Ju n 20 19 Fully-heavy tetraquarks

Ming-Sheng Liu ∗, Qi-Fang Lü †, Xian-Hui Zhong ‡, Qiang Zhao § 1) Department of Physics, Hunan Normal University, and Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Changsha 410081, China 2) Institute of High Energy Physics and Theoretical Physics Center for Science Facilities, Chinese Academy of Sciences, Beijing 100049, China 3) School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China and 4) Synergetic Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China


I. INTRODUCTION
Experimental searches for and theoretical studies of exotic hadrons beyond the conventional quark model are an important test of non-perturbative properties of the strong interaction theory QCD. Since the discovery of quark model [1] and QCD, the progresses on the experimental tools have brought to us a lot of novel phenomena in hadron physics. In particular, during the past 15 years there have been a sizeable number of candidates for QCD exotics [2][3][4][5][6][7][8]. Interestingly, but also puzzlingly, it shows that the number of exotic candidates is far less than what we have expected for the hadron spectroscopy where the internal effective degrees of freedom of a hadron may contain quarks and gluons beyond the conventional quark model prescription. Strong evidences for such exotic hadrons include some of those recently observed XYZ states, e.g. X(3872), Z c (3900), Z c (4020), Z b (10610), and Z b (10650) [2]. In particular, these charged quarkonium-like states, Z c and Z b , contain not only the hidden heavy flavor cc or bb, but also charged light flavors of ud or dū. Since at least four constituent quarks are confined inside these Z c or Z b states, it makes them the best candidates for QCD exotic hadrons.
Recently, the tetraquarks of fully-heavy systems, such as cccc and bbbb, have received considerable attention with the development of experiments. If there are stable tetraquark cccc and/or bbbb states, they are most likely to be observed at LHC [9]. In fact, a search for the tetraquark bbbb states is being carried out by the LHCb collaboration although no confirmed information has been observed [10]. Other study interests for physicists arise from the special aspects of the fully-heavy tetraquark systems [11]. They may favor to form genuine tetraquark configurations rather than loosely bound hadronic molecules, since the light mesons cannot be exchanged between two heavy mesons. Furthermore, it will be very easy to distinguish the fully-heavy tetraquark states from the states which have been observed because their masses should be far away from the mass regions of the observed states. Thus, besides some previous works on the fully-heavy tetraquark states [12][13][14][15][16][17], many new studies have been carried out in recent years [11,[18][19][20][21][22][23][24][25][26][27][28][29][30], although some of the conclusions are quite different from each other. In some works it is predicted that there exist stable bound tetraquark cccc states and/or bound tetraquark bbbb states with relatively smaller masses below the thresholds of heavy charmonium pairs [11,[21][22][23][24][25][26][27][28]. Thus, their decays into heavy quarkonium pairs through quark rearrangements will be hindered. In contrast, in some other works it is predicted that there should be no stable bound tetraquark cccc and bbbb states [12,16,18,29,30] because the predicted masses are large enough for them to decay into heavy quarkonium pairs. To some extent, a better understanding of the possible mass locations is not only crucial for understanding their underlying dynamics, but also useful for experimental searches for their existence.
In this work, we systemically study the mass spectra of the fully-heavy tetraquark Q 1 Q 2Q3Q4 systems with a potential model widely used in the literature [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49]. Our purpose is to understand two key issues based on the knowledge collected in the study of heavy quarkonium spectrum. The first one is what a quark potential model can tell about the fully-heavy tetraquark system. The second one is what the masses of the ground states could be located if the fully-heavy tetraquark states do exist.
At this moment, we do not consider any orbital or radial excitations of the fully-heavy tetraquarks. Instead, we would like to address where and how the fully-heavy tetraquarks would manifest themselves in their lowest states? For a spectrum of multiquark states, a correct identification of the ground state should be the first step towards a better understanding of the multiquark dynamics in the non-perturbative regime.
