Analysis of the $D\bar{D}^*K$ system with QCD sum rules

In this article, we construct the color singlet-singlet-singlet interpolating current with $I\left(J^P\right)=\frac{3}{2}\left(1^-\right)$ to study the $D\bar{D}^*K$ system through QCD sum rules approach. In calculations, we consider the contributions of the vacuum condensates up to dimension-16 and employ the formula $\mu=\sqrt{M_{X/Y/Z}^{2}-\left(2{\mathbb{M}}_{c}\right)^{2}}$ to choose the optimal energy scale of the QCD spectral density. The numerical result $M_Z=4.71_{-0.11}^{+0.19}\,\rm{GeV}$ indicates that there exists a resonance state $Z$ lying above the $D\bar{D}^*K$ threshold to saturate the QCD sum rules. This resonance state $Z$ may be found by focusing on the channel $J/\psi \pi K$ of the decay $B\longrightarrow J/\psi \pi \pi K$ in the future.


Introduction
Since the observation of the X(3872) by the Belle collaboration in 2003 [1], more and more exotic hadrons have been observed and confirmed experimentally, such as the charmonium-like X, Y , Z states, hidden-charm pentaquarks, etc [2,3,4]. Those exotic hadron states, which cannot be interpreted as the quark-antiquark mesons or three-quark baryons in the naive quark model [5], are good candidates of the multi-quark states [6,7]. The multi-quark states are color-neutral objects because of the color confinement, and provide an important platform to explore the low energy behaviors of QCD, as no free particles carrying net color charges have ever been experimentally observed. Compared to the conventional hadrons, the dynamics of the multi-quark states is poorly understood and calls for more works.
Some exotic hadrons can be understood as hadronic molecular states [8], which are analogous to the deuteron as a loosely bound state of the proton and neutron. The most impressive example is the original exotic state, the X(3872), which has been studied as the DD * molecular state by many theoretical groups [9,10]. Another impressive example is the P c (4380) and P c (4450) pentaquark states observed by the LHCb collaboration in 2015, which are good candidates for thē DΣ * c ,D * Σ c ,D * Σ * c molecular states [8]. In additional to the meson-meson type and meson-baryon type molecular state, there maybe also exist meson-meson-meson type molecular states, in other words, there maybe exist three-meson hadronic molecules.
In Refs. [11,12], the authors explore the possible existence of three-meson system DD * K molecule according to the attractive interactions of the two-body subsystems DK,DK, D * K, D * K and DD * with the Born-Oppenheimer approximation and the fixed center approximation, respectively. In this article, we study the DD * K system with QCD sum rules.
The QCD sum rules method is a powerful tool in studying the exotic hadrons [13,14,15,16], and has given many successful descriptions, for example, the mass and width of the Z c (3900) have been successfully reproduced as an axialvector tetraquark state [17,18]. In QCD sum rules, we expand the time-ordered currents into a series of quark and gluon condensates via the operator product expansion method. These quark and gluon condensates parameterize the non-perturbative properties of the QCD vacuum. According to the quark-hadron duality, the copious information about the hadronic parameters can be obtained on the phenomenological side [19,20].
In this article, the color singlet-singlet-singlet interpolating current with I J P = 3 2 (1 − ) is constructed to study the DD * K system. In calculations, the contributions of the vacuum condensates are considered up to dimension-16 in the operator product expansion and the energy-scale 2 is used to seek the ideal energy scale of the QCD spectral density.
The rest of this article is arranged as follows: in Sect.2, we derive the QCD sum rules for the mass and pole residue of the DD * K state; in Sect.3, we present the numerical results and discussions; Sect.4 is reserved for our conclusion.
2 QCD sum rules for the DD * K state In QCD sum rules, we consider the two-point correlation function, where the m, n, k are color indexes. The color singlet-singlet-singlet current operator J µ (x) has the same quantum numbers I J P = 3 2 (1 − ) as the DD * K system. On the phenomenological side, a complete set of intermediate hadronic states, which has the same quantum numbers as the current operator J µ (x), is inserted into the correlation function Π µν (p) to obtain the hadronic representation [19,20]. We isolate the ground state contribution Z from the pole term, and get the result: where the pole residue λ Z is defined by 0|J µ (0)|Z(p) = λ Z ε µ , the ε µ is the polarization vector of the vector hexaquark state Z. At the quark level, we calculate the correlation function Π µν (p) via the operator product expansion method in perturbative QCD. The u, d, s and c quark fields are contracted with the Wick theorem, and the following result is obtained: where the U ij (x), D ij (x), S ij (x) and C ij (x) are the full u, d, s and c quark propagators, respectively. We give the full quark propagators explicitly in the following, (the P ij (x) denotes the U ij (x) or and t n = λ n 2 , the λ n is the Gell-Mann matrix [20]. We compute the integrals in the coordinate space for the light quark propagators and in the momentum space for the charm quark propagators, and obtain the QCD spectral density ρ(s) via taking the imaginary part of the correlation function: ρ(s) = lim ε→0 ImΠ(s+iε) π [17]. In the operator product expansion, we take into account the contributions of vacuum condensates up to dimension-16, and keep the terms which are linear in the strange quark mass m s . We take the truncation k ≤ 1 for the operators of the order O(α k s ) in a consistent way and discard the perturbative corrections. Furthermore, the condensates qq αsGG π , qq 2 αsGG π and qq 3 αsGG π play a minor important role and are neglected. According to the quark-hadron duality, we match the correlation function Π(p 2 ) gotten on the hadron side and at the quark level below the continuum threshold s 0 , and perform Borel transform with respect to the variable P 2 = −p 2 to obtain the QCD sum rule: where the QCD spectral density the subscripts 0, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16 denote the dimensions of the vacuum condensates, the T 2 is the Borel parameter, the lengthy and complicated expressions are neglected for simplicity. However, for the explicit expressions of the QCD special densities, the interested readers can obtain them through emailing us. We derive Eq.(9) with respect to 1 T 2 , and eliminate the pole residue λ Z to extract the QCD sum rule for the mass:
