-
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] andTcc(3875)+ [2], dozens of hidden-charm (-bottom) exotic states have been discovered since the observation of the hidden-charm stateX(3872) in 2003 by the Belle Collaboration [3]; see reviews [4–8]. 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 thecˉc -g configuration [10, 11], or missing charmonium states, whose masses can be predicted by potential models but are drastically modified by thresholds [12–14], which is a kinematic effect called triangle singularity [15–17]. 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)+ , andZb(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 ofe+e−→K+D−sD∗0 andK+D∗−sD0 [26]. Experimental analysis indicated that the exotic state has a mass of(3982.5+1.8−2.6±2.1) MeV and a width of(12.8+5.3−4.4±3.0) MeV, which is close to theD−sD∗0 andD∗−sD0 thresholds. It is the first observed candidate of the charged hidden-charm tetraquark with strangeness, i.e.,[csˉcˉu] , which is namedZcs(3985)− . Next, the LHCb Collaboration observed an exotic state, i.e.,Zcs(4000)+ , with a mass of(4003±6+4−14) MeV and a width of(131±15±16) MeV in theJ/ψK+ invariant-mass spectrum [27]. Its quark composition is probably[cuˉcˉs] . In addition, three new candidates namedZcs(4220)+ ,X(4685) , andX(4630) were observed with high significance in theJ/ψK+ andJ/ψϕ final states [27]. Very recently, a near-threshold peaking structure referred to asX(3960) was discovered by the LHCb Collaboration in theD+sD−s 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 ofX(3960) as(3955±6±22) MeV and(48±17±10) MeV, respectively. The quantum number of this state is favored to beI(JPC)=0(0++) . In addition, a possible structureX0(4140) is observed in theD+sD−s invariant mass spectrum [28].On the theoretical side, the mass spectra of
QqˉQˉq states have been predicted by the potential model [29–34], the QCD sum rule [35, 36], lattice QCD [18, 37–39], effective field theory [40–45], and the chromomagnetic interaction (CMI) model [46]. Because the CMI model [46–56] 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 [57–67]. The ICMI model has been used to predict the mass spectra of open heavy-flavor tetraquark states [60–62], open and hidden heavy-flavor pentaquark states [63–65], 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 statesQqˉQˉq (Q=c,b , andq=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 stateQqˉQˉq . 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. -
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 antiquarkˉQ , a light quark q, and a light antiquarkˉq in the quark model, whereQ=c,b andq=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)6mimjλci⋅λcjσi⋅σj,
(2) and the chromoelectric interaction part
Vceij=αs4rijλci⋅λcj.
(3) Here, the parameter
mi represents the effective mass of the ith constituent quark,αs is the running coupling constant,rij=|rij|=|ri−rj| 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, andσi denotes the Pauli matrices on the spin space of the ith quark. In addition,λci should be replaced with−λc∗i 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=4∑i=1mi+Hcm+Hce,
(4) where the chromomagnetic interaction term can be expressed as
Hcm=−∑i<jvijλci⋅λcjσi⋅σj,
(5) and the chromoelectric interaction term can be expressed as
Hce=−∑i<jcijλci⋅λcj.
(6) The model parameters
vij andcij 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]:H0≡4∑i=1mi+Hce=−316∑i<jmijλci⋅λcj,
(7) with an introduced parameter
mij=mi+mj+16cij/3 , which is related to the effective massesmi andmj of the constituent quarks and the coupling strengthcij of the chromoelectric interaction. Then, the Hamiltonian of the ICMI model can be simplified asH=H0+Hcm.
(8) The above parameters, such as
vij andmij , 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 statesQ1q2ˉQ3ˉq4 whereQ=c,b andq=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 areJP=0+ ,1+ , and2+ . 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)10⟩10,|α2⟩≡|(Q1ˉQ3)11⊗(q2ˉq4)11⟩10,|α3⟩≡|(Q1ˉQ3)80⊗(q2ˉq4)80⟩10,|α4⟩≡|(Q1ˉQ3)81⊗(q2ˉq4)81⟩10,
(9) where the superscripts and subscripts denote the total color and spin of
Q1ˉQ3 , theq2ˉq4 subsystems, and the tetraquarkQ1q2ˉQ3ˉq4 systems, respectively. We know that the charge-neutral system has definiteC -parity. Thus, ifQ1 andQ3 , as well asq2 andq4 , 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)11⟩11,|β2⟩≡|(Q1ˉQ3)11⊗(q2ˉq4)10⟩11,|β3⟩≡|(Q1ˉQ3)11⊗(q2ˉq4)11⟩11,
|β4⟩≡|(Q1ˉQ3)80⊗(q2ˉq4)81⟩11,|β5⟩≡|(Q1ˉQ3)81⊗(q2ˉq4)80⟩11,|β6⟩≡|(Q1ˉQ3)81⊗(q2ˉq4)81⟩11.
