PRDPRVDAQPhysical Review DPhys. Rev. D2470-00102470-0029American Physical Society10.1103/PhysRevD.100.096013ARTICLESPhenomenological aspects of field theory, general methodsSpectrum of the fully-heavy tetraquark state QQQ¯′Q¯′SPECTRUM OF THE FULLY-HEAVY TETRAQUARK …GUANG-JUAN WANG, LU MENG, AND SHI-LIN ZHUhttps://orcid.org/0000-0001-9265-6741WangGuang-Juan1,2,*https://orcid.org/0000-0001-9791-7138MengLu2,†ZhuShi-Lin2,3,‡Center of High Energy Physics, Peking University, Beijing 100871, ChinaSchool of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, ChinaCollaborative Innovation Center of Quantum Matter, Beijing 100871, China
wgj@pku.edu.cn
lmeng@pku.edu.cn
zhusl@pku.edu.cn
20November20191November2019100909601330July2019Published by the American Physical Society2019authorsPublished 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 systematically calculate the mass spectra of the S-wave fully-heavy tetraquark states, bbb¯b¯, ccc¯c¯, and bbc¯c¯, in two nonrelativistic quark models. A tetraquark state may be an admixture of a 6c-6¯c state and a 3¯c-3c one, where 6c-6¯c(3¯c-3c) denotes the color configuration with a 6c (3¯c) diquark and a 6¯c (3c) antidiquark. For the tetraquark states bbb¯b¯ and ccc¯c¯ with JPC=0++, the 6c-6¯c state is lower than the 3¯c-3c one in both the two quark models, while the order of the bbc¯c¯ states depend on models. The 6c-6¯c and 3¯c-3c mixing effects are induced by the hyperfine interactions between the diquark and antidiquark, while the contributions from the one-gluon-exchange Coulomb or the linear confinement potentials vanish for the QQQ¯′Q¯′ system. With the couple-channel effects, we obtain the similar mass spectra. The numerical results show that the ground QQQ¯′Q¯′ (Q=b, c and Q′=b, c) tetraquark states are located above the corresponding scattering states, which indicates that there may not exist a bound state in the scheme of the two quark models.
National Natural Science Foundation of China10.13039/501100001809115750081162113100111975033INTRODUCTION
Since 2003, numerous exotic structures have been observed in experiments [1–11], amongst which many states cannot be accommodated into the traditional quark model. In the literature, there are many possible explanations. The most prominent ones are the molecules (loosely bound states of two hadrons), the tetraquarks (compact bound states), the hybrids (composed of gluons and quarks), etc. For a recent review, see Refs. [12–16].
A fully-heavy tetraquark state is a topic of great interest. The interactions between the heavy quarks may be dominated by the short-range one-gluon-exchange (OGE) potential rather than the long-range potentials. Thus, they are good candidates of the compact tetraquark states. Unlike a meson or a baryon where the color configuration of the quarks is unique, i.e., qiq¯jδij or εijkqiqjqk, the color structure for the tetraquark is much richer. For the tetraquark states, the four quarks can neutralize the color in two ways, 6c⊗6¯c=1c and 3¯c⊗3c=1c. In this work, we label the two color configurations |(QQ)6cQ¯Q¯6¯c⟩ and |(QQ)3¯cQ¯Q¯3c⟩ as 6c-6¯c and 3¯c-3c, respectively. In Refs. [17–19], the authors investigate the tetraquark states in the 3¯c-3c configuration. In Refs. [20,21], the authors pointed out that the 6c-6¯c configuration is also very important to form the tetraquark states. The fully-heavy tetraquark state is a golden system to investigate the inner color configuration of the multiquark states. For the above reasons, the fully-heavy tetraquark states have inspired both the experimental and theoretical attention.
Recently, the CMS Collaboration observed the ϒ(1S) pair production and indicated a bbb¯b¯ signal around 18.4 GeV with a global significance of 3.6σ[22–24]. Later, the LHCb searched the invariant mass distribution of ϒ(1s)μ+μ- and did not observe the tetraquark state Xbbb¯b¯[25]. The tension between CMS and LHCb data requires more experimental and theoretical studies of the beauty-full tetraquarks.
The mass spectroscopy has been a major platform to probe the dynamics of the tetraquarks. Since 1975, there have been many theoretical works about the mass spectroscopy of the fully-heavy quark states [26–32]. The existence of the fully-heavy quark states is still controversial. Recent interests have followed the experimental developments in the past several years. The mass spectra have been calculated in different schemes, for instance, a diffusion Monte Carlo method [33], the nonrelativistic effective field theory [34], the QCD sum rules [35–37], covariant Bethe-Salpeter equations [38], various quark models [39–41], and other phenomenological models [42–46]. The lowest bbb¯b¯ and ccc¯c¯ states are estimated to be in the mass ranges 18–20 GeV and 5–7 GeV, respectively. In contrast, the authors of Ref. [47] investigated the mass spectra of the QQQ¯Q¯ states in the chromomagnetic interaction (CMI) model and concluded that no stable QQQ¯Q¯ states exist. Later, several other approaches, such as the nonrelativistic chiral quark model [48,49], the lattice QCD [50], and other models [51,52] also do not support the existence of the bound QQQ¯Q¯ states.
