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Investigation of the tetraquark states QqˉQˉq in the improved chromomagnetic interaction model

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Tao Guo, Jianing Li, Jiaxing Zhao and Lianyi He. Investigation of the tetraquark states QqˉQˉq in the improved chromomagnetic interaction model[J]. Chinese Physics C. doi: 10.1088/1674-1137/accb87
Tao Guo, Jianing Li, Jiaxing Zhao and Lianyi He. Investigation of the tetraquark states QqˉQˉq in the improved chromomagnetic interaction model[J]. Chinese Physics C.  doi: 10.1088/1674-1137/accb87 shu
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Received: 2023-02-10
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Investigation of the tetraquark states QqˉQˉq in the improved chromomagnetic interaction model

  • 1. School of Mathematics and Physics, Chengdu University of Technology, Chengdu 610059, China
  • 2. Department of Physics, Tsinghua University, Beijing 100084, China
  • 3. SUBATECH, Université de Nantes, IMT Atlantique, IN2P3/CNRS, 4 rue Alfred Kastler, 44307 Nantes cedex 3, France

Abstract: In the framework of the improved chromomagnetic interaction model, we complete a systematic study of the S-wave tetraquark states QqˉQˉq (Q=c,b, and q=u,d,s) with different quantum numbers: JPC=0+(+), 1+(±), and 2+(+). The mass spectra of tetraquark states are predicted, and the possible decay channels are analyzed by considering both the angular momentum and C-parity conservation. The recently observed hidden-charm tetraquark states with strangeness, such as Zcs(3985), X(3960), and Zcs(4220)+, can be well explained in our model. Additionally, according to the wave function of each tetraquark state, we find that the low-lying states of each QqˉQˉq configuration have a large overlap to the QˉQ and qˉq meson basis, instead of the Qˉq and qˉQ meson basis. This indicates that one can search these tetraquark states in future experiments via the channel of QˉQ and qˉq mesons.

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    I.   INTRODUCTION
    • Exotic hadrons, especially the heavy flavor exotic hadrons, provide a unique tool to study the nature of the strong force and the low-energy properties of quantum chromodynamics (QCD). In addition to the open-charm tetraquark states, such as X0,1(2900)0 [1] and Tcc(3875)+ [2], dozens of hidden-charm (-bottom) exotic states have been discovered since the observation of the hidden-charm state X(3872) in 2003 by the Belle Collaboration [3]; see reviews [48]. The most fascinating and unknown aspect is the inner structure of these exotic hadrons. Theoretically, these exotic states can be mainly explained as multiquark states, which can be molecule states or compact tetraquark states [5, 6, 9], hybrid states with the cˉc-g configuration [10, 11], or missing charmonium states, whose masses can be predicted by potential models but are drastically modified by thresholds [1214], which is a kinematic effect called triangle singularity [1517]. A clear probe to distinguish multiquark states from hybrid states or charmonia is the charged hidden-charm (-bottom) exotic state [18].

      In recent years, many hidden-charm (-bottom) exotic states with non-zero electric charge, such as Zc(3900)+ [19, 20], Zc(4025)+ [21], X(4100)+ [22], Zc(4430)+ [23, 24], Zb(10610)+, and Zb(10652)+ [25], have been observed in experiments. From the decay, one can infer that their quark constitutes are [cuˉcˉd] or [buˉbˉd]. In the meantime, charged hidden-charm tetraquark states with strangeness have also been found experimentally. In 2020, the BESIII Collaboration reported a charged hidden-charm exotic structure with strangeness based on the processes of e+eK+DsD0 and K+DsD0 [26]. Experimental analysis indicated that the exotic state has a mass of (3982.5+1.82.6±2.1) MeV and a width of (12.8+5.34.4±3.0) MeV, which is close to the DsD0 and DsD0 thresholds. It is the first observed candidate of the charged hidden-charm tetraquark with strangeness, i.e., [csˉcˉu], which is named Zcs(3985). Next, the LHCb Collaboration observed an exotic state, i.e., Zcs(4000)+, with a mass of (4003±6+414) MeV and a width of (131±15±16) MeV in the J/ψK+ invariant-mass spectrum [27]. Its quark composition is probably [cuˉcˉs]. In addition, three new candidates named Zcs(4220)+, X(4685), and X(4630) were observed with high significance in the J/ψK+ and J/ψϕ final states [27]. Very recently, a near-threshold peaking structure referred to as X(3960)was discovered by the LHCb Collaboration in the D+sDs invariant mass spectrum [28]. It is very likely a hidden-charm and hidden-strange tetraquark state, i.e., [csˉcˉs]. The best fit gives the mass and width of X(3960) as (3955±6±22) MeV and (48±17±10) MeV, respectively. The quantum number of this state is favored to be I(JPC)=0(0++). In addition, a possible structure X0(4140) is observed in the D+sDs invariant mass spectrum [28].

      On the theoretical side, the mass spectra of QqˉQˉq states have been predicted by the potential model [2934], the QCD sum rule [35, 36], lattice QCD [18, 3739], effective field theory [4045], and the chromomagnetic interaction (CMI) model [46]. Because the CMI model [4656] only considers the short-range chromomagnetic interaction between constituent quarks and antiquarks, it is more suitable to describe the tightly bound states. For the heavy flavor exotic state, which contains more than one light quark (antiquark) probably has a large size, the chromoelectric contribution should be included. This comes to the improved chromomagnetic interaction (ICMI) model [5767]. The ICMI model has been used to predict the mass spectra of open heavy-flavor tetraquark states [6062], open and hidden heavy-flavor pentaquark states [6365], and heavy-flavor dibaryons [66, 67]. In this work, we investigated the masses, possible decay channels, and inner structures of charged and charge-neutral and open- and hidden-strange tetraquark states QqˉQˉq (Q=c,b, and q=u,d,s) via the ICMI model firstly.

