Exotic fully-heavy $Q\bar QQ\bar Q$ tetraquark states in $\mathbf{8}_{[Q\bar{Q}]}\otimes \mathbf{8}_{[Q\bar{Q}]}$ color configuration

We have systematically calculated the mass spectra for S-wave and P-wave fully-charm $c\bar{c}c\bar{c}$ and fully-bottom $b\bar{b}b\bar{b}$ tetraquark states in the $\mathbf{8}_{[Q\bar{Q}]}\otimes \mathbf{8}_{[Q\bar{Q}]}$ color configuration, by using the moment QCD sum rule method. The masses for the fully-charm $c\bar cc\bar c$ tetraquark states are predicted about $6.3-6.5$ GeV for S-wave channels and $7.0-7.2$ GeV for P-wave channels. These results suggest the possibility that there are some $\mathbf{8}_{[c\bar{c}]}\otimes \mathbf{8}_{[c\bar{c}]}$ components in LHCb's di-$J/\psi$ structures. For the fully-bottom $b\bar{b}b\bar{b}$ system, their masses are calculated around 18.2 GeV for S-wave tetraquark states while 18.4-18.8 GeV for P-wave ones, which are below the $\eta_b\eta_b$ and $\Upsilon(1S)\Upsilon(1S)$ two-meson decay thresholds.


I. INTRODUCTION
The existence of multiquark states was first suggested by Gell-Mann and Zweig at the birth of quark model [1,2].Since 2003, plenty of charmoniumlike exotic states and P c states have been observed [3][4][5][6][7][8][9][10][11][12] , many of which are unexpected and can not be fitted into the conventional quark model.To understand the nature of these new resonances, many exotic hadron configurations have been proposed such as hadron molecules, compact multiquarks, hybrid mesons and so on [13][14][15][16][17][18].Among these theoretical models, the loosely bound hadron molecule and compact multiquark are two especially appealing configurations.For the charmoniumlike XYZ and P c states, it is complicated and difficult to distinguish these two different hadron configurations experimentally and theoretically since the existence of light quarks.
Nevertheless, the authors of Ref. [78] discussed the interaction between two J/ψ mesons via the exchange of soft gluons, which hadronise into two light mesons at large distance.By studying the correlated ππ and K K exchanges, they found that it is possible for two J/ψ mesons to form a bound state.In Ref. [34], the authors studied the dicharmonia states in 1 [cc] ⊗ 1 [cc] configuration with J P C = 0 ++ and predicted their masses around 6.0-6.7 GeV in the method of QCD sum rules.The existence of di-charmonia bound states were also studied in Ref. [61], in which the authors investigated the η c η c , J/ψJ/ψ bound states in both color structures by using a non-relativistic quark model.The 0 ++ di-charmonia states in the 8 [cc] ⊗ 8 [cc] color structure were also investigated in Ref. [54] by the Laplace QCD sum rule method.
In our previous works in Refs.[23,79,80], we have studied the fully-heavy tetraquark states in diquark-antidiquark configuration with both 6 In this work, we shall further investigate the possibility of fully-heavy tetraquark states in meson-meson configuration with color structure by using the method of QCD moment sum rules [81,82].
This paper is organized as follows.In Sec.II, we construct the interpolating currents for meson-meson tetraquark states.In Sec.III, we evaluate the correlation functions for these interpolating currents.We extract the masses for these tetraquark states by performing the moment QCD sum rule analyses in Sec.IV.The last section is a brief summary.

II. INTERPOLATING CURRENTS
The color structure of a meson-meson operator [Q Q][Q Q] can be written via the SU(3) symmetry in which the color singlet structures come from the we can construct the S-wave and P-wave • The S-wave interpolating currents are in which we only obtain and 2 ++ in the S-wave channel.The tensor current J µν can couple to both the J P C = 0 ++ and 2 ++ quantum numbers, but not the J P C = 1 ++ channel since the Lorentz symmetry restriction.
• The P-wave interpolating currents are where only one P-wave [Q Q] operator is contained in these interpolating currents.One should note that these tetraquark interpolating currents with color structure can be written as combinations of the diquarkantidiquark operators through the Fierz transformation and the color rearrangement.Their decay properties should be the same with the [QQ][ Q Q] tetraquark states as discussed in Refs.[23,74].Thus we shall not discuss the decay behaviors for these

