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 to 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 ijk q T j CΓq k have five Dirac tensor structures, scalar Cγ 5 , pseudoscalar C, vector Cγ µ γ 5 , axialvector 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 axialvector light diquark states have been studied with the QCD sum rules [1,2,3], the scalar and axialvector heavy-light diquark states have also been studied with the QCD sum rules [4]. 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 [5,6,7], the corresponding C ⊗C-type and Cγ µ γ 5 ⊗γ 5 γ µ C-type 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 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 [8]. The calculations based on the random instanton liquid model indicate that the most strongly correlated diquarks exist in the scalar and tensor channels [9]. 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 hiddenbottom 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 provides a powerful theoretical tool in studying the hadronic properties, and has been applied extensively to study the masses, decay constants, hadronic form-factors, coupling constants, etc [10,11]. 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. [12], 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 P C = 0 ++ with statistical significances 4.0σ and 4.5σ, respectively [13]. The X(4500) and X(4700) are excellent candidates 1 E-mail: zgwang@aliyun.com.
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 section 2; in section 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; section 4 is reserved for our conclusion.
2 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 Πs s/du (p) in the QCD sum rules, where where the i, j, k, m, n are color indexes, the 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 Πs s/du (p) to obtain the hadronic representation [10,11]. After isolating the ground state contributions of the scalar cscs 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 Πs s/du (p) in perturbative QCD. We contract the u, d, s and c quark fields in the correlation functions Πs s/du (p) with Wick theorem, and obtain the results: where the S ij (x), U ij (x), D ij (x) and C ij (x) 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 [11]. Then we compute the integrals both in the coordinate space and the momentum space, and obtain the correlation functions Πs s/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. [17].
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 together to form a compact tetraquark state [7,17,20,21]. 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 hidden-bottom 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 [22]. 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.
3 QCD sum rules for the Cσ αβ ⊗σ αβ C-type tetraquark states including the first radial excited states Now we take into account both the ground state contribution and the first radial excited state contribution at the hadronic side of the QCD sum rules [23]. Firstly, we introduce the notations τ = 1 T 2 , D n = − d dτ n , and use the subscripts 1 and 2 to denote the ground state and the first radial excited state of the Cσ αβ ⊗ σ αβ C-type tetraquark states, respectively. Then the QCD sum rules can be written as the subscript QCD denotes the QCD side of the correlation functions Πs s/du (τ ). We differentiate both sides of the QCD sum rules in Eq.(21) with respect to τ and obtain Then we solve the two equations, and obtain the QCD sum rules, where i = j. We differentiate both sides of the QCD sum rules in Eq.(23) with respect to τ and obtain The squared masses M 2 i satisfy the following equation, i = 1, 2, j, k = 0, 1, 2, 3. We solve the equation in Eq. (25) and obtain two solutions The ground state contributions are separated from the first radial excited state contributions unambiguously, and we obtain the QCD sum rules for the ground states and the first radial excited states, respectively. Again, 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. The resulting Borel parameters T 2 and continuum threshold parameters s 0 ss/du are for the central values of the continuum threshold parameters, the operator product expansion is convergent, the criterion 2 is satisfied. Now we take into account the uncertainties of all the input parameters, and obtain the masses and pole residues of the Cσ αβ ⊗ σ αβ C-type tetraquark states,    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 The Z(4430) is assigned to be the first radial excitation of the Z c (3900) according to the analogous decays,