      The remainder of this paper is organized as follows. A brief introduction of the ICMI model and the wave functions of the tetraquark QqˉQˉq systems in the color-spin space is presented in Sec. II. In Sec. III, by substituting the effective masses and coupling strengths into the ICMI model, we calculate the mass spectra and wave functions of the S-wave tetraquark state QqˉQˉq. In addition, the related analysis of possible decay channels and the inner structures is presented in this section. We summarize the paper in Sec. IV.

    II.   MODEL DESCRIPTION
    • Analog to the meson and baryon, in this paper we study the heavy flavor tetraquark state QqˉQˉq. The state can be considered as the composition of a heavy quark Q, a heavy antiquark ˉQ, a light quark q, and a light antiquark ˉq in the quark model, where Q=c,b and q=u,d,s. At the leading order, the strong interaction between constituent quarks (antiquarks) can be estimated using the one-gluon-exchange (OGE) potential. For S-wave tetraquark states, the spin-orbit angular momentum coupling part disappears. The total potential can be reduced to two parts [68, 69]

      VOGEij=Vcmij+Vceij,

      (1)

      i.e., the chromomagnetic interaction part

      Vcmij=αsπδ(rij)6mimjλciλcjσiσj,

      (2)

      and the chromoelectric interaction part

      Vceij=αs4rijλciλcj.

      (3)

      Here, the parameter mi represents the effective mass of the ith constituent quark, αs is the running coupling constant, rij=|rij|=|rirj| represents the spatial distance between the i-th and j-th quarks, λci (c=1,2,...,8) denotes the Gell-Mann matrices acting on the color space of the ith quark, and σi denotes the Pauli matrices on the spin space of the ith quark. In addition,λci should be replaced with λci if the subscript i (or j) denotes an antiquark. By integrating the spatial wave function part, we can obtain the ICMI model, which consists of the total mass of the constituent quarks, chromomagnetic interaction, and chromoelectric interaction. Therefore, the effective Hamiltonian of a tetraquark system in the ICMI model is expressed as [57]

      H=4i=1mi+Hcm+Hce,

      (4)

      where the chromomagnetic interaction term can be expressed as

      Hcm=i<jvijλciλcjσiσj,

      (5)

      and the chromoelectric interaction term can be expressed as

      Hce=i<jcijλciλcj.

      (6)

      The model parameters vij and cij incorporate the effects of the effective mass of the constituent quarks, the spatial configuration of the tetraquark system, and the running coupling constant. Considering the symmetry in the color-spin space, the chromoelectric interaction term and constituent quark mass term can be consolidated into one term [58, 63]:

      H04i=1mi+Hce=316i<jmijλciλcj,

      (7)

      with an introduced parameter mij=mi+mj+16cij/3, which is related to the effective masses mi and mj of the constituent quarks and the coupling strength cij of the chromoelectric interaction. Then, the Hamiltonian of the ICMI model can be simplified as

      H=H0+Hcm.

      (8)

      The above parameters, such as vij and mij, can be obtained by fitting the conventional hadron spectra.

      Aiming to solve the eigen equations with the given Hamiltonian (8), we need to construct the wave function of the QqˉQˉq system first. For the tetraquark states Q1q2ˉQ3ˉq4 where Q=c,b and q=u,d,s, there are two types of decomposition of the wave function in the color space based on the SU(3) group theory. They physically correspond to two different configurations in color space: the diquark-antidiquark configuration labeled as |(Q1q2)(ˉQ3ˉq4) and the meson-meson configuration labeled as |(Q1ˉQ3)(q2ˉq4) (or |(Q1ˉq4)(q2ˉQ3)). Taking into account the symmetry characteristics, these two configurations can be properly connected by a linear transformation. It is convenient to see that the total spin of the S-wave tetraquark states can be 0, 1, and 2; thus, the possible quantum numbers are JP=0+, 1+, and 2+. Now, we can construct the color-spin wave function for given tetraquark states. We only show the results for the |(Q1ˉQ3)(q2ˉq4) basis.

      For the scalar tetraquark states with JP=0+, the color-spin basis wave functions |αi (i=1,2,3,4) can be built as follows:

      |α1|(Q1ˉQ3)10(q2ˉq4)1010,|α2|(Q1ˉQ3)11(q2ˉq4)1110,|α3|(Q1ˉQ3)80(q2ˉq4)8010,|α4|(Q1ˉQ3)81(q2ˉq4)8110,

      (9)

      where the superscripts and subscripts denote the total color and spin of Q1ˉQ3, the q2ˉq4 subsystems, and the tetraquark Q1q2ˉQ3ˉq4 systems, respectively. We know that the charge-neutral system has definite C-parity. Thus, if Q1 and Q3, as well as q2 and q4, are of the same flavor, all the above color-spin bases have a positive charge conjugation, i.e., JPC=0++.

      For the axial vector tetraquark states with quantum number JP=1+, the color-spin basis wave functions |βi (i=1,2,...,6) can be built as follows:

      |β1|(Q1ˉQ3)10(q2ˉq4)1111,|β2|(Q1ˉQ3)11(q2ˉq4)1011,|β3|(Q1ˉQ3)11(q2ˉq4)1111,

      |β4|(Q1ˉQ3)80(q2ˉq4)8111,|β5|(Q1ˉQ3)81(q2ˉq4)8011,|β6|(Q1ˉQ3)81(q2ˉq4)8111.

      (10)

      If Q1 and Q3, as well as q2 and q4, in the tetraquark Q1q2ˉQ3ˉq4 systems are of the same flavor, the tetraquark state has definite C-parity. The bases |β3 and |β6 do not change under the symmetry operation of charge conjugation, which gives JPC=1++, while |β1, |β2, |β4, and |β5 change sign under the operation of charge conjugation, which gives JPC=1+.

      For JP=2+ states, the color-spin basis wave functions |γi (i=1,2) are given by

      |γ1|(Q1ˉQ3)11(q2ˉq4)1112,|γ2|(Q1ˉQ3)81(q2ˉq4)8112.

      (11)

      Similarly, each basis introduced above has definite charge conjugation if Q1 and Q3, as well as q2 and q4, are of the same flavor. All the above color-spin bases of the tetraquark systems have positive C-parity.

