵ ϵ† ϵ A ϵ† ϵ ϵ† ϵ A ϵ† ϵ ϵ† ϵ 7 ¼ð 3 · 1Þð 4 · 2Þ; 8 ¼ð 3 × 1Þ · ð 4 × 2Þ; 9 ¼ Tð 3 × 1; 4 × 2Þ; 1 † † 1 † † 2 † † 1 † † A10 − ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ; A11 T ϵ ; ϵ T ϵ ; ϵ T ϵ ; ϵ T ϵ ; ϵ ; ¼ 3 ð 3 · 1Þð 4 · 2Þþ3 ð 3 · 4Þð 1 · 2Þ ¼ 3 ð 3 1Þ ð 4 2Þþ3 ð 3 4Þ ð 1 2Þ 1 † † 1 † † 1 † † A12 ϵ ϵ T ϵ ; ϵ ϵ ϵ T ϵ ; ϵ − ϵ ϵ T ϵ ; ϵ ; ¼ 6 ð 3 · 4Þ ð 1 2Þþ6 ð 1 · 2Þ ð 3 4Þ 3 ð 3 · 1Þ ð 4 2Þ X X † † † 0 † † † A13 ¼ ðϵ · ϵ Þðϵ · ϵ Þðϵ · ϵ Þ; A13 ¼ ðϵ · ϵ Þðϵ · ϵ Þðϵ · ϵ Þ; X 3 1 4m 2a 4n 2b X4 2 3m 1a 3n 1b † † † 0 † † † A14 ¼ ðϵ · ϵ Þ½ðϵ × ϵ Þ · ðϵ × ϵ Þ; A14 ¼ ðϵ · ϵ Þ½ðϵ × ϵ Þ · ðϵ × ϵ Þ; X 4m 2a 3 1 4n 2b X 3m 1a 4 2 3n 1b A ϵ† ϵ ϵ† ϵ ϵ† ϵ A0 ϵ† ϵ ϵ† ϵ ϵ† ϵ 15 ¼ ð 4m · 2aÞTð 3 × 1; 4n × 2bÞ; 15 ¼ ð 3m · 1aÞTð 4 × 2; 3n × 1bÞ; X X 1 † † † 1 † † † A16 ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ; ¼ 27 ½ð 3m × 1Þ · ð 4 × 2aÞð 3n · 2bÞþ27 ½ð 3m × 1Þ · 2b½ 3n · ð 4 × 2aÞ X X 1 † † † 1 † † † A0 ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ ; 16 ¼ 27 ½ð 4m × 2Þ · ð 3 × 1aÞð 4n · 1bÞþ27 ½ð 4m × 2Þ · 1b½ 4n · ð 3 × 1aÞ X X 1 † † † 1 † † † A17 T ϵ ϵ ; ϵ T ϵ ϵ ; ϵ T ϵ ϵ ; ϵ ϵ T ϵ ; ϵ ¼ 27 ð 3m × 1 3nÞ ð 4 × 2a 2bÞþ27 ð 3m × 1 4 × 2aÞ ð 3n 2bÞ X 1 † † † T ϵ ϵ ; ϵ T ϵ ; ϵ ϵ ; þ 27 ð 3m × 1 2bÞ ð 3n 4 × 2aÞ X X 1 † † † 1 † † † A0 T ϵ ϵ ; ϵ T ϵ ϵ ; ϵ T ϵ ϵ ; ϵ ϵ T ϵ ; ϵ 17 ¼ 27 ð 4m × 2 4nÞ ð 3 × 1a 1bÞþ27 ð 4m × 2 3 × 1aÞ ð 4n 1bÞ X 1 † † † T ϵ ϵ ; ϵ T ϵ ; ϵ ϵ ; þ 27 ð 4m × 2 1bÞ ð 4n 3 × 1aÞ X X 1 † † † 1 † † † A18 ϵ ϵ ϵ ϵ T ϵ ; ϵ ϵ ϵ ϵ T ϵ ; ϵ ϵ ¼ 54 ½ð 3m × 1Þ · ð 4 × 2aÞ ð 3n 2bÞþ54 ½ð 3m × 1Þ · 2b ð 3n 4 × 2aÞ 1 X 1 X ϵ† ϵ T ϵ† ϵ ; ϵ† ϵ ϵ† ϵ† ϵ T ϵ† ϵ ; ϵ ; þ 54 ð 3n · 2bÞ ð 3m × 1 4 × 2aÞþ54 ½ 3n · ð 4 × 2aÞ ð 3m × 1 2bÞ 1 X 1 X A0 ϵ† ϵ ϵ† ϵ T ϵ† ; ϵ ϵ† ϵ ϵ T ϵ† ; ϵ† ϵ 18 ¼ 54 ½ð 4m × 2Þ · ð 3 × 1aÞ ð 4n 1bÞþ54 ½ð 4m × 2Þ · 1b ð 4n 3 × 1aÞ X X 1 † † † 1 † † † ϵ ϵ T ϵ ϵ ; ϵ ϵ ϵ ϵ ϵ T ϵ ϵ ; ϵ ; þ 54 ð 4n · 1bÞ ð 4m × 2 3 × 1aÞþ54 ½ 4n · ð 3 × 1aÞ ð 4m × 2 1bÞ X X A ϵ† ϵ ϵ† ϵ ϵ† ϵ A0 ϵ† ϵ ϵ† ϵ ϵ† ϵ 19 ¼ ð 3m · 1Þð 4 · 2aÞð 3n · 2bÞ; 19 ¼ ð 4m · 2Þð 3 · 1aÞð 4n · 1bÞ: ðB15Þ

