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 ispin- 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. ð7Þ The paper is organized as follows: a brief introduction to S the tetraquark spectrum is given in Sec. II. In Sec. III, the In the above equations, i stands for the spin of the ith L numerical results and discussions are presented. A short quark, and ij stands for the relative orbital angular summary is given in Sec. IV. momentum between the ith and jth quark. If the interaction λ λ occurs between two quarks orP antiquarks, the i · j operator is defined as λ · λ ≡ 8 λaλa, while if the II. MASS SPECTRUM i j a¼1 i j interaction occurs between a quark andP an antiquark, the A. Hamiltonian λ λ λ λ ≡ 8 −λaλa i · j operator is defined as i · j a¼1 i j , where We adopt a NRPQM to calculate the mass spectrum of λa is the complex conjugate of the Gell-Mann matrix λa. the sss¯s¯ system. In this model, the Hamiltonian is given by The parameters bij and αij denote the strength of the

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TABLE I. Quark model parameters used in this work. 1 χ11 ↑↑↑↓ ↑↑↓↑ − ↑↓↑↑ − ↓↑↑↑ 11 ¼ 2 ð þ Þ; ð14Þ ms (GeV) 0.60 α ss 0.77 11 χ22 ¼ ↑↑↑↑: ð15Þ σss (GeV) 0.60 b (GeV2) 0.135 C0 (GeV) −0.519 In the spatial space, we define the relative Jacobi coordinates with the single-partial coordinates ri (i ¼ 1, 2, 3, 4), confinement and strong coupling of the one-gluon- exchange potential, respectively. ξ1 ≡ r1 − r2; ð16Þ The five parameters ms, αss, σss, bss, and C0 have been determined by fitting the mass spectrum of the strangeo- ξ2 ≡ r3 − r4; ð17Þ nium in our previous work [2]. The quark model parameters adopted in this work are collected in Table I. m1r1 þ m2r2 m3r3 þ m4r4 ξ3 ≡ − ; ð18Þ m1 þ m2 m3 þ m4 B. Configurations classified in the quark model r r r r R ≡ m1 1 þ m2 2 þ m3 3 þ m4 4 To calculate the spectroscopy of a qqq¯ q¯ (q ∈ s; c; b ) : ð19Þ f g m1 þ m2 þ m3 þ m4 system, first we construct the configurations in the product space of flavor, color, spin, and spatial parts. Note that ξ1 and ξ2 stand for the relative Jacobi coordinates 66¯ In the color space, there are two color-singlet bases j ic between two quarks q1 and q2 (or antiquarks q¯ 1 and q¯ 2), 33¯ ¯ ¯ and j ic, and their wave functions are given by and two antiquarks q3 and q4 (or quarks q3 and q4), respectively. While ξ3 stands for the relative Jacobi coor- 1 ¯ ¯ j66¯i ¼ pffiffiffi ½ðrb þ brÞðb¯ r¯ þr¯ b¯Þþðgr þ rgÞðg¯ r¯ þr¯ g¯Þ dinate between diquark qq and antidiquark q q. Using the c 2 6 above Jacobi coordinates, it is easy to obtain basis functions þðgb þ bgÞðb¯ g¯ þg¯ b¯Þ that have well-defined symmetry under permutations of the ¯ ¯ pairs (12) and (34) [37]. þ 2ðrrÞðr¯ r¯Þþ2ðggÞðg¯ g¯Þþ2ðbbÞðb bÞ; ð8Þ In the Jacobi coordinate system, the spatial wave func- Ψ ξ ξ ξ R ¯ ¯ 1 tion NLMð 1; 2; 3; Þ for a qqq q system with principal 33¯ ffiffiffi − ¯ ¯ −¯ ¯ − − ¯ ¯ −¯ ¯ j ic ¼ p ½ðbr rbÞðb r r bÞ ðrg grÞðg r r gÞ quantum number N and orbital angular momentum quan- 2 3 tum numbers LM may be expressed as the linear combi- ¯ ¯ Φ R ψ ξ ψ ξ ψ ξ þðbg − gbÞðb g¯ −g¯ bÞ: ð9Þ nation of ð Þ α1 ð 1Þ α2 ð 2Þ α3 ð 3Þ,

In the spin space, there are six spin bases, which are ΨNLMðξ1; ξ2; ξ3; RÞ χS12S34 X denoted by S . Where S12 stands for the spin quantum Φ R ψ ξ ψ ξ ψ ξ ¼ Cα1;α2;α3 ½ ð Þ α1 ð 1Þ α2 ð 2Þ α3 ð 3ÞNLM; ð20Þ numbers for the diquark ðq1q2Þ (or antidiquark ðq¯ 1q¯ 2Þ), α1;α2;α3 while S34 stands for the spin quantum number for the ¯ ¯ antidiquark ðq3q4Þ (or diquark ðq3q4Þ). S is the total spin where Cα α α stands for the combination coefficients, ¯ ¯ 1; 2; 3 quantum number of the tetraquark qqq q system. The spin ΦðRÞ is the c.m. motion wave function. In the quantum wave functions χS12S34 with a determined S (S stands for α ≡ SSz z z number set i fnξi ;lξi ;mξi g, nξi is the principal quantum S the third component of the total spin ) can be explicitly number, lξi is the angular momentum, and mξi is its third expressed as follows: ψ ξ component projection. The wave functions αi ð iÞ, which account for the relative motions, can be written as 1 χ00 ¼ ð↑↓↑↓ − ↑↓↓↑ − ↓↑↑↓ þ ↓↑↓↑Þ; ð10Þ 00 2 ψ ξ ξ ξˆ α ð iÞ¼Rnξ lξ ð iÞYlξ mξ ð iÞ; ð21Þ rffiffiffiffiffi i i i i i 11 1 χ00 ¼ ð2↑↑↓↓ − ↑↓↑↓ − ↑↓↓↑ ξˆ 12 where Ylξ mξ ð iÞ is the spherical harmonic function, and i i ξ Rnξ lξ ð iÞ is the radial part. It is seen that for an excited − ↓↑↑↓ − ↓↑↓↑ þ 2↓↓↑↑Þ; ð11Þ i i rffiffiffi state, there are three spatial excitation modes corresponding 1 01 to three independent internal wave functions ψ α ðξiÞ (i ¼ 1, χ11 ¼ ð↑↓↑↑ − ↓↑↑↑Þ; ð12Þ i 2 2, 3), which are denoted as ξ1, ξ2, and ξ3, respectively, in rffiffiffi the present work. One point should be emphasized that 1 10 considering the fact that the sss¯s¯ system is composed of χ11 ¼ ð↑↑↑↓ − ↑↑↓↑Þ; ð13Þ 2 equal mass constituent quarks and antiquarks, we adopt a

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TABLE II. Configurations for the tetraquark qqq¯ q¯ system up to the 2S-wave states.

JPðCÞ Configuration Wave function PC þþ 1 1S 00 ¯ c J ¼ 0 1 S0þþ 66¯ ψ χ 66 ð Þc 000 00 j i PC þþ 1 1S 11 ¯ c J ¼ 0 1 S0þþ 33¯ ψ χ j33i ð Þc 000 00 PC þ− 3 1S 11 ¯ c J ¼ 1 1 S1þ− 33¯ ψ χ j33i ð Þc 000 11 PC 2þþ 15 ψ 1S χ11 33¯ c J ¼ S2þþð33¯ Þ 000 22 j i c qffiffi PC −− 3 ¯ c J ¼ 0 1 P0−− 66¯ ξ ξ 1 ξ1 10 ξ1 10 ξ1 10 ξ2 01 ξ2 01 ξ2 01 j66i ð Þcð 1; 2Þ ðψ χ − ψ χ þ ψ χ − ψ χ þ ψ χ − ψ χ Þ qffiffi6 011 1−1 010 10 01−1 11 011 1−1 010 10 01−1 11 PC −− 3 ¯ c J ¼ 0 1 P0−− 33¯ ξ ξ 1 ξ1 01 ξ1 01 ξ1 01 ξ2 10 ξ2 10 ξ2 10 j33i ð Þcð 1; 2Þ ðψ χ − ψ χ þ ψ χ − ψ χ þ ψ χ − ψ χ Þ qffiffi6 011 1−1 010 10 01−1 11 011 1−1 010 10 01−1 11 PC −þ 3 ξ ξ ξ ξ ξ ξ ¯ c J ¼ 0 1 P0−þ 66¯ ξ ξ 1 1 10 1 10 1 10 2 01 2 01 2 01 j66i ð Þcð 1; 2Þ ðψ χ − ψ χ þ ψ χ þ ψ χ − ψ χ þ ψ χ Þ qffiffi6 011 1−1 010 10 01−1 11 011 1−1 010 10 01−1 11 PC −þ 3 ξ ξ ξ ξ ξ ξ ¯ c J ¼ 0 1 P0−þ 33¯ ξ ξ 1 1 01 1 01 1 01 2 10 2 10 2 10 j33i ð Þcð 1; 2Þ ðψ χ − ψ χ þ ψ χ þ ψ χ − ψ χ þ ψ χ Þ qffiffi6 011 1−1 010 10 01−1 11 011 1−1 010 10 01−1 11 PC −þ 3 ξ ξ ξ ¯ c J ¼ 0 1 P0−þ 33¯ ξ 1 3 11 3 11 3 11 j33i ð Þcð 3Þ 3ðψ 011χ1−1 − ψ 010χ10 þ ψ 01−1χ11Þ