To investigate the existence of the fully-heavy tetraquark states, we systematically calculate the mass spectra of the bbb¯b¯, ccc¯c¯ and bbc¯c¯(ccb¯b¯) in two nonrelativistic quark models. In general, a tetraquark state should be an admixture of the two color configurations, 6c-6¯c and 3¯c-3c. In this work, with the couple-channel effects, we perform the dynamical calculation of the mass spectra of the tetraquark states and investigate the inner structures of the ground states.
We organize the paper as follows. In Sec. II, we introduce the formalism to calculate their mass spectra, including two nonrelativistic quark models, the construction of the wave functions, and the analytical expressions of the Hamiltonian matrix elements. In Sec. III, we present the numerical results and discuss the couple-channel effects between the 3¯c-3c and 6c-6¯c configurations. In Sec. IV, we compare our results with those in other models and give a brief summary.
FORMALISMHamiltonian
The nonrelativistic Hamiltonian of a Q1Q2Q¯3Q¯4 tetraquark state reads H=∑i=14pj22mj+∑i<jVij+∑imi=p22u+VI+h12+h34with VI=V13+V14+V23+V24,hij=pij22uij+Vij+mi+mj,pij=mipj-mjpimi+mj,uij=mimjmi+mj,mij=mi+mj,u=m12m34m12+m34,Pij=pi+pj,p=m12P34-m34P12m12+m34,where pi and mi are the momentum and mass of the ith quark. The kinematic energy of the center-of-mass system has been excluded by the constraint ∑i=14pi=0. Vij is the potential between the ith and jth quarks. The uij, mij, pij, and Pij are the reduced mass, total mass, relative momentum, and total momentum of the (ij) pair of quarks, respectively. The u and p are the reduced mass and relative momentum between the (12) and (34) quark pairs. h12, h34 and VI represent the (12) quark pair inner interaction, (34) quark pair interaction and interaction between the two pairs.
Since the heavy quark mass is large, the relativistic effect is less important. We use a nonrelativistic quark model to describe the interaction between two heavy quarks. The quark model proposed in Ref. [53] contains one gluon exchange plus a phenomenological linear confinement interaction and the Vij reads Vij(rij)=λi2λj2(Vcoul+Vconf+Vhyp+Vcons)=λi2λj2(αsrij-3b4rij-8παs3mimjsi·sje-σ2r2σ3π3/2+Vcons),where λ is the color matrix (replaced by -λ* for an antiquark). si is the spin operator of the ith quark. rij is the relative position of the ith and jth quarks. Vcoul, Vconf, and Vhyp represent the OGE color Coulomb, the linear confinement, and the hyperfine interactions, respectively. The OGE interaction leads to a contact hyperfine effect and an infinite hyperfine splitting. In Eq. (7), the smearing effect has been considered in Vhyp, where σ parametrizes the size of the effect.
The αs is the running coupling constant in the perturbative QCD, αs(Q2)=12π(33-2Nf)ln(A+Q2/B2).In this work, we take the square of the invariant mass of the interacting quarks as the scale Q2. The values of the parameters are listed in Table I. They are determined by fitting the mass spectra of the mesons as listed in Table II.
I10.1103/PhysRevD.100.096013.t1
The values of parameters in quark model I [53] and model II [54].
Model Imc[GeV]mb[GeV]b[GeV2]σ[GeV]Vcons[GeV]AB[GeV]1.7765.1020.180.8970.62100.31Model IIprcmc[GeV]mb[GeV]κκ′λ[GeV2]Λ[GeV]A[GeVB-1]B101.8365.2270.50691.86090.16530.83211.65530.2204II10.1103/PhysRevD.100.096013.t2
The mass spectra of the heavy quarkonia in units of MeV. The Mex, MthI, and MthII refer to the mass spectra of mesons from experiments [55], in model I [53], and in model II [54], respectively.
To investigate the model dependence of the mass spectrum, we also consider another nonrelativistic quark model proposed in Ref. [54]. The potential reads Vij(rij)=-316∑i<jλiλj(-κ(1-exp(-rij/rc))rij+λrijp-Λ+8π3mimjκ′(1-exp(-rij/rc))exp(-rij2/r02)π3/2r03si·sj),where r0=A(2mimjmi+mj)-B is related to the reduced mass of the two quarks (ij). In this model, all the mass information is included in the hyperfine potential, which is expected to play a more important role than that in model I. The parameters of the potentials are listed in Table I. With these parameters, we calculate the mass spectra of the mesons and list them in Table II.
