Analysis of the $QQ\bar{Q}\bar{Q}$ tetraquark states with QCD sum rules

In this article, we study the $J^{PC}=0^{++}$ and $2^{++}$ $QQ\bar{Q}\bar{Q}$ tetraquark states with the QCD sum rules, and obtain the predictions $M_{X(cc\bar{c}\bar{c},0^{++})} =5.99\pm0.08\,\rm{GeV}$, $M_{X(cc\bar{c}\bar{c},2^{++})} =6.09\pm0.08\,\rm{GeV}$, $M_{X(bb\bar{b}\bar{b},0^{++})} =18.84\pm0.09\,\rm{GeV}$ and $M_{X(bb\bar{b}\bar{b},2^{++})}=18.85\pm0.09\,\rm{GeV}$, which can be confronted to the experimental data in the future. Furthermore, we illustrate that the diquark-antidiquark type tetraquark state can be taken as a special superposition of a series of meson-meson pairs and embodies the net effects.


Introduction
The observations of the charmonium-like and bottomonium-like states have provided us with a good opportunity to study the exotic states and understand the strong interactions, especially those charged states Z c (3885), Z c (3900), Z c (4020), Z c (4025), Z c (4200), Z(4430), Z b (10610), Z b (10650), they are excellent candidates for the multiquark states [1]. If they are tetraquark states, they consist two heavy quarks and two light quarks, we have to deal with both the heavy and light degrees of freedom of the dynamics. On the other hand, if there exist tetraquark configurations consist of four heavy quarks, the dynamics is much simple. There have been several works on the mass spectrum of the QQQQ tetraquark states, such as the non-relativistic potential models [2,3,4,5], the Bethe-Salpeter equation [6], the constituent diquark model with spin-spin interaction [7,8], the constituent quark model with color-magnetic interaction [9], the moment QCD sum rules [10], etc. In this article, we study the tetraquark states consist of four heavy quarks with the Borel QCD sum rules.
The QCD sum rules is a powerful theoretical tool in studying the ground state tetraquark states and molecular states, and has given many successful descriptions of the masses and hadronic coupling constants [11]. In this article, we study the J P C = 0 ++ and 2 ++ QQQQ tetraquark states, which may be observed in the e + e − and pp collisions, for example, e + e − → J/ψcc, pp →cccc. The ATLAS, CMS and LHCb collaborations have measured the cross section for double charmonium production [12], the CMS collaboration has observed the Υ pair production [13].
The quarks have color SU (3) symmetry, we can construct the tetraquark states according to the routine quark → diquark → tetraquark, where the 1 c , 3 c (3 c ), 6 c and 8 c denote the color singlet, triplet (antitriplet), sextet and octet, respectively. The one-gluon exchange leads to attractive (repulsive) interaction in the color antitriplet (sextet) channel, which favors (disfavors) the formation of diquark states in the color antitriplet (sextet) [14]. The diquarks ε ijk q T j CΓq ′ k in color antitriplet have five structures in Dirac spinor space, where the i, j and k are color indexes, CΓ = Cγ 5 , C, Cγ µ γ 5 , Cγ µ and Cσ µν for the scalar, pseudoscalar, vector, axialvector and tensor diquarks, respectively. The stable diquark configurations are the scalar (Cγ 5 ) and axialvector (Cγ µ ) diquark states from the QCD sum rules [15,16]. The double-heavy diquark states ε ijk Q T j Cγ 5 Q k cannot exist due to the Pauli principle. In this article, we take the double-heavy diquark states ε ijk Q T j Cγ µ Q k as basic constituents to construct the tetraquark states.
The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the QQQQ tetraquark states in section 2; in section 3, we present the numerical results and discussions; section 4 is reserved for our conclusion.
At the phenomenological side, we can insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators J µν (x) and J(x) into the correlation functions Π µναβ (p) and Π(p) to obtain the hadronic representation [17,18]. After isolating the ground state contributions of the scalar and tensor QQQQ tetraquark states (denoted by X), we get the following results, where g µν = g µν − pµpν p 2 , the pole residues λ X are defined by 0|J µν (0)|X(p) = λ X ε µν (λ, p) , the ε µν (λ, p) is the polarization vector of the tensor tetraquark states, λ ε * αβ (λ, p)ε µν (λ, p) = g αµ g βν + g αν g βµ 2 − g αβ g µν 3 .
