PHYSICAL REVIEW D 100, 016006 (2019)
All-heavy tetraquarks
† ‡ Ming-Sheng Liu,1,4,* Qi-Fang Lü,1,4, Xian-Hui Zhong,1,4, and Qiang Zhao2,3,4,§ 1Department of Physics, Hunan Normal University, and Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Changsha 410081, China 2Institute of High Energy Physics and Theoretical Physics Center for Science Facilities, Chinese Academy of Sciences, Beijing 100049, China 3School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 4Synergetic Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China
(Received 21 January 2019; published 16 July 2019)
In this work, we study the mass spectra of the all-heavy tetraquark systems, i.e., ccc¯c¯, bbb¯b¯, bbc¯c=cc¯ b¯b¯, bcc¯c=cc¯ b¯c¯, bcb¯b=bb¯ b¯c¯, and bcb¯c¯, within a potential model by including the linear confining potential, Coulomb potential, and spin-spin interactions. It shows that the linear confining potential has important contributions to the masses and is crucial for our understanding of the mass spectra of the all-heavy ¯ ¯ tetraquark systems. For the all-heavy tetraquarks Q1Q2Q3Q4, our explicit calculations suggest that ¯ ¯ no bound states can be formed below the thresholds of any meson pairs ðQ1Q3Þ − ðQ2Q4Þ or ¯ ¯ ðQ1Q4Þ − ðQ2Q3Þ. Thus, we do not expect narrow all-heavy tetraquark states to be existing in experiments.
DOI: 10.1103/PhysRevD.100.016006
I. INTRODUCTION constituent quarks are confined inside these Zc or Zb states, it makes them the best candidates for QCD exotic hadrons. Experimental searches for and theoretical studies of Recently, the tetraquarks of all-heavy systems, such as exotic hadrons beyond the conventional quark model are ¯¯ ¯ ¯ an important test of nonperturbative properties of the strong cccc and bbbb, have received considerable attention with interaction theory QCD. Since the discovery of quark model the development of experiments. If there are stable tetra- quark ccc¯c¯ and/or bbb¯b¯ states, they are most likely to be [1] and QCD, the progresses on the experimental tools have ¯ ¯ brought to us a lot of novel phenomena in hadron physics. In observed at LHC [9]. In fact, a search for the tetraquark bbbb particular, during the past 15 years, there have been a states is being carried out by the LHCb Collaboration sizeable number of candidates for QCD exotics [2–8]. although no confirmed information has been observed Interestingly, but also puzzlingly, it shows that the number [10]. Other study interests for physicists arise from the of exotic candidates is far less than what we have expected special aspects of the all-heavy tetraquark systems [11]. for the hadron spectroscopy, where the internal effective They may favor to form genuine tetraquark configurations degrees of freedom of a hadron may contain quarks and rather than loosely bound hadronic molecules, since the light gluons beyond the conventional quark model prescription. mesons cannot be exchanged between two heavy mesons. Strong evidences for such exotic hadrons include some Furthermore, it will be very easy to distinguish the all-heavy of those recently observed XYZ states, e.g., Xð3872Þ, tetraquark states from the states which have been observed because their masses should be far away from the mass Zcð3900Þ, Zcð4020Þ, Zbð10610Þ, and Zbð10650Þ [2].In particular, these charged quarkoniumlike states, Z and Z , regions of the observed states. Thus, besides some previous c b – contain not only the hidden heavy flavor cc¯ or bb¯, but also works on the heavy tetraquark states [12 17],manynew – charged light flavors of ud¯ or du¯. Since at least four studies have been carried out in recent years [11,18 30], although some of the conclusions are quite different from each other. In some works, it is predicted that there exist *[email protected] † stable bound tetraquark ccc¯c¯ states and/or bound tetraquark [email protected] ‡ ¯ ¯ [email protected] bbbb states with relatively smaller masses below the thresh- §[email protected] olds of heavy charmonium pairs [11,21–28]. Thus, their decays into heavy quarkonium pairs through quark rear- Published by the American Physical Society under the terms of rangements will be hindered. In contrast, in some other the Creative Commons Attribution 4.0 International license. works, it is predicted that there should be no stable bound Further distribution of this work must maintain attribution to ¯ ¯ the author(s) and the published article’s title, journal citation, tetraquark ccc¯c¯ and bbbb states [12,16,18,29,30] because and DOI. Funded by SCOAP3. the predicted masses are large enough for them to decay into
2470-0010=2019=100(1)=016006(12) 016006-1 Published by the American Physical Society LIU, LÜ, ZHONG, and ZHAO PHYS. REV. D 100, 016006 (2019)