In this work, we concentrate on the S-wave tetraquark states and do not include the tensor and spin-orbital interactions in the two quark models. In Table II, we notice that both models are able to reproduce the mass spectra of the heavy quarkonia. In the following, we will extend the two quark models to study the fully-heavy tetraquarks.
Wave function
In a Q1Q2Q¯3Q¯4 tetraquark state, there are three sets of Jacobi coordinates as illustrated in Fig. 1. Each of them contains three independent Jacobi coordinates, and they can be transformed into others as follows: rjk=rj-rk=r+cjkar12+cjkbr34,r=m1r1+m2r2m1+m2-m3r3+m4r4m3+m4,r′=m1r1+m3r3m1+m3-m2r2+m4r4m2+m4=(m1m3-m2m4)r+MTu12r12-MTu34r34(m1+m4)(m2+m3),r′′=m1r1+m4r4m1+m4-m2r2+m3r3m2+m3=(m1m4-m2m3)r+MTu12r12-MTu34r34(m1+m3)(m2+m4),where MT=∑i=14mi is the total mass of the four quarks. The transformation coefficients cjka(b) are listed in Table III. The superscripts a and b represent the quark cluster and antiquark cluster, respectively.
To simplify the calculation, we use the first coordinate configuration to construct the wave function considering the symmetry of the inner quarks. The wave function of a tetraquark state is ψJJz=∑[φnaJa(r12,βa)⊗φnbJb(r34,βb)⊗ϕNLab(r,β)]JJz,φnaJaMa=[ϕnala(r12,βa)χsa]MaJaχfχca,where the ψ is the total wave function of the tetraquark state, and φ denotes that of the cluster (a) or (b). J (Jz) is the total angular momentum (the third direction component) of a tetraquark state. The ∑ is the sum over all the possible wave functions which may couple to the definite angular momentum J. na(b) and N specify the radial dependence. The sa(b), la(b) and Ja(b) are the spin, orbital and total angular momentum of the cluster a (b). Lab is the orbital angular momentum between the two clusters. The χs, χf, χc are the wave functions in the spin, the isospin, and the color space, respectively. ϕ is the spatial wave function and is expressed by the Gaussian basis [56], ϕnalama(r12,βa)=ilar12la4π(2la+1)!!(naβa2π)3/4×(2naβa2)la/2e-r122βa2na/2Ylama(Ω12),with βa being the oscillating parameter.
In this work, we concentrate on the S-wave tetraquark states. Their wave functions are expanded by the basis which satisfies the relation la+lb+Lab=0. The states with higher orbital excitations contribute to the ground state through the tensor or the spin-orbital potentials. These contributions are higher order effects and neglected in this work. Thus, for the lowest S-wave tetraquark states, we only consider the wave functions with la=lb=Lab=0. The wave function of the tetraquark state in Eq. (11) is simplified as ψSSz=∑α,na,nb,nabχαϕna(r12,βa)ϕnb(r34,βb)ϕnab(r,β),χα=[χsa⊗χsb]S[χfa⊗χfb][χca⊗χcb]1,where S is the total spin of the tetraquark state and 1 represents the color-singlet representation. For the spatial wave functions, we have omitted the orbital angular momentum in the Gaussian wave function ϕ.
The wave functions are constrained by the Pauli principle. The S-wave diquark (antidiquark) with two identical quarks (antiquarks) has two possible configurations as listed in Table IV. Then, for the ccc¯c¯, bbb¯b¯, and bbc¯c¯ tetraquark states, the possible color-flavor-spin functions read
where the superscript and subscript denote the spin and color representations.
IV10.1103/PhysRevD.100.096013.t4
The configurations of the diquark (antiquark) constrained by Pauli principle. “S” and “A” represent symmetry and antisymmetry.
JP=1+QQJP=0+QQS-wave(L=0)SS-wave(L=0)SFlavorSFlavorSSpin(S=1)SSpin(S=0)AColor(3¯c)AColor(6c)SHamiltonian matrix elements
With the wave function constructed in Sec. II B, we calculate the Hamiltonian matrix elements. For the quark model I, the matrix element of ⟨h12⟩ reads ⟨χαiϕn(r12)ϕλ(r34)ϕk(r)|h12|χαjϕm(r12)ϕν(r34)ϕk′(r)⟩=δαiαjNλ,νNk,k′⟨ϕn(r12,βa)|h12|ϕm(r12,βa)⟩=δαiαjNλ,νNk,k′(⟨T12+m1+m2⟩+⟨V12⟩),with Nk,k′=(2kk′k+k′)3/2,⟨T12+m1+m2⟩=Nm,n(3mnβa22u12(m+n)+m1+m2),⟨V12(r12)⟩=⟨Vcoul⟩+⟨Vconf⟩+⟨Vhyp⟩+⟨Vcons⟩,⟨Vcoul⟩=IC4παsβa(2π)3/2m+nNm,n,⟨Vconf⟩=-34IC8πb(2π)3/2βam+nNm,n,⟨Vhyp⟩=-ICM8παs3mimjσ3π3/2(2mnm+n+2σ2/βa2)32,⟨Vcons⟩=ICVconsNm,n,where n,λ,k,m,ν,k′ specify the radial dependence. The IC and ICM are the color factor and the color electromagnetic factor in Tables V and VI, respectively. χαi,αj denote the color-flavor-spin configurations as illustrated in Eq. (12). Since the potential h12 is diagonal in the color-flavor-spin space, it does not induce the coupling of different χαi,αj channels and the ⟨h12⟩ is proportional to δαiαj. The derivation of ⟨h34⟩ is similar to that of ⟨h12⟩.
