PRDPRVDAQPhysical Review DPhys. Rev. D2470-00102470-0029American Physical Society10.1103/PhysRevD.103.016016ARTICLESPhenomenological aspects of field theory, general methodsFully strange tetraquark sss¯s¯ spectrum and possible experimental evidenceFULLY-STRANGE TETRAQUARK SSS¯S¯ …FENG-XIAO LIU et al.https://orcid.org/0000-0003-1718-8081LiuFeng-Xiao1,2LiuMing-Sheng1,2ZhongXian-Hui1,2,*ZhaoQiang3,4,2,†Department of Physics, Hunan Normal University, and Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Changsha 410081, ChinaSynergetic Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal University, Changsha 410081, ChinaInstitute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, ChinaUniversity of Chinese Academy of Sciences, Beijing 100049, China
zhongxh@hunnu.edu.cn
zhaoq@ihep.ac.cn
26January20211January2021103101601621August20205January2021Published by the American Physical Society2021authorsPublished by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.
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 f0(2200) and f2(2340) may favor the assignments of ground states T(sss¯s¯)0++(2218) and T(sss¯s¯)2++(2378), respectively, and the newly observed X(2500) at BESIII may be a candidate of the lowest mass 1P-wave 0-+ state T(sss¯s¯)0-+(2481). Signals for the other 0++ ground state T(sss¯s¯)0++(2440) may also have been observed in the ϕϕ invariant mass spectrum in J/ψ→γϕϕ at BESIII. The masses of the JPC=1--Tsss¯s¯ states are predicted to be in the range of ∼2.44–2.99GeV, which indicates that the ϕ(2170) 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.
National Natural Science Foundation of China10.13039/50110000180911775078U183217311705056114255251152150512070131001Deutsche Forschungsgemeinschaft10.13039/501100001659Chinesisch-Deutsche Zentrum für Wissenschaftsförderung10.13039/501100010007CRC 110INTRODUCTION
From the Review of Particle Physics (RPP) of Particle Data Group [1], above the mass range of 2.0 GeV one can see that there are several unflavored qq¯ isoscaler states, such as f0(2200), f2(2150), f2(2300), f2(2340), etc., dominantly decaying into ϕϕ, ηη, and/or KK¯ final states. The decay modes indicate that these states might be good candidates for conventional ss¯ meson resonances. Recently, we carried out a systematical study of the mass spectrum and strong decay properties of the ss¯ system in Ref. [2]. It shows that these states cannot be easily accommodated by the conventional ss¯ meson spectrum. While they may be candidates for tetraquark sss¯s¯ (Tsss¯s¯) states, it is easy to understand that they can fall apart into ϕϕ and ηη final states through quark rearrangements, or easily decay into KK¯ final states through a pair of ss¯ annihilations and then a pair of light quark creations. The mass analysis with the relativistic quark model in Ref. [3] supports the f0(2200) and f2(2340) to be assigned as the Tsss¯s¯ ground states with 0++ and 2++, respectively. However, a relativized quark model calculation [4] only favors f2(2300) to be a Tsss¯s¯ state.
Some other candidates of the Tsss¯s¯ states from experiment are also suggested in the literature. For example, the vector meson resonance ϕ(2170) listed in RPP [1] is suggested to be a 1--Tsss¯s¯ state based on the mass analysis of QCD sum rules [5–9] and flux-tube model [10]. The newly observed X(2239) resonance in the e+e-→K+K- process at BESIII [11] is suggested to be a candidate of the lowest 1--Tsss¯s¯ state in a relativized quark model [4]. Moreover, the newly observed resonances X(2500) observed in J/ψ→γϕϕ[12] and X(2060) observed in J/ψ→ϕηη′[13] at BESIII are suggested to be 0-+ and 1+-Tsss¯s¯ states, respectively, according to the QCD sum rule studies [14,15]. The assignment of X(2500) is consistent with that in Ref. [4].
With the recent experimental progresses, more quantitative studies on the Tsss¯s¯ states can be carried out and their evidences can also be searched for in experiments. Very recently, the LHCb Collaboration reported their results on the observations of tetraquark ccc¯c¯ (Tccc¯c¯) states [16]. A broad structure above the J/ψJ/ψ threshold ranging from 6.2 to 6.8 GeV and a narrower structure Tccc¯c¯(6900) are observed with more than 5σ of significance level. There are also some vague structure around 7.2 GeV to be confirmed. These observations could be evidences for genuine Tccc¯c¯ states [17–21].
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 narrowness of Tccc¯c¯(6900) suggests that there should be more profound mechanism that “slows down” the fall-apart decays of such a tetraquark system. Although this may be related to the properties of the static potential of heavy quark systems, more direct evidences are still needed to disentangle the dynamical features between the heavy and light flavor systems. As an analogy of the Tccc¯c¯ system, there might exist stable Tsss¯s¯ states above the ϕϕ threshold and can likely be observed in the di-ϕ mass spectrum. On the other hand, flavor mixings could be important for the light flavor systems and pure sss¯s¯ states may not exist. To answer such questions, systematic calculations of the sss¯s¯ system should be carried out. The BESIII experiments can provide a large data sample for the search of the Tsss¯s¯ states in J/ψ and ψ(2S) decays. In theory, although there have been some predictions of the Tsss¯s¯ spectrum within the quark model [3,4,10] and QCD sum rules [17–20], most of the studies focus on some special states in a diquark-antidiquark picture. About the status of the tetraquark states, some recent review works can be referenced [22,23]. In this study, we intend to provide a systematical calculation of the mass spectrum of the 1S, 1P, and 2S-wave Tsss¯s¯ states without the diquark-antidiquark approximation in a nonrelativistic potential quark model (NRPQM).
