Tetraquark state candidates: Y(4260), Y(4360), Y(4660), and Zc(4020/4025)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_c(4020/4025)$$\end{document}

In this article, we construct the axialvector-diquark–axialvector-antidiquark type tensor current to interpolate both the vector- and the axialvector-tetraquark states, then calculate the contributions of the vacuum condensates up to dimension 10 in the operator product expansion, and we obtain the QCD sum rules for both the vector- and the axialvector-tetraquark states. The numerical results support assigning the Zc(4020/4025)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_c(4020/4025)$$\end{document} to be the JPC=1+-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J^{PC}=1^{+-}$$\end{document} diquark–antidiquark type tetraquark state, and assigning the Y(4660) to be the JPC=1--\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J^{PC}=1^{--}$$\end{document} diquark–antidiquark type tetraquark state. Furthermore, we take the Y(4260) and Y(4360) as the mixed charmonium–tetraquark states, and we construct the two-quark–tetraquark type tensor currents to study the masses and pole residues. The numerical results support assigning the Y(4260) and Y(4360) to be the mixed charmonium–tetraquark states.

There have been several tentative assignments for the Y (4260), Y (4360), Y (4660), and Z c (4020), such as tetraquark states, molecular states, re-scattering effects, etc., for more literature on the X , Y , Z mesons, one can consult the recent reviews [9,10]. In this article, we will focus on the scenario of tetraquark states based on the QCD sum rules.
The diquarks q T j C q k have five structures in Dirac spinor space, where C = Cγ 5 , C, Cγ μ γ 5 , Cγ μ , and Cσ μν for the scalar, pseudoscalar, vector, axialvector, and tensor diquarks, respectively. The structures Cγ μ and Cσ μν are symmetric, while the structures Cγ 5 , C, and Cγ μ γ 5 are antisymmetric. The attractive interactions of one-gluon exchange favor formation of the diquarks in color antitriplet, flavor antitriplet and spin singlet [11,12], while the favored configurations are the scalar-(Cγ 5 ) and axialvector-(Cγ μ ) diquark states [13][14][15]. The calculations based on the QCD sum rules indicate that the heavy-light scalar-and axialvector-diquark states have almost degenerate masses [13,14]. We can construct the diquark-antidiquark type hidden charm tetraquark states [16][17][18], the Cγ 5 ⊗ γ 5 C type and Cγ μ ⊗ γ μ C type currents couple potentially to the lowest scalar tetraquark states with the masses about 3.82 GeV [19] and 3.85 GeV [20], respectively. If the contribution of an additional P-wave to the mass is about 0.5 GeV, we can construct the vector currents to study the vector-tetraquark states, the estimated masses are about 4.35 GeV, which happens to be the value of the mass of the Y (4360) [20]. In Refs. [21,22] We can also construct the type currents to study the vector-tetraquark states [23,24]. One can consult Ref. [25] for more interpolating currents for the vector-tetraquark states without introducing additional Pwave. In Refs. [23,24], we observe that the C ⊗ γ μ C type and Cγ 5 ⊗ γ 5 γ μ C type tetraquark states have degenerate (or slightly different) masses based on the QCD sum rules, the ground state masses of the vector-tetraquark states with the symbolic quark constituentccqq are about 4.95 GeV, which is much larger than the mass of the Y (4660). In Ref. [26], Albuquerque and Nielsen take the Cγ 5 ⊗ γ 5 γ μ C type current to study the Y (4660) with the QCD sum rules and obtain the value M Y (4660) = 4.65 GeV, which is in excellent agreement with the mass of the Y (4660). Although both in Refs. [23,24] and in Ref. [26], the standard values of the vacuum condensates are taken, in Refs. [23,24], the QCD spectral densities are calculated at the energy scale μ = 1 GeV and the value m c (μ = 1GeV) = 1.35 GeV is taken; while in Ref. [26], the vacuum condensates are taken at the energy scale μ = 1 GeV and the M S mass m c (m c ) = 1.23 GeV is taken, the energy scales of the QCD spectral densities are not specified. In Ref. [27], we suggest the formula μ = M 2 X/Y /Z − (2M c ) 2 with the effective mass M c to determine the energy scales of the QCD spectral densities of the hidden charmed tetraquark states, and we evolve the vacuum condensates and the M S mass to the energy scale μ using the C ⊗ γ μ C type current; we obtain the mass 4.66 or 4.70 GeV for the Y (4660).
