Scalar Hidden-Charm Tetraquark States with QCD Sum Rules∗

Zun-Yan Di(狄尊燕)Zhi-Gang Wang(王志刚)† Jun-Xia Zhang(张君霞)and Guo-Liang Yu(于国梁)

1Department of Physics,North China Electric Power University,Baoding 071003,China

2School of Nuclear Science and Engineering,North China Electric Power University,Beijing 102206,China

1 Introduction

In recent years,a number of charged charmoniumlike(bottomonium-like)exotic states have been observed,such as theZc(3900),Zc(4020),Zc(4050),Zc(4055),Zc(4200),Zc(4240),Zc(4250),Z(4430),Zb(10610),andZb(10650).[1]Ifthosecharged charmonium-like(bottomonium-like)states are resonances indeed,their quark constituents must beirrespective of the diquark-antidiquark type or mesonmeson type substructures.

Those exotic states cannot be accommodated within the naive quark model,and represent a new facet of QCD and provide a new opportunity for a deeper understanding of the non-perturbative QCD.The QCD sum rules method is a powerful tool in studying the hidden-charm(bottom)tetraquark or molecular states and hidden-charm pentaquark states.[2−4]In the QCD sum rules,the operator product expansion is used to expand the time-ordered currents into a series of quark and gluon condensates,which parameterize the non-perturbative properties of the QCD vacuum.Based on the quark-hadron duality,we can obtain copious information about the hadronic parameters at the phenomenological side.[5−6]

The diquarkshave fi ve structures in Dirac spinor space,whereandCσµνfor the scalar,pseudoscalar,vector,axialvector and tensor diquarks,respectively.In this expression,qjdenotes the quark field;i,jandkare color indexes;Cis the charge conjugation matrix;and the superscript T denotes the transpose of the Dirac indexes.The attractive interactions of one-gluon exchange favor formation of the diquarks in color antitriplet, fl avor antitriplet and spin singlet,[7]while the favored con fi gurations are the scalar(Cγ5)and axialvector(Cγµ)diquark states.[8−9]We can construct the diquark-antidiquark type hidden charm tetraquark states,[10]

to study the scalar tetraquark states having larger masses.

In this article,we constructC⊗Candtype currents to explore the charged scalar hidden-charm tetraquark states by calculating the contributions of the vacuum condensates up to dimension-10 in a consistent way.

This article is organized as follows:in Sec.2,we derive the QCD sum rules to extract the masses and pole residues of the charged scalarcu¯c¯d(cu¯c¯s)tetraquark states;in Sec.3,we present the numerical results and discussions;section 4 is reserved for conclusion.

2 QCD Sum Rules for the Scalar Hidden-Charm Tetraquark States

In QCD sum rules,we consider the two-point correlation functions Π1,2,3,4(p),

where theJ1,2,3,4(x)are the interpolating currents with the same quantum numbers as the tetraquark states we want to study.Those currents are constructed in the diquark model and can be written as,

On the phenomenological side,we insert a complete set of intermediate hadronic states with the same quantum numbers as the current operatorsJ1,2,3,4(x)into the correlation functions Π1,2,3,4(p)to obtain the hadronic representations.[5−6]After isolating the ground state contributions of the scalar tetraquark states,we get the following results,

At the quark level,the two-point correlation functions Π1,2,3,4(p)can be evaluated via the operator product expansion method.We contract theu,d,candsquark fields with the wick theorem and obtain the following results:

where theUij(x),Dij(x),Sij(x),andCij(x)are the fullu,d,s,andcquark propagators,respectively.For simplicity,theUij(x)andDij(x)can be written asPij(x),

andis the Gell-Mann matrix,Dα=In Eqs.(13)–(14),we retain the termsand>originate from the Fierz re-arrangement of the>to absorb the gluons emitted from the heavy quark lines so as to extract the mixed condensates and four-quark condensatesrespectively.We compute the integrals both in the coordinate and momentum spaces by taking into account the contributions of the vacuum condensates up to dimension-10.Then,we obtain the QCD spectral densities from the imaginary parts of the correlations.

