The two-body strong decays of the fully-charm tetraquark states

We study the hadronic coupling constants in the two-body strong decays of the fully-charm tetraquark states with the $J^{PC}=0^{++}$, $1^{+-}$ and $2^{++}$ via the QCD sum rules based on rigorous quark-hadron duality. Then we obtain the hadronic coupling constants and partial decay widths therefore total decay widths, which support assigning the $X(6552)$ as the first radial excitation of the scalar tetraquark state. And other predictions can be used to diagnose exotic states in the future.


Introduction
In 2020, the LHCb collaboration explored the di-J/ψ invariant mass spectrum using the pp collision data at centre-of-mass energies of √ s = 7, 8 and 13 TeV, which corresponds to an integrated luminosity of 9 fb −1 [1].They observed a narrow structure at about 6.9 GeV and a broad structure above the di-J/ψ threshold ranging from 6.2 to 6.8 GeV, and they also observed some vague structures around 7.2 GeV.It is the first time to observe the fully-heavy tetraquark candidates.
In Refs.[6,14], we study the mass spectrum of the scalar, axialvector, vector and tensor fullyheavy tetraquark states in the framework of the QCD sum rules.Subsequently, we explore the mass spectrum of the first radial excitations of the fully-charm tetraquark states, and resort to the Regge trajectories to obtain the masses of the second radial excitations, and make possible assignments of the LHCb's new resonances [16].In Ref. [23], we re-study the mass spectrum of the ground state and first/second/third radial excited diquark-antidiquark type fully-charm tetraquark states, then we take account of the new CMS and ATLAS experimental data and try to make possible assignments of the X(6600), X(6900) and X(7300) consistently.
Because we cannot assign a hadron by the mass alone, we should explore the decays at least.In the present work, we explore the two-body strong decays of the ground states and first radial excitations of the scalar, axialvector and tensor fully-charm tetraquark states in the framework of the QCD sum rules based on rigorous quark-hadron duality, which works well in studying the hadronic coupling constants [33,34,35,36,37], and make more robust assignments based on the masses and widths together.
The article is organized as follows: we obtain the QCD sum rules for the hadronic coupling constants of the fully-charm tetraquark states in Section 2; in Section 3, we present the numerical results and discussions; and Section 4 is reserved for our conclusion.

