A survey of heavy-antiheavy hadronic molecules
AA survey of heavy-antiheavy hadronic molecules
Xiang-Kun Dong , , ∗ Feng-Kun Guo , , † and Bing-Song Zou , , ‡ CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China School of Physics, Central South University, Changsha 410083, China
Many efforts have been made to reveal the nature of the overabundant resonant structures ob-served by the worldwide experiments in the last two decades. Hadronic molecules attract specialattention because many of these seemingly unconventional resonances are located close to the thresh-old of a pair of hadrons. To give an overall feature of the spectrum of hadronic molecules composedof a pair of heavy-antiheavy hadrons, namely, which pairs are possible to form molecular states,we take charmed hadrons for example to investigate the interaction between them and search forpoles by solving the Bethe-Salpeter equation. We consider all possible combinations of hadron pairsof the S -wave singly-charmed mesons and baryons as well as the narrow P -wave charmed mesons.The interactions, which are assumed to be meson-exchange saturated, are described by constantcontact terms which are resummed to generate poles. It turns out that if a system is attractivenear threshold by the light meson exchange, there is a pole close to threshold corresponding to abound state or a virtual state, depending on the strength of interaction and the cutoff. In total, 229molecular states are predicted. The observed near-threshold structures with hidden-charm, like thefamous X (3872) and P c states, fit into the spectrum we obtain. We also highlight a Λ c ¯Λ c boundstate that has a pole consistent with the cross section of the e + e − → Λ c ¯Λ c precisely measured bythe BESIII Collaboration. CONTENTS
I. Introduction 1II. Lagrangian from heavy quark spin symmetry 2A. Heavy mesons 2B. Heavy baryons 5III. Potentials 6A. Conventions 6B. Potentials from light vector exchange 6C. Potentials from vector charmonia exchange 7IV. Molecular states from constant interactions 7A. Poles 7B. Discussions of selected systems 81. D ( ∗ ) ¯ D ( ∗ ) : X (3872), Z c (3900) and theirpartners 82. D ( ∗ ) s ¯ D ( ∗ ) s virtual states 143. D ( ∗ ) ¯ D ( ∗ ) s : Z cs as virtual states 154. D ( ∗ ) ¯ D , : Y (4260) and related states 155. Λ c ¯Λ c : analysis of the BESIII data andmore baryon-antibaryon bound states 166. ¯ D ( ∗ ) Σ ( ∗ ) c : P c states 177. ¯ D ( ∗ ) Ξ ( (cid:48) ) c : P cs and related states 17V. Summary and discussion 18Acknowledgments 19 ∗ [email protected] † [email protected] ‡ [email protected] A. Vertex factors for direct processes 19B. List of the potential factor F I. INTRODUCTION
The formation of hadrons from quarks and gluons isgoverned by quantum chromodynamics (QCD), which atlow energy is nonperturbative. Therefore, it is difficult totackle with the hadron spectrum model-independently.The traditional quark model [1, 2], where hadrons areclassified as mesons, composed of q ¯ q and baryons, com-posed of qqq , provides quite satisfactory description ofthe hadrons observed in last century. The last twodecades witnessed the emergence of plenty of states orresonant structures in experiments, many of which donot fit the hadron spectrum predicted by the naive quarkmodel and thus are candidates of the so-called exoticstates. Many efforts have been devoted to understandthe nature of these states but most of them still remaincontroversial (see Refs. [3–18] for recent reviews).The observation that most of these structures are lo-cated near the threshold of a pair of heavy-antiheavyhadrons may shed light on identifying the nature of thesestructures. To name a few famous examples, let us men-tion the X (3872), aka χ c (3872) [19] and Z c (3900) ± [20–22] around the D ¯ D ∗ threshold, the Z c (4020) ± [23,24] near the D ∗ ¯ D ∗ threshold, the Z b (10610) ± and Z b (10650) ± [25, 26] near the B ¯ B ∗ and B ∗ ¯ B ∗ thresholds, a r X i v : . [ h e p - ph ] J a n the Z cs (3985) − [27] near the ¯ D s D ∗ and ¯ D ∗ s D thresholds,and the P c states [28] near the ¯ D ( ∗ ) Σ c thresholds, re-spectively. These resonant structures were widely inves-tigated in many models, including assigning them to bethe molecular states of the corresponding systems. Werefer to Ref. [8] for a comprehensive review of hadronicmolecules.Although these near-threshold resonant structureshave been explored by many works using various meth-ods, our understanding of these structures will greatlybenefit from a whole and systematic spectrum of heavy-antiheavy hadronic molecules based on one single model.In this paper, we provide such a spectrum of hadronicmolecules composed of a pair of heavy-antiheavy hadronsincluding S, P -wave heavy mesons and S -wave heavybaryons. The interactions between these hadron pairsare assumed to be contact terms saturated by mesonexchanges as in, e.g., Refs. [29–31]. In order to havea unified treatment in all systems, we will not considercoupled-channel effects. In Ref. [31] a similar study, fo-cusing on the possible bound states composed of ¯ D ( ∗ ) and Σ ( ∗ ) c , are performed. Also note that such interactionshave been obtained in many other works, e.g. Refs. [32–37], to look for possible bound states associated with thenear-threshold resonant structures. These works useddifferent models and conventions and in some cases theresults are inconsistent with each other.In this work we consider the resonance saturation ofthe contact interaction due to the one-boson-exchange(light pseudoscalar and vector mesons) together withheavy quark spin symmetry (HQSS), chiral symmetryand SU(3) flavor symmetry, to estimate the potentialsof the heavy-antiheavy systems, and special attentionis paid to the signs of coupling constants to get self-consistent results. Heavy quark flavor symmetry (HQFS)implies that the potentials between bottomed hadronpairs are the same as those between charmed ones andthus we take charmed hadrons for example. The obtainedpotentials are used to solve the Bethe-Salpeter equationin the on-shell form to determine the pole positions ofthese heavy-antiheavy hadron systems and a systematicspectrum of these hadronic molecules is obtained.Besides the possible molecular states, the interactionbetween a pair of heavy-antiheavy hadrons at threshold iscrucial to understand the line shape of the invariant massdistribution of related channels, as discussed in Ref. [38].It is known that unitarity of the S matrix requires thethreshold to be a branch point of the scattering ampli-tude. Therefore, the square of amplitude modulus alwaysshows a cusp at the threshold and a nontrivial structuremay appear in the line shape of the invariant mass dis-tribution. The detailed structure of the cusp, a peak, adip or buried in the background, depends on the inter-action of the two relevant particles near the threshold.More specifically, the cusp at the threshold of a heavy-antiheavy hadron pair will generally show up as a peakin the invariant mass distribution of a heavy quarkoniumand light hadron(s), which couple to the heavy-antiheavy hadron pair, if the interaction is attractive at thresholdand a nearby virtual state comes into existence (for sub-tlety and more detailed discussions, we refer to Ref. [38]).If the attraction is strong and a bound state is formed,the peak will be located at the mass of the bound statebelow threshold. Therefore, the potentials and the polepositions obtained in this work are of great significancefor the study of near-threshold line shapes.Note that the leading order interaction, which is just aconstant, between a heavy-antiheavy hadron pair, whenresummed, works well for both purposes mentioned aboveif we only focus on the near-threshold pole and line shape.Therefore, we only keep the leading order interaction,which is saturated by the vector meson exchange as dis-cussed in the following, and its resummation.This paper is organized as follows. In Section II, theLagrangians of heavy hadron and light meson couplingare presented. In Section III, the potentials of differentsystems are obtained. We calculate the pole positions ofdifferent systems and compare them with experimentaland other theoretical results in Section IV. Section V isdevoted to a brief summary. Some details regarding thepotentials are relegated to Appendices A, B and C. II. LAGRANGIAN FROM HEAVY QUARKSPIN SYMMETRY
For hadrons containing one or more heavy quarks,additional symmetries emerge in the low energy effec-tive field theory for QCD due to the large heavy quarkmass [39, 40]. On the one hand, the strong suppres-sion of the chromomagnetic interaction, which is propor-tional to σ · B /m Q ∼ Λ QCD /m Q with Λ QCD ∼ − s Q is decoupled with the angular mo-mentum s (cid:96) of light quarks in the limit of m Q → ∞ .Therefore, HQSS emerges, which means that the inter-action is invariant under a transformation of s Q . Onthe other hand, the change of the velocity of the heavyquark in a singly-heavy hadron during the interaction,∆ v = ∆ p/m Q ∼ Λ QCD /m Q , vanishes in the limit of m Q → ∞ , and the heavy quark behaves like a static colorsource, independent of the quark flavor. Therefore, it isexpected that the potentials between bottomed hadronpairs are the same as those of the charmed ones, and inturn it is sufficient to focus on the charm sector. A. Heavy mesons