The potentials between the quarks, such as the linear con-fining potential, color Coulomb potential and spin-spin interactions, are adopted the standard forms of the potential models. The model parameters are determined by fitting the mass spectra of charmonium, bottomonium and B c meson. In our calculations, we find both the confining potential and color Coulomb potential are very crucial for understanding the masses of the fully-heavy tetraquarks. The linear confining potential as well as the kinetic energy contributes a quite large positive mass term to the fully-heavy tetraquarks Q 1 Q 2Q3Q4 , which leads to a large mass far above the threshold of the meson pair Q 1Q3 -Q 2Q4 or Q 1Q4 -Q 2Q3 , although color Coulomb potential contributes a very large negative mass term. As a consequence, we find no bound fully-heavy tetraquarks Q 1 Q 2Q3Q4 below the threshold of any meson pairs Q 1Q3 - The paper is organized as follows: a brief introduction to the framework is given in Sec. II. In Sec. III, the numerical results and discussions are presented. A short summary is given in Sec. IV.
To calculate the spectroscopy of a Q 1 Q 2Q3Q4 system, first we construct the configurations in the space of Flavor ⊗Color⊗Spin. Considering the Pauli principle and color confinement for the four-quark system Q 1 Q 2Q3Q4 , we have 12 configurations as follows: where { } and [ ] denote the symmetric and antisymmetric flavor wave functions of the two quarks (antiquarks) subsystems, respectively. The subscripts and superscripts are the spin quantum numbers and representations of the color SU(3) group, respectively. A symmetric spatial wavefunction is implied for the ground states under investigation. In Table I all possible configurations and corresponding quantum numbers for the cccc and bbbb systems are listed. For the bcbc systems the J = 1 states can have both C = ±, which can be constructed by the linear combinations of |6 , |7 , |9 and |10 : where configurations |6 ′ and |9 ′ have C = −1, and |7 ′ and |10 ′ have C = +1.

B. Hamiltonian for the multiquark system
The following nonrelativistic Hamiltonian is adopted for the calculation of the masses of the fully-heavy Q 1 Q 2Q3Q4 system: where m i and T i stand for the constituent quark mass and kinetic energy of the i-th quark, respectively; T G stands for the center-of-mass (c.m.) kinetic energy of the Q 1 Q 2Q3Q4 system; r i j ≡ |r i − r j | is the distance between the i-th quark and j-th quark, and V i j (r i j ) stands for the effective potential between the i-th and j-th quark. In this work, we adopt a widely used potential form for V i j (r i j ) [31][32][33][34][35][43][44][45][46][47], i.e.
where V OGE i j stands for the one-gluon-exchange (OGE) potential which describes the short-range quark-quark interactions, while V Con f i j (r i j ) stands for the confinement potential which describes the long-range interaction behaviors. The form of V OGE i j is given by where σ i are the Pauli matrices, and α i j stands for the strong coupling strength between two quarks. If the interaction occurs between two quarks or antiquarks, the λ i · λ j operator appearing in Eq. (7) is defined as λ i · λ j ≡ 8 a=1 λ a i λ a j , while if the interaction occurs between a quark and antiquark, the where λ a * is the complex conjugate of the Gell-Mann matrix λ a . The OGE potential V OGE i j is composed of the Coulomb type potential V OGE coul ∝ 1/r i j which provides the short-range attraction, and the color-magnetic interaction V OGE CM ∝ σ i · σ j which provides mass splittings. The form of V Con f i j (r i j ) is given by System where the parameter b i j denotes the strength of the confinement potential.
There are eleven parameters m c , m b , α cc , α bb , α bc , σ cc , σ bb , σ bc , b cc , b bb , b bc to be determined in the calculations. In Ref. [46], the masses of cc spectrum are calculated by using the three-point difference central method [50] from the center (r = 0) towards outside (r → ∞) point by point. The parameters m c , α cc , σ cc , b cc have been determined. In this work, we use the same method to determine the parameters m b , α bb , σ bb , b bb by fitting the masses of bb spectrum, and determine the parameters α bc , σ bc , b bc by fitting the masses of B c , B * c and B c (2S ). The parameter set is listed in Table II. The corresponding theoretical results for the masses of heavy quarkonia are shown in Table III. The B * c has not been observed in experiment, and our theoretical result is very close to other theoretical predictions [32,34,42,51].