PC −− 3 1 ξ1 10 ξ1 10 ξ2 01 ξ2 01 ¯ c J ¼ 1 1 P1−− 66¯ ξ ξ ψ χ − ψ χ − ψ χ ψ χ j66i ð Þcð 1; 2Þ 2 ð 011 10 010 11 011 10 þ 010 11Þ PC −− 3 1 ξ1 01 ξ1 01 ξ2 10 ξ2 10 ¯ c J ¼ 1 1 P1−− 33¯ ξ ξ ψ χ − ψ χ − ψ χ ψ χ j33i ð Þcð 1; 2Þ q2 ðffiffiffiffi011 10 010qffiffiffiffi11 011 10qþffiffi 010 11Þ PC −− 5 ¯ c J ¼ 1 1 P1−− 33¯ ξ 1 ξ3 11 3 ξ3 11 3 ξ3 11 j33i ð Þcð 3Þ 10ψ 011χ20 − 10ψ 010χ21 þ 5ψ 01−1χ22

PC −− 1 ξ3 11 ¯ c J ¼ 1 1 P1−− 33¯ ξ ψ χ 33 ð Þcð 3Þ 011 00 j i PC −− 1 ξ3 00 ¯ c J ¼ 1 1 P1−− 66¯ ξ ψ χ 66 ð Þcð 3Þ 011 00 j i PC −þ 3 1 ξ1 10 ξ1 10 ξ2 01 ξ2 01 ¯ c J ¼ 1 1 P1−þ 66¯ ξ ξ ψ χ − ψ χ ψ χ − ψ χ 66 ð Þcð 1; 2Þ 2 ð 011 10 010 11 þ 011 10 010 11Þ j i PC −þ 3 1 ξ1 01 ξ1 01 ξ2 10 ξ2 10 ¯ c J ¼ 1 1 P1−þ 33¯ ξ ξ ψ χ − ψ χ ψ χ − ψ χ j33i ð Þcð 1; 2Þ q2 ðffiffi 011 10 010 11 þ 011 10 010 11Þ PC −þ 3 ξ ξ ¯ c J ¼ 1 1 P1−þ 33¯ ξ 1 3 11 3 11 j33i ð Þcð 3Þ ðψ χ − ψ χ Þ qffiffi2 011 10 010 11 PC −− 3 ¯ c J ¼ 2 1 P2−− 66¯ ξ ξ 1 ξ1 10 ξ2 01 j66i ð Þcð 1; 2Þ ðψ χ − ψ χ Þ qffiffi2 011 11 011 11 PC −− 3 ¯ c J ¼ 2 1 P2−− 33¯ ξ ξ 1 ξ1 01 ξ2 10 j33i ð Þcð 1; 2Þ ðψ χ − ψ χ Þ qffiffi2 011 11 qffiffi011 11 PC −− 5 ¯ c 2 1 −− ¯ 1 ξ3 11 2 ξ3 11 33 J ¼ P2 ð33Þ ðξ3Þ ψ χ − ψ χ j i c qffiffi3 011 21 3 010 22 PC −þ 3 ξ ξ ¯ c J ¼ 2 1 P2−þ 66¯ ξ ξ 1 1 10 2 01 j66i ð Þcð 1; 2Þ ðψ χ þ ψ χ Þ qffiffi2 011 11 011 11 PC −þ 3 ¯ c 2 1 −þ ¯ 1 ξ1 01 ξ2 10 33 J ¼ P2 33 ξ1;ξ2 j i ð Þcð Þ 2ðψ 011χ11 þ ψ 011χ11Þ

PC −þ 3 ξ3 11 ¯ c J ¼ 2 1 P2−þ 33¯ ξ ψ χ j33i ð Þcð 3Þ 011 11 PC −− 5 ξ3 11 ¯ c 3 1 −− ¯ ψ χ 33 J ¼ P3 ð33Þ ðξ3Þ 011 22 j i c qffiffi PC þ− 1 ξ ξ ¯ c J ¼ 0 2 S0þ− 66¯ ξ ξ 1 1 00 2 00 j66i ð Þcð 1; 2Þ ðψ χ − ψ χ Þ qffiffi2 100 00 100 00 PC þ− 1 ξ ξ ¯ c J ¼ 0 2 S0þ− 33¯ ξ ξ 1 1 11 2 11 j33i ð Þcð 1; 2Þ ðψ χ − ψ χ Þ qffiffi2 100 00 100 00 PC þþ 1 ξ ξ ¯ c J ¼ 0 2 S0þþ 66¯ ξ ξ 1 1 00 2 00 j66i ð Þcð 1; 2Þ ðψ χ þ ψ χ Þ qffiffi2 100 00 100 00 PC þþ 1 ¯ c 0 2 þþ ¯ 1 ξ1 11 ξ2 11 33 J ¼ S0 33 ξ1;ξ2 j i ð Þcð Þ 2ðψ 100χ00 þ ψ 100χ00Þ

PC þþ 1 ξ3 00 ¯ c J ¼ 0 2 S0þþ 66¯ ξ ψ χ j66i ð Þcð 3Þ 100 00 PC þþ 1 ξ3 11 ¯ c J ¼ 0 2 S0þþ 33¯ ξ ψ χ j33i ð Þcð 3Þ q100ffiffi 00 PC þ− 3 ¯ c 1 2 þ− ¯ 1 ξ1 11 ξ2 11 33 J ¼ S1 33 ξ1;ξ2 j i ð Þcð Þ 2ðψ 100χ11 − ψ 100χ11Þ

PC þ− 3 ξ3 11 ¯ c J ¼ 1 2 S1þ− 33¯ ξ ψ χ j33i ð Þcð 3Þ q100ffiffi 11 PC þþ 3 ξ ξ ¯ c J ¼ 1 2 S1þþ 33¯ ξ ξ 1 1 11 2 11 j33i ð Þcð 1; 2Þ ðψ χ þ ψ χ Þ qffiffi2 100 11 100 11 PC þ− 5 ξ ξ ¯ c J ¼ 2 2 S2þ− 33¯ ξ ξ 1 1 11 2 11 j33i ð Þcð 1; 2Þ ðψ χ − ψ χ Þ qffiffi2 100 22 100 22 PC þþ 5 ¯ c 2 2 þþ ¯ 1 ξ1 11 ξ2 11 33 J ¼ S2 33 ξ1;ξ2 j i ð Þcð Þ 2ðψ 100χ22 þ ψ 100χ22Þ