To construct a Lagrangian that is invariant under theheavy quark spin transformation and chiral transforma-tion, it is convenient to represent the ground states ofcharmed mesons as the following superfield [41–43] H ( Q ) a = 1 + /v (cid:104) P ∗ ( Q ) µa γ µ − P ( Q ) a γ (cid:105) , (1)where a is the SU(3) flavor index, P ( Q ) = ( D , D + , D + s ) , P ∗ ( Q ) µ = ( D ∗ µ , D ∗ + µ , D ∗ + sµ ) , (2)and v µ = p µ /M is the four-velocity of the heavy mesonsatisfying v · v = 1. The heavy field operators containa factor √ M H and have dimension 3/2. The superfieldthat creates heavy mesons is constructed as¯ H ( Q ) a = γ H ( Q ) † a γ . (3)The superfields that annihilate or create mesons contain-ing an antiheavy quark are not ¯ H ( Q ) a or H ( Q ) a but thefollowing ones [44]: H ( ¯ Q ) a = C (cid:16) C H ( Q ) a C − (cid:17) T C − = (cid:104) P ∗ ( ¯ Q ) aµ γ µ − P ( ¯ Q ) a γ (cid:105) − /v , (4)¯ H ( ¯ Q ) a = γ H ( ¯ Q ) † a γ , (5)with P ( ¯ Q ) = ( ¯ D , D − , D − s ) , P ∗ ( ¯ Q ) µ = ( ¯ D ∗ µ , D ∗− µ , D ∗− sµ ) . (6) C is the charge conjugation operator and C = iγ γ isthe charge conjugation matrix, where we have taken thephase convention for charge conjugation as C P ( Q ) a C − = P ( ¯ Q ) a and C P ∗ ( Q ) a C − = − P ∗ ( ¯ Q ) a .The P -wave heavy mesons have two spin multiplets,one with s (cid:96) = 1 / S while the other with s (cid:96) = 3 / T [45, 46], S ( Q ) a = 1 + /v (cid:104) P (cid:48) ( Q ) µ a γ µ γ − P ∗ ( Q )0 a (cid:105) , (7) T ( Q ) µa = 1 + /v (cid:20) P ∗ ( Q ) µν a γ ν − (cid:114) P ( Q )1 aν γ (cid:18) g µν − γ ν ( γ µ − v µ ) (cid:19) (cid:21) . (8)Analogous with Eqs. (3,4,5), we have¯ S ( Q ) a = γ S ( Q ) † a γ , (9)¯ T ( Q ) µa = γ T ( Q ) µ † a γ , (10) S ( ¯ Q ) a = (cid:104) P (cid:48) ( ¯ Q ) µ a γ µ γ − P ∗ ( ¯ Q )0 a (cid:105) − /v , (11) T ( ¯ Q ) µa = (cid:34) P ( ¯ Q ) µν a γ ν − (cid:114) P ( ¯ Q )1 aν × γ (cid:18) g µν −
13 ( γ µ − v µ ) γ ν (cid:19)(cid:21) − /v , (12)¯ S ( ¯ Q ) a = γ S ( ¯ Q ) † a γ , (13)¯ T ( ¯ Q ) aµ = γ T ( ¯ Q ) † aµ γ . (14)The mesons in the T multiplet are P ( Q )1 = ( D (2420) , D (2420) + , D s (2536) + ) ,P ( Q )2 = ( D (2460) , D (2460) + , D s (2573) + ) , (15)which can couple to D ∗ π/K only in D -wave in the heavyquark limit. While the P -wave charmed mesons with s (cid:96) = 1 / D ∗ π/K in S -wave without vio-lating HQSS, there are issues in identifying them. The D ∗ (2300) and D (2430) listed in the Review of Par-ticle Physics (RPP) [47] could be candidates for thecharm-nonstrange ones. However, on the one hand, theyhave rather large widths such that they would have de-cayed before they can be bound together with anotherheavy hadron [48, 49]; on the other hand, they were ex-tracted using the Breit-Wigner parameterization whichhas deficiencies in the current case [50] and has beendemonstrated [51] to lead to resonance parameters forthe D ∗ (2300) in conflict with the precise LHCb data ofthe B − → D + π − π − process [52]. For the ones withstrangeness, the lowest positive-parity D ∗ s (2317) and D s (2460) are widely considered as molecular states of DK and D ∗ K [53–58], see Ref. [59] for a recent reviewcollecting evidence for such an interpretation. This mul-tiplet S , therefore, will not be considered in the rest ofthis work. For studies of three-body hadronic molecularstates involving the D ∗ s (2317) as a DK subsystem, werefer to Refs. [60–63].The light pseudoscalar meson octet can be introducedusing the nonlinear realization of the spontaneous chiralsymmetry breaking of QCD as Σ = ξ and ξ = e i Π / ( √ F π ) with F π = 92 MeV the pion decay constant andΠ = π √ + η √ π + K + π − − π √ + η √ K K − ¯ K − (cid:113) η . (16)The effective Lagrangian for the coupling of heavymesons and light pseudoscalar mesons is constructed byimposing invariance under both heavy quark spin trans-formation and chiral transformation [41, 44, 46], L P P Π = ig (cid:68) H ( Q ) b / A ba γ ¯ H ( Q ) a (cid:69) + ik (cid:68) T ( Q ) µb / A ba γ ¯ T ( Q ) aµ (cid:69) + i ˜ k (cid:68) S ( Q ) b / A ba γ ¯ S ( Q ) a (cid:69) + (cid:104) ih (cid:68) S ( Q ) b / A ba γ ¯ H ( Q ) a (cid:69) + i ˜ h (cid:68) T ( Q ) µb A µba γ ¯ S ( Q ) a (cid:69) + i h Λ χ (cid:68) T ( Q ) µb (cid:0) D µ / A (cid:1) ba γ ¯ H ( Q ) a (cid:69) + i h Λ χ (cid:68) T ( Q ) µb (cid:0) /D A µ (cid:1) ba γ ¯ H ( Q ) a (cid:69) + h.c. (cid:21) + ig (cid:68) ¯ H ( ¯ Q ) a / A ab γ H ( ¯ Q ) b (cid:69) + ik (cid:68) ¯ T ( ¯ Q ) µa / A ab γ T ( ¯ Q ) bµ (cid:69) + i ˜ k (cid:68) ¯ S ( ¯ Q ) a / A ab γ S ( ¯ Q ) b (cid:69) + (cid:20) ih (cid:68) ¯ H ( ¯ Q ) a / A ab γ S ( ¯ Q ) b (cid:69) + i ˜ h (cid:68) ¯ S ( ¯ Q ) a A µab γ T ( ¯ Q ) µb (cid:69) + i h Λ χ (cid:28) ¯ H ( ¯ Q ) a (cid:16) / A ← D (cid:48) µ (cid:17) ab γ T ( ¯ Q ) µb (cid:29) + i h Λ χ (cid:28) ¯ H ( ¯ Q ) a (cid:16) A µ ← /D (cid:48) (cid:17) ab γ T ( ¯ Q ) µb (cid:29) + h.c. (cid:21) , (17)where D µ = ∂ µ + V µ , D (cid:48) µ = ∂ µ − V µ , (cid:104)· · · (cid:105) denotes trac-ing over the Dirac γ matrices, Λ χ (cid:39) πF π is the chiralsymmetry breaking scale, and V µ = 12 (cid:0) ξ † ∂ µ ξ + ξ∂ µ ξ † (cid:1) , (18) A µ = 12 (cid:0) ξ † ∂ µ ξ − ξ∂ µ ξ † (cid:1) (19) are the vector and axial currents which contain an evenand odd number of pseudoscalar mesons, respectively.The coupling of heavy mesons and light vector mesonscan be introduced by using the hidden local symmetryapproach [64–66], and the Lagrangian reads [42, 43, 67] L P P V = iβ (cid:68) H ( Q ) b v µ ( V µ − ρ µ ) ba ¯ H ( Q ) a (cid:69) + iλ (cid:68) H ( Q ) b σ µν F µν ( ρ ) ba ¯ H ( Q ) a (cid:69) + iβ (cid:68) S ( Q ) b v µ ( V µ − ρ µ ) ba ¯ S ( Q ) a (cid:69) + iλ (cid:68) S ( Q ) b σ µν F µν ( ρ ) ba ¯ S ( Q ) a (cid:69) + iβ (cid:68) T ( Q ) λb v µ ( V µ − ρ µ ) ba ¯ T ( Q ) aλ (cid:69) + iλ (cid:68) T ( Q ) λb σ µν F µν ( ρ ) ba ¯ T ( Q ) aλ (cid:69) + (cid:104) iζ (cid:68) H ( Q ) b γ µ ( V µ − ρ µ ) ba ¯ S ( Q ) a (cid:69) + iµ (cid:68) H ( Q ) b σ λν F λν ( ρ ) ba ¯ S ( Q ) a (cid:69) + iζ (cid:68) T ( Q ) µb ( V µ − ρ µ ) ba ¯ H ( Q ) a (cid:69) + µ (cid:68) T ( Q ) µb γ ν F µν ( ρ ) ba ¯ H ( Q ) a (cid:69) + h.c. (cid:105) − iβ (cid:68) ¯ H ( ¯ Q ) a v µ ( V µ − ρ µ ) ab H ( ¯ Q ) b (cid:69) + iλ (cid:68) ¯ H ( ¯ Q ) a σ µν F µν ( ρ ) ab H ( ¯ Q ) b (cid:69) − iβ (cid:68) ¯ S ( ¯ Q ) a v µ ( V µ − ρ µ ) ab S ( ¯ Q ) b (cid:69) + iλ (cid:68) ¯ S ( ¯ Q ) a σ µν F µν ( ρ ) ab S ( ¯ Q ) b (cid:69) − iβ (cid:68) ¯ T ( ¯ Q ) aλ v µ ( V µ − ρ µ ) ab T ( ¯ Q ) λb (cid:69) + iλ (cid:68) ¯ T ( ¯ Q ) aλ σ µν F µν ( ρ ) ab T ( ¯ Q ) λb (cid:69) + (cid:104) iζ (cid:68) ¯ S ( ¯ Q ) a γ µ ( V µ − ρ µ ) ab H ( ¯ Q ) a (cid:69) + iµ (cid:68) ¯ S ( ¯ Q ) a σ λν F λν ( ρ ) ab H ( ¯ Q ) b (cid:69) − iζ (cid:68) ¯ H ( ¯ Q ) a ( V µ − ρ µ ) ab T ( ¯ Q ) µb (cid:69) + µ (cid:68) ¯ H ( ¯ Q ) a γ ν F µν ( ρ ) ab T ( ¯ Q ) µb (cid:69) + h.c. (cid:105) , (20)with F µν = ∂ µ ρ ν − ∂ ν ρ µ + [ ρ µ , ρ ν ], and ρ = i g V √ V = i g V √ ω √ + ρ √ ρ + K ∗ + ρ − ω √ − ρ √ K ∗ K ∗− ¯ K ∗ φ , (21)which satisfies C V C − = − V T .Remind that in the following we are only interested inthe potential near threshold and will not consider cou-pled channels. Therefore, the Lagrangian that results inpotentials proportional to the transferred momentum q will have little contributions. At the leading order of thechiral expansion, the light pseudoscalar mesons as Gold-stone bosons only couple in derivatives, as demonstratedin Eq. (17), so all pseudoscalar exchanges have sublead-ing contributions near threshold in comparison with theconstant contact term that can generate a near-thresholdpole after resummation. Moreover, coupled channels arenot taken into account here, and we do not consider the s (cid:96) = 1 / β , β and ζ terms in Eq. (20). Expanding these terms we obtain L P P V = −√ βg V (cid:16) P ( Q ) a P ( Q ) † b − P ( ¯ Q ) b P ( ¯ Q ) † a (cid:17) v µ V µab + √ βg V (cid:16) P ∗ ( Q ) νa P ∗ ( Q ) † bν − P ∗ ( ¯ Q ) νb P ∗ ( ¯ Q ) † aν (cid:17) v µ V µab − √ β g V (cid:16) P ( Q ) ν a P ( Q ) † bν − P ( ¯ Q ) ν b P ( ¯ Q ) † aν (cid:17) v µ V µab + √ β g V (cid:16) P ( Q ) αβ a P ( Q ) † bαβ − P ( ¯ Q ) αβ b P ( ¯ Q ) † aαβ (cid:17) v µ V µab + (cid:104) √ ζ g V (cid:16) P ( Q ) µν a P ( Q ) ∗† bν + P ( ¯ Q ) µν b P ( ¯ Q ) ∗† aν (cid:17) V abµ − iζ g V √ (cid:15) αβγδ (cid:16) P ( Q ) α a P ( Q ) ∗† βb + P ( ¯ Q ) α b P ( ¯ Q ) ∗† βa (cid:17) v γ V δab − ζ g V √ (cid:16) P ( Q )1 aµ P ( Q ) † b − P ( ¯ Q )1 bµ P ( ¯ Q ) † a (cid:17) V µab + h . c . (cid:105) . (22)Assuming vector meson dominance, the coupling con-stants g V and β were estimated to be 5 . . β ≈ − β = − . D D V is the same as that of DDV and ζ ≈ . K → Kρ . B. Heavy baryons
In the heavy quark limit, the ground states of heavybaryons
Qqq form an SU(3) antitriplet with J P =
12 + denoted by B ( Q )¯3 and two degenerate sextets with J P =( , ) + denoted by ( B ( Q )6 , B ( Q ) ∗ ) [70], B ( Q )¯3 = + c Ξ + c − Λ + c c − Ξ + c − Ξ c , (23) B ( Q )6 = Σ ++ c √ Σ + c √ Ξ (cid:48) + c √ Σ + c Σ c √ Ξ (cid:48) c √ Ξ (cid:48) + c √ Ξ (cid:48) c Ω c , (24) B ( Q ) ∗ = Σ ∗ ++ c √ Σ ∗ + c √ Ξ ∗ + c √ Σ ∗ + c Σ ∗ c √ Ξ ∗ c √ Ξ ∗ + c √ Ξ ∗ c Ω ∗ c . (25)Here we do not consider the P -wave heavy baryons sincethey are not well established experimentally. The twosextets are collected into the superfield S µ , S ( Q ) µ = B ( Q ) ∗ µ − √ γ µ + v µ ) γ B ( Q )6 , (26)¯ S ( Q ) µ = ¯ B ( Q ) ∗ µ + 1 √ B ( Q )6 γ ( γ µ + v µ ) , (27)where B µ is the Rarita-Schwinger vector-spinorfield [71]. The fields that annihilate anti-baryons are ob-tained by taking the charge conjugation of B ( Q )¯3 , B ( Q )6 and B ( Q ) ∗ , B ( ¯ Q )3 = − c Ξ − c − Λ − c c − Ξ − c − ¯Ξ c , (28) B ( ¯ Q )¯6 = Σ −− c √ Σ − c √ Ξ (cid:48)− c √ Σ − c ¯Σ c √ ¯Ξ (cid:48) c √ Ξ (cid:48)− c √ ¯Ξ (cid:48) c ¯Ω c , (29) B ( ¯ Q ) ∗ ¯6 = Σ ∗−− c √ Σ ∗− c √ Ξ ∗− c √ Σ ∗− c ¯Σ ∗ c √ ¯Ξ ∗ c √ Ξ ∗− c √ ¯Ξ ∗ c ¯Ω ∗ c , (30)where we have used the phase conventions such that C B ( Q ) C − = B ( ¯ Q ) . The corresponding superfields nowread S ( ¯ Q ) µ = B ( ¯ Q ) ∗ ¯6 µ − √ γ µ + v µ ) γ B ( ¯ Q )¯6 , (31)¯ S ( ¯ Q ) µ = ¯ B ( ¯ Q ) ∗ ¯6 µ + 1 √ B ( ¯ Q )¯6 γ ( γ µ + v µ ) . (32) The Lagrangian for the coupling of heavy baryons andlight mesons is constructed as [72] L B = L B + L S + L int , (33) L B ¯3 = 12 tr (cid:104) ¯ B ( Q )¯3 ( iv · D ) B ( Q )¯3 (cid:105) + iβ B tr (cid:104) ¯ B ( Q )¯3 v µ ( V µ − ρ µ ) B ( Q )¯3 (cid:105) , (34) L S = − tr (cid:104) ¯ S ( Q ) α ( iv · D − ∆ B ) S ( Q ) α (cid:105) + 32 g ( iv κ ) (cid:15) µνλκ tr (cid:104) ¯ S ( Q ) µ A ν S ( Q ) λ (cid:105) + iβ S tr (cid:104) ¯ S ( Q ) µ v α ( V α − ρ α ) S ( Q ) µ (cid:105) + λ S tr (cid:104) ¯ S ( Q ) µ F µν S ( Q ) ν (cid:105) , (35) L int = g tr (cid:104) ¯ S ( Q ) µ A µ B ( Q )¯3 (cid:105) + iλ I (cid:15) µνλκ v µ tr (cid:104) ¯ S ( Q ) ν F λκ B ( Q )¯3 (cid:105) + h . c ., (36)where D µ B = ∂ µ B + V µ B + B V Tµ , and ∆ B = m − m ¯3 is the mass difference between the anti-triplet and sextetbaryons. The coupling constants β B and β S are esti-mated in Ref. [72] where β S = − β B = 1 .
