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

The European Physical Journal C, Nov 2016

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.

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

Eur. Phys. J. C Analysis of the tensor-tensor type scalar tetraquark states with QCD sum rules Zhi-Gang Wang 0 Jun-Xia Zhang 0 0 Department of Physics, North China Electric Power University , Baoding 071003 , People's Republic of China 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. - 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 εi jk q Tj C qk have five Dirac tensor structures, scalar C γ5, pseudoscalar C , vector C γμγ5, axial-vector C γμ and tensor C σμν . The structures C γμ and C σμν are symmetric, while the structures C γ5, C and C γμγ5 are antisymmetric. The scalar and axial-vector light diquark states have been studied with the QCD sum rules [1–4], 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 C γ5 ⊗ γ5C -type and C γμ ⊗ γ μC type currents to study the lowest scalar light tetraquark states, hidden-charm or hidden-bottom tetraquark states [7– 18], the corresponding C ⊗ C -type and C γμγ5 ⊗ γ5γ μC type scalar tetraquark states have much larger masses. The C σαβ ⊗ 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 UA(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 C γ5 ⊗ γ5C -type, C γμ ⊗ γ μC -type or C σαβ ⊗ 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– 23]. In this article, we construct the C σαβ ⊗ 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 csc¯s¯ state based on lacking of the observed D D¯ and D∗ D¯ ∗ decays, and attribute the single known decay mode J /ψ ω to the ω–φ mixing effect. Recently, the LHCb collaboration observed two new particles X (4500) and X (4700) in the J /ψ φ 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 csc¯s¯ tetraquark states. In Refs. [27,28], we study the C γμ ⊗ γ μC -type, C γμγ5 ⊗ γ5γ μC -type, C γ5 ⊗ γ5C type, and C ⊗ C -type scalar csc¯s¯ tetraquark states with the QCD sum rules. The numerical results support assigning the X (3915) to be the 1S C γ5 ⊗ γ5C -type or C γμ ⊗ γ μC type csc¯s¯ tetraquark state, assigning the X (4500) to be the 2S C γμ ⊗ γ μC -type csc¯s¯ tetraquark state, assigning the × Tr σ αβ C n n(−x )σ μν C Sm mT (−x )C , d¯u ( p) = i εi jk εimnεi j k εi m n where Si j (x ), Ui j (x ), Di j (x ) and Ci j (x ) are the full s, u, d and c quark propagators, respectively, i δi j x δi j ms Si j (x ) = 2π 2x 4 − 4π 2x 2 − X (4700) to be the 1S C γμγ5 ⊗ γ5γ μC -type csc¯s¯ tetraquark state. For other possible assignments of the X (4500) and X (4700), one can consult Refs. [29–34]. In this article, we study the C σαβ ⊗ 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 C σαβ ⊗ 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 C σαβ ⊗ 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 C σαβ ⊗ σ αβ C -type tetraquark states; Sect. 4 is reserved for our conclusion. 2 QCD sum rules for the Cσαβ ⊗ σ αβ C-type tetraquark states without including the first radial excited states In the following, we write down the two-point correlation functions s¯s/d¯u ( p) in the QCD sum rules: s¯s/d¯u ( p) = i d4x ei p·x 0|T Js¯s/d¯u (x ) Js¯†s/d¯u (0) |0 , 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/d¯u (x ) into the correlation functions s¯s/d¯u ( p) to obtain the hadronic representation [21–23]. After isolating the ground state contributions of the scalar csc¯s¯/cuc¯d¯ tetraquark states Xs¯s/d¯u , we get the results, s¯s/d¯u ( p) = Xs¯s/d¯u ( p) = λs¯s/d¯u . In the