Fully-heavy tetraquarks: $bb\bar{c}\bar{c}$ and $bc\bar{b}\bar{c}$

In the framework of a nonrelativistic chiral quark model, we continue to study the mass spectra of the fully-heavy $bb\bar{c}\bar{c}$ and $bc\bar{b}\bar{c}$ tetraquarks. In the present calculations, two structures, meson-meson [$\bar{Q}Q$][$\bar{Q}Q$] and diquark-antidiquark [$QQ$][$\bar{Q}\bar{Q}$] ($Q$ = $c$ or $b$), and their mixing, along with all possible color, spin configurations are considered. The calculations suggest that no bound state can be formed for $bb\bar{c}\bar{c}$ and $bc\bar{b}\bar{c}$ systems. However, resonances are possible because of the color structure. Several resonances are predicted and their stabilities are checked using the real scaling method.


I. INTRODUCTION
In the past years, experimental searches for the exotic states beyond the conventional quark model have made great progress.The observation of XY Z states, such as X(3872) [1], Y (4260) [2,3], Z c (3900) [4][5][6][7], Z b (10610) [8], P + c (4380), P + c (4450) [9] and so on, provided us a good opportunity to extend our knowledge of the heavy flavor spectroscopy.Especially those charged quarkonium-like states with heavy flavor mesons as the decay products, make them the best candidates for the exotic hadrons.
Recently, the tetraquarks composed of four heavy quarks QQ Q Q (Q = c or b), have received great attention.Experimentally, the heavy-flavor states provide some advantages, because they can be explored with the help of the efficient triggers such as J/ψ.LHCb collaboration is hunting for the bb bb tetraquark state and the existence information needs the further confirmation [10].In the theoretical aspect, there are also many studies on the fully-heavy tetraquarks.For example, in some work [11][12][13][14][15][16][17][18], it is suggested that there exist stable bound bb bb and cccc states with relatively smaller masses below the thresholds of the corresponding meson pairs.But some other work argues to the contrary that there should no bound bb bb or cccc tetraquark states because of the lager masses than the thresholds to decay [19][20][21][22][23][24].Refs.[25,26] also studied the weak decay properties about tetraquarks bbcc and bcq q.Although some of the opinions are quite different from each other, the researches on the exotic states are quite important for our understanding the underlying dynamics of the exotic states and the nature of strong interactions of QCD.
If the bb bb and cccc tetraquark states do exist in nature, we have strong reason to believe that there exist more other heavy-flavor tetraquark states.In our previous work [24], we focused on the full-bottom tetraquarks bb bb in the framework of the chiral quark model.In present work, we would like to extend the study to the tetraquarks bbcc and bc bc.Although they are still missing in experiment, the study of the mass spectra of these two systems will offer information for the further experimental explorations.Theoretically, in Ref. [27], Liu et * xychen@jit.edu.cnal. studied the mass spectra of the fully-heavy tetraquark systems including bbcc and bc bc within a potential model and no bound states with masses below the corresponding thresholds were found.Recent studies by Wu et al.
showed that bc bc bound state was found to be possible, but bbcc state was not a bound state [22].In Ref. [28], Richard et al. also observed that bound bc bc state might be more favorable than bb bb and cccc.Ref. [15] showed that tensor tetraquark bc bc (2 ++ ) can be observed in both B c B c and J/ψΥ(1S) modes.It is well-known that the color magnetic interaction (CMI) of the one-gluonexchange plays an important role in the hadron spectrum and hadron-hadron interactions.Compared with bb bb tetraquark state, CMI is beneficial to form compact tetraquarks for bc bc.Considering the higher thresholds of bc + bc, bbcc may also a possible tetraquark state.Our purpose is firstly to check whether there are stable bound states in the bbcc and bc bc systems, if not, secondly, we aim to look for the possible resonances.
In this work, we calculated the mass spectra of the bbcc and bc bc systems in a nonrelativistic chiral quark model systematically.For bbcc state, the possible quantum numbers are I(J P ) = 0(0 + ), 0(1 + ) and 0(2 + ).
For bc bc state, it should have definite Cparity, and the allowed quantum numbers are I(J P C ) = 0(0 ++ ), 0(1 +− ), 0(1 ++ ) and 0(2 ++ ).For the interaction between the heavy quarks, the short-distance one-gluonexchange effects play an important role now.In the calculations, the meson-meson [ QQ][ QQ] and diquarkantidiquark [QQ][ Q Q] structures, and the mixing of them are considered, respectively, along with all possible color, spin configurations.To distinguish genuine resonances, we employ the Gaussian expansion method [29] supplemented by the real scaling method (stabilization) [30,31].The real scaling method was often used for analyzing electron-atom and electron-molecule scattering [32] and was applied in the quark model calculation recently [24,30,31].
The paper is organized as follows.In Sec.II, the chiral quark model and the wave functions of the four-body system will be introduced briefly.In Sec.III, the numerical results and discussion are presented.A short summary is given in Sec.IV.