Now we briefly outline the operator product expansion for the correlation functions Π µναβ (p) and Π(p) in perturbative QCD. We contract the heavy quark fields in the correlation functions Π µναβ (p) and Π(p) with Wick theorem, and obtain the results: where the S ij (x) is the full Q quark propagator, and t n = λ n 2 , the λ n is the Gell-Mann matrix [18]. Then we compute the integrals both in the coordinate and momentum spaces to obtain the correlation functions Π µναβ (p) and Π(p), therefore the QCD spectral densities through dispersion relation. The calculations are straightforward but tedious.
We take the quark-hadron duality below the continuum thresholds s 0 and perform Borel transform with respect to the variable P 2 = −p 2 to obtain the QCD sum rules: where ρ(s, z, t, r) = ρ S (s, z, t, r) and ρ T (s, z, t, r) for the scalar and tensor tetraquark states, respectively, the explicit expressions are given in the Appendix, We derive Eq.(11) with respect to τ = 1 T 2 , then eliminate the pole residues λ X , and obtain the QCD sum rules for the masses of the scalar and tensor QQQQ tetraquark states, In the moment QCD sum rules, the moments M n (P 2 0 ) at the phenomenological side are defined by where the X ′ denotes the first radial excited state of the X, the P 2 0 is a particular value for the parameter P 2 = −p 2 . We can extract the mass M X according to the ratio r(n, P 2 0 ) at large values of n, where the small values δ n (P 2 0 ) ≈ δ n+1 (P 2 0 ). In Refs. [19,20], we observe that .6 (or 9.6) for the central values of the pole residues for X = Z c (3900), X ′ = Z(4430) (or X = X(3915), X ′ = X(4500)) in the scenario of tetraquark states. So the n has to be postponed to very large values [10]. In the present work, the contributions of the high resonances and continuum states are depressed by the weight function exp − s T 2 . The differences between the predicted masses in the present work and in Ref. [10] originate from the different currents or quark structures.

Numerical results and discussions
We take the gluon condensate to be the standard value [17,18,21], and take the M S masses m c (m c ) = (1.275 ± 0.025) GeV and m b (m b ) = (4.18 ± 0.03) GeV from the Particle Data Group [22]. We take into account the energy-scale dependence of the M S masses from the renormalization group equation, , Λ = 213 MeV, 296 MeV and 339 MeV for the flavors n f = 5, 4 and 3, respectively [22].
The values of the thresholds are 2M ηc = 5966.8 MeV, 2M J/ψ = 6193.8 MeV, 2M η b = 18798.0 MeV, 2M Υ = 18920.6 MeV from the Particle Data Group [22]. The masses of the 0 ++ and 2 ++ QQQQ tetraquark states from the phenomenological quark models lie above or below those thresholds [2,3,4,5,6,7,8,9,10]. In Ref. [23], we study the vector and axialvector B c mesons with the QCD sum rules and obtain the masses M B * c = 6.337 ± 0.052 GeV and M Bc1 = 6.730 ± 0.061 GeV at the typical energy scale µ = 2 GeV. The B c mesons have two heavy quarks, and the mass M B * c = 6.337 ± 0.052 GeV lies slightly above the threshold 2M J/ψ = 6193.8 MeV, so we expect the ideal energy scale to extract masses of the cccc tetraquark states from the QCD sum rules is about µ = 2 GeV, it is indeed the case.    [22]. Now we revisit the mass gaps of the tetraquark states. In Ref. [19], we tentatively assign the Z c (3900) and Z(4430) to be the ground state and the first radial excited state of the axial-vector tetraquark states with J P C = 1 +− , respectively, and reproduce the experimental values of the masses with the QCD sum rules. In Ref. [20], we tentatively assign the X(3915) and X(4500) to be the ground state and the first radial excited state of the scalar cscs tetraquark states with J P C = 0 ++ , respectively, and We search for the Borel parameters T 2 and continuum threshold parameters s 0 to satisfy the two criteria of the QCD sum rules: pole dominance at the phenomenological side and convergence of the operator product expansion at the QCD side. Furthermore, we