44 or 2 . β S = − β B = 1 .
74 from the vec-tor meson dominance assumption. However, there is asign ambiguity (only the absolute value of the ρN N cou-pling was determined from the
N N scattering [73]). Itturns out that the sign choice in Ref. [72] yields potentialsof the anti-charmed meson and charmed baryon systemswith an opposite sign compared to the ones obtained bySU(4) relations [35]. It also conflicts with the famous P c states [28], which are believed to be molecular statesof ¯ D ( ∗ ) Σ ( ∗ ) c with isospin 1 / β S , thesesystems will be repulsive (see below). These issues canbe fixed by choosing the signs of β B and β S opposite tothose taken in Ref. [72], just like what Ref. [74] did.For the coupling of antiheavy baryons and lightmesons, by taking the charge conjugation transformationof the above ones, we have L (cid:48)B = L (cid:48) B + L (cid:48) S + L (cid:48) int , (37) L (cid:48) B ¯3 = 12 tr (cid:104) ¯ B ( ¯ Q )3 ( iv · D ) T B ( ¯ Q )3 (cid:105) − iβ B tr (cid:104) ¯ B ( ¯ Q )3 v µ ( V µ − ρ µ ) T B ( ¯ Q )3 (cid:105) , (38) L (cid:48) S = − tr (cid:104) ¯ S ( ¯ Q ) α ( iv · D − ∆ B ) T S ( ¯ Q ) α (cid:105) + 32 g ( iv κ ) (cid:15) µνλκ tr (cid:104) ¯ S ( ¯ Q ) µ A Tν S ( ¯ Q ) λ (cid:105) − iβ S tr (cid:104) ¯ S ( ¯ Q ) µ v α ( V α − ρ α ) T S ( ¯ Q ) µ (cid:105) + λ S tr (cid:104) ¯ S ( ¯ Q ) µ ( F µν ) T S ( ¯ Q ) ν (cid:105) , (39) L (cid:48) int = g tr (cid:104) ¯ S ( ¯ Q ) µ A Tµ B ( ¯ Q )3 (cid:105) + iλ I (cid:15) µνλκ v µ tr (cid:104) ¯ S Tν ( F λκ ) T B ( ¯ Q )3 (cid:105) + h . c ., (40)with the transpose acting on the SU(3) flavor matrix.Notice that the spinor for an antibaryon is u instead of v since here the fields of heavy baryons and heavy an-tibaryons are treated independently.Similar with the Lagrangian for heavy mesons, we canfocus on the vector exchange contributions, and only thefollowing terms are relevant, L BBV = iβ B tr (cid:104) ¯ B ( Q )¯3 v µ ( V µ − ρ µ ) B ( Q )¯3 (cid:105) − iβ B tr (cid:104) ¯ B ( ¯ Q )3 v µ ( V µ − ρ µ ) T B ( ¯ Q )3 (cid:105) + iβ S tr (cid:104) ¯ S ( Q ) ν v µ ( V µ − ρ µ ) S ( Q ) ν (cid:105) − iβ S tr (cid:104) ¯ S ( ¯ Q ) ν v µ ( V µ − ρ µ ) T S ( ¯ Q ) ν (cid:105) . (41) III. POTENTIALSA. Conventions
In this paper, we take the following charge conjugationconventions:
C | D (cid:105) = (cid:12)(cid:12) ¯ D (cid:11) , C | D ∗ (cid:105) = − (cid:12)(cid:12) ¯ D ∗ (cid:11) , C | D (cid:105) = (cid:12)(cid:12) ¯ D (cid:11) , C | D (cid:105) = − (cid:12)(cid:12) ¯ D (cid:11) , C | B ¯3 (cid:105) = (cid:12)(cid:12) ¯ B ¯3 (cid:11) , C (cid:12)(cid:12)(cid:12) B ( ∗ )6 (cid:69) = (cid:12)(cid:12)(cid:12) ¯ B ( ∗ )6 (cid:69) , (42)which are consistent with the Lagrangians in Section II.Within these conventions, the flavor wave functions of theflavor-neutral systems that are charge conjugation eigen-states, including | DD ∗ (cid:105) c , | DD (cid:105) c , | D ∗ D (cid:105) c , | DD (cid:105) c , | D ∗ D (cid:105) c , | D D (cid:105) c and | B B ∗ (cid:105) c , can be expressed as | A A (cid:105) c = 1 √ (cid:0)(cid:12)(cid:12) A ¯ A (cid:11) ± ( − J − J − J c c c (cid:12)(cid:12) A ¯ A (cid:11)(cid:1) , (43)where J i is the spin of | A i (cid:105) , J is the total spin of thesystem | A A (cid:105) c , c i is defined by C | A i (cid:105) = c i (cid:12)(cid:12) ¯ A i (cid:11) , andthe plus and minus between the two terms are for boson-boson and fermion-fermion systems, respectively. Thesesystems satisfy C | A A (cid:105) c = c | A A (cid:105) c with c = ± | u (cid:105) = (cid:12)(cid:12)(cid:12)(cid:12) , (cid:29) , | d (cid:105) = (cid:12)(cid:12)(cid:12)(cid:12) , − (cid:29) , (cid:12)(cid:12) ¯ d (cid:11) = (cid:12)(cid:12)(cid:12)(cid:12) , (cid:29) , | ¯ u (cid:105) = − (cid:12)(cid:12)(cid:12)(cid:12) , − (cid:29) . (44)Consequently, we have (cid:12)(cid:12)(cid:12) D ( ∗ )+ (cid:69) = (cid:12)(cid:12)(cid:12)(cid:12) , (cid:29) , (cid:12)(cid:12)(cid:12) D ( ∗ ) − (cid:69) = (cid:12)(cid:12)(cid:12)(cid:12) , − (cid:29) , (cid:12)(cid:12)(cid:12) D ( ∗ )0 (cid:69) = − (cid:12)(cid:12)(cid:12)(cid:12) , − (cid:29) , (cid:12)(cid:12)(cid:12) ¯ D ( ∗ )0 (cid:69) = (cid:12)(cid:12)(cid:12)(cid:12) , (cid:29) , (cid:12)(cid:12)(cid:12) D ( ∗ )+ s (cid:69) = | , (cid:105) , (cid:12)(cid:12)(cid:12) D ( ∗ ) − s (cid:69) = | , (cid:105) , (cid:12)(cid:12) Λ + c (cid:11) = | , (cid:105) , (cid:12)(cid:12) Λ − c (cid:11) = − | , (cid:105) , (cid:12)(cid:12)(cid:12) Ξ ( (cid:48) ∗ )+ c (cid:69) = (cid:12)(cid:12)(cid:12)(cid:12) , (cid:29) , (cid:12)(cid:12)(cid:12) Ξ ( (cid:48) ∗ ) − c (cid:69) = (cid:12)(cid:12)(cid:12)(cid:12) , − (cid:29) , (cid:12)(cid:12)(cid:12) Ξ ( (cid:48) ∗ )0 c (cid:69) = (cid:12)(cid:12)(cid:12)(cid:12) , − (cid:29) , (cid:12)(cid:12)(cid:12) ¯Ξ ( (cid:48) ∗ )0 c (cid:69) = − (cid:12)(cid:12)(cid:12)(cid:12) , (cid:29) , (cid:12)(cid:12)(cid:12) Σ ( ∗ )++ c (cid:69) = | , (cid:105) , (cid:12)(cid:12)(cid:12) Σ ( ∗ ) −− c (cid:69) = | , − (cid:105) , (cid:12)(cid:12)(cid:12) Σ ( ∗ )+ c (cid:69) = | , (cid:105) , (cid:12)(cid:12)(cid:12) Σ ( ∗ ) − c (cid:69) = − | , (cid:105) , (cid:12)(cid:12)(cid:12) Σ ( ∗ )0 c (cid:69) = | , − (cid:105) , (cid:12)(cid:12)(cid:12) ¯Σ ( ∗ )0 c (cid:69) = | , (cid:105) , (cid:12)(cid:12)(cid:12) Ω ( ∗ )0 c (cid:69) = | , (cid:105) , (cid:12)(cid:12)(cid:12) ¯Ω ( ∗ )0 c (cid:69) = | , (cid:105) . (45)The isospin states of D ∗ , D and D ∗ are the same as thoseof D . The flavor wave functions of the systems consideredbelow with certain isospins can be easily computed usingClebsch-Gordan coefficients with these conventions.The potential we calculate is V = −M with M the2 → V means an attraction interaction. This convention isthe same as the widely used one in the on-shell Bethe-Salpeter equation T = V + V GT [75], and it is also thesame as the nonrelativistic potential in the Schr¨odingerequation up to a mass factor.