III. QCD SUM RULES
In this section, we investigate the two-point correlation functions of the interpolating currents constructed above.For the scalar and pseudo-scalar currents, the correlation function are while for the vector and axial-vector currents The correlation function Π µν (p 2 ) in Eq. ( 5) can be divided into two part where Π 0 p 2 and Π 1 p 2 represent the spin-0 and spin-1 invariant functions, respectively.For the tensor currents J µν (x) in Eq. ( 2), The correlation function Π µν,ρσ (p 2 ) in Eq. ( 7) can be expressed as where The invariant function Π 2 p 2 relates to the spin-2 intermediate state, and the " • • • " represents the contribution from the spin-0 state.At the hadron level, the invariant functions can be expressed through the dispersion relation where b n is the subtraction constant.In QCD sum rules, the imaginary part of the correlation functions are usually simplified as the following "pole plus continuum" spectral function H ) + QCD continuum and higher states , (11) in which the δ function represents the lowest-lying state.The parameters f H and m H are the coupling constant and mass of the lowest-lying hadronic resonance H respectively with the polarization vector ǫ µ and polarization tensor ǫ µν .
To extract the lowest-lying resonance, we first define the moments by taking derivatives of the correlation function We then rewrite the moments by applying the above equation to the Eq.( 10) and obtain where δ n (Q 2 0 ) represents higher excited states and continuum contribution, and it is a function of n and Q 2 0 .It should be noted that for a specific value of Q 2 0 , δ n (Q 2 0 ) will tend to zero as n going to infinity.Considering the ratio of the moments to remove the unknown coupling constant where the relation ) will be satisfied when n is large enough, and we can extract the hadron mass as At the quark-gluon level, we can evaluate the invariant functions Π(p 2 ) via the operator product expansion (OPE) method.The Wilson coefficients can be calculated by adopting the following heavy quark propagator in momentum space where Q represents the charm quark or bottom quark field.The superscripts a, b are the color indices and p = p µ γ µ .
In this work, we will only evaluate the perturbative term and gluon condensate term in the correlation function, the contributions from higher non-perturbative terms such as tri-gluon condensate are small enough to be neglected.

IV. NUMERICAL ANALYSIS
We now perform the QCD moment sum rule analyses by adopting the following values of heavy quark masses and gluon condensate [84][85][86] As mentioned in Sec.III, there remain two important parameters n and Q 2 0 in the extracted hadron mass in Eq. ( 16).In the original literatures on moment sum rules, the authors set Q 2 0 = 0, which may lead to a bad OPE convergence.To avoid such bad behavior, we follow the Refs.[23,79,80] to choose Q 2 0 > 0 and introduce ξ = Q 2 0 /(4m c ) 2 to perform sum rule analysis.The parameters n and ξ are related to each other through the following respects: (1) a large enough n will reduce the higher excited states and continuum region contributions, but it will also decrease the convergence of OPE series.(2) a large ξ (or Q 2 0 ) can compensate the OPE convergence (see Fig. 1), but may cause a bad convergence of δ n (Q 2 0 ) and make it difficult to obtain the parameters of the lowest lying resonance.One needs to find suitable working regions for these two parameters to establish stable sum rules.
We take the interpolating current J 1 (x) with J P C = 0 ++ as an example to show the numerical analysis details.The correlation function of this current is evaluated as the following where We shall not evaluate the dimension-6 trigluon condensate G 3 (∼g 3 s ) and dimension-8 G 4 (∼g 4 s ) condensate in the OPE series.The G 3 term gives negligible contribution to the correlation functions even at ξ = 0 for the charmonium system [87] and four-charm tetraquark system [55].For the dimension-8 G 4 condensate, it was proven in the charmonium moment sum rules that this term was much larger suppressed comparing to the dimension-4 gluon condensate G 2 at ξ = 0, and thus can also be neglected for the mass sum rule analysis [81,88].To obtain convergent OPE series, we require that the contribution of the gluon condensate be smaller than the perturbative term, and obtain the upper bound n max = 47, 61, 77, 91 for ξ = 2, 3, 4, 5 respectively.We show the ratio |Π GG /Π P ert | in Fig. 1 to display the convergence of the OPE series with respect to n and ξ, which indicates that the OPE convergence becomes better with increasing of ξ and decreasing of n.
In Fig. 2, we show the variation of the extracted mass with n for different value of ξ, and get stable mass prediction plateaus (n, ξ) = (32, 2), (42, 3), (52,4), (62,5).To choose the values of ξ, one should consider both the existence of the mass plateaus and the stability of the hadron mass for growing ξ.Both of these two criteria can be satisfied for ξ ≥ 2, as shown in Fig. 2. Accordingly, the mass of such a cccc tetraquark state is finally predicted to be in which the errors come from the uncertainties of ξ and n, charm quark mass and the gluon condensate.The same numerical analyses can be done for the other interpolating currents in Eqs.(2-3).Then we obtain the masses for all cccc tetraquark states in 8 [cc] ⊗ 8 [cc] configuration in Table I.In this mass spectra, we predict three S-wave cccc tetraquarks with J P C = 0 ++ , 1 +− and 2 ++ and four P-wave cccc tetraquarks with J P C = 0 −+ , 0 −− , 1 −− and 1 −+ .The masses are predicted to be around 6.3−6.5 GeV for the S-wave states while 7.0−7.2GeV for the P-wave states.Comparing to the mass spectra obtained in Ref. [23], the masses for S-wave fully-charm tetraquark states are consistent with each other in both the [cc][cc] and [cc][cc] configurations.However, the P-wave [cc][cc] tetraquarks are predicted to be 200-300 MeV higher than those in the diquark-antidiquark configuration [23].In Table I, we also list the masses for some S-wave cccc tetraquark states in 8 [cc] ⊗ 8 [cc] configuration obtained by the Laplace QCD sum rule [54] and a non-relativistic quark model [61].Our results for these cccc tetraquark states are in good agreement with those in Refs.[54,61].One notes that the two interpolating currents in the same channel (J P C = 0 ++ , 0 −+ , 1 −+ , 1 −− ) lead to almost the same hadron masses.To specify if these two currents couple to the same physical state or not, we calculate their cross correlation functions of two different currents with the same quantum number, e.g., the J 1 (x) and J 2 (x) with Our calculations show that all these cross correlation functions are large enough and comparable to the diagonal correlators, implying that they couple to the same physical states.Since the two interpolating currents in the same channel give almost the same hadron masses, we don't reanalyze the mass sum rules by using their mixing current, avoiding more errors from the uncertain mixing angle.
We can also study the fully-bottom tetraquark states in For the fully-bottom system, we define ξ = Q 2 0 /(m b ) 2 and find that the two criteria of mass plateaus and ξ stability can be achieved for ξ = 0.2 − 0.8.By requiring that the contribution of the gluon condensate be smaller than the perturbative term, we obtain the upper bound on the parameter n max = 119, 121, 123, 125 for ξ = 0.2, 0.4, 0.6, 0.8 respectively.We show the variation of the extracted mass with n for different value of ξ in Fig.
Applying the similar moment sum rule analyses, we obtain the mass spectra for these b bb b tetraquark states and list them in Table II.Accordingly, the S-wave b bb b tetraquark states are obtained to be around 18.2 GeV while the P-wave states are about 18.4-18.8GeV.Such results are several hundreds MeV below the masses of diquark-antidiquark bb bb tetraquarks predicted in Ref. [23].As shown in Table II, our results for the S-wave 8 [b b] ⊗ 8 [b b] tetraquarks are much smaller than those predicted in the non-relativistic quark model [61], but in roughly agreement with the results in Laplace QCD sum rules [54].