take the relation S 0 S/T = M S/T + (0.4 ∼ 0.6) GeV as an additional constraint to obey. The resulting Borel parameters, continuum threshold parameters, energy scales, pole contributions are shown explicitly in    Table, we can see that the pole dominance at the phenomenological side is well satisfied. In the Borel windows, the dominant contributions come from the perturbative terms, the contributions of the gluon condensate are about −10%, the operator product expansion is well convergent. Now the two criteria of the QCD sum rules are all satisfied, we expect to make reasonable predictions. In Ref. [24], we tentatively assign the D * s3 (2860) to be a D-wave cs meson, and study the mass and decay constant of the D * s3 (2860) with the QCD sum rules by calculating the contributions of the vacuum condensates up to dimension-6 in the operator product expansion. In calculations, we observe that only the perturbative term, gluon condensate and three-gluon condensate have contributions. At the Borel window, the contributions are about (107 − 109)%, −(7 − 9)% and ≪ 1%, respectively, see the first diagram in Fig.3 [24], the three-gluon condensate can be neglected safely. In the present case, the contributions of the gluon condensate are about −10%, just like in the case of the QCD sum rules for the D * s3 (2860), so neglecting the three-gluon condensate cannot impair the predictive ability. As the dominant contributions come from the perturbative terms, perturbative O(α s ) corrections amount to multiplying the pertubative terms by a factor κ, which can be absorbed into the pole residues and cannot impair the predicted masses remarkably.
We take into account all uncertainties of the input parameters, and obtain the values of the ground state masses and pole residues, which are also shown explicitly in Table 1 and Figs.3-4. From Table 1, we can see that the additional constraint is also satisfied. In Figs.3-4, we plot the masses and pole residues with variations of the Borel parameters at much larger intervals than the Borel windows shown in Table 1. From Figs.3-4, we can see that the predicted masses and pole residues are rather stable with variations of the Borel parameters, the uncertainties originate from the Borel parameters in the Borel windows are very small.
From Table 1, we can see that the mass splitting between the 0 ++ and 2 ++ bbbb tetraquark states is much smaller than that for the cccc tetraquark states. The heavy quark effective Lagrangian   can be written as where D µ ⊥ = D µ − v µ v · D, the D µ is the covariant derivative, and the h v is the effective heavy quark field. The heavy quark spin symmetry breaking terms appear at the order O(1/m Q ) [25]. The M S masses are m c (m c ) = (1.275 ± 0.025) GeV and m b (m b ) = (4.18 ± 0.03) GeV from the Particle Data Group [22], the heavy quark spin symmetry breaking effects in the c-quark systems are much larger than that in the b-quark systems.
In 2002, the SELEX collaboration reported the first observation of a signal for the doublecharm baryon state Ξ + cc in the decay mode Ξ + cc → Λ + c K − π + [26], and confirmed later by the same collaboration in the decay mode Ξ + cc → pD + K − with the measured mass M Ξ = (3518.9 ± 0.9) MeV [27]. In Ref. [28], we study the 1 2 + doubly heavy baryon states Ω QQ and Ξ QQ by subtracting the contributions from the corresponding 1 2 − doubly heavy baryon states with the QCD sum rules, and obtain the value M Ξcc = 3.57 ± 0.14 GeV. If exciting additional quark-antiquark pair qq with J P C = 0 ++ costs energy about 1 GeV, then M X(cccc,0 ++ /2 ++ ) ≈ 2M Ξcc − 1 GeV = 6.14 ± 0.14 GeV, which is consistent with the present prediction.
The predicted masses are The decays are kinematically allowed, but the available spaces are small. The decays are kinematically forbidden, but the decays can take place, we can search for the X(cccc, 0 ++ /2 ++ ) and X(bbbb, 0 ++ /2 ++ ) in the mass spectrum of the µ + µ − µ + µ − in the future.
