Vector tetraquark state candidates: $Y(4260/4220)$, $Y(4360/4320)$, $Y(4390)$ and $Y(4660/4630)$

In this article, we construct the $C \otimes \gamma_\mu C$ and $C\gamma_5 \otimes \gamma_5\gamma_\mu C$ type currents to interpolate the vector tetraquark states, then carry out the operator product expansion up to the vacuum condensates of dimension-10 in a consistent way, and obtain four QCD sum rules. In calculations, we use the formula $\mu=\sqrt{M^2_{Y}-(2{\mathbb{M}}_c)^2}$ to determine the optimal energy scales of the QCD spectral densities, moreover, we take the experimental values of the masses of the $Y(4260/4220)$, $Y(4360/4320)$, $Y(4390)$ and $Y(4660/4630)$ as input parameters and fit the pole residues to reproduce the correlation functions at the QCD side. The numerical results support assigning the $Y(4660/4630)$ to be the $C \otimes \gamma_\mu C$ type vector tetraquark state $c\bar{c}s\bar{s}$, assigning the $Y(4360/4320)$ to be $C\gamma_5 \otimes \gamma_5\gamma_\mu C$ type vector tetraquark state $c\bar{c}q\bar{q}$, and disfavor assigning the $Y(4260/4220)$ and $Y(4390)$ to be the pure vector tetraquark states.


Introduction
In 2005, the BaBar collaboration studied the initial-state radiation process e + e − → γ ISR π + π − J/ψ and observed the Y (4260) in the π + π − J/ψ invariant-mass spectrum [1]. Later the Y (4260) was confirmed by the Belle and CLEO collaborations [2,3], the Belle collaboration also observed an evidence for a very broad structure Y (4008) in the π + π − J/ψ mass spectrum. In 2014, the BES collaboration searched for the production of e + e − → ωχ cJ with J = 0, 1, 2, and observed a resonance in the ωχ c0 cross section, the measured mass and width of the resonance are 4230 ± 8 ± 6 MeV and 38 ± 12 ± 2 MeV, respectively [4].
In 2016, the BES collaboration measured the cross sections of the process e + e − → π + π − h c , and observed two structures, the Y (4220) has a mass of 4218.4 ± 4.0 ± 0.9 MeV and a width of 66.0 ± 9.0 ± 0.4 MeV respectively, and the Y (4390) has a mass of 4391.6 ± 6.3 ± 1.0 MeV and a width of 139.5 ± 16.1 ± 0.6 MeV respectively [5].
Also in 2016, the BES collaboration precisely measured the cross section of the process e + e − → π + π − J/ψ at center-of-mass energies from 3.77 to 4.60 GeV and observed two resonant structures. The first resonance has a mass of 4222.0 ± 3.1 ± 1.4 MeV and a width of 44.1 ± 4.3 ± 2.0 MeV, the second one has a mass of 4320.0 ± 10.4 ± 7.0 MeV and a width of 101.4 +25. 3 −19.7 ± 10.2 MeV [6]. The first resonance agrees with the Y (4260) while the second resonance agrees with the Y (4360) according to the uncertainties, the Y (4008) resonance previously observed by the Belle experiment is not confirmed [2].
We can construct the diquark-antidiquark type currents to interpolate the scalar hidden-charm tetraquark states [47,48,49] or construct the diquarkantidiquark type currents to interpolate the vector hidden-charm tetraquark states [17,18,19,20,21,22]. One can consult Ref. [50] for more interpolating currents for the vector tetraquark states without introducing an additional P-wave between the diquark and antidiquark. We can also choose the mixed charmonium-tetraquark currents to study the vector mesons Y (4260) and Y (4360) [22,51].
