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Fully Strange Tetraquark Sss¯S¯ Spectrum and Possible Experimental Evidence

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Fully Strange Tetraquark Sss¯S¯ Spectrum and Possible Experimental Evidence PHYSICAL REVIEW D 103, 016016 (2021) Fully strange tetraquark sss¯s¯ spectrum and possible experimental evidence † Feng-Xiao Liu ,1,2 Ming-Sheng Liu,1,2 Xian-Hui Zhong,1,2,* and Qiang Zhao3,4,2, 1Department of Physics, Hunan Normal University, and Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Changsha 410081, China 2Synergetic Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China 3Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China 4University of Chinese Academy of Sciences, Beijing 100049, China (Received 21 August 2020; accepted 5 January 2021; published 26 January 2021) In this work, we construct 36 tetraquark configurations for the 1S-, 1P-, and 2S-wave states, and make a prediction of the mass spectrum for the tetraquark sss¯s¯ system in the framework of a nonrelativistic potential quark model without the diquark-antidiquark approximation. The model parameters are well determined by our previous study of the strangeonium spectrum. We find that the resonances f0ð2200Þ and 2340 2218 2378 f2ð Þ may favor the assignments of ground states Tðsss¯s¯Þ0þþ ð Þ and Tðsss¯s¯Þ2þþ ð Þ, respectively, and the newly observed Xð2500Þ at BESIII may be a candidate of the lowest mass 1P-wave 0−þ state − 2481 0þþ 2440 Tðsss¯s¯Þ0 þ ð Þ. Signals for the other ground state Tðsss¯s¯Þ0þþ ð Þ may also have been observed in PC −− the ϕϕ invariant mass spectrum in J=ψ → γϕϕ at BESIII. The masses of the J ¼ 1 Tsss¯s¯ states are predicted to be in the range of ∼2.44–2.99 GeV, which indicates that the ϕð2170Þ resonance may not be a good candidate of the Tsss¯s¯ state. This study may provide a useful guidance for searching for the Tsss¯s¯ states in experiments. DOI: 10.1103/PhysRevD.103.016016 I. INTRODUCTION and f2ð2340Þ to be assigned as the Tsss¯s¯ ground states with 0þþ and 2þþ, respectively. However, a relativized From the Review of Particle Physics (RPP) of Particle quark model calculation [4] only favors f2 2300 to be a Data Group [1], above the mass range of 2.0 GeV one can ð Þ T ¯¯ see that there are several unflavored qq¯ isoscaler states, ssss state. Some other candidates of the Tsss¯s¯ states from experi- such as f0ð2200Þ, f2ð2150Þ, f2ð2300Þ, f2ð2340Þ, etc., ment are also suggested in the literature. For example, dominantly decaying into ϕϕ, ηη, and/or KK¯ final states. the vector meson resonance ϕð2170Þ listed in RPP [1] is The decay modes indicate that these states might be good −− suggested to be a 1 T ¯¯ state based on the mass analysis candidates for conventional ss¯ meson resonances. Recently, ssss of QCD sum rules [5–9] and flux-tube model [10]. The we carried out a systematical study of the mass spectrum 2239 þ − → þ − and strong decay properties of the ss¯ system in Ref. [2].It newly observed Xð Þ resonance in the e e K K process at BESIII [11] is suggested to be a candidate of shows that these states cannot be easily accommodated by 1−− the conventional ss¯ meson spectrum. While they may be the lowest Tsss¯s¯ state in a relativized quark model [4]. Moreover, the newly observed resonances Xð2500Þ candidates for tetraquark sss¯s¯ (T ¯¯) states, it is easy to ssss ψ → γϕϕ 2060 understand that they can fall apart into ϕϕ and ηη final observed in J= [12] and Xð Þ observed in ψ → ϕηη0 0−þ states through quark rearrangements, or easily decay into J= [13] at BESIII are suggested to be and 1þ− KK¯ final states through a pair of ss¯ annihilations and then a Tsss¯s¯ states, respectively, according to the QCD sum 2500 pair of light quark creations. The mass analysis with the rule studies [14,15]. The assignment of Xð Þ is con- sistent with that in Ref. [4]. relativistic quark model in Ref. [3] supports the f0ð2200Þ With the recent experimental progresses, more quanti- tative studies on the Tsss¯s¯ states can be carried out and their *[email protected] evidences can also be searched for in experiments. Very † [email protected] recently, the LHCb Collaboration reported their results on the observations of tetraquark ccc¯ c¯ (Tccc¯ c¯ ) states [16].A Published by the American Physical Society under the terms of broad structure above the J=ψJ=ψ threshold ranging from the Creative Commons Attribution 4.0 International license. 