      The wave function Ψ of the tetraquark state QqˉQˉq with a given quantum number JP can be expressed as the superposition of the bases shown above:

      Ψ=Ncsi=1ci|κi,

      (12)

      where Ncs represents the number of the color-spin basis and ci represents the amplitude for various color-spin bases, which satisfies the normalization condition Ncsi=1|ci|2=1. Here, |κi=|αi, or |βi, or |γi depends on the quantum numbers. With this wave function, we can obtain the matrix form of the Hamiltonian (8), i.e., Ψ|H|Ψ [60]. The mass spectra of tetraquark states can be obtained by diagonalizing this matrix, and the probability |ci|2 can be used to analyze the possible decay channels of the tetraquark states.

    III.   RESULTS AND DISCUSSION
    • The parameters, such as mij and vij, in the ICMI model can be extracted by fitting the masses, especially the low-lying conventional hadrons, which have been observed in experiments. In this work, we adopt the parameters obtained in Ref. [60]. Now, we have calculated the mass spectra and wave functions of the tetraquark QqˉQˉq (Q=c,b, and q=u,d,s) systems with various quantum numbers JP=0+(+), 1+(±), and 2+(+). The tetraquark state has definite C-parity as long as Q(q) and ˉQ(ˉq) are the same flavor. The C-parity of the tetraquark state is determined by the basis, as discussed in the previous section. The tetraquark states can strongly decay into a pair of mesons. The C-parity is conserved in these processes. For the DˉD (DˉD) pair, its C-parity can be estimated as (1)L+S, where S represents the total spin and L represents the relative angular momentum. For other meson pairs, such as DˉDand DsˉDs, the C-parity can be either positive or negative. The mass spectra of the tetraquark states are shown in Figs. 13. Additionally, for comparison, we plot all the possible meson-meson thresholds in each figure. The superposition amplitudes {ci} of the corresponding color-spin wave functions for each tetraquark state are listed in Tables 13.

      Figure 1.  (color online) Mass spectra of S-wave tetraquark states (a) cnˉcˉn (n=u,d), (b) csˉcˉn, and (c) csˉcˉs with different quantum numbers. The black dashed and red dot-dashed lines indicate all possible meson-meson thresholds.

      Figure 2.  (color online) Mass spectra of S-wave tetraquark states (a) bnˉbˉn (n=u,d), (b) bsˉbˉn, and (c) bsˉbˉs with different quantum numbers. The black dashed and red dot-dashed lines indicate all possible meson-meson thresholds.

      SystemJPCMass{ci} for |(Q1ˉQ3)(q2ˉq4) basis{ci} for |(Q1ˉq4)(q2ˉQ3) basis
      bnˉbˉn0+(+)10821.9(−0.05, 0.01, 0.98, 0.18)(0.30, −0.88, −0.15, 0.35)
      (a)10688.5(0.03, −0.17, −0.18, 0.97)(−0.82, −0.29, 0.47, 0.17)
      10221.4(0.01, 0.98, −0.04, 0.17)(−0.44, −0.21, −0.74, −0.46)
      9533.9(0.99, −0.01, 0.06, −0.02)(0.21, −0.32, 0.46, −0.80)
      1+()10812.2(0.002, −0.06, 0, −0.20, 0.98, 0)(0.36, 0.36, −0.77, −0.16, −0.16, 0.32)
      10739.2(−0.13, −0.02, 0, 0.97, 0.20, 0)(0.53, 0.53, 0.49, −0.27, −0.27, −0.25)
      10164.9(0.99, −0.01, 0, 0.12, 0.02, 0)(0.23, 0.23, 0.30, 0.44, 0.44, 0.64)
      9595.6(−0.003, −0.998, 0, −0.01, −0.06, 0)(−0.20, −0.20, 0.27, −0.46, −0.46, 0.65)
      1+(+)10713.7(0, 0, −0.15, 0, 0, 0.99)(0.62, −0.62, 0, −0.33, 0.33, 0)
      10224.2(0, 0, −0.99, 0, 0, −0.15)(−0.33, 0.33, 0, −0.62, 0.62, 0)
      2+(+)10757.2(−0.14, 0.99)(−0.89, −0.45)
      10226.9(−0.99, −0.14)(0.45, −0.89)
      bsˉbˉn0+10883.6(0.07, −0.01, −0.97, −0.23)(−0.26, 0.88, 0.13, −0.37)
      (b)10776.5(0.04, −0.19, −0.22, 0.96)(−0.82, −0.25, 0.49, 0.15)
      10335.7(0.02, 0.98, −0.05, 0.19)(−0.46, −0.22, −0.73, −0.46)
      9886.2(0.99, −0.01, 0.08, −0.03)(0.23, −0.33, 0.46, −0.79)
      1+10874.7(0.02, −0.07, −0.001, −0.31, 0.95, 0.01)(−0.30, −0.28, 0.82, 0.13, 0.13, −0.36)
      10827.2(0.14, 0.04, 0.01, −0.94, −0.31, −0.04)(0.59, 0.54, 0.40, −0.30, −0.28, −0.21)
      10803.1(0.01, 0.002, −0.17, −0.03, −0.02, 0.99)(0.59, −0.64, −0.01, −0.33, 0.36, 0.01)
      10338.9(0.004, 0, 0.99, 0, −0.001, 0.16)(0.34, −0.34, 0.001, 0.62, −0.62, 0.003)
      10280.1(0.99, −0.01, −0.003, 0.13, 0.03, −0.002)(−0.24, −0.24, −0.31, −0.43, −0.44, −0.64)
      9947.9(−0.01, −0.99, 0, −0.02, −0.08, 0)(−0.21, −0.21, 0.28, −0.46, −0.46, 0.65)
      2+10847.8(−0.14, 0.99)(−0.89, −0.46)
      10342.2(−0.99, −0.14)(0.46, −0.89)
      bsˉbˉs0++10952.6(0.09, 0.001, −0.96, −0.27)(−0.22, 0.89, 0.12, −0.39)
      (c)10864.3(0.06, −0.22, −0.26, 0.94)(0.82, 0.21, −0.52, −0.13)
      10460.5(0.03, 0.98, −0.06, 0.21)(−0.47, −0.23, −0.71, −0.47)
      10185.2(0.99, −0.02, 0.10, −0.04)(−0.25, 0.35, −0.46, 0.78)
      1+10945.9(0.05, −0.09, 0, −0.48, 0.87, 0)(0.18, 0.18, −0.88, −0.08, −0.08, 0.41)
      10913.0(−0.15, −0.07, 0, 0.86, 0.48, 0)(0.60, 0.60, 0.24, −0.33, −0.33, −0.14)
      10406.3(0.99, −0.03, 0, 0.15, 0.03, 0)(0.24, 0.24, 0.32, 0.43, 0.43, 0.65)
      10246.7(−0.02, −0.99, 0, −0.03, −0.11, 0)(−0.23, −0.23, 0.29, −0.46, −0.46, 0.63)
      1++10891.9(0, 0, −0.19, 0, 0, 0.98)(0.61, −0.61, 0, −0.36, 0.36, 0)
      10464.4(0, 0, −0.98, 0, 0, −0.19)(−0.36, 0.36, 0, −0.61, 0.61, 0)
      2++10937.3(−0.16, 0.98)(−0.88, 0.48)
      10468.3(−0.98, −0.16)(−0.48, −0.88)

