Heavy-flavored tetraquark states with the $QQ\bar{Q}\bar{Q}$ configuration

In the framework of the color-magnetic interaction, we systematically investigate the mass spectrum of the tetraquark states composed of four heavy quarks with the $QQ\bar Q\bar Q$ configuration in this work. We also show their strong decay patterns. Stable or narrow states in the $bb\bar{b}\bar{c}$ and $bc\bar{b}\bar{c}$ systems are found to be possible. We hope the studies shall be helpful to the experimental search for heavy-full exotic tetraquark states.

Q 1 Q 2Q3Q4 tetraquark system, which may provide important information to further experimental exploration.
In Ref. [64], Iwasaki studied the hidden-charm tetraqark state composed of a pair of charmed quark and anti-quark as well as a pair of light quark and anti-quark, which has the ccqq configuration. The dimeson configuration (Q 2q2 ) is stable against strong decay into two mesons [65]. Chao performed a systematical investigation of the cccc tetraquark system in a quark-gluon model for the first time in Ref. [66]. In Ref. [67], the authors used the Born-Oppenheimer approximation for heavy quarks in the MIT bag and found that the heavy-quark system c 2c2 is stable against breakup into two cc pairs. But in a potential model calculation [68], the authors suggested that for identical quarks, there is no stable QQQQ state. Similar opinions were shared by the authors of Ref. [69]. However, in Ref. [70], Lloyd and Vary obtained bound tetraquark states by adopting a parameterized Hamiltonian to compute the spectrum of the cccc tetraquark states. The tetraquark spectrum was also studied with a generalization of the hyperspherical harmonic formalism in Ref. [71]. In addition, the calculation of the chromomagnetic interaction for the Q 2Q2 system was performed in Ref. [72]. In understanding the nature of the Y (4260) meson in Ref. [73], as a byproduct, a lattice study indicates that the J P C = 1 −− cccc state is possible. If the P-wave cccc state exists where the orbital angular momentum contributes some repulsion, the lower ground tetraquark states should also exist. Recently, there are further discussions about the properties of fully-heavy tetraquarks [74][75][76][77][78][79][80][81][82][83][84].
In this work, we calculate the mass splittings of the Q 1 Q 2Q3Q4 tetraquark system in a simple quark model systematically. A typical feature for such tetraquarks is that the isospin is always zero and the flavor wave function is always symmetric for identical quarks. One may only focus on the color-spin part when the Pauli principle is employed to exclude some configurations. For the interaction between the heavy quarks, the short-range gluon exchange force is a dominant source. In the one-gluon-exchange potential model, the color-spin or color-magnetic interaction part is responsible for the mass splitting of the ground hadrons with the same flavor content. In the present study, we will adopt the color-magnetic interaction (CMI) to perform the calculations, which violates the heavy quark symmetry.
We organize the paper as follows. After the introduction, we present the formalism of calculation in Sec. II. We give the numerical results and discuss the possible decay modes in Sec. III. We summarize our results in the final section.

II. FORMALISM
The color-magnetic interaction in the model reads where λ i 's are the Gell-Mann matrices and σ i 's are the Pauli matrices. The above Hamiltonian was deduced from the one-gluon-exchange interaction [85]. The effective coupling constants C ij incorporate effects from the spatial wave function and the quark masses, which depend on the system. We determine their values in the next section. This Hamiltonian leads to a mass formula for the studied system where m i is the effective mass of the i-th constituent quark, which includes the constituent quark mass and effects from other terms such as color-electric interaction and color confinement.
To calculate the matrix elements for the color-spin interaction, one may construct the color ⊗ spin wave functions explicitly and calculate them by definition. In studying multiquark systems, a simpler way was used in Refs. [86][87][88]. One just calculates the matrix elements in color space and in spin space separately with the Hamiltonaians Then H CM is obtained by combining the results of H C and H S with the common coefficient C ij .