V10.1103/PhysRevD.100.096013.t5
The color matrix element IC=⟨λi2λj2⟩ for the (ij) pair of quarks. The subscripts denote the color representation of the cluster.
The color magnetic factor ⟨ICM⟩αiαj=⟨χαi|λi2λj2si·sj|χαj⟩ for the (ij) quark pairs. The χαi,j denotes the color-flavor-spin wave functions in Eqs. (13)–(15).
Unlike the h12 and h34, the VI(rij) with i=1, 2 and j=3, 4, which is the interaction between the diquark and antidiquark, may lead to the mixing between different color-spin-flavor configurations, i.e., χαi and χαj. The ⟨VI(rij)⟩ reads ⟨χαiϕn(r12,βa)ϕλ(r34,βb)ϕk(r,β)|VI(rij)|χαjϕm(r12,γa)ϕν(r34,γb)ϕk′(r,γ)⟩=⟨Vcoul(rij)⟩αiαj+⟨Vconf(rij)⟩αiαj+⟨Vhyp(rij)⟩αiαj,where β(a,b) and γ(a,b) are the oscillating parameters. The implicit forms of the notations are ⟨Vcoul(rij)⟩=ICN˜m,nN˜λ,νN˜k,k′2αsπ2kβ2+k′γ2+4aij2,⟨Vconf(rij)⟩=ICN˜m,nN˜λ,νN˜k,k′(-3bzijπ),⟨Vhyp(rij)⟩=ICMN˜m,nN˜λ,νN˜k,k′×(-8αs3mimj(4τij2+2kβ2+k′γ2)3/2π),where N˜m,n=(2mnβaγamγa2+nβa2)3/2,aij2=(cija)22(mβa2+nγa2)+(cijb)22(λβb2+νγb2),τij2=aij2+14σ2,zij2=aij2+12(kβ2+k′γ2).With the above analytical expressions, we calculate the mass spectrum of the fully-heavy tetraquark states QQQ¯′Q¯′. The numerical results are given in the next section.
NUMERICAL RESULTS
The wave function of a tetraquark state is composed of all wave functions which are subjected to the conditions discussed in Sec. II B. The number of the basis N3 increases from the minimum required to a large limit. We take the ccc¯c¯ tetraquark state with JPC=1+- as an example to investigate the dependence of the results on the number of the basis. Its wave function is expanded with N3=13, 23, 33, 43, and 53 basis, respectively. The corresponding eigenvalues obtained through the variational method are displayed in Fig. 2. The mass spectrum tends to be stable when N3 is larger than 23. Therefore, we expand the wave functions of the tetraquark states with the 23 Gaussian basis in the following calculation.
210.1103/PhysRevD.100.096013.f2
The dependence of the mass spectrum on the number of Gaussian basis N3. The line and dashed line represent the numerical results in model I and model II, respectively.
A tetraquark state <inline-formula><mml:math display="inline"><mml:mi>Q</mml:mi><mml:mi>Q</mml:mi><mml:msup><mml:mover accent="true"><mml:mi>Q</mml:mi><mml:mo stretchy="false">¯</mml:mo></mml:mover><mml:mo>′</mml:mo></mml:msup><mml:msup><mml:mover accent="true"><mml:mi>Q</mml:mi><mml:mo stretchy="false">¯</mml:mo></mml:mover><mml:mo>′</mml:mo></mml:msup></mml:math></inline-formula> with <inline-formula><mml:math display="inline"><mml:msup><mml:mi>J</mml:mi><mml:mrow><mml:mi>P</mml:mi><mml:mi>C</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><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>
A tetraquark state QQQ¯′Q¯′ with JPC=0++ contains two color-flavor-spin configurations χ1 and χ2 as listed in Eq. (13). Its wave function reads ψJJzIIz=∑α1Aα1ϕα1χ1+∑α2Bα2ϕα2χ2=∑α1Aα1ϕα1(βa,βb,β)|(QQ)3¯c(Q¯Q¯)3c⟩+∑α2Bα2ϕα2(γa,γb,γ)|(QQ)6c(Q¯Q¯)6¯c⟩,where α1,2={na,la,nb,lb,N,L}, β(a,b) and γ(a,b) are the oscillating parameters for the 3¯c-3c and 6c-6¯c tetraquark states. Aα1 and Bα2 are the expanding coefficients.