After getting the QCD spectral densities,we take the quark-hadron duality below the continuum thresholdss0and perform the Borel transformation with respect to the variableP2=−p2to obtain the QCD sum rules,

the explicit expressions of the spectral densitiesρ1,2,3,4(s)are given in the appendix.The subscripts 0,3,4,5,6,7,8,10 denote the dimensions of the vacuum condensates.We take into account the vacuum condensates,which are vacuum expections of the operators of the orderswithk≤1 consistently.

We derive Eq.(17)with respect to 1/T2,then eliminate the pole residuesλZ1,2,3,4,and obtain the expressions for the masses of the scalar tetraquark states,

Once the masses are obtained,we can take them as input parameters and obtain the pole residues from the QCD sum sules in Eq.(17).

3 Numerical Results and Discussions

In this section,we perform the numerical analysis,and choose the reasonable QCD parameters for the quark masses and vacuum condensates.The vacuum condensates are taken to be the standard values

at the energy scaleThe quark condensates and mixed quark condensates evolve with the renormalization group equation,

In addition,we take the valuesmu(µ=1 GeV)=md(µ=1 GeV)=mq(µ=1 GeV)=0.006 GeV from the Gell-Mann-Oakes-Renner relation,and choose themassmc(mc)=(1.275±0.025)GeV andms(µ=2 GeV)=(0.095±0.005)GeV from the Particle Data Group,[1]and take into account the energy-scale dependence of themasses from the renormalization group equation,

for the flavorsnf=5,4 and 3,respectively.[1]

The energy scaleµis considered as a variable.In Refs.[3–4],the energy scale dependence of the QCD sum rules for the hidden-charm tetraquark states and molecular states is studied in details for the first time,and an energy scale formula,

with the effectivec-quark mass Mcis suggested.The formula works well for theX(3872),Zc(3885/3900),X∗(3860),Y(3915),Zc(4020/4025),Z(4430),X(4500),Y(4630/4660),X(4700)in the scenario of tetraquark states.In this article,we take the updated value of the effectivec-quark mass Mc=1.82 GeV to determine the energy scales of the QCD spectral densities.[12]

The mass gaps between the ground states and the first radial excited states are usually taken as(0.4−0.6)GeV.For examples,theZ(4430)is tentatively assigned to be the first radial excitation of theZc(3900)according to the analogous decays,and the mass differences576 MeV,theX(3915)andX(4500)are assigned to be the ground state and the first radial excited state of the axialvector-diquark-axialvectorantidiquark type scalartetraquark states,respectively,and their mass difference is588 MeV.[16]The relation

serves as another constraint on the masses of the hiddencharm tetraquark states.In calculations,we observe that the values of the massesMZdecrease with increase of the energy scalesµfrom QCD sum rules in Eq.(19).While Eq.(21)indicates that the value of the massesMZincrease when the energy scalesµincrease.There exist optimal energy scales,which lead to reasonable massesMZ.

We should obey two criteria to choose the Borel parametersT2and threshold parameterss0in numerical calculations.The first criterion is the pole dominance on the phenomenological side.The pole contribution(PC)is defined by,

The second criterion is the convergence of the operator product expansion.To judge the convergence,we calculate the contributionsDiin the operator product expansion with the formula,

where the indexidenotes the dimension of the vacuum condensates.

In Fig.1,the contributions of the pole terms are plotted with variations of the Borel parametersT2for different values of the threshold parameterss0,where the values of energy scales are taken asµ=4.00 GeV,2.90 GeV,4.05 GeV and 2.95 GeV for the tetraquark statesZ1,Z2,Z3,Z4,respectively.From the fi gure,we can see that the continuum thresholds≤5.75 GeV,≤5.05 GeV for the tetraquark statesZ1,Z2,Z3,Z4respectively are too small to satisfy the pole dominance condition to result in reasonable Borel windows.

Fig.1 The pole contributions with variations of the Borel parameters T2and threshold parameters s0.

