Analysis of the $Z_c(4020)$, $Z_c(4025)$, $Y(4360)$ and $Y(4660)$ as vector tetraquark states with QCD sum rules

In this article, we distinguish the charge conjugations of the interpolating currents, calculate the contributions of the vacuum condensates up to dimension-10 in the operator product expansion, and study the masses and pole residues of the $J^{PC}=1^{-\pm}$ hidden charmed tetraquark states with the QCD sum rules. We suggest a formula $\mu=\sqrt{M^2_{X/Y/Z}-(2{\mathbb{M}}_c)^2}$ with the effective mass ${\mathbb{M}}_c=1.8\,\rm{GeV}$ to estimate the energy scales of the QCD spectral densities of the hidden charmed tetraquark states, which works very well. The numerical results disfavor assigning the $Z_c(4020)$, $Z_c(4025)$, $Y(4360)$ as the diquark-antidiquark (with the Dirac spinor structure $C-C\gamma_\mu$) type vector tetraquark states, and favor assigning the $Z_c(4020)$, $Z_c(4025)$ as the diquark-antidiquark type $1^{+-}$ tetraquark states. While the masses of the tetraquark states with symbolic quark structures $c\bar{c}s\bar{s}$ and $c\bar{c}(u\bar{u}+d\bar{d})/\sqrt{2}$ favor assigning the $Y(4660)$ as the $1^{--}$ diquark-antidiquark type tetraquark state, more experimental data are still needed to distinguish its quark constituents. There are no candidates for the positive charge conjugation vector tetraquark states, the predictions can be confronted with the experimental data in the future at the BESIII, LHCb and Belle-II.


Introduction
Recently, the BESIII collaboration studied the process e + e − → (D * D * ) ± π ∓ at a center-of-mass energy of 4.26 GeV using a 827 pb −1 data sample obtained with the BESIII detector at the Beijing Electron Positron Collider, and observed a structure Z ± c (4025) near the (D * D * ) ± threshold in the π ∓ recoil mass spectrum [1]. The measured mass and width of the Z ± c (4025) are (4026.3 ± 2.6 ± 3.7) MeV and (24.8 ± 5.6 ± 7.7) MeV, respectively [1]. Later, the BESIII collaboration studied the process e + e − → π + π − h c at center-of-mass energies from 3.90 GeV to 4.42 GeV, and observed a distinct structure Z c (4020) in the π ± h c mass spectrum, the measured mass and width of the Z c (4020) are (4022.9 ± 0.8 ± 2.7) MeV and (7.9 ± 2.7 ± 2.6) MeV, respectively [2]. No significant signal of the Z c (3900) was observed in the π ± h c mass spectrum [2], the Z c (3900) and Z c (4020) maybe have different quantum numbers.
At first sight, the S-wave D * D * systems have the quantum numbers J P C = 0 ++ , 1 +− , 2 ++ , while the S-wave π ± h c systems have the quantum numbers J P C = 1 −− , so the Z c (4025) and Z c (4020) are different particles. On the other hand, it is also possible for the P-wave D * D * (h c π) systems to have the quantum numbers J P C = 1 −− (1 +− ). We cannot exclude the possibility that the Z c (4025) and Z c (4020) are the same particle with the quantum numbers J P C = 1 −− or 1 +− . There have been several tentative assignments of the Z c (4025) and Z c (4020), such as the re-scattering effects [3], molecular states [4], tetraquark states [5], etc. The Z c (4025) and Z c (4020) are charged charmonium-like states, their quark constituents must be ccud or ccdū irrespective of the diquark-antidiquark type or meson-meson type substructures.