At first, we do not consider the mixture between the 3¯c-3c and 6c-6¯c tetraquark states and solve the Schödinger equation with the variational method. We obtain their mass spectra and display them in the left panel of Fig. 3.
310.1103/PhysRevD.100.096013.f3
The mass spectrum of the 0++ tetraquark states QQQ¯′Q¯′ without and with the coupling between the 3¯c-3c and 6c-6¯c configurations. The blue lines and red dotted dashed lines represent the results in model I and II, respectively. In every diagram, the left half and the right half are the mass spectrum without and with mixing between 3¯c-3c and 6c-6¯c configurations, respectively. The corresponding states are connected by the black dashed lines.
For the ccc¯c¯ and bbb¯b¯ systems, the 6c-6¯c states are located lower than the 3¯c-3c ones as illustrated in Fig. 3. In the OGE model, the interactions between the two quarks within a color-sextet diquark are repulsive due to the color factor in Table V, while those in the 3¯c one are attractive. However, the interactions between the 6c diquark and 6¯c antidiquark are attractive and much stronger than that between the 3¯c diquark and 3c antidiquark. There exists a 6c-6¯c tetraquark state, if the attraction between diquark and antidiquark wins against the repulsion within the diquak (antidiquark). If the attractive potentials are strong enough, the 6c-6¯c state stays even lower than the 3¯c-3c one. That is what happens to the ccc¯c¯, bbb¯b¯ tetraquark states with JPC=0++ in the two quark models. For the bbc¯c¯ (ccb¯b¯) state, the 6c-6¯c state is lower in model I, while the 3¯c-3c state is lower in model II.
In general, a tetraquark state is a mixture of the 3¯c-3c and 6c-6¯c states as illustrated in Eq. (23). With the couple-channel effects of the 3¯c-3c and 6c-6¯c color configurations, we obtain the mass spectrum of the 0++ states and list them in Table VII. The spectra obtained with 3¯c-3c and 6c-6¯c mixing are given in Fig. 3. The mixing effect will pull down the lower state and raise the higher state. The two quark models lead to similar mass spectra for the ccc¯c¯, bbb¯b¯, and bbc¯c¯ (ccb¯b¯) tetraquark states with the differences up to tens of MeV. However, the proportions of the components in the two quark models are quite different. The mixing between the 3¯c-3c and 6c-6¯c states are stronger in model II. The reasons are explained as follows.
VII10.1103/PhysRevD.100.096013.t7
The mass spectra of ccc¯c¯, bbb¯b¯, and bbc¯c¯ (b¯b¯cc) tetraquark states with JPC=0++. β(a,b) and γ(a,b) represent the oscillating parameters of the 3¯c-3c and 6c-6¯c tetraquark states, respectively.
In model I and model II, we find that only the hyperfine interactions contribute to the couple-channel effects of the 3¯c-3c configuration and the 6c-6¯c one, while the contributions from the confinement and Coulomb potentials vanish. We illustrate the underlying dynamics as follows. The matrices of h12 and h34 are diagonal due to the orthogonality of the wave functions of different configurations. However, the Vcoul+Vlinear+Vhyp in VI, which describes the interactions between the diquark and antidiquark, may result in the couple-channel effects of different configurations. For an S-wave tetraquark state with two identical quarks (antiquarks), such as QQQ¯1Q¯2 (Q1Q2Q¯Q¯), the spin wave functions of different possible configurations are orthogonal, which is constrained by the Fermi statistic. Since the OGE Coulomb and linear confinement potentials do not contain spin operators, they do not contribute to the couple-channel effects due to the orthogonality of the spin wave functions. And only the hyperfine potential contributes. That is what happens to the QQQ¯′Q¯′ state in this work.
For a tetraquark state without identical quarks and antiquarks, i.e., Q1Q2Q¯3Q¯4 (Q1≠Q2 and Q3≠Q4), the spin wave functions of different configurations may be the same. The four quarks form a color singlet state and the color matrix element is (∑n4λn)2|χi,j⟩=0,where χi and χj represent two different color configurations and they are the eigenvectors of λ1+λ2 and λ3+λ4. Considering their orthogonality, one obtains ⟨χi|(λ1+λ2)2|χj⟩=0,⟨χi|(λ3+λ4)2|χj⟩=0.Then the color factors of the (13), (14), (23), and (24) pairs of quarks cancel out, ⟨χi|(λ1+λ2)(λ3+λ4)|χj⟩=0.Moreover, if the coupling constants are the same for the four quark pairs, the contributions from the OGE Coulomb and the linear confinement potentials will cancel out completely. In model I, the contributions from the color interactions do not cancel out exactly due to different αs. However, partial cancellations are still expected. In model II, the OGE Coulomb and linear confinement potentials do not depend on the mass of the interacting quarks. Thus, the couple-channel effects arising from the OGE Coulomb and linear confinement potentials cancel out. The mixing between different color-flavor-spin configurations only comes from the hyperfine potential, which is inversely proportional to the interacting quark mass. Thus, the mixing in the ccc¯c¯ state is generally larger than that in the bbb¯b¯ state.