In Table 1, we present the existing predictions of the masses of the vector tetraquark states based on the QCD sum rules. In Refs. [18,19,20,50,51], the vacuum condensates are taken at the energy scale µ = 1 GeV while the M S mass m c (m c ) is taken at the energy scale µ = m c (m c ), the energy scales of the QCD spectral densities are not specified. In Refs. [21,22,49], we take the formula µ = M 2 X/Y /Z − (2M c ) 2 with the effective charmed quark mass M c to determine the energy scales of the QCD spectral densities of the hidden-charm tetraquark states, and evolve the vacuum condensates and the M S mass m c (m c ) to the optimal energy scales µ to extract the tetraquark masses; the hidden-bottom tetraquark states can be studied analogously [52].
In this article, we assume that the Y (4260/4220), Y (4360/4320), Y (4390) and Y (4660/4630) are vector tetraquark states, and restudy the C ⊗ γ µ C and Cγ 5 ⊗ γ 5 γ µ C type vector tetraquark states with the QCD sum rules in details by taking into account the vacuum condensates up to dimension 10 in a consistent way in the operator product expansion, and use the energy scale formula µ = M 2 X/Y /Z − (2M c ) 2 to determine the optimal energy scales of the QCD spectral densities. In the QCD sum rules for the tetraquark states, 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. The higher dimensional vacuum condensates play an important role in determining the Borel windows therefore the ground state masses and pole residues, though they maybe play a less important role in the Borel windows. We should take them into account consistently.
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, , 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, In the non-relativistic diquark-antidiquark model, one often introduces an explicit P-wave between the diquark and antidiquark in the ground state Cγ 5 ⊗ γ 5 C type or Cγ 5 ⊗ γ µ C type or Cγ µ ⊗ γ ν C type tetraquark state [12,13,14,15]. In this article, we choose the C ⊗ γ µ C type and Cγ 5 ⊗ γ 5 γ µ C type vector currents to interpolate the vector tetraquark states, the net effects of the relative P-waves are embodied in the underlined γ 5 in the Cγ 5 γ 5 ⊗ γ µ C type and Cγ 5 ⊗ γ 5 γ µ C type currents or in the underlined γ α in the Cγ α γ α ⊗ γ µ C type currents.
The tetraquark states are spatial extended objects, not point-like objects [16], in the QCD sum rules [17,18,19,20,21,22] and the effective Hamiltonian approach [12,13,14,15], the finite size effects are neglected, which leads to some uncertainties, while in the potential models, an explicit spatial extended potential between the diquark and antiquark is introduced [53]. Now we perform Fierz re-arrangement to the vector currents J 1 µ (x) and J 3 µ (x) both in the color and Dirac-spinor spaces, and obtain the following results, the Fierz re-arrangement of the J 2 µ (x) and J 4 µ (x) can be obtained analogously. The diquarkantidiquark type current with special quantum numbers couples potentially to a special tetraquark state, while the current can be re-arranged to a current as a special superposition of color singletsinglet type currents, which couple potentially to the meson-meson pairs or molecular states. 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.
According to the current-meson couplings, where the e µ are the polarization vectors of the J/ψ and h c , we can obtain the conclusion, if the Y (4260/4220), Y (4360/4320), Y (4390) and Y (4660/4630) are the C ⊗ γ µ C type or Cγ 5 ⊗ γ 5 γ µ C type tetraquark states, there are no heavy quark spin-flips in the decays to the final states J/ψ and h c , which is consistent with the experimental data [4,5,6,8,9,10]. At the hadronic side, 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 [54,55]. After isolating the ground state contributions of the vector tetraquark states which are supposed to be the Y (4260/4220), Y (4360/4320), Y (4390) and Y (4660/4630), we get the following results, where the pole residues λ Y are defined by the ε µ are the polarization vectors of the vector tetraquark states Y (4260/4220), Y (4360/4320), Y (4390) and Y (4660/4630), etc.
In the following, we take the currents J 1 µ (x) and J 3 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 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 [55], 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.(15), we retain the terms s j σ µν s i originate from the Fierz re-arrangement of the s isj to absorb the gluons emitted from other quark lines to extract the mixed condensate sg s σGs [52].
