Covalent hadronic molecules induced by shared light quarks

Hua-Xing Chen

School of Physics,Southeast University,Nanjing 210094,China

Abstract After examining Feynman diagrams corresponding to the andhadronic molecular states,we propose a possible binding mechanism induced by shared light quarks.This mechanism is similar to the covalent bond in chemical molecules induced by shared electrons.We use the method of QCD sum rules to calculate its corresponding light-quark-exchange diagrams,and the obtained results indicate a model-independent hypothesis: the light-quark-exchange interaction is attractive when the shared light quarks are totally antisymmetric so they obey the Pauli principle.We build a toy model with four parameters to formulize this picture and estimate binding energies of some possibly-existing covalent hadronic molecules.A unique feature of this picture is that the binding energies of thehadronic molecules are much larger than those of the (I)JP=(0)1+ones,while the hadronic molecules have similar binding energies.

Keywords: hadronic molecule,covalent bond,QCD sum rules

1.Introduction

Since the discovery of theX(3872) by Belle in 2003 [1],lots of charmonium-likeXYZstates were discovered in the past two decades [2].Some of these structures may contain four quarks and are good candidates for hidden-charm tetraquark states.In recent years the LHCb Collaboration continually observed sixPc/Pcsstates [3–6],which contain five quarks and are good candidates for hidden-charm pentaquark states.Although there is still a long way to fully understand how the strong interaction binds these quarks and antiquarks together,the above exotic structures have become one of the most intriguing research topics in hadron physics.Their theoretical and experimental studies are significantly improving our understanding of the non-perturbative behaviors of the strong interaction at the low energy region.We refer to the reviews[7–22] and references therein for detailed discussions.

Some of theXYZandPc/Pcsstates can be interpreted as hadronic molecular states,which consist of two conventional hadrons[23–32].For example,thePcstates were proposed to be thehadronic molecular states in [33–39] bound by the one-meson-exchange interaction ΠM,as depicted in figure 1(a).Besides,we know from QCD that there can be the double-gluon-exchange interaction ΠGbetweenandas depicted in figure 1(b).

In this paper we propose another possible interaction betweeninduced by the light-quark-exchange term ΠQ,as depicted in figure 1(c).This term indicates thatare exchanging and so sharing two light up/down quarks,as depicted in figure 2.It can induce an interaction betweeneither attractive or repulsive.Note that the two interactions,ΠMat the hadron level and ΠQat the quark-gluon level,can overlap with each other.The quark-exchange effect has been studied in[40]by Hoodbhoy and Jaffe to explain the European Muon Collaboration(EMC)effect in three-nucleon systems,and later used in [41,42] to study some other nuclei.We also refer to [43] for some relevant discussions.

In this paper we shall systematically examine the Feynman diagrams corresponding to thehadronic molecular states.We shall apply the method of QCD sum rules to investigate the lightquark-exchange term ΠQ,and study its contributions to these states.Based on the obtained results,we shall study the binding mechanism induced by shared light quarks.This mechanism is somewhat similar to the covalent bond in chemical molecules induced by shared electrons,so we call such hadronic molecules ‘covalent hadronic molecules’.

Figure 1.Feynman diagrams between (*) and Σcorresponding to : (a) the one-meson-exchange interaction ΠM,(b) the double-gluonexchange interaction ΠG,and c) the light-quark-exchange interaction ΠQ.Here q denotes a light up/down quark.

Figure 2.Possible binding mechanism induced by shared light quarks,described by the light-quark-exchange term ΠQ.Here q denotes a light up/down quark.

Based on the obtained results,we shall further propose a model-independent hypothesis:the light-quark-exchange interaction is attractive when the shared light quarks are totally antisymmetric so that they obey the Pauli principle.We shall apply this hypothesis to predict some possiblyexisting covalent hadronic molecules.We shall also build a toy model to formulize this picture and estimate their binding energies.Our model has four parameters,which are fixed by considering thePc/Pcsand the recently observed[44,45]as possible covalent hadronic molecules.