following, we briefly outline the operator product expansion for the correlation functions s¯s/d¯u ( p) in perturbative QCD. We contract the u, d, s and c quark fields in the correlation functions s¯s/d¯u ( p) with Wick theorem, and obtain the results: and t n = λ2n , the λn is the Gell-Mann matrix, Dα = ∂α − i gs Gnαt n [23]. Then we compute the integrals both in the coordinate space and the momentum space, and obtain the correlation functions s¯s/d¯u ( 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 d4ke−ik·x k − mc (k2 − mc2)2 3(k2 − mc2)4 4(k2 − mc2)5 thresholds ss¯0s/d¯u and perform Borel transform with respect to the variable P2 = − p2 to obtain the QCD sum rules: 1−y 1−y yi zi × 10s2 − 12smc2 + 3mc4 1−y 1−y dz yz (1 − y − z) dz yz s − mc2 2s − mc2 dz s − mc2 , 1−y × s − mc2 1−y 1−y 1−y 1−y dz (1 − y − z)3 dz (1 − y − z)2 × (1 − y − z) s − mc2 × 2s − mc2 , 1−y 1−y 1−y dz yz(1−y−z) dz yz s − mc2 dy y(1 − y) 3s − 2mc2 1−y 1−y 1−y s2 + 28(y + z) 2s − mc2 + 6 δ s − mc2 dy y(1 − y) 3s − 2mc2 1−y dz (1 − y − z) mc2 2 + s δ s − mc2 3s − 2mc2 s s2 × (1 − y − z) 1 + T 2 + 2T 4 1−y 1−y 1−y 1−y 1−y dz (1 − y − z) 1−y 1−y Table 1 The input parameters in the QCD sum rules, the values in the bracket denote the energy scales μ = 1, 2 GeV and mc, respectively −(0.24 ± 0.01 GeV)3 [21–23,36] (0.8 ± 0.1) q¯q (1 GeV) [21–23,36] m20 q¯q (1 GeV) [21–23,36] m20 s¯s (1 GeV) [21–23,36] (0.8 ± 0.1) GeV2 [21–23,36] (0.33 GeV)4 [21–23,36] (1.275 ± 0.025) GeV [37] (0.095 ± 0.005) GeV [37] dy s2 δ s − mc2 , (18) y f = mc2 = (y+yzz)mc2 , mc2 = y(1m−c2 y) , yyi f dy → 01 dy, z1i−y dz → 01−y dz, where the δ functions δ s − mc2 and δ s − mc2 appear. We differentiate Eq. (9) with respect to T12 , then eliminate the pole residues λs¯s/d¯u , and obtain the QCD sum rules for the ground state masses Ms¯s/d¯u of the C σαβ ⊗ σ αβ C -type scalar hidden-charm tetraquark states, ysy−mmc2 c2 , Ms¯2s/d¯u = − 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: q¯ gs σ Gq (μ) = q¯ gs σ Gq (Q) s¯gs σ Gs (μ) = s¯gs σ Gs (Q) 2 where t = log μ2 , b0 = 331−22πn f , b1 = 1532−4π192n f , b2 = 2857− 50933 n f + 32275 n2f 128π3 , = 213, 296 and 339 MeV for the flavors n f = 5, 4 and 3, respectively [37]. Furthermore, we set mu = md = 0. In the diquark–antidiquark type tetraquark system Qq Q¯ q¯ , 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 the Q¯ -quark serves as another static well potential and combines with the light antiquark q¯ to form a heavy antidiquark D¯ in color triplet; the D and D¯ combine to form a compact tetraquark state [16–18,35,38,39]. For such diquark–antidiquark type tetraquark systems, we suggest an energy-scale formula μ = M X2/Y/Z − (2MQ )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 Qq Q¯ q¯ , the MQ are the effective heavy quark masses. In this article, we choose the updated value Mc = 1.82 GeV [40]. Now we search for the Borel parameters T 2 and continuum threshold parameters ss¯0s/d¯u 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 ss¯0s/d¯u , 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 Cσαβ ⊗ 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 [41]. First of all, we introduce the notations τ = T12 , Dn = − ddτ n, and use the subscripts 1 and 2 to denote the ground state and the first radial excited state of the C σαβ ⊗σ αβ C -type tetraquark states, respectively. Then the QCD sum rules can be written as D2 − M 2j D D3 − M 2j D2 b = DD32⊗⊗DD00−−DD2⊗⊗DD , D3 ⊗ D − D2 ⊗ D2 c = D2 ⊗ D0 − D ⊗ D , D j ⊗ Dk = D j QCD(τ ) Dk The squared masses Mi2 satisfy the following