In model II, all the flavor dependence is packaged into the hyperfine interaction, which is different from model I. The hyperfine interaction in model II should play a more important role than that in model I. Therefore, the couple-channel effect in model II is stronger as illustrated in Fig. 3.
In model II, since the r0 in the hyperfine interaction is the function of the reduced mass between the two quarks, its value for bc¯ is in proximity to that of cc¯. Then, the mixing in ccc¯c¯ and bbc¯c¯ are similar as illustrated in Table VII. One may question the additional dependence of the mixing on the number of the expanding basis. For instance, when we use 2×33 bases to expand the wave function of the ccc¯c¯ state in model I, we find there are 11.4%3¯c-3c and 88.6%6c-6¯c components in the tetraquark state. The percents change slightly with the number of the basis.
In Ref. [49], the authors pointed out that the state |(QQ)3¯c(Q¯Q¯)3c⟩ (Q=c, b) is located lower than the |(QQ)6c(Q¯Q¯)6¯c⟩ state, which contradicts with our results. The inconsistency was due to their use of particular wave functions. The authors used the same oscillating parameters for the 3¯c-3c and 6c-6¯c states. Moreover, the oscillating parameters are proportional to the reduced masses of the interacting quarks. With their wave function, we reproduced their results. However, if we remove the two constrains on the wave functions, we find the lowest state with a dominant 6c-6¯c component as listed in Table VIII, which is lower than that in Ref. [49].
VIII10.1103/PhysRevD.100.096013.t8
The comparison of the mass spectra of 0++ccc¯c¯ and bbb¯b¯ from Ref. [49] and our results using the same quark model. In the right table, we remove the constrains on the wave functions used in Ref. [49].
Ref. [49]Without constrainsJPC=0++w=0.325M [GeV]3¯c⊗3c6c⊗6¯cM [GeV]3¯c⊗3c6c⊗6¯cccc¯c¯βa=βb=0.49, β=0.69647066%34%βa=βb=0.4, β=0.6641733%67%γa=γb=0.49, γ=0.69655934%66%γa=γb=0.4, γ=0.7650967%33%bbb¯b¯βa=βb=0.88, β=1.241926866%34%βa=βb=0.7, β=0.91922618%82%γa=γb=0.88, γ=1.241930634%66%γa=γb=0.7, γ=0.91926882%18%The tetraquark states with <inline-formula><mml:math display="inline"><mml:msup><mml:mi>J</mml:mi><mml:mi>P</mml:mi></mml:msup><mml:mo>=</mml:mo><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> and <inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo>+</mml:mo><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>
Constrained by the Fermi statistics, the tetraquark states QQQ¯′Q¯′ (Q and Q′ may be the same flavors) with JP=1+- and 2++ only contain one color component, i.e., 3¯c-3c. We list the mass spectra of the S-wave states and their radial excitations in Table IX. The mass spectra in the two models are quite similar to each other. The results from model II are slightly higher than those in model I.
IX10.1103/PhysRevD.100.096013.t9
The mass spectra of the ccc¯c¯, bbb¯b¯, and bbc¯c¯ states with JPC=1+- and 2++ in units of GeV.
Model InSJPC=1+-JPC=2++Model IInSJPC=1+-JPC=2++ccc¯c¯βa=0.41S6.4256.432βa=0.51S6.4506.479βb=0.42S6.8566.864βb=0.52S6.8946.919β=0.63S6.9156.919β=0.63S7.0367.058bbb¯b¯βa=0.71S19.24719.249βa=1.01S19.31119.325βb=0.72S19.59419.596βb=1.02S19.81319.823β=0.93S19.68119.682β=1.13S20.06520.077bbc¯c¯βa=0.71S12.86412.868βa=0.71S12.92412.940βb=0.52S13.25913.262βb=0.52S13.32113.334β=0.73S13.29713.299β=0.73S13.36413.375
The tetraquark states with JP=1+- and 2++ have the same configurations except the total spin. Therefore, the mass difference arises from the hyperfine potential, which is quite small compared with the OGE Coulomb and linear confinement potentials. Thus, the mass spectra of these two kinds of states are almost the same.