The NRPQM is based on the Hamiltonian proposed by the Cornell model [24], which contains a linear confinement and a one-gluon-exchange potential for quark-quark and quark-antiquark interactions. With the NRPQM, we have successfully described the ss¯, cc¯, and bb¯ meson spectra [2,25,26], and sss, ccc, and bbb baryon spectra [27,28]. Furthermore, we adopted the NRPQM for the study of both 1S and 1P-wave all-heavy tetraquark states with a Gaussian expansion method (GEM) [21,29]. In this work, we continue to extend this method to study the Tsss¯s¯ spectrum by constructing the full tetraquark configurations without the diquark-antidiquark approximation. With the parameters determined in our study of the ss¯ spectrum [2], we obtain a relatively reliable prediction of the mass spectrum for 36 Tsss¯s¯ states, i.e., 4 1S-wave ground states, 20 1P-wave orbital excitations, and 12 2S-wave radial excitations.
The paper is organized as follows: a brief introduction to the tetraquark spectrum is given in Sec. II. In Sec. III, the numerical results and discussions are presented. A short summary is given in Sec. IV.
MASS SPECTRUMHamiltonian
We adopt a NRPQM to calculate the mass spectrum of the sss¯s¯ system. In this model, the Hamiltonian is given by H=(∑i=14mi+Ti)-TG+∑i<jVij(rij),where mi and Ti stand for the constituent quark mass and kinetic energy of the ith quark, respectively; TG stands for the center-of-mass (c.m.) kinetic energy of the tetraquark system, rij≡|ri-rj| is the distance between the ith and jth quark, and Vij(rij) stands for the effective potential between them. In this work, the Vij(rij) adopts a widely used form [24–26,30–36]Vij(rij)=Vijconf(rij)+Vijsd(rij),where the confinement potential adopts the standard form of the Cornell potential [24], which includes the spin-independent linear confinement potential VijLin(rij)∝rij and Coulomb-like potential VijCoul(rij)∝1/rij, Vijconf(rij)=-316(λi·λj)(bijrij-43αijrij+C0).The constant C0 stands for the zero point energy. While the spin-dependent potential Vijsd(rij) is the sum of the spin-spin contact hyperfine potential VijSS, the spin-orbit potential VijSS, and the tensor term VijT, Vijsd(rij)=VijSS+VijT+VijLS,with VijSS=-αij4(λi·λj){π2·σij3e-σij2rij2π3/2·163mimj(Si·Sj)},VijLS=-αij16λi·λjrij3(1mi2+1mj2+4mimj){Lij·(Si+Sj)}-αij16λi·λjrij3(1mi2-1mj2){Lij·(Si-Sj)},VijT=-αij4(λi·λj)·1mimjrij3{3(Si·rij)(Sj·rij)rij2-Si·Sj}.In the above equations, Si stands for the spin of the ith quark, and Lij stands for the relative orbital angular momentum between the ith and jth quark. If the interaction occurs between two quarks or antiquarks, the λi·λj operator is defined as λi·λj≡∑a=18λiaλja, while if the interaction occurs between a quark and an antiquark, the λi·λj operator is defined as λi·λj≡∑a=18-λiaλja*, where λa* is the complex conjugate of the Gell-Mann matrix λa. The parameters bij and αij denote the strength of the confinement and strong coupling of the one-gluon-exchange potential, respectively.
The five parameters ms, αss, σss, bss, and C0 have been determined by fitting the mass spectrum of the strangeonium in our previous work [2]. The quark model parameters adopted in this work are collected in Table I.
I10.1103/PhysRevD.103.016016.t1
Quark model parameters used in this work.
ms (GeV)0.60αss0.77σss (GeV)0.60b (GeV2)0.135C0 (GeV)-0.519Configurations classified in the quark model
To calculate the spectroscopy of a qqq¯q¯ (q∈{s,c,b}) system, first we construct the configurations in the product space of flavor, color, spin, and spatial parts.
In the color space, there are two color-singlet bases |66¯⟩c and |3¯3⟩c, and their wave functions are given by |66¯⟩c=126[(rb+br)(b¯r¯+r¯b¯)+(gr+rg)(g¯r¯+r¯g¯)+(gb+bg)(b¯g¯+g¯b¯)+2(rr)(r¯r¯)+2(gg)(g¯g¯)+2(bb)(b¯b¯)],|3¯3⟩c=123[(br-rb)(b¯r¯-r¯b¯)-(rg-gr)(g¯r¯-r¯g¯)+(bg-gb)(b¯g¯-g¯b¯)].
In the spin space, there are six spin bases, which are denoted by χSS12S34. Where S12 stands for the spin quantum numbers for the diquark (q1q2) (or antidiquark (q¯1q¯2)), while S34 stands for the spin quantum number for the antidiquark (q¯3q¯4) (or diquark (q3q4)). S is the total spin quantum number of the tetraquark qqq¯q¯ system. The spin wave functions χSSzS12S34 with a determined Sz (Sz stands for the third component of the total spin S) can be explicitly expressed as follows: χ0000=12(↑↓↑↓-↑↓↓↑-↓↑↑↓+↓↑↓↑),χ0011=112(2↑↑↓↓-↑↓↑↓-↑↓↓↑-↓↑↑↓-↓↑↓↑+2↓↓↑↑),χ1101=12(↑↓↑↑-↓↑↑↑),χ1110=12(↑↑↑↓-↑↑↓↑),χ1111=12(↑↑↑↓+↑↑↓↑-↑↓↑↑-↓↑↑↑),χ2211=↑↑↑↑.
