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

The European Physical Journal C, Jun 2018

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.

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

Eur. Phys. J. C Vector tetraquark state candidates: Y (4260/4220), Y (4360/4320), Y (4390) and Y (4660/4630) Zhi-Gang Wang 0 0 Department of Physics, North China Electric Power University , Baoding 071003 , People's Republic of China In this article, we construct the C ⊗ γμC and C γ5 ⊗ γ5γμC 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 μ = MY2 − (2Mc)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 ⊗ γμC type vector tetraquark state cc¯ss¯, assigning the Y (4360/4320) to be C γ5 ⊗γ5γμC type vector tetraquark state cc¯qq¯ , and disfavor assigning the Y (4260/4220) and Y (4390) to be the pure vector tetraquark states. 1 Introduction In 2005, the BaBar collaboration studied the initial-state radiation process e+e− → γI S R 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− → ωχc J with J = 0, 1, 2, and observed a resonance in the ωχc0 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 −hc, 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 ]. In Ref. [ 7 ], Gao, Shen and Yuan perform a combined fit for the cross sections of e+e− → ωχc0, π +π −hc, π +π − J /ψ , D0 D∗−π + + c.c. measured by the BESIII experiment, and determine a mass 4219.6 ± 3.3 ± 5.1 MeV and a total width 56.0 ± 3.6 ± 6.9 MeV for the Y (4220). In 2006, the BaBar collaboration observed an evidence for a broad structure at 4.32 GeV in the π +π −ψ mass spectrum in the process e+e− → π +π −ψ [ 8 ]. In 2007, the Belle collaboration studied the initial-state radiation process e+e− → γI S R π +π −ψ , and observed two structures Y (4360) and Y (4660) in the π +π −ψ invariant-mass spectrum [ 9,10 ]. In 2008, the Belle collaboration studied the initial-state radiation process e+e− → γI S R c+ c− and observed a clear peak Y (4630) in the c+ c− invariant-mass spectrum [11]. The Y (4360) and Y (4660/4630) were confirmed by the BaBar collaboration [ 12 ]. There have been several assignments for those Y states, such as the tetraquark states [ 13–24 ], hybrid states [ 25–28 ], hadro-charmonium states [ 29–32 ], molecular states [ 33–36 ], kinematical effects [ 37–40 ], baryonium states [41], etc. The Y (4260), which is the milestone of the Y states, has been extensively studied. In Ref. [ 13 ], Maiani et al. assign the Y (4260) to be the first orbital excitation of a diquark–antidiquark state [cs]S=0[c¯s¯]S=0 based on the effective Hamiltonian with the spin-spin and spin-orbit interactions [42], where the subscript S denotes the diquark spins. In the type-II diquark– antidiquark model [ 14 ], where the spin-spin interactions between the quarks and antiquarks are neglected, L. Maiani et al interpret the Y (4360) and Y (4660) as the first radial excitations of the Y (4008) and Y (4260) respectively, and interpret the Y (4008), Y (4260), Y (4290/4220) and Y (4630) as the four ground states with the angular momentum L = 1. One can consult Ref. [ 15 ] for detailed reviews of the effective Hamiltonian approach. In Ref. [ 16 ], Ali et al. analyze the hidden-charm P-wave tetraquarks and the newly excited charmed c states in the diquark model using the effective Hamiltonian incorporating the dominant spin-spin, spin-orbit and tensor interactions, and observe that the preferred assignments of the ground state tetraquark states with L = 1 are the Y (4220), Y (4330), Y (4390), Y (4660) rather than the Y (4008), Y (4260), Y (4360), Y (4660). In