In Fig.2,the contributions of different condensate terms in the operator product expansion are plotted with the Borel parametersT2for the continuum thresholdsand 5.25 GeV at the energy scales,µ=4.00 GeV,2.90 GeV,4.05 GeV and 2.95 GeV for the tetraquark statesZ1,Z2,Z3,Z4,respectively.From the figure,we can see the contributionsDiexplicitly and choose reasonable Borel parameters.We take theZ1state as an example to illustrate the procedure.In that case,we observe that the perturbative term and the⟨¯qq>term play an important role,while the other condensate terms play a minor important role.At the valueT2≤4.9 GeV2,the perturbative term andterm decrease monotonously and quickly with increase of theT2,which cannot lead to stable masses and pole residues.At the valueT2=(5.0–5.4)GeV2,the convergent behavior in the operator product expansion is very good and the perturbative term makes the main contribution.We present the optimal energy scalesµ,ideal Borel parametersT2,continuum threshold parameterss0and pole contributions in Table 1.From the Table,we can see that the two criteria of the QCD sum rules can be satisfied.

We take into account all uncertainties of the input parameters,and obtain the masses and pole residues of the hidden-charm tetraquark states,which are shown explicitly in Figs.3–4 and Table 1.From Figs.3–4,we can see that the Borel platforms exist.On the other hand,from Table 1,we can see that the energy scale formulais well satisfied.We expect to make reasonable predictions,the present predictions can be confronted with the experimental data in the future.

Table 1 The energy scales,Borel parameters,continuum threshold parameters,pole contributions,masses and pole residues for the scalar tetraquark states.

Fig.2 The contributions of different terms in the operator product expansion with variations of the Borel parameters T2,where the 0,3,4,5,6,7,8,10 denote the dimensions of the vacuum condensates.

In the following,we discuss the possible hadronic decay patterns of thescalar tetraquark states.Being composed of a diquark and antidiquark pair,a hidden-charm tetraquark state can decay very easily into a pair of open-charmDmesons or one charmonium state plus a light meson through quark rearrangement.The strong decays are Okubo-Zweig-Iizuka super-allowed.Under the restrictions of the symmetries and the masses of the studied scalar tetraquark states obtained in Table 1,the possible two-body strong decay channels are

which are kinematically allowed.Theoretically,the mass of the ground state is lighter than the counterpart of its excited state,and the corresponding phase space is larger,thus the decay width of the resonance to the ground state is wider.This means that the ground state decay modes of the resonance can take place more easily.Besides,the excited sate is unstable,which brings some difficulty to the observation of the decay process for the resonance state.Consequently,the dominant decay modes of the scalar tetraquark states areWe can search for those scalar hidden-charm tetraquark states in those decay channels.

Fig.3 The masses with variations of the Borel parameters T2.

Fig.4 The pole residues with variations of the Borel parameters T2.

4 Conclusion

In this article,we study the pseudoscalar-diquarkpseudoscalar-antidiquark type and vector-diquark-vectorantidiquark type scalar hidden-charmtetraquark states with the QCD sum rules by calculating the contributions of the vacuum condensates up to dimension-10 in the operator product expansion.In numerical calculations,we use the energy scale formula

to determine the ideal energy scales of the QCD spectral densities and search for the optimal Borel parametersT2and continuum thresholdss0to satisfy the two criteria of the QCD sum rules(i.e.pole dominance on the phenomenological side and convergence of the operator product expansion).We obtain the masses and pole residues of the scalar hidden-charmtetraquark states.The predicted masses are around(5.43–5.45)GeV for theC⊗Ctype tetraquark states and(4.64–4.67)GeV for thetype ones,which can be confronted with the experimental data in the future.Moreover,we discuss the possible hadronic decay patterns of the two types of tetraquark states,and list their dominant decays. As the predicted masses of thetype tetraquark states are lighter than the counterparts of theC⊗Ctype ones and the two types of tetraquark states have the same dominant decays,the widths of thetype tetraquark states are narrower.Therefore,thetype tetraquark states can be observed more easily,which can be testi fi ed in the future at the BESIII,LHCb,and Belle-II.

Appendix

The explicit expressions of the QCD spectral densitiesρ1,2,3,4(s),

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