      Table 2.  Masses (in MeV) and color-spin wave functions (represented by the superposition amplitudes {ci}) of the S-wave hidden-bottom tetraquark states with quantum numbers JPC=0+(+), 1+(±), and 2+(+). The labels (a), (b), and (c) correspond to Fig. 2.

      Figure 3.  (color online) Mass spectra of S-wave tetraquark states (a) bnˉcˉn (n=u,d), (b) bsˉcˉn, and (c) bsˉcˉs with different quantum numbers. The black dashed and red dot-dashed lines indicate all possible meson-meson thresholds.

      SystemJPCMass{ci} for |(Q1ˉQ3)(q2ˉq4) basis{ci} for |(Q1ˉq4)(q2ˉQ3) basis
      cnˉcˉn0+(+)4146.5(0.06, −0.16, −0.96, −0.23)(−0.21, 0.90, 0.25, 0.29)
      (a)3970.9(0.06, −0.72, −0.05, 0.69)(−0.37, −0.19, 0.82, 0.39)
      3748.2(0.11, 0.68, −0.27, 0.67)(−0.86, −0.24, −0.26, −0.38)
      3109.3(−0.99, 0.02, −0.09, 0.10)(−0.30, 0.31, −0.44, 0.79)
      1+()4125.2(−0.01, −0.06, 0, −0.45, 0.89, 0)(0.20, 0.20, −0.88, −0.11, −0.11, 0.35)
      4036.0(−0.30, −0.08, 0, 0.85, 0.42, 0)(0.54, 0.54, 0.24, −0.39, −0.39, −0.25)
      3731.7(−0.95, 0.04, 0, −0.27, −0.13, 0)(−0.34, −0.34, −0.32, −0.36, −0.36, −0.63)
      3226.8(−0.02, −0.99, 0, −0.06, −0.09, 0)(−0.24, −0.24, −0.26, −0.45, −0.45, 0.64)
      1+(+)4002.1(0, 0, −0.56, 0, 0, 0.83)(0.42, −0.42, 0, −0.57, 0.57, 0)
      3813.1(0, 0, −0.83, 0, 0, −0.56)(−0.57, 0.57, 0, −0.42, 0.42, 0)
      2+(+)4107.5(−0.28, 0.96)(−0.81, 0.58)
      3852.2(−0.96, −0.28)(−0.58, −0.81)
      csˉcˉn0+4220.8(0.08, −0.19, −0.95, −0.24)(0.19, −0.90, −0.29, 0.29)
      (b)4078.5(0.08, −0.73, −0.01, 0.68)(0.33, 0.21, −0.83, −0.39)
      3857.2(−0.19, −0.65, 0.29, −0.68)(−0.85, −0.25, −0.20, −0.43)
      3455.1(−0.98, 0.05, −0.13, 0.17)(−0.38, 0.30, −0.43, 0.77)
      1+4203.2(−0.03, 0.07, 0.01, 0.56, −0.83, −0.01)(−0.13, −0.12, 0.90, 0.07, 0.06, −0.39)
      4133.2(0.32, 0.14, −0.01, −0.78, −0.52, 0.01)(0.55, 0.53, 0.13, −0.44, −0.43, −0.18)
      4105.5(−0.001, −0.002, −0.60, 0.02, 0.001, 0.80)(0.39, −0.40, 0.01, −0.58, 0.59, −0.003)
      3923.8(0.02, −0.002, 0.80, −0.001, 0.003, 0.60)(0.59, −0.58, 0.01, 0.40, −0.39, 0.02)
      3846.6(−0.95, 0.09, 0.01, −0.27, −0.15, 0.01)(0.33, 0.35, 0.33, 0.33, 0.34, 0.66)
      3575.6(0.04, 0.99, 0.00, 0.09, 0.14, 0.001)(−0.28, −0.28, 0.25, −0.45, −0.45, 0.62)
      2+4209.2(−0.29, 0.96)(−0.80, −0.60)
      3967.2(−0.96, −0.29)(0.60, −0.80)
      csˉcˉs0++4302.7(0.11, −0.24, −0.94, −0.22)(−0.17, 0.88, 0.34, −0.28)
      (c)4195.1(0.10, −0.75, 0.05, 0.65)(0.27, 0.25, −0.84, −0.40)
      3976.5(−0.35, −0.60, 0.27, −0.67)(−0.79, −0.29, −0.09, −0.54)
      3739.3(−0.93, 0.11, −0.21, 0.29)(−0.53, 0.28, −0.41, 0.69)
      1+4289.2(0.07, −0.08, 0, −0.65, 0.75, 0)(0.05, 0.05, −0.90, −0.02, −0.02, 0.43)
      4232.3(−0.34, −0.22, 0, 0.69, 0.60, 0)(0.51, 0.51, 0.03, −0.48, −0.48, −0.10)
      3975.2(0.92, −0.25, 0, 0.27, 0.12, 0)(0.30, 0.30, 0.37, 0.25, 0.25, 0.75)
      3864.9(−0.16, −0.94, 0, −0.18, −0.24, 0)(−0.38, −0.38, 0.22, −0.45, −0.45, 0.50)
      1++4215.2(0, 0, −0.66, 0, 0, 0.75)(0.34, −0.34, 0, −0.62, 0.62, 0)
      4039.1(0, 0, −0.75, 0, 0, −0.66)(−0.62, 0.62, 0, −0.34, 0.34, 0)
      2++4309.7(−0.33, 0.94)(−0.78, −0.63)
      4092.6(−0.94, −0.33)(0.63, −0.78)