B. Potentials from light vector exchange
With the Lagrangian and conventions presented abovewe are ready to calculate the potentials of different sys-tems. We will use the resonance saturation model to getthe approximate potentials as constant contact terms.Note that the resonance saturation has been known tobe able to well approximate the low-energy constants(LECs) in the higher order Lagrangians of chiral per-turbation theory [29, 76], and it turns out therein thatwhenever vector mesons contribute they dominate thenumerical values of the LECs at the scale around the ρ -meson mass, which is called the modern version of vectormeson dominance.The general form of M for a process A ( p ) ¯ A ( p ) → A ( k ) ¯ A ( k ) by the vector meson exchange reads i M = ig v µ − i ( g µν − q µ q ν /m ) q − m + i(cid:15) v ν ig ≈ − i g g m , (46)where g and g account for the vertex information for A and ¯ A , respectively, q = p − k , and we have neglectedterms suppressed by O ( (cid:126)q /m ). For different parti-cles g and g are collected in Appendix A. It is worthmentioning that the spin information of the componentparticles is irrelevant here since the exchanged vectorsonly carry the momentum information, see Eqs. (22,41).Hence for a given system with different total spins, thepotentials at threshold are the same. With the vertex fac- TABLE I. Values of the coupling parameters used in the cal-culations. g V β β ζ β B β S − . − . V ≈ − F ˜ β ˜ β g V m m m , (47)where m , m and m ex are the masses of the two heavyhadrons and the exchanged particle, respectively. ˜ β and˜ β are the coupling constants for the two heavy hadronswith the vector mesons, and, given explicitly, ˜ β i = β forthe S -wave charmed mesons, ˜ β i = − β for the P -wavecharmed mesons, ˜ β i = β B for the anti-triplet baryons,and ˜ β i = − β S / F is a grouptheory factor accounting for the light-flavor SU(3) infor-mation, and in our convention a positive F means anattractive interaction. The values of F are listed in Ta-bles VIII and IX in Appendix B for all combinations ofheavy-antiheavy hadron pairs.For a system which can have different C -parities, likethose in Eqs. (43), the potential is expressed as V = V d ± ( − J − J − J c c c V c (48)with V d the potential from the direct process, e.g., D ¯ D ∗ → D ¯ D ∗ and V c from the cross one, e.g., D ¯ D ∗ → D ∗ ¯ D . V d ’s for these systems are covered by Eq. (47),while for the cross processes, it turns out that V c ’s for | DD ∗ (cid:105) c , | DD (cid:105) c , | D D (cid:105) c and | B B ∗ (cid:105) c systems vanish atthreshold. Explicit calculation shows that for the otherthree systems, | DD (cid:105) c , | D ∗ D (cid:105) c , and | D ∗ D (cid:105) c , V c ≈ F c F ζ g V m m m − ∆ m , (49)where F is the same as in Eq. (47), ∆ m = ( m − m ) ,and the additional factor F c accounts for the spin infor-mation, which are shown in Appendix C. However, V c is much smaller than V d and has little influence on thepole positions compared with the cutoff dependence (seebelow).In Table I, we collect the numerical values of the cou-pling parameters used in our calculations. C. Potentials from vector charmonia exchange
In principle, the
J/ψ , as well as the excited ψ vec-tor charmonia, can also be exchanged between charmedand anti-charmed hadrons. Being vector mesons, theircouplings to the charmed mesons have the same spin- momentum behavior as that of the light vectors. Accord-ing to Eq. (47), such contributions should be suppresseddue to the much larger masses of the ψ states than thoseof the light vectors by a factor of m ρ /m ψ ∼ .
1, up to thedifference of coupling constants. Therefore, the exchangeof light mesons, if not vanishing, dominates the potentialsat threshold. While for the systems where contributionsfrom light vectors vanish (or the ρ and ω exchanges can-cel each other), the vector charmonia exchange, as thesub-leading term, will play an important role in the near-threshold potentials.To be more precise, let us take the J/ψ exchange forexample, for which the Lagrangian reads L DDJ/ψ = ig DDJ/ψ ψ µ (cid:0) ∂ µ D † D − D † ∂ µ D (cid:1) , (50)with g DDJ/ψ ≈ .
64 [77]. The resulting potential in thenonrelativistic limit is V ∼ − g DDJ/ψ m m m , (51)which is about 40% of the potential from the φ exchangebetween D s and ¯ D s . The contributions from the othervector charmonia will be similar since their masses areof the same order. Notice that for all charmed and anti-charmed hadron systems, the vector charmonia exchangeyields attractive interactions. Unfortunately, it is noteasy to quantitatively estimate their contributions be-cause the masses of these charmonia are much larger thanthe energy scale of interest, and there is no hierarchyamong them to help selecting the dominant ones. Nev-ertheless, it could be possible to use, e.g., the Z c (3900),as a benchmark to estimate the overall contribution ofthe charmonia exchange. Given the controversy regard-ing its pole position [20, 22, 78–80], we refrain from doingso here (for further discussion, see Section IV B 1). IV. MOLECULAR STATES FROM CONSTANTINTERACTIONSA. Poles
Now that we have obtained the constant interactionsbetween a pair of heavy-antiheavy hadrons, we can givea rough picture of the spectrum of possible molecularstates. We search for poles of the scattering amplitude bysolving the single channel Bethe-Salpeter equation whichfactorizes into an algebraic equation for a constant po-tential, T = V − V G , (52)where G is the one loop two-body propagator. Here weadopt the dimensional regularization (DR) to regularizethe loop integral [81], G ( E ) = 116 π (cid:26) a ( µ ) + log m µ + m − m + s s log m m + kE log (2 kE + s ) − m + m (2 kE − s ) − m + m (cid:27) , (53)where s = E , m and m are the particle masses, and k = 12 E λ / ( E , m , m ) (54)is the corresponding three-momentum with λ ( x, y, z ) = x + y + z − xy − yz − xz for the K¨all´en trianglefunction. Here µ , chosen to be 1 GeV, denotes the DRscale, and a ( µ ) is a subtraction constant. The branch cutof k from the threshold to infinity along the positive real E axis splits the whole complex energy plane into twoRiemann sheets (RSs) defined as Im( k ) > k ) < G ( E ) = (cid:90) l dl π ω + ω ω ω e − l / Λ E − ( ω + ω ) + i(cid:15) , (55)with ω i = (cid:112) m i + l . The cutoff Λ is usually in the rangeof 0 . ∼ . a ( µ ) in DR isdetermined by matching the values of G from these twomethods at threshold. We will use the DR loop with theso-determined subtraction constant for numerical calcu-lations.For a single channel, if the interaction is attractiveand strong enough to form a bound state, the pole willbe located below threshold on the first RS. If it is notstrong enough, the pole will move onto the second RSas a virtual state, still below threshold. In Tables II,III, IV, V, VI and VII, we list all the pole positions ofthe heavy-antiheavy hadron systems which have attrac-tive interactions, corresponding to the masses of hadronicmolecules. For better illustration, these states, togetherwith some hadronic molecule candidates observed in ex-periments, are also shown in Figs. 1, 2, 3, 4, 5 and 6.In total, we obtain a spectrum of 229 hadronic moleculesconsidering constant contact interactions, saturated bythe light vector mesons, with the coupled-channel effectsneglected. B. Discussions of selected systems
It is worthwhile to notice that the overwhelming ma-jority of the predicted spectrum is located in the energyregion that has not been experimentally explored in de-tail. Searching for these states at BESIII, Belle-II, LHCband other planned experiments will be important to es-tablish a clear pattern of the hidden-charm states and to
TABLE II. Pole positions of heavy-antiheavy hadron systemswith (
I, S ) = (0 , E th in the second column is the thresholdin MeV. The number 0.5 (1.0) in the third (fourth) columnmeans that the cutoff Λ = 0 . .
0) GeV for Eq. (55) is usedto determine the subtraction constant a ( µ ) in Eq. (53). In thelast two columns, the first number in the parenthesis refersto the RS where the pole is located while the second numbermeans the distance between the pole position and the corre-sponding threshold, namely, E th − E pole , in MeV. System E th J P C
Pole (0.5) Pole (1.0) D ¯ D ++ (1, 1.31) (1, 35.8) D ¯ D ∗ + ± (1, 1.56) (1, 36.2) D s ¯ D s ++ (2, 35.5) (2, 4.72) D ∗ ¯ D ∗ , ++ , + − (1, 1.82) (1, 36.6) D s ¯ D ∗ s + ± (2, 31.0) (2, 3.15) D ∗ s ¯ D ∗ s , ++ , + − (2, 26.7) (2, 1.92) D ¯ D −± (1, 2.2) (1, 36.7) D s ¯ D s −± (2, 21.3) (2, 0.713) D ¯ D , ++ , + − (1, 3.01) (1, 36.7) D ¯ D , , + ± (1, 3.06) (1, 36.6) D ¯ D , , ++ , (1 , + − (1, 3.1) (1, 36.6) D s ¯ D s , ++ , + − (2, 11.7) (1, 0.074) D s ¯ D s , , + ± (2, 11.3) (1, 0.104) D s ¯ D s , , ++ , (1 , + − (2, 10.9) (1, 0.139)Λ c ¯Λ c − + , −− (1, 1.98) (1, 33.8)Σ c ¯Σ c − + , −− (1, 11.1) (1, 60.8)Ξ c ¯Ξ c − + , −− (1, 4.72) (1, 42.2)Σ ∗ c ¯Σ c −± , −± (1, 11.0) (1, 60.1)Σ ∗ c ¯Σ ∗ c , − + , (1 , −− (1, 10.9) (1, 59.5)Ξ c ¯Ξ (cid:48) c −± , −± (1, 4.79) (1, 41.9)Ξ c ¯Ξ ∗ c −± , −± (1, 4.84) (1, 41.6)Ξ (cid:48) c ¯Ξ (cid:48) c − + , −− (1, 4.87) (1, 41.5)Ξ ∗ c ¯Ξ (cid:48) c −± , −± (1, 4.91) (1, 41.3)Ξ ∗ c ¯Ξ ∗ c , − + , (1 , −− (1, 4.95) (1, 41.0)Ω c ¯Ω c − + , −− (1, 4.17) (1, 38.0)Ω ∗ c ¯Ω c −± , −± (1, 4.22) (1, 37.8)Ω ∗ c ¯Ω ∗ c , − + , (1 , −− (1, 4.26) (1, 37.6)understand how QCD organizes the hadron spectrum.In the following, we discuss a few interesting systemsthat have experimental candidates. D ( ∗ ) ¯ D ( ∗ ) : X (3872) , Z c (3900) and their partners Within the mechanism considered here, the interac-tions of D ¯ D , D ¯ D ∗ and D ∗ ¯ D ∗ are the same. For the me-son pairs to be isospin scalars, the attractions are strongenough to form bound states with similar binding ener- m ( G e V ) D ¯ D D ∗ ¯ D D ∗ ¯ D D s ¯ D s D ∗ s ¯ D s D ∗ s ¯ D s D ¯ D D s ¯ D s Λ c ¯Λ c Σ c ¯Σ c Ξ c ¯Ξ c Σ c ¯Σ ∗ c Σ ∗ c ¯Σ ∗ c Ξ c ¯Ξ c Ξ c ¯Ξ ∗ c Ξ c ¯Ξ c Ξ ∗ c ¯Ξ c Ξ ∗ c ¯Ξ ∗ c Ω c ¯Ω c Ω c ¯Ω ∗ c Ω ∗ c ¯Ω ∗ c ψ (4230) bound virtual − + − − − + − − − + − − − + − − FIG. 1. The spectrum of hadronic molecules consisting of a pair of charmed-anticharmed hadrons with (
I, S ) = (0 , −− ,1 − + and 3 − + are exotic quantum numbers. The colored rectangle, green for a bound state and orange for a virtual state, coversthe range of the pole position for a given system with cutoff Λ varies in the range of [0 . , .