¯ ¯ heavy quarkonium pairs. To some extent, a better under- confinement for the four-quark system Q1Q2Q3Q4,we standing of the possible mass locations is not only crucial for have 12 configurations as follows: understanding their underlying dynamics, but also useful for 6 ¯ ¯ 6¯ 0 6 ¯ ¯ 6¯ 0 experimental searches for their existence. j1i¼j½Q1Q2 1½Q3 Q4 1i0; j2i¼jfQ1Q2g0fQ3 Q4g0i0; In this work, we systemically study the mass spectra of ¯ ¯ 3 3¯ ¯ ¯ 3 0 4 3¯ ¯ ¯ 3 0 the all-heavy tetraquark Q1Q2Q3Q4 systems with a poten- j i¼jfQ1Q2g1fQ3 Q4g1i0; j i¼j½Q1Q2 0½Q3 Q4 0i0; – 6 ¯ ¯ 6¯ 0 6 ¯ ¯ 6¯ 0 tial model widely used in the literature [31 49].Our j5i¼j½Q1Q2 1½Q3 Q4 1i1; j6i¼j½Q1Q2 1fQ3 Q4g0i1; purpose is to understand two key issues based on the 7 6 ¯ ¯ 6¯ 0 8 3¯ ¯ ¯ 3 0 knowledge collected in the study of heavy quarkonium j i¼jfQ1Q2g0½Q3 Q4 1i1; j i¼jfQ1Q2g1fQ3 Q4g1i1; spectrum. The first one is what a quark potential model can 3¯ ¯ ¯ 3 0 3¯ ¯ ¯ 3 0 j9i¼jfQ1Q2g1½Q3 Q4 0i1; j10i¼j½Q1Q2 0fQ3 Q4g1i1; tell about the all-heavy tetraquark system. The second one 6 ¯ ¯ 6¯ 0 3¯ ¯ ¯ 3 0 is where the masses of the ground states could be located if j11i¼j½Q1Q2 1½Q3 Q4 1i2; j12i¼jfQ1Q2g1fQ3 Q4g1i2; the all-heavy tetraquark states do exist. At this moment, we do not consider any orbital or radial where fg and ½ denote the symmetric and antisymmetric excitations of the all-heavy tetraquarks. Instead, we would flavor wave functions of the two quarks (antiquarks) sub- like to address where and how the all-heavy tetraquarks systems, respectively. The subscripts and superscripts are the would manifest themselves in their lowest states. For a spin quantum numbers and representations of the color spectrum of multiquark states, a correct identification of the SU(3) group, respectively. A symmetric spatial wave func- ground state should be the first step towards a better tion is implied for the ground states under investigation. ¯ understanding of the multiquark dynamics in the non- It should be emphasized that for the bcbc¯ systems the perturbative regime. J ¼ 1 states can have both C ¼ 1, which can be con- The potentials between the quarks, such as the linear structed by the linear combinations of j6i, j7i, j9i and j10i, confining potential, color Coulomb potential, and spin-spin 1 interactions, are adopted the standard forms of the potential 0 6 ¯ 6¯ 0 6 ¯ 6¯ 0 j6 i¼pffiffiffi ðjðbcÞ1ðbc¯Þ0i1 − jðbcÞ0ðbc¯Þ1i1Þ; ð1Þ models. The model parameters are determined by fitting the 2 mass spectra of charmonium, bottomonium, and Bc meson. 1 ¯ ¯ In our calculations, we find both the confining potential and j70i¼pffiffiffi ðjðbcÞ6ðb¯c¯Þ6i0 þjðbcÞ6ðb¯c¯Þ6i0Þ; ð2Þ color Coulomb potential are very crucial for understanding 2 1 0 1 0 1 1 the masses of the all-heavy tetraquarks. The linear confining 1 potential as well as the kinetic energy contributes a quite large 90 ffiffiffi 3¯ ¯ ¯ 3 0 − 3¯ ¯ ¯ 3 0 ¯ ¯ j i¼p ðjðbcÞ1ðbcÞ0i1 jðbcÞ0ðbcÞ1i1Þ; ð3Þ positive mass term to the all-heavy tetraquarks Q1Q2Q3Q4, 2 which leads to a large mass far above the threshold of the ¯ ¯ ¯ ¯ 1 meson pair Q1Q3 − Q2Q4 or Q1Q4 − Q2Q3, although the 0 3¯ ¯ 3 0 3¯ ¯ 3 0 j10 i¼pffiffiffi ðjðbcÞ1ðbc¯Þ0i1 þjðbcÞ0ðbc¯Þ1i1Þ; ð4Þ color Coulomb potential contributes a very large negative 2 mass term. As a consequence, we find no bound all-heavy ¯ ¯ 60 90 −1 70 tetraquarks Q1Q2Q3Q4 below the threshold of any meson where configurations j i and j i have C ¼ , and j i ¯ − ¯ ¯ − ¯ and j100i have C ¼þ1. Since the permutation symmetries pairs Q1Q3 Q2Q4 or Q1Q4 Q2Q3. ¯ The paper is organized as follows: a brief introduction to are lost for the bc and bc¯ subsystems, in this work, we use the framework is given in Sec. II. In Sec. III, the numerical () to denote no permutation symmetries for these quark pair results and discussions are presented. A short summary is subsystems. given in Sec. IV. In Table I, all possible configurations and corresponding quantum numbers for the ccc¯c¯, bbb¯b¯, bbc¯c¯, bcc¯c¯, bcb¯b¯, ¯ II. FRAMEWORK and bcbc¯ systems are listed. A. Quark model classification B. Hamiltonian for the multiquark system In the charm and bottom quark sector, there are nine ¯ ¯ ¯¯ ¯ ¯ ¯¯ The following nonrelativistic Hamiltonian is adopted for different all-heavy Q1Q2Q3Q4 systems: cccc, bbbb, bccc, ¯ ¯ the calculation of the masses of the all-heavy Q1Q2Q3Q4 bcb¯b¯, bbc¯c¯, bcb¯c¯, ccb¯c¯, bbb¯c¯, and ccb¯b¯. Note that ccb¯c¯, system: bbb¯c¯, and ccb¯b¯ are the antiparticles of bcc¯c¯, bcb¯b¯, and ¯¯ bbcc, respectively. Thus, we need only consider six X4 X ¯¯ ¯ ¯ ¯¯ ¯ ¯ ¯¯ ¯ ¯ systems, cccc, bbbb, bccc, bcbb, bbcc, and bcbc,in H ¼ mi þ Ti − TG þ VijðrijÞ; ð5Þ our calculations. i 1 i 016006-2 ALL-HEAVY TETRAQUARKS PHYS. REV. D 100, 016006 (2019) TABLE I. Configurations of all-heavy tetraquarks. System