Threshold Structures in Hadron Spectrum and Hadronic Molecules

Numerous candidates for exotic hadrons have been detected experimentally in the past two decades,predominantly near the threshold of a pair of hadrons.This study aims to investigate the overall behavior of nearthreshold line shapes in invariant mass distributions.It is noteworthy that the threshold cusp might manifest as a peak only in channels with attractive interaction.The assertion is made that there should be near-threshold structures for any heavy-quark and heavy-antiquark hadron pairs exhibiting attractive interaction at the threshold,as observed in the invariant mass distribution of heavy quarkonium and light hadrons that couple to the open-flavor hadron pair.Furthermore,we have conducted an analysis of potential hadronic molecules comprising pairs of heavy hadrons,utilizing the Bethe-Salpeter equation with constant interactions derived from the one-boson-exchange model.Observed candidates for these hadronic molecules are in good agreement with our predicted spectrum.

A survey of heavy–heavy hadronic molecules

The spectrum of hadronic molecules composed of heavy-antiheavy charmed hadrons has been obtained in our previous work.The potentials are constants at the leading order,which are estimated from resonance saturation.The experimental candidates of hadronic molecules,say X(3872),Y(4260),three P_(c)states and P_(cs)(4459),fit the spectrum well.The success in describing the pattern of heavy-antiheavy hadronic molecules stimulates us to give more predictions for the heavy-heavy cases,which are less discussed in literature than the heavy-antiheavy ones.Given that the heavy-antiheavy hadronic molecules,several of which have strong experimental evidence,emerge from the dominant constant interaction from resonance saturation,we find that the existence of many heavy-heavy hadronic molecules is natural.Among these predicted heavy-heavy states we highlight the DD;molecule and the D^(*)∑_(c)^(*)molecules,which are the partners of the famous X(3872)and Pc states.Quite recently,LHCb collaboration reported a doubly charmed tetraquark state,T_(cc),which is in line with our results for the DD^(*)molecule.With the first experimental signal of this new kind of exotic states,the upcoming update of the LHCb experiment as well as other experiments will provide more chances of observing the heavy-heavy hadronic molecules.

Heavy quark spin symmetry and heavy flavor hadronic molecules

We argue that the heavy quark spin symmetry can lead to important consequences for heavy flavor hadronic molecules.It can be used to predict new heavy flavor hadronic molecules and hence provides a method to identify the nature of some newly observed exotic hadrons.For example,if the Y（4660） were an S-wave ψ f 0 （980） shallow bound state,then the mass,width and line shape of its spin partner are predicted.

Covalent hadronic molecules induced by shared light quarks

After examining Feynman diagrams corresponding to the ■, ■, ■, and ■ hadronic 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 the(I)JP=(0)1+■ hadronic molecules are much larger than those of the (I)JP=(0)1+■ ones, while the(I)JP=(1/2)1/2+■ hadronic molecules have similar binding energies.

Isospin-conserving hadronic decay of the D_(s1)(2460) into D_(s)π^(+)π^(-)

The internal structure of the charm-strange mesons D_(s0)^(*)(2317)and D_(s1)(2460)are the subject of intensive studies.Their widths are small because they decay dominantly through isospinbreaking hadronic channels D_(s0)^(*)(2317)^(+)→D_(s)^(+)π^(0) and D _(s1)(2460)^(+)→D_(s)^(*+)π^(0).The D_(s1)(2460)can also decay into the hadronic final states D_(s)^(+)ππ,conserving isospin.In that case there is,however,a strong suppression from phase space.We study the transition D_(s1)(2460)^(+)→D_(s)^(+)π^(+)π^(-)in the scenario that the D_(s1)(2460)is a D^(*)K hadronic molecule.The ππ final state interaction is taken into account through dispersion relations.We find that the ratio of the partial widths of the Γ(D_(s1)(2460)^(+)→D_(s)^(+)π^(+)π^(-)) / Γ(D_(s1)(2460)^(+)→D_(s)^(*+)π^(0))obtained in the molecular picture is consistent with the existing experimental measurement.More interestingly,we demonstrate that theπ+π−invariant mass distribution shows a double bump structure,which can be used to disentangle the hadronic molecular picture from the compact state picture for the D_(s1)(2460)^(+).Predictions on the B_(s1)^(0)→B_(s)^(0)π^(+)π^(-)are also made.