In 2013, the BESIII collaboration studied the process e + e − → π + π − J/ψ and observed the Z c (3900) in the π ± J/ψ mass spectrum with the mass (3899.0 ± 3.6 ± 4.9) MeV and width (46 ± 10 ± 20) MeV, respectively [6]. Later the Z c (3900) was confirmed by the Belle and CLEO collaborations [7,8]. Also in 2013, the BESIII collaboration studied the process e + e − → π ∓ DD * ± and observed the Z c (3885) in the (DD * ) ± mass spectrum with the mass (3883.9 ± 1.5 ± 4.2) MeV and width (24.8 ± 3.3 ± 11.0) MeV, respectively [9]. The angular distribution of the πZ c (3885) system favors assigning the Z c (3885) with J P = 1 + [9]. We tentatively identify the Z c (3900) and Z c (3885) as the same particle according to the uncertainties of the masses and widths [10], one can consult Ref. [10] for more articles on the Z c (3900). The possible quantum numbers of the Z c (3900) or Z c (3885) are J P C = 1 +− . There is a faint possibility that the Z c (3900) and Z c (4020) are the same axial-vector meson with J P C = 1 +− according to the masses.
In 2007, the Belle collaboration measured the cross section for the process e + e − → π + π − ψ ′ between threshold and √ s = 5.5 GeV using a 673 fb −1 data sample collected with the Belle detector at KEKB, and observed two structures Y (4360) and Y (4660) in the π + π − ψ ′ invariant mass distributions at (4361 ± 9 ± 9) MeV with a width of (74 ± 15 ± 10) MeV and (4664 ± 11 ± 5) MeV with a width of (48 ± 15 ± 3) MeV, respectively [11]. The quantum numbers of the Y (4360) and Y (4660) are J P C = 1 −− , which are unambiguously listed in the Review of Particle Physics now [12]. In 2008, the Belle collaboration studied the exclusive process e + e − → Λ + c Λ − c and observed a clear peak Y (4630) in the Λ + c Λ − c invariant mass distribution just above the Λ + c Λ − c threshold, and determined the mass and width to be 4634 +8
In this article, we study the diquark-antidiquark type vector tetraquark states in details with the QCD sum rules, and explore possible assignments of the Z c (4020), Z c (4025), Y (4360) and Y (4660) in the tetraquark scenario. In Ref. [10], we extend our previous works on the axial-vector tetraquark states [22], distinguish the charge conjugations of the interpolating currents, calculate the contributions of the vacuum condensates up to dimension-10 and discard the perturbative corrections in the operator product expansion, study the Cγ 5 − Cγ µ type axial-vector hidden charmed tetraquark states with the QCD sum rules. We explore the energy scale dependence of the charmed tetraquark states in details for the first time, and tentatively assign the X(3872) and Z c (3900) (or Z c (3885)) as the J P C = 1 ++ and 1 +− tetraquark states, respectively [10]. In calculations, we observe that the tetraquark masses decrease monotonously with increase of the energy scales, the energy scale µ = 1.5 GeV is the lowest energy scale to reproduce the experimental values of the masses of the X(3872) and Z c (3900), and serves as an acceptable energy scale for the charmed mesons in the QCD sum rules [10].
In Ref. [23], we study the Cγ µ − C and Cγ µ γ 5 − Cγ 5 type tetraquark states with the QCD sum rules by carrying out the operator product expansion to the vacuum condensates up to dimension-10 and setting the energy scale to be µ = 1 GeV. In Refs. [5,18,24], the authors carry out the operator product expansion to the vacuum condensates up to dimension-8 to study the vector tetraquark states with the QCD sum rules, but do not show the energy scales or do not specify the energy scales at which the QCD spectral densities are calculated. In Refs. [5,18,23,24], some higher dimension vacuum condensates involving the gluon condensate, mixed condensate and four-quark condensate are neglected, which maybe impair the predictive ability. The terms associate with 1 T 2 , 1 T 4 , 1 T 6 in the QCD spectral densities manifest themselves at small values of the Borel parameter T 2 , we have to choose large values of the T 2 to warrant convergence of the operator product expansion and appearance of the Borel platforms. In the Borel windows, the higher dimension vacuum condensates play a less important role. In summary, the higher dimension vacuum condensates play an important role in determining the Borel windows therefore the ground state masses and pole residues, so we should take them into account consistently.
