Analysis of the tensor–tensor type scalar tetraquark states with QCD sum rules

In this article, we study the ground states and the first radial excited states of the tensor–tensor type scalar hidden-charm tetraquark states with the QCD sum rules. We separate the ground state contributions from the first radial excited state contributions unambiguously, and obtain the QCD sum rules for the ground states and the first radial excited states, respectively. Then we search for the Borel parameters and continuum threshold parameters according to four criteria and obtain the masses of the tensor–tensor type scalar hidden-charm tetraquark states, which can be confronted with the experimental data in the future.


Introduction
The attractive interaction induced by one-gluon exchange favors formation of diquark states in color antitriplet and disfavors formation of diquark states in color sextet. The antitriplet diquark states ε i jk q T j C q k have five Dirac tensor structures, scalar Cγ 5 , pseudoscalar C, vector Cγ μ γ 5 , axial-vector Cγ μ and tensor Cσ μν . The structures Cγ μ and Cσ μν are symmetric, while the structures Cγ 5 , C and Cγ μ γ 5 are antisymmetric. The scalar and axial-vector light diquark states have been studied with the QCD sum rules [1][2][3][4], the scalar and axial-vector heavy-light diquark states have also been studied with the QCD sum rules [5,6]. The calculations based on the QCD sum rules indicate that the scalar and axialvector diquark states are more stable than the corresponding pseudoscalar and vector diquark states, respectively. We usually construct the Cγ 5 ⊗ γ 5 C-type and Cγ μ ⊗ γ μ Ctype currents to study the lowest scalar light tetraquark states, hidden-charm or hidden-bottom tetraquark states [7][8][9][10][11][12][13][14][15][16][17][18], the corresponding C ⊗ C-type and Cγ μ γ 5 ⊗ γ 5 γ μ Ctype scalar tetraquark states have much larger masses. The Cσ αβ ⊗ σ αβ C-type scalar hidden-charm or hidden-bottom tetraquark states have not been studied with the QCD sum a e-mail: zgwang@aliyun.com rules, so it is interesting to study them with the QCD sum rules.
The instantons play an important role in understanding the U A (1) anomaly and in generating the spectrum of light hadrons [19]. The calculations based on the random instanton liquid model indicate that the most strongly correlated diquarks exist in the scalar and tensor channels [20]. The heavy-light tensor diquark states, although they differ from the light tensor diquark states due to the appearance of the heavy quarks, maybe play an important role in understanding the rich exotic hadron states, we should explore this possibility, the lowest hidden-charm and hidden-bottom tetraquark states maybe of the Cγ 5 ⊗ γ 5 C-type, Cγ μ ⊗ γ μ C-type or Cσ αβ ⊗ σ αβ C-type.
The QCD sum rules method provides a powerful theoretical tool in studying the hadronic properties, and it has been applied extensively to the study of the masses, decay constants, hadronic form-factors, coupling constants, etc. [21][22][23]. In this article, we construct the Cσ αβ ⊗ σ αβ C-type currents to study the scalar hidden-charm tetraquark states. There exist some candidates for the scalar hidden-charm tetraquark states. In Ref. [24], Lebed and Polosa propose that the X (3915) is the ground state scalar cscs state based on lacking of the observed DD and D * D * decays, and attribute the single known decay mode J/ψω to the ω-φ mixing effect. Recently, the LHCb collaboration observed two new particles X (4500) and X (4700) in the J/ψφ mass spectrum with statistical significances 6.1σ and 5.6σ , respectively, and determined the quantum numbers to be J PC = 0 ++ with statistical significances 4.0σ and 4.5σ , respectively [25,26]. The X (4500) and X (4700) are excellent candidates for the cscs tetraquark states. In Refs. [27,28], we study the Cγ μ ⊗ γ μ C-type, Cγ μ γ 5 ⊗ γ 5 γ μ C-type, Cγ 5 ⊗ γ 5 Ctype, and C ⊗ C-type scalar cscs tetraquark states with the QCD sum rules. The numerical results support assigning the X (3915) to be the 1S Cγ 5 ⊗ γ 5 C-type or Cγ μ ⊗ γ μ Ctype cscs tetraquark state, assigning the X (4500) to be the 2S Cγ μ ⊗ γ μ C-type cscs tetraquark state, assigning the X (4700) to be the 1S Cγ μ γ 5 ⊗ γ 5 γ μ C-type cscs tetraquark state. For other possible assignments of the X (4500) and X (4700), one can consult Refs. [29][30][31][32][33][34]. In this article, we study the Cσ αβ ⊗ σ αβ C-type hidden-charm tetraquark states with the QCD sum rules, and explore whether or not the X (3915), X (4500) and X (4700) can be assigned to be the Cσ αβ ⊗ σ αβ C-type tetraquark states.