PC þþ 5 ξ3 11 ¯ c J ¼ 2 2 S2þþ 33¯ ξ ψ χ j33i ð Þcð 3Þ 100 22

016016-4 FULLY-STRANGE TETRAQUARK SSS¯ S¯ … PHYS. REV. D 103, 016016 (2021) single set of Jacobi coordinates in this study as an approxi- ϕ ðωξ l; ξ Þ nξi lξi i i mation. In fact, the four-body wave function describing a  1 lξþ2−nξ 3 2 2lξ 2nξ 1 !! 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¯¯ ð i þ i þ Þ p lξ scalar ssss state contains a small contribution of internal μ ω 4 ffiffiffi μ ω ξ i ¼ð ξi ξilÞ p 2 ð ξi ξil iÞ angular momentum. This contribution is neglected in our πnξ !½ð2lξ þ 1Þ!! i i calculations. To precisely treat an N-body system, one can 1 2 3 − μξ ωξ lξ 2 × e 2 i i i F −nξ ;lξ þ ; μξ ωξ lξ ; ð23Þ involve several different sets of Jacobi coordinates as i i 2 i i i those done in Refs. [38–43], or adopt a single set of Jacobi coordinates X ¼ðξi; ξ2; …; ξN−1Þ with nondiagonal −XAXT – − 3 μ ω ξ 2 Gaussians e as those done in Refs. [44 47], where A where Fð nξi ;lξi þ 2 ; ξi ξil i Þ is the confluent hyper- is a symmetric matrix. geometric function. It should be pointed out that if there are Taking into account the Pauli principle and color con- no radial excitations, the expansion method with harmonic finement for the four-quark system qqq¯ q¯, we have four oscillator wave functions (HOEM) is just the same as the configurations for 1S-wave ground states, 20 configura- Gaussian expansion method adopted in the litera- tions for the 1P-wave orbital excitations, and 12 configu- ture [38,39]. rations for the 2S-wave radial excitations. The spin-parity For an sss¯s¯ system, if we ensure that the spatial wave quantum numbers, notations, and wave functions for function with Jacobi coordinates can transform into the these configurations are presented in Table II. With the single particle coordinate system, the harmonic oscillator wave functions for all the configurations, the mass matrix frequencies ωξ l (i ¼ 1, 2, 3) can be related to the harmonic i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi elements of the Hamiltonian can be worked out. oscillator stiffness factor Kl with ωξ l 2Kl=μξ , pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ¼ 1 To work out the matrix elements in the coordinate ω 2 μ ω 4 μ ξ ξ2l ¼ Kl= ξ2 , and ξ3l ¼ Kl= ξ3 . Considering space, we expand the radial part Rnξ lξ ð iÞ with a series i i the reduced masses μξ ¼ μξ ¼ m =2, μξ ¼ m for of harmonic oscillator functions [21,27], 1 2 spffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 s ω ω ω 4 Tðsss¯s¯Þ, one has ξ1l ¼ ξ2l ¼ ξ3l ¼ Kl=ms. It indi- Xn cates that the harmonic oscillator frequencies ωξ l for ξ C ϕ ω ξ i Rnξ lξ ð iÞ¼ ξ l nξ lξ ð ξ l; iÞ; ð22Þ i i i i i i Tðsss¯s¯Þ are not independent. According toQ the relation l¼1 3 ωξ l ¼ ωξ l ¼ ωξ l ¼ ωl, the expansion of 1 R ðξ Þ 1 2 3 i¼ nξi lξi i with can be simplified as

Y3 Xn Xn Xn R ðξ Þ¼ Cξ lCξ l0 Cξ l00 ϕ ðωξ l; ξ1Þϕ ðωξ l0 ; ξ2Þϕ ðωξ l00 ; ξ3Þδll0 δll00 nξi lξi i 1 2 3 nξ1 lξ1 1 nξ2 lξ2 2 nξ3 lξ3 3 i¼1 l l0 l00 Xn Clϕ ωl; ξ1 ϕ ωl; ξ2 ϕ ωl; ξ3 : 24 ¼ nξ1 lξ1 ð Þ nξ2 lξ2 ð Þ nξ3 lξ3 ð Þ ð Þ l

Then we introduce oscillator length parameters dl that can be related to the harmonic oscillator frequencies ωl 2 with 1=dl ¼ msωl. Following the method of Refs. [38,39], we let the dl parameters form a geometric progression

l−1 dl ¼ d1a ðl ¼ 1; …;nÞ; ð25Þ where n is the number of harmonic oscillator functions and a is the ratio coefficient. There are three parameters fd1;dn;ng to be determined through the variation method. It is found that with the parameter set f0.085 fm; 3.399 fm; 15g for the sss¯s¯ system, we can obtain stable solutions. The numerical results should be independent of the parameter d1. To confirm this point, as done in the literature [40–42], we scale the parameter d1 of the basis functions as d1 → αd1. The mass of a Tsss¯s¯ state should be stable at a resonance energy insensitive to the scaling para- FIG. 1. Predicted masses of 12 2S-wave Tsss¯ s¯ configurations as meter α. As an example, we plot the masses of 12 2S-wave a function of the scaling factor α.

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Tsss¯s¯ configurations as a function of the scaling factor α in can be obtained by diagonalizing the mass matrix of Fig. 1. It is found that the numerical results are nearly different configurations with the same JPC numbers. independent of the scaling factor α. The stabilization of other states predicted in this work has also been examined III. RESULTS AND DISCUSSIONS by the same method. With the mass matrix elements ready for each configu- Our predictions of the Tsss¯s¯ mass spectrum with the ration, the mass of the tetraquark configuration and its HOEM are given in Table III, where the components of spacial wave function can be determined by solving a different configurations for a physical state can be seen. generalized eigenvalue problem. The details can be found For example, the two 0þþ ground states are mixing 1 in our previous works [27,29]. Finally, the physical states states between two different configurations S0þþ 66¯ and ð Þc

TABLE III. Predicted mass spectrum for the sss¯ s¯ system with the HOEM.