In this article, we extend our previous works [10] to study the vector tetraquark states, distinguish the charge conjugations of the interpolating currents, calculate the contributions of the vacuum condensates up to dimension-10 and discard the perturbative corrections, study the masses and pole residues of the C − Cγ µ type vector hidden charmed tetraquark states with the QCD sum rules. Furthermore, we explore the energy scale dependence in details so as to obtain some useful formulae, and make tentative assignments of the Z c (4020), Z c (4025), Y (4360) and Y (4660) as the J P C = 1 −+ or 1 −− tetraquark states. The scalar and axial-vector heavy-light diquark states have almost degenerate masses from the QCD sum rules [25], the Cγ µ − C and Cγ µ γ 5 − Cγ 5 type tetraquark states have degenerate (or slightly different) masses [23], as the pseudoscalar and vector heavy-light diquark states have slightly different masses.
The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the vector tetraquark states in section 2; in section 3, we present the numerical results and discussions; section 4 is reserved for our conclusion.

QCD sum rules for the vector tetraquark states
In the following, we write down the two-point correlation functions Π µν (p) in the QCD sum rules, where J µ (x) = J 1 µ (x), J 2 µ (x), J 3 µ (x), t = ±1, the i, j, k, m, n are color indexes, the C is the charge conjugation matrix. Under charge conjugation transform C, the currents J µ (x) have the properties, which originate from the charge conjugation properties of the pseudoscalar and axial-vector diquark states, We choose the neutral currents J 1 µ (x) and J 2 µ (x) with t = − to interpolate the J P C = 1 −− diquark-antidiquark type tetraquark states Y (4660) and Y (4360), respectively. There are two structures in π + π − invariant mass distributions at about 0.6 GeV and 1.0 GeV in the π + π − ψ ′ mass spectrum, which maybe due to the scalar mesons f 0 (600) and f 0 (980), respectively [11]. In the two-quark scenario, f 0 (600) = (uu + dd)/ √ 2 and f 0 = ss in the ideal mixing limit, while in the tetraquark scenario, the f 0 (600) and f 0 (980) have the symbolic quark structures udūd and (usūs + dsds)/ √ 2, respectively. The Y (4660) couples to the current J 1 µ (x) while the Y (4360) couples to the current J 2 µ (x). However, we cannot exclude the possibility that the Y (4660) has the symbolic quark structure cc(uū + dd)/ √ 2, in that case the decay Y (4660) → f 0 (600)ψ ′ is Okubo-Zweig-Iizuka (OZI) allowed. We choose the charged vector current J 3 µ (x) with t = ± to interpolate the Z c (4020) and Z c (4025), the results for the scalar and tensor currents will be presented elsewhere. At present time, we cannot exclude the possibility that the Z c (4020) and Z c (4025) are the same vector particle.
We can insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators J µ (x) into the correlation functions Π µν (p) to obtain the hadronic representation [26,27]. After isolating the ground state contributions of the vector tetraquark states, we get the following results, where the pole residues λ Y /Z are defined by the ε µ are the polarization vectors of the vector tetraquark states Z c (4020), In the following, we take the current J µ (x) = J 1 µ (x) as an example and briefly outline the operator product expansion for the correlation functions Π µν (p) in perturbative QCD. We contract the c and s quark fields in the correlation functions Π µν (p) with Wick theorem, obtain the results: where the ∓ correspond to C = ± respectively, the S ij (x) and C ij (x) are the full s 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 [27], then compute the integrals both in the coordinate and momentum spaces, and obtain the correlation functions Π µν (p) therefore the spectral densities at the level of quark-gluon degrees of freedom. In Eq.(10), we retain the terms s j σ µν s i and s j γ µ s i originate from the Fierz re-arrangement of the s isj to absorb the gluons emitted from the heavy quark lines to form s j g s G a αβ t a mn σ µν s i and s j γ µ s i g s D ν G a αβ t a mn so as to extract the mixed condensate and four-quark condensates sg s σGs and g 2 s ss 2 , respectively. One can consult Ref. [10] for some technical details in the operator product expansion.
