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Exotic Double-Charm Molecular States with Hidden Or Open Strangeness and Around 4.5 ∼ 4.7 Gev

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Exotic Double-Charm Molecular States with Hidden Or Open Strangeness and Around 4.5 ∼ 4.7 Gev 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 Hadron 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) à In this work, we investigate the interactions between the charmed-strange meson (Ds;Ds )inH-doublet à and the (anti-)charmed-strange meson (Ds1;Ds2)inT-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 HsTs and HsTs systems, we try to find the bound state solutions for the corresponding systems. We predict the possible hidden-charm hadronic molecular states with hidden à ¯ PC 0−− 0−þ à ¯ à 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 pentaquark molecules [10 16]. different from conventional mesons and baryons, 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 ¯ à 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¯ à 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 ¯ à – 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 – hadrons, it is natural to consider them as the candidates Belle [26]. In Refs. [27 29], a dynamic calculation of þ à ¯ 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 quark 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 Æ vÞ==2 denotes the projection operator, and around 4.5 ∼ 4.7 GeV mass range. Further suggestion of Æ 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., 094006-2 EXOTIC DOUBLE-CHARM MOLECULAR STATES WITH HIDDEN … PHYS. REV. D 102, 094006 (2020) 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 ; vector meson field strength tensor FμνðρÞ are given by Ãμ − μpffiffiffiffiffiffiffiffi h0jDs jcs¯ð1 Þi ¼ ϵ mDà ; 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 Ãμν þ μν 2 h0jD 2 jcs¯ð2 Þi ¼ ζ m à ; s Ds2 ϵμ 0 1 1 respectively.
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