Analysis of the vector hidden-charm-hidden-strange tetraquark states with implicit P-waves via the QCD sum rules

We take the scalar, pseudoscalar, axialvector, vector and tensor diquarks as the basic building blocks to construct the four-quark currents with implicit P-waves, and investigate the hidden-charm-hidden-strange tetraquark states with the J PC = 1 −− and 1 − + via the QCD sum rules in a comprehensive and consistent way, and revisit the assignments of the X/Y states, especially the Y (4500), X (4630), Y (4660), Y (4710) and Y (4790), in the tetraquark picture.


Introduction
In recent years, a number of charmonium-like states have been observed, they cannot be accommodated suitably in the traditional quark model.In this work, we will focus on the Y states and tetraquark scenario.The Y (4260) observed by the BaBar collaboration [1], the Y (4220), Y (4390) and Y (4320) observed by the BESIII collaboration [2,3], and the Y (4360), Y (4660) and Y (4630) observed by the Belle collaboration [4,5,6] are excellent candidates for the vector tetraquark states.
In 2022, the BESIII collaboration observed two resonant structures in the K + K − J/ψ invariant mass spectrum, one is the Y (4230) and the other is the Y (4500), which was observed for the first time with the Breit-Wigner mass and width 4484.7 ± 13.3 ± 24.1 MeV and 111.1 ± 30.1 ± 15.2 MeV, respectively [7].
In 2023, the BESIII collaboration observed three enhancements in the D * − D * 0 π + invariant mass spectrum, the first and third resonances are the Y (4230) and Y (4660), respectively, while the second resonance has the Breit-Wigner mass and width 4469.1 ± 26.2 ± 3.6 MeV and 246.3 ± 36.7 ± 9.4 MeV, respectively, and is roughly compatible with the Y (4500) [8].
Also in 2023, the BESIII collaboration observed three resonance structures in the D * + s D * − s invariant mass spectrum, the two significant structures are consistent with the ψ(4160) and ψ(4415), respectively, while the third structure is new, and has the Breit-Wigner mass and width 4793.3 ± 7.5 MeV and 27.1 ± 7.0 MeV, respectively, therefore is named as Y (4790) [9].Also in 2023, the BESIII collaboration observed a new resonance, Y (4710), in the K + K − J/ψ invariant mass spectrum with a significance over 5σ, the measured Breit-Wigner mass and width are 4708 +17 −15 ± 21 MeV and 126 +27 −23 ± 30 MeV, respectively [10].In the picture of tetraquark states and in the theoretical framework of the QCD sum rules, we can construct the color antitriplet-triplet type, sextet-antisextet type, singlet-singlet type and octet-octet type configurations to explore the properties of the Y states, such as the masses and strong decays [11].We usually take the diquarks in color antitriplet 3c as the elementary constituents, as one-gluon exchange leads to attractive interactions in this channel [12,13].
The heavy-light diquarks ε abc q T b CΓQ c in color antitriplet 3c have five structures, where the a, b and c are color indexes, and CΓ = Cγ 5 , C, Cγ µ γ 5 , Cγ µ and Cσ µν /Cσ µν γ 5 for the scalar (S), pseudoscalar (P ), vector (V ), axialvector (A) and tensor (T ) diquarks, respectively.The Cγ 5 and Cγ µ type diquarks have the spin-parity J P = 0 + and 1 + , respectively [14], while the C and Cγ µ γ 5 type diquarks have the spin-parity J P = 0 − and 1 − , respectively, the implicit P-waves are embodied in the underlined γ 5 in the Cγ 5 γ 5 and Cγ µ γ 5 .Under parity transform P , the tensor diquarks have the following properties, where j, k = 1, 2, 3, the four vectors x µ = (t, x) and xµ = (t, − x).We can see explicitly that the tensor diquarks have both the J P = 1 + and 1 − components, and we project out the J P = 1 + and 1 − components explicitly by multiplying them with suitable operators when constructing the interpolating currents, and represent them as A and V , respectively.Therefore, we can take the S, P , V , A, A and V diquarks as the elementary building blocks to construct the vector tetraquark states (or four-quark currents) with an implicit P-wave, or introduce an explicit P-wave between the S-wave diquark pairs to construct the vector tetraquark states (or four-quark currents).