Discussion
A tetraquark state can be expressed in another set of color representations as illustrated in the Appendix, |(Q1Q2)3¯c(Q¯3Q¯4)3c⟩=13|(Q1Q¯3)1c(Q2Q¯4)1c⟩-23|(Q1Q¯3)8c(Q2Q¯4)8c⟩=-13|(Q1Q¯4)1c(Q2Q¯3)1c⟩+23|(Q1Q¯4)8c(Q2Q¯3)8c⟩,|(Q1Q2)6c(Q¯3Q¯4)6¯c⟩=23|(Q1Q¯3)1c(Q2Q¯4)1c⟩+13|(Q1Q¯3)8c(Q2Q¯4)8c⟩=23|(Q1Q¯4)1c(Q2Q¯3)1c⟩+13|(Q1Q¯4)8c(Q2Q¯3)8c⟩.To investigate the inner structure of the tetraquark, we calculate its proportions in the new set and the root mean square radii of the state, which are listed in Table X. The ground states contain the 8c⊗8c configuration. In model I, the proportion of the 8c⊗8c configuration is considerable, which supports that the solution is a confined state rather than a scattering state of two mesons. In model II, though the 1c⊗1c configuration is dominant, the root mean square radii are of the size of nucleons. Thus, they are also unlikely to be scattering states.
X10.1103/PhysRevD.100.096013.t10
The proportion of the color configurations and the root mean square radii of the ccc¯c¯, bbb¯b¯, and bbc¯c¯ (b¯b¯cc) tetraquark states with JPC=0++. ⟨rij2⟩ and ⟨r(′)2⟩ are the root mean square radii corresponding to the second Jacobi coordinate in Fig. 1.
We also take the ccc¯c¯ as an example to study the density distributions of r2ρ(r), r2ρ(r′), r122ρ(r12), and r132ρ(r13). The ρ(r) and ρ(r12) are defined as follows: ρ(r)=∫|ψ(r12,r34,r)|2dr→12dr→34dr→^,ρ(r12)=∫|ψ(r12,r34,r)|2dr→dr→34dr→^12.The definitions of the ρ(r13) and ρ(r′) are similar. The dependence of the density distributions on the extension of the basis function is displayed in Fig. 4. We find that the distributions are confined in the spatial space and tend to be stable with different number of the expanding basis, which indicates the state may be a confined state instead of a scattering state.
410.1103/PhysRevD.100.096013.f4
The dependence of density distributions on the number of the basis functions.
We present the mass spectra of the tetraquark states and the mass thresholds of possible scattering states in Fig. 5. As illustrated in this figure, the bbb¯b¯, ccc¯c¯, and bbc¯c¯ states with JPC=0++ are the lowest states. But they are still located above the corresponding meson-meson mass thresholds, which indicates that there may not exist bound states in the two quark models.
510.1103/PhysRevD.100.096013.f5
The mass spectra of the ccc¯c¯, bbb¯b¯, and bbc¯c¯(ccb¯b¯) tetraquark states. The blue line and red dotted dashed line represent the results in models I and II, respectively.
We also investigate the constituent quark mass dependence of the tetraquark spectra. We vary the quark mass and display the results in Fig. 6. The figure shows that both the tetraquark mass and the ηQηQ threshold increase with the quark mass. The QQQ¯Q¯ is always located above the mass thresholds of the ηQηQ and no bound tetraquark states exist.
610.1103/PhysRevD.100.096013.f6
The quark mass dependence of the 0++ tetraquark states QQQ¯Q¯ in model II. In this figure, we use the ηQ to denote the meson state QQ¯ with JPC=0-+.
SUMMARY
In this work, we have systematically calculated the mass spectra of the tetraquark states ccc¯c¯, bbb¯b¯, and bbc¯c¯ in two nonrelativistic quark models, which contain the OGE Coulomb, linear confinement, and hyperfine potentials.
For a QQQ¯′Q¯′ (Q and Q′ may be the same flavors) state with JPC=0++, it can be formed by a 6c diquark and a 6¯c antidiquark, or a 3¯c diquark and a 3c antidiquark. For the tetraquark states ccc¯c¯ and bbb¯b¯, the 6c-6¯c states are located lower than the 3¯c-3c ones due to the strong attractions between the diquark and the antidiquark. For the bbc¯c¯ (ccb¯b¯), the mass of the 6c-6¯c state is lower than that of the 3¯c-3c one in model I, while the 3¯c-3c one is lower in model II. Our calculation shows that the 6c-6¯c color configuration is important and sometimes even dominant in the formation of fully-heavy tetraquark states. One should be cautious about neglecting the 6c-6¯c color configurations in calculating the tetraquark states.
The 6c-6¯c configuration couples with the 3¯c-3c one through the interactions between the diquark and antidiquark. For a QQQ¯′Q¯′ state, we prove that only the hyperfine potential contributes to the mixing between the two configurations, while the contributions from the OGE Coulomb and the linear confinement potentials cancel out exactly.
In Table XI, we summarize our numerical results and those from the CMI model [43,44,47], a nonrelativistic effective field theory and a relativized diquark and antidiquark model [34], a diffusion Monte-Carlo method [33], a constituent quark model with the hyperspherical formalism [41], the nonrelativistic potential model [49], and the QCD sum rule [37,57]. In this table, we notice that the numerical results in the two nonrelativistic quark models are similar to each other. The results show that the lowest states are the ones with JPC=0++. These ground states are located about 300–450 MeV above the lowest scattering states, which indicates that there may not exist bound tetraquark states ccc¯c¯, bbb¯b¯, and bbc¯c¯(ccb¯b¯) in the scheme of the two nonrelativistic quark models.