JPðCÞ Configuration ¯ ¯ 0þþ 6 6¯ 0 3¯ 3 0 ccc c jfccg0fc¯ c¯g0i0 jfccg1fc¯ c¯g1i0 ··· 1þ− 3¯ 3 0 jfccg1fc¯ c¯g1i1 ··· ··· 2þþ 3¯ 3 0 jfccg1fc¯ c¯g1i2 ··· ··· ¯ ¯ 0þþ 6 ¯ ¯ 6¯ 0 3¯ ¯ ¯ 3 0 bbb b jfbbg0fb bg0i0 jfbbg1fb bg1i0 ··· 1þ− 3¯ ¯ ¯ 3 0 jfbbg1fb bg1i1 ··· ··· 2þþ 3¯ ¯ ¯ 3 0 jfbbg1fb bg1i2 ··· ··· ¯ ¯ 0þ 6 6¯ 0 3¯ 3 0 bbc c jfbbg0fc¯ c¯g0i0 jfbbg1fc¯ c¯g1i0 ··· 1þ 3¯ 3 0 jfbbg1fc¯ c¯g1i1 ··· ··· 2þ 3¯ 3 0 jfbbg1fc¯ c¯g1i2 ··· ··· ¯ ¯ 0þ 6 6¯ 0 3¯ 3 0 bcc c jðbcÞ0fc¯ c¯g0i0 jðbcÞ1fc¯ c¯g1i0 ··· 1þ 6 6¯ 0 3¯ 3 0 3¯ 3 0 jðbcÞ1fc¯ c¯g0i1 jðbcÞ1fc¯ c¯g1i1 jðbcÞ0fc¯ c¯g1i1 2þ 3¯ 3 0 jðbcÞ1fc¯ c¯g1i2 ··· ··· ¯ ¯ 0þ 6 ¯ ¯ 6¯ 0 3¯ ¯ ¯ 3 0 bcb b jðbcÞ0fb bg0i0 jðbcÞ1fb bg1i0 ··· 1þ 6 ¯ ¯ 6¯ 0 3¯ ¯ ¯ 3 0 3¯ ¯ ¯ 3 0 jðbcÞ1fb bg0i1 jðbcÞ1fb bg1i1 jðbcÞ0fb bg1i1 2þ 3¯ ¯ ¯ 3 0 jðbcÞ1fb bg1i2 ··· ··· ¯ ¯ 0þþ 6 ¯ 6¯ 0 6 ¯ 6¯ 0 bcb c jðbcÞ1ðb c¯Þ1i0 jðbcÞ0ðb c¯Þ0i0 ··· 3¯ ¯ 3 0 3¯ ¯ 3 0 jðbcÞ1ðb c¯Þ1i0 jðbcÞ0ðb c¯Þ0i0 ··· þ− 6 6¯ 0 1 6 6¯ 0 6 6¯ 0 1 ¯ ¯ pffiffi ¯ ¯ − ¯ ¯ ··· jðbcÞ1ðb cÞ1i1 2 jðbcÞ1ðb cÞ0i1 jðbcÞ0ðb cÞ1i1 3¯ 3 0 1 3¯ 3 0 3¯ 3 0 ¯ ¯ pffiffi ¯ ¯ − ¯ ¯ ··· jðbcÞ1ðb cÞ1i1 2 jðbcÞ1ðb cÞ0i1 jðbcÞ0ðb cÞ1i1 þþ 1 6 6¯ 0 6 6¯ 0 1 3¯ 3 0 3¯ 3 0 1 pffiffi ¯ ¯ ¯ ¯ pffiffi ¯ ¯ ¯ ¯ ··· 2 jðbcÞ1ðb cÞ0i1 þjðbcÞ0ðb cÞ1i1 2 jðbcÞ1ðb cÞ0i1 þjðbcÞ0ðb cÞ1i1 2þþ 6 ¯ 6¯ 0 3¯ ¯ 3 0 jðbcÞ1ðb c¯Þ1i2 jðbcÞ1ðb c¯Þ1i2 ··· P ¯ ¯ λ λ λ λ ≡ 8 −λaλa the center-of-mass (c.m.) kinetic energy of the Q1Q2Q3Q4 i · j operator is defined as i · j a¼1 i j , where ≡ r − r a a system; rij j i jj is the distance between the ith quark λ is the complex conjugate of the Gell-Mann matrix λ . OGE and jth quark; and VijðrijÞ stands for the effective potential The OGE potential Vij is composed of the Coulomb type between the ith and jth quark. In this work, we adopt a potential VOGE ∝ ðλ · λ Þð1=r Þ, which provides the short- – – coul i j ij widely used potential form for VijðrijÞ [31 35,43 47], i.e., range interaction, and the color-magnetic interaction VOGE ∝ ðλ · λ Þðσ · σ Þ which provides mass splittings. OGE Conf CM i j i j VijðrijÞ¼Vij ðrijÞþVij ðrijÞ; ð6Þ Conf The form of Vij ðrijÞ is given by OGE where Vij stands for the one-gluon-exchange (OGE) 3 Conf − λ λ potential which describes the short-range quark-quark Vij ðrijÞ¼ 16 ð i · jÞ · brij; ð8Þ Conf interactions, while Vij ðrijÞ stands for the confinement potential which describes the long-range interaction behav- where the parameter b denotes the strength of the confine- OGE iors. The form of Vij is given by ment potential. There are nine parameters mc, mb, αcc, αbb, αbc, σcc, σbb, 3 −σ2 r2 α 1 π σ e ij ij 4 σbc, and b to be determined in the calculations. In VOGE ¼ ij ðλ · λ Þ − · ij · ðσ · σ Þ ; ¯ ¯ ij 4 i j r 2 π3=2 3m m i j Refs. [46,50], the masses of cc and bc spectrum are ij i j calculated by using the three-point difference central ð7Þ method [51] from the center (r ¼ 0) towards outside (r → ∞) point by point. The parameters mc, αcc, σcc, b, where σ are the Pauli matrices, and α stands for the strong i ij mb, αbc, σbc have been determined. In this work, we use the coupling strength between two quarks. If the interaction same method to determine the parameters αbb, σbb,by λ λ ¯ occurs between two quarks or antiquarks, theP i · j operator fitting the masses of bb spectrum. The parameter set is λ λ ≡ 8 λaλa appearing in Eq. (7) is defined as i · j a¼1 i j , while listed in Table II. The corresponding theoretical results for if the interaction occurs between a quark and antiquark, the the masses of heavy quarkonia bb¯ are shown in Table III. 016006-3 LIU, LÜ, ZHONG, and ZHAO PHYS. REV. D 100, 016006 (2019) TABLE II. Quark model parameters used in this work. TABLE IV. Color matrix elements. mc (GeV) 1.483 hλ1 · λ2ihλ3 · λ4ihλ1 · λ3ihλ2 · λ4ihλ1 · λ4ihλ2 · λ3i mb (GeV) 4.852 ˆ 4 343 −10 3 −10 3 −10 3 −10 3 α hζ1jOjζ1i ======cc 0.5461 ˆ −8 3 −8 3 −4 3 −4 3 −4 3 −4 3 α hζ2jOjζ2i = = = ffiffiffi = ffiffiffi = = bb 0.4311 p p pffiffiffi pffiffiffi ζ ˆ ζ 00−2 2 −2 2 2 2 2 2 αbc 0.5021 h 1jOj 2i σcc (GeV) 1.1384 σbb (GeV) 2.3200 σ bc (GeV) 1.3000 In the spin space, one has six spin wave functions, b (GeV2) 0.1425 00 ¯ ¯ χ0 ¼jðQ1Q2Þ0ðQ3Q4Þ0i0; ð13Þ χ11 ¯ ¯ TABLE III. The masses (MeV) of bottomonium mesons. 0 ¼jðQ1Q2Þ1ðQ3Q4Þ1i0; ð14Þ Experimental data are taken from PDG [2]. 