In this article, we construct the color singlet-singlet-singlet interpolating current with IJP=3/21- to study the DD¯⁎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 μ=MX/Y/Z2-2Mc2 to choose the optimal energy scale of the QCD spectral density. The numerical result MZ=4.71-0.11+0.19GeV indicates that there exists a resonance state Z lying above the DD¯⁎K threshold to saturate the QCD sum rules. This resonance state Z may be found by focusing on the channel J/ψπK of the decay B→J/ψππK in the future.

National Natural Science Foundation of China117750791. 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. [2–4]. 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 multiquark states [6, 7]. The multiquark 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 multiquark 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 [9–17]. Another impressive example is the Pc(4380) and Pc(4450) pentaquark states observed by the LHCb collaboration in 2015, which are good candidates for the D¯Σc∗, D¯∗Σc, D¯∗Σc∗ molecular states [8]. In additional to the meson-meson type and meson-baryon type molecular state, there may also exist meson-meson-meson type molecular states; in other words, there may exist three-meson hadronic molecules.

In [18, 19], the authors explore the possible existence of three-meson system DD¯∗K molecule according to the attractive interactions of the two-body subsystems DK, D¯K, 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 [20–25] and has given many successful descriptions; for example, the mass and width of the Zc(3900) have been successfully reproduced as an axial vector tetraquark state [26, 27]. 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 nonperturbative 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 [28, 29].

In this article, the color singlet-singlet-singlet interpolating current with IJP=3/21- 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 formula μ=MX/Y/Z2-2Mc2 is used to seek the ideal energy scale of the QCD spectral density.

The rest of this article is arranged as follows: in Section 2, we derive the QCD sum rules for the mass and pole residue of the DD¯∗K state; in Section 3, we present the numerical results and discussions; Section 4 is reserved for our conclusion.

2. QCD Sum Rules for the <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M34"><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mover accent="false"><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>∗</mml:mi></mml:mrow></mml:msup><mml:mi>K</mml:mi></mml:math></inline-formula> State

In QCD sum rules, we consider the two-point correlation function, (1)Πμνp=i∫d4xeip·x0TJμxJν†00,where(2)Jμx=u¯mxiγ5cmxc¯nxγμdnxu¯kxiγ5skx,and the m, n, and k are color indexes. The color singlet-singlet-singlet current operator Jμ(x) has the same quantum numbers IJP=3/21- 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 Jμ(x), is inserted into the correlation function Πμνp to obtain the hadronic representation [28, 29]. We isolate the ground state contribution Z from the pole term, and get the result:(3)Πμνp=λZ2MZ2-p2-gμν+pμpνp2+⋯=Πp2-gμν+pμpνp2+⋯,where the pole residue λZ is defined by 〈0|Jμ(0)|Z(p)〉=λZεμ, the εμ is the polarization vector of the vector hexaquark state Z.

At the quark level, we calculate the correlation function Πμνp 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:(4)Πμνp=-i∫d4xeip·xTrγμDnn′xγνCn′n-xTriγ5Cmm′xiγ5Um′m-xTriγ5Skk′xiγ5Uk′k-x-TrγμDnn′xγνCn′n-xTriγ5Cmm′xiγ5Um′k-xiγ5Skk′xiγ5Uk′m-x,where the Uij(x), Dij(x), Sij(x), and Cij(x) are the full u, d, s, and c quark propagators, respectively. We give the full quark propagators explicitly in the following, (the Pij(x) denotes the Uij(x) or Dij(x)), (5)Pijx=iδijx2π2x4-δijq¯q12-δijx2q¯gsσGq192-igsGαβntijnxσαβ+σαβx32π2x2-18q¯jσαβqiσαβ+⋯,(6)Sijx=iδijx2π2x4-δijms4π2x2-δijs¯s12+iδijxmss¯s48-δijx2s¯gsσGs192+iδijx2xmss¯gsσGs1152-igsGαβntijnxσαβ+σαβx32π2x2-18s¯jσαβsiσαβ+⋯,(7)Cijx=i2π4∫d4ke-ik·xk+mck2-mc2δij-gstijnGαβnk+mcσαβ+σαβk+mc4k2-mc22-gs2tntmijGαβnGμνnfαβμν+fαμβν+fαμνβ4k2-mc25+⋯,(8)fλαβ=k+mcγλk+mcγαk+mcγβk+mc,fαβμν=k+mcγαk+mcγβk+mcγμk+mcγνk+mc,and tn=λn/2; the λn is the Gell-Mann matrix [29]. 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)=limε→0ImΠ(s+iε)/π [26]. 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 ms. We take the truncation k≤1 for the operators of the order O(αsk) in a consistent way and discard the perturbative corrections. Furthermore, the condensates 〈q¯q〉〈αsGG/π〉, 〈q¯q〉2〈αsGG/π〉, and 〈q¯q〉3〈αsGG/π〉 play a minor important role and are neglected.

According to the quark-hadron duality, we match the correlation function Π(p2) gotten on the hadron side and at the quark level below the continuum threshold s0 and perform Borel transform with respect to the variable P2=-p2 to obtain the QCD sum rule:(9)λZ2exp-MZ2T2=∫4mc2s0dsρsexp-sT2,where the QCD spectral density is(10)ρs=ρ0s+ρ3s+ρ4s+ρ5s+ρ6s+ρ8s+ρ9s+ρ10s+ρ11s+ρ12s+ρ13s+ρ14s+ρ16s,and the subscripts 0, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, and 16 denote the dimensions of the vacuum condensates, the T2 is the Borel parameter, and 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 (9) with respect to 1/T2 and eliminate the pole residue λZ to extract the QCD sum rule for the mass:(11)MZ2=∫4mc2s0dsd/d-1/T2ρsexp-s/T2∫4mc2s0dsρsexp-s/T2.