JPðCÞ Configuration hHi (MeV) Mass (MeV) Eigenvector þþ 1 0 1 S0þþ 66¯ 2365 −105 2218 −0.58 −0.81 ð Þc ð Þ 1 −105 2293 2440 −0 81 −0 58 1 S0þþ 33¯ ð . . Þ ð Þc þ− 3 1 1 S1þ− 33¯ ð 2323 Þ 2323 1 ð Þc 2þþ 5 2378 1 S2þþ 33¯ ð Þ 2378 1 ð Þc −− 3 0 1 P0−− 66¯ ξ ξ 2635 154 2507 −0.77 0.64 ð Þcð 1; 2Þ ð Þ 3 154 2694 2821 0 64 0 77 1 P0−− 33¯ ξ ξ ð . . Þ ð Þcð 1; 2Þ ! ! ! −þ 3 2616 −35 −111 2481 0 61 −0 11 0 78 0 1 P0−þ 66¯ ξ ξ ð . . . Þ ð Þcð 1; 2Þ 3 −35 2685 56 2635 −0 56 −0 76 0 34 1 P0−þ 33¯ ξ ξ ð . . . Þ ð Þcð 1; 2Þ 3 −111 56 2576 2761 ð 0.56 −0.64 −0.53 Þ 1 P0−þ 33¯ ξ ð Þcð 3Þ −− 3 0 1 0 1 0 1 1 1 P1−− 66¯ ξ ξ 2585 −154 −89 −46 90 2445 ð −0.80 −0.38 −0.43 −0.14 0.11 Þ ð Þcð 1; 2Þ 3 B −154 2694 42 22 −76 C B 2567 C B 0 18 0 57 −0 78 −0 05 0 15 C 1 P1−− 33¯ ξ ξ ð . . . . . Þ ð Þcð 1; 2Þ B C B C B C 5 B −89 42 2584 −829C B 2627 C B ð 0.03 0.11 0.11 −0.97 −0.18 Þ C 1 P1−− 33¯ ξ @ A @ A @ A ð Þcð 3Þ 1 −46 22 −8 2636 −51 2766 ð −0.42 0.57 0.43 0.00 0.56 Þ 1 P1−− 33¯ ξ ð Þcð 3Þ 90 −76 29 −51 2889 2984 0 38 −0 43 −0 07 −0 19 0 79 1 ð . . . . . Þ 1 P1−− 66¯ ξ ð Þcð 3Þ ! ! ! −þ 3 2628 95 25 2564 0 83 −0 51 −0 21 1 1 P1−þ 66¯ ξ ξ ð . . . Þ ð Þcð 1; 2Þ 3 95 2712 12 2632 −0 09 0 25 −0 96 1 P1−þ 33¯ ξ ξ ð . . . Þ ð Þcð 1; 2Þ 3 25 12 2633 2778 ð 0.55 0.82 0.12 Þ 1 P1−þ 33¯ ξ ð Þcð 3Þ −− 3 ! ! ! 2 1 P2−− 66¯ ξ ξ 2620 −217 −50 2446 ð 0.79 0.60 0.12 Þ ð Þcð 1; 2Þ 3 −217 2725 24 2657 −0 03 0 23 −0 97 1 P2−− 33¯ ξ ξ ð . . . Þ ð Þcð 1; 2Þ 5 −50 24 2665 2907 ð −0.61 0.76 0.20 Þ 1 P2−− 33¯ ξ ð Þcð 3Þ −þ 3 ! ! ! 2 1 P2−þ 66¯ ξ ξ 2638 138 −33 2537 ð 0.82 −0.56 0.13 Þ ð Þcð 1; 2Þ 3 138 2733 −16 2669 0 00 0 23 0 97 1 P2−þ 33¯ ξ ξ ð . . . Þ ð Þcð 1; 2Þ 3 −33 −16 2673 2837 ð −0.58 −0.79 0.19 Þ 1 P2−þ 33¯ ξ ð Þcð 3Þ −− 5 3 1 P3−− 33¯ ξ ð 2719 Þ 2719 1 ð Þcð 3Þ þ− 1 0 2 S0þ− 66¯ ξ ξ 2848 −27 2841 −0.97 −0.26 ð Þcð 1; 2Þ ð Þ 1 −27 2942 2949 −0 26 0 97 2 S0þ− 33¯ ξ ξ ð . . Þ ð Þcð 1; 2Þ 0 1 0 1 0 1 þþ 1 2781 0 2 S0þþ 66¯ ξ ξ 2859 −53 −61 −18 ð −0.61 −0.52 −0.16 −0.57 Þ ð Þcð 1; 2Þ 1 B −53 2903 −20 −49 C B 2876 C B 0 67 0 02 0 03 −0 74 C 2 S0þþ 33¯ ξ ξ B C B ð . . . . Þ C ð Þcð 1; 2Þ @ A @ A @ A 1 −61 −20 3218 −40 2948 ð 0.39 −0.85 0.07 0.34 Þ 2 S0þþ 66¯ ξ ð Þcð 3Þ 1 −18 −49 −40 2856 3232 ð 0.15 0.02 −0.98 0.09 Þ 2 S0þþ 33¯ ξ ð Þcð 3Þ þ− 3 1 2 S1þ− 33¯ ξ ξ 2954 0 2867 01 ð Þcð 1; 2Þ ð Þ 3 0 2867 2954 10 2 S1þ− 33¯ ξ ð Þ ð Þcð 3Þ þþ 3 1 2 S1þþ 33¯ ξ ξ ð 2920 Þ 2920 1 ð Þcð 1; 2Þ þ− 5 2977 2 2 S2þ− 33¯ ξ ξ ð Þ 2977 1 ð Þcð 1; 2Þ 2þþ 5 2 S2þþ 33¯ ξ ξ 2952 −28 2878 ð −0.35 −0.94 Þ ð Þcð 1; 2Þ 5 −28 2888 2963 −0 94 0 35 2 S2þþ 33¯ ξ ð . . Þ ð Þcð 3Þ

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TABLE IV. The average contributions of each part of the Hamiltonian to the sss¯ s¯ configurations with the HOEM. hTi stands for the contribution of the kinetic energy term. hVLini and hVCouli stand for the contributions from the linear confinement potential and Coulomb-type potential, respectively. hVSSi, hVT i, and hVLSi stand for the contributions from the spin-spin interaction term, the tensor potential term, and the spin-orbit interaction term, respectively. The second number in every column is calculated with the GEM.

JPðCÞ Configuration Mass hTi hVLinihVCouli hVSSi hVT i hVLSi þþ 1 −774 0 1 S0þþ 66¯ 2365 807 930 40.07 ð Þc 1 1 S0þþ 33¯ 2293 884 890 −812 −29.72 ð Þc þ− 3 1 1 S1þ− 33¯ 2323 851 906 −797 0.00 ð Þc þþ 5 −767 2 1 S2þþ 33¯ 2378 793 937 52.33 ð Þc −− 3 0 1 P0−− 66¯ ξ ξ 2635 827 1077 −659 4.42 34.92 −11.64 ð Þcð 1; 2Þ 3 1 P0−− 33¯ ξ ξ 2694 902 1094 −644 −0.95 9.02 −27.06 ð Þcð 1; 2Þ −þ 3 −675 −4 20 −12 60 0 1 P0−þ 66¯ ξ ξ 2616 862 1056 26.93 . . ð Þcð 1; 2Þ 3 1 P0−þ 33¯ ξ ξ 2685 921 1083 −651 8.30 −9.39 −28.16 ð Þcð 1; 2Þ 3 1 P0−þ 33¯ ξ 2576 976 1015 −703 9.95 −20.88 −62.65 ð Þcð 3Þ −− 3 1 1 P1−− 66¯ ξ ξ 2585 895 1037 −688 5.06 −20.11 −6.70 ð Þcð 1; 2Þ 3 1 P1−− 33¯ ξ ξ 2694 902 1094 −644 −0.95 −4.51 −13.53 ð Þcð 1; 2Þ 5 1 P1−− 33¯ ξ 2584 972 1018 −702 43.08 −14.60 −93.87 ð Þcð 3Þ 1 1 P1−− 33¯ ξ 2636 876 1068 −665 −5.24 00 ð Þcð 3Þ 1 1 P1−− 66¯ ξ 2889 847 1210 −564 33.27 0 0 ð Þcð 3Þ −þ 3 1 1 P1−þ 66¯ ξ ξ 2628 845 1066 −667 26.33 2.03 −6.08 ð Þcð 1; 2Þ 3 1 P1−þ 33¯ ξ ξ 2712 881 1106 −636 8.27 4.33 −13.00 ð Þcð 1; 2Þ 3 1 P1−þ 33¯ ξ 2633 883 1064 −668 9.07 8.73 −26.18 ð Þcð 3Þ −− 3 2 1 P2−− 66¯ ξ ξ 2620 845 1066 −667 4.59 3.63 6.05 ð Þcð 1; 2Þ 3 1 P2−− 33¯ ξ ξ 2725 859 1120 −628 −0.58 0.83 12.40 ð Þcð 1; 2Þ 5 1 P2−− 33¯ ξ 2665 845 1087 −652 35.76 11.32 −24.26 ð Þcð 3Þ −þ 3 −662 −0 39 2 1 P2−þ 66¯ ξ ξ 2638 832 1074 25.87 . 5.91 ð Þcð 1; 2Þ 3 1 P2−þ 33¯ ξ ξ 2733 854 1123 −626 8.22 −0.82 12.2 ð Þcð 1; 2Þ 3 1 P2−þ 33¯ ξ 2673 830 1097 −646 8.52 −1.56 23.43 ð Þcð 3Þ −− 5 3 1 P3−− 33¯ ξ 2719 779 1131 −625 31.92 −2.80 41.96 ð Þcð 3Þ þ− 1 −363 − 529 0 2 S0þ− 66¯ ξ ξ 2848/2927 703/844 1131/1225 = 14.19/24.74 ð Þcð 1; 2Þ 1 2 S0þ− 33¯ ξ ξ 2942/2931 936/941 1246/1246 −599=− 618 −4.09=0.80 ð Þcð 1; 2Þ þþ 1 −552 − 585 0 2 S0þþ 66¯ ξ ξ 2859/2874 848/863 1177/1206 = 24.92/27.90 ð Þcð 1; 2Þ 1 2 S0þþ 33¯ ξ ξ 2903/2918 957/954 1236/1235 −639=− 620 −12.68=− 12.40 ð Þcð 1; 2Þ 1 −469 − 517 2 S0þþ 66¯ ξ 3218/3148 898/922 1399/1364 = 27.06/17.52 ð Þcð 3Þ 1 2 S0þþ 33¯ ξ 2856/2841 897/899 1186/1197 −586=− 622 −3.69=4.74 ð Þcð 3Þ þ− 3 1 2 S1þ− 33¯ ξ ξ 2954/2943 921/926 1254/1256 −591=− 614 7.97/12.47 ð Þcð 1; 2Þ 3 2 S1þ− 33¯ ξ 2867/2851 884/895 1192/1201 −578=− 621 7.36/14.61 ð Þcð 3Þ þþ 3 1 c S1þþ 33¯ ξ ξ 2920/2934 939/926 1247/1252 −632=− 610 4.36/3.51 ð Þcð 1; 2Þ 2þ− 5 −578 − 607 2 S2þ− 33¯ ξ ξ 2977/2965 893/905 1270/1271 = 30.15/34.34 ð Þcð 1; 2Þ þþ 5 −616 − 601 2 2 S2þþ 33¯ ξ ξ 2952/2964 903/898 1267/1272 = 35.64/33.36 ð Þcð 1; 2Þ 5 −564 − 612 2 S2þþ 33¯ ξ 2888/2871 859/868 1203/1220 = 27.60/33.30 ð Þcð 3Þ