      Table 1.  Masses (in MeV) and color-spin wave functions (represented by the superposition amplitudes {ci}) of the S-wave hidden-charm tetraquark states with quantum numbers JPC=0+(+), 1+(±), and 2+(+). The labels (a), (b), and (c) correspond to Fig. 1.

      SystemJPMass{ci} for |(Q1ˉQ3)(q2ˉq4) basis{ci} for |(Q1ˉq4)(q2ˉQ3) basis
      bnˉcˉn0+7471.9(−0.06, 0.07, 0.97, 0.23)(0.24, −0.90, −0.18, 0.33)
      (a)7303.8(0.06, −0.46, −0.17, 0.87)(−0.65, −0.21, 0.68, 0.26)
      7069.5(0.05, 0.89, −0.16, 0.43)(−0.68, −0.23, −0.55, −0.43)
      6406.2(0.99, −0.02, 0.08, −0.06)(0.25, −0.32, 0.45, −0.80)
      1+7461.8(−0.01, −0.06, 0.02, −0.27, 0.95, 0.14)(0.41, 0.21, −0.80, −0.17, −0.13, 0.32)
      7401.3(−0.18, −0.03, −0.12, 0.83, 0.16, 0.49)(0.73, 0.13, 0.41, −0.46, −0.07, −0.26)
      7322.6(0.15, 0.05, −0.35, −0.43, −0.23, 0.79)(0.17, −0.72, −0.11, −0.21, 0.62, 0.12)
      7088.7(0.14, 0.02, 0.93, −0.03, −0.07, 0.34)(0.43, −0.47, 0.06, 0.63, −0.44, 0.07)
      7030.0(0.96, −0.03, −0.10, 0.23, 0.09, −0.08)(0.23, 0.38, 0.33, 0.34, 0.44, 0.63)
      6477.1(0.01, 0.99, −0.01, 0.04, 0.08, −0.03)(0.20, 0.24, −0.26, 0.46, 0.45, −0.65)
      2+7424.9(−0.22, 0.98)(−0.85, −0.53)
      7105.1(−0.98, −0.22)(0.53, −0.85)
      bsˉcˉn0+7544.2(−0.08, 0.07, 0.96, 0.26)(0.21, −0.90, −0.18, 0.35)
      (b)7404.5(0.08, −0.49, −0.19, 0.85)(−0.63, −0.19, 0.72, 0.25)
      7182.0(0.09, 0.87, −0.18, 0.45)(−0.69, −0.23, −0.51, −0.46)
      6756.7(0.99, −0.03, 0.10, −0.09)(0.29, −0.32, 0.45, −0.78)
      1+7535.8(−0.01, 0.08, −0.02, 0.38, −0.91, −0.13)(−0.33, −0.15, 0.84, 0.14, 0.10, −0.36)
      7499.5(−0.19, −0.05, −0.12, 0.79, 0.25, 0.52)(0.77, 0.14, 0.32, −0.49, −0.08, −0.22)
      7421.1(0.17, 0.08, −0.37, −0.42, −0.27, 0.76)(0.13, −0.71, −0.08, −0.19, 0.66, 0.09)
      7203.3(0.16, 0.04, 0.91, −0.03, −0.08, 0.36)(0.44, −0.47, 0.07, 0.64, −0.41, 0.07)
      7145.4(0.96, −0.05, −0.12, 0.23, 0.10, −0.09)(0.22, 0.40, 0.33, 0.32, 0.43, 0.64)
      6827.9(0.02, 0.99, −0.02, 0.05, 0.11, −0.05)(0.21, 0.28, −0.27, 0.45, 0.45, −0.63)
      2+7524.4(−0.23, 0.97)(−0.84, −0.54)
      7221.2(−0.97, −0.23)(0.54, −0.84)
      bsˉcˉs0+7615.1(−0.11, 0.08, 0.95, 0.27)(0.19, −0.89, −0.20, 0.37)
      (c)7507.2(0.12, −0.56, −0.17, 0.80)(−0.55, −0.18, 0.77, 0.26)
      7297.1(0.17, 0.82, −0.20, 0.51)(−0.72, −0.27, −0.41, − 0.50)
      7049.8(0.97, −0.07, 0.16, −0.16)(0.39, −0.33, 0.44, −0.74)
      1+7609.7(−0.07, 0.10, −0.05, 0.56, −0.82, −0.03)(−0.15, −0.08, 0.88, 0.04, 0.08, −0.44)
      7588.7(−0.21, −0.09, −0.15, 0.66, 0.44, 0.54)(0.80, 0.15, 0.12, −0.55, −0.10, −0.13)
      7519.9(0.19, 0.12, −0.43, −0.41, −0.29, 0.72)(0.10, −0.65, −0.07, −0.19, 0.72, 0.07)
      7322.4(0.19, 0.08, 0.88, −0.02, −0.08, 0.42)(0.49, −0.49, 0.07, 0.63, −0.34, 0.06)
      7269.2(0.94, −0.13, −0.13, 0.26, 0.10, −0.11)(0.20, 0.41, 0.36, 0.26, 0.38, 0.67)
      7121.2(0.07, 0.97, −0.05, 0.10, 0.18, −0.09)(0.24, 0.38, −0.27, 0.44, 0.46, −0.58)
      2+7617.1(−0.27, 0.96)(−0.82, −0.57)
      7344.9(−0.96, −0.27)(0.57, −0.82)

      Table 3.  Masses (in MeV) and color-spin wave functions (represented by the superposition amplitudes {ci}) of the S-wave mixed-charm-bottom tetraquark states with quantum numbers JP=0+, 1+, and 2+. The labels (a), (b), and (c) correspond to Fig. 3.