In order to consider the constraint from the Pauli principle, we use a diquark-antidiquark picture to analyze the configurations. Here the notation diquark just means two quarks. In the spin space, the allowed wave functions read where the subscripts on the right hand side denote the spin of the Q 1 Q 2 , spin of theQ 3Q4 , and that of the system, respectively. According to the SU (3) group theory, the diquark in the color space belongs to the representation 6 c or 3 c , while the anti-diquark's representation is6 c or 3 c . Then one has two kinds of color-singlet state Combine the spin and color wave functions, we get twelve possible color ⊗ spin wave functions for the Q 1 Q 2Q3Q4 system where the used notation is |(Q 1 Q 2 ) color spin (Q 3Q4 ) color spin spin . We insert a symbol δ ij in the wave functions to reflect the constraint from the Pauli principle. When the i-th quark and the j-th quark are the same, δ ij = 0. Otherwise, δ ij = 1.
Replacing Q i with c or b quark, one gets nine possibilities for the flavor content, six of which need to be studied: bbbb, cccc, bbcc, bbbc, cccb, and bcbc. The other three cases ccbb, cbbb, and bccc correspond to the antiparticles of bbcc, bbbc, and cccb, respectively. So their formulas are not independent. Here, considering the Pauli principle, one may categorize the six systems into three sets: (1) bbbb, cccc, and bbcc, where δ 12 = δ 34 = 0; (2) bbbc and ccbc, where δ 12 = 0, δ 34 = 1; and (3) bcbc where δ 12 = δ 34 = 1. The number of independent wave functions for them is 4, 6, and 12, respectively. We present the resulting CMI matrix elements system by system. By the way, the usually mentioned "good" diquark (spin=0, color=3 c ) exists only in the systems containing the (bc) substructure since the most attractive (bb)3 c spin-0 and (cc)3 c spin-0 objects are forbidden by the Pauli principle.
A. The bbbb and cccc systems The color-spin structures of these two systems are the same. The only difference lies in the quark mass. So we put them together for discussions.
The bbbb system is a neutral state. Its possible quantum numbers are I G (J P C ) = 0 + (2 ++ ), 0 − (1 +− ), or 0 + (0 ++ ). The number of states is constrained by the Pauli principle. For the case J = 2, the wave function is φ 2 χ 1 and the obtained H CM is given by 16 3 (C bb + C bb ). The color-magnetic interaction is certainly repulsive. For the case J = 1, the wave function is φ 2 χ 2 and H CM = 16 3 (C bb − C bb ). Since the quark-quark interaction and the quark-antiquark interaction is related with the G-parity transformation, the CMI for this J = 1 state is expected to be weak. For the case J = 0, there are two possible wave functions φ 2 χ 3 and φ 1 χ 6 . Although the color-spin structures are different, their mixing is allowed by the color-magnetic interaction. The obtained symmetric CMI matrix in the (φ 2 χ 3 , φ 1 χ 6 ) T base is The cccc system has similar expressions. For the case J = 2, H CM = 16 3 (C cc + C cc ) with the color-spin wave function φ 2 χ 1 . For the case J = 1, H CM = 16 3 (C cc − C cc ) with the wave function φ 2 χ 2 . For J = 0, the CMI matrix is The above compact tetraquark system has the same quark content as the molecular states composed of two bottomonium (charmonium) mesons. However, the interaction between heavy quarks is dominantly through the short-range gluon exchange. Once the interaction is strong enough to bind the two mesons, the resulting object very probably tends to form a compact structure instead of a loosely-bound molecule.
