Analysis of the $\overline{\Xi}_{cc}\Xi_{cc}$ hexaquark molecular state with the QCD sum rules

In this work, we construct the color-singlet-color-singlet type six-quark pseudoscalar current to investigate the $\overline{\Xi}_{cc}\Xi_{cc}$ hexaquark molecular state with the QCD sum rules, the predicted mass $M_X \sim 7.2\,\rm{GeV}$ supports assigning the $X(7200)$ to be the $\overline{\Xi}_{cc}\Xi_{cc}$ hexaquark molecular state with the quantum numbers $J^{PC}=0^{-+}$. The $X(7200)$ can decay through fusions of the $\bar{c}c$ and $\bar{q}q$ pairs, we can search for the $X(7200)$ and explore its properties in the $J/\psi J/\psi$, $\chi_{c1}\chi_{c1}$, $D^*\bar{D}^*$ and $D_1\bar{D}_1$ invariant mass spectrum in the future.


Introduction
In 2017, the LHCb collaboration observed the doubly charmed baryon state Ξ ++ cc in the Λ + c K − π + π + invariant mass distribution [1]. The observation of the Ξ ++ cc makes a great progress on the spectroscopy of the doubly charmed baryon states, tetraquark states and pentaquark states.
In 2020, the LHCb collaboration investigated the J/ψJ/ψ invariant mass distributions and observed a narrow structure about 6.9 GeV and a broad structure just above the J/ψJ/ψ threshold at p T > 5.2 GeV with the global significances larger than 5σ [2]. The Breit-Wigner mass and width of the X(6900) are M X = 6905 ± 11 ± 7 MeV and Γ X = 80 ± 19 ± 33 MeV, respectively. Furthermore, they also observed some vague structures around 7.2 GeV, which coincides with the Ξ cc Ξ cc threshold 7242.4 MeV [2]. The energy is sufficient to create a baryon-antibaryon pair Ξ cc Ξ cc containing the valence quarks ccqccq.
In the dynamical diquark model, the first radial excited states of the D-wave tetraquark states with the valence quarks cccc have the masses about 7.2 GeV, the insignificant enhancement X(7200) may be a combination of some 2P and (or) 2D diquark-antidiquark type cccc states with threshold effects of the transitions Ξ cc Ξ cc → J/ψJ/ψ [3]. The assignment of the Y (4630) serves as a benchmark, in this model, the Y (4630) can be assigned to be a diquark-antidiquark type tetraquark state, fragmentation of the color flux-tube connecting the diquark-antidiquark pair cqcq leads to the lowest-lying baryon-antibaryon pair Λ + c Λ − c [4]. Just like the Y (4630), fragmentation of the color flux-tube connecting the diquark-antidiquark pair cccc in the X(7200) leads to the lowest-lying baryon-antibaryon pair Ξ cc Ξ cc , then translates to the J/ψJ/ψ pair.
In the V-baryonium tetraquark scenario and the string-junction scenario, which share the same feature, the exotic X, Y and Z states are genuine tetraquarks rather than molecular states, they have a baryonic vertex (or string-junction) attaching a cc-diquark in color antitriplet, which is connected by a string to an anti-baryonic vertex (or string-junction) attaching acc-antidiquark in color triplet. In the V-baryonium tetraquark scenario, the un-conformed structure X(7200) is assigned to be the first radially excited state of the X(6900), the decays to the baryon-antibaryon pair Ξ cc Ξ cc can occur via breaking the string and creating a quark-antiquark pair [5]. In the string junction scenario, the X(6900) and X(7200) are assigned to be the 2S tetraquark states with the quantum numbers J P C = 0 ++ and 2 ++ , respectively [6].
If the Ξ ( * ) cc Ξ ( * ) cc system and D ( * )D( * ) system are related to each other via heavy antiquarkdiquark symmetry, several molecular states can be predicted based on the contact-range effective field theory, the Ξ cc Ξ cc molecular states which have the quantum numbers J P C = 0 −+ and (or) 1 −− maybe contribute to the X(7200) [7].