P P 2;mþn 2;aþb Að0Þ Here, we define ¼ m;n;a;b C1m;1n C1a;1b . For these operators k , they should be sandwiched by the spin-orbital wave 2Sþ1 Að0Þ functions j LJi, we present the relevant operator matrix elements k ½J in Table X.

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Að0Þ 1 … 19 ¯ TABLE X. The relevant operator matrix elements k ½Jðk¼ ; ; Þ for the S-wave HsTs systems. Að0Þ Að0Þ k ½J¼hfj k jii pffiffi ! pffiffi ! ð0Þ 8 4 2 7 Að0Þ 1 1 1 A 1 1 1 − pffiffiffiffi 0 − pffiffiffiffi 1 ½ ¼diagð ; Þ 2 ½ ¼diagð ; Þ ð0Þ 135 27 35 ð0Þ 27 10 A 2 pffiffi A 2 pffiffi ð0Þ ð0Þ 2 2 5 ½ ¼ 4 2 4 6 ½ ¼ 7 1 A ½2¼diagð1;1Þ A ½2¼diagð ; Þ − pffiffiffiffi − pffiffiffiffi − 3 4 27 27 27 35 27! 27 10 126ffiffi ! pffiffiffi p A7½0¼diagð1;1Þ A8½0¼diagð2;−1Þ 4 −2 2 0ffiffiffi 2 3 ffiffi 3 A 0 2 −1 A9½0¼ p A11½0¼ p 10½ ¼diagð3; 3Þ 2 2 2 2 − 3 4 pffiffi ! pffiffiffi ! 2 A 1 1 1 1 0 7½ ¼diagð ; ; Þ 0 ffiffiffi − 2 0 A 0 pffiffi 15 A 1 1 1 −1 p 12½ ¼ 8 2 1 8½ ¼diagð ; ; Þ A9½1¼ − 2 10 − 15 −15 1 1 1 A10½1¼diagð−3;−3;−3Þ 001 pffiffi 0 pffiffi 1 2 2 2 ! 1 3 − − 0 0 − pffiffi pffiffi A7½2¼diagð1;1;1;1Þ 3 ffiffi 3 30 2 10 2 p B pffiffi C A 2 −1 2 1 −1 A 1 2 2 1 4 3 3 8½ ¼diagð ; ; ; Þ 11½ ¼ − 00 A 1 B− pffiffi C 1 2 1 1 3 12½ ¼@ 30 2 105 70 A A 2 − − − 4 pffiffi pffiffi 10½ ¼diagð 3;3; 3; 3Þ 00−3 3 3 3 1 pffiffi − 0 1 0 10 2 70 105 1 0 1 ffiffi ffiffiffiffi pffiffi pffiffi pffiffi p p 8 2 2 4 2 1 2 ffiffi2 14ffiffi − pffiffi 0 − pffiffiffiffi 0 −pffiffiffiffi 0 pffiffiffiffi 0 p 0 − p 15 3 5 3 35 10 3 35 B ffiffi 5 5 C B pffiffi C B C B p C B 2 2 4 4 C B 4 ffiffi C B ffiffi2 2ffiffi C B − pffiffi 0 pffiffi C B 000− p C p 00−p 3 5 3 3 7 21 7 A9½2¼B 5 7 C A11½2¼B C A12½2¼B 1 1 C B C B 4 C B 00− pffiffi C 00ffiffiffiffi −10 