1 S0þþ 33¯ due to a strong contribution of the confine- contribute a large positive value to the mass, while the ð Þc ment potential to the nondiagonal elements. To see Coulomb-type potential hVCouli has a large cance- the contributions from each part of the Hamiltonian to lation with these two terms. The spin-spin interaction the mass of different configurations, we also present our term hVSSi, the tensor potential term hVTi, and/or the results in Table IV. It is found that both the kinetic energy spin-orbit interaction term hVLSi have also sizable con- term hTi and the linear confinement potential term hVLini tributions to some configurations. Thus, as a reliable

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FIG. 2. Mass spectrum for the sss¯ s¯ system. The solid and dotted lines stand for the results predicted by the GEM and HOEM, respectively. calculation, both the spin-independent and spin-dependent problem, the spatial wave functions containing the radial potentials should be reasonably included for the sss¯s¯ excitations are expanded with the Gaussian functions, system. For clarity, our predicted Tsss¯s¯ spectrum is plotted while the spatial wave functions containing no excitations in Fig. 2. are adopted the single Gaussian function as an approxi- mation. We have tested the single Gaussian approxima- 1 A. Discussions of the numerical method tion in the calculations of the ground STðsss¯s¯Þ states, the numerical values are reasonably consistent with those cal- Herein we discuss the differences of numerical results culated with a series of Gaussian functions. The differences between the HOEM used in present work and the GEM of the numerical results between these two methods are often adopted in the literature. For the 1S-, 1P-wave T ¯¯ ðssssÞ about 10 MeV. states, etc., there are no radial excitations. Thus, the GEM is Our numerical results for the 2S-wave T ¯¯ states with the same as the HOEM. For the first radial excited 2S-wave ðssssÞ the GEM are listed in Tables IV and V. From Table IV,itis T ¯¯ states, the HOEM is different from the GEM because ðssssÞ found that the numerical values for the 0þ− configuration the trail harmonic oscillator wave functions are different 1 þþ 1 2 S0þ− 66¯ ξ ξ and 0 configuration 2 S0þþ 66¯ ξ cal- from the Gaussian functions. ð Þcð 1; 2Þ ð Þcð 3Þ To see the differences between the two expansion culated with the HOEM are significantly different from methods, we also give our predictions of the 2S-wave those obtained with the GEM. For these two configura- tions, the predicted mass differences by the HOEM and T ¯¯ states based on the GEM. It should be mentioned ðssssÞ Q GEM can reach up to ∼70 MeV. However, for the other 3 R ξ that by fully expanding i¼1 nξ lξ ð iÞ with the GEM, 2 i i S-wave Tðsss¯s¯Þ configurations, the numerical values of one cannot distinguish the ξ1 and ξ2 excited modes 2 these two methods are comparable with each other. The which are defined for the S configurations presented differences of the predicted masses between these two in Table II. Then we cannot numerically work out the methods are about 10–20 MeV. It should be mentioned that þ− 1 masses for the following states of 0 (2 S0þ− 66¯ ξ ξ Coul ð Þcð 1; 2Þ the Coulomb-type potential hV i for the 2S-wave states 1 þ− 3 þ− and 2 S0þ− 33¯ ξ ξ ), 1 (2 S1þ− 33¯ ξ ξ ), and 1 ð Þcð 1; 2Þ ð Þcð 1; 2Þ seems to be sensitive to the numerical methods as shown 5 (2 S2þ− 33¯ ξ ξ ) listed in Table II. To overcome this in Table IV. ð Þcð 1; 2Þ

016016-8 FULLY-STRANGE TETRAQUARK SSS¯ S¯ … PHYS. REV. D 103, 016016 (2021)

TABLE V. Predicted mass spectrum for the 2S-wave sss¯ s¯ system with the GEM. þ− 1 0 2 S0þ− 66¯ ξ ξ 2927 −20 2909 −0.75 −0.67 ð Þcð 1; 2Þ ð Þ 1 −20 2931 2949 −0 67 0 75 2 S0þ− 33¯ ξ ξ ð . . Þ ð Þcð 1; 2Þ 0 1 0 1 0 1 þþ 1 2874 −48 134 −17 2796 0 2 S0þþ 66¯ ξ ξ ð −0.75 −0.41 0.23 −0.45 Þ ð Þcð 1; 2Þ 1 B −48 2918 −9 −37 C B 2835 C B −0 47 0 12 0 29 0 82 C 2 S0þþ 33¯ ξ ξ B ð . . . . Þ C ð Þcð 1; 2Þ @ A @ A @ A 1 134 −9 3148 −31 2943 ð 0.25 −0.90 −0.15 0.33 Þ 2 S0þþ 66¯ ξ ð Þcð 3Þ 1 −17 −37 −31 2841 3208 ð −0.38 0.08 −0.92 0.09 Þ 2 S0þþ 33¯ ξ ð Þcð 3Þ þ− 3 1 2 S1þ− 33¯ ξ ξ 2943 0 2851 01 ð Þcð 1; 2Þ ð Þ 3 0 2851 2943 10 2 S1þ− 33¯ ξ ð Þ ð Þcð 3Þ þþ 3 1 2 S1þþ 33¯ ξ ξ ð 2934 Þ 2934 1 ð Þcð 1; 2Þ 2þ− 5 2965 2 S2þ− 33¯ ξ ξ ð Þ 2965 1 ð Þcð 1; 2Þ 2þþ 5 2 S2þþ 33¯ ξ ξ 2964 −33 2860 ð −0.30 −0.95 Þ ð Þcð 1; 2Þ 5 −33 2871 2975 −0 95 0 30 2 S2þþ 33¯ ξ ð . . Þ ð Þcð 3Þ