(10) If
Q1 andQ3 , as well asq2 andq4 , in the tetraquarkQ1q2ˉQ3ˉq4 systems are of the same flavor, the tetraquark state has definiteC -parity. The bases|β3⟩ and|β6⟩ do not change under the symmetry operation of charge conjugation, which givesJPC=1++ , while|β1⟩ ,|β2⟩ ,|β4⟩ , and|β5⟩ change sign under the operation of charge conjugation, which givesJPC=1+− .For
JP=2+ states, the color-spin basis wave functions|γi⟩ (i=1,2 ) are given by|γ1⟩≡|(Q1ˉQ3)11⊗(q2ˉq4)11⟩12,|γ2⟩≡|(Q1ˉQ3)81⊗(q2ˉq4)81⟩12.
(11) Similarly, each basis introduced above has definite charge conjugation if
Q1 andQ3 , as well asq2 andq4 , are of the same flavor. All the above color-spin bases of the tetraquark systems have positiveC -parity.The wave function Ψ of the tetraquark state
QqˉQˉq with a given quantum numberJP can be expressed as the superposition of the bases shown above:Ψ=Ncs∑i=1ci|κi⟩,
(12) where
Ncs represents the number of the color-spin basis andci 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. -
The parameters, such as
mij andvij , 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 tetraquarkQqˉQˉq (Q=c,b , andq=u,d,s ) systems with various quantum numbersJP=0+(+) ,1+(±) , and2+(+) . The tetraquark state has definiteC -parity as long asQ(q) andˉQ(ˉq) are the same flavor. TheC -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. TheC -parity is conserved in these processes. For theDˉD (D∗ˉD∗ ) pair, itsC -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 asDˉD∗ andDsˉD∗s , theC -parity can be either positive or negative. The mass spectra of the tetraquark states are shown in Figs. 1–3. 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 1–3.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.System JPC Mass {ci} for|(Q1ˉQ3)(q2ˉq4)⟩ basis{ci} for|(Q1ˉq4)(q2ˉQ3)⟩ basisbnˉbˉn 0+(+) 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) bsˉbˉn 0+ 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) bsˉbˉs 0++ 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 numbersJPC=0+(+) ,1+(±) , and2+(+) . 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.System JPC Mass {ci} for|(Q1ˉQ3)(q2ˉq4)⟩ basis{ci} for|(Q1ˉq4)(q2ˉQ3)⟩ basiscnˉcˉn 0+(+) 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) csˉcˉn 0+ 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) csˉcˉs 0++ 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 numbersJPC=0+(+) ,1+(±) , and2+(+) . The labels (a), (b), and (c) correspond to Fig. 1.System JP Mass {ci} for|(Q1ˉQ3)(q2ˉq4)⟩ basis{ci} for|(Q1ˉq4)(q2ˉQ3)⟩ basisbnˉcˉn 0+ 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) bsˉcˉn 0+ 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) bsˉcˉs 0+ 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 numbersJP=0+ ,1+ , and2+ . The labels (a), (b), and (c) correspond to Fig. 3. -
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 theηcπ threshold, and its wave function has a large fraction on theηcπ basis (c1=−0.99 ). The 3748.2 MeV is above theηcπ andDˉD thresholds. Considering the angular momentum andC -parity conservation (for charge-neutral states), this state is allowed to decay strongly into the mesonsηcπ orDˉD . 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 theηcπ ,DˉD , andJ/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 mesonsηcπ ,DˉD ,J/ψρ , andD∗ˉD∗ .(2) There are six S-wave tetraquark states with
JP=1+ . Charge-neutral tetraquark states can be separated into two classes according to theC -parity. There are four S-wave tetraquark states withJPC=1+− . The lowest state, with a mass of 3226.8 MeV, is slightly lower than theJ/ψπ threshold. Its wave function has a large fraction on theJ/ψπ basis (c2=−0.99 ). It can decay intoJ/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 theηcρ threshold. The large fractionci indicates that it can decay intoηc and ρ easily. Additionally, it can decay into the mesonsJ/ψπ (decay intoDˉD via S-wave breaks theC -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 areJ/ψπ ,ηcρ , andDˉD∗ . There are two other S-wave tetraquark states with positiveC -parity, i.e.,JPC=1++ . Their masses are 3813.1 and 4002.1 MeV. The state with 3813.1 MeV is near theJ/ψρ andDˉD∗ threshold. Thus, it can decay into mesonsπ+π−J/ψ via the quantum off-shell decay processρ→π+π− . The tetraquark statesQqˉQˉq have no isospin symmetry. However, considering the mixing, different tetraquark states, such ascuˉcˉu andcuˉdˉd , can form a state with definite isospin. In particular, the state|X⟩=(|cuˉcˉu⟩−|cuˉdˉd⟩)/√2 withI=1 is in good agreement with the experimental results ofX(3872) withI(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 theD∗ˉD∗ threshold. It can decay intoD∗ˉD∗ via the off-shell process. Therefore, the possible decay channels areD∗ˉD∗ ,DˉD ,J/ψρ , andηcπ .