The matrix elements of λ i · λ j and σ i · σ j can be worked out in the color and spin spaces with explicit color and spin wavefunctions [52][53][54][55][56].

C. Matrix elements in coordinate space
In this work, the wave function of the ground state in the coordinate space is taken as the single Gaussian function, it reads where r i = |r i |, ω is the effective harmonic oscillator frequency parameter. To describe the relative motion between two quarks, we define the following Jacobi coordinates,  According to the Jacobi coordinates, the coordinate space wave function of Eq. (9) is rewritten as where On the other hand, in terms of the Jacobi coordinates the kinetic energy of the relative motion for the tetraquark system can be expressed as Then, the kinetic energy matrix element is worked out: (16) Furthermore, we should calculate the matrix elements of 1/r i j , e −σ 2 i j r 2 i j , r i j . Using Eq. (9), we can obtain: where m i j = m i m j m i +m j .

III. NUMERICAL RESULTS AND DISCUSSIONS
When all the matrix elements have been worked out, we can express the mass of the tetraquark configuration, m(ω) = H , as a function of the effective harmonic oscillator frequency parameter ω. The parameter ω is to be determined by using the variation method. The mass for a physical state should satisfy the relation, With this relation, the effective harmonic oscillator frequency parameter ω is determined. Then, the mass of the tetraquark configuration and its spacial wave function can be determined.
A. The cccc and bbbb systems The predicted mass spectrum for the cccc system has been given in Table IV and also shown in Fig. 1 a. From Table IV, it is found that the configurations mixing between MeV. The other two states J PC = 1 +− and J PC = 2 ++ are also located in a similar mass region, i.e. ∼ 6.5 GeV, and the mass splitting between them is about 20 MeV. As shown in Fig. 1 a, the two J PC = 0 ++ states are about 500 MeV and 300 MeV above the mass thresholds of η c η c and J/ψJ/ψ, respectively. It suggests that the J PC = 0 ++ states are unstable, and they can easily decay into the η c η c and J/ψJ/ψ final states through quark rearrangements. The J PC = 1 +− state lies about 430 MeV above the mass threshold of η c J/ψ, while J PC = 2 ++ is about 340 MeV above the mass threshold of J/ψJ/ψ, they can also easily decay into η c J/ψ and J/ψJ/ψ, respectively, through the quark rearrangements.
As a comparison, our predicted masses and some other typical results from other works are collected in Table V. It shows that our predicted masses for the cccc system are roughly compatible with the nonrelativistic quark model predictions of Refs. [12,16], where both confining and coulomb potentials are considered. It is also interesting to find that similar results are given by the QCD sum rules [11]. In contrast, the masses predicted by us are much larger than those predicted in Refs. [17,21,22,24,25,27]. These methods which obtained small masses have some common features: either no confining potentials were included [17,21,22,24] or a diquark picture was adopted in the calculations [25,27]. Recently, Wu et al. also obtained a large mass ∼ 6.8−7.0 GeV for the cccc system with heavier constituent c-quark mass 1.72 GeV adopted [29].
We further analyze the contributions from each part of the Hamiltonian for the cccc system. The results are listed in Table VII. It shows that the kinetic energy T , the confining  potential V Con f i j (r i j ) , and the coulomb potential V OGE coul have the same order of magnitude. In particular, the contributions from the confining potential are sizeable and apparently cannot be neglected. Note that the confining potential contributes a positive energy to the system. Thus, neglecting this contribution will lead to much lower masses for the fully-heavy system. In Refs. [17,21,22,24], the confining potential was neglected. Thus, a relatively small mass was obtained in these works.
In order to examine the role played by the confining potential in the spectrum of heavy quark system, we compare the contributions from the OGE and confining potential for the η c meson, i.e. V OGE coul ≃ −637 MeV and V Con f (r) ≃ 233 MeV, which are consistent with our previous study in Ref. [46]. The ratio between the confining potential V Con f and color Coulomb potential V OGE coul can reach up to This explicit result suggests that the neglect of confining potential cannot be justified for the cc system.
As a general conclusion, we find that the confining potential has significant contributions to masses of the cccc system, and are the same order of magnitude as the color Coulomb potential. This will enhance the masses of the cccc system and does not support the existence of a bound tetraquark of cccc with narrow widths.