2.Correlation functions of molecules

2.1.correlation function

In this subsection we investigate the correlation function of themolecule.Firstly,we investigate the correlation functions of theD-meson and thebaryon.Their corresponding interpolating currents are

Figure 3.Feynman diagrams corresponding to: are correlation functions of the D- meson and thebaryon,respectively.The correlation function of the molecule satisfies

Its correlation function in the coordinate space is

2.2.correlation function

Their correlation functions in the coordinate space are:

while there exists an extra term,

2.3.I=correlation function

with the correlation function in the coordinate space

Figure 4.Feynman diagrams corresponding to:are correlation functions of themeson and thebaryon,respectively.The correlation function of the molecule satisfies

2.4.I=correlation function

For completeness,we investigate the correlation function of theI=3/2cΣ molecule in this subsection.Its corresponding interpolating current is

with the correlation function to be

3.QCD sum rule studies of molecules

In QCD sum rule analyses we consider the two-point correlation function in the momentum space:

whereJ(x)is an interpolating current.We generally assume it to be a composite current,coupling to the molecular stateX≡∣YZ〉through

We write Π(q2) in the form of dispersion relation as

At the quark-gluon level we calculate Π(q2) using the method of operator product expansion (OPE) up to certain order.According to equations (8),(15),(22),and (28),we further separate it into

and define their imaginary parts to be ρ0(s) and ρQ(s).Here we have omitted the other term ΠG(q2),since the light-quarkexchange term ΠQ(q2) is much larger.

At the hadron level we evaluate the spectral density by inserting intermediate hadron states

where we have adopted a parametrization of one pole dominance for the ground stateXtogether with a continuum contribution.

GivenX≡∣YZ〉to be a molecular state,its massMXcan be expanded as

Then we insert equation (33) into (31),and expand it as

The former term is contributed by the non-correlatedYandZ,and the latter term is contributed by their interactions.Compared to equation (32),we further obtain:

We perform the Borel transformation to the above correlation functions at both hadron and quark-gluon levels.After assuming contributions from the continuum to be approximated by the OPE spectral densities ρ0(s) and ρQ(s)above a threshold values0,we arrive at two sum rule equations:

There are two free parameters in equations (38) and(39):the Borel massMBand the threshold values0.Differentiating equation (38) with respect towe obtain

GivenM0=MY+MZ,this equation can be used to relateMBands0,so that there is only one free parameter left.

Dividing equation (39) by (38),we obtain

This equation can be used to calculate ΔM.

we calculate its non-local correlation function in the coordinate space to be:

?Because we are using local currents in QCD sum rule analyses,

?Because the term ΠQ(x) is color-unconfined,its contribution decreases asrincreases:

We may build a model and use the light-quark-exchange potentialV(r) to derive the binding energy ofX,but this will not be done in the present study.Other than this,we shall calculate ΔMand qualitatively study several hadronic molecules possibly bound by this potential,through which we shall propose a model-independent hypothesis for such molecules.

The binding mechanism induced by the light-quarkexchange potentialV(r) is somewhat similar to the covalent bond in chemical molecules induced by shared electrons,so we call such hadronic molecules ‘covalent hadronic molecules’.

3.1.I=1/2 Σc sum rules

is given in appendix A.

To perform numerical analyses,we use the following values for various QCD sum rule parameters in the present study [2,57–65]:

where the running mass in theMSscheme is used for the charm quark.

There are two free parameters in equations(38)and(39):the Borel massMBand the threshold values0.We use equation (40) to constrain them by setting [2]:

The derived relation betweenMBands0is depicted in figure 5,which will be used in the following calculations.

Figure 5.Relation between the Borel mass MB and the threshold value s0,constrained by equation (40).

Figure 6.Convergence (solid) and Pole-Contribution (dashed) as functions of the Borel mass MB.

The second criterion is to ensure the validity of one-pole parametrization,by requiring the pole contribution to be larger than 40%:

3.2.sum rules

As shown in equation (8),the light-quark-exchange term ΠQ(x) does not contribute to the correlation function of themolecule,so thecovalent molecule does not exist.

Figure 7.The mass correctionas a function of the Borel mass MB.

3.3.sum rules

3.4.I=3/2 Σc sum rules

4.More hadronic molecules

4.1.molecules

4.2.molecules

Figure 8.Mass correctionsas functions of the Borel mass MB.The curves from top to bottom correspond to and respectively.

4.3.molecules

4.4.molecules

Figure 9.Mass correctionsas functions of the Borel mass MB.The curves from top to bottom correspond to(shortdashed),(middle-dashed),(long-dashed),and(solid),respectively.