The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the ground state Cσ αβ ⊗ σ αβ C-type tetraquark states in Sect. 2; in Sect. 3, we derive the QCD sum rules for the masses and pole residues of the ground state and the first radial excited state of the Cσ αβ ⊗ σ αβ C-type tetraquark states; Sect. 4 is reserved for our conclusion.

QCD sum rules for the Cσ αβ ⊗ σ αβ C-type tetraquark states without including the first radial excited states
In the following, we write down the two-point correlation functions ss/du ( p) in the QCD sum rules: where where the i, j, k, m, n are color indices, C is the charge conjugation matrix. At the hadronic side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators Js s/du (x) into the correlation functions ss/du ( p) to obtain the hadronic representation [21][22][23]. After isolating the ground state contributions of the scalar cscs/cucd tetraquark states Xs s/du , we get the results, where the pole residues λs s/du are defined by 0|Js s/du (0)| Xs s/du ( p) = λs s/du . In the following, we briefly outline the operator product expansion for the correlation functions ss/du ( p) in perturbative QCD. We contract the u, d, s and c quark fields in the correlation functions ss/du ( p) with Wick theorem, and obtain the results: are the full s, u, d and c quark propagators, respectively, and t n = λ n 2 , the λ n is the Gell-Mann matrix, D α = ∂ α − ig s G n α t n [23]. Then we compute the integrals both in the coordinate space and the momentum space, and obtain the correlation functions ss/du ( p) at the quark level, therefore the QCD spectral densities through dispersion relation. In this article, we calculate the contributions of the vacuum condensates up to dimension 10 in a consistent way, for technical details, one can consult Ref. [35].
Once the analytical QCD spectral densities are obtained, we take the quark-hadron duality below the continuum thresholds s 0 ss/du and perform Borel transform with respect to the variable P 2 = −p 2 to obtain the QCD sum rules: where       We differentiate Eq. (9) with respect to 1 T 2 , then eliminate the pole residues λs s/du , and obtain the QCD sum rules for the ground state masses Ms s/du of the Cσ αβ ⊗ σ αβ C-type scalar hidden-charm tetraquark states, The input parameters are shown explicitly in Table 1. The quark condensates, mixed quark condensates, and M S masses evolve with the renormalization group equation; we take into account the energy-scale dependence according to the following equations: 4 9 , ss (μ) = ss (Q) α s (Q) α s (μ) 4 9 , 12 25 m s (μ) = m s (2GeV) α s (μ) α s (2GeV) 4 9 , α s (μ) where t = log μ 2 2 , = 213, 296 and 339 MeV for the flavors n f = 5, 4 and 3, respectively [37]. Furthermore, we set m u = m d = 0.