3. Numerical Results and Discussions

In this section, we perform the numerical analysis. To extract the numerical values of MZ, we take the values of the vacuum condensates 〈q¯q〉=-(0.24±0.01GeV)3, 〈s¯s〉=(0.8±0.1)〈q¯q〉, 〈q¯gsσGq〉=m02〈q¯q〉, 〈s¯gsσGs〉=m02〈s¯s〉, m02=(0.8±0.1)GeV2, 〈αsGG/π〉=(0.33GeV)4 at the energy scale μ=1GeV [28–30], choose the MS¯ masses mc(mc)=(1.275±0.025)GeV, ms(μ=2GeV)=(0.095-0.003+0.009)GeV from the Particle Data Group [2], and neglect the up and down quark masses, i.e., mu=md=0. Moreover, we consider the energy-scale dependence of the input parameters on the QCD side from the renormalization group equation,(12)q¯qμ=q¯q1GeVαs1GeVαsμ4/9,s¯sμ=s¯s1GeVαs1GeVαsμ4/9,q¯gsσGqμ=q¯gsσGq1GeVαs1GeVαsμ2/27,s¯gsσGsμ=s¯gsσGs1GeVαs1GeVαsμ2/27,msμ=ms2GeVαsμαs2GeV4/9,mcμ=mcmcαsμαsmc12/25,αsμ=1b0t1-b1b02logtt+b12log2t-logt-1+b0b2b04t2,where t=logμ2/Λ2, b0=33-2nf/12π, b1=153-19nf/24π2, b2=2857-5033/9nf+325/27nf2/128π3, Λ=213MeV, 296MeV and 339MeV for the flavors nf=5, 4 and 3, respectively [2].

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 On(αsk) 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 [26, 31–34], 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=MX/Y/Z2-2Mc2 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 [33, 34]. Hence, the previous works can be extended to study the DD¯∗K state. We employ the energy-scale formula μ=MX/Y/Z2-2Mc2 with the updated value of the effective c-quark mass Mc=1.85GeV 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 cc¯du¯su¯. However, in the scenario of four-quark states, the Zc(3900) and Z(4430) can be tentatively assigned to be the ground state and the first radial excited state of the axial vector four-quark states, respectively [35], 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 [36, 37]. By comparison, the energy gap is about 0.6GeV 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 0.6GeV between the ground state and the first radial excited state of the hidden-charm six-quark states and take the relation s0=MZ+(0.4-0.6)GeV as a constraint to obey.

In (11), there are two free parameters: the Borel parameter T2 and the continuum threshold parameter s0. The extracted hadron mass is a function of the Borel parameter T2 and the continuum threshold parameter s0. 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(13)PC=∫4mc2s0dsρsexp-s/T2∫4mc2∞dsρsexp-s/T2.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:(14)Dn=∫4mc2s0dsρnsexp-s/T2∫4mc2s0dsρsexp-s/T2,where the n is the dimension of the vacuum condensates.

In Figure 1, we show the variation of the PC with respect to the Borel parameter T2 for different values of the continuum threshold parameter s0 at the energy scale μ=2.9GeV. From the figure, we can see that the value s0≤5.0GeV is too tiny to obey the pole dominance criterion and result in sound Borel window for the state Z. To warrant the Borel platform for the mass mZ, we take the value T2=(2.8-3.2)GeV2. In the above Borel window, if we choose the value s0=(5.1-5.3)GeV, the PC is about (39-63)%. The pole dominance condition is well satisfied.

The pole contribution with variation of the Borel parameter T2.

In Figure 2, we draw the absolute contribution values of the vacuum condensates |D(n)| at central values of the above input parameters. From the figure, we can observe that the contribution of the perturbative term D(0) is not the dominant contribution; the contributions of the vacuum condensates with dimensions 3, 6, 8, 9, and 11 are very great. If we take the contribution of the vacuum condensate with dimension 11 as a milestone, the absolute contribution values of the vacuum condensates |D(n)| decrease quickly with the increase of the dimensions n, and the operator product expansion converges nicely.

The absolute contributions of the vacuum condensates with dimension n in the operator product expansion.

Thus, we obtain the values T2=(2.8-3.2)GeV2, s0=(5.1-5.3)GeV and μ=2.9GeV for the state Z. Considering all uncertainties of the input parameters, we get the values of the mass and pole residue of the state Z:(15)MZ=4.71-0.11+0.19GeV,λZ=4.60-0.69+1.15×10-4GeV8,which are shown explicitly in Figures 3 and 4. Obviously, the energy-scale formula μ=MX/Y/Z2-(2Mc)2 and the relation s0=MZ+(0.4-0.6)GeV are also well satisfied. The central value MZ=4.71GeV is about 337MeV above the threshold MK+D+D¯∗=497.6+1865+2010=4372.6MeV, 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 [38, 39]. Their studies indicated that the underlying two-particle component (in the present case, corresponding to three-particle component) plays 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:(16)Z→J/ψπK,ηcρ770K,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 easiest channel J/ψπK in the experiment.

The mass with variation of the Borel parameter T2.

The pole residue with variation of the Borel parameters T2.

4. Conclusion

In this article, we construct the color singlet-singlet-singlet interpolating current operator with IJP=3/21- 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=MX/Y/Z2-2Mc2 to take the optimal energy scale of the QCD spectral density, and seek the ideal Borel parameter T2 and continuum threshold s0 by obeying two criteria of QCD sum rules for multiquark states. Finally, we obtain the mass and pole residue of the corresponding hexaquark molecular state Z. The predicted mass, MZ=4.71-0.11+0.19GeV, 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.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by National Natural Science Foundation, Grant Number 11775079.

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