In the spatial space, we define the relative Jacobi coordinates with the single-partial coordinates ri (i=1, 2, 3, 4), ξ1≡r1-r2,ξ2≡r3-r4,ξ3≡m1r1+m2r2m1+m2-m3r3+m4r4m3+m4,R≡m1r1+m2r2+m3r3+m4r4m1+m2+m3+m4.Note that ξ1 and ξ2 stand for the relative Jacobi coordinates between two quarks q1 and q2 (or antiquarks q¯1 and q¯2), and two antiquarks q¯3 and q¯4 (or quarks q3 and q4), respectively. While ξ3 stands for the relative Jacobi coordinate between diquark qq and antidiquark q¯q¯. Using the above Jacobi coordinates, it is easy to obtain basis functions that have well-defined symmetry under permutations of the pairs (12) and (34) [37].
In the Jacobi coordinate system, the spatial wave function ΨNLM(ξ1,ξ2,ξ3,R) for a qqq¯q¯ system with principal quantum number N and orbital angular momentum quantum numbers LM may be expressed as the linear combination of Φ(R)ψα1(ξ1)ψα2(ξ2)ψα3(ξ3), ΨNLM(ξ1,ξ2,ξ3,R)=∑α1,α2,α3Cα1,α2,α3[Φ(R)ψα1(ξ1)ψα2(ξ2)ψα3(ξ3)]NLM,where Cα1,α2,α3 stands for the combination coefficients, Φ(R) is the c.m. motion wave function. In the quantum number set αi≡{nξi,lξi,mξi}, nξi is the principal quantum number, lξi is the angular momentum, and mξi is its third component projection. The wave functions ψαi(ξi), which account for the relative motions, can be written as ψαi(ξi)=Rnξilξi(ξi)Ylξimξi(ξi^),where Ylξimξi(ξi^) is the spherical harmonic function, and Rnξilξi(ξi) is the radial part. It is seen that for an excited state, there are three spatial excitation modes corresponding to three independent internal wave functions ψαi(ξi) (i=1, 2, 3), which are denoted as ξ1, ξ2, and ξ3, respectively, in the present work. One point should be emphasized that considering the fact that the sss¯s¯ system is composed of equal mass constituent quarks and antiquarks, we adopt a single set of Jacobi coordinates in this study as an approximation. In fact, the four-body wave function describing a scalar sss¯s¯ state contains a small contribution of internal angular momentum. This contribution is neglected in our calculations. To precisely treat an N-body system, one can involve several different sets of Jacobi coordinates as those done in Refs. [38–43], or adopt a single set of Jacobi coordinates X=(ξi,ξ2,…,ξN-1) with nondiagonal Gaussians e-XAXT as those done in Refs. [44–47], where A is a symmetric matrix.
Taking into account the Pauli principle and color confinement for the four-quark system qqq¯q¯, we have four configurations for 1S-wave ground states, 20 configurations for the 1P-wave orbital excitations, and 12 configurations for the 2S-wave radial excitations. The spin-parity quantum numbers, notations, and wave functions for these configurations are presented in Table II. With the wave functions for all the configurations, the mass matrix elements of the Hamiltonian can be worked out.
II10.1103/PhysRevD.103.016016.t2
Configurations for the tetraquark qqq¯q¯ system up to the 2S-wave states.
To work out the matrix elements in the coordinate space, we expand the radial part Rnξilξi(ξi) with a series of harmonic oscillator functions [21,27], Rnξilξi(ξi)=∑ℓ=1nCξiℓϕnξilξi(ωξiℓ,ξi),with ϕnξilξi(ωξiℓ,ξi)=(μξiωξiℓ)34[2lξ+2-nξ(2lξi+2nξi+1)!!πnξi![(2lξi+1)!!]2]12(μξiωξiℓξi)lξi×e-12μξiωξiℓξi2F(-nξi,lξi+32,μξiωξiℓξi2),where F(-nξi,lξi+32,μξiωξiℓξi2) is the confluent hypergeometric function. It should be pointed out that if there are no radial excitations, the expansion method with harmonic oscillator wave functions (HOEM) is just the same as the Gaussian expansion method adopted in the literature [38,39].
For an sss¯s¯ system, if we ensure that the spatial wave function with Jacobi coordinates can transform into the single particle coordinate system, the harmonic oscillator frequencies ωξiℓ (i=1, 2, 3) can be related to the harmonic oscillator stiffness factor Kℓ with ωξ1ℓ=2Kℓ/μξ1, ωξ2ℓ=2Kℓ/μξ2, and ωξ3ℓ=4Kℓ/μξ3. Considering the reduced masses μξ1=μξ2=ms/2, μξ3=ms for T(sss¯s¯), one has ωξ1ℓ=ωξ2ℓ=ωξ3ℓ=4Kℓ/ms. It indicates that the harmonic oscillator frequencies ωξiℓ for T(sss¯s¯) are not independent. According to the relation ωξ1ℓ=ωξ2ℓ=ωξ3ℓ=ωℓ, the expansion of ∏i=13Rnξilξi(ξi) can be simplified as ∏i=13Rnξilξi(ξi)=∑ℓn∑ℓ′n∑ℓ′′nCξ1ℓCξ2ℓ′Cξ3ℓ′′ϕnξ1lξ1(ωξ1ℓ,ξ1)ϕnξ2lξ2(ωξ2ℓ′,ξ2)ϕnξ3lξ3(ωξ3ℓ′′,ξ3)δℓℓ′δℓℓ′′=∑ℓnCℓϕnξ1lξ1(ωℓ,ξ1)ϕnξ2lξ2(ωℓ,ξ2)ϕnξ3lξ3(ωℓ,ξ3).