    • A.   Hidden-charm tetraquark states with cnˉcˉn, csˉcˉn, and csˉcˉs configurations

      1.   The cnˉcˉn configuration
    • The mass spectra of the tetraquark states with the cnˉcˉn configuration are shown in Fig. 1(a). The superposition amplitudes of the corresponding wave functions are listed in Table 1(a). Hereinafter, the notation n represents a u or d quark. In our ICMI model, u and d quarks have the same mass.

      (1) There are four S-wave tetraquark states with JP=0+ (JPC=0++ for charge-neutral states, such as [cuˉcˉu] and [cdˉcˉd]). Their masses are 3109.3, 3748.2, 3970.9, and 4146.5 MeV. The lowest state, i.e., 3109.3 MeV, lies slightly below the ηcπ threshold, and its wave function has a large fraction on the ηcπ basis (c1=0.99). The 3748.2 MeV is above the ηcπ and DˉD thresholds. Considering the angular momentum and C-parity conservation (for charge-neutral states), this state is allowed to decay strongly into the mesons ηcπ or DˉD. In this paper, we discuss only the S-wave decay and neglect the high-order decays, such as the P- and D-wave, which usually give a small contribution [63]. The 3970.9 MeV is above the ηcπ, DˉD, and J/ψρ thresholds. Therefore, compared with 3748.2 MeV, this state has one more possible decay mode: J/ψρ. The remaining state with a large mass, i.e., 4146.5 MeV, can decay into mesons ηcπ, DˉD, J/ψρ, and DˉD.

      (2) There are six S-wave tetraquark states with JP=1+. Charge-neutral tetraquark states can be separated into two classes according to the C-parity. There are four S-wave tetraquark states with JPC=1+. The lowest state, with a mass of 3226.8 MeV, is slightly lower than the J/ψπ threshold. Its wave function has a large fraction on the J/ψπ basis (c2=0.99). It can decay into J/ψ and π easily owing to the large fraction and small mass difference. Similarly, the state with a mass of 3731.7 MeV is close to the ηcρ threshold. The large fraction ci indicates that it can decay into ηc and ρ easily. Additionally, it can decay into the mesons J/ψπ (decay into DˉD via S-wave breaks the C-parity). The other two states, i.e., 4036.0 and 4125.2 MeV, lie above all meson-meson thresholds. Considering the conservation law in the decay process, the possible decay channels are J/ψπ, ηcρ, and DˉD. There are two other S-wave tetraquark states with positive C-parity, i.e., JPC=1++. Their masses are 3813.1 and 4002.1 MeV. The state with 3813.1 MeV is near the J/ψρ and DˉD threshold. Thus, it can decay into mesons π+πJ/ψ via the quantum off-shell decay process ρπ+π. The tetraquark states QqˉQˉq have no isospin symmetry. However, considering the mixing, different tetraquark states, such as cuˉcˉu and cuˉdˉd, can form a state with definite isospin. In particular, the state |X=(|cuˉcˉu|cuˉdˉd)/2 with I=1 is in good agreement with the experimental results of X(3872) with I(JPC)=0(1++) [3]. However, our predicted mass of 3813.1 MeV is almost 60 MeV lower than 3872 MeV. If this is true, X(3872) is probably a mixed state of excited charmonium (χc1(2P)) and tetraquark |X states, which is similar to a mixed molecule-charmonium state [70,71]. Our results are consistent with the conclusions of some previous theoretical studies [72,73]. The 4002.1 MeV is slightly below the DˉD threshold. It can decay into DˉD via the off-shell process. Therefore, the possible decay channels are DˉD, DˉD, J/ψρ, and ηcπ.

      (3) There are two S-wave tetraquark states with JP=2+ (JPC=2++ for charge-neutral states), whose masses are 3852.2 and 4107.5 MeV. The mass of 3852.2 MeV is very close to the J/ψρ threshold, and its wave function has a large component of the J/ψρ basis (c1=0.96). The 4107.5 MeV is above the DˉD and J/ψρ thresholds and may be allowed to decay into DˉD and J/ψρ.

    • 2.   The csˉcˉn configuration
    • The mass spectra of the tetraquark states with the csˉcˉn configuration are shown in Fig. 1(b). For comparison, we also show the corresponding meson-meson thresholds in this figure. The masses and wave functions amplitudes {ci} are listed in Table 1(b). For the csˉcˉn state, the C-parity is not a good quantum number. Thus, the quantum numbers of the S-wave tetraquark state csˉcˉn are JP=0+, 1+, and 2+.

      (1) There are four JP=0+ states, whose masses are 3455.1, 3857.2, 4078.5, and 4220.8 MeV. The state with the smallest mass, i.e., 3455.1 MeV, is close to the ηcK threshold, and its wave function has a large component of the ηcK basis (c1=0.98). Thus, it can decay into ηcK easily. The mass of 3857.2 MeV is above the ηcK and DˉDs thresholds and can naturally decay into mesons ηcK and DˉDs. The mass of 4078.5 MeV is below the DˉDs threshold but above other possible meson-meson thresholds. The possible decay channels are ηcK, DˉDs, and J/ψK. The state with the largest mass, i.e., 4220.8 MeV, lies above all possible meson-meson thresholds and can decay to ηcK, DˉDs, J/ψK, and DˉDs.