(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 theJ/ψρ threshold, and its wave function has a large component of theJ/ψρ basis (c1=0.96 ). The 4107.5 MeV is above theD∗ˉD∗ andJ/ψρ thresholds and may be allowed to decay intoD∗ˉD∗ andJ/ψρ . -
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 thecsˉcˉn state, theC -parity is not a good quantum number. Thus, the quantum numbers of the S-wave tetraquark statecsˉcˉn areJP=0+ ,1+ , and2+ .(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 theηcK threshold, and its wave function has a large component of theηcK basis (c1=0.98 ). Thus, it can decay intoηcK easily. The mass of 3857.2 MeV is above theηcK andDˉDs thresholds and can naturally decay into mesonsηcK andDˉDs . The mass of 4078.5 MeV is below theD∗ˉD∗s threshold but above other possible meson-meson thresholds. The possible decay channels areηcK ,DˉDs , andJ/ψK∗ . The state with the largest mass, i.e., 4220.8 MeV, lies above all possible meson-meson thresholds and can decay toηcK ,DˉDs ,J/ψK∗ , andD∗ˉD∗s .(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 theJ/ψK basis (c2=0.99 ). Because its mass is very close to theJ/ψK threshold, it can decay intoJ/ψK with a high probability. Similarly, the mass of 3846.6 MeV can decay into anηcK pair owing to the large fractionc1=0.95 . The mass of 3923.8 MeV is above theJ/ψK andηcK∗ thresholds and close to theD∗ˉDs andDˉD∗s thresholds. We can see that its wave function has a large|β3⟩ component (c3=0.80 ), indicating that the mesonηcK∗ component occupies a large proportion. To study the weights of this state to theD∗ˉDs andDˉD∗s 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 mesonsD∗ˉDs andDˉD∗s . The results nicely explain the nature of the exotic resonant structureZcs(3985)− [26]. However, another exotic stateZcs(4000)+ , whose mass is very close toZcs(3985)− but with a larger decay width, cannot be classified in our calculations. The mass of 4105.5 MeV is only below theD∗ˉD∗s threshold and thus can naturally decay into the mesonsJ/ψK ,ηcK∗ ,J/ψK∗ ,D∗ˉDs , andDˉD∗s . 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 contentcuˉcˉs is probably the experimentally observedZcs(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 theJ/ψK∗ threshold, and its wave function has a large component of theJ/ψK∗ basis (c1=−0.96 ). Thus, it can be searched in theJ/ψK∗ channel. The higher state lies above theJ/ψK∗ andD∗ˉD∗s thresholds, and its dominant decay modes are mesonsJ/ψK∗ andD∗ˉD∗s . -
The tetraquark
csˉcˉs configuration has definiteC -parity. The possible quantum numbers areJPC=0++ ,1+− ,1++ , and2++ . The mass spectra of the tetraquark state with thecsˉcˉs configuration are shown in Fig. 1(c) with corresponding meson-meson thresholds. For thesˉs system, there is no pure spin singlet state, owing to the mixing betweenuˉu anddˉd . For the spin-triplet state, the mixing angle is opportune to form a very nearly puresˉs state, which is named the ϕ meson. Thus, we only plot theηcϕ andJ/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 onηc and the spin singlet statesˉs , becausesˉs content mixes withuˉu anddˉd . Thus, this state may be a superposition ofηcη andηcη′ . 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 theDsˉDs threshold and thus can naturally decay into mesonsDsˉDs . 