Then we introduce oscillator length parameters dℓ that can be related to the harmonic oscillator frequencies ωℓ with 1/dℓ2=msωℓ. Following the method of Refs. [38,39], we let the dℓ parameters form a geometric progression dℓ=d1aℓ-1(ℓ=1,…,n),where n is the number of harmonic oscillator functions and a is the ratio coefficient. There are three parameters {d1,dn,n} to be determined through the variation method. It is found that with the parameter set {0.085fm,3.399fm,15} 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 parameter α. As an example, we plot the masses of 12 2S-wave Tsss¯s¯ configurations as a function of the scaling factor α in Fig. 1. It is found that the numerical results are nearly independent of the scaling factor α. The stabilization of other states predicted in this work has also been examined by the same method.
110.1103/PhysRevD.103.016016.f1
Predicted masses of 12 2S-wave Tsss¯s¯ configurations as a function of the scaling factor α.
With the mass matrix elements ready for each configuration, the mass of the tetraquark configuration and its spacial wave function can be determined by solving a generalized eigenvalue problem. The details can be found in our previous works [27,29]. Finally, the physical states can be obtained by diagonalizing the mass matrix of different configurations with the same JPC numbers.
RESULTS AND DISCUSSIONS
Our predictions of the Tsss¯s¯ mass spectrum with the HOEM are given in Table III, where the components of different configurations for a physical state can be seen. For example, the two 0++ ground states are mixing states between two different configurations1S0++(66¯)c and S10++(3¯3)c due to a strong contribution of the confinement potential to the nondiagonal elements. To see the contributions from each part of the Hamiltonian to the mass of different configurations, we also present our results in Table IV. It is found that both the kinetic energy term ⟨T⟩ and the linear confinement potential term ⟨VLin⟩ contribute a large positive value to the mass, while the Coulomb-type potential ⟨VCoul⟩ has a large cancelation with these two terms. The spin-spin interaction term ⟨VSS⟩, the tensor potential term ⟨VT⟩, and/or the spin-orbit interaction term ⟨VLS⟩ have also sizable contributions to some configurations. Thus, as a reliable calculation, both the spin-independent and spin-dependent potentials should be reasonably included for the sss¯s¯ system. For clarity, our predicted Tsss¯s¯ spectrum is plotted in Fig. 2.
III10.1103/PhysRevD.103.016016.t3
Predicted mass spectrum for the sss¯s¯ system with the HOEM.
The average contributions of each part of the Hamiltonian to the sss¯s¯ configurations with the HOEM. ⟨T⟩ stands for the contribution of the kinetic energy term. ⟨VLin⟩ and ⟨VCoul⟩ stand for the contributions from the linear confinement potential and Coulomb-type potential, respectively. ⟨VSS⟩, ⟨VT⟩, and ⟨VLS⟩ 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.
Mass spectrum for the sss¯s¯ system. The solid and dotted lines stand for the results predicted by the GEM and HOEM, respectively.
Discussions of the numerical method
Herein we discuss the differences of numerical results between the HOEM used in present work and the GEM often adopted in the literature. For the 1S-, 1P-wave T(sss¯s¯) states, etc., there are no radial excitations. Thus, the GEM is the same as the HOEM. For the first radial excited 2S-wave T(sss¯s¯) states, the HOEM is different from the GEM because the trail harmonic oscillator wave functions are different from the Gaussian functions.
To see the differences between the two expansion methods, we also give our predictions of the 2S-wave T(sss¯s¯) states based on the GEM. It should be mentioned that by fully expanding ∏i=13Rnξilξi(ξi) with the GEM, one cannot distinguish the ξ1 and ξ2 excited modes which are defined for the 2S configurations presented in Table II. Then we cannot numerically work out the masses for the following states of 0+- (21S0+-(66¯)c(ξ1,ξ2) and 21S0+-(3¯3)c(ξ1,ξ2)), 1+- (23S1+-(3¯3)c(ξ1,ξ2)), and 1+- (25S2+-(3¯3)c(ξ1,ξ2)) listed in Table II. To overcome this problem, the spatial wave functions containing the radial excitations are expanded with the Gaussian functions, while the spatial wave functions containing no excitations are adopted the single Gaussian function as an approximation. We have tested the single Gaussian approximation in the calculations of the ground 1ST(sss¯s¯) states, the numerical values are reasonably consistent with those calculated with a series of Gaussian functions. The differences of the numerical results between these two methods are about 10 MeV.
Our numerical results for the 2S-wave T(sss¯s¯) states with the GEM are listed in Tables IV and V. From Table IV, it is found that the numerical values for the 0+- configuration 21S0+-(66¯)c(ξ1,ξ2) and 0++ configuration 21S0++(66¯)c(ξ3) calculated with the HOEM are significantly different from those obtained with the GEM. For these two configurations, the predicted mass differences by the HOEM and GEM can reach up to ∼70MeV. However, for the other 2S-wave T(sss¯s¯) configurations, the numerical values of these two methods are comparable with each other. The differences of the predicted masses between these two methods are about 10–20 MeV. It should be mentioned that the Coulomb-type potential ⟨VCoul⟩ for the 2S-wave states seems to be sensitive to the numerical methods as shown in Table IV.
V10.1103/PhysRevD.103.016016.t5
Predicted mass spectrum for the 2S-wave sss¯s¯ system with the GEM.
In brief, most of the predictions are consistent with each other between the HOEM and GEM. The uncertainties from the numerical methods do not change our main predictions of the T(sss¯s¯) spectrum. Although some numerical results for the JPC=0+- and 0++2S states show a significant numerical method dependence (see Fig. 2), the GEM may give a slightly more accurate numerical result based on our tests of the charmonium spectrum. In the following, our discussions of the 2S states are based on the GEM calculations.
<inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula>-wave states
There are four 1S-wave Tsss¯s¯ states with JPC=0++, 1+-, 2++ in the quark model. Their masses are predicted to be in the range of 2.21–2.44 GeV. In contrast, the 2S wave includes 12 states. Except for the highest mass state T(sss¯s¯)0++(3208), their masses lie in a relative narrow range of 2.78–2.98 GeV. Apart from the conventional quantum numbers, i.e., JPC=0++, 1++, 1+-, 2++, the 2S-wave can access exotic quantum numbers, i.e., JPC=0+-, 2+-.
<inline-formula><mml:math display="inline"><mml:msup><mml:mn>0</mml:mn><mml:mrow><mml:mo>+</mml:mo><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> states
In the 1S-wave mutiplets, the two 0++ ground states include T(sss¯s¯)0++(2218) and T(sss¯s¯)0++(2440). Their mass splitting reaches up to about 200 MeV. These two states have a strong mixing between the two color structures |66¯⟩c and |3¯3⟩c. Their masses are much larger than the mass threshold of ϕϕ. Thus, they may easily decay into ϕϕ pair through quark rearrangements. The mass of the lowest 0++Tsss¯s¯ in our model is close to the prediction of 2203 MeV in the relativistic diquark-antidiquark model [3]. However, it turns out to be much higher than the predicted value 1716 MeV by the relativized quark model with a diquark-antidiquark approximation [4]. There might be some crucial dynamics missing in the diquark-antidiquark approximation. As a test of the diquark-antidiquark approximation, we adopt the approximation as done in Ref. [4] and calculate the mass of the 0++Tsss¯s¯ state with the same potential model parameters. We obtain a mass of 1758 MeV, which is comparable with the prediction of Ref. [4], but is obviously smaller than the results without the diquark-antidiquark approximation.
In the 2S-wave sector, there are four 0++ states, T(sss¯s¯)0++(2796), T(sss¯s¯)0++(2835), T(sss¯s¯)0++(2943), and T(sss¯s¯)0++(3208), predicted in the NRPQM. A strong mixing between the two color structures |66¯⟩c and |3¯3⟩c is also found among these states. In particular, the radial excitation modes (ξ1,ξ2) and (ξ3) strongly mix with each other. The highest state T(sss¯s¯)0++(3208) is nearly a pure configuration of S10++(66¯)c(ξ3), with the color structure |66¯⟩c and the radial excitation between diquark (ss) and antidiquark (s¯s¯). The special color structure of T(sss¯s¯)0++(3208) leads to a rather large mass gap Δ≃265MeV from the nearby T(sss¯s¯)0++(2943). These 2S-wave 0++sss¯s¯ states may easily decay into ϕϕ, ϕϕ(1680) final states through quark rearrangements. They may also easily decay into Ks0Ks0 and K+K- final states through the ss¯ annihilation and a pair of nonstrange qq¯ creation. One also notices that these states may directly decay into ΞΞ¯ baryon pair with a light qq¯ pair creation.
Some evidences for T(sss¯s¯)0++(2218) and T(sss¯s¯)0++(2440) may have been seen in the previous experiments. Recently, Kozhevnikov carried out a dynamical analysis of the resonance contributions to J/ψ→γX→γϕϕ[48] with the data from BESIII [12]. Two 0++ resonances with masses at ∼2.2 and ∼2.4GeV were extracted from the data. Evidence for a scalar around 2.2 GeV in the ϕϕ mass spectra in Bs0→J/ψϕϕ[49] was also reported by Ref. [50]. Considering the mass and decay mode, these two scalar structures may be good candidates for T(sss¯s¯)0++(2218) and T(sss¯s¯)0++(2440).
It should be mentioned that f0(2200) is listed in RPP [1] as a well-established state. It has been seen in the Ks0Ks0, K+K-, and ηη, and may be assigned to T(sss¯s¯)0++(2218). Some qualitative features can be expected: (i) the 0++Tsss¯s¯ state can decay into ηη, η′η′, and ηη′ through quark rearrangements via the ss¯ component in the η and η′ mesons. An approximate branching ratio fraction can be examined: BR(ηη):BR(η′η′):BR(ηη′)≃sin4αP:cos4αP:2sin2αPcos2αP≃0.24:0.25:0.50, with αP≡arctan2+θP≃44.7° and without including the phase space factors. (ii) The 0++ states may also easily decay into Ks0Ks0 and K+K- final states through annihilating a pair of ss¯ and creating a pair of light qq¯. (iii) It is interesting to note that no conventional 0++ss¯ states are predicted around 2.2 GeV in most literatures [2].
To establish the 0++ ground states T(sss¯s¯)0++(2218) and T(sss¯s¯)0++(2440), a combined study of decay channels, such as ϕϕ, Ks0Ks0, K+K-, ηη, η′η′, and ηη′, should be necessary. The 2S-wave 0++ states can be probed in these meson pair decay channels including higher channels such as ϕϕ(1680), and some baryon pair decay channels such as ΞΞ¯.
<inline-formula><mml:math display="inline"><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:mo>+</mml:mo><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> states
There is only one 2++ state T(sss¯s¯)2++(2378) in the 1S-wave states. This state lies between the two 0++ ground states and has a pure |3¯3⟩c color structure. T(sss¯s¯)2++(2378) may have large decay rates into the ϕϕ, ηη, and η′η′ final states through quark rearrangements and/or into K(*)K¯(*) final states through the annihilation of ss¯ and creation of a pair of nonstrange qq¯. It should be mentioned that with the diquark-antidiquark approximation, the mass of the 2++ state is predicted to be 2192 MeV, which is about 200 MeV lower than the four-body calculation results.
The f2(2340) resonance listed in RPP [1] may be assigned to T(sss¯s¯)2++(2378). Besides the measured mass 2345-40+50MeV, the observed decay modes ϕϕ and ηη are consistent with the expectation of the tetraquark scenario. On the other hand, as a conventional ss¯ state, the f2(2340) cannot be easily accommodated by the quark model expectation [2]. The relativistic quark model calculation of Ref. [3] also supports the f2(2340) to be assigned as the Tsss¯s¯ ground state with 2++. To confirm this assignment, the other main decay modes of T(sss¯s¯)2++(2378) such as ηη′, η′η′, K(*)K¯(*) should be investigated in experiment.