In the extreme case that the molecules exist, the configuration mixing is possible. For the S-wave ΥΥ (ψψ) state, the allowed quantum numbers are I G (J P C ) = 0 + (2 ++ ) or 0 + (0 ++ ) since the wave function in spin space should be symmetric for identical mesons. The quantum numbers for the S-wave state composed of two η b (η c ) mesons are only 0 + (0 ++ ). For the state composed of one η b (η c ) and one Υ (ψ), the quantum numbers are just I G (J P C ) = 0 − (1 +− ). Therefore, the allowed J P C appear both in the molecule and diquark-antidiquark pictures. The number of states is also equivalent. Considering that the interaction between heavy quarks is through the gluon exchange force, one does not expect large mass difference between these two configurations. The compact structure may contribute significantly to the properties of the molecules with the same quantum numbers.
If the tetraquark states have larger masses, the quark rearrangements into the meson-meson channels with the same quantum numbers may happen. We will discuss such possible decay patterns after their masses are estimated. The two systems are related through the C-parity transformation and they have the same color-spin matrix elements. The possible quantum numbers of them are I(J P ) = 0(2 + ), 0(1 + ), or 0(0 + ). Again the Pauli principle results in four states for the bbcc (or ccbb) system. The color-spin wave functions are φ 2 χ 1 and φ 2 χ 2 for the case of J = 2 and J = 1, respectively. The corresponding H CM 's are 8 3 (C bb + C cc + 2C bc ) and 8 3 (C bb + C cc − 2C bc ). For the case J = 0, the allowed wave functions are the same as those of the previous systems, φ 2 χ 3 and φ 1 χ 6 . Consider their mixing, one gets These systems have the same quark content as the B

C. The bbbc and cbbb systems
The two systems have the same matrix elements. Their quantum numbers are also I(J P ) = 0(2 + ), 0(1 + ), or 0(0 + ). We here focus on the bbbc system. Now the number of the vector states is 3 and that of the scalar states is 2. For the J = 2 case, the color-spin wave function is again φ 2 χ 1 . The resulting color-magnetic matrix element is H CM = 8 3 (C bb + C bc + C bb + C bc ). For the case J = 1, three possible wave functions φ 2 χ 2 , φ 2 χ 4 , and φ 1 χ 5 are allowed. The last one has different color structure from the other two. Although their mixing occurs, from the obtained matrix [base: one observes that the mixing strength for φ 2 χ 2 and φ 2 χ 4 and that for φ 2 χ 2 and φ 1 χ 5 may be both small. The remaining case is for J = 0, where possible wave functions are φ 2 χ 3 and φ 1 χ 6 . Now, one has .

D. The cccb and bccc systems
The situation is similar to the bbbc and cbbb systems. By exchanging b and c there, one easily gets relevant matrix elements. For comparison, we focus on the cccb system. For the case J = 2, one has H CM = 8 3 (C cc +C bc +C bc +C cc ). For the case J = 1, the matrix for the color-spin interaction reads For the case J = 0, the matrix is .
The signs for the non-diagonal matrix elements seem to be inconsistent with the previous systems after the replacements b → c and c → b. Actually they do not affect the final results. For detailed argument, one may consult Eq. (2) of Ref. [89] and relevant explanations there.
The meson-meson states that these tetraquarks can rearrange into are ψB c , ψB * c , η c B c , and η c B * c .

E. The bcbc system
The Pauli principle does not give any constraints on the wave functions now. The two wave functions for J = 2 and the four wave functions for J = 0 will mix, respectively. However, one should be careful in discussing the mixing with the six wave functions for J = 1 because the system is neutral and may have C-parity.