In the following, we perform Fierz re-arrangement to the tensor current J µν and scalar current J both in the color and Dirac-spinor spaces to obtain the results, Now we can see that the diquark-antidiquark type current can be changed to a current as a special superposition of color singlet-singlet type currents, which couple potentially to the meson-meson pairs. The diquark-antidiquark type tetraquark state can be taken as a special superposition of a series of meson-meson pairs, and embodies the net effects. The decays to its components (mesonmeson pairs) are Okubo-Zweig-Iizuka super-allowed, but the re-arrangements in the color-space are highly non-trivial.
We take the current J as an example to illustrate that the scalar tetraquark state can embody the net effects of all the meson-meson pairs. At the phenomenological side, we can insert a complete set of intermediate hadronic states with the same quantum numbers as the current operator J(x) into the correlation function Π(p) to obtain the hadronic representation [17,18]. After isolating the lowest meson-meson pairs, we get the following result, where the decay constants f ηc , f J/ψ , f χc0 and f χc1 are defined by the ε µ are the polarization vectors of the J/ψ and χ c1 . We can rewrite the correlation function Π(p) into the following form through dispersion relation, In this article, we choose the value s 0 < 4M 2 χc0 , 4M 2 χc1 , the meson pairs χ c0 χ c0 and χ c1 χ c1 have no contributions, the QCD sum rules can be written as In Fig.7, we plot the mass M X(cccc,0 ++ ) comes from Eqs.(29-30) with variation of the Borel parameter T 2 , from the figure, we can see that the values of the M X(cccc,0 ++ ) are rather stable with variation of the Borel parameter at the energy scale µ ≥ 1.8 GeV. Now we choose the special value T 2 = 4.4 GeV 2 , and plot the mass M X(cccc,0 ++ ) with variation of the energy scale µ in Fig.8. From the figure, we can see that the mass M X(cccc,0 ++ ) increases monotonously and quickly with increase of the energy scale µ at the region µ ≤ 1.8 GeV, and decreases monotonously and slowly at the region µ ≥ 2.0 GeV. In the range µ = (1.8 − 2.0) GeV, the predicted mass M X(cccc,0 ++ ) is rather stable, M X(cccc,0 ++ ) = 5.76 GeV, which is below the threshold 2M ηc = 5966.8 MeV. On the other hand, the pole residue λ X(cccc,0 ++ ) increases monotonously and quickly with increase of the energy scale µ at the region µ ≥ 1.6 GeV, no stable QCD sum rules can be obtained, see Fig.9. So the QCD sum rules cannot be saturated by the meson pairs η c η c , J/ψJ/ψ plus a scalar tetraquark state X(cccc, 0 ++ ).
In this article, the diquark-antidiquark type tetraquark state is taken as a special superposition of a series of meson-meson pairs, and embodies the net effects. The decays to its components (meson-meson pairs) are Okubo-Zweig-Iizuka super-allowed, but the re-arrangements in the colorspace are highly non-trivial. In other words, the lowest states X(QQQQ, 0 ++ /2 ++ ) can saturate the QCD sum rules satisfactorily.

Conclusion
In this article, we study the J P C = 0 ++ and 2 ++ QQQQ tetraquark states with the QCD sum rules by constructing the diquark-antidiquark type currents and calculating the contributions of the vacuum condensate up to dimension 4 in the operator product expansion. We obtain the predictions M X(cccc,0 ++ ) = 5.99 ± 0.08 GeV, M X(cccc,2 ++ ) = 6.09 ± 0.08 GeV, M X(bbbb,0 ++ ) = 18.84 ± 0.09 GeV, M X(bbbb,2 ++ ) = 18.85 ± 0.09 GeV, which can be confronted to the experimental data in the future. Furthermore, we illustrate that the diquark-antidiquark type tetraquark state can be taken as a special superposition of a series of meson-meson pairs and embodies the net effects. We can search for the J P C = 0 ++ and 2 ++ QQQQ tetraquark states in the mass spectrum of the µ + µ − µ + µ − .