For the 2S-wave sector, there are two 2++sss¯s¯ states T(sss¯s¯)2++(2860) and T(sss¯s¯)2++(2975) predicted in our model, which are dominated by the S52++(3¯3)c(ξ3) and S52++(3¯3)c(ξ1,ξ2) configurations, respectively. Their masses are predicted to be above the thresholds of ϕϕ, ϕϕ(1680) and Ξ(1530)Ξ¯. Therefore, experimental search for their signals in these decay channels should be helpful for understanding these tensor tetraquarks.
<inline-formula><mml:math display="inline"><mml:msup><mml:mn>1</mml:mn><mml:mrow><mml:mo>+</mml:mo><mml:mo>-</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> states
In the 1S-wave multiplets, T(sss¯s¯)1+-(2323) is the only state with C=-1 and has a pure |3¯3⟩c color structure. Its mass is about 100 MeV larger than the lowest 1S-wave state T(sss¯s¯)0++(2218). Its mass is about 200–300 MeV larger than that predicted by the QCD sum rules [15] and the relativized quark model [4] in the diquark picture. The mass of the 1+- state may be notably underestimated in the diquark picture. As a comparison, we calculate the mass of the 1+- state in the diquark picture using the same potential model parameter set adopted in present work, and obtain a mass of 1936 MeV, which is 300 MeV smaller than the NRQPM prediction 2236 MeV.
T(sss¯s¯)1+-(2323) may easily decay into ηϕ and η′ϕ through the quark rearrangements. The decay of ψ′/J/ψ→ϕηη′ can access this state in ηϕ and η′ϕ channels. It should be mentioned that some hints of T(sss¯s¯)1+-(2323) may have been found in the η′ϕ invariant mass spectrum around 2.3–2.4 GeV by observing the J/ψ→ϕηη′ reaction at BESIII recently [13].
For the 2S-wave sector, there are two states, i.e., 1+-sss¯s¯T(sss¯s¯)1+-(2851) and T(sss¯s¯)1+-(2943) predicted in the quark model. There are no configuration mixings in these two states. T(sss¯s¯)1+-(2851) has a pure |3¯3⟩c color structure, and the radial excitation occurs between diquark (ss) and antidiquark (s¯s¯) (i.e., the ξ3 mode). In contrast, although T(sss¯s¯)1+-(2943) has also a pure |3¯3⟩c color structure, its radial excitation occurs in the diquark (ss) and antidiquark (s¯s¯). Apart from the ηϕ and η′ϕ decay channels, it may favor decays into a pseudoscalar plus a radially excited vector [i.e., ηϕ(1680) and η′ϕ(1680)], or a radially excited pseudoscalar plus a vector [i.e., η(1295)ϕ and η(1405)ϕ], through the quark rearrangements.
It should be mentioned that in Refs. [9,15] the authors suggest that the new structure X(2063) observed in the J/ψ→ϕηη′ at BESIII [13] could be a 1+-Tsss¯s¯ candidate according to the QCD sum rule calculation. However, the observed mass of X(2063) is too small to be comparable with our quark model predictions.
<inline-formula><mml:math display="inline"><mml:msup><mml:mn>0</mml:mn><mml:mrow><mml:mo>+</mml:mo><mml:mo>-</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:mo>+</mml:mo><mml:mo>-</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> states
In the 2S-wave multiplets, there are two 0+- states, T(sss¯s¯)0+-(2909) and T(sss¯s¯)0+-(2949), predicted in the NRPQM. There is a strong configuration mixing between S10+-(66¯)c(ξ1,ξ2) and S10+-(3¯3)c(ξ1,ξ2). There is only one 2+- state T(sss¯s¯)2+-(2965) corresponding to the configuration S52+-(3¯3)c(ξ1,ξ2). The 0+- and 2+- are exotic quantum numbers which cannot be accommodated by the conventional qq¯ scenario. The P-wave decays into the ηh1(1P) and η′h1(1P) channels could be useful for the search for these states in experiments.
<inline-formula><mml:math display="inline"><mml:mn>1</mml:mn><mml:mi>P</mml:mi></mml:math></inline-formula>-wave states
There are twenty 1P-wave Tsss¯s¯ states predicted in the NRPQM. Apart from the conventional quantum numbers, i.e., JPC=0-+, 1--, 2-+, 2--, 3--, the P wave can access exotic quantum numbers, i.e., JPC=0--,1-+. The masses of the 1P-wave Tsss¯s¯ states scatter in a wide range of about 2.4–3.0 GeV. The masses of the low-lying 1P-wave states may highly overlap with the heaviest 1S-wave state T(sss¯s¯)0++(2440).
<inline-formula><mml:math display="inline"><mml:msup><mml:mn>0</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> states
There are three 0-+ states, T(sss¯s¯)0-+(2481), T(sss¯s¯)0-+(2635), and T(sss¯s¯)0-+(2761), predicted in the NRPQM. They are mixed states with two color structures |66¯⟩c and |3¯3⟩c and also mixed states between two orbital excitations (ξ1,ξ2) and ξ3 modes. They can decay into ϕh1(1P) via an S wave, or ϕϕ via a P wave, through the quark rearrangements.