Once analytical results are obtained, we can take the quark-hadron duality below the continuum threshold s 0 and perform Borel transform with respect to the variable P 2 = −p 2 to obtain the following QCD sum rules: where the 0, 3, 4, 5, 6, 7, 8, 10 denote the dimensions of the vacuum condensates, the explicit expressions of the spectral densities ρ i (s) are presented in the Appendix. In this article, we carry out the operator product expansion to the vacuum condensates up to dimension-10 and discard the perturbative corrections, and assume vacuum saturation for the higher dimension vacuum condensates. The higher dimension vacuum condensates are always factorized to lower condensates with vacuum saturation in the QCD sum rules, factorization works well in large N c limit. In reality, N c = 3, some (not much) ambiguities maybe come from the vacuum saturation assumption. The condensates αs π GG , ss αs π GG , ss 2 αs π GG , sg s σGs 2 and g 2 s ss 2 are the vacuum expectations of the operators of the order O(α s ). The four-quark condensate g 2 s ss 2 comes from the terms sγ µ t a sg s D η G a λτ , s j D † µ D † ν D † α s i and s j D µ D ν D α s i , rather than comes from the perturbative corrections of ss 2 (see Ref. [10] for the technical details). The condensates g 3 s GGG , αsGG π 2 , αsGG π sg s σGs have the dimensions 6, 8, 9 respectively, but they are the vacuum expectations of the operators of the order O(α s ) respectively, and discarded. 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. Furthermore, the numerical values of the condensates g 3 s GGG , αsGG π 2 , αsGG π sg s σGs are very small, and they are neglected safely. Differentiate Eq.(12) with respect to 1 T 2 , then eliminate the pole residues λ Y /Z , we obtain the QCD sum rules for the masses of the vector tetraquark states, We can obtain the QCD sum rules for the vector tetraquark states ccud and cc(uū + dd)/ √ 2 with the simple replacements, the QCD sum rules for the ccud and cc(uū + dd)/ √ 2 degenerate in the isospin limit.
In Ref. [10], we observe that the energy scale µ = (1.1 − 1.6) GeV is an acceptable energy scale of the QCD spectral densities in the QCD sum rules for the hidden and open charmed mesons, as it can reproduce the experimental values M D = 1.87 GeV and M J/ψ = 3.1 GeV with suitable Borel parameters. However, such energy scale and truncation in the operator product expansion cannot reproduce the experimental values of the decay constants f D and f J/ψ . In calculation, we observe that the masses of the axial-vector tetraquark states decrease monotonously with increase of the energy scales of the QCD spectral densities, the energy scale µ = 1.5 GeV is the lowest energy scale to reproduce the experimental values of the masses of the X(3872) and Z c (3900) (or Z c (3885)), and serves as an acceptable energy scale (not the universal energy scale) for the tetraquark states [10]. On the other hand, it is hard to obtain the true values of the pole residues λ X/Y /Z of the tetraquark states, so we focus on the masses to study the tetraquark states, and the predictions of the pole residues maybe not as robust. If the Z c (4020) and Z c (4025) are the vector tetraquark states, we can choose the threshold parameters √ s 0 = (4.3 − 4.8) GeV and energy scales µ = (1.5 − 3.0) GeV tentatively, and search for the ideal parameters, such as the threshold parameters, energy scales and Borel parameters.