If J = 2, both the diquark and the antidiquark have angular momentum 1. The state should have definite C-parity + and the quantum numbers are I G (J P C ) = 0 + (2 ++ ). With the base (φ 1 χ 1 , φ 2 χ 1 ) T , one may get the CMI matrix If J = 0, both the diquark and the antidiquark have the same angular momentum. The quantum numbers for the system are I G (J P C ) = 0 + (0 ++ ). The obtained CMI matrix is where the base is ( If J = 1, the states φ 1 χ 2 and φ 2 χ 2 have negative C-parities. All the other four wave functions φ 1 χ 4 , φ 2 χ 4 , φ 1 χ 5 , and φ 2 χ 5 are not invariant under C-parity transformation. But we may construct four states which are invariant under C-parity transformations. The basic procedure is similar to that given in Ref. [90]. Explicitly, the two C = + states are and the two C = − states are Only states with the same quantum numbers may mix. So we have two color-spin matrices. For the states with with the base ([φχ] 66 where the base is ( There are two kinds of molecule configurations for the bcbc system. In the bottomonium+charmonium case, the allowed quantum numbers are I G (J P C ) = 0 + (0 ++ ) for the η b η c system, 0 − (1 +− ) for the η b ψ system, 0 − (1 +− ) for the Υη c , and 0 + ([2, 1, 0] ++ ) for the Υψ system. In the meson-antimeson case, those for

A. Parameters
We need to determine six coefficients C bb , C cc , C bc , C bb , C cc , and C bc in discussing the mass splittings for various Q 1 Q 2Q3Q4 systems. Their masses may be further estimated with the Hamiltonian in Eq. (2) once the effective quark masses m c and m b are used.
From the mass splitting between Υ and , one extracts C bb = 2.9 MeV. Similarly, the value of C cc = 5.3 MeV is obtained from the mass splitting m J/ψ − m ηc = 114 MeV. Since the excited B * c meson has not been observed yet, we just estimate the value of C bc to be 3.3 MeV from m B * c − m Bc = 70 MeV calculated with a quark model [92]. Although the Ξ ++ cc baryon was confirmed recently by the LHCb Collaboration [93] after the first observation at SELEX [94,95], the available heavy baryon masses are still not enough for us to extract the value of C cc . Here we perform our calculation with the approximation C QQ = C QQ (Q = c, b). Since there is no dynamics in the present model, the choice of the approximation to determine C QQ is not unique. However, the results induced by the change of C QQ should not be large [96,97]. For comparison, we also adopt the approximation Cnn ≈ 2 3 and check the extreme case C QQ = 0, where C nn = 18.4 MeV is extracted from the light baryon masses [98]. By using the mass difference between ρ and π, one gets C nn = 29.8 MeV. The latter approximation certainly gives a smaller C QQ . If the interactions within the diquarks are effectively attractive (repulsive), the approximation 3 should result in heavier (lighter) tetraquark states. Here, the effective interaction within diquarks reflects the repulsion or attraction effect from the enhancement or cancellation between the quark-quark (and antiquark-antiquark) interaction in the case of channel coupling. For comparison, we use these two approximations in our estimation. The values of the quark-quark coupling parameters estimated with them are listed in Table I. To determine the masses of the tetraquarks, we adopt two approaches in the present work: 1) One estimates the meson masses with the effective heavy quark masses m b = 5052.9 MeV and m c = 1724.8 MeV. These values were adopted in understanding the strange properties of tetraquark states [98,99]  It is easy to get the numerical results for the CMI matrix elements with the above two sets of parameters. Adopting the approximation C QQ = C QQ , we obtain the CMI matrices, their eigenvalues and corresponding eigenvectors, and estimated masses with two different approaches. These results are presented in Table II. In the approximation C QQ = Cnn Cnn C QQ , the estimated masses of the tetraquark states are slightly different from those in the former approximation. In the following discussions, we mainly use the masses estimated with the parameters in Set I. Since the hadron masses estimated with the effective quarks are usually like an upper limit [98][99][100]103], we focus on the results estimated with reference thresholds. We assume that these masses are all reasonable values. To have an impression for the spectrum, we plot relative positions for the bbbb (cccc) states in Fig. 1 (a) [(b)]. The solid/black lines are for the approximation C QQ = C QQ and the dashdotted/blue lines are for the approximation C QQ = Cnn Cnn C QQ . We also show the results in the extreme case C QQ = 0 with the dashed/red lines. The uncertainty caused by the change of C QQ is less than 20 (37) MeV in the bbbb (cccc) case.  The mass splitting between the scalar tetraquarks is around 120 MeV for the bottom system and 220 MeV for the charmed system. One finds that the mixing between different color structures is important here, which enlarges the mass difference between these two states. If one does not consider the mixing, the masses for the bottom case are 18913 MeV and 18875 MeV. Both states are below the ΥΥ threshold and above the η b Υ threshold. Once the mixing is considered, the higher state (6 c bb dominates) becomes a state above the ΥΥ threshold while the lower one (3 c bb dominates) becomes a state below the η b Υ threshold. Certainly the mass shift affects decay properties. The charmed case is similar.