Once analytical expressions of the QCD spectral densities are obtained, we can take the quarkhadron duality below the continuum thresholds s 0 and perform Borel transform with respect to the variable P 2 = −p 2 to obtain the following four QCD sum rules: where ρ(s) = ρ 1 (s), ρ 2 (s), ρ 3 (s) and ρ 4 (x), the subscripts i = 0, 3, 4, 5, 6, 7, 8, 10 denote the dimensions of the vacuum condensates, the explicit expressions of the QCD 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 dimensional vacuum condensates. The higher dimensional vacuum condensates are always factorized to lower dimensional vacuum 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 higher dimensional vacuum condensates have not been well studied yet.
In the present QCD sum rules, we search for the ideal Borel parameters T 2 and continuum threshold parameters s 0 to obey the following four criteria: 1. Pole dominance at the phenomenological side; 2. Convergence of the operator product expansion; 3. Appearance of the Borel platforms; 4. Satisfying the energy scale formula, using try and error.
In Refs. [21,49,52], we study the energy scale dependence of the QCD sum rules in details and suggest an energy scale formula µ = M 2 X/Y /Z − (2M Q ) 2 with the effective heavy quark mass M Q to determine the energy scales of the QCD spectral densities of the hidden-charm and hiddenbottom tetraquark states, which also works well for the hidden-charm pentaquark states [58]. In this article, we take the updated value M c = 1.82 GeV [22].
The resulting Borel parameters or Borel windows T 2 , continuum threshold parameters s 0 , ideal energy scales of the QCD spectral densities, pole contributions of the ground state tetraquark states, and contributions of the vacuum condensates of dimension 7, 8 and 10 in the operator product expansion are shown explicitly in Table 2. From the table, we can see that the first two criteria of the QCD sum rules are 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 vector tetraquark states, which are shown explicitly in Figs.1-2 and Table 3. From Figs.1-2, we can see that there appear platforms in the Borel windows, the criterion 3 is satisfied. From Table 3, we can see that the criterion 4 is also satisfied. Now the four criteria of the QCD sum rules are all satisfied, and we expect to make reliable predictions.
In Fig.1, we also present the experimental values of the masses of the Y (4260/4220), Y (4360/4320), Y (4390) and Y (4660/4630) [5,57]. From the figure, we can see that the experimental values of the masses M Y (4660/4630) and M Y (4360/4320) can be well reproduced, the present predications support assigning the Y (4660/4630) to be the C ⊗ γ µ C type tetraquark state ccss, and assigning the Y (4360/4320) to be the Cγ 5 ⊗ γ 5 γ µ C type tetraquark state ccqq. The mass M Y (4660/4630) lies just below the upper bound of the predicted mass of the C ⊗ γ µ C type vector tetraquark state ccqq, which disfavors assigning the Y (4660/4630) to be the C ⊗ γ µ C type vector tetraquark state ccqq, however, such an assignment is not excluded.
If the relative P-waves between the diquark and antidiquark cost a universal energy for all the tetraquark states, then the C ⊗ γ µ C type tetraquark states have larger masses than the corresponding Cγ 5 ⊗ γ 5 γ µ C type tetraquark states, as C ⊗ γ µ C = Cγ 5 γ 5 ⊗ γ µ C ⊕ Cγ α γ α ⊗ γ µ C and Cγ 5 ⊗ γ 5 γ µ C = Cγ 5 ⊗ γ 5 γ µ C, the Cγ µ diquark states have slightly larger masses than the corresponding Cγ 5 diquark states from the QCD sum rules [43,44]. In other words, the bad diquarks have slightly larger masses than the good diquarks, those effects are also accounted for in the effective Hamiltonian [13,15].