The exotic X, Y , Z and P states always lie near two-particle thresholds, such as In Eq.(4), we have neglected the two-particle scattering state contributions, just like in the QCD sum rules for the tetraquark (molecular) states. In Ref. [18], Lucha, Melikhov and Sazdjian obtain the conclusion that "A possible exotic tetraquark state may appear only in N c -subleading contributions to the QCD Green functions" based on the naive large-N c counting. An explicit analysis by S. Narison et al at the real word N c = 3 shows the opposite due to the induced loop factor missed in a naive N c counting rule [19]. In fact, as pointed in Ref. [20], without excluding the contributions of factorizable Feynman diagrams in the color space to the QCD sum rules by hand, we cannot obtain the conclusion that the factorizable parts of the operator product expansion series cannot have any relationship to the possible tetraquark bound states. For the baryon, tetraquark, pentaquark states with string junctions [21], the standard large-N c counting rules for the ordinary mesons cannot be trivially extrapolated to the exotic hadrons but should be modified for being properly applied at the real word N c = 3. All in all, we can examine the predictions of the multiquark states based on the QCD sum rules in different channels in the future.
In the QCD side, we carry out the operator product expansion up to the vacuum condensates of dimension 10 in a consistent way. There are four heavy quark propagators and two light quark propagators in the correlation function Π(p 2 ) after accomplishing the Wick's contractions. 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 G µν g s G αβ g s G ρσ g s G λτq qqq, which is of dimension 14, we should take account of the vacuum condensates of dimensions up to dimension 14. In the QCD sum rules for the hidden-charm or hidden-bottom tetraquark (molecular) states, pentaquark (molecular) states, we usually take the truncation O(α k s ) with k ≤ 1 [10,13,22,23]. If we also take the truncation k ≤ 1 to discard the quark-gluon operators of the orders O(α >1 s ) in the present work, the operator product expansion is terminated at the vacuum condensates of dimension 10.
In the QCD sum rules for the triply-charmed diquark-diquark-diquark type hexaquark states and triply-charmed dibaryon states, there are three light quark propagators and three heavy quark propagators in the correlation functions after accomplishing the Wick's contractions [15,24]. Again, if each heavy quark line emits a gluon and each light quark line contributes quark-antiquark pair, we obtain a quark-gluon operator g s G µν g s G αβ g s G λτq qqqqq, which is of dimension 15. If we take the truncations k ≤ 1, the operator product expansion is terminated at the vacuum condensates of dimension 13. The operator g s G µν g s G αβ g s G λτq qqqqq leads to the vacuum condensates αsGG π qq 2 qg s σGq , g 3 s GGG qq 3 and qg s σGq 3 , we calculate the vacuum condensate qg s σGq 3 , and observe that its small contribution can be neglected safely [15,24]. The vacuum condensates g 3 s GGG qq 3 and αsGG π qq 2 qg s σGq receive additional suppressions due to the small contributions of the gluon condensate and three-gluon condensate. In Ref. [25], we re-explore the mass spectrum of the ground state triply-heavy baryon states with the QCD sum rules by taking account of the three-gluon condensate g 3 s GGG for the first time, and observe that the contributions of the three-gluon condensate g 3 s GGG are tiny indeed. In summary, the truncations k ≤ 1 work very good.
In the present wok, we take account of the vacuum condensates qq , qg s σGq , qq 2 , qq qg s σGq , s ), respectively, and are neglected, direct calculations indicate that those contributions are tiny indeed [26].
We obtain the spectral density at the quark level through dispersion relation, take the quarkhadron duality below the continuum threshold s 0 , and perform Borel transform in regard to the variable P 2 = −p 2 to obtain the QCD sum rules: where the ρ QCD (s) is the spectral density at the quark level. We derive Eq.(6) with respect to τ = 1 T 2 , then eliminate the pole residue λ X , and obtain the QCD sum rules for the mass of the pseudoscalar hexaquark molecular state Ξ cc Ξ cc , 3 Numerical results and discussions  [32]. In addition, we take account of the energy-scale dependence of all the input parameters [33], , Λ = 213 MeV, 296 MeV and 339 MeV for the quark flavor numbers n f = 5, 4 and 3, respectively [32]. In the present work, we explore the hidden-charm hexaquark molecular state Ξ cc Ξ cc and choose n f = 4, then evolve all the input parameters to a typical energy scale µ to extract the hexaquark molecule mass. Furthermore, we present the predictions based on the updated parameters obtained by S. Narison, m c (m c ) = (1.266 ± 0.006) GeV and αsGG π = 0.021 ± 0.001 GeV 4 [34], and in this case the energy scales of other vacuum condensates are taken at µ = 1 GeV.