In Refs.[15,16], we introduce a relative P-wave between the diquark pair explicitly to construct the local four-quark currents, and explore the vector tetraquark states in the framework of the QCD sum rules systematically, and obtain the lowest vector tetraquark masses up to now.The predictions support assigning the Y (4220/4260), Y (4320/4360), Y (4390) and Z(4250) as the lowest vector tetraquark states with a relative P-wave between the diquark pair.
While in the type-II diquark model [17], L. Maiani et al assign the Y (4008), Y (4260), Y (4290/4220) and Y (4630) as the four ground states with L = 1 based on the effective Hamiltonian with the spin-spin and spin-orbit interactions but neglecting the q − q spin-spin interactions.In Ref. [18], A. Ali et al incorporate the dominant spin-spin, spin-orbit and tensor interactions, and observe that the preferred assignments of the ground states with L = 1 are the Y (4220), Y (4330), Y (4390), Y (4660), however, the mass-splitting effects among the multiplet are too large.
In Ref. [19], we take the S, P , V , A, A and V diquarks as the elementary building blocks to construct four-quark currents with an implicit P-wave, and explore the hidden-charm tetraquark states with the J P C = 1 −− and 1 −+ in the framework of the QCD sum rules comprehensively, and revisit the assignments of the Y states in the picture of tetraquark states.
In this work, we take the S, P , V , A, A and V diquarks as the elementary building blocks to construct the local four-quark currents with an implicit P-wave to realize the negative parity, and investigate the hidden-charm-hidden-strange tetraquark states with the J P C = 1 −− and 1 −+ in a comprehensive and consistent way.We take account of the light-flavor SU (3)-breaking effects and try to make possible assignments of the X/Y states combined with Ref. [19] in a consistent way.
The article is arranged as follows: we obtain the QCD sum rules for the vector hidden-charmhidden-strange tetraquark states in section 2; in section 3, we present the numerical results and discussions; section 4 is reserved for our conclusion.2 QCD sum rules for the vector hidden-charm-hidden-strange tetraquark states

Structures
We write down the two-point correlation functions Π µν (p) and Π µναβ (p) firstly, where the currents the i, j, k, m, n are color indexes, the C is the charge conjugation matrix, the subscripts ± represent the positive and negative charge conjugation, respectively, the superscripts P , S, A( A) and V ( V ) represent the pseudoscalar, scalar, axialvector and vector diquarks or antidiquarks, respectively.The currents in Eq.( 4) have one Lorentz index and are made of the S, P , A and V diquarks.The currents in Eq.( 5) also have one Lorentz index but are made of the A, V , A and V diquarks, thus involve the tensor diquarks.The currents in Eq.( 6) have two Lorentz indexes which are antisymmetric, and are made of the S, P , A and V diquarks, thus also involve the tensor diquarks.The current in Eq.( 7) has two Lorentz indexes which are also antisymmetric but are made of the A diquarks not the T diquarks.
Under parity transform P , the currents J µ (x) and J µν (x) have the properties, where Jµν (x) = J S V −,µν (x), J S V +,µν (x), J P A −,µν (x), J P A +,µν (x), the coordinates x µ = (t, x) and xµ = (t, − x).We rewrite Eq.( 8) more explicitly to make sense of the parity eigenstates, where i, j = 1, 2, 3.The currents J µ (x) and J µν (x) have both negative and positive-parity components, which couple potentially to the hidden-charm-hidden-strange tetraquark states with the negative and positive-parity, respectively, we distinguish their contributions explicitly by choosing the pertinent tensor structures.Under charge-conjugation transform C, the currents J µ (x) and J µν (x) have the properties, the interpolating currents have definite charge conjugation, and couple potentially to the tetraquark states with definite charge conjugation.The current J P A −,µ (x) has been studied in Refs.[22,23], while the currents J SV −,µ (x) and J P A +,µ (x) have been studied in Ref. [22] and Ref. [23], respectively.In this work, we update the old calculations [22,23], and extend our previous work on the mass spectrum of the hidden-charm tetraquark states with the J P C = 1 −± [19] to the mass spectrum of the hidden-charm-hidden-strange tetraquark states with the J P C = 1 −± by taking account of the light flavor SU (3) breaking effects, and perform a comprehensive and consistent analysis.