      (2) There are six S-wave tetraquark states with JP=1+. Their masses are 3575.6, 3846.6, 3923.8, 4105.5, 4133.2, and 4203.2 MeV. The state with the smallest mass, i.e., 3575.6 MeV, which wave function has a large component of the J/ψK basis (c2=0.99). Because its mass is very close to the J/ψK threshold, it can decay into J/ψK with a high probability. Similarly, the mass of 3846.6 MeV can decay into an ηcK pair owing to the large fraction c1=0.95. The mass of 3923.8 MeV is above the J/ψK and ηcK thresholds and close to the DˉDs and DˉDs thresholds. We can see that its wave function has a large |β3 component (c3=0.80), indicating that the meson ηcK component occupies a large proportion. To study the weights of this state to the DˉDs and DˉDs bases, we convert the tetraquark configuration |(Q1ˉQ3)(q2ˉq4) to |(Q1ˉq4)(q2ˉQ3). As shown in Table 1, the amplitude of the state (3923.8 MeV) under this set of |(Q1ˉq4)(q2ˉQ3) bases is (0.59,0.58,0.01,0.4,0.39,0.02). This suggests that the first two components are approximately the same and relatively large, indicating that the tetraquark state can be allowed to decay into the mesons DˉDs and DˉDs. The results nicely explain the nature of the exotic resonant structure Zcs(3985) [26]. However, another exotic state Zcs(4000)+, whose mass is very close to Zcs(3985) but with a larger decay width, cannot be classified in our calculations. The mass of 4105.5 MeV is only below the DˉDs threshold and thus can naturally decay into the mesons J/ψK, ηcK, J/ψK, DˉDs, and DˉDs. The remaining two states 4133.2 and 4203.2 MeV lie above all possible meson-meson thresholds so that all decay modes are possible. In addition, it is worth noting that the mass and decay channel of the tetraquark state 4203.2 MeV with quark content cuˉcˉs is probably the experimentally observed Zcs(4220)+ [27].

      (3) For JP=2+, masses of two S-wave tetraquark states are 3967.2 and 4209.2 MeV. The lower state lies slightly below the J/ψK threshold, and its wave function has a large component of the J/ψK basis (c1=0.96). Thus, it can be searched in the J/ψK channel. The higher state lies above the J/ψK and DˉDs thresholds, and its dominant decay modes are mesons J/ψK and DˉDs.

    • 3.   The csˉcˉs configuration
    • The tetraquark csˉcˉs configuration has definite C-parity. The possible quantum numbers are JPC=0++, 1+, 1++, and 2++. The mass spectra of the tetraquark state with the csˉcˉs configuration are shown in Fig. 1(c) with corresponding meson-meson thresholds. For the sˉs system, there is no pure spin singlet state, owing to the mixing between uˉu and dˉd. For the spin-triplet state, the mixing angle is opportune to form a very nearly pure sˉs state, which is named the ϕ meson. Thus, we only plot the ηcϕ and J/ψϕ thresholds. The superposition amplitudes {ci} of the corresponding tetraquark wave functions are listed in Table 1(c).

      (1) There are four S-wave tetraquark states with JPC=0++, whose masses are 3739.3, 3976.5, 4195.1, and 4302.7 MeV. From the wave function of the lowest state, i.e., 3739.3 MeV, we can see that it has a large component based on ηc and the spin singlet state sˉs, because sˉs content mixes with uˉu and dˉd. Thus, this state may be a superposition of ηcη and ηcη. We will not discuss the decay channels of these mixed states in this paper. Considering that its mass is largely below the meson-meson thresholds, it would be a stable tetraquark state. The mass of 3976.5 MeV is close to and above the DsˉDs threshold and thus can naturally decay into mesons DsˉDs. Now, we convert the |(Q1ˉQ3)(q2ˉq4) configuration into the |(Q1ˉq4)(q2ˉQ3) configuration. Under this set of color-spin bases, the superposition amplitude {ci} of this tetraquark state is (0.79,0.29,0.09,0.54). It can be found that the amplitude c1 of the DsˉDs component is -0.79. Thus, the main decay channel of this state is mesons DsˉDs. These analyses indicate that this state probably is the newly observed exotic hadron state X(3960) with JPC=0++. The mass of 4195.1 MeV is above the DsˉDs and J/ψϕ thresholds but below the DsˉDs threshold. Thus, the possible decay channels are DsˉDs and J/ψϕ. This state may be the observed X0(4140) at LHCb [28]. The last state with a mass of 4302.7 MeV is above all possible meson-meson thresholds and can be allowed to decay into DsˉDs, DsˉDs, and J/ψϕ.

      (2) There are four S-wave tetraquark states with JPC=1+, whose masses are 3864.9, 3975.2, 4232.3, and 4289.2 MeV. Without the spin singlet sˉs meson, the lowest state with a mass of 3864.9 MeV is probably a stable tetraquark state. The state of 3975.2 MeV, with a large ηcϕ component, can decay into ηcϕ via the off-shell process. The two highest states, i.e., 4232.3 and 4289.2 MeV are located above all possible meson-meson thresholds. Considering the conservation of angular momentum and C-parity in the decay process, they can decay into DsˉDs and DsˉDs via the S wave, because the DsˉDs pair can form a negative C-parity state via (DsˉDsDsˉDs)/2. Additionally, there are two S-wave tetraquark states with JPC=1++ and masses 4039.1 and 4215.2 MeV, respectively. The state with a mass of 4039.1 MeV has large fractions on both J/ψϕ and DsˉDs bases. Satisfying the C-parity conversion, it can decay into J/ψϕ and a mixed state of DsˉDs and its antiparticle (DsˉDs+DsˉDs)/2. The state of 4215.2 MeV is very close to the DsˉDs threshold. It can decay into J/ψϕ, while the P wave decaying into the final DsˉDs pair is allowed.