01 ¯ ¯ χ1 ¼jðQ1Q2Þ0ðQ3Q4Þ1i1; ð15Þ ηb hb χb0 χb1 χb2 Meson ϒ η ϒð2SÞ ð2SÞ ð1PÞ ð1PÞ ð1PÞ ð1PÞ b 10 ¯ ¯ χ1 ¼jðQ1Q2Þ1ðQ3Q4Þ0i1; ð16Þ Ours 9460 9390 10024 10005 9941 9859 9933 9957 Expt. 9460 9399 10023 9999 9899 9859 9893 9912 11 ¯ ¯ χ1 ¼jðQ1Q2Þ1ðQ3Q4Þ1i1; ð17Þ 11 ¯ ¯ C. Matrix elements in color and spin spaces χ2 ¼jðQ1Q2Þ1ðQ3Q4Þ1i2: ð18Þ In order to obtain the mass of a tetraquark state from the nonrelativistic Hamiltonian defined in Eq. (5), first one According to the SU(2) Clebsch-Gordan coefficients, we needs to calculate the matrix elements of hλ · λ i and easily obtain the expressions of the spin wave functions as i j follows: hσi · σji in the color and spin spaces, respectively. In the color space, one has two kinds of a color-singlet 1 state, χ00 ↑↓↑↓ − ↑↓↓↑ − ↓↑↑↓ ↓↑↓↑ 0 ¼ 2 ð þ Þ; ð19Þ rffiffiffiffiffi ¯ 6 ¯ ¯ 6¯ 0 ζ1 ¼j66i¼jðQ1Q2Þ ðQ3Q4Þ i ; ð9Þ 1 χ11 2↑↑↓↓ − ↑↓↑↓ − ↑↓↓↑ 0 ¼ 12ð − ↓↑↑↓ − ↓↑↓↑ 2↓↓↑↑ ¯ 3¯ ¯ ¯ 3 0 þ Þ; ð20Þ ζ2 ¼j33i¼jðQ1Q2Þ ðQ3Q4Þ i : ð10Þ rffiffiffi 1 χ01 ↑↓↑↑ − ↓↑↑↑ According to the SU(3) Clebsch-Gordan coefficients, one 1 ¼ 2ð Þ; ð21Þ easily obtains the expressions of the color wave functions – rffiffiffi as follows [52 55]: 1 χ10 ↑↑↑↓ − ↑↑↓↑ 1 ¼ 2ð Þ; ð22Þ 1 ¯ ¯ ζ1 ¼ pffiffiffi ½ðrb þ brÞðb r¯ þr¯ bÞþðgr þ rgÞðg¯ r¯ þr¯ g¯Þ 2 6 1 χ11 ↑↑↑↓ ↑↑↓↑ − ↑↓↑↑ − ↓↑↑↑ 1 ¼ 2 ð þ Þ; ð23Þ þðgb þ bgÞðb¯ g¯ þg¯ b¯Þ 2 ¯ ¯ 2 ¯ ¯ 2 ¯ ¯ 11 þ ðrrÞðr rÞþ ðggÞðg gÞþ ðbbÞðb bÞ ; ð11Þ χ2 ¼ ↑↑↑↑; ð24Þ with these spin wave functions one can work out the matrix 1 ¯ ¯ σ σ ζ2 ¼ pffiffiffi ½ðbr − rbÞðb r¯ −r¯ bÞ − ðrg − grÞðg¯ r¯ −r¯ g¯Þ elements of h i · ji [56], which have been listed in 2 3 Table V. þðbg − gbÞðb¯ g¯ −g¯ b¯Þ ; ð12Þ D. Matrix elements in the coordinate space with these color wave functions one can work out the The trail wave function of the four-quark states without matrix elements hλi · λji [56], which have been summa- any spatial excitations in the coordinate space is expanded rized in Table IV. by a series of Gaussian functions, 016006-4 ALL-HEAVY TETRAQUARKS PHYS. REV. D 100, 016006 (2019) TABLE V. Spin matrix elements. where ψðωl; ri; rjÞ ≡ ϕðωl; riÞϕðωl; rjÞ, mij ¼ mimj=ðmi þ mjÞ. hσ1 ·σ2ihσ3 ·σ4ihσ1 ·σ3ihσ2 ·σ4ihσ1 ·σ4ihσ2 ·σ3i To separate out the center-of-mass kinetic energy χ00 Oˆ χ00 −3 −3 0000 h 0 j j 0 i TPG and finally work out the kinetic energy matrix element h χ11jOˆ j χ11i 11−2 −2 −2 −2 4 − 0 0 pffiffiffi pffiffiffi ffiffiffi ffiffiffi h i¼1 Ti TGi, we need to redefine the coordinates by the 00 ˆ 11 − 3 − 3 p p h χ0 jOj χ0 i 00 3 3 following Jacobi coordinates: 01 ˆ 01 −3 h χ1 jOj χ1 i 1 0000 10 ˆ 10 −3 h χ1 jOj χ1 i 1 0000 ξ1 ≡ r1 − r2; ð30Þ 11 ˆ 11 −1 −1 −1 −1 h χ1 jOj χ1 i 11 01 ˆ 10 −1 −1 h χ jOj χ i 00 11 ξ2 ≡ r3 − r4; 1 1 pffiffiffi pffiffiffi pffiffiffi pffiffiffi ð31Þ h χ01jOˆ j χ11i 00− 2 2 − 2 2 1 1 pffiffiffi pffiffiffi pffiffiffi pffiffiffi χ10 ˆ χ11 2 − 2 − 2 2 m1r1 þ m2r2 m3r3 þ m4r4 h 1 jOj 1 i 00 ξ ≡ − 11 ˆ 11 3 ; ð32Þ h χ2 jOj χ2 i 11 1111 m1 þ m2 m3 þ m4 m1r1 þ m2r2 þ m3r3 þ m4r4 ξ4 ≡ ; ð33Þ Y4 Xn 3=4 m1 þ m2 þ m3 þ m4 1 1 2 ψ r1; r2; r3; r4 C l − r ; ð Þ¼ i π 2 exp 2 2 i i¼1 l¼1 bil bil with these one can rewritten the Eq. (26) as ð25Þ Xn Y4 μ ω 3=4 μ ω ψ ξ ξ ξ ξ C i l − i l ξ2 ð 1; 2; 3; 4Þ¼ l π exp 2 i ; where the parameters bil are related to the harmonic l i¼1 2 oscillator frequencies ωl with 1=b ¼ m ωl.Withthe il i ð34Þ ansatz of the mass independent frequency ωl for a 2 quark of mass m [57],i.e.,1=b l ¼ 1=bl ¼ m ωl i i u where μ1 ≡ m1m2=ðm1 þ m2Þ, μ2 ≡ m3m4=ðm3 þ m4Þ, (m ¼ 313 MeV), the trail wave function of the four quark u μ3 ≡ðm1þm2Þðm3 þm4Þ=M, μ4 ¼M≡m1 þm2 þm3þm4. states can be simplified to be With the trail wave function defined in Eq. (34), the kinetic energy matrix element is worked out to be Xn Y4 3=4 miωl miωl 2 ψðr1; r2; r3; r4Þ¼ Cl exp − r π 2 i X4 9 Xn Xn ω ω 4 l i¼1 ð l l0 Þ Ti − TG ¼ ClCl0 ω ω : ð35Þ 4 4 lþ l0 7 Xn Y i¼1 l l0 ð 2 Þ ≡ Cl ϕðωl; riÞ; ð26Þ l 1 i¼ III. RESULTS AND DISCUSSIONS which is often adopted in the calculations of the multiquark In this work, we adopt the variation principle to solve systems [58,59]. the Schrödinger equation. Following the method used in In the coordinate space, we need work out the matrix −σ2 r2 Ref. [60], the oscillator length bl are set to be elements of h1=riji, he ij ij i, and hriji. Combing the trail l−1 wave functions defined in Eq. (26), we obtain bl ¼ b1a ðl ¼ 1; …;nÞ; ð36Þ rffiffiffiffiffiffiffi 3=2 1 mij ðωlωl0 Þ where n is the number of Gaussian functions, and a is the hψðωl; ri; rjÞ ψðωl0 ; ri; rjÞi ¼ 2 ω ω ; π lþ l0 5=2 rij ð 2 Þ ratio coefficient. There are three parameters fb1;bn;ng to be determined through variation method. It is found that ð27Þ when we take b1 ¼ 0.1 fm, bn ¼ 4 fm, n ¼ 15, we will obtain stable solutions for the four-quark systems. 