equation: λ21 exp −τ M12 + λ22 exp −τ M22 the subscript QCD denotes the QCD side of the correlation functions s¯s/d¯u (τ ). We differentiate both sides of the QCD sum rules in Eq. (21) with respect to τ and obtain λ21 M12 exp −τ M12 + λ22 M22 exp −τ M22 = D 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 i = 1, 2, j, k = 0, 1, 2, 3. We solve Eq. (25) and obtain two solutions, 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 pole contributions are the pole dominance condition is well satisfied, the criterion 1 is satisfied. The contributions come from the vacuum condensates of dimension 10 D10 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 C σαβ ⊗ σ αβ C -type tetraquark states, Md¯u,1S = 3.82 ± 0.16 GeV, Ms¯s,1S = 3.84 ± 0.16 GeV, λd¯u,1S = (5.20 ± 1.35) × 10−2 GeV5 , λs¯s,1S = (4.87 ± 1.25) × 10−2 GeV5 , Md¯u,2S = 4.38 ± 0.09 GeV, λd¯u,2S = (2.12 ± 0.31) × 10−1 GeV5, Ms¯s,2S = 4.40 ± 0.09 GeV, λs¯s,2S = (2.14 ± 0.33) × 10−1 GeV5 , 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/s¯s 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 Md¯u,2S − Md¯u,1S = 0.56 GeV , Ms¯s,2S − Ms¯s,1S = 0.56 GeV. Z (4430) is assigned to be the first radial excitation of Zc(3900) according to the analogous decays, and the mass differences MZ(4430) − MZc(3900) = 576 MeV and Mψ − MJ/ψ = 589 MeV [42–44]. The energy gaps Md¯u,2S − Md¯u,1S, Ms¯s,2S − Ms¯s,1S, MZ(4430) − MZc(3900) are compatible with each other. The widths Zc(3900) = 46 ± 10 ± 20 MeV [45] and Z(4430) = 172 ± 13+−3374 MeV [46] are not broad, the QCD sum rules for the ground state Zc(3900) alone or without including the first radial excited state Z (4430) work well [35]. If both the ground state Zc(3900) and the first radial excited state Z (4430) are included in, the continuum threshold parameter √s0 = 4.8 ± 0.1 GeV = MZ(4430) + (0.2 ∼ 0.4) GeV, the lower bound of the √s0 − MZ(4430) is about 0.2 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 √s0 − Md¯u/s¯s,2S are about 0.3 GeV, which indicates that the widths of the first radial excited states of the C σαβ ⊗ σ αβ C type 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 √s0 − Md¯u/s¯s,1S = 0.5 ± 0.1 GeV without including the first radial excited states Xd¯u/s¯s,2S explicitly, however, for such large continuum thresholds, the contributions of the Xd¯u/s¯s,2S are already included in due to their large widths. So the QCD sum rules in which only the ground state C σαβ ⊗ 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 MX (3915) = 3918.4 ± 1.9 MeV slightly [37], the X (3915) cannot be a pure C σαβ ⊗σ αβ C -type csc¯s¯ tetraquark state. The predicted mass Ms¯s,2S = 4.40 ± 0.09 GeV overlaps with the experimental value MX (4500) = 4506 ± 11+−1125 MeV slightly [25,26], the X (4500) cannot be a pure C σαβ ⊗σ αβ C -type csc¯s¯ tetraquark state. As the central values of the Ms¯s,1S and Ms¯s,2S differ from the central values of the MX (3915) and MX (4500) significantly, it is difficult to assign the MX (3915) and MX (4500) to be the C σαβ ⊗ σ αβ C -type csc¯s¯ 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. 4 Conclusion In this article, we study the ground states and the first radial excited states of the C σαβ ⊗ 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 C σαβ ⊗ σ αβ C -type hidden-charm tetraquark states. 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Zhi-Gang Wang, Jun-Xia Zhang. Analysis of the tensor–tensor type scalar tetraquark states with QCD sum rules, The European Physical Journal C, 2016, 650, DOI: 10.1140/epjc/s10052-016-4514-x