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. j1i¼j½Q1Q21½Q3 Q41i0; 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 i¼j½Q1Q20½Q3 Q40i0; – 6 ¯ ¯ 6¯ 0 6 ¯ ¯ 6¯ 0 tial model widely used in the literature [31 49].Our j5i¼j½Q1Q21½Q3 Q41i1; j6i¼j½Q1Q21fQ3 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 Q41i1; j i¼jfQ1Q2g1fQ3 Q4g1i1; spectrum. The first one is what a quark potential model can 3¯ ¯ ¯ 3 0 3¯ ¯ ¯ 3 0 j9i¼jfQ1Q2g1½Q3 Q40i1; j10i¼j½Q1Q20fQ3 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 j11i¼j½Q1Q21½Q3 Q41i2; 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 ¼1, 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.