þþ In brief, most of the predictions are consistent with each calculate the mass of the 0 Tsss¯s¯ state with the other between the HOEM and GEM. The uncertainties same potential model parameters. We obtain a mass of from the numerical methods do not change our main 1758 MeV, which is comparable with the prediction of predictions of the Tðsss¯s¯Þ spectrum. Although some numeri- Ref. [4], but is obviously smaller than the results without cal results for the JPC ¼ 0þ− and 0þþ 2S states show a the diquark-antidiquark approximation. significant numerical method dependence (see Fig. 2), the In the 2S-wave sector, there are four 0þþ states, 2796 2835 2943 GEM may give a slightly more accurate numerical result Tðsss¯s¯Þ0þþ ð Þ, Tðsss¯s¯Þ0þþ ð Þ, Tðsss¯s¯Þ0þþ ð Þ, and 3208 based on our tests of the charmonium spectrum. In the Tðsss¯s¯Þ0þþ ð Þ, predicted in the NRPQM. A strong 2S 66¯ 33¯ following, our discussions of the states are based on the mixing between the two color structures j ic and j ic GEM calculations. is also found among these states. In particular, the radial excitation modes ðξ1; ξ2Þ and (ξ3) strongly mix with each S 3208 B. -wave states other. The highest state Tðsss¯s¯Þ0þþ ð Þ is nearly a pure 1 ¯ 1 PC 0þþ þþ ¯ 66 There are four S-wave Tsss¯s¯ states with J ¼ , configuration of S0 ð66Þ ðξ3Þ, with the color structure j ic − c 1þ , 2þþ in the quark model. Their masses are predicted to and the radial excitation between diquark ðssÞ and anti- – 2 ¯¯ 3208 be in the range of 2.21 2.44 GeV. In contrast, the S wave diquark ðssÞ. The special color structure of Tðsss¯s¯Þ0þþ ð Þ includes 12 states. Except for the highest mass state leads to a rather large mass gap Δ ≃ 265 MeV from the T ¯¯ 0þþ 3208 2943 2 0þþ ¯¯ ðssssÞ ð Þ, their masses lie in a relative narrow range nearby Tðsss¯s¯Þ0þþ ð Þ.These S-wave ssss states of 2.78–2.98 GeV. Apart from the conventional quantum may easily decay into ϕϕ, ϕϕð1680Þ final states through − numbers, i.e., JPC ¼ 0þþ, 1þþ, 1þ , 2þþ, the 2S-wave can quark rearrangements. They may also easily decay into K0K0 − − s s access exotic quantum numbers, i.e., JPC ¼ 0þ , 2þ . and KþK− final states through the ss¯ annihilation and a pair of nonstrange qq¯ creation. One also notices that these states ++ 1. 0 states may directly decay into ΞΞ¯ baryon pair with a light qq¯ pair In the 1S-wave mutiplets, the two 0þþ ground states creation. 2218 2440 Some evidences for T ¯¯ 0þþ ð2218Þ and T ¯¯ 0þþ ð2440Þ include Tðsss¯s¯Þ0þþ ð Þ and Tðsss¯s¯Þ0þþ ð Þ. Their mass ðssssÞ ðssssÞ splitting reaches up to about 200 MeV. These two states may have been seen in the previous experiments. Recently, have a strong mixing between the two color structures j66¯i Kozhevnikov carried out a dynamical analysis of the reso- c ψ → γ → γϕϕ and j33¯ i . Their masses are much larger than the mass nance contributions to J= X [48] with the data c 0þþ ∼2 2 threshold of ϕϕ. Thus, they may easily decay into ϕϕ pair from BESIII [12].Two resonances with masses at . ∼2 4 through quark rearrangements. The mass of the lowest 0þþ and . GeV were extracted from the data. Evidence for a ϕϕ 0 → T ¯¯ in our model is close to the prediction of 2203 MeV in scalar around 2.2 GeV in the mass spectra in Bs ssss ψϕϕ the relativistic diquark-antidiquark model [3]. However, J= [49] was also reported by Ref. [50]. Considering the it turns out to be much higher than the predicted value mass and decay mode, these two scalar structures may be 2218 2440 1716 MeV by the relativized quark model with a diquark- good candidates for Tðsss¯s¯Þ0þþ ð Þ and Tðsss¯s¯Þ0þþ ð Þ. antidiquark approximation [4]. There might be some It should be mentioned that f0ð2200Þ is listed in RPP [1] 0 0 crucial dynamics missing in the diquark-antidiquark appro- as a well-established state. It has been seen in the Ks Ks, þ − ηη 2218 ximation. As a test of the diquark-antidiquark approxima- K K , and , and may be assigned to Tðsss¯s¯Þ0þþ ð Þ. þþ tion, we adopt the approximation as done in Ref. [4] and Some qualitative features can be expected: (i) the 0 Tsss¯s¯

016016-9 LIU, LIU, ZHONG, and ZHAO PHYS. REV. D 103, 016016 (2021) state can decay into ηη, η0η0, and ηη0 through quark mass is about 100 MeV larger than the lowest 1S-wave ¯ η η0 2218 – rearrangements via the ss component in the and state Tðsss¯s¯Þ0þþ ð Þ. Its mass is about 200 300 MeV mesons. An approximate branching ratio fraction can be larger than that predicted by the QCD sum rules [15] and ηη ∶ η0η0 ∶ ηη0 ≃ 4 α ∶ 4 α ∶ examined: BRð Þ BRð Þ BRð Þ sin P cospffiffiffiP the relativized quark model [4] in the diquark picture. The 2 2 1þ− 2sin αP cos αP ≃ 0.24∶0.25∶0.50, with αP ≡ arctan 2 þ mass of the state may be notably underestimated in the θP ≃ 44.7° and without including the phase space factors. diquark picture. As a comparison, we calculate the mass of þþ 0 0 1þ− (ii) The 0 states may also easily decay into KsKs and the state in the diquark picture using the same potential KþK− final states through annihilating a pair of ss¯ and model parameter set adopted in present work, and obtain a creating a pair of light qq¯. (iii) It is interesting to note mass of 1936 MeV, which is 300 MeV smaller than the that no conventional 0þþ ss¯ states are predicted around NRQPM prediction 2236 MeV. − 2323 ηϕ η0ϕ 2.2 GeV in most literatures [2]. Tðsss¯s¯Þ1þ ð Þ may easily decay into and 0þþ 2218 0 To establish the ground states Tðsss¯s¯Þ0þþ ð Þ and through the quark rearrangements. The decay of ψ =J=ψ → 2440 ϕηη0 ηϕ η0ϕ Tðsss¯s¯Þ0þþ ð Þ, a combined study of decay channels, such can access this state in and channels. It should 0 0 þ − 0 0 0 þ− 2323 as ϕϕ, Ks Ks, K K , ηη, η η , and ηη , should be necessary. be mentioned that some hints of Tðsss¯s¯Þ1 ð Þ may have The 2S-wave 0þþ states can be probed in these meson been found in the η0ϕ invariant mass spectrum around 2.3– pair decay channels including higher channels such as 2.4 GeV by observing the J=ψ → ϕηη0 reaction at BESIII ϕϕð1680Þ, and some baryon pair decay channels such recently [13]. as ΞΞ¯ . For the 2S-wave sector, there are two states, i.e., − 1þ ¯¯ − 2851 − 2943 ssssTðsss¯s¯Þ1þ ð Þ and Tðsss¯s¯Þ1þ ð Þ predicted 2. 2 ++ states in the quark model. There are no configuration mixings in ¯ 2þþ 2378 1 these two states. T ¯¯ 1þ− ð2851Þ has a pure j33i color There is only one state Tðsss¯s¯Þ2þþ ð Þ in the S- ðssssÞ c wave states. This state lies between the two 0þþ ground structure, and the radial excitation occurs between diquark 33¯ 2378 ðssÞ and antidiquark ðs¯s¯Þ (i.e., the ξ3 mode). In contrast, states and has a pure j ic color structure. Tðsss¯s¯Þ2þþ ð Þ − 2943 33¯ may have large decay rates into the ϕϕ, ηη, and η0η0 final although Tðsss¯s¯Þ1þ ð Þ has also a pure j ic color states through quark rearrangements and/or into KðÞK¯ ðÞ structure, its radial excitation occurs in the diquark ðssÞ ¯¯ ηϕ η0ϕ final states through the annihilation of ss¯ and creation of a and antidiquark ðssÞ. Apart from the and decay pair of nonstrange qq¯. It should be mentioned that with the channels, it may favor decays into a pseudoscalar plus a ηϕ 1680 η0ϕ 1680 diquark-antidiquark approximation, the mass of the 2þþ radially excited vector [i.e., ð Þ and ð Þ], or a η 1295 ϕ state is predicted to be 2192 MeV, which is about 200 MeV radially excited pseudoscalar plus a vector [i.e., ð Þ η 1405 ϕ lower than the four-body calculation results. and ð Þ ], through the quark rearrangements. The f2ð2340Þ resonance listed in RPP [1] may be It should be mentioned that in Refs. [9,15] the authors suggest that the new structure X 2063 observed in the assigned to T ¯¯ 2þþ ð2378Þ. Besides the measured mass ð Þ ðssssÞ 0 þ− 50 J=ψ → ϕηη at BESIII [13] could be a 1 T ¯¯ candidate 2345þ MeV, the observed decay modes ϕϕ and ηη are ssss −40 according to the QCD sum rule calculation. However, the consistent with the expectation of the tetraquark scenario. observed mass of Xð2063Þ is too small to be comparable On the other hand, as a conventional ss¯ state, the f2ð2340Þ with our quark model predictions. cannot be easily accommodated by the quark model expectation [2]. The relativistic quark model calculation 4. 0 + − and 2 + − states of Ref. [3] also supports the f2ð2340Þ to be assigned as the þþ þ− Tsss¯s¯ ground state with 2 . To confirm this assignment, In the 2S-wave multiplets, there are two 0 states, 2378 ηη0 − 2909 − 2949 the other main decay modes of Tðsss¯s¯Þ2þþ ð Þ such as , Tðsss¯s¯Þ0þ ð Þ and Tðsss¯s¯Þ0þ ð Þ, predicted in the η0η0, KðÞK¯ ðÞ should be investigated in experiment. NRPQM. There is a strong configuration mixing between þþ 1 1 þ− For the 2S-wave sector, there are two 2 sss¯s¯ states S0þ− 66¯ ξ ξ and S0þ− 33¯ ξ ξ . There is only one 2 ð Þcð 1; 2Þ ð Þcð 1; 2Þ T ¯¯ 2þþ 2860 T ¯¯ 2þþ 2975 − 2965 ðssssÞ ð Þ and ðssssÞ ð Þ predicted in our state Tðsss¯s¯Þ2þ ð Þ corresponding to the configuration 5 5 þ− þ− model, which are dominated by the S2þþ 33¯ ξ and þ− ¯ 0 2 ð Þcð 3Þ S2 ð33Þ ðξ1;ξ2Þ. The and are exotic quantum 5 c S2þþ 33¯ ξ ξ configurations, respectively. Their masses ð Þcð 1; 2Þ numbers which cannot be accommodated by the conven- are predicted to be above the thresholds of ϕϕ, ϕϕð1680Þ tional qq¯ scenario. The P-wave decays into the ηh1ð1PÞ ¯ 0 and Ξð1530ÞΞ. Therefore, experimental search for their and η h1ð1PÞ channels could be useful for the search for signals in these decay channels should be helpful for these states in experiments. understanding these tensor tetraquarks. C. 1P-wave states 3. 1 + − states There are twenty 1P-wave Tsss¯s¯ states predicted in the 1 − 2323 In the S-wave multiplets, Tðsss¯s¯Þ1þ ð Þ is the only NRPQM. Apart from the conventional quantum numbers, −1 33¯ PC 0−þ 1−− 2−þ 2−− 3−− state with C ¼ and has a pure j ic color structure. Its i.e., J ¼ , , , , , the P wave can access