In Refs. [28,29], the molecule currents, are chosen to study the Y (4260) and Y (4660) in the QCD sum rules, and it is observed that the Y (4660) can be assigned to be the ψ f 0 (980) molecular state [28], and the Y (4260) cannot be assigned to be the J/ψ f 0 (980) molecular state [29]. Again the parameters are taken as in Refs. [23,24] and in Ref. [26], respectively. In Ref. [30], Dias et al. take the Y (4260) as a mixed charmonium-tetraquark state and choose the current J μ (x), where to study its mass and decay width with the QCD sum rules, and observe that at the mixing angle around θ ≈ (53.0 ± 0.5) • , the mass of the Y (4260) can be reproduced but the decay width is far below the experimental value.
In this article, we take the axialvector-(Cγ μ ) diquark states as the basic constituents [13][14][15], construct the type tensor current without introducing the additional P-wave to interpolate both the vector-and the axialvector-tetraquark states, and study the Y (4260), Y (4360), Y (4660/4630), and Z c (4020/4025) with the QCD sum rules by calculating the operator product expansion up to the vacuum condensates of dimension 10. The tensor current is expected to couple to the vector-tetraquark state with smaller mass compared to the Cγ α ⊗ ∂ μ γ α C, Cγ 5 ⊗ ∂ μ γ 5 C, C ⊗ γ μ C, Cγ 5 ⊗ γ 5 γ μ C type axialvector currents, so as to reproduce the mass of the Y (4260) as the vector-tetraquark state. Furthermore, we study the Z 0 c (4020/4025) as the axialvector-tetraquark state consists of an axialvector-diquark pair, which is expected to have slight larger mass than the Cγ 5 ⊗ γ μ C type tetraquark state [13][14][15]. In Ref. [31], we choose the Cγ 5 ⊗γ μ C type current to study the axialvector-tetraquark states, and we obtain the mass M Z c (3900) = 3.91 +0.11 −0.09 GeV for the Z c (3900) with the assignment J PC = 1 +− .
The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the Y (4260), Y (4360), Y (4660), and Z c (4020) as pure tetraquark states in Sect. 2; in Sect. 3, we derive the QCD sum rules for the masses and pole residues of the Y (4260) and Y (4360) as mixed charmonium-tetraquark states; Sect. 4 is reserved for our conclusion.

QCD sum rules for the Y (4260), Y (4360), Y (4660), and Z c (4020) as pure tetraquark states
In the following, we write down the two-point correlation function μναβ ( p) in the QCD sum rules, where the i, j, k, m, n are color indexes, the C is the charge conjugation matrix. The charged partner η μν (x), couples to the Z + c (4020/4025) potentially. In the isospin limit, the currents η μν (x) and η μν (x) couple to the tetraquark states with degenerate masses.
At the hadronic side, we can insert a complete set of intermediate hadronic states with the same quantum numbers as the current operator η μν (x) into the correlation function μναβ ( p) to obtain the hadronic representation [32][33][34]. After isolating the ground state contributions of the axialvector-and vector-tetraquark states, we get the following results: where the Z denotes the axialvector-tetraquark state the ε μ are the polarization vectors of the vector-and axialvector-tetraquark states with the following property: We can rewrite the correlation function μναβ ( p) into the following form according to Lorentz covariance: Now we project out the components Z ( p 2 ) and Y ( p 2 ) by introducing the operators P where In the following, we carry out the operator product expansion for the correlation function μναβ ( p) up to the vacuum condensates of dimension 10, and project out the components at the QCD side, and we obtain the QCD spectral densities through dispersion relation, where we take into account the contributions of the terms D 0 , D 3 , D 4 , D 5 , D 6 , D 7 , D 8 , and D 10 , The explicit expressions of the QCD spectral densities ρ Z (s) and ρ Y (s) are given in the appendix. The four-quark condensate g 2 s qq 2 comes from the terms qγ μ t a qg s D η G a λτ , s ), respectively, and neglected. We take the truncations n ≤ 10 and k ≤ 1 in a consistent way, the operators of the orders O(α k s ) with k > 1 are discarded. In Tables 1 and 2 Once the analytical expressions of the QCD spectral densities ρ Z (s) and ρ Y (s) are obtained, we can take the quarkhadron duality below the continuum thresholds s 0 and perform a Borel transform with respect to the variable P 2 = − p 2 to obtain the QCD sum rules: We differentiate Eqs. (20) and (21) with respect to 1 T 2 , eliminate the pole residues λ Z and λ Y , and we obtain the QCD sum rules for the masses of the axialvector-and vector-tetraquark states, We take the standard values of the vacuum condensates, qq = −(0.24 ± 0.01 GeV) 3 , qg s σ Gq = m 2 0 qq , In the article, we take the M S mass m c (m c ) = (1.275 ± 0.025) GeV from the Particle Data Group [36], and take into account the energy-scale dependence of the M S mass from the renormalization group equation, 12 25 , where t = log μ 2 2 , = 213, 296 and 339 MeV for the flavors n f = 5, 4, and 3, respectively [36].