At the hadron side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the currents J µ (x) and J µν (x) into the correlation functions Π µν (p) and Π µναβ (p) to reach the hadronic representation [35,36,37], and isolate the lowest hidden-charmhidden-strange tetraquark contributions, where the pole residues are defined by the ε µ/α are the polarization vectors, we use the superscripts/subscripts ± to represent the positive and negative parity, respectively, and introduce the notation Z to represent the tetraquark states with the positive parity.We choose the components Π − (p 2 ) and p 2 Π − (p 2 ) to investigate the hidden-charm-hidden-strange tetraquark states with the J P C = 1 −− and 1 −+ , respectively, so as not to get polluted QCD sum rules.
In fact, there also exist two-meson scattering state contributions, if they have the same quantum numbers as the intermediate tetraquark states, because the quantum field theory does not forbid the current-two-meson couplings.In the QCD sum rules, we take the local currents, the four valence quarks lie almost at the same point, which requires that the four-quark states, irrespective of the color antitriplet-triplet type or singlet-singlet type, have the average spatial sizes of the same magnitude as the traditional mesons.The physical mesons with two valence quarks are spatial extended objects and have average spatial sizes r 2 ∼ 0.5 fm.If the couplings between the local currents and two-meson scattering states are strong enough, the average spatial sizes of the two-meson scattering states are larger or about 1 fm, and cannot be interpolated by the local currents in the QCD sum rules.In the local limit, the spatial separations are small enough, the charmed meson pairs lose themselves and merge into four-quark states, while the net effects of the current-two-meson couplings can be safely absorbed into the pole residues [38,39,40].
At the QCD side of the Π µν (p) and Π µναβ (p), there are two heavy/light quark propagators.If each heavy quark line emits a gluon and each light quark line contributes a quark-antiquark pair, we obtain a quark-gluon operator g s Gg s Gssss of dimension 10, therefore we should calculate the condensates ss , αsGG π , sg s σGs , ss 2 , g 2 s ss 2 , ss αsGG π , ss sg s σGs , sg s σGs 2 and ss 2 αsGG π , which are vacuum expectations of the quark-gluon operators of the order O(α k s ) with k ≤ 1 [23,38,41,42].We recalculate the higher dimensional condensates with the identity where the λ a is the Gell-Mann matrix, and reach slightly different expressions for the spectral densities of the J P A −,µ (x), J P A +,µ (x) and J SV −,µ (x) [22,23].The condensate g 2 s ss 2 comes from the matrix elements sγ µ t a sg s D η G a λτ , sj D † µ D † ν D † α s i and sj D µ D ν D α s i , rather than come from the radiative corrections for the condensate ss 2 , where 2 , its contributions are very small and are neglected in most of the QCD sum rules.
In this work, we use the energy scale formula µ = M 2 X/Y /Z − (2M c ) 2 − 2M s to determine the best energy scales of the QCD spectral densities [23,27], where the M c and M s are the effective c and s-quark masses, respectively.We take the updated value M c = 1.82 GeV [23,27], and take account of the light-flavor SU (3)-breaking effects by introducing an effective s-quark mass M s = 0.12 GeV for the P-wave states, while for the S-wave states, M s = 0.20 GeV [32,34,46].The energy scale formula can enhance the pole contributions remarkably and improve the convergent behaviors of the operator product expansion remarkably, therefore it is more easier to acquire the lowest tetraquark (molecule) masses.