the effective Hamiltonian, an approximately common rest frame for all the components is assumed, while in the dynamical diquark picture, the diquark–antidiquark pair forms promptly at the production point, and rapidly separates due to the kinematics of the production process, then they create a color flux tube or string between them [ 17 ]. Another possible assignment of the Y (4260) is the ccg ¯ hybrid state [ 25,26 ], the lattice calculations indicate that the vector charmonium hybrid has a mass about 4285 MeV, which is quite close to that of the Y (4260) [27]. Furthermore, lattice results strongly indicate that the quarks should form a spin singlet in the low-lying vector hybrid, which in rough agreement with the spin-flip suppression in the annihilation e+e− → Y (4260). An alternative approach is to describe the X , Y , Z mesons as bound states in the Born–Oppenheimer potentials (usually lattice-QCD computed gluon-induced potentials) for a heavy quark and a heavy antiquark [ 28 ]. For more literatures on the Q Q¯ potentials involving or not involving of the gluonic excitations, one can consult Refs. [43,44]. In Ref. [ 29 ], Li and Voloshin suggest that the Y (4260) and Y (4360) are a mixture, with mixing close to maximal, of two states of the hadro-charmonium, one containing a spin-triplet cc¯ pair and the other containing a spin-singlet cc¯ pair. While the Y (4660) can be assigned to be the ψ f0(980) hadrocharmonium or molecular state (the two scenarios overlap in this case) [ 31,32 ]. For more literatures on the hadrocharmonium states, one can consult Ref. [30]. In the scenario of molecular states, the Y (4260) and Zc(3900) are assigned to be the D¯ D1(2420) + D D¯ 1(2420) and D¯ D∗ + D D¯ ∗ molecular states respectively [ 33,34 ], which are compatible with the processes e+e− → J /ψ π + π −, hcπ +π − measured by the BESIII and Belle collaborations. In Ref. [35], Wang et al. confront both the hadronic C γ5 ⊗ γ5C, C γμ ⊗ γ μC, C ⊗ γμC, C γ5 ⊗ γ5γμC, C γ5 ⊗ ∂μγ5C, C γα ⊗ ∂μγ αC, C γμ ⊗ γν C − C γν ⊗ γμC, to interpolate the scalar hidden-charm tetraquark states [55– 60] or construct the diquark–antidiquark type currents molecule and the hadro-charmonium interpretations of the Y (4260) with the available experimental data, and conclude that the data support the Y (4260) being dominantly a D¯ D1(2420) + D D¯ 1(2420) hadronic molecule while they challenge the hadro-charmonium interpretation. The assignment of the Y (4260) as the tetraquark state or molecular state means different decay ratio Y (4260) → γ X (3872) [45,46]. Precisely measuring the decay ratio can shed light on the nature of the Y (4260), Zc(3900) and X (3872). In other interpretations, the three charmonium-like states Y (4008), Y (4260) and Y (4360) are Fango-like interference phenomena or coupled-channel effects rather than genuine resonances [ 37–40 ]. While Qiao assigns the Y (4260) to be a baryonium state c+ c− [41]. In this article, we tentatively assume that there exist four exotic vector Y states, Y (4260/4220), Y (4360/4320), Y (4390) and Y (4660/4630), Y (4220) → ωχc0, J /ψ π +π −, hcπ +π − [ 4–6 ], Y (4320) → J /ψ π +π −, ψ π +π − [ 6,8–10 ], Y (4390) → hcπ +π − [5], Y (4660) → ψ π +π −, + − [ 9–11 ], c c and will focus on the scenario of tetraquark states based on the QCD sum rules. The diquarks εi jk q Tj C qk have five structures in Dirac spinor space, where C = C γ5, C , C γμγ5, C γμ and C σμν for the scalar, pseudoscalar, vector, axialvector and tensor diquarks, respectively. The attractive interactions of onegluon exchange