In Fig.1, the masses of the vector ccud tetraquark states are plotted with variations of the Borel parameters T 2 , energy scales µ, and continuum threshold parameters √ s 0 . From the figure, we can see that the masses decrease monotonously with increase of the energy scales, the parameters √ s 0 ≤ 4.5 GeV and µ ≤ 1.5 GeV can be excluded, as the predicted masses M Z ≫ (or >) √ s 0 = 4.5 GeV for the values of the Borel parameters at a large interval. We have to choose larger threshold parameters or (and) energy scales, the resulting masses are larger than 4.3 GeV for the parameters √ s 0 ≥ 4.5 GeV and µ = 3.0 GeV. The predictions based on the QCD sum rules disfavor assigning the Z c (4020) and Z c (4025) as the diquark-antidiquark type vector tetraquark states. We cannot satisfy the relation √ s 0 = M Z + 0.5 GeV with reasonable M Z compared to the experimental data. The BESIII collaboration observed the Z ± c (4025) and Z ± c (4020) in the following processes [1,2], where we present the possible quantum numbers J P C of the (D * D * ) ± and (h c π) ± systems in the brackets. If the Z ± c (4025) and Z ± c (4020) are the same particle, the quantum numbers are J P C = 1 −− , 0 ++ , 1 +− , 2 ++ . On the other hand, the Z ± c (4025)π ∓ and Z ± c (4020)π ∓ systems have the quantum numbers J P C = 1 −− , then the survived quantum numbers of the Z ± c (4025) and Z ± c (4020) are J P C = 1 −− , 1 +− and 2 ++ . The predictions based on the QCD sum rules reduce the possible quantum numbers of the Z c (4025) and Z c (4020) to J P C = 1 +− and 2 ++ .
The strong decays take place through relative D-wave, and are kinematically suppressed in the phase-space. The 2 ++ assignment is disfavored, but not excluded. In the following, we list out the possible strong decays of the Z ± c (4025) and Z ± c (3900) in the case of the J P C = 1 +− assignment.
where the (ππ) P denotes the P-wave ππ systems have the same quantum numbers of the ρ. We take the Z c (4025) and Z c (4020) as the same particle in the J P C = 1 +− assignment, and will denote them as Z c (4025). In Ref. [10], we observe that the Z c (3900) couples to the axial-vector current J µ 1 +− . Now we perform Fierz re-arrangement both in the color and Dirac-spinor spaces and obtain the following result, the components such asciγ 5 cdγ µ u,cγ µ cdiγ 5 u, etc couple to the meson-meson pairs, the strong decays are OZI super-allowed, we take the decays to the (ππ) ± P final states as OZI super-allowed according to the decays ρ → ππ. The BESIII collaboration observed no evidence of the Z c (3900) in the process e + e − → π + π − h c at center-of-mass energies from 3.90 GeV to 4.42 GeV [2]. We expect to observe the Z ± c (3900) in the h c (1P)π ± final states when a large amount of events are accumulated. The Z c (4025) and Z c (3900) have the same quantum numbers and analogous strong decays but different masses and quark configurations. Now we take a short digression to discuss the interpolating currents consist of four quarks. The diquark-antidiquark type current with special quantum numbers couples to a special tetraquark state, while the current can be re-arranged both in the color and Dirac-spinor spaces, and changed to a current as a special superposition of color singlet-singlet type currents. The color singletsinglet type currents couple 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 (meson-meson pairs) are OZI super-allowed, the kinematically allowed decays take place easily.
We can also search for the neutral partner Z 0 c (4025)(1 +− ) in the following strong and electromagnetic decays, where the (πππ) P denotes the P-wave πππ systems with the same quantum numbers of the ω.
The strong decays cannot take place. The 0 ++ assignment is excluded. Now, we explore the possibility of assigning the Y (4360) and Y (4660) as the diquark-antidiquark type vector tetraquark states. We consult the often used energy scale µ = m 2 D − m 2 c ≈ 1 GeV in the QCD sum rules for the D mesons, and suggest a formula to estimate the energy scales of the QCD spectral densities in the QCD sum rules for the hidden charmed tetraquark states, where the effective mass of the c-quark M c = 1.8 GeV. The heavy tetraquark system could be described by a double-well potential with two light quarks q ′q lying in the two wells respectively. In the heavy quark limit, the c (and b) quark can be taken as a static well potential, which binds the light quark q to form a diquark in the color antitriplet channel. The heavy tetraquark state are characterized by the effective heavy quark masses M Q (or constituent quark masses) and the virtuality V = M 2 X/Y /Z − (2M Q ) 2 (or bound energy not as robust). It is natural to take the energy scale µ = V . The energy scales are estimated to be µ = 1.5 GeV for the X(3872) and Z c (3900) [10], µ = 3.0 GeV for the Y (4660), and µ = 2.5 GeV for the Y (4360). The formula also works well for the scalar hidden charmed (and double charmed) tetraquark states, and we can use the formula to improve the predictions [30]. Furthermore, we study the possible applications in the QCD sum rules for the molecular states [31]. From Fig.1, we can see that the energy scales µ = 2.5 GeV and 3.0 GeV lead to slightly different masses for the threshold parameters √ s 0 = 4.7 GeV or larger than 4.7 GeV. In this article, we set the energy scale µ = 3.0 GeV to study the vector tetraquark states.