In the diquark-antidiquark type tetraquark system QqQq , the Q-quark serves as a static well potential and combines with the light quark q to form a heavy diquark D in color antitriplet, while theQ-quark serves as another static well potential and combines with the light antiquarkq to form a heavy antidiquarkD in color triplet; the D andD combine to form a compact tetraquark state [16][17][18]35,38,39]. For such diquark-antidiquark type tetraquark systems, we suggest an energy-scale formula μ = M 2 X/Y /Z − (2M Q ) 2 to determine the energy scales of the QCD spectral densities, where the X , Y and Z are the hidden-charm or hiddenbottom tetraquark states QqQq , the M Q are the effective heavy quark masses. In this article, we choose the updated value M c = 1.82 GeV [40]. Now we search for the Borel parameters T 2 and continuum threshold parameters s 0 ss/du according to the four criteria: 1. Pole dominance at the hadron side; 2. Convergence of the operator product expansion; 3. Appearance of the Borel platforms; 4. Satisfying the energy scale formula.
We cannot obtain reasonable Borel parameters T 2 and continuum threshold parameters s 0 ss/du , if the energy gap between the ground state and the first radial excited state is about 0.3 − 0.7 GeV.
at the energy scale μ = 2.50 GeV. The central values of the predicted masses satisfy the energy-scale formula, the criterion 4 is satisfied.
In Fig. 1, we plot the predicted masses Md u/ss with variations of the Borel parameters T 2 . From the figure, we can see that the plateaus are rather flat, the criterion 3 is satisfied.
The four criteria are all satisfied, we expect to make reliable predictions.
The energy gaps between the ground states and the first radial excited states are Z (4430) is assigned to be the first radial excitation of Z c (3900) according to the analogous decays, 2 GeV, which is large enough to take into account the contribution of the Z (4430) [44]. In the present case, the lower bound of the √ s 0 − Md u/ss,2S are about 0.3 GeV, which indicates that the widths of the first radial excited states of the Cσ αβ ⊗ σ αβ Ctype tetraquark states are rather large. According to the energy gaps between the ground states and the first radial excited states, the continuum threshold parameters should be chosen as large as √ s 0 − Md u/ss,1S = 0.5±0.1 GeV without including the first radial excited states Xd u/ss,2S explicitly, however, for such large continuum thresholds, the contributions of the Xd u/ss,2S are already included in due to their large widths. So the QCD sum rules in which only the ground state Cσ αβ ⊗ σ αβ C-type tetraquark states are taken into account cannot work.
The predicted mass Ms s,1S = 3.84 ± 0.16 GeV overlaps with the experimental value M X (3915) = 3918.4 ± 1.9 MeV slightly [37], the X (3915) cannot be a pure Cσ αβ ⊗σ αβ C-type cscs tetraquark state. The predicted mass Ms s,2S = 4.40 ± 0.09 GeV overlaps with the experimental value M X (4500) = 4506 ± 11 +12 −15 MeV slightly [25,26], the X (4500) cannot be a pure Cσ αβ ⊗σ αβ C-type cscs tetraquark state. As the central values of the Ms s,1S and Ms s,2S differ from the central values of the M X (3915) and M X (4500) significantly, it is difficult to assign the M X (3915) and M X (4500) to be the Cσ αβ ⊗ σ αβ C-type cscs tetraquark states. The Xs s, 1S and Xs s,2S are new particles, the present predictions can be confronted to the experimental data in the future.

Conclusion
In this article, we study the ground states and the first radial excited states of the Cσ αβ ⊗ σ αβ C-type hidden-charm tetraquark states with the QCD sum rules by calculating the contributions of the vacuum condensates up to dimension 10 in a consistent way. We separate the ground state contributions from the first radial excited state contributions unambiguously, and obtain the QCD sum rules for the ground states and the first radial excited states, respectively. Then we search for the Borel parameters and continuum threshold parameters according to the four criteria: (1) pole dominance at the hadron side; (2) convergence of the operator product expansion; (3) appearance of the Borel platforms; (4) satisfying the energy-scale formula. Finally, we obtain the masses and pole residues of the Cσ αβ ⊗ σ αβ C-type hidden-charm tetraquark states. The masses can be confronted to the experimental data in the future, while the pole residues can be used to study the relevant processes with the three-point QCD sum rules or the light-cone QCD sum rules.