QCD sum rules for the hadronic coupling constants
Let us write down the three-point correlation functions firstly, where the currents interpolate the charmonia η c , J/ψ, χ c1 and h c , respectively, the currents interpolate the fully-charm tetraquark states with the spins 0, 1 and 2, respectively [14,16,23], the i, j, k, m and n are color indexes.We take those correlation functions to study the hadronic coupling constants , and G X (′) 2 J/ψJ/ψ , respectively.At the hadron side of the correlation functions, see Eqs.( 7)-( 9), we insert a complete set of intermediate hadronic states with the same quantum numbers as the currents, and isolate the ground state contributions in the charmonium channels, and the ground state plus first radial excited state contributions in the tetraquark channels explicitly [38,39], where the decay constants or pole residues are defined by, , the hadronic coupling constants are defined by, the ξ µ , ζ µ , ε µ and ε µν are the polarization vectors of the corresponding charmonium/tetraquark states.As the currents J hc αβ (x) and J 1 αβ (x) couple potentially to the charmonia/tetraquarks with both the J P C = 1 +− and 1 −− , we introduce the projector P µναβ A (p) to project out the states with the J P C = 1 +− [14,16,23].In this work, we explore the hadronic coupling constants with the components Π i (p ′2 , p 2 , q 2 ) with i = 1, • • • , 7 to avoid contaminations.
Generally speaking, if a quark current and a hadron have the same quantum numbers, then they have non-vanishing couplings with each other.For example, the currents J 0 (x), J 1 αβ (x) and Figure 1: The lowest order Feynman diagrams for the hadronic coupling constants G X0ηcηc and G χc0ηcηc , respectively, where the dashed lines denote the Cutkosky's cuts.
J 2 αβ (x) couple potentially to the 1S, 2S, • • • states of the fully-charm tetraquarks, we take into account the 1S and 2S states in the two-point QCD sum rules, and obtain the masses and poles residues by solving an univariate quadratic equation.For the technical details, one can consult Refs.[16,23].Now, we obtain the hadronic spectral densities ρ H (s ′ , s, u) through triple dispersion relation, where we add the subscript H to stand for the components Π i (p ′2 , p 2 , q 2 ) with i = 1 − 7 at the hadron side.Although the variables p ′ , p and q have the relation p ′ = p + q, it is feasible to take the p ′2 , p 2 and q 2 as free parameters to determine the spectral densities, and we can obtain a nonzero imaginary part indeed for all the variables p ′2 , p 2 and q 2 .At the QCD side, we take account of the perturbative terms and gluon condensate contributions, and obtain the QCD spectral densities of the components Π i (p ′2 , p 2 , q 2 ) through double dispersion relation, as we cannot obtain a nonzero imaginary part for the variable p ′2 .In Fig. 1, we plot the lowest order Feynman diagrams for the hadronic coupling constants G X0ηcηc and G χc0ηcηc as an example.We calculate those diagrams with the Cutkosky's rules, although the left diagram for the tetraquark state X 0 does not allow a nonzero imaginary part with respect to the s ′ + iǫ 3 , it does not forbid putting the s ′ on the mass-shell at the hadron side; while the right diagram for the traditional charmonium χ c0 does forbid putting the s ′ on the mass-shell at the hadron side.Thus, in the QCD sum rules for the traditional mesons, we have to put one of the variables p ′2 , p 2 and q 2 off mass-shell.At the hadron side, there is triple dispersion relation, see Eq.( 27), while at the QCD side, there is only double dispersion relation, see Eq.( 29), they do not match with each other channel by channel without performing some trick.We accomplish the integral over ds ′ firstly at the hadron side, then match the hadron side with the QCD side of the components Π i (p ′2 , p 2 , q 2 ) of the correlation functions bellow the continuum thresholds s 0 and u 0 to obtain rigorous quark-hadron duality [33,34], Now we write down the hadron representation explicitly, where we introduce the parameters C i with i = 1 − 7 to represent all the contributions concerning the higher resonances in the s ′ channel.We set p ′2 = 2p 2 in the components Π H (p ′2 , p 2 , q 2 ), and perform double Borel transform in regard to P 2 = −p 2 and Q 2 = −q 2 , respectively.The spectral densities ρ H (s ′ , s, u) and ρ QCD (s, u) are physical, while the variables p ′2 , p 2 and q 2 in Eq.( 31) are free variables, as we perform the operator product expansion at the large space-like regions −p 2 → ∞ and −q 2 → ∞.Generally speaking, we can set p ′2 = αp 2 or αq 2 with α to be a finite quantity.Considering the mass poles at s , s = m 2 ηc,J/ψ,χc,hc and u = m 2 ηc,J/ψ,χc,hc have the relations s ′ = 4s = 4u approximately, we can set α = 1 − 4. In calculations, we observe that the optimal value α = 2.
Then we set T 2 1 = T 2 2 = T 2 to obtain seven QCD sum rules, where we introduce the notations, the other six QCD sum rules are given explicitly in the Appendix.There are three unknown parameters in each QCD sum rules in Eq.( 39) and Appendix, we set the hadronic coupling constants /ψJ/ψ , and take the C i as free parameters and adjust the suitable values to obtain flat Borel platforms for the hadronic coupling constants [33,34,35,36,37].In calculations, we observe that there appear endpoint divergences  we can set δC 2 = 0 and approximately.Finally, we obtain the values of the hadronic coupling constants, 15 GeV, G X1hcηc = 0.30 +0.04 −0.03 GeV −1 , G X2ηcηc = 0.94 +0.17 −0.16 GeV −1 , G X2J/ψJ/ψ = 9.69 +1. 25  −1.02GeV.
Now we set m X0 = 6.20 GeV, m X ′ 0 = 6.57GeV, m X1 = 6.24GeV, m X ′ 1 = 6.64 GeV, m X2 = 6.27GeV, and m X ′ 2 = 6.69 GeV based on the calculations of the QCD sum rules [23].Then we obtain the partial decay widths directly, Then we obtain the total decay widths, by saturating the total widths with only the supposed dominant decays, which take place through the Okubo-Zweig-Iizuka super-allowed fall-apart mechanism.In fact, such approximations would underestimate the total widths slightly, as there maybe exist other non-dominant decays, for example, the two-body strong decays X 0 → D D and D * D * can take place through annihilating a cc pair and creating a q q pair, which are Okubo-Zweig-Iizuka forbidden.
The predicted width Γ X0 = 16.6 +6.0 −2.4 MeV is much smaller than the preliminary data Γ X(6220) = 0.31±0.12+0.07 −0.08 GeV from the ATLAS collaboration [3].The predicted width Γ X ′ 0 = 118.1 +32.2 −22.3 MeV is in very good agreement with the experimental data Γ X(6552) = 124 +32 −26 ± 33 MeV from the CMS collaboration [4], which indicates that the X(6552) can be assigned as the first radial excitation of the scalar tetraquark state.The resonances from the LHCb, ATLAS and CMS collaborations are observed in the J/ψJ/ψ or J/ψψ ′ invariant mass spectrum, the charge conjugation should be positive, the assignments as the tetraquark states with the J P C = 1 +− are not favored.
In Refs.[31,32], Agaev et al take the X(6600) and X(6900) as the ground state A Ā and P P type fully-charm scalar tetraquark states, respectively, and study the masses and two-body strong decays with the QCD sum rules, where the A and P denote the axialvector and pseudoscalar diquarks, respectively.While in Ref. [23], the X(6220) and X(6552) are assigned as the ground state and first radial excited state of the A Ā type scalar tetraquark states, respectively.By studying the two-body strong decays in the present work, we obtain further support to assign the X(6552) as the first radial excitation of the scalar tetraquark state.
All the predictions can be confronted to the experimental data in the future and serve as valuable guides for the high energy experiments.

Conclusion
In this work, we study the hadronic coupling constants in the two-body strong decays of the fully-charm tetraquark states with the J P C = 0 ++ , 1 +− and 2 ++ via the three-point correlation functions.We carry out the operator product expansion up to the gluon condensate and obtain the spectral representation through double-dispersion, then match the hadron side with the QCD side to obtain rigorous quark-hadron duality.We obtain the hadronic coupling constants and acquire the partial decay widths therefore the total decay widths of the fully-charm tetraquark states with the J P C = 0 ++ , 1 +− and 2 ++ , respectively.Our predictions support assigning the X(6552) as the first radial excitation of the scalar tetraquark state.Other predictions play a valuable role in diagnosing the X, Y and Z states and can be confronted to the experimental data in the future.

Figure 2 :
Figure 2: The central values of the hadronic coupling constants with variations of the Borel parameters T 2 , where the (I) and (II) denote the G X0ηcηc and G X0J/ψJ/ψ , respectively.