In Fig.2, the contributions of the pole terms are plotted with variations of the threshold parameters √ s 0 and Borel parameters T 2 at the energy scale µ = 3.0 GeV. From the figure, we can see that the values √ s 0 ≤ 4.8 GeV are too small to satisfy the pole dominance condition and result in reasonable Borel windows. In Fig.3, the contributions of different terms in the operator product expansion are plotted with variations of the Borel parameters T 2 for the threshold parameters √ s 0 = 5.1 GeV and 5.0 GeV in the channels ccss and cc(uū + dd)/ √ 2 respectively at the energy scale µ = 3.0 GeV. From the figure, we can see that the contributions of the vacuum condensates of dimensions-0, 5, 6 change quickly with variations of the Borel parameters at the region T 2 < 3.2 GeV 2 , which does not warrant platforms for the masses. In this article, the value T 2 ≥ 3.2 GeV 2 is chosen tentatively, in that case the convergent behavior in the operator product expansion is very good, as the perturbative terms make the main contributions. The Borel parameters, continuum threshold parameters and the pole contributions are shown explicitly in Table 1. The two criteria (pole dominance and convergence of the operator product expansion) of the QCD sum rules are fully satisfied, so we expect to make reasonable predictions. While in the QCD sum rules for the light tetraquark states, the two criteria are difficult to satisfy [32].
Taking into account all uncertainties of the input parameters, finally we obtain the values of the masses and pole residues of the vector tetraquark states, which are shown explicitly in Figs.4-5 and Table 1. The prediction M ccss(1 −− ) = 4.70 +0.14 −0.10 GeV is consistent with the experimental data M Y (4660) = (4664 ± 11 ± 5) MeV within uncertainties [12], and the prediction M ccud(1 −− ) or       [12]. The present predictions favor assigning the Y (4660) as the J P C = 1 −− diquark-antidiquark type tetraquark state, the masses M ccss and M cc(uū+dd)/ √ 2 are both consistent with the experimental data M Y (4660) within uncertainties. By precisely measuring the π + π − mass spectrum in the final state π + π − ψ ′ , we can distinguish the f 0 (600) and f 0 (980), therefore disentangle the quark constituents of the Y (4660). On the other hand, we can also take the Y (4360) as the cc and cc(uū + dd)/ √ 2 mixed state, as the cc component can reduce the mass so as to reproduce the experimental value at about 4.4 GeV.
From Table 1, we can also see that there is energy gap about (70 − 90) MeV between the central values of the C = + and C = − vector tetraquark states, which can be confronted with the experimental data in the future. In Ref. [10], we observe that there is a small energy gap less than 40 MeV between the central values of the C = + and C = − axial-vector tetraquark states, which is consistent with the value 10 MeV from the constituent diquark model [33].