5.Covalent hadronic molecule

In the previous section we applied QCD sum rules to study the binding mechanism induced by shared light quarks.This mechanism is somewhat similar to the covalent bond in chemical molecules induced by shared electrons,so we call such hadronic molecules ‘covalent hadronic molecules’.Recalling that the two shared electrons must spin in opposite directions (and so totally antisymmetric obeying the Pauli principle) in order to form a chemical covalent bond,our QCD sum rule results indicate a similar behavior:the lightquark-exchange interaction is attractive when the shared light quarks are totally antisymmetric so that obey the Pauli principle(these quarks may spin in the same direction given their flavor structure capable of being antisymmetric).

In this section we qualitatively study the above hypothesis.Its logical chain is quite straightforward.We assume the two light quarksqAinsideYandqBinsideZare totally antisymmetric.Hence,qAandqBobey the Pauli principle,so that they can be exchanged and shared.By doing this,wave functions ofYandZoverlap with each other,so that they are attracted and there can be the covalent hadronic moleculeX=|YZ〉.This picture has been depicted in figure 2.We believe it is better and more important than our QCD sum rule results,given it is model-independent and more easily applicable.

We apply it to study several examples as follows. The two exchanged light quarks have the same color and so the symmetric color structure; besides, we assume their orbital structure to be S-wave and so also symmetric; consequently,we only need to investigate their spin and flavor structures.

c.covalent molecule(q=u/d).After including the isospin symmetry,the exchange can take place between up and down quarks.Let us exchangeq2insideandq3inside Σc.As discussed above,they have the symmetric spin structure,so they can be totally antisymmetric as long as their flavor structure is antisymmetric:

It is well known that there are both the para-hydrogen and ortho-hydrogen,where the two protons spin in opposite directions and in the same direction,respectively.Similarly,there might be twoJ=covalent molecules,i.e.the two componentsofI=1 andofI=0.However,our QCD sum rule studies performed in section 4.3 can not differentiate these two hyperfine structures,because there we have summed over theD?andpolarizations.

It is useful to generally discuss how many light quarks at most are there in the lowest orbit (q=u/d):

Hence,two antiquarks can share (at most) two quarks,

with the quantum numbers either (I)JP=(0)1+or (1)0+.2.In thecovalent molecule,there is three light up/down quarks.We assume the two exchanged quarks to beq2andq3with the same color.There are also two possible configurations:

They satisfy that any two of the four light quarks are totally antisymmetric so they obey the Pauli principle.Hence,two quarks can share (at most) four quarks,with(I)JP=(0)2+/(1)1+/(2)0+.However,neithernor ΛbΛbsatisfies this condition.

We apply the above model-independent hypothesis to qualitatively predict more covalent hadronic molecules:

Table 1.Possibly-existing covalent hadronic molecules ∣(I )JP 〉induced by shared light up/down quarks,derived from the hypothesis that the light-quark-exchange interaction is attractive when the shared light quarks are totally antisymmetric so that they obey the Pauli principle.Here,q and s denote the light up/down and strange quarks,respectively; Q andQ′ denote two different heavy quarks;the symbols [AB]=AB-BA and{AB}=AB+BA denote the antisymmetric and symmetric SU(3)light flavor structures,respectively.The states with ?have been confirmed in QCD sum rule calculations of the present study,but the states with?and??have not.This is because the latter contain relatively-polarized components,while one needs to sum over polarizations of these components within QCD sum rule method and so can not differentiate these hyperfine structures.Moreover,the states with??contain light up/down quarks with the symmetric flavor/isospin structure,whose masses are(probably)considerably larger than their partners with the antisymmetric flavor/isospin structure.The states without any identification are still waiting to be carefully analysed in our future QCD sum rule studies.

6.A toy model to formulize covalent hadronic molecules

In the previous section we have qualitatively discussed the hypothesis:the light-quark-exchange interaction is attractive when the shared light quarks are totally antisymmetric so that they obey the Pauli principle.In this section we further build a toy model to quantitatively formulize it,and estimate binding energies of some possibly-existing covalent hadronic molecules.