II. QUARK MODEL AND WAVE FUNCTIONS
The chiral quark model has been successful both in describing the hadron spectra and hadron-hadron interactions.The details of the model can be found in Ref. [33,34].For bbcc and bc bc full-heavy system, the Hamiltonian of the chiral quark model consists of three parts: quark rest mass, kinetic energy, and potential energy: The potential energy consists of pieces describing quark confinement (C); one-gluon-exchange (G).The detailed forms of potentials are shown below (only central parts are presented) [33]: m i is the constituent mass of quark/antiquark, and µ ij is the reduced mass of two interacting quarks and p ij = (p i − p j )/2, p 1234 = (p 12 − p 34 )/2; r 0 (µ ij ) = s 0 /µ ij ; σ are the SU (2) Pauli matrices; λ, λ c are SU (3) flavor, color Gell-Mann matrices, respectively; and α s is an effective scale-dependent running coupling [34], All the parameters are determined by fitting the meson spectrum, from light to heavy; and the resulting values are listed in Table I.Table II gives the masses of some heavy mesons in the chiral quark model.The wave functions of four-quark states for the two structures, diquark-antidiquark and meson-meson, can be constructed in two steps.For each degree of freedom, first we construct the wave functions for two-body sub-clusters, then couple the wave functions of two subclusters to obtain the wave functions of four-quark states.
For the spin part, the wave functions for two-body subclusters are,  then the wave functions for four-quark states are obtained, where the superscript σi(i = 1 ∼ 6) of χ represents the index of the spin wave functions of four-quark states.The subscripts of χ are SM S , the total spin and the third projection of total spin of the system.S = 0, 1, 2, and only one component (M S = S) is shown for a given total spin S.
For the flavor part, the configurations of bbcc and bc bc states are demonstrated in Fig. 1 in diquark-antidiquark structure, and the wave functions for bbcc and bc bc systems take, respectively.The subscript d0 of χ represents the diquark-antidiquark structure and isospin (I = 0).For the color part, the wave functions of four-quark states must be color singlet [222] and it is obtained as Where, χ c1 d and χ c2 d represents the color antitriplet-triplet ( 3×3) and sextet-antisextet (6× 6) coupling, respectively.The detailed coupling process for the color wave functions can refer to our previous work [35].
For the spin part, the wave functions are the same as those of the diquark-antidiquark structure, Eq. (6).
For the flavor part, there are three wave functions, one function for bbcc system, and two functions for bc bc system, The subscript m0 of χ represents the meson-meson structure and isospin equals zero.Fig. 2 shows the mesonmeson structure of bbcc and bc bc systems.
For the color part, the wave functions of four-quark states in the meson-meson structure are, Where, χ c1 m and χ c2 m represents the color singlet-singlet (1×1) and color octet-octet (8×8) coupling, respectively.The details refer to our previous work [35].
As for the orbital wave functions, they can be constructed by coupling the orbital wave function for each relative motion of the system, where l 1 and l 2 is the angular momentum of two subclusters, respectively.Ψ Lr (r 1234 ) is the wave function of the relative motion between two sub-clusters with orbital angular momentum L r .L is the total orbital angular momentum of four-quark states.Here for the low-lying bbcc and bc bc state, all angular momentum (l 1 , l 2 , L r , L) are taken as zero.The used Jacobi coordinates are defined as, For diquark-antidiquark structure, the quarks are numbered as 1, 2, and the antiquarks are numbered as 3, 4; for meson-meson structure, the antiquark and quark in one cluster are marked as 1, 2, the other antiquark and quark are marked as 3, 4. In the two structure coupling calculation, the indices of quarks, antiquarks in diquarkantidiquark structure will be changed to be consistent with the numbering scheme in meson-meson structure.
In GEM, the spatial wave function is expanded by Gaussians [29]: where N nl are normalization constants, c n are the variational parameters, which are determined dynamically.The Gaussian size parameters are chosen according to the following geometric progression This procedure enables optimization of the expansion using just a small numbers of Gaussians.Finally, the complete channel wave function for the four-quark system for diquark-antidiquark structure is written as where A 1 is the antisymmetrization operator, for bbcc system, For meson-meson structure, the complete wave function is written as where A 2 is the antisymmetrization operator, for bbcc system, (1 − P 13 − P 24 + P 13 P 24 ).
Lastly, the eigenenergies of the four-quark system are obtained by solving a Schrödinger equation: where Ψ MI MJ IJ is the wave function of the four-quark states, which is the linear combinations of the above channel wave functions, Eq. ( 19) in the diquark-antidiquark structure or Eq. ( 21) in the meson-meson structure, or both wave functions of Eq. ( 19) and ( 21), respectively.