2ω ω 3 l l0 2 2 2 m ð Þ When all the matrix elements have been worked out, we −σ ij ωlþωl0 ψ ω r r ijrij ψ ω r r h ð l; i; jÞje j ð l0 ; i; jÞi ¼ ω ω ; can solve the generalized matrix eigenvalue problem, lþ l0 σ2 mij 2 þ ij Xn Xn ð28Þ − Cl 0 ðHll0 ElNll0 Þ l0 ¼ ; ð37Þ sffiffiffiffiffiffiffiffiffiffi l¼1 l0¼1 3=2 1 ðωlωl0 Þ hψðωl; ri; rjÞjrijjψðωl0 ; ri; rjÞi ¼ 2 ω ω ; π lþ l0 7=2 where mij ð 2 Þ ð29Þ Hll0 ¼hψðωlÞφζ χjHjψðωl0 Þφζ χi; ð38Þ 016006-5 LIU, LÜ, ZHONG, and ZHAO PHYS. REV. D 100, 016006 (2019) and Table VII. It shows that our predicted masses for the ccc¯c¯ system are roughly compatible with the nonrelativistic Nll0 ¼hψðωlÞφζ χjψðωl0 Þφζ χi; ð39Þ quark model predictions of Refs. [12,16], where both confining and Coulomb potentials are considered explicitly. Q 4 μiωl 3=4 μiωl 2 It is also interesting to find that similar results are with ψðωlÞ¼ ð Þ exp ½− ξ . φ, ζ,and χ i¼1 π 2 i given by the QCD sum rules [11]. In contrast, the masses stand for the flavor, color, and spin wave functions, respec- predicted by us are much larger than those predicted in tively. The physical state corresponds to the solution with a Refs. [17,21,22,24,25,27]. These methods which obtained minimum energy E . By solving this generalized matrix m small masses have some common features: either no eigenvalue problem, the mass of the tetraquark configuration confining potentials were explicitly included [17,21,22,24] and its spacial wave function can be determined. or a diquark picture was adopted in the calculations ¯ ¯ [25,27]. Recently, Wu et al. also obtained a large mass A. The ccc¯c¯ and bbbb systems ∼6.8–7.0 GeV for the ccc¯c¯ system with the heavier The predicted mass spectrum for the ccc¯c¯ system has constituent c-quark mass 1.72 GeV adopted [29]. been given in Table VI and also shown in Fig. 1(a). From We further analyze the contributions from each part of Table VI, it is found that there is little configuration mixing the Hamiltonian for the ccc¯c¯ system. The results are listed ¯ between the two JPC ¼ 0þþ states jfccg6fc¯c¯g6i0 and in Table IX. It shows that the averaged kinetic energy hTi, 0 0 0 Conf 3¯ 3 0 6 6¯ 0 the confining potential hV i, and the Coulomb potential jfccg fc¯c¯g i . The mass of the jfccg fc¯c¯g i configura- 1 1 0 0 0 0 VOGE have the same order of magnitude. In particular, the tion, 6518 MeV, is slightly larger than that of h coul i 3¯ ¯¯ 3 0 contributions from the confining potential are esizeable and jfccg1fccg1i0. The mass splitting between these two apparently cannot be neglected. Note that the confining PC 0þþ J ¼ states is about 31 MeV. The other two states potential contributes a positive energy to the system. Thus, PC 1þ− PC 2þþ J ¼ and J ¼ are also located in a similar neglecting this contribution will lead to much lower masses ∼6 5 mass region, i.e., . GeV, and the mass splitting between for the all-heavy system. In Refs. [17,21,22,24], the them is about 20 MeV. As shown in Fig. 1(a), the two confining potential was explicitly neglected. Although part PC J ¼ 0þþ states are about 500 MeV and 300 MeV above of the confining potential effects can be taken into account η η ψ ψ the mass thresholds of c c and J= J= , respectively. It by the effective constituent quark masses in the ground suggests that the JPC ¼ 0þþ states are unstable, and they states, our calculation shows that the impact from the η η ψ ψ might easily decay into the c c and J= J= final states inclusion of the confining potential seems not to be on − through quark rearrangements. The JPC ¼ 1þ state lies the constituent quark masses in the heavy quark sector, about 420 MeV above the mass threshold of ηcJ=ψ, while but rather on the relative strengths of the averaged JPC ¼ 2þþ is about 330 MeV above the mass threshold of matrix elements among the terms of the nonrelativistic J=ψJ=ψ; they might also easily decay into ηcJ=ψ and Hamiltonian. J=ψJ=ψ, respectively, through the quark rearrangements. In order to examine the role played by the confining As a comparison, our predicted masses and some potential in the spectrum of heavy quark system, we other typical results from other works are collected in compare the contributions from the OGE and confining TABLE VI. Predicted mass spectra for the ccc¯ c¯, bbb¯ b¯ and bbc¯ c¯ systems. JPðCÞ Configuration hHi (MeV) Mass (MeV) Eigenvector 0þþ 6 ¯ ¯ 6¯ 0 jfccg0fc cg0i0 6518 −0.2371 6518 ð1; 0Þ 3¯ ¯ ¯ 3 0 jfccg1fc cg1i0 −0.2371 6487 6487 ð0; 1Þ 1þ− 3¯ 3 0 jfccg1fc¯ c¯g1i1 (6500) 6500 1 2þþ 3¯ 3 0 jfccg1fc¯ c¯g1i2 (6524) 6524 1 0þþ 6 ¯ ¯ 6¯ 0 jfbbg0fb bg0i0 19338 −0.1102 19338 ð1; 0Þ 3¯ ¯ ¯ 3 0 jfbbg1fb bg1i0 −0.1102 19322 19322 ð0; 1Þ 1þ− 3¯ ¯ ¯ 3 0 jfbbg1fb bg1i1 (19329) 19329 1 2þþ 3¯ ¯ ¯ 3 0 jfbbg1fb bg1i2 (19341) 19341 1 0þ 6 ¯ ¯ 6¯ 0 jfbbg0fc cg0i0 13032 −0.1105 13032 ð1; 0Þ 3¯ ¯ ¯ 3 0 jfbbg1fc cg1i0 −0.1105 12953 12953 ð0; 1Þ 1þ 3¯ 3 0 jfbbg1fc¯ c¯g1i1 (12960) 12960 1 2þ 3¯ 3 0 jfbbg1fc¯ c¯g1i2 (12972) 12972 1 016006-6 ALL-HEAVY TETRAQUARKS PHYS. REV. D 100, 016006 (2019) (a) (b) (c) (d) (e) (f) FIG. 1. Mass spectra of all-heavy tetraquarks (solid lines) and their possible main decay channels (dashed lines). The unit of mass is MeV. η OGE ≃−637 conf potential for the c meson, i.e., hVcoul i MeV hV i ≃ 36%: ð40Þ and hVConfi ≃ 233 MeV, which are consistent with our OGE hVcoul i previous study in Ref. [46]. The ratio between the confining Conf OGE potential hV i and color Coulomb potential hVcoul i can This explicit result suggests that the neglect of confining reach up to potential cannot be justified for the cc¯ system. 