In 2016, the BESIII Collaboration observed a new resonance X(2500) with a mass of 2470-19+15-23+101MeV and a width of 230-35+64-33+56MeV in J/ψ→γϕϕ[12]. The preferred spin-parity numbers for X(2500) are JPC=0-+[12]. The X(2500) resonance may be a candidate for T(sss¯s¯)0-+(2481) in terms of mass, decay modes, and quantum numbers although X(2500) may favor the 41S0ss¯ state as suggested in our previous work [2]. In the recent work of Ref. [14], the authors also suggested X(2500) to be a 0-+Tsss¯s¯ state according to the QCD sum rule studies. A measurement of the branching fraction of B[X(2500)→ϕϕ] might provide a test of the nature of X(2500). The decay rate of T(sss¯s¯)0-+(2481) into ϕϕ through the quark rearrangements should be significantly larger than that via an ss¯ pair production for the 41S0ss¯ state.
<inline-formula><mml:math display="inline"><mml:msup><mml:mn>1</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mo>-</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> states
There are five 1-- states, T(sss¯s¯)1--(2445), T(sss¯s¯)1--(2567), T(sss¯s¯)1--(2627), T(sss¯s¯)1--(2766), and T(sss¯s¯)1--(2984), predicted in the NRPQM. Their masses scatter in a rather wide range of about 2.4–3.0 GeV. From Table III, it is found that there are obvious configuration mixings in these tetraquark states except that T(sss¯s¯)1--(2627) may nearly be a pure P11--(3¯3)c(ξ3) state. The lowest state T(sss¯s¯)1--(2445) is dominated by the P31--(66¯)c(ξ1,ξ2) configuration. Its orbital excitation mainly occurs within the diquark (ss) or antidiquark (s¯s¯). Meanwhile, the highest state T(sss¯s¯)1--(2984) is dominated by the P11--(66¯)c(ξ3) configuration, and the orbital excitation occurs between the diquark (ss) and antidiquark (s¯s¯).
The vector meson ϕ(2170) in RPP [1] is suggested to be a 1--Tsss¯s¯ state in the literature [5–10] since it is hard to be explained as a conventional meson state according to its measured decay modes [51,52]. Furthermore, the X(2239) resonance, which was observed in e+e-→K+K- by the BESIII Collaboration [11], was suggested to be a candidate of the lowest 1--Tsss¯s¯ state by comparing with the mass spectrum from the relativized quark model [4] and the QCD two-point sum rule method [53]. However, our calculations indicate that neither ϕ(2170) nor X(2239) can be assigned to a 1--Tsss¯s¯ state since their measured masses are much lower than our predictions. It should be mentioned that in recent studies ϕ(2170) was considered as a vector tetraquark state with content sus¯u¯ rather than as a state sss¯s¯[23,54].
The 1--Tsss¯s¯ states may have large decay rates into the f0(980)ϕ via an S wave, or into ηϕ and η′ϕ via a P wave. There are some experimental evidences for structures around 2.4 GeV observed in the f0(980)ϕ invariant mass spectrum from BABAR[55,56], Belle [57], BESII [58], and BESIII [59], which could be signals of the 1--sss¯s¯ tetraquark states [60]. For the heavier states T(sss¯s¯)1--(2766) and T(sss¯s¯)1--(2984), they can also decay into ΞΞ¯ baryon pair through a qq¯ pair production in vacuum. Thus, experimental search for these states in e+e-→ΞΞ¯ should be very interesting.
<inline-formula><mml:math display="inline"><mml:msup><mml:mn>1</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> states
There are three 1-+ states, T(sss¯s¯)1-+(2564), T(sss¯s¯)1-+(2632), and T(sss¯s¯)1-+(2778), predicted in the NRPQM. Note that 1-+ are exotic quantum numbers which cannot be accommodated by the conventional qq¯ scenario. Both the lowest mass state T(sss¯s¯)1-+(2564) and highest mass state T(sss¯s¯)1-+(2778) are mixed states between the two color structures |66¯⟩c and |3¯3⟩c, and their orbital excitations are dominated by the (ξ1,ξ2) mode. The middle state T(sss¯s¯)1-+(2632) is dominated by the P31-+(3¯3)c(ξ3) configuration of which the orbital excitation mainly occurs between the diquark (ss) and antidiquark (s¯s¯). It should be noted that a corresponding state in the relativized quark model [4] has a mass of 2581 MeV, which is about 50 MeV smaller than our prediction.
These 1-+Tsss¯s¯ states may easily decay into ϕh1(1P), η(′)f1(1420), ϕϕ channels through the quark rearrangements. They can be searched for in χcJ(1P)→ηϕϕ,ϕK*K with sufficient χcJ(1P) data samples at BESIII, although no obvious structures were found in previous observations [61,62].
<inline-formula><mml:math display="inline"><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> states
There are three 2-+ states, T(sss¯s¯)2-+(2537), T(sss¯s¯)2-+(2669), and T(sss¯s¯)2-+(2837), predicted in the NRPQM. Both the lowest mass state T(sss¯s¯)2-+(2537) and highest mass state T(sss¯s¯)2-+(2837) are mixed states between the two color structures |66¯⟩c and |3¯3⟩c. Their orbital excitations are dominated by the (ξ1,ξ2) mode. The middle state T(sss¯s¯)2-+(2669) is dominated by the P32-+(3¯3)c(ξ3) configuration of which the orbital excitation occurs between the diquark (ss) and antidiquark (s¯s¯). A corresponding state in the relativized quark model [4] has a mass of 2619 MeV, which is about 50 MeV smaller than our prediction.
These 2-+Tsss¯s¯ states may easily fall apart into ϕh1(1P) and ηf2′(1525) in an S wave, or into ϕϕ in a P wave through the quark rearrangements. For the high mass state T(sss¯s¯)2-+(2837), the strong decay mode ΞΞ¯ also opens. These states can be searched for in χc2(1P)→ηT(sss¯s¯)2-+→ηηf2′(1525)→ηηKK¯ at BESIII with the sufficient χc2(1P) data samples.