From the diagrams (a) and (b) in Fig. 1, the estimated tetraquark masses are all above the lowest meson-meson threshold. This observation is consistent with those in Refs. [74,79,82]. The lowest bbbb mass in [79] and ours are similar. From these diagrams, the masses obtained with parameters in Set II are all lower than those in Set I. This means that the interactions within the diquarks are effectively repulsive, which can be verified from the Hamiltonian expressions. If stable multiquark states need attractive diquarks, these bbbb and cccc compact tetraquarks would tend to become meson-meson states because the quark-antiquark interaction is usually attractive (see the diagonal matrix elements in the Hamiltonian expressions) and these tetraquarks should not be stable.
If the studied states do exist, finding out their decay properties are helpful to the search at experiments. Possible rearrangement decay modes are easy to be understood from Fig. 1. Since the feature for the cccc system is very similar to the bottom case, we here only concentrate on the latter one. For the J = 2 tetraquark, the present model estimation gives a mass around the ΥΥ threshold. If the approximation C QQ ≈ 2 3 C QQ is more appropriate, the state is blow the threshold and it should have a relatively narrow width through D-wave decay into η b η b . It is a basic feature that high spin multiquark states have dominantly D-wave decay modes and should not be very broad [99,100,103]. For the J = 1 tetraquark, its mass is 30 MeV above the η b Υ threshold. From the quantum numbers, its rearrangement decay channel is only this η b Υ. For the J = 0 tetraquarks, the higher one can decay into both ΥΥ and η b η b through both S-and D-wave interactions while the lower one decays only into η b η b . If we use their masses to denote these states, probably the ordering of the widths is 18954 > 18890 ∼ 18834 > 18921.
From Fig. 1 (a)-(b), the feature that all the states can decay is consistent with the feature that the effective interaction within the diquarks is repulsive. If we want to find relatively stable compact tetraquarks in Fig. 1, good candidates should be those states for which dashed/red lines are above solid/black lines. Although the bbbb and cccc systems do not satisfy this condition, we will see that such systems exist.

C. The bbcc and ccbb systems
We present the numerical results for the ccbb states in Table III. The bbcc states are antiparticles of the ccbb states and have the same results. Now the mass splitting between different spins is less than 140 MeV. This number lies between the splittings for the bbbb case and the cccc case and thus the mixing effect is in the middle of the two. To understand the decay properties easily, we plot the spectrum for the ccbb system and relevant meson-meson thresholds in Fig. 1 (c).    Basically, the behaviors for the rearrangement decays are similar to those for the bbbb and cccc systems. The main difference lies in the C parity. The former states have definite C parities while the ccbb states not. Without the condition of C parity conservation, the 2 + ccbb tetraquark can also decay into the B + c B * + c channel through D-wave. Up to now, experiments confirm only the ground B c meson. It means that the axial vector tetraquark decaying into B c B * c may not be observed in the near future. In the case that the B * c meson is confirmed with enough data, the search for the 1 + ccbb (or bbcc) tetraquark is also possible. However, the interactions within the diquarks are effectively repulsive and these tetraquarks should not be stable.  We show the results for the bbbc (and cbbb) states in Table IV and the spectrum for bbbc in Fig. 1 (d). The maximum mass splitting is around 130 MeV. Comparing with the former systems, the Pauli principle works only for one diquark now, which leads to one more 1 + tetraquark.