A discussion on the anomalous threshold enhancement of J/ψ–ψ(3770)couplings and X(6900)peak

The attractive interaction between J/ψandψ(3770)has to be strong enough if X(6900)is of the molecule type.We argue that sinceψ(3770)decays predominantly into a DD^(¯)pair,the interactions between J/ψandψ(3770)may be significantly enhanced owing to the three point DD^(¯)loop diagram.The enhancement originates from the anomalous threshold located at t=−1.288 GeV^(2),whose effect propagates into the s-channel partial wave amplitude in the vicinity of √s≃6.94 GeV.This effect may be helpful in the formation of the X(6900)peak.

Hadron molecules

We discuss a possible interpretation of the open charm mesons D＊s0 （2317）,D s1 （2460） and the hidden charm mesons X（3872）,Y（3940） and Y（4140） as hadron molecules.Using a phenomenological Lagrangian approach we review the strong and radiative decays of the D＊ s0 （2317） and D s1 （2460） states.The X（3872） is assumed to consist dominantly of molecular hadronic components with an additional small admixture of a charmonium configuration.Determing the radiative （γJ/ψ and γψ（2s）） and strong （J/ψ2π and J/ψ3π） decay modes we show that the present experimental observation is consistent with the molecular structure assumption of the X（3872）.Finally,we give evidence for molecular interpretations of the Y（3940） and Y（4140） related to the observed strong decay modes J/ψ＋ω or J/ψ＋φ,respectively.

对3.9GeV附近0^(++)和2^(++)(类)粲偶素的理解

在过去的20年里,各个实验组在粲偶素能量区间(约3.1~4.5 GeV)观测到了大量的(类)粲偶素,这些粲偶素的发现不仅极大地丰富了强子谱,也将传统的cc介子拓展到了奇特强子态领域.在3.9 GeV能量附近有许多(类)粲偶素的实验候选者,如:X(3915),Z(3930),χc0(3930),χc2(3930)以及X(3960)等,然而它们的本质是什么依然存在争议.本文提出以下猜想:(1)J/φω中观测到的X(3915)与χc2(3930)是同一个态;(2) Ds+Ds-道中观测到的X(3960)是Ds+Ds-体系的S波强子分子态;(3)B+→D+D-K+过程里的JPC=0++组分与X(3960)具有相同来源,质量大约为3.94GeV,并非如《粒子物理综述》中所列来源于X(3915).通过对LHCb实验组在B+→D((s))+D((s))-K+过程以及Belle和BaBar实验组在双光子对撞过程给出的实验数据进行联合拟合,作者发现上述猜想与当前的实验结果都是一致的.本工作不仅对(类)粲偶素能谱提供了新的探索思路,也加深了研究人员对含粲强子之间的相互作用的理解.

Investigating S-wave bound states composed of two pseudoscalar mesons

In this study,we systematically investigated two-pseudoscalar meson systems with the Bethe-Salpeter equation in the ladder and instantaneous approximations.By solving the Bethe-Salpeter equation numerically with the kernel containing the one-particle exchange diagrams,we found that the K■,DK,B■,D■,B■,BD,D■,BK,and■systems withI=0 can exist as bound states.We also studied the contributions from heavy meson(J/ψandΥ)exchanges and found that the contributions from heavy meson exchanges cannot be ignored.

Note on X(3872) production at hadron colliders and its molecular structure

The production of the X（3872） as a hadronic molecule in hadron colliders is clarified. We show that the conclusion of Bignamini et al., Phys. Rev. Lett. 103（2009） 162001, that the production of the X（3872） at high pT implies a non-molecular structure, does not hold. In particular, using the well understood properties of the deuteron wave function as an example, we identify the relevant scales in the production process.

Comment on ‘Note on X(3872) production at hadron colliders and its molecular structure’

We briefly comment on the paper by Albaladejo et al., Chinese Phys. C 41 121001, rejecting its conclusions.