We should choose suitable continuum threshold s 0 to avoid contamination from the first radial excited state. In the scenario of the tetraquark states, the possible assignments of the exotic states Z c (3900), Z c (4430), X(3915), X(4500), Z c (4020), Z c (4600), X(4140) and X(4685) are presented plainly in Table 1 according to the (possible) quantum numbers, decay modes and energy gaps. From the table, we can obtain the conclusion tentatively that the energy gaps between the ground states and first radial excited states of the hidden-charm tetraquark states are about 0.58 GeV. We can choose the continuum threshold parameter as √ s 0 = M X + 0.6 ± 0.1 GeV. If the mass of the Ξ cc Ξ cc hexaquark molecular state and the energy scale of the spectral density at the quark  Table 2: The energy scale of the QCD spectral density, Borel parameter, pole contribution, mass and pole residue of the hexaquark molecular state Ξ cc Ξ cc , where the superscript * denotes the c-quark mass and gluon condensate are taken from Ref. [34].
level satisfies energy scale formula M X = µ 2 + (4M c ) 2 , the lower bound of the mass M X ≥ (1 GeV) 2 + (4 × 1.85 GeV) 2 = 7.47 GeV > 2M Ξcc . Now let us suppose that the multiquark states X, Y , Z and P have N Q + N q valence quarks, where the N Q and N q are the numbers of the heavy quarks and light quarks, respectively. Generally speaking, if N Q ≤ N q , we can apply the energy scale formula µ = M 2 X/Y /Z/P − (N Q M Q ) 2 to enhance the pole contributions and improve the convergent behavior of the operator product expansion [10,13,15,22,23,24]. In the present case, N Q = 4 > N q = 2, the energy scale formula is not applicable.
After trial and error, we obtain the continuum threshold parameter √ s 0 = 7.8 ± 0.1 GeV, and Borel parameters which are shown in Table 2  Then we take account of all uncertainties of the parameters, and obtain the values of the mass and pole residue of the pseudoscalar hexaquark molecular state Ξ cc Ξ cc , which are shown explicitly in Table 2 and Fig.1. From Fig.1, we can see that the predicted mass is rather stable with variation of the Borel parameter, the uncertainty comes from the Borel parameter is rather small. From Table 2, we can see that the predicted mass M X is almost independent on the energy scale of the QCD spectral density, while the pole residue depends heavily on the energy scale of the QCD spectral density, which is qualitatively consistent with the evolution behavior of the current operator from the re-normalization group equation, J(x, µ) = L γJ J(x, µ 0 ) and λ X (µ) = L γJ λ X (µ 0 ), where L = αs(µ0) αs(µ) , and the γ J is the anomalous dimension of the current operator J(x). The predicted mass M X ∼ 7.2 GeV is compatible with the vague structure around 7.2 GeV in the J/ψJ/ψ invariant mass spectrum observed by the LHCb collaboration [2], and supports assigning the X(7200) as the Ξ cc Ξ cc hexaquark molecular state with the quantum numbers J P C = 0 −+ . On the other hand, direct calculations based on the QCD sum rules [44] do not support assigning the X(7200) as the first radial excited state of the diquark-antidiquark-type (3 c 3 c -type) cccc tetraquark state claimed in Refs. [3,5,6].

Conclusion
In the present work, we construct the color-singlet-color-singlet type six-quark pseudoscalar current to interpolate the Ξ cc Ξ cc hexaquark molecular state, then accomplish the operator product expansion by calculating the vacuum condensates up to dimension 10, which are vacuum expectation values of the quark-gluon operators of the orders O(α k s ) with k ≤ 1, and obtain the QCD sum rules for the mass and pole residue. The predicted mass M X ∼ 7.2 GeV supports assigning the X(7200) to be the Ξ cc Ξ cc hexaquark molecular state with the quantum numbers J P C = 0 −+ . Moreover, direct calculations based on the QCD sum rules do not support assigning the X(7200) as the first radial excited state of the diquark-antidiquark-type (3 c 3 c -type) cccc tetraquark state. The decays of the X(7200) can take place through fusions of thecc andqq pairs, we can search for the Ξ cc Ξ cc hexaquark molecular state and explore its properties in the J/ψJ/ψ, χ c1 χ c1 , D * D * and D 1D1 invariant mass spectrum at the BESIII, LHCb, Belle II, CEPC, FCC and ILC in the future.