favor formation of the diquarks in color antitriplet, flavor antitriplet and spin singlet [47,48], while the favored configurations are the scalar (C γ5) and axialvector (C γμ) diquark states based on the QCD sum rules [49– 54]. The C γ5 type and C γμ type diquark states can be reduced to the non-relativistic spin-0 and spin-1 diquark states respectively in the effective Hamiltonian approach [ 15 ]. We can construct the diquark–antidiquark type currents (1) (2) (3) to interpolate the vector hidden-charm tetraquark states [ 18– 24 ]. One can consult Ref. [61] for more interpolating currents for the vector tetraquark states without introducing an additional P-wave between the diquark and antidiquark. We can cc¯ss¯ cc¯ss¯ cc¯ss¯/cc¯qq¯ cc¯qq¯ cc¯qq¯ cc¯qq¯ cc¯qq¯ ⊕ cc¯ cc¯qq¯ ⊕ cc¯ cc¯qq¯ ⊕ cc¯ 6 8 (7) 10 10 8 (7) 6 10 10 8 (7) [ 20 ] [ 22 ] [ 23 ] [ 24 ] [61] [ 21 ] [ 24 ] [ 24 ] [62] also choose the mixed charmonium-tetraquark currents to study the vector mesons Y (4260) and Y (4360) [ 24,62 ]. In Table 1, we present the existing predictions of the masses of the vector tetraquark states based on the QCD sum rules. In Refs. [ 20–22,61,62 ], the vacuum condensates are taken at the energy scale μ = 1 GeV while the M S mass mc(mc) is taken at the energy scale μ = mc(mc), the energy scales of the QCD spectral densities are not specified. In Refs. [ 23,24,60 ], we take the formula μ = M X2/Y/Z − (2Mc)2 with the effective charmed quark mass Mc to determine the energy scales of the QCD spectral densities of the hiddencharm tetraquark states, and evolve the vacuum condensates and the M S mass mc(mc) to the optimal energy scales μ to extract the tetraquark masses; the hidden-bottom tetraquark states can be studied analogously [63,64]. 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 ⊗ γμC and C γ5 ⊗ γ5γμC 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 X2/Y/Z − (2Mc)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 T12 , T14 , T16 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 Sect. 2; in Sect. 3, we present the numerical results and discussions; Sect. 4 is reserved for our conclusion. 2 QCD sum rules for the vector tetraquark states In the following, we write down the two-point correlation functions μν ( p) in the QCD sum rules, d4x ei p·x 0|T Jμ(x ) Jν†(0) |0 , μν ( p) = i Jμ1(x ) = Jμ2(x ) = Jμ3(x ) = Jμ4(x ) = εi jk εimn √2 εi jk εimn εi jk εimn √2 εi jk εimn sT j (x )C ck (x )s¯m (x )γμC c¯T n(x ) − sT j (x )C γμck (x )s¯m (x )C c¯T n(x ) , uT j (x )C ck (x )u¯m (x )γμC c¯T n(x ) 2 + d T j (x )C ck (x )d¯m (x )γμC c¯T n(x ) − uT j (x )C γμck (x )u¯m (x )C c¯T n(x ) − d T j (x )C γμck (x )d¯m (x )C c¯T n(x ) , sT j (x )C γ5ck (x )s¯m (x )γ5γμC c¯T n(x ) + sT j (x )C γμγ5ck (x )s¯m (x )γ5C c¯T n(x ) , uT j (x )C γ5ck (x )u¯m (x )γ5γμC c¯T n(x ) 2 + d T j (x )C γ5ck (x )d¯m (x )γ5γμC c¯T n(x ) + uT j (x )C γμγ5ck (x )u¯m (x )γ5C c¯T n(x ) + d T j (x )C γμγ5ck (x )d¯m (x )γ5C c¯T n(x ) , (6) where μν ( p) = 1μν ( p), 2μν ( p), 3μν ( p), 4μν ( p), Jμ(x ) = Jμ1(x ), Jμ2(x ), Jμ3(x ), Jμ4(x ), the i , j , k, m, n are color indexes, the C is the charge conjugation matrix. Under charge conjugation transform C , the currents Jμ(x ) have the properties, C Jμ(x )C −1 = − Jμ(x ). In the non-relativistic diquark–antidiquark model, one often introduces an explicit P-wave between the diquark and antidiquark in the ground state C γ5 ⊗γ5C type or C γ5 ⊗γμC type or C γμ ⊗ γν C type tetraquark state [ 13–16 ]. In this arti(4) (5) (7) cle, we choose the C ⊗ γμC type and C γ5 ⊗ γ5γμC type vector currents to