This formula will be explained in section 6.1 in details,and similar formulae are used for some other covalent hadronic molecules.There are altogether four parameters:

which are estimated by considering thePc/Pcsand the recently observedas possible covalent hadronic molecules.

6.1.Parameters

As discussed in the previous section,there exists an attractive interaction when exchanging up and down quarks with the configuration:

These two light up/down quarks have the same color and their relative orbital structure isS-wave.Besides,they have the antisymmetric flavor structure and the symmetric spin structure,so with the quantum numbers (I)JP=(0)1+.We use the attractive bond energyAto describe this attraction,which is estimated to beA～30 MeV for each bond,withNAthe number of such bonds.It is illustrated in figure 10 using the solid curve.

Figure 10.(Weakly-)Attractive and repulsive covalent hadronic bonds as well as their combinations.In the present study we take into account the attractive bond and its combination with the repulsive bond,whose bond energies are estimated to be A ～30 MeV and A-R ～13 MeV,respectively.

As discussed in the previous section,the two exchanged light up/down quarks can form another configuration of(I)JP=(1)0+:

However,its induced interaction is weaker,so we do not take this configuration into account in the present study.It is illustrated in figure 10 using the dashed curve.

There exists an(I)JP=(0)0+up–down quark pair inside the proton/neutron with the configuration:

One can not exchangeq1(norq2)with another light up/down quarkq3,at the same time keeping: (a) any two of the three light quarks are totally antisymmetric,and (b) the proton/neutron remains unchanged.As an example,we exchangeq1?q3and keep (b),but then (a) is not satisfied:Therefore,the above up–down quark pair is in some sense‘saturated’.This suggests that the (I)JP=(0)0+up–down quark pairs inside protons/neutrons can not be exchanged,so they are capable of forming repulsive cores in the nucleus.

We use the repulsive bond energyRto describe the repulsion between two repulsive cores,which is estimated to beR～17 MeV for each bond,withNRthe number of such bonds.It is illustrated in figure 10 using the dotted curve.Besides,we shall find in sections 6.3 and 6.4 that the two charm quarks can also form such repulsive cores inD(?)D(?)andhadronic molecules,etc.

The third parameter is the residual energy ?,which is estimated to be ?～6 MeV for each component hadron,withNthe number of components.We use it to describe the part of kinetic energy that can not be absorbed intoAandR.

The fourth parameter κ relates to the spin splitting.We use the following term to describe the interaction between thecandquarks when investigatinghadronic molecules:

However,we do not include such terms when investigatingD(?)D(?)andhadronic molecules,since the interaction between two charm quarks has been (partly) taken into account in the repulsive bond energyR.Note that the spin splitting effect in hadronic molecules still needs to be updated with future experiments,since we do not well understand it at this moment.See section 6.7 for more discussions.

6.2.Nucleus

Taking the proton/neutron as the combination of an(I)JP=(0)0+up–down quark pair together with another up/down quark,we can estimate binding energies of the2H,3H,3He,and4He,as illustrated in figure 11.In the present study we do not investigate other nuclei consisting of more nucleons,because there are at most two up and two down quarks in the lowest orbit.See section 6.6 for relevant studies on the hypernucleus.

The2H contains two shared light up/down quarks with the configuration of (I)JP=(0)1+:

We estimate its binding energy to be

The3H and3He both contain three shared light up/down quarks with the configuration of (I)JP=(1/2)1/2+:

We estimate their binding energies to be

Figure 11.Illustration of the 2H,3H,3He,and 4He in our model.The shape of 4He is a tetrahedron other than a square.

The4He contains four shared light up/down quarks with the configuration of (I)JP=(0)0+:

We estimate its binding energy to be

6.3.molecules

The (I)JP=(0)0+DDhadronic molecule does not exist due to the Bose–Einstein statistics.So do theD?D?molecules of(I)JP=(0)0+and (I)JP=(0)2+.Actually,we can construct their corresponding currents,and explicitly prove them to be zero.