0] GeV. Thresholds are marked bydotted horizontal lines. The rectangle closest to, but below, the threshold corresponds to the hadronic molecule in that system.In some cases where the pole positions of two systems overlap, small rectangles are used with the left (right) one for the systemwith the higher (lower) threshold. The blue line (band) represents the center value (error) of the mass of the experimentalcandidate of the corresponding molecule. The averaged central value and error of the ψ (4230) mass are taken from RPP [47]. D ¯ D D ¯ D D ¯ D D s ¯ D s D s ¯ D s D s ¯ D s bound virtual m ( G e V ) D ¯ DD ¯ D ∗ D s ¯ D s D ∗ ¯ D ∗ D s ¯ D ∗ s D ∗ s ¯ D ∗ s ˜ X (3872) X (3872) + + + + + − + + + − + + + − + + FIG. 2. The spectrum of hadronic molecules consisting of a pair of charmed-anticharmed hadrons with (
I, S ) = (0 , + − areexotic quantum numbers. The parameters of the X (3872) and ˜ X (3872) are taken from RPP [47] and Ref. [82], respectively.See the caption for Fig.1. gies, see Table II and Fig. 2, while for the isovector pairs,the contributions from the ρ and ω exchanges cancel eachother, see Table VIII in Appendix B.The X (3872) observed by the Belle Collaboration [19]is widely suggested as an isoscalar D ¯ D ∗ molecule with J P C = 1 ++ [84–87] (for reviews, see, e.g., Refs. [3, 8, 12,15]). Actually such a hadronic molecule was predicted10 years before the discovery by T¨ornvist considering theone-pion exchange [88]. Our results show that the lightvector exchange leads to a near-threshold isoscalar boundstate that can be identified with the X (3872) as well, to-gether with a negative C -parity partner of X (3872) withthe same binding energy (see also Refs. [89, 90]). There isexperimental evidence of such a negative C -parity state, named as ˜ X (3872), reported by the COMPASS Collab-oration [82]. A recent study of the D ( ∗ )( s ) ¯ D ( ∗ )( s ) molecularstates using the method of QCD sum rules also finds both1 ++ and 1 + − D ¯ D ∗ states [91].The potential predicting the X (3872) as an isoscalar D ¯ D ∗ bound state, also predicts the existence of isoscalar D ¯ D and D ∗ ¯ D ∗ bound states. By imposing only HQSS,there are two independent contact terms in the D ( ∗ ) ¯ D ( ∗ ) interactions for each isospin [92, 93], which can be defined It should be called h c (3872) according to the RPP nomenclature. m ( G e V ) ¯ D Σ c ¯ D Σ ∗ c ¯ D ∗ Σ c ¯ D ∗ Σ ∗ c P c (4312) P c (4440) P c (4457) bound virtual / − / − / − m ( G e V ) ¯ D Σ c ¯ D Σ c ¯ D Σ ∗ c ¯ D Σ ∗ c bound virtual / + / + / + / + FIG. 3. The spectrum of hadronic molecules consisting of a pair of charmed-anticharmed hadrons with (
I, S ) = ( ,
0) and unitbaryon number. The left orange band and right green band for each pole represent that the pole moves from a virtual state onthe second RS to a bound state on the first RS when the cutoff Λ changes from 0.5 to 1.0 GeV. The parameters of these three P c states are taken from Ref. [28], whose J P have not been determined experimentally. See the caption for Fig. 1. m ( G e V ) ¯ D Ξ c ¯ D Ξ c ¯ D ∗ Ξ c ¯ D Ξ ∗ c ¯ D ∗ Ξ c ¯ D ∗ Ξ ∗ c P cs bound virtual / − / − / − m ( G e V ) ¯ D Ξ c ¯ D Ξ c ¯ D Ξ c ¯ D Ξ c ¯ D Ξ ∗ c ¯ D Ξ ∗ c bound virtual / + / + / + / + FIG. 4. The spectrum of hadronic molecules consisting of a pair of charmed-anticharmed hadrons with (
I, S ) = (0 ,
1) and unitbaryon number. The parameters of P cs (4459) are taken from Ref. [83], whose J P have not been determined experimentally.See the caption for Fig. 3. as [8] C H ¯ H, = (cid:28) , , (cid:12)(cid:12)(cid:12)(cid:12) H I (cid:12)(cid:12)(cid:12)(cid:12) , , (cid:29) ,C H ¯ H, = (cid:28) , , (cid:12)(cid:12)(cid:12)(cid:12) H I (cid:12)(cid:12)(cid:12)(cid:12) , , (cid:29) , (56)where H I is the interaction Hamiltonian, and | s (cid:96) , s (cid:96) , s (cid:96) (cid:105) denotes the charmed meson pair with s (cid:96) being the totalangular momentum of the light degrees of freedom in thetwo-meson system and s (cid:96) ,(cid:96) for the individual mesons.Such an analysis leads to the prediction of a 2 ++ D ∗ ¯ D ∗ tensor state as the HQSS partner of the X (3872) con- sidering the physical charmed meson masses [93–95] andthree partners with 0 ++ , + − and 2 ++ in the strict heavyquark limit [96, 97] that depend on the same contact termas the X (3872). The resonance saturation by the lightvector mesons in fact leads to a relation C H ¯ H, = C H ¯ H, ,and consequently 6 S -wave D ( ∗ ) ¯ D ( ∗ ) bound states.The existence of an isoscalar D ¯ D bound state hasbeen predicted by various phenomenology models [85,89, 93, 94, 98, 99], and more recently by lattice QCDcalculations [100]. Despite attempts [101–103] to dig outhints for such a state from the available experimentaldata [104–106], no clear evidence has yet been found.However, this could be because its mass is below the D ¯ D m ( G e V ) Λ c ¯Σ c Λ c ¯Σ ∗ c Σ c ¯Σ c Ξ c ¯Ξ c Σ c ¯Σ ∗ c Σ ∗ c ¯Σ ∗ c Ξ c ¯Ξ c Ξ c ¯Ξ ∗ c Ξ c ¯Ξ c Ξ ∗ c ¯Ξ c Ξ ∗ c ¯Ξ ∗ c bound virtual − + − − − + − − − + − − − − FIG. 5. The spectrum of hadronic molecules consisting of a pair of charmed-anticharmed hadrons with (
I, S ) = (1 , m ( G e V ) Λ c ¯Ξ c Λ c ¯Ξ c Ξ c ¯Σ c Λ c ¯Ξ ∗ c Ξ c ¯Σ ∗ c Ξ c ¯Ω c Ξ c ¯Ω ∗ c bound virtual − − − m ( G e V ) Σ c ¯Ξ c Σ ∗ c ¯Ξ c Σ c ¯Ξ ∗ c Σ ∗ c ¯Ξ ∗ c Ξ c ¯Ω c Ξ ∗ c ¯Ω c Ξ c ¯Ω ∗ c Ξ ∗ c ¯Ω ∗ c bound virtual − − − − FIG. 6. The spectrum of hadronic molecules consisting of a pair of charmed-anticharmed hadrons with (
I, S ) = ( , TABLE III. Pole positions of heavy-antiheavy hadron systemswith (
I, S ) = (0 , C -parities yield slightly differentpole positions. System E th J P C
Pole (0.5) Pole (1.0) D ¯ D − + (1, 1.78) (1, 34.9)1 −− (1, 2.53) (1, 38.4) D ∗ ¯ D − + (1, 2.55) (1, 37.4)0 −− (1, 2.29) (1, 36.3)1 − + (1, 2.36) (1, 36.6)1 −− (1, 2.49) (1, 37.1)2 − + (1, 2.36) (1, 36.6)2 −− (1, 2.49) (1, 37.1) D ∗ ¯ D − + (1, 2.54) (1, 37.1)1 −− (1, 2.4) (1, 36.5)2 − + (1, 2.68) (1, 37.7)2 −− (1, 2.26) (1, 35.9)3 − + (1, 2.89) (1, 38.6)3 −− (1, 2.05) (1, 34.9) D s ¯ D s − + (2, 24.2) (2, 1.4)1 −− (2, 19.6) (2, 0.402) D ∗ s ¯ D s − + (2, 17.5) (2, 0.179)0 −− (2, 19.2) (2, 0.402)1 − + (2, 18.7) (2, 0.402)1 −− (2, 17.9) (2, 0.227)2 − + (2, 18.7) (2, 0.342)2 −− (2, 17.9) (2, 0.402) D ∗ s ¯ D s − + (2, 17.4) (2, 0.177)1 −− (2, 18.3) (2, 0.402)2 − + (2, 16.6) (2, 0.402)2 −− (2, 19.2) (2, 0.418)3 − + (2, 15.4) (2, 0.023)3 −− (2, 20.6) (2, 0.402)threshold so that no easily detectable decay modes areavailable.As for the isoscalar 2 ++ D ∗ ¯ D ∗ bound state, it can de-cay into D ¯ D in D − wave, and the width was predicted tobe in the range from a few to dozens of MeV [97, 107].No evidence has been found so far. One possible reasonis that the coupling to ordinary charmonia could eithermove the 2 ++ pole deep into the complex energy planeand thus invisible [108] or make the D ∗ ¯ D ∗ interaction inthe 2 ++ sector unbound [18, 110]. For more discussions For a discussion of the intricate interplay between a meson-mesonchannel with multiple quark model states, see Ref. [109]. Mixing of two energy levels will push them further apart. Thus,
TABLE IV. Pole positions of heavy-antiheavy hadron systemswith (
I, S ) = (1 / ,
0) and unit baryon number. See the cap-tion for Table II.
System E th J P Pole (0.5) Pole (1.0)¯ D Σ c − (2, 2.04) (1, 7.79)¯ D Σ ∗ c − (2, 1.84) (1, 8.1)¯ D ∗ Σ c , ) − (2, 1.39) (1, 8.95)¯ D ∗ Σ ∗ c , , ) − (2, 1.23) (1, 9.26)¯ D Σ c , ) + (2, 0.417) (1, 11.5)¯ D Σ c , ) + (2, 0.366) (1, 11.7)¯ D Σ ∗ c , , ) + (2, 0.34) (1, 11.8)¯ D Σ ∗ c , , , ) + (2, 0.294) (1, 12.0) TABLE V. Pole positions of heavy-antiheavy hadron systemswith (
I, S ) = (0 ,
1) and unit baryon number. See the captionfor Table II.