016016-10 FULLY-STRANGE TETRAQUARK SSS¯ S¯ … PHYS. REV. D 103, 016016 (2021) exotic quantum numbers, i.e., JPC ¼ 0−−; 1−þ. The masses two-point sum rule method [53]. However, our calculations ϕ 2170 2239 of the 1P-wave Tsss¯s¯ states scatter in a wide range of indicate that neither ð Þ nor Xð Þ can be assigned −− about 2.4–3.0 GeV. The masses of the low-lying 1P-wave to a 1 Tsss¯s¯ state since their measured masses are states may highly overlap with the heaviest 1S-wave much lower than our predictions. It should be mentioned 2440 ϕ 2170 state Tðsss¯s¯Þ0þþ ð Þ. that in recent studies ð Þ was considered as a vector tetraquark state with content sus¯ u¯ rather than as a state 1. 0 − + states sss¯s¯ [23,54]. −− − The 1 T ¯¯ states may have large decay rates into the 0 þ − 2481 ssss There are three states, Tðsss¯s¯Þ0 þ ð Þ, 0 f0ð980Þϕ via an S wave, or into ηϕ and η ϕ via a P wave. T ¯¯ 0−þ 2635 , and T ¯¯ 0−þ 2761 , predicted in the ðssssÞ ð Þ ðssssÞ ð Þ There are some experimental evidences for structures NRPQM. They are mixed states with two color structures around 2.4 GeV observed in the f0ð980Þϕ invariant mass 66¯ 33¯ j ic and j ic and also mixed states between two orbital spectrum from BABAR [55,56], Belle [57], BESII [58], and ξ ξ ξ excitations ð 1; 2Þ and 3 modes. They can decay into BESIII [59], which could be signals of the 1−− sss¯s¯ tetra- ϕ 1 ϕϕ h1ð PÞ via an S wave, or via a P wave, through the −− 2766 quark states [60]. For the heavier states Tðsss¯s¯Þ1 ð Þ and quark rearrangements. ¯ T ¯¯ 1−− ð2984Þ, they can also decay into ΞΞ baryon pair In 2016, the BESIII Collaboration observed a new ðssssÞ ¯ 2500 2470þ15þ101 through a qq pair production in vacuum. Thus, exper- resonance Xð Þ with a mass of −19 −23 MeV þ − ¯ 64 imental search for these states in e e → ΞΞ should be and a width of 230þ þ56 MeV in J=ψ → γϕϕ [12]. The −35 −33 very interesting. preferred spin-parity numbers for Xð2500Þ are JPC ¼ 0−þ [12]. The Xð2500Þ resonance may be a candidate for 3. − + states − 2481 1 Tðsss¯s¯Þ0 þ ð Þ in terms of mass, decay modes, and − 1 1 þ − 2564 quantum numbers although Xð2500Þ may favor the 4 S0 There are three states, Tðsss¯s¯Þ1 þ ð Þ, ¯ − 2632 − 2778 ss state as suggested in our previous work [2]. In the recent Tðsss¯s¯Þ1 þ ð Þ, and Tðsss¯s¯Þ1 þ ð Þ, predicted in the work of Ref. [14], the authors also suggested Xð2500Þ NRPQM. Note that 1−þ are exotic quantum numbers 0−þ to be a Tsss¯s¯ state according to the QCD sum rule which cannot be accommodated by the conventional studies. A measurement of the branching fraction of qq¯ scenario. Both the lowest mass state T ¯¯ 1−þ ð2564Þ B 2500 → ϕϕ ðssssÞ ½Xð Þ might provide a test of the nature of − 2778 and highest mass state Tðsss¯s¯Þ1 þ ð Þ are mixed states Xð2500Þ. The decay rate of T ¯¯ 0−þ ð2481Þ into ϕϕ ðssssÞ between the two color structures j66¯i and j33¯ i , and their through the quark rearrangements should be significantly c c orbital excitations are dominated by the ðξ1; ξ2Þ mode. ¯ 41 larger than that via an ss pair production for the S0 − 2632 The middle state Tðsss¯s¯Þ1 þ ð Þ is dominated by the ss¯ state. 3 P1−þ 33¯ ξ configuration of which the orbital excitation ð Þcð 3Þ 2. 1 −− states mainly occurs between the diquark ðssÞ and antidiquark ¯¯ −− ðssÞ. It should be noted that a corresponding state in the 1 −− 2445 There are five states, Tðsss¯s¯Þ1 ð Þ, relativized quark model [4] has a mass of 2581 MeV, which −− 2567 −− 2627 −− 2766 Tðsss¯s¯Þ1 ð Þ, Tðsss¯s¯Þ1 ð Þ, Tðsss¯s¯Þ1 ð Þ, and is about 50 MeV smaller than our prediction. −− 2984 −þ Tðsss¯s¯Þ1 ð Þ, predicted in the NRPQM. Their masses These 1 Tsss¯s¯ states may easily decay into ϕh1ð1PÞ, – ð0Þ scatter in a rather wide range of about 2.4 3.0 GeV. η f1ð1420Þ, ϕϕ channels through the quark rearrange- From Table III, it is found that there are obvious con- ments. They can be searched for in χcJð1PÞ → ηϕϕ; ϕK K figuration mixings in these tetraquark states except that with sufficient χ ð1PÞ data samples at BESIII, although no 1 cJ T ¯¯ 1−− ð2627Þ may nearly be a pure P1−− 33¯ ξ state. ðssssÞ ð Þcð 3Þ obvious structures were found in previous observa- −− 2445 The lowest state Tðsss¯s¯Þ1 ð Þ is dominated by the tions [61,62]. 3 P1−− 66¯ ξ ξ configuration. Its orbital excitation mainly ð Þcð 1; 2Þ occurs within the diquark ðssÞ or antidiquark ðs¯s¯Þ. 4. 2 − + states −− 2984 − Meanwhile, the highest state Tðsss¯s¯Þ1 ð Þ is dominated 2 þ − 2537 1 There are three states, Tðsss¯s¯Þ2 þ ð Þ, −− ¯ by the P1 ð66Þ ðξ3Þ configuration, and the orbital excitation − 2669 − 2837 c Tðsss¯s¯Þ2 þ ð Þ, and Tðsss¯s¯Þ2 þ ð Þ, predicted in the occurs between the diquark ðssÞ and antidiquark ðs¯s¯Þ. NRPQM. Both the lowest mass state T sss¯s¯ 2−þ ð2537Þ The vector meson ϕð2170Þ in RPP [1] is suggested to be ð Þ − 2837 −− and highest mass state Tðsss¯s¯Þ2 þ ð Þ are mixed states a 1 T ¯¯ state in the literature [5–10] since it is hard to be ssss 66¯ 33¯ explained as a conventional meson state according to its between the two color structures j ic and j ic. Their ξ ξ measured decay modes [51,52]. Furthermore, the Xð2239Þ orbital excitations are dominated by the ð 1; 2Þ mode. The − 2669 resonance, which was observed in eþe− → KþK− by the middle state Tðsss¯s¯Þ2 þ ð Þ is dominated by the 3 P2−þ 33¯ ξ configuration of which the orbital excitation BESIII Collaboration [11], was suggested to be a candidate ð Þcð 3Þ −− ¯¯ of the lowest 1 Tsss¯s¯ state by comparing with the mass occurs between the diquark ðssÞ and antidiquark ðssÞ.A spectrum from the relativized quark model [4] and the QCD corresponding state in the relativized quark model [4] has a