In previous work, we described the hidden charm (or bottom) four-quark systems qq QQ by a double-well potential [20,27,[37][38][39][40][41][42]. In the four-quark system qq QQ, the Qquark serves as a static well potential and combines with the light quark q to form a heavy diquark D i q Q in color antitriplet q + Q → D i q Q [20,27,[37][38][39], or combines with the light antiquarkq to form a heavy meson in color singlet (mesonlike state in color octet)q + Q →q Q (q λ a Q) [40][41][42]; thē Q-quark serves as another static well potential and combines with the light antiquarkq to form a heavy antidiquark D iq Q in color tripletq +Q → D iq Q [20,27,[37][38][39], or combines with the light quark q to form a heavy meson in color singlet (meson-like state in color octet) q +Q →Qq (Qλ a q) [40][41][42], where the i is color index, the λ a is Gell-Mann matrix. Then the two heavy quarks Q andQ stabilize the four-quark systems qq QQ, just as in the case of the (μ − e + )(μ + e − ) molecule in QED [43]. In Refs. [20,27,31,[37][38][39][40][41][42], we study the acceptable energy scales of the QCD spectral densities for the hidden charm (bottom) four-quark systems qq QQ with the QCD sum rules in detail for the first time, and suggest the formula Fig. 1 The predicted masses with variations of the Borel parameters T 2 and the energy scales μ to determine the energy scales, where the X , Y , Z denote the four-quark systems, and the M Q denotes the effective heavy quark masses. In Refs. [31,[37][38][39], we obtain the optimal value of the effective mass for the diquarkantidiquark type tetraquark states, M c = 1.8 GeV. Recently, we re-checked the numerical calculations and found that there exists a small error involving the mixed condensates. The Borel windows are modified slightly and the numerical results are also improved slightly after the small error is corrected, the conclusions survive, the optimal value of the effective mass is M c = 1.82 GeV for the diquark-antidiquark type tetraquark states. In this article, we choose the value M c = 1.82 GeV. First of all, we assume that the Y (4260) and Y (4360) are the ground state vector-tetraquark states, the energy gap between the ground states and the first radial excited states is about (0.4-0.6) GeV, just like that of the conventional mesons. In case I, the Y (4260) is the ground state vectortetraquark state; in case II, the Y (4360) is the ground state vector-tetraquark state.
In Fig. 1, we plot the masses of the vector-tetraquark states with variations of the Borel parameters T 2 and energy scales μ for the continuum threshold parameters s 0 Y (4260) = 23 GeV 2 and s 0 Y (4360) = 24 GeV 2 , respectively. According to the formula in Eq. (26), the energy scales μ Y (4260) = 2.2 GeV and μ Y (4360) = 2.4 GeV are the optimal energy scales. From Fig. 1, we can see that the masses decrease monotonously with increase of the energy scales at the value T 2 > 2.7 GeV 2 . However, it is impossible to reproduce the experimental values even if much larger energy scales are taken, the QCD sum rules do not support assigning the Y (4260) and Y (4360) to be the vector-tetraquark states.