For the hadron mass, it is independent of the energy scale because of its observability. However, in calculations, the perturbative corrections are neglected, the operators of the orders O n (α k s ) with k > 1 or the dimensions n > 16 are discarded, and some higher dimensional vacuum condensates are factorized into lower dimensional ones therefore the corresponding energy-scale dependence is modified. We have to take into account the energy-scale dependence of the QCD sum rules.
In Refs. [17,22,23,24], the energy-scale dependence of the QCD sum rules is studied in detail for the hidden-charm tetraquark states and molecular states, and an energy scale formula µ = 2 is come up with to determine the optimal energy scale. This energy-scale formula enhances the pole contribution remarkably, improves the convergent behaviors in the operator product expansion, and works well for the exotic hadron states. In this article, we explore the DD * K state Z through constructing the color singlet-singlet-singlet type current based on the color-singlet qq substructure. For the two-meson molecular states, the basic constituent is also the color-singlet qq substructure [24]. Hence, the previous works can be extended to study the DD * K state. We employ the energy-scale formula µ = M 2 X/Y /Z − (2M c ) 2 with the updated value of the effective c-quark mass M c = 1.85 GeV to take the ideal energy scale of the QCD spectral density. At the present time, no candidate is observed experimentally for the hexaquark state Z with the symbolic quark constituent ccdūsū. However, in the scenario of four-quark states, the Z c (3900) and Z(4430) can be tentatively assigned to be the ground state and the first radial excited state of the axialvector four-quark states, respectively [25], while the X(3915) and X(4500) can be tentatively assigned to be the ground state and the first radial excited state of the scalar four-quark states, respectively [26]. By comparison, the energy gap is about 0.6 GeV between the ground state and the first radial excited state of the hidden-charm four-quark states. Here, we suppose the energy gap is also about 0.6 GeV between the ground state and the first radial excited state of the hiddencharm six-quark states, and take the relation √ s 0 = M Z + (0.4 − 0.6) GeV as a constraint to obey.
In Eq.(11), there are two free parameters: the Borel parameter T 2 and the continuum threshold parameter s 0 . The extracted hadron mass is a function of the Borel parameter T 2 and the continuum threshold parameter s 0 . To obtain a reliable mass sum rule analysis, we obey two criteria to choose suitable working ranges for the two free parameters. One criterion is the pole dominance on the phenomenological side, which requires the pole contribution (PC) to be about (40 − 60)%. The PC is defined as: The other criterion is the convergence of the operator product expansion. To judge the convergence, we compute the contributions of the vacuum condensates D(n) in the operator product expansion with the formula: where the n is the dimension of the vacuum condensates. In Fig. 1 6 MeV, which indicates that the Z is probably a resonance state. For some exotic resonances, the authors have combined the effective range expansion, unitarity, analyticity and compositeness coefficient to probe their inner structure in Refs. [27,28]. Their studies indicated that the underlying two-particle component (in the present case, corresponding to three-particle component) play an important or minor role, in other words, there are the other hadronic degrees of freedom inside the corresponding resonance. Hence, a resonance state embodies the net effect. Considering the conservation of the angular momentum, parity and isospin, we list out the possible hadronic decay patterns of the hexaquark state Z: Z −→ J/ψπK, η c ρ(770)K, DD * K.
To search for the X(3872), Belle, BaBar and LHCb have collected numerous data in the decay B −→ J/ψππK. Thus, the hexaquark state Z may be found by focusing on the most easy channel J/ψπK in the experiment.

Conclusion
In this article, we construct the color singlet-singlet-singlet interpolating current operator with I J P = 3 2 (1 − ) to study the DD * K system through QCD sum rules approach by taking into account the contributions of the vacuum condensates up to dimension-16 in the operator product expansion. In numerical calculations, we saturate the hadron side of the QCD sum rule with a hexaquark molecular state, employ the energy-scale formula µ = M 2 X/Y /Z − (2M c ) 2 to take the optimal energy scale of the QCD spectral density, and seek the ideal Borel parameter T 2 and continuum threshold s 0 by obeying two criteria of QCD sum rules for multi-quark states. Finally, we obtain the mass and pole residue of the corresponding hexaquark molecular state Z. The predicted mass, M Z = 4.71 +0.19 −0.11 GeV, which lies above the DD * K threshold, indicates that the Z is probably a resonance state. This resonance state Z may be found by focusing on the channel J/ψπK of the decay B −→ J/ψππK in the future.