00− 0 14 14 7 @ p A @ pffiffi 3 A @ pffiffi A 14ffiffi 2ffiffi 3 4 2 4 4 2 23 1 13 − p −p 0 −7 − pffiffiffiffi pffiffi 0 pffiffiffiffi pffiffi pffiffi 5 7 3 35 3 7 3 3 35 21 7 14 7 294 0 pffiffi pffiffiffiffi 1 0 pffiffi pffiffiffiffi 1 ð0Þ 3 6 21 3 6 21 A 1 1 1 1 1 0 pffiffi pffiffi pffiffi 0 pffiffi −pffiffi pffiffi 13½ ¼diagð ; ; ; Þ 5 2 5 5 2 5 2 5 5 2 B pffiffi pffiffi C B pffiffi pffiffi C ð0Þ 3 3 1 B 3 3 3 3 C B 3 3 3 3 C A14½1¼diagð ; ; ;−1Þ pffiffi − pffiffi − pffiffi pffiffi − −pffiffi − pffiffi 2 2 2 B5 2 10 5 5 7C B 5 2 10 5 5 7C A 1 B pffiffi pffiffi C A0 1 B pffiffi pffiffi C Að0Þ 1 1 1 5 − 1 15½ ¼ 6 3 1 2 15½ ¼ 6 3 1 2 16½ ¼diagð18;18;54; 27Þ B pffiffi pffiffi pffiffiffiffi C B−pffiffi −pffiffi −pffiffiffiffiC @ 5 5 2 35 A @ 5 5 2 35A ð0Þ 1 1 1 pffiffiffiffi pffiffi pffiffiffiffi pffiffi A 1 1 21 3 2 48 21 3 2 48 19½ ¼diagð6;6;2; Þ pffiffi − pffiffi pffiffiffiffi pffiffi − pffiffi −pffiffiffiffi 5 2 5 7 35 35 5 2 5 7 35 35 0 pffiffi pffiffi 1 0 pffiffi pffiffi 1 0 pffiffi 1 2 2 2 2 2 2 7 7 − 0 pffiffiffiffi − 0 pffiffiffiffi 0 − pffiffi 0 pffiffi 45 45 15 21 45 45 15 21 90 2 30 6 B pffiffi C B pffiffi C B C 2 1 8 2 1 8 B 7 7 1 C B − 0 − pffiffiffiffiC B − 0 − pffiffiffiffiC − pffiffi 0 − pffiffiffiffi B 45 15 15 21C B 45 15 15 21C B 90 2 180 30 21C A 1 B pffiffi C A0 1 B pffiffi C A 1 B pffiffi C 17½ ¼ 1 2 5 17½ ¼ 1 2 5 18½ ¼ 1 5 B 00− − pffiffi C B 00− pffiffi C B 00 pffiffi C @ 9 27 7 A @ 9 27 7 A @ 108 27 7 A pffiffi pffiffi pffiffi pffiffi pffiffi pffiffi 2 8 2 5 92 2 8 2 5 92 7 1 5 4 pffiffiffiffi − pffiffiffiffi pffiffi pffiffiffiffi − pffiffiffiffi − pffiffi pffiffi − pffiffiffiffi − pffiffi 15 21 15 21 27 7 945 15 21 15 21 27 7 945 30 6 30 21 27 7 315 0 pffiffi 1 0 pffiffi pffiffi pffiffi 1 7 7 ð0Þ 3 2 7 7 0 − pffiffi 0 pffiffi A 2 1 1 1 1 0 − −pffiffiffiffi 90 2 30 6 13½ ¼diagð ; ; ; Þ 5 10 5 B C B pffiffi pffiffi C B 7 ffiffi 