interpolate the vector tetraquark states, the net effects of the relative P-waves are embodied in the underlined γ5 in the C γ5γ5 ⊗ γμC type and C γ5 ⊗ γ5γμC type currents or in the underlined γ α in the C γαγ α ⊗ γμC type currents. The tetraquark states are spatial extended objects, not point-like objects [ 17 ], in the QCD sum rules [ 18–24 ] and the effective Hamiltonian approach [ 13–16 ], 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 [65,66]. Now we perform Fierz re-arrangement to the vector currents Jμ1(x ) and Jμ3(x ) both in the color and Dirac-spinor spaces, and obtain the following results, Jμ1 = 2 √12 cγ μc s¯s − c¯c s¯γ μs ¯ Jμ3 = 2 √12 + i c¯γ μγ5s s¯i γ5c − i c¯i γ5s s¯γ μγ5c − i c¯γν γ5c s¯σ μν γ5s + i c¯σ μν γ5c s¯γν γ5s − i s¯γν c c¯σ μν s + i s¯σ μν c c¯γν s , c¯c s¯γ μs + c¯γ μc s¯s − c¯γ μs s¯c − c¯s s¯γ μc − i c¯σ μν γ5c s¯γν γ5s − i c¯γν γ5c s¯σ μν γ5s + i s¯γν γ5c c¯σ μν γ5s + i s¯σ μν γ5c c¯γν γ5s , (8) (9) the Fierz re-arrangement of the Jμ2(x ) and Jμ4(x ) can be obtained analogously. The diquark–antidiquark 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 singlet-singlet 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, 0|c¯(0)c(0)|χc0( p) = fχc0 Mχc0 , 0|c¯(0)γμc(0)| J /ψ ( p) = f J/ψ MJ/ψ eμ, 0|c¯(0)σμν γ5c(0)|hc( p) = i fhc eμ pν − eν pμ , 0|c¯(0)σμν γ5c(0)| J /ψ ( p) = i f JT/ψ εμναβ eα pβ , (10) where the eμ are the polarization vectors of the J /ψ and hc, we can obtain the conclusion, if the Y (4260/4220), Y (4360/4320), Y (4390) and Y (4660/4630) are the C ⊗γμC type or C γ5⊗γ5γμC type tetraquark states, there are no heavy quark spin-flips in the decays to the final states J /ψ and hc, which is consistent with the experimental data [ 4–6,8–11 ]. At the hadronic side, we can insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators Jμ(x ) into the correlation functions μν ( p) to obtain the hadronic representation [67–69]. 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, μν ( p) = = λ2Y M 2 Y − p2 ( p2) −gμν + pμ pν p2 + · · · , −gμν + pμ pν p2 + · · · , where the pole residues λY are defined by 0| Jμ(0)|Y ( p) = λY εμ, the εμ 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 Jμ1(x ) and Jμ3(x ) as an example, and briefly outline the operator product expansion for the correlation functions μν ( p) in perturbative QCD. We contract the c and s quark fields in the correlation functions μν ( p) with Wick theorem, obtain the results: (11) (12) 1μν ( p) = i εi jk εimnεi j k εi m n d4x ei p·x × Tr C kk (x )C S j j T (x )C × Tr γν C n n(−x )γμC Sm mT (−x )C + Tr γμC kk (x )γν C S j j T (x )C × Tr C n n(−x )C Sm mT (−x )C + Tr γμC kk (x )C S j j T (x )C × Tr γν C n n(−x )C Sm mT (−x )C + Tr C kk (x )γν C S j j T (x )C 2 2 3μν ( p) = × Tr C n n(−x )γμC Sm mT (−x )C (13) i εi jk εimnεi j k εi m n d4x ei p·x × Tr γ5C kk (x )γ5C S j j T (x )C × Tr γ5γν C n n(−x )γμγ5C Sm mT (−x )C + Tr γμγ5C kk (x )γ5γν C S j j T (x )C × Tr γ5C n n(−x )γ5C Sm mT (−x )C − Tr γμγ5C kk (x )γ5C S j j T (x )C × Tr γ5γν C n n(−x )γ5C Sm mT (−x )C f λαβ = (k + mc)γ λ(k + mc)γ α(k + mc)γ β (k + mc), f αβμν = (k + mc)γ α(k + mc)γ β (k + mc)γ μ × (k + mc)γ ν (k + mc), and t n = λ2n , the λn is the Gell-Mann matrix [69], then compute the integrals both in the coordinate and momentum spaces, and obtain the correlation functions μν ( p) therefore the spectral densities at the level of quark-gluon degrees of freedom. In Eq. (15), we retain the terms s¯ j σμν si originate from the Fierz re-arrangement of the si s¯ j to absorb the gluons emitted from other quark lines to extract the mixed