XI10.1103/PhysRevD.100.096013.t11
The mass spectra (in units of GeV) of the tetraquark states ccc¯c¯, bbb¯b¯, and bbc¯c¯ in different frameworks. The Mth1 and Mth2 are the numerical results from the quark models I and II in this work, respectively.
The parameters of the two quark models are determined by the meson spectrum. The potentials in a four-body system may be slightly different from those which are widely used in the conventional meson and baryon systems. The different confinement mechanism may lead to different spectra. For example, the three-body force arising from the triple-gluon vertex may be non-negligible for the multiquark systems. In contrast, this force vanishes for the traditional qq¯ meson and qqq baryons. The fully-heavy tetraquark states can be searched for at CMS, LHCb, and BelleII. More experimental data may provide a deeper understanding of the interactions in the multiquark system.
ACKNOWLEDGMENTS
G. J. Wang is very grateful to X. Z. Weng, X. L. Chen, and W. Z. Deng for very helpful discussions. We also thank Professor Makoto Oka and Professor Emiko Hiyama for helpful suggestions. This project is supported by the National Natural Science Foundation of China under Grants No. 11575008, No. 11621131001, and No. 11975033.
FIERZ TRANSORMATION
The color wave function of the 3¯c-3c configuration in the tetraquark is |(Q1Q2)3¯c(Q¯3Q¯4)3c⟩=123εiabεief|Q1aQ2bQ¯3eQ¯4f⟩,which is normalized to be 1. The color rearrangement for the tetraquark is [58]δafδeb=13δaeδbf+12λnaeλnbf,where λn is the Gell-Mann Matrix. The color function is then decomposed as |(Q1Q2)3¯c(Q¯3Q¯4)3c⟩=123(δaeδbf-δafδbe)|Q1aQ2bQ¯3eQ¯4f⟩=123(δaeδbf-13δaeδbf-12λnaeλnbf)|Q1aQ2bQ¯3eQ¯4f⟩=123(23δaeδbf-12λnaeλnbf)|Q1aQ2bQ¯3eQ¯4f⟩=1313|(Q1aδaeQ¯3e)(Q2bδbfQ¯4f)⟩-23142|(Q1aλnaeQ¯3e)(Q2bλnbfQ¯4f)⟩=13|(Q1Q¯3)1c(Q2Q¯4)1c⟩-23|(Q1Q¯3)8c(Q2Q¯4)8c⟩,where |(Q1Q¯3)8c(Q2Q¯4)8c⟩=142|(Q1aλnaeQ¯3e)(Q2bλnbfQ¯4f)⟩.|(Q1Q¯3)1c(Q2Q¯4)1c⟩=13|(Q1aδaeQ¯3e)(Q2bδbfQ¯4f)⟩,with ⟨(Q1Q¯3)8c(Q2Q¯4)8c|(Q1Q¯3)8c(Q2Q¯4)8c⟩=1,⟨(Q1Q¯3)1c(Q2Q¯4)1c|(Q1Q¯3)1c(Q2Q¯4)1c⟩=1.Since the Q¯3 and Q¯4 form the antisymmetric antitriplet color representation in the 3c-3¯c configuration, one obtains |(Q1Q2)3¯c(Q¯3Q¯4)3c⟩=-|(Q1Q2)3¯c(Q¯4Q¯3)3c⟩=-13|(Q1Q¯4)1c(Q2Q¯3)1c⟩+23|(Q1Q¯4)8c(Q2Q¯3)8c⟩.The wave function of the 6c-6¯c color configuration is |(Q1Q2)6c(Q¯3Q¯4)6¯c⟩=146(δaiδbj+δajδbi)(δamδbn+δanδbm)|Q1iQ2jQ¯3mQ¯4n⟩=126(δimδjn+δjmδin)|Q1iQ2jQ¯3mQ¯4n⟩=126(δimδjn+13δnjδim+12λknjλkim)|Q1iQ2jQ¯3mQ¯4n⟩=2313|(Q1iδimQ¯3m)(Q2jδjnQ¯4n)⟩+13142|(Q1iλkimQ¯3m)(Q2jλkjnQ¯4n)⟩=23|(Q1Q¯3)1c(Q2Q¯4)1c⟩+13|(Q1Q¯3)8c(Q2Q¯4)8c⟩.Since the Q¯3 and Q¯4 are in the symmetric 6¯c representation, one obtains |(Q1Q2)6c(Q¯3Q¯4)6¯c⟩=|(Q1Q2)6c(Q¯4Q¯3)6¯c⟩=23|(Q1Q¯4)1c(Q2Q¯3)1c⟩+13|(Q1Q¯4)8c(Q2Q¯3)8c⟩.
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