016006-7 LIU, LÜ, ZHONG, and ZHAO PHYS. REV. D 100, 016006 (2019) TABLE VII. Our predicted masses (MeV) for the ccc¯ c¯ system compared with others. State Ours Ref. [29] Ref. [16] Ref. [11] Ref. [12] Ref. [13] Ref. [22] Ref. [17] Refs. [25,26] Ref. [27] Ref. [24] Ref. [21] 0þþ 6487 6797 6477 6460–6470 6437 6200 6192 6038–6115 5990 5969 5966 <6140 0þþ 6518 7016 6695 6440–6820 6383 1þ− 6500 6899 6528 6370–6510 6437 6101–6176 6050 6021 6051 2þþ 6524 6956 6573 6370–6510 6437 6172–6216 6090 6115 6223 As a general conclusion, we find that the confining In Table IX, the contributions from each part of the potential has significant contributions to the masses of the Hamiltonian for the bbb¯b¯ system are listed. It shows ccc¯c¯ system and are the same order of magnitude as the that the kinetic energy hTi ≃ 800 MeV, the confining Conf ≃ 400 color Coulomb potential. This will enhance the masses of potential hVij ðrijÞi MeV, and the coulomb poten- the ccc¯c¯ system and does not support the existence of a OGE ≃−1200 tial hVcoul i MeV, have the same order of magni- bound tetraquark of ccc¯c¯ with narrow widths. ¯ ¯ tude. As shown in Fig. 1(b), the mass splittings among The predicted mass spectrum for the bbbb system is very these JPC ¼ 0þþ, 1þ−, and 2þþ states follow a similar ¯¯ similar to that for the cccc one. The results are given in pattern as in the ccc¯c¯ system. Also similar to that for Table VI and shown in Fig. 1(b). The configuration the ccc¯c¯ system, the neglect of the confining potential 6 ¯ ¯ 6¯ 0 3¯ ¯ ¯ 3 0 ¯ ¯ mixing effects between jfbbg0fbbg0i0 and jfbbg1fbbg1i0 will lead to much lower masses for the bbbb system, are negligibly small, the higher mass state with mass and this may explain the low masses obtained in 6 ¯ ¯ 6¯ 0 ∼19338 MeV is the jfbbg0fbbg0i0 configuration, and the Refs. [17,21,22,24]. Although it is often argued that the 3¯ ¯ ¯ 3 0 lower one of ∼19322 MeV corresponds to jfbbg1fbbg1i0. confining potential contributions are perturbative for the Because of the heavier mass of the b quark, relatively smaller bottomonium system, explicit calculations seem not to support this phenomenon. In Ref. [46], we have studied mass splittings among these states are found. The pattern is ¯ also similar to that of the ccc¯c¯ system. Note that the pre- the bb spectrum and find that hVConfi ≃ 122 MeV and OGE ≃−970 η dicted masses are above the thresholds of the bottomo- hVcoul i MeV for the b meson. The ratio between – Conf OGE nium pairs for about 400 540 MeV. It suggests that hV i and hVcoul i can reach up to bound states of the bbb¯b¯ system with narrow widths are conf not favored. hV i ≃ 13% OGE : ð41Þ In Table VIII, we compare our results with other model hVcoul i calculations. It shows that our predicted masses are higher than most of the other predictions which are either For the four heavy quark system of bbb¯b¯, the increase of calculated without including the confining potential explic- the displacements between the two quarks (antiquarks) or itly [17,21,22,24] or based on the diquark picture [25]. quark-antiquark will experience larger confining forces. Similarly, based on the diquark picture, the lightest mass of Thus, the confining potential contributions cannot be bbb¯b¯ is estimated at 18.8 GeV by Ref. [28]. In these neglected in the calculations. As a consequence, our study − ¯ ¯ calculations, the tetraquark states of JPC ¼ 0þþ, 1þ ,or does not support the existence of the tetraquark bbbb 2þþ are either below or slightly above the thresholds of bound states with narrow widths. ηbηb, ηbϒð1SÞ or ϒð1SÞϒð1SÞ, respectively. Thus, they can Finally, it should be mentioned that for a simplicity, in become stable with narrow decay widths. In contrast, our our calculation, the variational wave functions of the calculations with the inclusion of the confining potential coordinate space are only adopted an s-wave form. − result in higher masses for the bbb¯b¯ system and do not Thus, the color wave functions for the JPC ¼ 1þ and favor the existence of such narrow tetraquark states. We 2þþ states is color 33¯ . However, the 33¯ color wave note that a rather large mass ∼20.2 GeV for the bbb¯b¯ functions for the JPC ¼ 1þ− and 2þþ states might slightly system is estimated by Ref. [29], where a heavier con- mix with the color 66¯ when one considers the orbital stituent b-quark mass 5.05 GeV is adopted. excitations in the coordinate space [18–20]. With a color TABLE VIII. Our predicted masses (MeV) for the bbb¯ b¯ system compared with others. State