7 1 ffiffiffiffiC ð0Þ 1 3 1 B 3 2 3 3 3 2C − p 0 − p A14½2¼diagð ; ; ;−1Þ − pffiffiffiffi − pffiffi B 90 2 180 30 21C 2 2 2 B 5 10 35 5 7C A0 1 B pffiffi C A 2 B pffiffi pffiffi C 18½ ¼ 1 5 Að0Þ 2 5 1 5 − 1 15½ ¼ 7 3 3 4 2 B 00 − pffiffi C 16½ ¼diagð54;18;54; 27Þ B−pffiffiffiffi pffiffiffiffi − pffiffi C @ ffiffi 108ffiffi 27 7 A @ 10 35 14 7 5 A p p ð0Þ 1 1 1 pffiffi pffiffi pffiffi 7 1 5 4 A 2 1 7 3 2 4 2 12 pffiffi − pffiffiffiffi pffiffi 19½ ¼diagð2;6;2; Þ − pffiffi pffiffi 30 6 30 21 27 7 315 5 5 7 7 5 35 0 pffiffi pffiffi pffiffi 1 0 pffiffi 1 0 pffiffi 1 3 2 7 7 2 2 2 2 2 2 0 −pffiffiffiffi − 0 − pffiffiffiffi pffiffi 0 − pffiffiffiffi − pffiffi 5 10 5 27 27 35 27 7 27 27 35 27 7 B pffiffi pffiffi C B pffiffi C B pffiffi C 3 2 3 3 3 2 1 2 1 2 B −pffiffiffiffi − pffiffi C B 0 0 pffiffi C B 0 0 pffiffi C B 5 10 35 5 7C B 45 15 7 C B 45 15 7 C A0 2 B pffiffi pffiffi C A 2 B pffiffi pffiffiffiffi C A0 2 B pffiffi pffiffiffiffiC 15½ ¼ 7 3 3 4 2 17½ ¼ 2 29 5 10 17½ ¼ 2 29 5 10 B−pffiffiffiffi −pffiffiffiffi − − pffiffi C B− pffiffiffiffi 0 C B− pffiffiffiffi 0 − C @ 10 35 14 7 5A @ 27 35 189 189 A @ 27 35 189 189 A pffiffi pffiffi pffiffi pffiffi pffiffiffiffi pffiffi pffiffiffiffi 7 3 2 4 2 12 2 2 5 10 28 2 2 5 10 28 − − pffiffi − pffiffi − pffiffi pffiffi − − pffiffi pffiffi − 5 5 7 7 5 35 27 7 15 7 189 135 27 7 15 7 189 135 0 pffiffi pffiffi 1 0 pffiffi pffiffi 1 7 7 7 7 ð0Þ 00− pffiffiffiffi 00− pffiffiffiffi − A ½3¼diagð1;1;1;1Þ B 54 10 54 C B 54 10 54 C 13 7 1 7 1 ð0Þ 3 1 B 0 − 0 − pffiffiffiffiC B 0 − 0 − pffiffiffiffiC A 3 −1 −1 B 180 15 14C B 180 15 14C 14½ ¼diagð ;2;2; Þ A 2 pffiffi pffiffiffiffi A0 2 pffiffi pffiffiffiffi 18½ ¼B 7 1 2 10 C 18½ ¼B 7 1 2 10 C ð0Þ 1 1 5 1 B− pffiffiffiffi 0 − C B− pffiffiffiffi 0 − − C A ½3¼diagð− ; ; ;− Þ @ 54 10 252 189 A @ 54 10 252 189 A 16 27 18 54 27 pffiffi pffiffiffiffi pffiffi pffiffiffiffi 7 1 2 10 1 