Similar to section 2,we investigate the (I)JP=(0)1+DD?hadronic molecule through its corresponding current:

Its correlation function

can be separated into (omitting the subscripts αβ for simplicity):

Figure 12.Feynman diagrams between two charmed mesons corresponding to: (a) the leading term contributed by two non-correlated charmed mesons,(b) the light-quark-exchange interaction (c) the heavy-quark-exchange interaction ,and (d) the interaction exchanging both light and heavy quarks.Here q denotes a light up/down quark.

where

Their corresponding Feynman diagrams (without condensates) are depicted in figure 12.We calculate them using QCD sum rules and find:

Therefore,our QCD sum rule results suggest that the two charm quarks inD(?)D(?)hadronic molecules are capable of forming repulsive cores.Accordingly,we can estimate binding energies ofhadronic molecules,as illustrated in figure 13(a).

The (I)JP=(0)1+DD?hadronic molecule contains two shared light antiquarks with the configuration of(I)JP=(0)1+:

We estimate its binding energy to be

suggesting it possible to interpret the recently observed[44,45] as the (I)JP=(0)1+DD?covalent hadronic molecule.

The(I)JP=hadronic molecules have similar binding energies:

However,in the above estimations we have not considered the spin splitting effect,and we have also not considered the long-range light-meson-exchange interaction.These uncertainties prevent us to well determine whether these hadronic molecules exist or not.

Figure 13.Illustration of the hadronic molecules DD?and of(I)JP=(0)1+,DDD?andof(I)JP=(1/2)1+,and DDD?D?andof (I)JP=(0)0+/(0)2+ in our model.

In the above estimations we have assumed that the charm and bottom quarks form the symmetric color representation6c,so that

We further useD(?)to compose multi-D(?)hadronic molecules,such as theDDD?andDDD?D?molecules,etc.As illustrated in figures 13(c),(e),their structures are similar to the3He and4He,respectively.We estimate their binding energies to be:

More examples can be found in table 2.

6.4.molecules

We estimate its binding energy to be

Table 2.Binding energies of some possibly-existing covalent hadronic molecules,estimated in our toy model through the simplified formula B=NAA+NSS-NRR-N?,with A ～30 MeV,S ～20 MeV,R ～17 MeV,and ?～6 MeV.We do not take into account the spin splitting effect described by the parameter κ ～13 MeV here.

Figure 14.Illustration of the hadronic molecules Σc and Σc of(I)JP=(0)1+/(0)2+ in our model.

6.5. molecules

The charm and anti-charm quarks inhadronic molecules can not be exchanged,so they are not capable of forming repulsive cores.Accordingly,we include the spin splitting effect described by the parameter κ,and estimate their binding energies to be

These hadronic molecules all contain three shared light up/down quarks with the configuration of(I)JP=(1/2)3/2+:

Figure 15.Illustration of the hadronic molecules cΣ of(I)JP=(1/2)1/2- and Σc of (I)JP=(0)1/2+ in our model.

We estimate its binding energy to be:

More examples can be found in table 2.

6.6.Molecules with strangeness

In the previous subsections we only consider the up and down quarks as exchanged light quarks,and in this subsection we further take the light strange quark into account.We introduce another parameterSto describe the attractive interaction induced by the shared strange and up/down quarks with the configuration of either (I)JP=(1/2)0+or (1/2)1+:

This parameter is estimated to beS～20 MeV for each bond,withNSthe number of such bonds.We still use the solid curve to illustrate it,but this solid curve is slightly thinner than that denoting the attractive bondA.

Taking the Λ hyperon as the combination of an(I)JP=(0)0+up–down quark pair together with a strange quark,we can estimate binding energies of some hypernuclei.Considering that there are at most two up,two down,and two strange quarks in the lowest orbit,we obtain:

Figure 16.Illustration of the ,,,andHe in our model.The shape of ppnn in the subfigures(c)and(d)is a tetrahedron other than a square.

We illustrate these hypernuclei in figure 16.More possiblyexisting covalent hadronic molecules with strangeness can be found in table 2.

6.7.Discussions on the spin splitting effect

Because we do not well understand the spin splitting effect in hadronic molecules at this moment,we still need to update it with future experiments,and there can be other approaches that better describe it.We takehadronic molecules as an example,and discuss several possible improvements.

Firstly,it is possible to use two different κ?s for equations (91) and (92).It is also possible to use some formulae other than equations (91) and (92),such as the spin–spin interaction

or the spin–orbit interaction,etc.

It can be seen from the recent LHCb experiment[6]that there might be two peaks near thethreshold,and we propose to further study them as well as thePc(4440)+andPc(4457)+in order to better understand the spin splitting effect of hadronic molecules in future experiments.