condensate s¯gs σ Gs [63,64]. Once analytical expressions of the QCD spectral densities are obtained, we can take the quark-hadron duality below the continuum thresholds s0 and perform Borel transform with respect to the variable P2 = − p2 to obtain the following four QCD sum rules: λ2Y exp ρ4(s) = ρ3(s) |ms →0, s¯s → q¯q , s¯gs σ Gs → q¯gs σ Gq , (15) (16) (17) (18) (19) 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 ρi (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 Nc limit. In reality, Nc = 3, some (not much) ambiguities maybe come from the vacuum saturation assumption. The higher dimensional vacuum condensates have not been well studied yet. We derive Eq. (17) with respect to τ = T12 , then eliminate the pole residues λY , and obtain the QCD sum rules for the masses of the vector tetraquark states, MY2 = − 4sm0c2 ds ddτ ρ(s) exp (−τ s) 4sm0c2 dsρ(s) exp (−τ s) . (20) 3 Numerical results and discussions We take the standard values of the vacuum condensates qq = −(0.24 ± 0.01 GeV)3, q¯ gs σ Gq = m20 q¯q , m2 ¯ 0 = (0.8 ± 0.1) GeV2, s¯s = (0.8 ± 0.1) q¯q , s¯gs σ Gs = m20 s¯s , αsπGG = (0.33 GeV)4 at the energy scale μ = 1 GeV [67–70], and choose the M S masses mc(mc) = (1.28 ± 0.03) GeV, ms (μ = 2 GeV) = 0.096+−00..000084 GeV from the Particle Data Group [71], and set mu = md = 0. Moreover, we take into account the energy-scale dependence of the input parameters on the QCD side, q¯q (μ) = q¯q (Q) s¯s (μ) = s¯s (Q) q¯ gs σ Gq (μ) = q¯ gs σ Gq (Q) s¯gs σ Gs (μ) = s¯gs σ Gs (Q) mc(μ) = mc(mc) ms (μ) = ms (2GeV) 12 αs (Q) 25 αs (μ) 12 αs (Q) 25 αs (μ) αs (μ) αs (2GeV) 2 αs (Q) 25 αs (μ) 2 αs (Q) 25 Table 2 The Borel parameters, continuum threshold parameters, pole contributions, contributions of the vacuum condensates of dimension 7, 8 and 10, where the superscripts 1, 2, 3 and 4 denote the currents Jμ1(x), Jμ2(x), Jμ3(x) and Jμ4(x), respectively cc¯ss¯1 cc¯qq¯2 cc¯ss¯3 cc¯qq¯4 T 2 (GeV2) 3.6–4.0 3.5–3.9 3.2–3.6 3.0–3.4 √s0 (GeV) where t = log μ22 , b0 = 331−22πn f , b1 = 1532−4π192n f , b2 = 2857− 50933 n f + 32275 n2f 128π3 , = 210, 292 and 332 MeV for the flavors n f = 5, 4 and 3, respectively [71], and evolve all the input parameters to the optimal energy scales μ to extract the masses of the vector hidden-charm tetraquark states. In the present QCD sum rules, we search for the ideal Borel parameters T 2 and continuum threshold parameters s0 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. [ 23,60,63,64 ], we study the energy scale dependence of the QCD sum rules in details and suggest an energy scale formula μ = M X2/Y/Z − (2MQ )2 with the effective heavy quark mass MQ 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 [72,73]. In this article, we take the updated value Mc = 1.82 GeV [ 24 ]. The resulting Borel parameters or Borel windows T 2, continuum threshold parameters s0, 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 and 2 and Table 3. From Figs. 1 and 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) [ 5,71 ]. From the figure, we can see that the experimental values of the masses MY (4660/4630) and MY (4360/4320) can be well reproduced, the present predications support assigning the Y (4660/4630) to be the C ⊗ γμC type tetraquark state cc¯ss¯, and assigning the Y (4360/4320) to be the C γ5 ⊗ γ5γμC type tetraquark state cc¯qq¯ . The mass MY (4660/4630) lies just below the upper bound of the predicted mass of the C ⊗γμC type vector tetraquark state cc¯qq¯ , which disfavors assigning the Y (4660/4630) to be the C ⊗γμC type vector tetraquark state cc¯qq¯ , 