In the conventional QCD sum rules [32][33][34], there are two criteria (pole dominance at the phenomenological side and convergence of the operator product expansion) for choosing the Borel parameters T 2 and continuum threshold parameters s 0 . Now we assume the tensor current couples potentially to the vector-tetraquark state Y (4660) and the axialvector-tetraquark state Z c (4020), and search for the Borel parameters T 2 and continuum threshold parameters s 0 . The resulting Borel parameters, continuum threshold parameters, energy scales, pole contributions, and contributions of the vacuum condensates of dimension 10 are shown in Table 1.

QCD sum rules for the Y (4260) and Y (4360) as mixed charmonium-tetraquark states
Now we take the Y (4260) and Y (4360) to be the mixed charmonium-tetraquark states, and we study the masses and pole residues with the QCD sum rules. First, let us write down the interpolating current, where the θ is the mixing angle, the i 3 qq is normalization factor [46]. The calculations can be carried out straightforwardly with the simple replacement in the correlation function μναβ ( p) in Eq. (1). The resulting QCD sum rules are where the ρ Y (s) is the QCD spectral density of the tetraquark component shown in Eq. (18), and In case I, we take the Y (4260) as the ground state mixed charmonium-tetraquark state, and choose the optimal energy scale μ = 2.2 GeV. In case II, we take the Y (4360) as the ground state mixed charmonium-tetraquark state, and choose the optimal energy scale μ = 2.4 GeV. Then we impose the two criteria (pole dominance at the phenomenological side and convergence of the operator product expansion) of the QCD sum rules on the Y (4260) and Y (4360), and search for the mixing angles θ , Borel parameters T 2 , and continuum threshold parameters s 0 . The resulting mixing angles, Borel parameters, continuum threshold parameters, energy scales, pole contributions, and contributions of the vacuum condensates of dimension 10 are shown in Table  2. From the table, we can see that the two criteria of the conventional QCD sum rules can be satisfied, so we expect to make reasonable predictions.
We take into account all uncertainties of the input parameters, and obtain the values of the masses (and pole residues) of the Y (4260) and Y (4360) as mixed charmonium-tetraquark states, which are shown explicitly in Fig. 3, The prediction M Y (4260) = (4.26 ± 0.11) GeV is consistent with the experimental value M Y (4260) = 4259 ± 8 +2 −6 MeV [1], which favors assigning the Y (4260) to be the mixed charmonium-tetraquark state. On the other hand, the prediction M Y (4360) = (4.36 ± 0.10) GeV is consistent with the experimental value M Y (4360) = (4361 ± 9 ± 9) MeV [2,3], which also favors assigning the Y (4360) to be the mixed charmonium-tetraquark state. In the two cases, cos 2 θ ≈ 0.99, the dominant components are the tetraquark states, 2 sin θ cos θ ≈ 0.20 or 0.19, the mixing effects are also considerable. In Ref. [26], the tetraquark component of the Y (4260) is about sin 2 θ ≈ 0.64, the conclusion is quite different from the present work. The difference maybe originate from the interpolating currents and the truncation of the operator product expansion.

Conclusion
In this article, we construct the axialvector-diquark-axialvector-antidiquark type tensor current to interpolate both the vector-and the axialvector-tetraquark states, then we calculate the contributions of the vacuum condensates up to dimension 10 in the operator product expansion, and we obtain the QCD sum rules for both the vector-and the axialvector-tetraquark states. In calculations, we use the formula μ = M 2 X/Y /Z − (2M c ) 2 suggested in our previous work to determine the energy scales of the QCD spectral densities, which works well. The numerical results support assigning the Z c (4020/4025) to be the J PC = 1 +− diquarkantidiquark type tetraquark state, and assigning the Y (4660) to be the J PC = 1 −− diquark-antidiquark type tetraquark state. Furthermore, we take the Y (4260) and Y (4360) as the Fig. 3 The masses with variations of the Borel parameters T 2 for the Y (4260) and Y (4360) as mixed charmonium-tetraquark states mixed charmonium-tetraquark states, introduce the mixing angle and construct the two-quark-tetraquark type tensor currents to study the masses and pole residues. The experimental values of the masses can be reproduced with suitable mixing angles, the QCD sum rules support assigning the Y (4260) and Y (4360) to be the mixed charmonium-tetraquark states.