On the other hand, the Y (4660) can be assigned to be the ψ ′ f 0 (980) hadro-charmonium or molecular state based on fitting the mass distribution of the process e + e − → ψ ′ π + π − [28] or the calculations of the QCD sum rules [29]. More experimental data are still needed to assign the Y (4660) unambiguously.
In this article, we recalculate the QCD sides of the correlation functions for the C ⊗ γ µ C type currents by taking into account the neglected terms due to the approximations involving the higher dimensional vacuum condensates in Ref. [21] and correct a small error in numerical calculations, the present predictions 4.66 ± 0.09/4.59 ± 0.08 GeV are more robust than the values 4.70 +0.14 −0.10 /4.66 +0.17 −0.10 GeV obtained in Ref. [21]. In Fig.3, we plot the mass and pole residue of the C ⊗ γ µ C type vector tetraquark state ccss with variation of the Borel parameter T 2 for truncations of the operator product expansion, D = 6, 7, 8 and 10. From the figure, we can see that the higher dimensional vacuum condensates play an important role in determining the Borel platforms.
The ground state Cγ 5 ⊗ γ 5 C type and Cγ µ ⊗ γ µ C type hidden-charm tetraquark states ccqq have the masses about 3.85 GeV from the QCD sum rules in which the vacuum condensates up to dimension 10 are taken into account in a consistent way [48,49], if an additional P-wave costs about 0.5 GeV, the ground state vector hidden-charm tetraquark states ccqq have the mass about 4.35 GeV, the present prediction 4.34±0.08 GeV is robust. In Ref. [19], Zhang and Huang introduce an explicit P-wave in the currents, and obtain the value 4.32 ± 0.20 GeV by taking into account the vacuum condensates up to dimension 6.
In Ref. [15], A. Ali et al study the hidden-charm P-wave tetraquarks and the newly excited charmed Ω c states with the effective Hamiltonian incorporating the dominant spin-spin, spin-orbit and tensor interactions, and observe that the Y (4220), Y (4330), Y (4390), Y (4660) can be assigned to be the four ground states with L = 1 by fitting the coefficients in the effective Hamiltonian to the experimental masses. In the effective Hamiltonian approach, the lowest state is the Y (4220), while we cannot obtain such low mass based on the QCD sum rules [17,18,19,20,21,22].
The mass M Y (4390) lies just below the upper bound of the predicted mass of the Cγ 5 ⊗ γ 5 γ µ C type vector tetraquark state ccqq, which disfavors assigning the Y (4390) to be the Cγ 5 ⊗ γ 5 γ µ C type vector tetraquark state ccqq, on the other hand, the mass M Y (4390) lies just below the lower bound of the predicted mass of the Cγ 5 ⊗ γ 5 γ µ C type vector tetraquark state ccss, which also disfavors assigning the Y (4390) to be the Cγ 5 ⊗ γ 5 γ µ C type vector tetraquark state ccss, however, such assignments are not completely excluded. If we take the energy scale µ = 3.4 GeV for the ccss tetraquark state or µ = 1.8 GeV for the ccqq tetraquark state, the experimental value of the M Y (4390) can be reproduced, however, such energy scales are not consistent with the QCD sum rules for other tetraquark states.