System E th J P Pole (0.5) Pole (1.0)¯ D Ξ c − (2, 2.14) (1, 7.53)¯ D Ξ (cid:48) c − (2, 1.82) (1, 8.05)¯ D ∗ Ξ c , ) − (2, 1.47) (1, 8.69)¯ D Ξ ∗ c − (2, 1.65) (1, 8.34)¯ D ∗ Ξ (cid:48) c , ) − (2, 1.21) (1, 9.21)¯ D ∗ Ξ ∗ c , , ) − (2, 1.08) (1, 9.51)¯ D Ξ c , ) + (2, 0.455) (1, 11.3)¯ D Ξ c , ) + (2, 0.4) (1, 11.5)¯ D Ξ (cid:48) c , ) + (2, 0.326) (1, 11.8)¯ D Ξ (cid:48) c , ) + (2, 0.28) (1, 12.0)¯ D Ξ ∗ c , , ) + (2, 0.262) (1, 12.1)¯ D Ξ ∗ c , , , ) + (2, 0.222) (1, 12.3)regarding the mixing of charmonia with meson-mesonchannels, we refer to Refs. [111–113].The Z c (3900) [20–22] was also suggested to be anisovector D ¯ D ∗ molecule with quantum numbers J P C =1 + − [95, 114] even though the light vector exchange van-ishes in this case. Recall that the vector charmonia ex-change will also yield an attractive interaction, as dis-cussed in Section III C, which can possibly lead to a vir-tual state below threshold. In fact, it has been suggestedthat the J/ψ -exchange is essential in the formation of the Z c (3900) in Ref. [115]. It was shown in Refs. [78, 116, 117]that a virtual state assignment for the Z c (3900) is con-sistent with the experimental data. Furthermore, it wasshown in Ref. [118] that the finite volume energy levels mixing of the D ∗ ¯ D ∗ state with a lower-mass χ c (2 P ) can effectivelyprovide a repulsive contribution to the D ∗ ¯ D ∗ interaction. Notice that the coupling of the D ¯ D ∗ to a lower channel, which is TABLE VI. Pole positions of heavy-antiheavy hadron systemswith (
I, S ) = (1 / , System E th J P Pole (0.5) Pole (1.0)Λ c ¯Ξ c , − (2, 1.29) (1, 8.42)Λ c ¯Ξ (cid:48) c , − (2, 1.05) (1, 8.93)Ξ c ¯Σ c , − (1, 5.98) (1, 46.4)Λ c ¯Ξ ∗ c , − (2, 0.92) (1, 9.23)Ξ c ¯Σ ∗ c , − (1, 6.01) (1, 46.1)Σ c ¯Ξ (cid:48) c , − (1, 6.03) (1, 45.9)Σ ∗ c ¯Ξ (cid:48) c , − (1, 6.06) (1, 45.6)Σ c ¯Ξ ∗ c , − (1, 6.05) (1, 45.6)Σ ∗ c ¯Ξ ∗ c , , , − (1, 6.08) (1, 45.2)Ξ c ¯Ω c , − (2, 8e-5) (1, 15.9)Ξ c ¯Ω ∗ c , − (1, 0.002) (1, 16.2)Ξ (cid:48) c ¯Ω c , − (1, 0.006) (1, 16.4)Ξ ∗ c ¯Ω c , − (1, 0.016) (1, 16.6)Ξ (cid:48) c ¯Ω ∗ c , − (1, 0.016) (1, 16.6)Ξ ∗ c ¯Ω ∗ c , , , − (1, 0.030) (1, 16.8) TABLE VII. Pole positions of heavy-antiheavy hadron sys-tems with (
I, S ) = (1 , System E th J P C
Pole (0.5) Pole (1.0)Λ c ¯Σ c , −± (1, 2.19) (1, 33.9)Λ c ¯Σ ∗ c , −± (1, 2.27) (1, 33.9)Σ c ¯Σ c − + , −− (1, 8.28) (1, 53.3)Ξ c ¯Ξ c − + , −− (2, 18.2) (2, 0.39)Σ ∗ c ¯Σ c , −± (1, 8.27) (1, 52.8)Σ ∗ c ¯Σ ∗ c , − + , (1 , −− (1, 8.25) (1, 52.3)Ξ c ¯Ξ (cid:48) c , −± (2, 16.5) (2, 0.19)Ξ c ¯Ξ ∗ c , −± (2, 15.6) (2, 0.11)Ξ (cid:48) c ¯Ξ (cid:48) c − + , −− (2, 14.9) (2, 0.061)Ξ ∗ c ¯Ξ (cid:48) c , −± (2, 14.0) (2, 0.020)Ξ ∗ c ¯Ξ ∗ c , − + , (1 , −− (2, 13.2) (2, 0.002)are also consistent with the lattice QCD results whichdid not report an additional state [119]. Similarly, the Z c (4020) ± [23, 24] with isospin-1 near the D ∗ ¯ D ∗ thresh-old can be a virtual state as well. It was recently ar-gued that a near-threshold virtual state needs to be un-derstood as a hadronic molecule [120]. Analysis of theBelle data on the Z b states [25, 26] using constant contact J/ψπ in this case, induces a finite width for the virtual state poleso that it behaves like a resonance, i.e. a pole in the complex planeoff the real axis. The same is true for all other poles generatedhere. M J/ψφ [MeV]020406080100120140160 C a n d i d a t e s / ( M e V ) D s ¯ D ∗ s D ∗ s ¯ D ∗ s D s ¯ D ∗ s D s ¯ D s D s ¯ D s D s ¯ D s D ∗ s ¯ D ∗ s D ∗ s ¯ D s D ∗ s ¯ D s D ∗ s ¯ D s D ∗ s ¯ D ∗ s D ∗ s ¯ D s LCHb fitLHCb data
FIG. 7. Thresholds of charm-strange meson pairs in theenergy range relevant for the B + → J/ψφK + . Here, D ∗ s denotes D ∗ s (2317), D s and D (cid:48) s denote D s (2536) and D s (2460), respectively, and D s denotes D s (2573). Thedata are taken from Ref. [131]. terms to construct the unitary T -matrix also supports the Z b states as hadronic molecules [121–125]. The molecularexplanation of the Z c and Z b states is further supportedby their decay patterns studied using a quark exchangemodel [126, 127]. Without a quantitative calculation, ascommented in Section III C, we postulate that there canbe 6 isovector hadronic molecules (with the same J P C asthe isoscalar ones) as virtual states of D ( ∗ ) ¯ D ( ∗ ) , whichwill show up as prominent threshold cusps (see Ref. [38]for a general discussion of the line shape behavior in thenear-threshold region). D ( ∗ ) s ¯ D ( ∗ ) s virtual states Here we find that the potential from the φ exchange isprobably not enough to form bound states of D ( ∗ ) s ¯ D ( ∗ ) s .Instead, virtual states are obtained, see Table II andFig. 2.On the contrary, based on two prerequisites that1) X (3872) is a marginally bound state of D ¯ D ∗ withbinding energy 0 ∼ D s ¯ D s can form a bound states with a bindingenergy of 2 . ∼ . χ c (3930) mass determined byLHCb [128, 129],Ref. [130] obtained D ( ∗ ) s ¯ D ( ∗ ) s bound systems with bindingenergies up to 80 MeV.The X (4140) first observed by the CDF Collabora-tion [132] was considered as a molecule of D ∗ s ¯ D ∗ s with J P C = 0 ++ or 2 ++ in Refs. [33, 133–138], which is, how-ever, disfavored by the results of LHCb [131, 139] wherethe J P C of X (4140) were suggested to be 1 ++ (and thus5the X (4140) was named as χ c (4140) in the latest versionof RPP). Actually in our calculation, it is not likely forthe D ∗ s ¯ D ∗ s to form such a deeply bound state, noticingthat the X (4140) is about 80 MeV below the thresholdof D ∗ s ¯ D ∗ s . Instead, it is interesting to notice that just atthe D ∗ s ¯ D ∗ s threshold there is evidence for a peak in the in-variant mass distribution of J/ψφ , see Fig. 7. Followingthe analysis in Ref. [38], if the interaction of the D ∗ s ¯ D ∗ s isattractive but not strong enough to form a bound state, apeak will appear just at the D ∗ s ¯ D ∗ s threshold in the invari-ant mass distribution of J/ψφ , and the peak is narrow ifthere is a nearby virtual state pole. A detailed study ofthis threshold structure can tell us whether the attrac-tion between a pair of charm-strange mesons is strongenough to form a bound state or not.The difference between the
J/ψφ and D s ¯ D ∗ s thresholdsis merely 36 MeV. Thus, the shallow D s ¯ D ∗ s virtual statewith 1 ++ could be responsible for the quick rise of the J/ψφ invariant mass distribution just above thresholdobserved in the LHCb data [131]. D ( ∗ ) ¯ D ( ∗ ) s : Z cs as virtual states No light vector can be exchanged here and the attrac-tive interaction from vector charmonia exchange is cru-cial, similar to the isovector D ¯ D ( ∗ ) systems. A virtualstate pole could exist below threshold. In particular, ifthe Z c (3900) exists as a virtual state, the same interac-tion would induce D ( ∗ ) ¯ D ( ∗ ) s virtual states.Recently, a near-threshold enhancement in the invari-ant mass distribution of D − s D ∗ + D ∗− s D was reportedby the BESIII Collaboration [27] and an exotic state Z cs (3985) − was claimed. This state has been widelyinvestigated [141–158], some of which regard it as amolecule of D − s D ∗ + D ∗− s D while some others objectsuch an explanation. In Ref. [145], it was found that avirtual or resonant pole together with a triangle singular-ity can well reproduce the line shape of the BESIII data,consistent with the analysis here. D ( ∗ ) ¯ D , : Y (4260) and related states It is possible for the isoscalar D ¯ D pair to form a boundstate with a binding energy from a few MeV to dozensof MeV. Note that this system can have J P C = 1 −± andthe 1 −− state is slightly more deeply bound than the 1 − + one which has exotic quantum numbers.The Y (4260) was discovered by the B A B AR Collabo-ration [159] with a mass of (4259 ± +2 − ) MeV and a This structure around the D ∗ s ¯ D ∗ s threshold drew the attention ofRef. [140] where two resonances, a narrow X (4140) and a broad X (4160), were introduced to fit the J/ψφ invariant mass distri-bution from the threshold to about 4250 MeV. There the broad X (4160) was considered as a D ∗ s ¯ D ∗ s molecule. width of 50 ∼
90 MeV and later confirmed by other ex-periments [160, 161]. Now it is called ψ (4230) due toa lower mass from the more precise BESIII data and acombined analysis in four channels, e + e − → ωχ c [162], π + π − h c [163], π + π − J/ψ [164] and D D ∗− π + + c.c. [165],yielding a mass of (4219 . ± . ± .
1) MeV and a width of(56 . ± . ± .
9) MeV [166]. This state is a good candi-date of exotic states (see reviews, e.g., Refs. [3, 8, 15]). Itwas argued that the isoscalar D ¯ D plays important rolesin the structure of Y (4260) in, e.g., Refs. [114, 167, 168].The binding energy of the isoscalar D ¯ D system with J P C = 1 −± via the vector meson exchange was cal-culated by solving the Schr¨odinger equation in spatialspace in Ref. [69], and the results are consistent withthis work except that the sign of V c , which has a minorimpact, is not correct there. Note that the mass of theisoscalar D ¯ D bound state obtained here is larger thanthe nominally mass of the ψ (4230), see Fig. 1, but themixing of the D ¯ D molecule with a D -wave vector char-monium [169] may solve this discrepancy.From the results in Table II and Fig. 1, the isoscalar D ¯ D bound state has quite some partners, either ofHQSS or of SU(3) flavor. In particular, several of themhave vector quantum numbers, including a D ∗ ¯ D boundstate with a mass about 4.39 ∼ D ∗ ¯ D boundstate with a mass about 4.43 ∼ D s ¯ D s , D ∗ s ¯ D s and D ∗ s ¯ D s .The current status of the vector charmonium spectrumaround 4.4 GeV is not clear, and the peak structures inexclusive and inclusive R -value measurements are differ-ent (for a compilation of the relevant data, see Ref. [170]).Thus, it is unclear which structure(s) can be identifiedas the candidate(s) of the D ∗ ¯ D bound states. Nev-ertheless, the Y (4360), aka ψ (4360), and ψ (4415) havebeen suggested to correspond to the D ¯ D ∗ and D ¯ D ∗ states, respectively [171–174]. A determination of thepoles around 4.4 GeV would require a thorough analysisof the full data sets including these open-charm channels,and the first steps have been done in Refs. [175, 176].As for the virtual states with hidden-strangeness, theyare expected to show up as narrow threshold cusps infinal states like J/ψf (980) and ψ (2 S ) f (980). Theycould play an important role in generating the Y (4660),aka ψ (4660), peak observed in the ψ (2 S ) f (980) → ψ (2 S ) π + π − invariant mass distribution [177, 178]. Al-though it was proposed [180, 181] that the Y (4630) struc-ture observed in the Λ c ¯Λ c spectrum [182] could be thesame state as the Y (4660) one, the much more preciseBESIII data [183], however, show a different behavior upto 4.6 GeV in the Λ c ¯Λ c invariant mass distribution (seebelow). Further complications come from the 1 −− struc-tures around 4.63 GeV reported in the D s ¯ D s + c.c. [184]and D s ¯ D s + c.c. [185] distributions, the former of which The Y (4660) was suggested to be a ψ (2 S ) f (980) bound state inRef. [179] to explain why it was seen only the ψ (2 S ) π + π − finalstate with the pion pair coming from the f (980). D ∗ s ¯ D s - D s ¯ D s interaction [186]. Suffice it to say that thesituation of the Y (4630) is not unambiguous. With moreprecise data that will be collected at BESIII and Belle-II, we suggest to search for line shape irregularities (ei-ther peaks or dips) at the D ∗ s ¯ D s and D ∗ s ¯ D s thresholdsin open-charm-strangeness final states such as D ( ∗ ) s ¯ D ( ∗ ) s and D s ¯ D s s .There are also hints in data for positive C -parity D s ¯ D s and D ∗ s ¯ D s virtual states (see Table III andFig. 1), whose thresholds are at 4503 MeV and 4647 MeV,respectively. As can be seen from Fig. 7, there is a peakaround 4.51 GeV and a dip around 4.65 GeV in the J/ψφ energy distribution measured by the LHCb Collabora-tion [131], and the energy difference between the dip andpeak approximately equals to the mass splitting betweenthe D ∗ s and D s . We also notice that the highest peak inthe same data appears at the D ∗ s (2317) ¯ D s threshold. All these channels, together with the D s ¯ D ∗ s and D ∗ s ¯ D ∗ s discussed in Section IV B 2, need to be considered in areliable analysis of the B + → J/ψφK + data, which ishowever beyond the scope of this paper.Notice that because the D and D s s have finitewidths, the molecular states containing one of them candecay easily through the decays of D or D s s . Thestructures at the thresholds of D ( ∗ ) s ¯ D s s , for whichvirtual states are predicted, will get smeared by thewidths of D s s . Thus, the D ( ∗ ) s ¯ D s threshold struc-tures should be broader and smoother than the D ( ∗ ) s ¯ D s ones since the width of the D s , (16 . ± .