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 amplitudec1 of theDsˉDs component is -0.79. Thus, the main decay channel of this state is mesonsDsˉDs . These analyses indicate that this state probably is the newly observed exotic hadron stateX(3960) withJPC=0++ . The mass of 4195.1 MeV is above theDsˉDs andJ/ψϕ thresholds but below theD∗sˉD∗s threshold. Thus, the possible decay channels areDsˉDs andJ/ψϕ . This state may be the observedX0(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 intoDsˉDs ,D∗sˉD∗s , andJ/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 singletsˉs 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 largeηcϕ component, can decay intoηcϕ 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 andC -parity in the decay process, they can decay intoD∗sˉD∗s andDsˉD∗s via the S wave, because theDsˉD∗s pair can form a negativeC -parity state via(DsˉD∗s−D∗sˉDs)/√2 . Additionally, there are two S-wave tetraquark states withJPC=1++ and masses 4039.1 and 4215.2 MeV, respectively. The state with a mass of 4039.1 MeV has large fractions on bothJ/ψϕ andDsˉD∗s bases. Satisfying theC -parity conversion, it can decay intoJ/ψϕ and a mixed state ofDsˉD∗s and its antiparticle(DsˉD∗s+D∗sˉDs)/√2 . The state of 4215.2 MeV is very close to theD∗sˉD∗s threshold. It can decay intoJ/ψϕ , while the P wave decaying into the finalD∗sˉD∗s 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 largeJ/ψϕ component, and it can decay intoJ/ψϕ . The mass of 4309.7 MeV is above all possible meson-meson thresholds and can be allowed to decay into mesonsJ/ψϕ andD∗sˉD∗s . -
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 , andbsˉbˉs . 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)+ andZb(10652)+ [25]. They are observed in the decay channels ofΥ(ns)π andhb(mp)π withn=1,2,3 andm=1,2 . Their quantum numbers are probablyJP=1+ but without the information of theC -parity. In our model, the states with masses of 10713.7 and 10739.2 MeV withJP=1+ are probably the observedZb(10610)+ andZb(10652)+ , respectively, as shown in Fig. 2(a). If so, these two states should have differentC -parities. Their wave function has a very large fraction on theβ4 andβ6 bases, which indicates these two states may be the diquark-antiquark bound states. -
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 , andbsˉcˉs , areJP=0+ ,1+ , and2+ . 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. -
In this work, we complete a systematic study of the S-wave tetraquark states
QqˉQˉq (Q=c,b andq=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 tetraquarkQqˉQˉq with quantum numbersJPC=0+(+) ,1+(±) , and2+(+) 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) , andZcs(4220)+ , can be well explained in our model. The tetraquark statecsˉcˉu with mass 3923.8 MeV and the quantum numberJP=1+ can be considered as a good candidate forZcs(3985)− . For thecsˉcˉs configuration, we find a tetraquark state with a mass of 3976.5 MeV and a quantum number ofJPC=0++ . The properties of this state are in good agreement withX(3960) . Meanwhile, according to the wave functions of each S-wave tetraquarkQqˉQˉq , 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 intoQˉQ andqˉq mesons instead ofQˉq andqˉQ mesons. In some sense, these states are probably the molecule states ofQˉQ andqˉq . Of course, other evidence, such as the mean radius and hadron level interactions, is needed to determine whether they are molecular states [74–78]. Our predictions regarding these exotic tetraquark statesQqˉQˉq can be examined in future experiments.System JP Mass Meson-Meson cnˉcˉn 0+ 3109.3 ηcπ cnˉcˉn 1+ 3226.8 J/ψπ cnˉcˉn 1+ 3731.7 ηcρ cnˉcˉn 2+ 3852.2 J/ψρ csˉcˉn 0+ 3455.1 ηcK csˉcˉn 1+ 3575.6 J/ψK csˉcˉn 1+ 3846.6 ηcK∗ csˉcˉn 2+ 3967.2 J/ψK∗ bnˉbˉn 0+ 9533.9 ηbπ bnˉbˉn 1+ 9595.6 Υπ bnˉbˉn 1+ 10164.9 ηbρ bnˉbˉn 1+ 10224.2 Υρ bnˉbˉn 2+ 10226.9 Υρ bsˉbˉn 0+ 9886.2 ηbK bsˉbˉn 1+ 9947.9 ΥK bsˉbˉn 1+ 10280.1 ηbK∗ bsˉbˉn 1+ 10338.9 ΥK∗ bsˉbˉn 2+ 10342.2 ΥK∗ bsˉbˉs 1+ 10406.3 ηbϕ bsˉbˉs 1+ 10464.4 Υϕ bsˉbˉs 2+ 10468.3 Υϕ bnˉcˉn 0+ 6406.2 Bcπ bnˉcˉn 1+ 6477.1 B∗cπ bnˉcˉn 1+ 7030.0 Bcρ bnˉcˉn 2+ 7105.1 B∗cρ bsˉcˉn 0+ 6756.7 BcK bsˉcˉn 1+ 6827.9 B∗cK bsˉcˉn 1+ 7145.4 BcK∗ bsˉcˉn 2+ 7221.2 B∗cK∗ bsˉcˉs 2+ 7344.9 B∗cϕ Table 4. Tetraquark states, which have a large overlap (
|ci|2>90% ) to theQˉQ andqˉq mesons. Their masses (in MeV), quantum numbers, and meson-meson contents are listed.B∗c withJP=1− has not been found in experiments.
Investigation of the tetraquark states QqˉQˉq in the improved chromomagnetic interaction model
- Received Date: 2023-02-10
- Available Online: 2023-06-15
Abstract: In the framework of the improved chromomagnetic interaction model, we complete a systematic study of the S-wave tetraquark states