In the picture of hidden-charm tetraquark states, we can tentatively assign the X(3915) and X(4500) as the 1S and 2S states with the J P C = 0 ++ [47, 48], assign the Z c (3900) and Z c (4430) as the 1S and 2S states with the J P C = 1 +− , respectively [17,49,50], assign the Z c (4020) and Z c (4600) as the 1S and 2S states with the J P C = 1 +− , respectively [51,52], and assign the X(4140) and X(4685) as the 1S and 2S states with the J P C = 1 ++ , respectively [53,54].The energy gaps between the 1S and 2S states are about 0.57 ∼ 0.59 GeV.In Ref. [19], we choose the continuum threshold parameters as √ s 0 = M Y +0.4 ∼ 0.6 GeV for the hidden-charm tetraquark states with the J P C = 1 −− and 1 −+ .Now we choose slightly larger parameters √ s 0 = M Y +0.50 ∼ 0.55±0.10GeV so as to include the ground state contributions as large as possible, and perform a consistent and detailed analysis.The relation √ s 0 = M Y + 0.50 ∼ 0.55 ± 0.10 GeV serves as a strict constraint.We search for the best Borel parameters and continuum threshold parameters via trial and error to satisfy the pole dominance and convergence of the operator product expansion.The pole contributions (PC) are defined by and the contributions of the condensates of dimension n are defined by Finally, we obtain the Borel windows, continuum threshold parameters, suitable energy scales of the QCD spectral densities and pole contributions, which are shown explicitly in Table 2. From the Table, we can see clearly that the pole contributions are about (40 − 60)%, the central values are larger than 50%, the pole dominance is well satisfied.In calculations, we observe that the main contributions come from the perturbative terms, the higher dimensional condensates play a minor important role, just like that in Ref. [19], and the contributions |D(10)| ≪ 1%, the operator product expansion converges very good.It is reliable to extract the masses and pole residues in the Borel windows.
We take account of all the uncertainties of the relevant parameters and obtain the masses and pole residues of the hidden-charm-hidden-strange tetraquark states with the J P C = 1 −− and 1 −+ , which are shown explicitly in Table 3. From Tables 2-3, we can observe that the modified energy scale formula µ = M 2 X/Y /Z − (2M c ) 2 − 2M s is satisfied very good and the relation √ s 0 = M Y + 0.50 ∼ 0.55 ± 0.10 GeV is also satisfied very good.
For the conventional charmonium and bottomonium states η c , J/ψ, η b , Υ, χ b0 , χ b1 , h b , χ b2 and B c , the energy gaps between the ground state and first radial excited state are 654 MeV, 589 MeV, 600 MeV, 563 MeV, 373 MeV, 363 MeV, 361 MeV, 356 MeV and 597 MeV, respectively, from The Review of Particle Physics [43].We can see clearly that the energy gaps are about 0.6 GeV for the charmonium states.On the other hand, the energy gap between the pseudoscalar charm-strange mesons D s and D s (2590) is 622 MeV from The Review of Particle Physics [43].We estimate that the energy gaps of the ground state and first radial excited state of the cscs states are also about 0.6 GeV, just like the pseudoscalar charm-strange mesons or charmonium states.If there are some contaminations from the first radial excited states, they are suppressed by the factor exp − s0 T 2 < 0.0001 ∼ 0.002, which is small enough.In Fig. 1, we plot the masses of the sc] S tetraquark states with the J P C = 1 −− with variations of the Borel parameters for example.From the figure, we observe that there appear very flat platforms in the Borel windows indeed, the QCD sum rules work well.In Fig. 1, we also present the experimental masses of the Y (4790), Y (4710) and Y (4660) [9,10,43].From the figure, we can see clearly that it is reasonable to assign the Y (4790), Y (4710) and Y (4660) as the tetraquark states with the J P C = 1 −− , respectively, also see Table 4.
In Table 5, we present the possible assignments of the hidden-charm tetraquark states with the J P C = 1 −− and 1 −+ obtained in Ref. [19], which support assigning the Y (4360/4390) to be the In 2021, the LHCb collaboration observed the X(4630) in the J/ψφ invariant mass spectrum, the measured Breit-Wigner mass and width are 4626 ± 16 +18 −110 MeV and 174 ± 27 +134 −73 MeV, respectively [55].Considering the large uncertainties, it is possible to assign the X(4630) as the [sc] S [sc] Ṽ + [sc] Ṽ [sc] S state with the J P C = 1 −+ , which has a mass 4.68 ± 0.09 GeV, see Table 4.In Ref. [56], we prove that it is feasible and reliable to investigate the multiquark states in the framework of the QCD sum rules, and obtain the prediction for the mass of the molecular state with the exotic quantum J P C = 1 −+ , M X = 4.67 ± 0.08 GeV, which was obtained before the LHCb data and is compatible with the LHCb data.The X(4630) maybe have two important Fock components.