The predicted mass spectrum for the bbbb system is very similar to that for the cccc one. The results are given in Table IV and shown in Fig. 1  configuration. Due to the heavier mass of the b quark, relatively smaller mass splittings among these states are found. The pattern is also similar to that of the cccc system. Note that the predicted masses are above the thresholds of the bottomonium pairs for about 350 ∼ 470 MeV. It suggests that bound states of the bbbb system with narrow widths are not favored.
In Table VI we compare our results with other model calculations. It shows that our predicted masses are higher than State Ours Ref. [29] Ref. [25,26] Ref. [22] Ref. [24] Ref. [21] Ref. [23] Ref. [11] Ref. [30] Ref. [   most of the other predictions which are calculated either without the confining potential considered [17,21,22,24] or based on the diquark picture [25]. Similarly, base on the diquark picture, the lightest mass of bbbb is estimated at 18.8GeV by Ref. [28]. In these calculations the tetraquark states of J PC = 0 ++ , 1 +− or 2 ++ are either below or slightly above the thresholds of η b η b , η b Υ(1S ) or Υ(1S )Υ(1S ), respectively. Thus, they can become stable with narrow decay widths. In contrast, our calculations with the inclusion of the confining potential result in higher masses for the bbbb system and do not favor the existence of such narrow tetraquark states. We note that a rather large mass ∼ 20.2 GeV for the bbbb system is estimated by Ref. [29] where a heavier constituent b-quark mass 5.05 GeV is adopted.
In Table VII the contributions from each part of the Hamiltonian for the bbbb system are listed. It shows that the kinetic energy T ≃ 700 MeV, the confining potential V Con f i j (r i j ) ≃ 500 MeV, and the coulomb potential V OGE coul ≃ −950 MeV, have the same order of magnitude. As shown in Fig. 1 b, the mass splittings among these J PC = 0 ++ , 1 +− and 2 ++ states follow a similar pattern as in the cccc system. Also similar to that for the cccc system, the neglect of the confining potential will lead to much lower masses for the bbbb system and this may explain the low masses obtained in Refs. [17,21,22,24]. Although it is often argued that the confining potential contributions are perturbative for the bottomonium system, explicit calculations seem not to support this phenomenon. In Ref. [46] we have studied the bb spectrum, and find that V Con f (r) ≃ 173 MeV and V OGE coul ≃ −767 MeV for the η b meson. The ratio between V Con f and V OGE coul can reach up to For the four heavy quark system of bbbb, the increase of the displacements between the two quarks (antiquarks) or quarkantiquark will experience larger confining forces. Thus, the confining potential contributions cannot be neglected in the calculations. As a consequence, our study does not support the existence of the tetraquark bbbb bound states with narrow widths. Finally, it should be mentioned that for a simplicity, in our calculation the variational wave functions of the coordinate space are only adopted an s-wave form. Thus, the color wave functions for the J PC = 1 +− and 2 ++ states is color33. However, the33 color wave functions for the J PC = 1 +− and 2 ++ states might slightly mix with the color 66 when one considers the orbital excitations in the coordinate space [18][19][20]. With a color mixing effect, the mass of the J PC = 1 +− and 2 ++ states might become slightly lower [18][19][20], which does not affect our conclusions.
B. The bbcc system The bbcc system is similar to the cccc and bbbb ones except that it does not have determined C parity and there is no contributions from the annihilation potential. The predicted mass spectrum for the bbcc system is also listed in Table IV and shown in Fig. 1 c. From Table IV configuration. The mass splitting between these two J P = 0 + states is about 88 MeV. The other two states with J P = 1 + and 2 + have a small mass splitting of about 10 MeV and are located around 12.95 GeV. The masses predicted by us are about 600 MeV systematically smaller than those predicted in the recent work [29], where relatively large constituent quark masses for the b quark 5.05 GeV and c quark 1.72 GeV are adopted.
As shown in Fig. 1 c, all these states are above their lowest open flavor decay channels for about 300 MeV. Therefore, they can decay into the B c B c , B * c B * c , or B c B * c final states via the quark rearrangement quite easily.