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 ⊗ γμC type tetraquark states have larger masses than the corresponding C γ5 ⊗ γ5γμC type tetraquark states, as C ⊗ γμC = [C γ5γ5 ⊗ γμC ] ⊕ [C γαγ α ⊗ γμC ] and C γ5 ⊗ γ5γμC = C γ5 ⊗ γ5γμC , the C γμ diquark states have slightly larger masses than the corresponding C γ5 diquark states from the QCD sum rules [49–51]. In other words, the bad diquarks have slightly larger masses than the good diquarks, those effects are also accounted for in the effective Hamiltonian [ 14,16 ]. On the other hand, the Y (4660) can be assigned to be the ψ f0(980) hadro-charmonium or molecular state based on fitting the mass distribution of the process e+e− → ψ π +π − [ 31 ] or the calculations of the QCD sum rules [ 32 ]. 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 ⊗ γμC type currents by taking into account the neglected terms due to the approximations involving the higher dimensional vacuum condensates in Ref. [ 23 ] 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−+00..1104/4.66+−00..1170 GeV obtained in Ref. [ 23 ]. In Fig. 3, we plot the mass and pole residue of the C ⊗ γμC type vector tetraquark state cc¯ss¯ 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 cc¯ss¯1 cc¯qq¯2 cc¯ss¯3 cc¯qq¯4 see that the higher dimensional vacuum condensates play an important role in determining the Borel platforms. The ground state C γ5 ⊗ γ5C type and C γμ ⊗ γ μC type hidden-charm tetraquark states cc¯qq¯ 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 [58–60], if an additional P-wave costs about 0.5 GeV, the ground state vector hidden-charm tetraquark states cc¯qq¯ have the mass about 4.35 GeV, the present prediction 4.34±0.08 GeV is robust. In Ref. [ 21 ], 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. [ 16 ], Ali et al. study the hidden-charm P-wave tetraquarks and the newly excited charmed c states with the effective Hamiltonian incorporating the dominant spinspin, 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 [ 18–24 ]. The mass MY (4390) lies just below the upper bound of the predicted mass of the C γ5 ⊗ γ5γμC type vector tetraquark state cc¯qq¯ , which disfavors assigning the Y (4390) to be the C γ5 ⊗ γ5γμC type vector tetraquark state cc¯qq¯ , on the other hand, the mass MY (4390) lies just below the lower bound of the predicted mass of the C γ5 ⊗γ5γμC type vector tetraquark state cc¯ss¯, which also disfavors assigning the Y (4390) to be the C γ5 ⊗ γ5γμC type vector tetraquark state cc¯ss¯, however, such assignments are not completely excluded. If we take the energy scale μ = 3.4 GeV for the cc¯ss¯ tetraquark state or μ = 1.8 GeV for the cc¯qq¯ tetraquark state, the experimental value of the MY (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. [ 33,34 ], the Y (4260) and Zc(3900) are assigned to be the D¯ D1(2420) + D D¯ 1(2420) and D¯ D∗ + D D¯ ∗ molecular states respectively based on the heavy meson (non-relativistic) effective field theory. In Ref. [36], we study the vector molecular states D D¯ 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 [74,75], and obtain the predications MD D¯ 1(1−−) = 4.36 ± 0.08 GeV and MD∗ D¯ 0∗(1−−) = 4.78±0.07 GeV. The QCD sum rules support assigning the Y (4390) (not the Y (4260/4220)) and Zc(3900) to be the D D¯ 1 and D D¯ ∗ S-wave molecular states, respectively [ 36,74,75 ]. While the lattice QCD supports assigning the Y (4260) to be a hybrid state [ 27 ]. Furthermore, there have been observed evidences for the X (3872) in the lattice calculations, though its interpretation was not specified [76], there