In this article, we construct the C − Cγ µ type diquark-antidiquark currents to interpolate the vector tetraquark states. The scalar and axial-vector heavy-light diquark states have almost degenerate masses from the QCD sum rules [25], the Cγ µ − C and Cγ µ γ 5 − Cγ 5 type tetraquark states have degenerate (or slightly different) masses [23]. On the other hand, we can also construct the Cγ α −∂ µ −Cγ α and Cγ 5 −∂ µ −Cγ 5 type diquark-antidiquark currents to interpolate the vector  tetraquark states, the Cγ α − Cγ α and Cγ 5 − Cγ 5 type diquark-antidiquark currents couple to the scalar tetraquark states with the masses about 3.85 GeV [29]. If the contribution of an additional P-wave to the mass is about 0.5 GeV, the masses of the vector tetraquark states couple to the Cγ α −∂ µ −Cγ α and Cγ 5 −∂ µ −Cγ 5 type interpolating currents are about 4.35 GeV, which happens to be the value of the mass M Y (4360) . In Refs. [19,34], Zhang and Huang take the Cγ 5 −∂ µ −Cγ 5 type diquark-antidiquark currents to study the Y (4360) and Y (4660) with the symbolic quark structures cc(uū + dd)/ J µ 1 −− ,ss = ǫ ijk ǫ imn √ 2 s j Cc ksm γ µ Cc n − s j Cγ µ c ksm Cc n , J µ 1 −+ ,ss = ǫ ijk ǫ imn √ 2 s j Cc ksm γ µ Cc n + s j Cγ µ c ksm Cc n , where the subscripts 1 −± anddu (ss) are added to show the J P C and light quark constituents, respectively. Then we obtain the OZI super-allowed decays by taking into account the couplings to the meson-meson pairs, Y (cucd)(1 −− ) → J/ψa 0 (980) , χ c0 ρ , χ c1 ρ , J/ψπ , J/ψa 1 (1260) , DD 1 (2420) , D * D 1 (2420) , Y (cucd)(1 −+ ) → η c π , η c a 1 (1260) , χ c1 π , h c ρ , J/ψb 1 (1235) , D * D 0 (2400) , D * D 1 (2420) , where the (ππ) S and (KK) S denote the S-wave ππ and KK pairs, respectively. The mass spectrum of the light scalar mesons is well understood in terms of diquark-antidiquark bound states, while the strong decays into two pseudoscalar mesons based on the quark rearrangement mechanism cannot lead to a satisfactory description of the experimental data. In Ref. [35], 't Hooft et al introduce the instanton induced effective six-fermion Lagrangian, and illustrate that such Lagrangian leads to the tetraquark-qq mixing, therefore provides an additional amplitude which brings the strong decays of the light scalar mesons in good agreements with the experimental data. In the present work, we discuss the OZI super-allowed strong decays of the tetraquark states based on the quark rearrangement mechanism or fall-apart mechanism, as there is no instanton induced effective six-fermion Lagrangian in the hidden-charm systems to describe the tetraquark-qq mixing beyond the usual QCD interactions. The present predictions can be confronted with the experimental data in the futures the BESIII, LHCb and Belle-II.

Conclusion
In this article, we study the Z c (4020), Z c (4025), Y (4360) and Y (4660) as the diquark-antidiquark type vector tetraquark states in details with the QCD sum rules. We distinguish the charge conjugations of the interpolating currents, calculate the contributions of the vacuum condensates up to dimension-10 and discard the perturbative corrections in the operator product expansion, and take into account the higher dimensional vacuum condensates consistently, as they play an important role in determining the Borel windows. Then we suggest a formula µ = M 2 X/Y /Z − (2M c ) 2 to estimate the energy scales of the QCD spectral densities of the hidden charmed tetraquark states, and study the masses and pole residues of the J P C = 1 −± tetraquark states in details. The formula µ = M 2 X/Y /Z − (2M c ) 2 works well. The masses of the ccud (1 −± ) tetraquark states disfavor assigning the Z c (4020), Z c (4025) and Y (4360) as the C − Cγ µ type vector tetraquark states, and favor assigning the Z c (4020), Z c (4025) as the diquark-antidiquark type 1 +− tetraquark states. While the masses of the ccss and cc(uū + dd)/ √ 2 tetraquark states favor assigning the Y (4660) as the C − Cγ µ type 1 −− tetraquark state, more experimental data are still needed to distinguish the quark constituents. There are no candidates for the C = + vector tetraquark states, the predictions can be confronted with the experimental data in the futures at the BESIII, LHCb and Belle-II. The pole residues can be taken as basic input parameters to study relevant processes of the vector tetraquark states with the three-point QCD sum rules.