From Fig. 1 (d), we may easily understand the rearrangement decay behaviors for the bbbc states. The two scalar states have similar properties to the states in the former systems. For the J = 2 tetraquark, there is one more D-wave decay channel compared to the ccbb case because the violation of the heavy quark spin symmetry results in the non-degeneracy for the thresholds of η b B * c and ΥB c . The interesting observation appears for the J P = 1 + states. The color-spin mixing affects the masses of the states relatively largely. The resulting observation is: the highest and the intermediate states are kinematically allowed to decay into ΥB c and η b B * c channels, while the lowest state has no rearrangement decay channel. From the relative positions for the solid/black, dashdotted/blue, and dashed/red lines, the interactions within the diquarks are effectively attractive for the lowest 1 + tetraquark, repulsive for the intermediate 1 + state and also attractive for the highest 1 + state. This feature is a result of balance between attraction/repulsion in bc, repulsion in bb, attraction between quarks and antiquarks, and channel coupling. From the effective interaction within the diquarks and the estimated mass, the lowest 1 + state is a good candidate of stable tetraquarks. Apparently, this state looks like an excited B c where a bb pair is excited (see discussions in Ref. [99] for other tetraquarks). By checking the CMI matrix elements, one finds that the stable state is possible mainly because of the attraction within the bc diquark. The situation is similar to the T cc state (ccqq) where the attractive ud diquark contributes dominantly [101,102].
With the observed bottomonium and B c states, in principle, resonance structures above the η b B c threshold can all be investigated. Once experiments collect enough B c data, interesting states in the ΥB c channel would probably be observed first.

E. The cccb and bccc systems
We may call such tetraquarks as "mirror" partners of the previous states. The color-spin structure is the same as the bbbc system but the decay feature relies on masses and may be different. We present the numerical results in Table V and draw the spectrum in Fig. 1 (e). The maximum mass splitting between the two scalar tetraquarks is around 180 MeV. Rearrangement decays for these tetraquarks are possible only when the mass is high enough.
The rearrangement decay properties for the J = 2 and J = 0 tetraqaurks are similar to those for the bbbc states. For the three axial vector tetraquarks, the low mass one has only one decay channel η c B * c , the intermediate one has two η c B * c and ψB c , and the high mass state has one more channel ψB * c . Contrary to the bbbc case, the effective interaction within the diquarks in the lowest 1 + state is repulsive, which is a result that the cc interaction is stronger than the bb interaction. If this state does exist, it should be less stable than the lowest 1 + bbbc.
Early experimental investigations on possible resonances in the cccb system should be through the channel ψB c , which means that four tetraquarks, a high mass scalar, two axial vectors, and one tensor, could be observed first.  We show the results for the twelve states in Table VI. Now the maximum mass splitting (240 MeV) again occurs between the scalar tetraquarks. In estimating the tetraquark masses, there are two reference thresholds we can use, Υψ and B + c B − c . As the investigations in other systems [99,100], the former threshold leads to lighter masses that can be treated as a lower limit on the theoretical side. We here adopt the masses estimated with the latter threshold.
Future measurements may answer whether the scheme is reasonable or not. Seven meson-meson channels are involved in discussing decay properties with the obtained results. We show the spectrum and these channels in Fig. 1 (f).   Since there is no constraint from the Pauli principle, the obtained spectrum is more complicated than other systems. All the masses are above the lowest meson-meson threshold and they should decay. At least four rearrangement decay channels for the 2 ++ states are D-wave channels. If the 2 ++ state is above the B * + c B * − c threshold, the S-wave channel is also opened. For the channels with J P C = 1 ++ , the S-wave Υψ is the lowest one. The decays of the two 1 ++ tetraquarks into this channel are both allowed. For the four 1 +− states, they all have S-wave decay channels Υη c and η b ψ. There is at least one allowed rearrangement decay channel, η b η c , for the four 0 ++ tetraquarks through S-wave interaction. It seems that all the bcbc tetraquarks are not stable.