was no evidence for the Zc(3900) in the lattice calculations [77]. There are also other assignments of the Y (4390), for example, the D∗(2010)D¯ 1(2420) molecular state [78]. Now we check the assignment of the Y (4660/4630) as the C ⊗ γμC type vector tetraquark state cc¯ss¯ and the assignment of the Y (4360/4320) as the C γ5 ⊗ γ5γμC type vector tetraquark state cc¯qq¯ by assuming the currents Jμ1(x ) and Jμ4(x ) both couple potentially to the four Y states Y (4260/4220), Y (4360/4320), Y (4390) and Y (4660/4630). We take the experimental values MY (4220) = 4.230 GeV, MY (4360) = 4.341 GeV, MY (4390) = 4.392 GeV and MY (4660) = 4.643 GeV as input parameters [ 5,71 ], and take the pole residues λY as free parameters to fit the following two QCD sum rules, Y = Y (4220), Y (4360), Y (4390), Y (4660) λ2Y exp = = s0 4mc2 s0 4mc2 s ds ρ1(s) exp − T 2 , s ds ρ4(s) exp − T 2 . Y = Y (4220), Y (4360), Y (4390), Y (4660) λ2Y exp 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 Jμ1(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 Jμ4(x ), if we take the ideal energy scale μ = 2.4 GeV, the current Jμ4(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 Jμ4(x ) couples potentially to the Y (4360), the coupling to the Y (4220) is not neglectful. In Refs. [ 16,79 ], the Y (4220) is assigned to be the lowest vector tetraquark state in the simple diquark–antidiquark model with the constituent cc¯qq¯ . In the present work, we can see that the lowest tetraquark states couple potentially to the current Jμ4(x ), not to the currents Jμ1(x ), Jμ2(x ) and Jμ3(x ), now we suppose that the Y (4260/4220) is a pure vector tetraquark state and saturates the QCD sum rules, MY2(4260) = − 4sm0c2 ds ddτ ρ4(s) exp (−τ s) 4sm0c2 dsρ4(s) exp (−τ s) and study the energy scale dependence of the extracted mass MY (4260) with the central values of the input parameters in Table 2. In Fig.5, we plot the extracted mass MY (4260) with variations of the Borel parameters T 2 and energy scales μ. From the figure, we can see that the mass MY (4260) decreases monotonously with increase of the energy scales μ, the platforms appear at about T 2 = 3 GeV2. Even at the large energy scale μ = 5 GeV, the extracted mass MY (4260) > (24) 4.230 GeV, so the Y (4260/4220) is unlikely to be a pure vector tetraquark state. If we take the energy scale formula μ = M X2/Y/Z − (2Mc)2 with the effective mass Mc = 1.82 GeV as a constraint, the QCD sum rules only support assigning the Y (4660/4630) and Y (4360/4320) to be the C ⊗ γμC type vector tetraquark cc¯ss¯ and C γ5 ⊗ γ5γμC type vector tetraquark state cc¯qq¯ respectively, and disfavor assigning the Y (4660/4630) (or Y (4390)) to be the C ⊗ γμC (or C γ5 ⊗ γ5γμC ) type vector tetraquark state cc¯qq¯ . 4 Conclusion In this article, we construct the C ⊗ γμC and C γ5 ⊗ γ5γμC 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 X2/Y/Z − (2Mc)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 ⊗ γμC type vector tetraquark state cc¯ss¯, assigning the Y (4360/4320) to be C γ5 ⊗ γ5γμC type vector tetraquark state cc¯qq¯ , and disfavor assigning the Y (4260/4220) and Y (4390) to be the pure vector tetraquark states. Acknowledgements This work is supported by National Natural Science Foundation, Grant Number 11775079. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP3. 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Zhi-Gang Wang. Vector tetraquark state candidates: Y(4260 / 4220), Y(4360 / 4320), Y(4390) and Y(4660 / 4630), The European Physical Journal C, 2018, 518, DOI: 10.1140/epjc/s10052-018-5996-5