7) MeV [47], ismuch larger than that of the D s , (0 . ± .
05) MeV [47]. Λ c ¯Λ c : analysis of the BESIII data and morebaryon-antibaryon bound states From Table II and Fig. 1, in the spectrum of theisoscalar 1 −− states, in addition to those made of a pairof charmed mesons, we predict more than 10 baryon-antibaryon molecules. The lowest one is the Λ c ¯Λ c boundstate, and the others are above 4.85 GeV. While thoseabove 4.85 GeV are beyond the current reach of BESIII(there is a BESIII data-taking plan in the energy regionabove 4.6 GeV [187]), there is strong evidence for the ex-istence of a Λ c ¯Λ c bound state in the BESIII data [183].The Λ c ¯Λ c system can form a bound state with a bind-ing energy in the range from a few MeV to dozens ofMeV, depending on the cutoff. Therefore, we predictthat there is a pole below the Λ c ¯Λ c threshold and thepole position can be extracted from the line shape of theΛ c ¯Λ c invariant mass distribution near threshold. The coincidence of the peak position with the threshold and thehighly asymmetric line shape suggests a D ∗ s (2317) ¯ D s virtual state.Such systems will be studied in a future work. The cross section of e + e − → Λ c ¯Λ c was first measuredusing the initial state radiation by Belle [182] and a vectorcharmonium-like structure Y (4630) was observed. TheBESIII Collaboration measured such cross sections atfour energy points just above threshold much more pre-cisely [183]. The energy dependence of the cross sectionsat these four points has a weird behavior: it is almostflat. This can be understood as the consequence of theSommerfeld factor, which makes the distribution nonva-nishing even exactly at threshold, and the existence ofa near-threshold pole, which counteracts the increasingtrend of the phase space multiplied by the Sommerfeldfactor to result in an almost flat distribution. Here we fitBESIII data to estimate where the pole is located.The Sommerfeld factor [188] accounting for the multi-photon exchange between the Λ + c and ¯Λ − c reads, S ( E ) = 2 πx − e − πx , (57)where x = αµ/k with α ≈ / µ = m Λ c /
2, and k isdefined in Eq. (54). The cross section of e + e − → Λ c ¯Λ c is now parameterized as σ ( E ) = N · S ( E ) · | f ( E ) | · ρ ( E ) E , (58)with N a normalization constant and ρ ( E ) = k/ (8 πE )the phase space. Here f ( E ) denotes the nonrelativisticscattering amplitude, and the S -wave one, f ( E ) = (cid:18) a − i (cid:112) µ ( E − m Λ c ) (cid:19) − , (59)is sufficient in the immediate vicinity of the threshold.Note that we take the scattering length a complex totake into account the couplings between the Λ c ¯Λ c andlower channels [38]. Finally, we have 3 parameters,Re(1 /a ), Im(1 /a ) and N , to fit four experimental data.The fitted results are shown in Fig. 8. We can seethat the best fit leads to a pole located close to the realaxis but above threshold. A pole below threshold, as wepredicted, is also possible, see the bottom one in Fig. 8.These fits, though with larger χ , are reasonable sincewe have only four points. The obtained a values fromthese fits yield poles several MeV below threshold withan imaginary part of dozens of MeV. Such poles are lo-cated on the first RS corresponding to a bound state,which moves from the real axis onto the complex planedue to the coupling to lower channels. There is anotherpole located at the symmetric position on the second RS,corresponding to a virtual state. Actually, with the scat-tering length approximation in Eq. (59), we cannot de-termine on which RS the pole is located since the poleson different RSs below threshold have the same behav-ior above threshold. More data are needed to pin downthe exact pole position corresponding to our predictedΛ c ¯Λ c bound state, which should be different from the Y (4630) or Y (4660). In Ref. [189], the BESIII [183] and7 Re E (MeV) σ ( pd ) data /a = − − i MeV1 /a = − − i MeV1 /a = − − i MeV2 m Λ c FIG. 8. Top: pole positions of Eq. (59) on the first RS withdifferent scattering length ( a ) values and the color representsthe χ (in a logarithmic form for better illustration) of the fitto BESIII data [183]. The pole on the second RS is at thesame position if we change the sign of Re(1 /a ), which doesnot change the fit. Bottom: examples of some fits, which yieldpoles below threshold at 4456 − i MeV (red), 4468 − i MeV(blue dash-dotted) and 4566 − i MeV (green dashed).
Belle [182] data are fitted together using an amplitudewith a pole around 4.65 GeV. While the Belle data ofthe Y (4630) peak can be well described, the much moreprecise BESIII data points in the near-threshold regioncannot. We conclude that the data from Λ c ¯Λ c thresh-old up to 4.7 GeV should contain signals of at least twostates: the Λ c ¯Λ c molecule and another one with a massaround 4.65 GeV.As for the isoscalar vector states above 4.85 GeV, thestructures could be more easily identified from data thanthose around 4.3 GeV. This is because the charmoniumstates in that mass region should be very broad whilethese hadronic molecules are narrower due to the smallbinding energies, corresponding to large spatial exten-sions. We expect the Σ c ¯Σ c , Ξ c ¯Ξ c and Σ c ¯Σ ∗ c below 5 GeVto be seen in the forthcoming BESIII measurements, andthe ones higher than 5 GeV can be searched for in futuresuper tau-charm facilities [190, 191]. There are isovector 1 −− baryon-antibaryon molecularstates above 4.7 GeV, see Table VII and Fig. 5. It is moredifficult to observe these states than the isoscalar onesin e + e − collisions since the main production mechanismof vector states should be driven by a vector ¯ cγ µ c cur-rent, which is an isoscalar, coupled to the virtual photon.However, they could be produced together with a pion,and thus can be searched for in future super tau-charmfacilities with center-of-mass energies above 5 GeV. ¯ D ( ∗ ) Σ ( ∗ ) c : P c states The ¯ D ( ∗ ) Σ ( ∗ ) c systems with isospin-1 / P c (4450) and P c (4380), were observed by LHCb [197]. In the updatedmeasurement [28], the P c (4450) signal splits into two nar-rower peaks, P c (4440) and P c (4457). There is no clearevidence for the previous broad P c (4380), and meanwhilea new narrow resonance P c (4312) shows up. Several mod-els have been applied by tremendous works to understandthe structures of these states, and the ¯ D ( ∗ ) Σ ( ∗ ) c molecularexplanation stands out as it can explain the three statessimultaneously, see e.g. Refs. [198–200]. Particularly inRef. [200], the LHCb data are described quite well bythe interaction constructed with heavy quark spin sym-metry and actually four P c states, instead of three, showup, corresponding to ¯ D Σ c , ¯ D Σ ∗ c and ¯ D ∗ Σ c molecules. Ahint of a narrow P c (4380) was reported in the analysisof Ref. [200]. The rest three P c states related to ¯ D ∗ Σ ∗ c predicted there have no signals up to now.In the vector meson saturation model considered here,the two contact terms constructed considering onlyHQSS [198, 200–202], corresponding to the total angu-lar momentum of the light degrees of freedom to be 1 / /
2, are the same, similar to the H ¯ H interaction dis-cussed in Section IV B 1. As a result, 7 ¯ D ( ∗ ) Σ ( ∗ ) c molecu-lar states [195] with similar binding energies are obtained,and the two ¯ D ∗ Σ c states with different total spins, corre-sponding to the P c (4440) and P c (4457), degenerate. Thedegeneracy will be lifted by considering the exchangeof pion and other mesons and keeping the momentum-dependent terms of the light vector exchange. ¯ D ( ∗ ) Ξ ( (cid:48) ) c : P cs and related states It is natural for the isoscalar ¯ D ∗ Ξ c to form boundstates if the above P c states are considered as the isospin-1 / D ( ∗ ) Σ ( ∗ ) c molecules since the interactions from thelight vector exchange are the same in these two cases, seeTable VIII. Note that such states have been predicted byvarious works [203–212].8Recently, Ref. [83] reported an exotic state named P cs (4459) in the invariant mass distribution of J/ψ
Λ inΞ − b → J/ψK − Λ. Even though the significance is only3.1 σ , several works [213–217] have explored the possibil-ity of P cs (4459) being a molecule of ¯ D ∗ Ξ c , and the find-ing here supports such an explanation that the structurecould be caused by two isoscalar ¯ D ∗ Ξ c molecules.Furthermore, Ref. [218] moved forward to the doublestrangeness systems and claimed that ¯ D ∗ s Ξ (cid:48) c and ¯ D ∗ s Ξ ∗ c may form bound states with J P = 3 / − and 5 / − , re-spectively. The φ exchange for such systems yields repul-sive interaction at leading order, see Table VIII, and thebound states obtained in Ref. [218] result from other con-tributions, including the exchange of pseudoscalar andscalar mesons, the subleading momentum dependencefrom the φ exchange and coupled-channel effects. V. SUMMARY AND DISCUSSION
