Strong decays of the vector tetraquark states with the masses about $4.5\,\rm{GeV}$ via the QCD sum rules

We suppose that there exist three vector hidden-charm tetraquark states with the $J^{PC}=1^{--}$ at the energy about $4.5\,\rm{GeV}$, and investigate the two-body strong decays systematically. We obtain thirty QCD sum rules for the hadronic coupling constants based on rigorous quark-hadron duality, then obtain the partial decay widths, therefore the total widths approximately, which are compatible with the experimental data of the $Y(4500)$ from the BESIII collaboration. The $Y(4500)$ may be one vector tetraquark state having three main Fock components, or consists of three different vector tetraquark states. We can search for the typical decays $ Y \to \frac{\bar{D}^0_1D^{0}-\bar{D}^{0}D^{0}_1}{\sqrt{2}}$, $\frac{\bar{D}^-_1D^{+}-\bar{D}^{-}D^{+}_1}{\sqrt{2}}$, $\frac{\bar{D}^0_0D^{*0}-\bar{D}^{*0}D^{0}_0}{\sqrt{2}}$, $ \frac{\bar{D}^-_0D^{*+}-\bar{D}^{*-}D^{+}_0}{\sqrt{2}}$, $\eta_c\omega$, $J/\psi\omega$ to diagnose the nature of the $Y(4500)$.


Introduction
In recent years, many charmonium-like states have been observed, especially the vector charmoniumlike states larger than 4.2 GeV, they cannot be accommodated suitably in the traditional quark model.In the present work, we will focus on the Y states and tetraquark picture.In 2022-2024, the BESIII collaboration observed three Y structures at the energy about 4.5 GeV [1,2,3].
If we use the mass to represent the resonant structure, the Y observed by the BESIII collaboration in 2022, 2023 and 2024 can be named as Y (4484), Y (4469) and Y (4544), respectively, they may be three different particles, or just one particle, the Y (4500).
Also in 2023, the BESIII collaboration studied the Born cross sections of the process e + e − → D * + s D * − s using the data samples at center-of-mass energies up to 4.95 GeV, and observed two significant structures with the Breit-Wigner masses 4186.5±9.0±30MeV and 4414.5±3.2±6.0MeV, respectively, and widths 55±17±53 MeV and 122.6±7.0±8.2MeV, respectively, the third structure supports a new state with the Breit-Wigner mass 4793.3 ± 7.5 MeV and width 27.1 ± 7.0 MeV, respectively [4].Furthermore, the BESIII collaboration measured the cross section of the process e + e − → K + K − J/ψ using the data samples with an integrated luminosity of 5.85 fb −1 at center-ofmass energies √ s = 4.61 − 4.95 GeV, and observed a new resonance with a mass 4708 +17 −15 ± 21 MeV and a width 126 +27 −23 ± 30 MeV with a significance over 5σ [5].There have been several explanations for the under-structures of the Y (4500) et al.In Refs.[6,7], the Y (4500) is taken as the 5S −4D mixed charmonium state to study the three-body strong decays Y (4500) → J/ψK + K − and π + D 0 D * − .In an unquenched quark model including coupled-channel effects, the Y (4500) and Y (4710) are assigned to be the 3 3 D 1 and 4 3 D 1 charmonium states, respectively [8].In Ref. [9], the Y (4500) is assigned to be the D s Ds1 molecular state with the quantum numbers J P C = 1 −− based on the heavy-quark spin symmetry and light-flavor SU (3) symmetry, such an assignment is also supported by calculating the masses via the QCD sum rules [10,11] and Bethe-Salpeter equation [12].
In the present work, we will focus on the picture of tetraquark states.In Refs.[13,14], we take the pseudoscalar, scalar, axialvector, vector, tensor (anti)diquarks as the elementary building blocks, and construct the vector and tensor four-quark currents without introducing explicit Pwaves to explore the mass spectrum of the ground state vector hidden-charm tetraquark states in the framework of the QCD sum rules comprehensively.At the energy about 4.5 GeV, we obtain three hidden-charm tetraquark states with the quantum numbers J states have the masses 4.53 ± 0.07 GeV, 4.48 ± 0.08 GeV and 4.50 ± 0.09 GeV, respectively [13].They are all compatible with the Y (4500) within uncertainties, there maybe exist three vector tetraquark states at the energy about 4.5 GeV, or one vector tetraquark state which has three significant Fock components.While the tetraquark states Ṽ with the quantum numbers J P C = 1 −− have the masses 4.80 ± 0.08 and 4.70 ± 0.08, respectively, which support assigning them to be the Y (4790) and Y (4710), respectively [14].For more works on the spectroscopy, one can consult Refs.[15,16,17,18,19,20,21,22,23,24,25,26].
In Ref. [27], we take the Y (4500) as the Ã tetraquark state, and explore the three-body decay Y (4500) → D * − D * 0 π + in the framework of the light-cone QCD sum rules.It is the first time to use the light-cone QCD sum rules to calculate the four-hadron coupling constants.And we observe that the process Y (4500) → D * − D * 0 π + is not the main decay mode, the partial decay width 6.43 +0.80  −0.76 MeV is too small to match the experimental data.The calculation is consistent with our naive expectation that the main decay channels are the two-body strong decays, and we explore the two-body strong decays comprehensively in this work.
The article is arranged as follows: we obtain the QCD sum rules for the hadronic coupling constants in section 2; in section 3, we present numerical results and discussions; section 4 is reserved for our conclusion.