      (3) For JPC=2++, we find two S-wave tetraquark states with masses of 4092.6 and 4309.7 MeV, respectively. The wave function of 4092.6 MeV has a large J/ψϕ component, and it can decay into J/ψϕ. The mass of 4309.7 MeV is above all possible meson-meson thresholds and can be allowed to decay into mesons J/ψϕ and DsˉDs.

    • B.   Hidden-bottom tetraquark states with bnˉbˉn, bsˉbˉn, and bsˉbˉs configurations

    • The hidden-bottom tetraquark states can be realized by replacing the charm quark in the hidden-charm tetraquark states with the bottom quark. By substituting the corresponding model parameters into Eq. (8), we obtain the mass spectra and wave functions of the S-wave hidden-bottom tetraquark states. These hidden-bottom tetraquark states can be divided into three configurations: bnˉbˉn, bsˉbˉn, and bsˉbˉs. Their mass spectra and corresponding meson-meson thresholds are presented in Fig. 2. The superposition amplitudes {ci} of the wave functions of these tetraquark states are listed in Table 2. The analysis method is similar to that for the previous hidden-charm tetraquark states. Thus, we will not discuss them one by one and only focus on two experimental observed states: Zb(10610)+ and Zb(10652)+ [25]. They are observed in the decay channels of Υ(ns)π and hb(mp)π with n=1,2,3 and m=1,2. Their quantum numbers are probably JP=1+ but without the information of the C-parity. In our model, the states with masses of 10713.7 and 10739.2 MeV with JP=1+ are probably the observed Zb(10610)+ and Zb(10652)+, respectively, as shown in Fig. 2(a). If so, these two states should have different C-parities. Their wave function has a very large fraction on the β4 and β6 bases, which indicates these two states may be the diquark-antiquark bound states.

    • C.   Tetraquark states with bnˉcˉn, bsˉcˉn, and bsˉcˉs configurations

    • The mixed charm-bottom tetraquark state can be obtained by substituting a charm quark in the hidden-charm tetraquark state with a bottom quark or by substituting a bottom quark in the hidden-bottom tetraquark state with a charm quark. The difference, however, is that these configurations break the charge conjugation symmetry. Hence, the quantum numbers for the charm-bottom tetraquark systems, i.e., bnˉcˉn, bsˉcˉn, and bsˉcˉs, are JP=0+, 1+, and 2+. Similarly, we can obtain their masses and wave functions by using the ICMI model. Our theoretical results are shown in Fig. 3 and Table 3. The analysis method is similar to that for the previous hidden-charm tetraquark states.

    IV.   SUMMARY
    • In this work, we complete a systematic study of the S-wave tetraquark states QqˉQˉq (Q=c,b and q=u,d,s) via the ICMI model, which includes both the chromomagnetic and chromoelectric interactions. The parameters in the ICMI model are obtained by fitting the known hadron spectra, and they are taken directly from the previous work [60]. The mass spectra, possible decay channels, and inner structures of the S-wave tetraquark QqˉQˉq with quantum numbers JPC=0+(+), 1+(±), and 2+(+) are presented and analyzed. For the charge-neutral tetraquark state, the charge conjugation is also considered when analyzing the possible decay channels.

      The recently observed hidden-charm tetraquark states, such as Zcs(3985), X(3960), and Zcs(4220)+, can be well explained in our model. The tetraquark state csˉcˉu with mass 3923.8 MeV and the quantum number JP=1+ can be considered as a good candidate for Zcs(3985). For the csˉcˉs configuration, we find a tetraquark state with a mass of 3976.5 MeV and a quantum number of JPC=0++. The properties of this state are in good agreement with X(3960). Meanwhile, according to the wave functions of each S-wave tetraquark QqˉQˉq, we find that the low-lying states in each configuration have a significant component of the |(Q1ˉQ3)1(q2ˉq4)1 basis (the probability |ci|2>90%), as shown in Table 4. This indicates they have a large probability to decay into QˉQ and qˉq mesons instead of Qˉq and qˉQ mesons. In some sense, these states are probably the molecule states of QˉQ and qˉq. Of course, other evidence, such as the mean radius and hadron level interactions, is needed to determine whether they are molecular states [7478]. Our predictions regarding these exotic tetraquark states QqˉQˉq can be examined in future experiments.

      SystemJPMassMeson-Meson
      cnˉcˉn0+3109.3ηcπ
      cnˉcˉn1+3226.8J/ψπ
      cnˉcˉn1+3731.7ηcρ
      cnˉcˉn2+3852.2J/ψρ
      csˉcˉn0+3455.1ηcK
      csˉcˉn1+3575.6J/ψK
      csˉcˉn1+3846.6ηcK
      csˉcˉn2+3967.2J/ψK
      bnˉbˉn0+9533.9ηbπ
      bnˉbˉn1+9595.6Υπ
      bnˉbˉn1+10164.9ηbρ
      bnˉbˉn1+10224.2Υρ
      bnˉbˉn2+10226.9Υρ
      bsˉbˉn0+9886.2ηbK
      bsˉbˉn1+9947.9ΥK
      bsˉbˉn1+10280.1ηbK
      bsˉbˉn1+10338.9ΥK
      bsˉbˉn2+10342.2ΥK
      bsˉbˉs1+10406.3ηbϕ
      bsˉbˉs1+10464.4Υϕ
      bsˉbˉs2+10468.3Υϕ
      bnˉcˉn0+6406.2Bcπ
      bnˉcˉn1+6477.1Bcπ
      bnˉcˉn1+7030.0Bcρ
      bnˉcˉn2+7105.1Bcρ
      bsˉcˉn0+6756.7BcK
      bsˉcˉn1+6827.9BcK
      bsˉcˉn1+7145.4BcK
      bsˉcˉn2+7221.2BcK
      bsˉcˉs2+7344.9Bcϕ

      Table 4.  Tetraquark states, which have a large overlap (|ci|2>90%) to the QˉQ and qˉq mesons. Their masses (in MeV), quantum numbers, and meson-meson contents are listed. Bc with JP=1 has not been found in experiments.

Reference (78)

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