V. CONCLUSION AND DISCUSSION
We have studied the fully-heavy color structure by using the moment QCD sum rule method.We construct the S-wave and P-wave interpolating tetraquark currents with various quantum numbers and calculate their two-point correlation functions containing perturbative term and gluon condensate term.Choosing suitable parameter working regions, we have established stable moment sum rules for all interpolating currents and extracted the mass spectra for the fully-charm and fully-bottom tetraquark states.
For the fully-charm cccc system, our results suggest that the S-wave tetraquark states with J P C = 0 ++ , 1 +− , 2 ++ lie around 6.3 − 6.5 GeV while the P-wave tetraquark states with J P C = 0 −+ , 0 −− , 1 −− , 1 −+ are about 7.0 − 7.2GeV.Especially, the masses for the cccc tetraquarks with J P C = 0 ++ and 2 ++ are consistent with the broad structure observed by LHCb [33].The P-wave fully-charm tetraquarks with J P C = 0 −+ and 1 −+ are predicted to be roughly in agreement with the mass of X(6900) within errors.Such results suggest the possibility that there are some 8 [cc] ⊗ 8 [cc] components in LHCb's di-J/ψ structures.More investigations are needed in both theoretical and experimental aspects to study the nature of these structures.
For the fully-bottom b bb b system, the numerical results show that the S-wave tetraquark states are about 18.2 GeV while the P-wave states are around 18.4 − 18.8 GeV.All these fully-bottom b bb b tetraquarks are predicted to below the η b η b and Υ(1S)Υ(1S) two-meson decay thresholds, indicating that these tetraquark states will be stable against the strong interaction.Such results are consistent with our previous prediction for the diquark-antidiquark bb bb tetraquarks in Ref. [23].More efforts are expected to search for such fully-bottom tetraquark states in the future experiments, such as LHCb, CMS and so on.

Figure 1 :
Figure 1: |Π GG /Π P ert | with respect to n for different values of ξ from J 1 (x) with J P C = 0 ++ .

Figure 2 :
Figure 2: Hadron mass for the fully-charm cccc tetraquark state with J P C = 0 ++ from J 1 (x) with respect to n for different value of ξ.

Figure 3 :
Figure 3: Hadron mass for fully-bottom b bb b tetraquark state with J P C = 0 ++ from J 1 (x) with respect to n for different value of ξ.

Table I :
The mass spectra for the fully-charm cccc tetraquark states in 8 [cc] ⊗ 8 [cc] color configuration.

Table II :
The mass spectra for the fully-bottom b bb b tetraquark states in 8 [b b] ⊗ 8 [b b] color configuration.The mass of such a b bb b tetraquark state is finally predicted as m b bb b = 18.15 +0.14 −0.10 GeV .