QCD sum rules for the hadronic coupling constants
Firstly, we write down the three-point correlation functions in the QCD sum rules, With the simple replacement AV → V A, we obtain the corresponding correlation functions for the current where the currents J ηc (x) = c(x)iγ 5 c(x) , the i, j, k, m, n are color indexes [13], the C is the charge conjugation matrix, the superscripts S, A and V represent the scalar, axialvector and vector diquarks (or antidiquarks), respectively, while the A and V represent the axialvector and vector components in the tensor diquarks (or antidiquarks), respectively.We insert a complete set of intermediate hadronic states having non-vanishing couplings with the interpolating currents into the three-point correlation functions [28,29], and isolate the ground state contributions explicitly, other ground state contributions are given explicitly in the Appendix, where With the simple replacements AV → V A and S V , we obtain the corresponding components Π(p ′2 , p 2 , q 2 ) for the currents J V A −,µ (0) and J S V −,µν (0), except for the component Π ηcωS V (p ′2 , p 2 , q 2 ), Where we introduce the following collective notations to simplify the formula, With the simple replacements AV → V A and S V , we obtain the corresponding collective notations λ for the currents J V A −,µ (0) and J S V −,µν (0), except for the λ ηcωS V , λ J/ψωS V , λ χc1ωS V , where and And we choose the standard definitions for the decay constants or pole residues, etc.The explicit definitions for other decay constants or pole residues are given in the Appendix.And we use the following definitions for the hadronic coupling constants, etc.The explicit definitions for other hadronic coupling constants are given in the Appendix.In Eq.( 47), there are both contributions come from the J P C = 1 +− and 1 −− tetraquark states (see Eq.( 108) in the Appendix, in which the X state has a mass 4.01 GeV [16]), and we cannot choose the pertinent structures to exclude the contaminations from the J P C = 1 +− tetraquark state X, so we take it into account at the hadron side of the QCD sum rules.The unknown parameters C DD AV , C D * D AV , C D * D * AV , etc, parameterize the complex interactions among the excited states in the p ′2 channels and the ground state conventional charmed meson pairs (or charmonium plus ω), for example, the spectral density ρ(s ′ , m 2 D, m 2 D ) is unknown, where the s ′ 0 is the continuum threshold parameter for the ground state.
We choose the components Π H (p ′2 , p 2 , q 2 ) to investigate the hadronic coupling constants G H , routinely, we acquire the hadronic spectral densities ρ H (s ′ , s, u) through triple dispersion relation, where the ∆ ′2 s , ∆ 2 s and ∆ 2 u are thresholds, we use the notation H to represent the components Π DD AV (p ′2 , p 2 , q 2 ), Π D * D AV (p ′2 , p 2 , q 2 ), • • • at the hadron side.
We match the hadron side with the QCD side bellow the continuum thresholds to acquire rigorous quark-hadron duality [30,31], where the s 0 and u 0 are the continuum thresholds, we accomplish the integral over ds ′ firstly, and introduce some unknown parameters, such as the C DD AV , C D * D AV , C D * D * AV , • • • , to parameterize the contributions involving the higher resonances and continuum states in the s ′ channel.We set p ′2 = p 2 in the correlation functions Π(p ′2 , p 2 , q 2 ), and carry out the double Borel transform in regard to the variables P 2 = −p 2 and Q 2 = −q 2 respectively, then set the Borel parameters T 2 1 = T 2 2 = T 2 to acquire thirty QCD sum rules, With the simple replacements AV → V A and S V , we obtain the corresponding QCD sum rules for the currents J V A −,µ (0) and J S V −,µν (0), except for the η c ω channel, where the explicit expressions of the QCD side of other QCD sum rules are given in the Appendix.We take the unknown parameters C DD AV , C D * D AV , C D * D * AV , • • • as free parameters, and adjust the suitable values to obtain flat Borel platforms for the hadronic coupling constants [30,31,32,36,37,38].
The two-body strong decays of the three Y states take place through either S-wave or P-wave, and the partial decay widths are proportional to | p| or | p| 3 , and depend on the masses of the final-state mesons, where the p is the three-momentum in center-of-mass of the Y states.From the Particle Data Group, we can see clearly that the uncertainties of the masses are very small except for the f 0 (500), we would take the central values of the masses for all the mesons in a unform manner.As for the D * 0 (2300), its nature is still under debate, we take the masses of the cousins D 0 and D 1 from the QCD sum rules.Although the uncertainty of the mass of the f 0 (500) is rather large, however, the partial decay width is rather small, taking the central value would not lead to much uncertainties of the total widths.Accordingly, we take the central values of the decay constants and continuum threshold parameters, which are determined from the two-point correlation function QCD sum rules.
Furthermore, we take the continuum threshold parameters s 0 GeV) 2 , s 0 χc0 = (3.9GeV) 2 , s 0 χc1 = (4.0GeV) 2 from the two-point QCD sum rules combined with the experimental data [41,44,45].On the one hand, we can reproduce central values of the experimental masses, on the other hand, we can exclude the contaminations from the higher resonances and continuum states.In calculations, we only take account of the uncertainties of the quark masses and vacuum condensates, which are absorbed into the decay constants, pole residues and hadronic coupling constants reasonably to avoid doubly counting.
Generally speaking, we expect that the two-point and three-point correlation function QCD sum rules have the same Borel parameters, as the same masses and decay constants are involved.In practical calculations, different QCD sum rules have different Borel parameters.In the twopoint correlation function QCD sum rules, the quarks in the current J A (x) are contracted with the antiquarks in the current J † A (0) or JA (0), while in the three-point correlation function QCD sum rules, the quarks in the current J A (x) are contracted with the antiquarks in the current J B (y) or J † C (0) or JC (0), the spectral representations are quite different, it is not odd that the Borel parameters are also different.We would obtain flat platforms, which minimize the uncertainties.
In calculations, we fit the free parameters as to obtain the Borel windows, which are shown explicitly in Table 1.We obtain uniform flat platforms T 2 max − T 2 min = 1 GeV 2 , where the max and min denote the maximum and minimum, respectively.In Fig. 1, we plot the hadronic coupling constant G D1D AV with variation of the Borel parameter at large interval as an example.In the Borel windows, there appear very flat platforms indeed, now we would extract the hadron coupling constants.
We estimate the uncertainties of the hadronic coupling constants routinely.For an input parameter ξ, ξ = ξ + δξ, the left side of the QCD sum rules can be written as After taking into account the uncertainties, we obtain the values of the hadronic coupling constants, which are shown explicitly in Table 1.Then we obtain the partial decay widths routinely, and show them explicitly in Table 2.
Finally, we saturate the total widths with the summary of the partial decay widths, The widths of the Y (4484), Y (4469) and Y (4544) are 111.1 ± 30.1 ± 15.2 MeV, 246.3 ± 36.7 ± 9.4 MeV and 116.1 ± 33.5 ± 1.7 MeV, respectively, from the BESIII collaboration [1, 2, 3], which are compatible with the present calculations in magnitude.Generally speaking, the partial decay widths shown in Table 2 are not independent, and have some correlations with each other, as the masses of the relevant mesons could affect the phase-space differently in different channels.As a first step screen, we take the central values of the hadron masses and estimate the uncertainties of the partial decay widths using δΓ ∝ δG 2 H = 2G H δG H , then we take square root of the sum of their squares, which maybe underestimate the uncertainties of the total widths.In addition, we have set 9.93 ± 0.84 1.92 ± 0.13 4.10 ± 0.27 4.51 ± 0.97 have the largest partial decay width 59.7 ± 5.5 MeV; while the decay, has zero partial decay width.For the Y V A state, the decays, have the largest partial decay width 66.3 ± 6.1 MeV; while the decay, has zero partial decay width.For the Y S V state, the decay, has the largest partial decay width 96.0 ± 34.8 MeV; while the decay, has the partial decay width 18.0 ± 6.3 MeV.We can search for the Y (4500) in those typical decays to diagnose the nature of the Y states.