To end this section,we further simplify our toy model by neglecting the spin splitting effect described by the parameter κ ～13 MeV,and estimate binding energies of some possiblyexisting covalent hadronic molecules through the simplified formula,

which still have four parametersA～30 MeV,S～20 MeV,R～17 MeV,and ?～6 MeV.The obtained results are summarized in table 2,

7.Summary and discussions

These results suggest their corresponding light-quarkexchange potentialsV(r) to be attractive,so there can be

The binding mechanism induced by shared light quarks is somewhat similar to the covalent bond in chemical molecules induced by shared electrons,so we call such hadronic molecules ‘covalent hadronic molecules’.Different from the chemical molecules,the internal structure of the hadrons and so the covalent hadronic molecules is much more complicated,since they contain not only the valence quarks but also the sea quarks and gluons.In the present study we consider the covalent hadronic molecules induced only by the shared valence quarks.

Recalling that the two shared electrons must spin in opposite directions (and so totally antisymmetric obeying the Pauli principle) in order to form a chemical covalent bond,our QCD sum rule results indicate a similar hypothesis:the light-quark-exchange interaction is attractive when the shared light quarks are totally antisymmetric so that obey the Pauli principle.

Its logical chain is quite straightforward.We assume the two light quarksqAinsideYandqBinsideZare totally antisymmetric.Hence,qAandqBobey the Pauli principle,so that they can be exchanged and shared.By doing this,wavefunctions ofYandZoverlap with each other,so that they are attracted and there can be a covalent hadronic moleculeX=∣YZ〉.This picture has been depicted in figure 2.We believe it is better and more important than our QCD sum rule results,given it is model-independent and more easily applicable.

We apply the above hypothesis to the reanalysis of thehadronic molecules,and the obtained results are generally consistent with our QCD sum rule results.However,there can be more hyperfine structures allowed/predicted by the hypothesis,similar to the case of para-hydrogen and ortho-hydrogen with the two protons spinning in opposite directions and in the same direction,respectively.These hyperfine structures can not be differentiated in our QCD sum rule studies,since there we need to sum over polarizations.

We also apply the above hypothesis to predict more possibly-existing covalent hadronic molecules,as summarized in table 1.We build a toy model to formulize this picture and estimate their binding energies.Our model has four parameters,which are fixed by considering thePc/Pcsand the recently observedas possible covalent hadronic molecules.Some simplified results neglecting the spin splitting effect are summarized in table 2.Note that these results are obtained from the light-quark-exchange interaction only,and there can be some other interactions among hadrons,e.g.the light-quark-exchange term ΠQ(x) does not contribute to themolecules,suggesting that thecovalent hadronic molecules do not exist,but there can still be themolecules possibly induced by the one-mesonexchange interaction.These interactions can also contribute to the binding energies given in table 2,which is one source of their theoretical uncertainties.

To end this paper,we note again that the one-mesonexchange interaction ΠMat the hadron level and the lightquark-exchange interaction ΠQat the quark-gluon level can overlap with each other.Hence,we attempt to understand the nuclear force based on the picture of covalent hadronic molecules very roughly:(a)the(I)JP=(0)0+up–down quark pairs inside protons/neutrons are in some sense‘saturated’,so they can not be exchanged and form repulsive cores in the nucleus;(b) the other up/down quarks inside protons/neutrons can be freely exchanged/shared/moving,inducing some interactions among nucleons;(c) in the multi-nucleon nucleus there can be many up/down quarks being shared,so its binding mechanism transfers into the ‘metallic’ hadronic bond.Based on these understandings,we estimate binding energies of some nuclei using our toy model,as summarized in table 2.Finally,we propose another possibly-existing binding mechanism similar to the ‘ionic’ bond,but it might only be observable in the quark-gluon plasma.

Acknowledgments

We thank Yan-Rui Liu and Li-Ming Zhang for their helpful discussions.This project is supported by the National Natural Science Foundation of China under Grants No.11722540 and No.12075019,the Jiangsu Provincial Double-Innovation Program under Grant No.JSSCRC2021488,and the Fundamental Research Funds for the Central Universities.

Appendix A.Spectral densities

Appendix B.Spin decompositions