<inline-formula><mml:math display="inline"><mml:msup><mml:mn>0</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mo>-</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> states
There are two states with exotic quantum numbers of 0--, T(sss¯s¯)0--(2507) and T(sss¯s¯)0--(2821), predicted in the NRPQM. These two states have a strong mixing between the two color structures |66¯⟩c and |3¯3⟩c. The orbital excitation is the (ξ1,ξ2) mode, i.e., the excitation occurs within the diquark (ss) or antidiquark (s¯s¯). These two states may have large decay rates into ϕf1(1285) and ϕf1(1420) in an S wave, or into ηϕ and η′ϕ in a P wave through the quark rearrangements. These 0-- exotic states may be produced by the reactions e+e-→η(′)X→η(′)η(′)ϕ or J/ψ→η(′)η(′)ϕ.
<inline-formula><mml:math display="inline"><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mo>-</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> states
There are three 2-- states, T(sss¯s¯)2--(2446), T(sss¯s¯)2--(2657), and T(sss¯s¯)2--(2907), predicted in the NRPQM. Both the lowest mass state T(sss¯s¯)2--(2446) and highest mass state T(sss¯s¯)2--(2907) are mixed states between the two color structures, |66¯⟩c and |3¯3⟩c, and their orbital excitations are dominated by the (ξ1,ξ2) mode. The middle state T(sss¯s¯)2--(2657) is dominated by the P32--(3¯3)c(ξ3) configuration, of which the orbital excitation occurs between the diquark (ss) and antidiquark (s¯s¯). A corresponding state in the relativized quark model [4] has a mass of 2622 MeV, which is consistent with our prediction. These 2-- states may easily decay into ϕf1(1285), ϕf1(1420), and ϕf2′(1525) in an S wave, or into ηϕ, η′ϕ in a P wave through the quark rearrangements. They can also be searched for in e+e-→η(′)X→η(′)η(′)ϕ or vector charmonium decays such as J/ψ→ηηϕ.
<inline-formula><mml:math display="inline"><mml:msup><mml:mn>3</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mo>-</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> state
There is only one 3-- state T(sss¯s¯)3--(2719) predicted in the NRPQM. This state has a pure color structure |3¯3⟩c and also a pure orbital excitation between the diquark (ss) and antidiquark (s¯s¯). Our predicted mass is about 60 MeV larger than that predicted by the relativized quark model [4] with a diquark approximation. The 3-- states may easily decay into ϕf2′(1525) in an S wave by the quark rearrangements. Since it has a high spin, it may be produced relatively easier in pp¯ or pp collisions.
SUMMARY
In this work, we calculate the mass spectra for the 1S, 1P, and 2S-wave Tsss¯s¯ states in a nonrelativistic potential quark model without the often-adopted diquark-antidiquark approximation. The 1S-wave ground states lie in the mass range of ∼2.21–2.44GeV, while the 1P- and 2S-wave states scatter in a rather wide mass range of ∼2.44–2.99GeV. For the 2S-wave states, except for the highest state T(sss¯s¯)0++(3208), all the other states lie in a relatively narrow range of ∼2.78–2.98GeV. We find that most of the physical states are mixed states with different configurations.
For the sss¯s¯ system, it shows that both the kinetic energy ⟨T⟩ and the linear confinement potential ⟨VLin⟩ contribute a large positive value to the mass, while the Coulomb-type potential ⟨VCoul⟩ has a large cancellation with these two terms. The spin-spin interaction ⟨VSS⟩, tensor potential ⟨VT⟩, and/or the spin-orbit interaction term ⟨VLS⟩ also have sizable contributions to some configurations.
Some Tsss¯s¯ states may have shown hints in experiment. For instance, the observed decay modes and masses of f0(2200) and f2(2340) listed in RPP [1] could be good candidates for the ground states T(sss¯s¯)0++(2218) and T(sss¯s¯)2++(2378), respectively. The newly observed X(2500) at BESIII may be a candidate for the lowest mass 1P-wave 0-+ state T(sss¯s¯)0-+(2481). Another 0++ ground state T(sss¯s¯)0++(2440) may have shown signals in the ϕϕ channel at BESIII [12,48]. Our calculation shows that ϕ(2170) may not favor a vector state of Tsss¯s¯, because of the much higher mass obtained in our model.
It should be stressed that as a flavor partner of Tccc¯c¯, the Tsss¯s¯ system may have very different dynamic features that need further studies. One crucial point is that the strange quark is rather light and the light flavor mixing effects could become non-negligible. It suggests that strong couplings between Tsss¯s¯ and open strangeness channels could be sizable. As a consequence, mixings between Tsss¯s¯ and Tsqs¯q¯ would be inevitable. For an S-wave strong coupling, it may also lead to configuration mixings which can be interpreted as hadronic molecules for a near-threshold structure. In such a sense, this study can set up a reference on the basis of orthogonal states. More elaborated dynamics can be investigated by including the hadron interactions in the Hamiltonian. For states with exotic quantum numbers, experimental searches for their signals can be carried out at BESIII and Belle-II.
ACKNOWLEDGMENTS
The authors thank Wen-Biao Yan for useful discussions on the BESIII results. Helpful discussions with Atsushi Hosaka and Qi-Fang Lü are greatly appreciated. This work was supported by the National Natural Science Foundation of China (Grants No. 11775078, No. U1832173, No. 11705056, No. 11425525, and No. 11521505). The work of Q. Z. was also supported in part, by the DFG and NSFC funds to the Sino-German CRC 110 “Symmetries and the Emergence of Structure in QCD” (NSFC Grant No. 12070131001), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB34030302), and National Key Basic Research Program of China under Contract No. 2015CB856700.
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