Ours Ref. [29] Refs. [25,26] Ref. [22] Ref. [24] Ref. [21] Ref. [23] Ref. [11] Ref. [30] Ref. [21] 0þþ 19322 20155 18840 18826 18754 18720 18690 18460–18490 18798 <18890 0þþ 19338 20275 18450–19640 1þ− 19329 20212 18840 18808 18320–18540 2þþ 19341 20243 18850 18916 18320–18530 016006-8 ALL-HEAVY TETRAQUARKS PHYS. REV. D 100, 016006 (2019) TABLE IX. The contributions from each part of the Hamil- than those predicted in the recent work [29], where ¯ ¯ tonian of the ccc¯ c¯ and bbb b systems in units of MeV. relatively large constituent quark masses for the b quark 5.05 GeV and c quark 1.72 GeV are adopted. JPC Configuration M T VConf VOGE VOGE h ih ihcoul ihCM i As shown in Fig. 1(c), all these states are above their 0þþ 6 6¯ 0 −811 jfccg0fc¯ c¯g0i0 6518 715 664 18 lowest open flavor decay channels for about 300 MeV. 3¯ 3 0 −834 −13 jfccg1fc¯ c¯g1i0 6487 756 646 Therefore, they can decay into the BcBc, BcBc,orBcBc 1þ− 3¯ 3 0 −825 jfccg1fc¯ c¯g1i1 6500 739 653 0 final states via the quark rearrangement quite easily. 2þþ 3¯ 3 0 −806 jfccg1fc¯ c¯g1i2 6524 708 667 23 ¯ ¯ 0þþ 6 ¯ ¯ 6¯ 0 −1203 C. The bcc¯c¯ and bcbb systems jfbbg0fb bg0i0 19338 768 356 9 3¯ ¯ ¯ 3 0 −1225 −6 ¯¯ ¯ ¯ jfbbg1fb bg1i0 19322 796 350 The states of both bccc and bcbb systems do not have 1þ− 3¯ ¯ ¯ 3 0 −1216 jfbbg1fb bg1i1 19329 785 353 0 determined C parity, and they share some common features 2þþ 3¯ ¯ ¯ 3 0 −1199 jfbbg1fb bg1i2 19341 763 357 12 in terms of heavy quark symmetry. The predicted mass spectra for these two configurations are listed in Table X and shown in Fig. 1(d) and Fig. 1(e). It shows that both the ¯ ¯ mixing effect, the mass of the JPC ¼ 1þ− and 2þþ states bcc¯c¯ and bcbb systems do not have a configuration ¯ ¯ might become slightly lower [18–20], which does not affect mixings effect between the color 6 ⊗ 6 and 3 ⊗ 3 con- our conclusions. figurations. For the bcc¯c¯ system, it shows that the higher P þ 6 6¯ 0 and lower mass states of J ¼ 0 are the jfbcg0fc¯c¯g0i0 3¯ 3 0 B. The bbc¯c¯ system and jfbcg1fc¯c¯g1i0 configuration, respectively. The bbc¯c¯ system is similar to the ccc¯c¯ and bbb¯b¯ ones The configuration mixing effects among these three P þ except that it does not have determined C parity, and there J ¼ 1 states are also rare which is shown in Table X. is no contributions from the annihilation potential. The The typical mass splitting is about 10 MeV, and the predicted mass spectrum for the bbc¯c¯ system is also listed predicted masses are about 500 MeV systematically smaller in Table VI and shown in Fig. 1(c). From Table VI, there is than those predicted in the recent work [29]. Again, we note 6 6¯ 0 that rather large constituent quark masses for the c and b no mixing effect between the jfbbg0fc¯c¯g0i0 and ¯ quarks are adopted in Ref. [29]. bb 3 c¯c¯ 3 0 jf g1f g1i0 configurations. The higher mass state As a consequence of the high masses predicted by P 0þ 6 ¯¯ 6¯ 0 (13032 MeV) of J ¼ state is the jfbbg0fccg0i0 our model, namely, the states of bcc¯c¯ system are about configuration, while the lower mass one (12953 MeV) 320–350 MeV above the mass threshold of BcJ=ψ, we find 3¯ ¯¯ 3 0 of the same quantum numbers is the jfbbg1fccg1i0 that these states can easily decay into the Bcηc, BcJ=ψ,or configuration. The mass splitting between these two BcJ=ψ final states via the quark rearrangements. Thus, we JP ¼ 0þ states is about 79 MeV. The other two states do not expect narrow states of bcc¯c¯ to be observed in with JP ¼ 1þ and 2þ have a small mass splitting of about experiment. ¯ ¯ 10 MeV and are located around 12.96 GeV. The masses For the bcbb system, its main properties are very similar predicted by us are about 600 MeV systematically smaller to that of the bcc¯c¯ system as shown in Table X and TABLE X. Predicted mass spectra for the bcc¯ c¯ and bcb¯ b¯ systems. JP Configuration hHi (MeV) Mass (MeV) Eigenvector 0þ 6 ¯ ¯ 6¯ 0 jðbcÞ0fc cg0i0 9763 −0.1739 9763 ð1; 0Þ 3¯ ¯ ¯ 3 0 jðbcÞ1fc cg1i0 −0.1739 9740 9740 ð0; 1Þ 1þ 6 6¯ 0 jðbcÞ1fc¯ c¯g0i1 !"#"# ¯ 9757 −0.0378 0.1004 9757 ð1; 0; 0Þ jðbcÞ3fc¯ c¯g3i0 1 1 1 −0.0378 9749 0.0179 9749 ð0; 1; 0Þ 3¯ ¯ ¯ 3 0 jðbcÞ0fc cg1i1 0.1004 0.0179 9746 9746 ð0; 0; 1Þ 2þ 3¯ 3 0 jðbcÞ1fc¯ c¯g1i2 (9768) 9768 1 0þ 6 ¯ ¯ 6¯ 0 jðbcÞ0fb bg0i0 16173 −0.1286 16173 ð1; 0Þ 3¯ ¯ ¯ 3 0 jðbcÞ1fb bg1i0 −0.1286 16158 16158 ð0; 1Þ ¯ 1þ 6 ¯ ¯ 6 0 !"