III. RESULTS AND DISCUSSIONS
In the present work, we calculated the mass spectra of the bbcc and bc bc systems with allowed quantum numbers in the nonrelativistic quark model.Two structures of four-quark states, meson-meson and diquarkantidiquark, and the mixing of them are investigated, respectively.All possible color, and spin configurations are also considered.For example, for meson-meson structure, two color configurations, color singlet-singlet (1 × 1) and octet-octet (8 × 8) are employed; for diquark-antidiquark structure, color antitriplet-triplet ( 3 × 3) and sextetantisextet (6 × 6) are taken into account.For bbcc state, the wave functions need to be antisymmetrized.All the allowed channels are demonstrated in Table III.For bc bc, TABLE III.The allowed channels of bbcc and bc bc systems in meson-meson (M-M) and diquark-antidiquark (D-A) struc- ).The wave functions with superscript "C", for example χ f 3,C m0 , are the charge conjugate of the corresponding wave function without superscript "C".bbcc I(J P ) 0(0 The matrix elements of color and spin operators.Oij = λi • λj for color, and Oij = σi • σj for spin.
there is no need to consider the antisymmetrization because of no identical quarks.Because the hamiltonian of the system is invariant under the charge conjugate, the C-parity is a good quantum number and is shown in the table for bc bc system.All possible channels are also showed in Table III.
For full-heavy flavor system, the large masses of b and c-quark prevent the appearance of the Goldstone boson exchanges, only gluon exchanges are included.It is helpful to understand the numerical results by analyzing qualitatively the properties of the interactions between quarks in the system.Table IV gives the matrix elements of color operators and spin operators (only the results for total spin S=0 are given here).With the help of these matrix elements, we can estimate roughly the binding energy of the system.For single meson, the matrix element of color operator is For tetraquark states, the matrix element of color operator is So the color matrix elements are exactly same for the tetraquark system and two-meson pairs and the pure color interaction cannot contribute the binding energy of the tetraquark system.For CMI (color magnetic interaction), all the matrix elements are given in the Table V.From the table, we can see that the difference of the CMI matrix elements between tetraquark system and two-meson pairs are not smaller than 0, so CMI cannot lead to deep bound state.It is worth to note that ∆ CMI for color-spin configuration 3 ⊗ 3, 0 ⊗ 0 and 8 ⊗ 8, 1 ⊗ 1 may be negative if x is large enough, which means that the bound states are more possible in QQqq systems.Some previous work, for example Ref. [36], obtained several bound states in these systems.
The numerical results of bbcc and bc bc systems are shown in Tables VI and VII, respectively.E cc represents the ground state energy for each state after considering the all possible color and spin channels (refer to Table III).For bc bc system, the states with different C-parity are separated.
From the Table VI, we found that the lowest energies of 0 + , 1 + and 2 + in the meson-meson structure are a little higher than the relevant thresholds.In the diquarkantidiquark structure, the energies are all much lager than those in the meson-meson structure.The effects of the two-structure mixing seem to be tiny.So we cannot find the bound states of bbcc tetraquark in the present calculation.For bc bc system, with the lower threshold b b + cc compared with bc + bc, it may be much harder to form a bound state.In Table VII, the lowest energies of the three structures of bc bc system are all larger than the Because the colorful clusters cannot fall apart, there may be a resonance even with the higher eigenenergy.To find the genuine resonances, the dedicated real scaling (stabilization) method is employed.To realize the real scaling method in our calculation, the Gaussian size parameters r n in Eq. ( 18) are multiplied by a factor α, αr n just for the meson-meson structure with color singletsinglet configuration.α takes the values between 0.9 and 1.6.With the increasing of α, all states will fall off towards its thresholds, but a compact resonance should be stable because it will not be affected by the boundary at a large distance.We illustrate the results for bbcc and bc bc states with all possible quantum numbers Figs.3-6.
In   From Fig. 4, we can see that the energy of the lowest resonance is about 13230 MeV and 13020 MeV for bbcc and bc bc state, respectively.
For bc bc states with 0(1 ++ ) and 0(1 +− ) in Fig. 5, the lowest possible resonance is at 12910 MeV for C-parity negative and 13020 MeV for C-parity positive.For bbcc states with I(J P ) = 0(1 + ) in Fig. 6, two possible resonances stay very close to each other, with the stable energies 13180 MeV and 13200 MeV, respectively.
From our calculation, we can see that there may be more resonance states with the higher energies, and these states may be too wide to be observed or too hard to be produced.We are interested in the genuine resonance state with as low as possible energy, so we only give the lowest resonances for each quantum number set.We hope these information will be helpful for the searching for the bbcc and bc bc states in experiment.