On the other hand, one finds that the interactions within the diquarks in several states are effectively attractive. This indicates that in the competition between the interactions of QQ and QQ, the stronger attraction in QQ may make such states relatively stable. The lowest 0 ++ state satisfies this condition and it has only one rearrangement decay channel. Maybe this state has relatively narrow width. The lower 1 ++ state has also similar features. Its S-wave decay channel is Υψ and this tetraquark is worthwhile study. Because of the existence of possible tetraquark states, searching for exotic phenomena with Υψ may help us to understand the strong interactions between heavy quarks. In Ref. [82], Richard et al also observe that bound bcbc might be more favorable than bbbb and cccc.

IV. DISCUSSIONS AND SUMMARY
In the chiral quark model, the interaction between the light quarks may also arise from the exchange of Goldstone bosons. For the pure heavy systems, such an interaction is absent. Therefore, one needs to consider only the gluonexchange interaction for the present Q 1 Q 2Q3Q4 systems. As a short range force, it helps to form compact tetraquarks rather than meson-meson molecules if bound four-quark states do exist. We have studied the mass splittings of these tetraquark states with the simple color-magnetic interaction in this work.
For the case (bcbc system) without constraint from the Pauli principle, the number of color-singlet tetraquarks with 6 c diquark is equal to that with3 c diquark. The two color structures may couple through the color-magnetic interaction. From Table VI, the coupling for the 2 ++ states is weak while that for other states is stronger. For the case with constraint from the Pauli principle, the number of color-singlet tetraquarks with the3 c diquark is bigger than that with 6 c diquark. Their mixing is generally significant (see Tables II-V). In both cases, the tetraquarks with 6 c diquark do not exist independently.
After the configuration mixing effects are considered, both the heaviest tetraquark and the lightest tetraquark for a system are the scalar states. The maximum mass differences are around 120 MeV, 220 MeV, 140 MeV, 130 MeV, 180 MeV, and 240 MeV for the bbbc, cccc, bbcc, bbbc, cccb, and bcbc systems, respectively. Other states fall in these mass difference ranges. Whether the tetraquarks decay through quark rearrangements and how many thresholds fall in these ranges rely on the tetraquark masses. For comparison, we estimate the masses with two approaches. We mainly discuss the tetraquark properties by using the masses estimated with the threshold of some meson-meson state, the second approach.
Since the parameter C QQ cannot be extracted with experimental data, we use two approximations to determine them. We also show results in the extreme case C QQ = 0. From the comparison study, we find that (1) the estimated tetraquark masses are affected slightly and (2) the plotted spectra can be easily used to judge which states contain effectively attractive diquarks. From the viewpoint that narrow compact tetraquarks should have attractive diquarks inside them and have as few S-wave decay channels as possible, we find that most QQQQ states are not stable. However, a stable 1 + bbbc state is observed and relatively narrow bcbc tetraquarks are also possible in the present model investigations. The latter system is also proposed to have bound states in Ref. [82]. Although the model we use is simple and it does not involve dynamics, the basic features of the obtained spectra might be somehow reasonable. To give more reliable results, the present model needs to be improved.
If experiments could observe one resonant state in the channel of two heavy quarkonia, its nature as a tetraquark is favored. More tetraquarks should also exist and searches for them are strongly called for. We hope the decay channels discussed in this paper are helpful for the experimental search.
To summarize, we have explored the mass splittings between the Q 1 Q 2Q3Q4 tetraquarks with a simple model. The mixing between different color structures are considered. We have estimated their masses with two approaches and discussed possible rearrangement decay channels. Stable or narrow tetraquarks in the bbbc and bcbc systems are worthwhile study in heavy meson-meson channels such as Υψ. Hopefully, the exotic tetraquark states composed of four heavy quarks may be observed at LHCb in the future.