7 1 2 10 1 ð0Þ 1 1 − − pffiffiffiffi − − pffiffiffiffi A19½3¼diagð1;6;2;1Þ 54 15 14 189 135 54 15 14 189 135 0 pffiffi 1 0 pffiffi 1 0 pffiffi pffiffi 1 3 1 4 3 3 1 4 3 4 2 2 5 4 0 pffiffi −pffiffi − 0 pffiffi pffiffi − − − pffiffi 5 2 5 5 5 2 5 5 135 105 189 315 3 B pffiffi pffiffi C B pffiffi pffiffi C B pffiffi pffiffi C 3 3 6 2 6 6 3 3 6 2 6 6 2 16 2 B pffiffi − − pffiffi − C B pffiffi − pffiffi − C B 0 − pffiffi C B 5 2 35 7 5 35 C B 5 2 35 7 5 35 C B 105 315 35 3 C A 3 B pffiffi pffiffi C A0 3 B pffiffi pffiffi C A 3 B pffiffi pffiffi C 15½ ¼ 1 6 2 4 3 15½ ¼ 1 6 2 4 3 17½ ¼ 2 5 1 5 B −pffiffi − pffiffi − pffiffi C B pffiffi pffiffi − − pffiffi C B − 0 − pffiffi C @ 5 7 5 7 7 5 A @ 5 7 5 7 7 5A @ 189 21 63 3 A pffiffi pffiffi pffiffi pffiffi pffiffi pffiffi pffiffi pffiffi 4 3 6 6 3 22 4 3 6 6 3 22 4 2 5 82 − − pffiffi − − − − pffiffi − − pffiffi − pffiffi pffiffi 5 35 7 5 35 5 35 7 5 35 315 3 35 3 63 3 945

(Table continued)

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TABLE X. (Continued)

Að0Þ Að0Þ k ½J¼hfj k jii 0 pffiffi pffiffi 1 0 pffiffi 1 0 pffiffi 1 4 2 2 5 4 1 5 1 1 5 1 − − − pffiffi 0 pffiffi − pffiffi 0 pffiffi − − pffiffi 135 105 189 315 3 30 2 54 45 3 30 2 54 45 3 B pffiffi pffiffi C B pffiffi C B pffiffi C 2 16 2 1 1 2 1 1 2 B 0 − pffiffi C B pffiffi 0 − pffiffi C B pffiffi 0 − pffiffi C B 105 315 35 3 C B 30 2 90 35 3C B 30 2 90 35 3 C A0 3 B pffiffi pffiffi C A 3 B pffiffi pffiffi C A0 3 B pffiffi pffiffi C 17½ ¼ 2 5 1 5 18½ ¼ 5 2 5 18½ ¼ 5 2 5 B 0 pffiffi C B − 0 − pffiffi C B 0 − − pffiffi C @ 189 21 63 3 A @ 54 189 126 3 A @ 54 189 126 3A pffiffi pffiffi pffiffi pffiffi pffiffi pffiffi 4 2 5 82 1 2 5 11 1 2 5 11 − pffiffi − pffiffi − pffiffi − pffiffi − pffiffi − pffiffi − − pffiffi − pffiffi pffiffi − 315 3 35 3 63 3 945 45 3 35 3 126 3 1890 45 3 35 3 126 3 1890

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