016016-11 LIU, LIU, ZHONG, and ZHAO PHYS. REV. D 103, 016016 (2021) mass of 2619 MeV, which is about 50 MeV smaller than IV. SUMMARY our prediction. 1 −þ In this work, we calculate the mass spectra for the S, These 2 Tsss¯s¯ states may easily fall apart into ϕh1ð1PÞ 1P, and 2S-wave T ¯¯ states in a nonrelativistic potential and ηf0 ð1525Þ in an S wave, or into ϕϕ in a P wave ssss 2 quark model without the often-adopted diquark-antidiquark through the quark rearrangements. For the high mass state approximation. The 1S-wave ground states lie in the − 2837 ΞΞ¯ Tðsss¯s¯Þ2 þ ð Þ, the strong decay mode also opens. mass range of ∼2.21–2.44 GeV, while the 1P- and χ 1 → η − → These states can be searched for in c2ð PÞ Tðsss¯s¯Þ2 þ 2S-wave states scatter in a rather wide mass range of ηη 0 1525 → ηη ¯ χ 1 f2ð Þ KK at BESIII with the sufficient c2ð PÞ ∼2.44–2.99 GeV. For the 2S-wave states, except for the 3208 data samples. highest state Tðsss¯s¯Þ0þþ ð Þ, all the other states lie in a relatively narrow range of ∼2.78–2.98 GeV. We find that 5. 0 −− states most of the physical states are mixed states with different configurations. There are two states with exotic quantum numbers of −− For the sss¯s¯ system, it shows that both the kinetic energy 0 , T ¯¯ 0−− ð2507Þ and T ¯¯ 0−− ð2821Þ, predicted in ðssssÞ ðssssÞ hTi and the linear confinement potential hVLini contribute a the NRPQM. These two states have a strong mixing large positive value to the mass, while the Coulomb-type 66¯ 33¯ between the two color structures j ic and j ic. The potential hVCouli has a large cancellation with these two orbital excitation is the ðξ1; ξ2Þ mode, i.e., the excitation terms. The spin-spin interaction hVSSi, tensor potential occurs within the diquark ðssÞ or antidiquark ðs¯s¯Þ. These hVTi, and/or the spin-orbit interaction term hVLSi also have two states may have large decay rates into ϕf1ð1285Þ and 0 sizable contributions to some configurations. ϕf1ð1420Þ in an S wave, or into ηϕ and η ϕ in a P wave Some T ¯¯ states may have shown hints in experiment. through the quark rearrangements. These 0−− exotic states ssss For instance, the observed decay modes and masses may be produced by the reactions eþe− → ηð0ÞX → of f0ð2200Þ and f2ð2340Þ listed in RPP [1] could be ηð0Þηð0Þϕ or J=ψ → ηð0Þηð0Þϕ. 2218 good candidates for the ground states Tðsss¯s¯Þ0þþ ð Þ 2378 and Tðsss¯s¯Þ2þþ ð Þ, respectively. The newly observed −− 6. 2 states Xð2500Þ at BESIII may be a candidate for the lowest mass − −− þ − þþ −− 1P-wave 0 state T ¯¯ 0 þ ð2481Þ. Another 0 ground There are three 2 states, T sss¯s¯ 2 ð2446Þ, ðssssÞ ð Þ 2440 ϕϕ −− 2657 −− 2907 state T ¯¯ 0þþ ð Þ may have shown signals in the Tðsss¯s¯Þ2 ð Þ, and Tðsss¯s¯Þ2 ð Þ, predicted in the ðssssÞ NRPQM. Both the lowest mass state T ¯¯ 2−− ð2446Þ channel at BESIII [12,48]. Our calculation shows that ðssssÞ ϕ 2170 −− 2907 ð Þ may not favor a vector state of Tsss¯s¯, because of and highest mass state Tðsss¯s¯Þ2 ð Þ are mixed states 66¯ 33¯ the much higher mass obtained in our model. between the two color structures, j ic and j ic, and their It should be stressed that as a flavor partner of Tccc¯ c¯ , the orbital excitations are dominated by the ðξ1; ξ2Þ mode. Tsss¯s¯ system may have very different dynamic features −− 2657 The middle state Tðsss¯s¯Þ2 ð Þ is dominated by the 3 that need further studies. One crucial point is that the P2−− 33¯ ξ configuration, of which the orbital excitation ð Þcð 3Þ strange quark is rather light and the light flavor mixing occurs between the diquark ðssÞ and antidiquark ðs¯s¯Þ.A effects could become non-negligible. It suggests that strong corresponding state in the relativized quark model [4] has a couplings between Tsss¯s¯ and open strangeness channels mass of 2622 MeV, which is consistent with our prediction. could be sizable. As a consequence, mixings between Tsss¯s¯ 2−− ϕ 1285 These states may easily decay into f1ð Þ, and Tsqs¯ q¯ would be inevitable. For an S-wave strong ϕ 1420 ϕ 0 1525 ηϕ f1ð Þ, and f2ð Þ in an S wave, or into , coupling, it may also lead to configuration mixings which η0ϕ in a P wave through the quark rearrangements. They can be interpreted as hadronic molecules for a near-thresh- − can also be searched for in eþe → ηð0ÞX → ηð0Þηð0Þϕ or old structure. In such a sense, this study can set up a vector charmonium decays such as J=ψ → ηηϕ. reference on the basis of orthogonal states. More elaborated dynamics can be investigated by including the hadron 7. 3 −− state interactions in the Hamiltonian. For states with exotic −− quantum numbers, experimental searches for their signals 3 −− 2719 There is only one state Tðsss¯s¯Þ3 ð Þ predicted in can be carried out at BESIII and Belle-II. 33¯ the NRPQM. This state has a pure color structure j ic and also a pure orbital excitation between the diquark ðssÞ and ACKNOWLEDGMENTS antidiquark ðs¯s¯Þ. Our predicted mass is about 60 MeV larger than that predicted by the relativized quark model [4] The authors thank Wen-Biao Yan for useful discussions with a diquark approximation. The 3−− states may easily on the BESIII results. Helpful discussions with Atsushi 0 decay into ϕf2ð1525Þ in an S wave by the quark rearrange- Hosaka and Qi-Fang Lü are greatly appreciated. This work ments. Since it has a high spin, it may be produced was supported by the National Natural Science Founda- relatively easier in pp¯ or pp collisions. tion of China (Grants No. 11775078, No. U1832173,

016016-12 FULLY-STRANGE TETRAQUARK SSS¯ S¯ … PHYS. REV. D 103, 016016 (2021)

No. 11705056, No. 11425525, and No. 11521505). The No. 12070131001), the Strategic Priority Research work of Q. Z. was also supported in part, by the DFG and Program of Chinese Academy of Sciences (Grant NSFC funds to the Sino-German CRC 110 “Symmetries No. XDB34030302), and National Key Basic Research and the Emergence of Structure in QCD” (NSFC Grant Program of China under Contract No. 2015CB856700.

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