There are no candidates for the Y (4260/4220) in the present calculations. In Refs. [30,31], the Y (4260) and Z c (3900) are assigned to be theDD 1 (2420) + DD 1 (2420) andDD * + DD * molecular states respectively based on the heavy meson (non-relativistic) effective field theory. In Ref. [33], we study the vector molecular states DD 1 (2420) and D * D * 0 (2400) with the QCD sum rules by taking into account the vacuum condensates up to dimension-10 in the operator product expansion in a consistent way, and use the energy scale formula for the molecular states to determine the optimal energy scales of the QCD spectral densities [59], and obtain the predications M DD1(1 −− ) = 4.36 ± 0.08 GeV and M D * D * 0 (1 −− ) = 4.78 ± 0.07 GeV. The QCD sum rules support assigning the Y (4390) (not the Y (4260/4220)) and Z c (3900) to be the DD 1 and DD * S-wave molecular states, respectively [33,59]. While the lattice QCD supports assigning the Y (4260) to be a hybrid state [24]. Furthermore, there have been observed evidences for the X(3872) in the lattice calculations, though its interpretation was not specified [60], there was no evidence for the Z c (3900) in the lattice calculations [61]. There are also other assignments of the Y (4390), for example, the D * (2010)D 1 (2420) molecular state [62]. Now we check the assignment of the Y (4660/4630) as the C ⊗ γ µ C type vector tetraquark state ccss and the assignment of the Y (4360/4320) as the Cγ 5 ⊗ γ 5 γ µ C type vector tetraquark state ccqq by assuming the currents J 1 µ (x) and J 4 µ (x) both couple potentially to the four Y       GeV as input parameters [5,57], and take the pole residues λ Y as free parameters to fit the following two QCD sum rules, In Table 4, we present the central values of the fitted pole residues. In Fig.4, we plot the correlation functions with the central values of the fitted pole residues compared to the operator product expansion. From the figure, we can see that the QCD sides of the correlation functions can be well reproduced. From Table 4, we can see that the current J 1 µ (x) couples dominantly to the Y (4660) both at the energy scales µ = 2.9 GeV and µ = 2.4 GeV, the couplings to the Y (4220), Y (4360) and Y (4390) can be neglected safely. For the current J 4 µ (x), if we take the ideal energy scale µ = 2.4 GeV, the current J 4 µ (x) couples dominantly to the Y (4360), the couplings to the Y (4220), Y (4390) and Y (4660) can be neglected safely; on the other hand, if we take larger energy scale µ = 2.9 GeV, the current J 4 µ (x) couples potentially to the Y (4360), the coupling to the Y (4220) is not neglectful.
In Refs. [15,63], the Y (4220) is assigned to be the lowest vector tetraquark state in the simple diquark-antidiquark model with the constituent ccqq. In the present work, we can see that the lowest tetraquark states couple potentially to the current J 4 µ (x), not to the currents J 1 µ (x), J 2 µ (x) and J 3 µ (x), now we suppose that the Y (4260/4220) is a pure vector tetraquark state and saturates the QCD sum rules, and study the energy scale dependence of the extracted mass M Y (4260) with the central values of the input parameters in Table 2. In Fig.5, we plot the extracted mass M Y (4260) with variations of the Borel parameters T 2 and energy scales µ. From the figure, we can see that the mass M Y (4260) decreases monotonously with increase of the energy scales µ, the platforms appear at about T 2 = 3 GeV 2 . Even at the large energy scale µ = 5 GeV, the extracted mass M Y (4260) > 4.230 GeV, so the Y (4260/4220) is unlikely to be a pure vector tetraquark state.

Conclusion
In this article, we construct the C ⊗ γ µ C and Cγ 5 ⊗ γ 5 γ µ C type currents to interpolate the vector tetraquark states, then calculate the contributions of the vacuum condensates up to dimension-10 in the operator product expansion in a consistent way, and obtain four QCD sum rules. In calculations, we use the formula µ = M 2 X/Y /Z − (2M c ) 2 to determine the optimal energy scales of the QCD spectral densities, explore the energy scale dependence of the QCD sum rules in details, moreover, we take the experimental values of the masses of the Y (4260/4220), Y (4360/4320),  Y (4390) and Y (4660/4630) as input parameters and fit the pole residues to reproduce the correlation functions at the QCD side. The numerical results support assigning the Y (4660/4630) to be the C ⊗ γ µ C type vector tetraquark state ccss, assigning the Y (4360/4320) to be Cγ 5 ⊗ γ 5 γ µ C type vector tetraquark state ccqq, and disfavor assigning the Y (4260/4220) and Y (4390) to be the pure vector tetraquark states.