6900 Further distribution of this work must maintain attribution to 6.2 to 6.8 GeV and a narrower structure Tccc¯ c¯ ð Þ are the author(s) and the published article’s title, journal citation, observed with more than 5σ of significance level. There are and DOI. Funded by SCOAP3. also some vague structure around 7.2 GeV to be confirmed. 2470-0010=2021=103(1)=016016(14) 016016-1 Published by the American Physical Society LIU, LIU, ZHONG, and ZHAO PHYS. REV. D 103, 016016 (2021) 4 These observations could be evidences for genuine T ¯ ¯ X X ccc c − states [17–21]. H ¼ mi þ Ti TG þ VijðrijÞ; ð1Þ i¼1 i<j The observations of the Tccc¯ c¯ states above the J=ψJ=ψ threshold at LHCb may provide an important clue for the underlying dynamics for the ccc¯ c¯ system. In particular, the where mi and Ti stand for the constituent quark mass and kinetic energy of the ith quark, respectively; TG stands for narrowness of Tccc¯ c¯ ð6900Þ suggests that there should be “ ” the center-of-mass (c.m.) kinetic energy of the tetraquark more profound mechanism that slows down the fall-apart ≡ r − r decays of such a tetraquark system. Although this may be system, rij j i jj is the distance between the ith and related to the properties of the static potential of heavy quark jth quark, and VijðrijÞ stands for the effective potential systems, more direct evidences are still needed to disentangle between them. In this work, the VijðrijÞ adopts a widely the dynamical features between the heavy and light flavor used form [24–26,30–36] systems. As an analogy of the Tccc¯ c¯ system, there might exist V r Vconf r Vsd r ; stable Tsss¯s¯ states above the ϕϕ threshold and can likely be ijð ijÞ¼ ij ð ijÞþ ij ð ijÞ ð2Þ observed in the di-ϕ mass spectrum. On the other hand, flavor mixings could be important for the light flavor systems and where the confinement potential adopts the standard form pure sss¯s¯ states may not exist. To answer such questions, of the Cornell potential [24], which includes the spin- Lin ∝ systematic calculations of the sss¯s¯ system should be carried independent linear confinement potential Vij ðrijÞ rij Coul ∝ 1 out. The BESIII experiments can provide a large data sample and Coulomb-like potential Vij ðrijÞ =rij, ψ ψ 2 for the search of the Tsss¯s¯ states in J= and ð SÞ decays. In theory, although there have been some predictions of the 3 4 α conf ij V r − λ λ b r − C0 : T ¯¯ spectrum within the quark model [3,4,10] and QCD ij ð ijÞ¼ ð i · jÞ ij ij þ ð3Þ ssss 16 3 rij sum rules [17–20], most of the studies focus on some special states in a diquark-antidiquark picture. About the status of The constant C0 stands for the zero point energy. While the the tetraquark states, some recent review works can be sd spin-dependent potential V ðrijÞ is the sum of the spin- referenced [22,23]. In this study, we intend to provide a ij spin contact hyperfine potential VSS, the spin-orbit poten- systematical calculation of the mass spectrum of the 1S, 1P, ij tial VSS, and the tensor term VT , and 2S-wave Tsss¯s¯ states without the diquark-antidiquark ij ij approximation in a nonrelativistic potential quark model sd SS T LS (NRPQM). Vij ðrijÞ¼Vij þ Vij þ Vij ; ð4Þ The NRPQM is based on the Hamiltonian proposed by the Cornell model [24], which contains a linear confinement and with a one-gluon-exchange potential for quark-quark and quark- 3 −σ2 r2 antiquark interactions. With the NRPQM, we have success- α π σ e ij ij 16 SS − ij λ λ ij S S ¯ ¯ ¯ Vij ¼ ð i · jÞ · 3=2 · ð i · jÞ ; ð5Þ fully described the ss, cc,andbb meson spectra [2,25,26], 4 2 π 3mimj and sss, ccc, and bbb baryon spectra [27,28]. Furthermore, we adopted the NRPQM for the study of both 1S and 1P- α λ · λ 1 1 4 LS − ij i j L S S wave all-heavy tetraquark states with a Gaussian expansion Vij ¼ 3 2 þ 2 þ f ij · ð i þ jÞg 16 r m m mimj method (GEM) [21,29]. In this work, we continue to extend ij i j αij λi · λj 1 1 this method to study the Tsss¯s¯ spectrum by constructing the − − L S − S 16 3 2 2 f ij · ð i jÞg; ð6Þ full tetraquark configurations without the diquark-antidi- rij mi mj quark approximation. With the parameters determined in our study of the ss¯ spectrum [2], we obtain a relatively reliable α 1 3ðS · r ÞðS · r Þ VT ¼ − ij ðλ · λ Þ · i ij j ij − S · S : prediction of the mass spectrum for 36 T ¯¯ states, i.e., 4 1S- ij 4 i j 3 2 i j ssss mimjrij rij wave ground states, 20 1P-wave orbital excitations, and 12 2S-wave radial excitations.
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