Conclusion
In the present work, we suppose that there exist three vector hidden-charm tetraquark states with the quantum numbers J P C = 1 −− at the energy about 4.5 GeV, and investigate the two-body strong decays systematically.We carry out the operator product expansion up to the vacuum condensates of dimension 5 and neglect the small gluon condensate contributions, then we acquire thirty QCD sum rules for the hadronic coupling constants based on the rigorous quark-hadron duality proposed in our previous work.Finally we obtain the partial decay widths, and therefore the total decay widths as a summary approximately, which are about 200 MeV and are compatible with the experimental data of the Y (4500) from the BESIII collaboration.The Y (4500) may be one vector tetraquark state having three main Fock components, or consists of three vector tetraquark states Y (4484), Y (4469) and Y (4544).By searching for the typical decay modes Y → , η c ω, J/ψω, we can diagnose the nature of the Y (4500).

Figure 1 :
Figure 1: The hadron-coupling constant G D1D AV via the Borel parameter.

Table 1 :
The Borel parameters and hadronic coupling constants.
approximately, and attribute all the uncertainties originating from the input parameters to the hadronic coupling constants δGH GH , we obtain, Γ Y AV = 241.6 ± 36.0 MeV , Γ Y V A = 210.6 ± 37.6 MeV , Γ Y S V = 229.4± 159.6 MeV ,

Table 2 :
The partial decay widths.whichoverestimate the uncertainties.From Table2, we can obtain the typical decay modes.For the Y AV state, the decays,