If only the mass is concerned, the Y (4660) can be assigned as the A tetraquark state, see Tables 4-5.In other words, the Y (4660) maybe have several important Fock components, we have to study the strong decays in details to diagnose its nature.For example, if we assign After Ref. [19] has been published, the Y (4500) was observed by the BESIII collaboration [7,8].At the energy about 4.5 GeV, we obtain three hidden-charm tetraquark states with the J dc] S tetraquark states have the masses 4.53 ± 0.07 GeV, 4.48 ± 0.08 GeV and 4.50 ± 0.09 GeV, respectively [19].In Ref. [57], we take the Y (4500) as the In Ref. [16], we introduce an explicit P-wave to construct the four-quark currents to study the hidden-charm tetraquark states with the J P C = 1 −− , and the calculations support assigning the Y (4260/4220) as the Cγ All in all, there are enough rooms to accommodate the existing Y states above 4.2 GeV.However, due to uncertainties of the predictions, we cannot distinguish the quark structures based on the masses alone in the current accuracy, comprehensive and detailed studies on the strong decays are still needed.Again, for the example, at the energy about 4.5 GeV, there exist three hidden-charm tetraquark states with the J P C = 1 −− and Isospin I = 0, i.e. [19].We have studied the strong decays Y → D D, D * D * , D D * , D * D, ωJ/ψ, ωη c , ωχ c0 , ωχ c1 , J/ψf 0 (500) with the three-point QCD sum rules in a systematic way, and obtained very interesting results, which would be presented in another work after carefully checking.By precisely measuring the ratios among the partial decay widths, we can distinguish the quark structures.Analogously, we can study the strong decays s Ds , φJ/ψ, φη c , φχ c0 , φχ c1 , J/ψf 0 (980) in the present case.We can confront those hidden-charm-hidden-strange tetraquark states to the experimental data at the BESIII, LHCb, Belle II, CEPC, FCC, ILC in the future, which maybe play an important role in interpreting the exotic X, Y , Z states.For example, we can search for the Y states with the J P C = 1 −− and 1 −+ in the two-body or three-body strong decays, just as what have been done previously.The masses and pole residues of the ground state hidden-charm-hidden-strange tetraquark states.

Conclusion
In this work, we take the scalar, pseudoscalar, axialvector, vector and tensor diquarks as the elementary building blocks to construct local four-quark currents with an implicit P-wave, and study the mass spectrum of the hidden-charm-hidden-strange tetraquark states with the J P C = 1 −− and 1 −+ in the framework of the QCD sum rules consistently and comprehensively, and revisit the assignments of the X/Y states in the picture of tetraquark states.A tetraquark state, in other words, the Y (4660) maybe have several important Fock components, we have to study the strong decays exclusively in details to diagnose its nature.In summary, we can assign all the exotic Y states above 4.2 GeV in a consistent way, although the assignments are not definite in the current accuracy.Furthermore, the predictions favor assigning the X(4630) as the [sc] S [sc] Ṽ + [sc] Ṽ [sc] S state with the J P C = 1 −+ .We can confront all the hidden-charm/hidden-charm-hidden-strange tetraquark states with the J P C = 1 −− and 1 −+ to the experimental data in the future.

Table 1 :
The masses from the QCD sum rules with different quark structures, where the OPE denotes truncations of the operator product expansion up to the vacuum condensates of dimension n, the No denotes the vacuum condensates of dimension n ′ are not included.

Table 2 :
The Borel parameters, continuum threshold parameters, energy scales of the QCD spectral densities and pole contributions for the ground state hidden-charm-hidden-strange tetraquark states.