C. The bccc and bcbb systems
The states of both bccc and bcbb systems do not have determined C parity and they share some common features in terms of heavy quark symmetry. The predicted mass spectra for these two configurations are listed in Table VIII and shown in Fig. 1 d and Fig. 1 e. It shows that both bccc and bcbb systems have sizeable configuration mixings between the color 6 ⊗6 and 3 ⊗3 configurations. For the bccc system the mixing occurs between the |{bc} configuration, respectively. The configuration mixing effects among these three J P = 1 + states are also sizeable which are shown in Table VIII. The typical mass splitting is about 20 MeV and the predicted masses are about 500 MeV systematically smaller than those predicted in the recent work [29]. Again, we note that rather large constituent quark masses for the c and b quarks are adopted in Ref. [29].
As a consequence of the high masses predicted by our model, namely, the states of bccc system are about 280 − 350 MeV above the mass threshold of B * c J/ψ, we find that these states can easily decay into the B c η c , B c J/ψ or B * c J/ψ final states via the quark rearrangements. Thus, we do not expect narrow states of bccc to be observed in experiment. For the bcbb system its main properties is very similar to that of the bccc system as shown in Table VIII and Fig. 1 e. Instead of repeating the features seen in the bccc system, we only note the main features arising from the heavy constituent quark masses. Namely, the mass splittings among the multiplets with the same quantum numbers are expected to be smaller than that for the bccc system. For instance, the mass splitting among the J P = 1 + states is about 10 MeV.
As shown in Fig. 1 e, our results show that the states of the bcbb system are about 300 − 350 MeV above the mass threshold of B * c Υ. Thus, these states with different quantum numbers can also easily decay into the B c η b , B c Υ or B * c Υ final states via the quark rearrangement. Narrow states made of the bcbb are not favored in our model.

D. The bcbc system
The bcbc system has no constrains from the Pauli principle, and there are 12 different configurations allowed by this system, namely, four J PC = 0 ++ states, four J PC = 1 +− states, two J PC = 1 ++ states, and two J PC = 2 ++ states. The predicted mass spectrum is listed in Table IX and shown in Fig. 1 f.
A main feature of the bcbc system is that the configuration mixing appears to play an important role. For example, the highest mass J PC = 0 ++ state is a mixed state containing comparable components from three configurations |(bc) 6 0 (bc)¯6 0 0 0 , |(bc)¯3 1 (bc) 3 1 0 0 and |(bc)¯3 0 (bc) 3 0 0 0 . As a consequence, the predicted masses for these tetraquark states are in the range of 12930 ± 80 MeV. We note that our predicted masses are about 600 MeV systematically smaller than those predicted in Ref. [29] and about 400 − 500 MeV systematically larger than those predicted with diquark picture in Ref. [24]. Also, these states of bcbc are about 200 − 300 MeV above the mass threshold of B * c B * c . It suggests that these tetraquark states may easily decay into the B c B * c , B * c B * c , η b J/ψ or ΥJ/ψ channels via quark rearrangements. Thus, they are expected to be broad in width.

IV. SUMMARY
In this work we study the mass spectra of the fully-heavy cccc, bbbb, bbcc/ccbb, bccc/ccbc, bcbb/bbbc, and bcbc systems in the potential quark model with the linear confining potential, Coulomb potential, and spin-spin interactions included. We find that the linear confining potential contributes large positive energies to the eigenvalues of the ground states of these tetraquark systems. This is different from some existing calculations in the literature in which the neglect of the confining potential contributions leads to relatively low masses for the fully-heavy systems and some of those can be lower than the two-body decay thresholds. In our case all these states are found to have masses above the corresponding two meson decay thresholds via the quark rearrangement. This implies that narrow fully-heavy tetraquark states may not exist in reality. Nevertheless, our explicit calculations suggest that the confining potential still plays an important role in the heavy flavor multiquark system and it is crucial to include it in dynamical calculations in order to gain a better understanding of the multiquark dynamics. More experimental information from the Belle-II and LHCb analyses would be able to clarify these issues in the near future.