#"# jðbcÞ1fb bg0i1 16167 0 0023 0 0744 16167 1 0 0 3¯ ¯ ¯ 3 0 . . ð ; ; Þ jðbcÞ1fb bg1i1 0 0023 16164 −0 0011 16164 0 1 0 3¯ ¯ ¯ 3 0 . . ð ; ; Þ jðbcÞ0fb bg1i1 0.0744 −0.0011 16157 16157 ð0; 0; 1Þ 2þ 3¯ ¯ ¯ 3 0 jðbcÞ1fb bg1i2 (16176) 16176 1 016006-9 LIU, LÜ, ZHONG, and ZHAO PHYS. REV. D 100, 016006 (2019) TABLE XI. Predicted mass spectra for the bcb¯ c¯ system. JPC Configuration hHi (MeV) Mass (MeV) Eigenvector 0þþ 6 ¯ ¯ 6¯ 0 jðbcÞ1ðb cÞ1i0 0 1 2 3 2 3 6 ¯ 6¯ 0 12901 −0.0264 −94 −0.1540 12835 0.82; 0; 0.58; 0 jðbcÞ0ðb c¯Þ0i0 ð Þ 3¯ ¯ 3 0 B −0.0264 12956 −0.1517 −93 C 6 12864 7 6 ð0; 0.71; 0; 0.70Þ 7 jðbcÞ1ðb c¯Þ1i0 @ A 4 5 4 5 ¯ −94 −0.1517 12968 −0.0105 13035 ð−0.58; 0; 0.82; 0Þ jðbcÞ3ðb¯ c¯Þ3i0 0 0 0 −0.1540 −93 −0.0105 12958 13050 ð0; −0.70; 0; 0.71Þ 1þ− 6 ¯ 6¯ 0 jðbcÞ1ðb c¯Þ1i1 1 6 6¯ 0 6 6¯ 0 0 1 2 3 2 3 pffiffi ¯ ¯ − ¯ ¯ 12923 0 −93 0 12852 0 80 0 0 60 0 2 jðbcÞ1ðb cÞ0i1 jðbcÞ0ðb cÞ1i1 ð . ; ; . ; Þ 3¯ ¯ 3 0 B 0 12946 0 −93 C 6 12864 7 6 ð0; 0.75; 0; 0.66Þ 7 jðbcÞ1ðb c¯Þ1i1 @ A 4 5 4 5 1 3¯ ¯ 3 0 3¯ ¯ 3 0 −93 0 12976 0 13047 ð−0.60; 0; 0.80; 0Þ pffiffi jðbcÞ ðb c¯Þ i − jðbcÞ ðb c¯Þ i 2 1 0 1 0 1 1 0 −93 0 12970 13052 ð0; −0.66; 0; 0.75Þ þþ 1 6 ¯ 6¯ 0 6 ¯ 6¯ 0 1 pffiffi jðbcÞ ðb c¯Þ i þjðbcÞ ðb c¯Þ i 2 1 0 1 0 1 1 12953 −93 12870 ð0.74; 0.67Þ 1 3¯ 3 0 3¯ 3 0 pffiffi ¯ ¯ ¯ ¯ 2 jðbcÞ1ðb cÞ0i1 þjðbcÞ0ðb cÞ1i1 −93 12973 13056 ð−0.67; 0.74Þ 2þþ 6 ¯ ¯ 6¯ 0 jðbcÞ1ðb cÞ1i2 12962 −92 12884 ð0.76; 0.65Þ 3¯ ¯ ¯ 3 0 jðbcÞ1ðb cÞ1i2 −92 12992 13070 ð−0.65; 0.76Þ Fig. 1(e). Instead of repeating the features seen in the bcc¯c¯ IV. SUMMARY system, we only note the mass splittings among the multip- In this work, we study the mass spectra of the all-heavy lets with the same quantum numbers are expected to be ccc¯c¯ bbb¯b¯ bbc¯c=cc¯ b¯b¯ bcc¯c=cc¯ b¯c¯ bcb¯b=bb¯ b¯c¯ similar to the bcc¯c¯ system. For instance, the mass splitting , , , , , and bcb¯c¯ systems in the potential quark model with the linear among the JP ¼ 1þ bcb¯b¯ states is also about 10 MeV. As shown in Fig. 1(e), our results show that the states of confining potential, Coulomb potential, and spin-spin interactions included. We find that the linear confining the bcb¯b¯ system are about 370–390 MeV above the mass potential contributes large positive energies to the eigen- threshold of B ϒ. Thus, these states with different quantum c values of the ground states of these tetraquark systems. numbers can also easily decay into the B η , B ϒ,orB ϒ c b c c This is different from some existing calculations in the final states via the quark rearrangement. Narrow states literature in which the neglect of the confining potential made of the bcb¯b¯ are not favored in our model. contributions leads to relatively low masses for the all- heavy systems and some of those can be lower than the D. The bcb¯c¯ system two-body decay thresholds. In our case, all these states are The bcb¯c¯ system has no constraints from the Pauli found to have masses above the corresponding two meson principle, and there are 12 different configurations allowed decay thresholds via the quark rearrangement. This implies by this system, namely, four JPC ¼ 0þþ states, four JPC ¼ that narrow all-heavy tetraquark states may not exist in 1þ− states, two JPC ¼ 1þþ states, and two JPC ¼ 2þþ reality. Nevertheless, our explicit calculations suggest states. The predicted mass spectrum is listed in Table XI that the confining potential still plays an important and shown in Fig. 1(f). role in the heavy flavor multiquark system, and it is crucial For the bcb¯c¯ system, there are strong configuration to include it in dynamical calculations in order to gain a mixings between the 6 ⊗ 6¯ and 3 ⊗ 3¯ configurations with better understanding of the multiquark dynamics. More the same JPC numbers. For example, the highest mass JPC ¼ experimental information from the Belle-II and LHCb 0þþ state is a mixed state containing comparable components analyses would be able to clarify these issues in the near 6 ¯ 6¯ 0 3¯ ¯ 3 0 future. from two configurations jðbcÞ0ðbc¯Þ0i0 and jðbcÞ0ðbc¯Þ0i0. As a consequence, the predicted masses for these tetraquark ACKNOWLEDGMENTS states are in the range of 12950 120 MeV. We note that our predicted masses are about 600 MeV systematically smaller This work is supported by the National Natural Science than those predicted in Ref. [29] and about 300–600 MeV Foundation of China (Grants No. 11775078, No. U1832173, systematically larger than those predicted with diquark No. 11705056, No. 11425525, and No. 11521505). Q. Z. is picture in Ref. [24]. Also, these states of bcb¯c¯ are about also supported in part, by the DFG and NSFC funds to the 200–400 MeVabove the mass threshold of BcBc. It suggests Sino-German CRC 110 “Symmetries and the Emergence of that these tetraquark states may easily decay into the BcBc, Structure in QCD” (NSFC Grant No. 11261130311), BcBc, ηbJ=ψ,orϒJ=ψ channels via quark rearrangements. National Key Basic Research Program of China under Thus, they are expected to be broad in width. Contract No. 2015CB856700. 016006-10 ALL-HEAVY TETRAQUARKS PHYS. REV. D 100, 016006 (2019) [1] M. Gell-Mann, A schematic model of baryons and mesons, [24] A. 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