IV. SUMMARY
In the framework of the chiral quark model, we do a systematical calculation for the mass spectra of bbcc and bc bc systems with allowed quantum numbers using the Gaussian expansion method.The meson-meson structure, the diquark-antidiquark structure and the mixing of them are investigated severally.In our calculation all these states are found to have masses above the corresponding two meson decay thresholds, leaving no space for a bound state.These results are consistent with our qualitative analysis of properties of the interactions between quarks.With the help of the real scaling method, we try to look for the possible resonances in bbcc and bc bc systems.For bbcc system, the energies of the possible resonances are 13140 MeV, 13180 MeV and 13230 MeV for 0(0 + ), 0(1 + ) and 0(2 + ) state, respectively.For bc bc system, the resonance energies are little lower than bbcc state, which takes 12860 MeV, 13020 MeV, and 13020 MeV for 0(0 ++ ), 0(1 ++ ) and 0(2 ++ ) states, and 12910 MeV for 0(1 +− ), respectively.Hopefully, these information about the exotic tetraquark states composed of four heavy quarks may be useful for the search in experiments in the future.

FIG. 6 .
FIG.6.The stabilization plots of the energies of bbcc state for I(J P ) = 0(1 + ) with the respect to the scaling factor α.

TABLE I .
Model parameters, determined by fitting the meson spectrum from light to heavy.

TABLE II .
The masses of some heavy mesons (in units of MeV).M cal and Mexp represents the theoretical and the experimental masses, respectively.

TABLE VI .
The results of bbcc state in pure meson-meson structure, diquark-antidiquark structure, and in considering the mixing of two structures, respectively."E theo th " represents the theoretical thresholds.The masses are all in units of MeV.

TABLE VII .
The results of bc bc state in pure meson-meson structure, diquark-antidiquark structure, and in considering the mixing of two structures, respectively."E theo th " represents the theoretical thresholds.The masses are all in units of MeV.