Decay properties of the Zc(3900)through the Fierz rearrangement

We systematically construct all the tetraquark currents/operators of J^PC=1^+− with the quark configurations[cq][cq],[cq][qc],and[cc][qq](q=u/d),and derive their relations through the Fierz rearrangement of the Dirac and color indices.Using the transformations of[qc][qc]→[cc][qq]and[cq][qc],we study decay properties of the Zc(3900)as a compact tetraquark state;while using the transformation of[cq][qc]→[cc][qq],we study its decay properties as a hadronic molecular state.

Hidden-charm pentaquark states through current algebra:from their production to decay

There may be sevenD^((*))Σ_(c)^((*))hadronic molecular states.We construct their corresponding interpolating currents and calculate their masses and decay constants using QCD sum rules.Based on these results,we calculate their relative production rates inΛ^(0)_(b)decays using current algebra,that is,B(Λ^(0)_(b)→P_(c)K^(-)):B(Λ^(0)_(b)→P′_(c)K^(-)),where P_(c)and P′_(c)are two different states.We also study their decay properties via Fierz rearrangement and further calculate these ratios in the J/ψp mass spectrum,that is,B(Λ^(0)_(b)→P_(c)K^(-)→J/ψpK^(-)):B(Λ^(0)_(b)→P′_(c)K^(-)→J/ψpK^(-)).Our results suggest that theD*Σ^(*)_(c)molecular states of JP=1/2^(-)and 3/2^(-)may be observed in future experiments.

Decay properties of the X(3872)through the Fierz rearrangement

We systematically construct all the tetraquark currents of J^(PC)=1^(++) with the quark configurations[cq][cq],[cq][qc],and[cc][qq](q=u/d).Their relations are derived using the Fierz rearrangement of the Dirac and color indices,through which we study decay properties of the X(3872)under both the compact tetraquark and hadronic molecule interpretations.We conduct a search for the X(3872)→χ_(c0)π,η_(c)ππ,and χ_(c1)ππ decay processes in particle experiments.

Isospin eigenstates of the color singlet-singlet-type pentaquark states

In this study,we constructed color singlet-singlet-type five-quark currents with isospins(I,I_(3))=(1/2,1/2)and(3/2,1/2)unambiguously to explore theDΣ_(c),DΣ^(∗)_(c),¯D∗Σc,andD^(∗)Σ^(∗)_(c)pentaquark states via the quantum chromodynamics sum rules for the first time,where¯D,Σc,···represent the color singlet clusters with the same quantum numbers as the corresponding physical mesons or baryons.The numerical results support assigning P_(c)(4312),P_(c)(4380),P_(c)(4440),and P_(c)(4457)as theDΣ_(c),DΣ^(∗)_(c),¯D∗Σc,andD^(∗)Σ^(∗)_(c)pentaquark states,respectively,with the isospin I=1/2.The correspondingDΣ_(c),DΣ^(∗)_(c),¯D∗Σc,andD^(∗)Σ^(∗)_(c)pentaquark states with the isospin I=3/2 have slightly larger masses.The observations of the high pentaquark candidates in the J=Δinvariant mass spectrum would shed light on the nature of the P_(c)states and contribute in distinguishing the scenarios of the color antitriplet-antitriplet-antitriplet-type and color singlet-singlet-type pentaquark states.

Interpretation of theη_(1)(1855)as a KK_(1)(1400)+c.c.molecule

An exotic state with J^(PC)=1^(−+),denoted byη1(1855),was observed by BESIII Collaboration recently in J/ψ→γηη′.The fact that its mass is just below the threshold of KK_(1)(1400)stimulates us to investigate whether this exotic state can be interpreted as a KK_(1)(1400)+c.c.molecule or not.Using the one boson exchange model,we show that it is possible for KK_(1)(1400)with J^(PC)=1^(−+)to bind together by taking the momentum cutoffΛ&2 GeV and yield the same binding energy as the experimental value whenΛ≈2.5 GeV.In this molecular picture,the predicted branch ratio Br(η1(1855)→ηη′)≈15%is consistent with the experimental results,which again supports the molecular explanation ofη1(1855).Relevant systems,namely KK_(1)(1400)with JPC=1−−and KK_(1)(1270)with J^(PC)=1^(−±),are also investigated,some of which can be searched for in the future experiments.