The whole spectrum of hadronic molecules of a pairof charmed and anticharmed hadrons, considering all the S -wave singly-charmed mesons and baryons as well asthe s (cid:96) = 3 / P -wave charmed mesons, is systematicallyobtained using S -wave constant contact potentials satu-rated by the exchange of vector mesons. The coupling ofcharmed heavy hadrons and light mesons are constructedby implementing HQSS, chiral symmetry and SU(3) fla-vor symmetry.The spectrum predicted here should be regarded asthe leading approximation of the spectrum for heavy-antiheavy molecular states, and gives only a general over-all feature of the heavy-antiheavy hadronic molecularspectrum. Specific systems may differ from the pre-dictions here due to the limitations of our treatment.We considered neither the effects of coupled channels,nor the spin-dependent interactions, which arises frommomentum-dependent terms that are of higher order inthe very near-threshold region, nor the contribution fromthe exchange of pseudoscalar and scalar mesons, northe mixing with charmonia. Nevertheless, the spectrumshows a different pattern than that considering only theone-pion exchange (see, e.g., Ref. [196]), which does notallow the molecular states in systems such as D ¯ D andΣ c ¯ D , where the one-pion exchange is forbidden withoutcoupled channels, to exist.In total 229 hidden-charm hadronic molecules (boundor virtual) are predicted, many of which deserve atten-tions:1) The pole positions of the isoscalar D ¯ D ∗ with posi-tive and negative C -parity are consistent with themolecular explanation of X (3872) and ˜ X (3872), re-spectively. There is a shallow bound state in theisoscalar D ¯ D system, consistent with the recent lat-tice QCD result [100].2) The spectrum of the ¯ D ( ∗ ) Σ ( ∗ ) c systems is consis-tent with the molecular explanations of famous P c states: the P c (4312) as an isospin-1 / D Σ c molecule, and P c (4440) and P c (4457) as isospin-1 / D ∗ Σ c molecules. With the resonance sat-uration from the vector mesons, the two ¯ D ∗ Σ c molecules are degenerated. In addition, there isan isospin-1 / D Σ ∗ c molecule, consistent with thenarrow P c (4380) advocated in Ref. [200], and threeisospin-1 / D ∗ Σ ∗ c molecules, consistent with the re-sults in the literature.3) There are two isoscalar ¯ D ∗ Ξ c molecules, which maybe related to the recently announced P cs (4459). Inaddition, more negative-parity isoscalar P cs -typemolecules are predicted: one in ¯ D Ξ c , one in ¯ D Ξ (cid:48) c ,one in ¯ D Ξ ∗ c , two in ¯ D ∗ Ξ (cid:48) c , and three in ¯ D ∗ Ξ ∗ c .4) Instead of associating the X (4140) with a D ∗ s ¯ D ∗ s molecule like some other works did, our results pre-fer the D ∗ s ¯ D ∗ s to form a virtual state. The peak inthe invariant mass distribution of J/ψφ measuredby LHCb just at the D ∗ s ¯ D ∗ s threshold is consistentwith this scenario, according to the discussion inRef. [38].5) The isoscalar D ( ∗ ) ¯ D can form negative-paritybound states with both positive and negative C parities. The D ¯ D bound state is the lowest one inthis family, and the 1 −− one is consistent with thesizeable D ¯ D molecular component in the ψ (4230).6) Λ c ¯Λ c bound states with J P C = 0 − + and 1 −− arepredicted. The vector one should be responsibleto the almost flat line shape of the e + e − → Λ c ¯Λ c cross section in the near-threshold region observedby BESIII [183].7) Light vector meson exchanges either vanish due tothe cancellation between ρ and ω or are not allowedin the isovector D ( ∗ ) ¯ D ( ∗ ) systems and D ( ∗ ) ¯ D ( ∗ ) s sys-tems. However, the vector charmonia exchangesmay play an important role as pointed out inRef. [115], and the Z c (3900 , Z cs (3985)could well be the D ( ∗ ) ¯ D ∗ and D ∗ ¯ D s – D ¯ D ∗ s virtualstates.When the light vector meson exchange is allowed, theresults reported here are generally consistent with the re-sults from a more complete treatment of the one-bosonexchange model (e.g., by solving the Schr¨odinger equa-tion). For example, the binding energy of the isoscalar D ¯ D [98] and D ¯ D [69] bound states from the ρ and ω exchanges fit the spectrum well; a similar pattern ofmolecular states related to the X (3872) was obtained inRef. [99] and the light vector exchange was found neces-sary to bind D ¯ D ∗ together; the ¯ D ∗ Σ c bound states cor-responding to the P c (4440) and P c (4457) were obtainedvia one boson exchange in Ref. [219], and the degener-acy of the two states with J = 1 / / ρ and ω exchange. We should also notice that there9can be systems whose contact terms receive importantcontributions from the scalar-meson exchanges.We expect that there should be structures in the near-threshold region for all the heavy-antiheavy hadron pairsthat have attractive interactions at threshold. The struc-ture can be either exactly at threshold, if the attraction isnot strong enough to form a bound state, or below thresh-old, if a bound state is formed. Moreover, the structuresare not necessarily peaks, and they can be dips in invari-ant mass distributions, depending on the pertinent pro-duction mechanism as discussed in our recent work [38].When the predicted states have ordinary quantumnumbers as those of charmonia, the molecular states mustmix with charmonia, and the mixing can have an impor-tant impact on the spectrum. Yet, in the energy regionhigher than 4.8 GeV, where plenty of states are predictedas shown in Figs. 1 and 2, normal charmonia should bevery broad due to the huge phase space while the molec-ular states should be relatively narrow due to the largedistance between the consistent hadrons. Thus, narrowstructures to be discovered in this energy region shouldbe mainly due to the molecular structures, being eitherbound or virtual states.Among the 229 structures predicted here, only a mi-nority is in the energy region that has been studied in de-tail. The largest data sets from the current experimentshave the following energy restrictions: direct productionof the 1 −− sector in e + e − collisions goes up to 4.6 GeV atBESIII; the hidden-charm XY Z states produced throughthe weak process b → c ¯ cs in B → K decays should be be-low 4.8 GeV; the hidden-charm P c pentaquarks producedin Λ b → K decays should be below 5.1 GeV. To findmore states in the predicted spectrum, we need to haveboth data in these processes with higher statistics anddata at other experiments such as the prompt produc-tion at hadron colliders, PANDA, electron-ion collisionsand e + e − collisions above 5 GeV at super tau-charm fa-cilities.The potentials in the bottom sector are the same asthose in the charm sector, if using the nonrelativisticfield normalization, due to the HQFS, and we expectthe same number of molecular states in the analogoussystems therein. Because of the much heavier reducedmasses of hidden-bottom systems, the virtual states inthe charm sector will move closer to the thresholds oreven become bound states in the bottom sector, and thebound states in the charm sector will be more deeplybound in the bottom sector. There may even be excitedstates for some deeply bound systems. For these deeplybound systems, the constant contact term approxima-tion considered here will not be sufficient. However, dueto the large masses, such states are more difficult to beproduced than those in the charm sector. ACKNOWLEDGMENTS
We would like to thank Chang-Zheng Yuan for use-ful discussions, and thank Fu-Lai Wang for a commu-nication regarding Ref. [218]. This work is supportedin part by the Chinese Academy of Sciences (CAS) un-der Grant No. XDB34030000 and No. QYZDB-SSW-SYS013, by the National Natural Science Foundation ofChina (NSFC) under Grant No. 11835015, No. 12047503and No. 11961141012, by the NSFC and the DeutscheForschungsgemeinschaft (DFG, German Research Foun-dation) through the funds provided to the Sino-GermanCollaborative Research Center TRR110 “Symmetriesand the Emergence of Structure in QCD” (NSFC GrantNo. 12070131001, DFG Project-ID 196253076), andby the CAS Center for Excellence in Particle Physics(CCEPP).
Appendix A: Vertex factors for direct processes
The vertex factors in Eq. (46) for different particles arecalculated in the following: • D g = −√ βg V m D ( P ( Q ) a V ab P ( Q ) Tb ) , (A1) g = √ βg V m D ( P ( ¯ Q ) a V ab P ( ¯ Q ) Tb ) . (A2) • D ∗ g = √ βg V m D ∗ (cid:15) · (cid:15) ∗ ( P ∗ ( Q ) a V ab P ∗ ( Q ) Tb ) ≈ −√ βg V m D ∗ ( P ∗ ( Q ) a V ab P ∗ ( Q ) Tb ) , (A3) g = −√ βg V m D ∗ (cid:15) · (cid:15) ∗ ( P ∗ ( ¯ Q ) a V ab P ∗ ( ¯ Q ) Tb ) ≈ √ βg V m D ∗ ( P ∗ ( ¯ Q ) a V ab P ∗ ( ¯ Q ) Tb ) . (A4) • D g = −√ β g V m D (cid:15) · (cid:15) ∗ ( P ( Q )1 a V ab P ( Q ) T b ) , ≈ √ β g V m D ( P ( Q )1 a V ab P ( Q ) T b ) , (A5) g = √ β g V m D (cid:15) · (cid:15) ∗ ( P ( ¯ Q )1 a V ab P ( ¯ Q ) T b ) , ≈ −√ β g V m D ( P ( ¯ Q )1 a V ab P ( ¯ Q ) T b ) . (A6) • D ∗ g = √ β g V m D ∗ (cid:15) µν (cid:15) ∗ µν ( P ∗ ( Q )2 a V ab P ∗ ( Q ) T b ) ≈ √ β g V m D ∗ ( P ∗ ( Q )2 a V ab P ∗ ( Q ) T b ) , (A7) g = −√ βg V m D ∗ (cid:15) µν (cid:15) ∗ µν ( P ∗ ( ¯ Q )2 a V ab P ∗ ( ¯ Q ) T b ) ≈ √ βg V m D ∗ ( P ∗ ( ¯ Q )2 a V ab P ∗ ( ¯ Q ) T b ) . (A8)0 • B ¯3 g = 1 √ β B g V ¯ u ( k ) u ( p )tr (cid:104) ( B ( Q ) T ¯3 V B ( Q )¯3 (cid:105) ≈ √ β B g V m B ¯3 tr (cid:104) ( B ( Q ) T ¯3 V B ( Q )¯3 (cid:105) , (A9) g = − √ β B g V ¯ u ( k ) u ( p )tr (cid:104) B ( ¯ Q ) T V T B ( ¯ Q )3 (cid:105) ≈ −√ β B g V m B ¯3 tr (cid:104) B ( ¯ Q ) T V T B ( ¯ Q )3 (cid:105) . (A10) • B g = − β S g V √ u ( k ) γ ( γ µ + v µ ) γ u ( p ) × tr (cid:104) B ( Q ) T V B ( Q )6 (cid:105) ≈ −√ m B β S g V tr (cid:104) B ( Q ) T V B ( Q )6 (cid:105) , (A11) g = 13 β S g V √ u ( k ) γ ( γ µ + v µ ) γ u ( p ) × tr (cid:104) B ( ¯ Q ) T V T B ( ¯ Q )6 (cid:105) ≈ √ m B β S g V tr (cid:104) B ( ¯ Q ) T V T B ( ¯ Q )6 (cid:105) . (A12) • B ∗ g = β S g V √ u ∗ µ ( k ) u ∗ µ ( p )tr (cid:104) B ∗ ( Q ) T V B ∗ ( Q )6 (cid:105) ≈ −√ m B ∗ β S g V tr (cid:104) B ∗ ( Q ) T V B ∗ ( Q )6 (cid:105) , (A13) g = − β S g V √ u ∗ µ ( k ) u ∗ µ ( p )tr (cid:104) B ∗ ( ¯ Q ) T V T B ∗ ( ¯ Q )6 (cid:105) ≈ √ m B ∗ β S g V tr (cid:104) B ∗ ( ¯ Q ) T V T B ∗ ( ¯ Q )6 (cid:105) . (A14)In the above deductions we have used (cid:15) · (cid:15) ∗ = − (cid:15) µν · (cid:15) ∗ µν = 1, ¯ u ( k ) u ( p ) = 2 m and ¯ u ∗ µ u ∗ µ = − m atthreshold. Note that the factors such as P ( Q ) a V ab P ( Q ) Tb ,tr (cid:104) B ( Q ) T ¯3 V B ( Q )¯3 (cid:105) in the above expressions contain onlythe SU(3) flavor information and the properties of thecorresponding fields have already been extracted. Appendix B: List of the potential factor F The details of interactions between all combinations ofheavy-antiheavy hadron pairs are listed in Tables VIIIand IX.
Appendix C: Amplitude calculation for crossprocesses
In the following we show the deduction of Eq. (49). • D ¯ D and D s ¯ D s V = i (cid:18) i − ζ g V √ (cid:19) (cid:18) i ζ g V √ (cid:19) m D m D × (cid:15) ∗ µ − i ( g µν − q µ q ν /m ) q − m + i(cid:15) (cid:15) ν F ≈ F F c ζ g V m D m D m − ∆ m (C1)with F c = 4 / • D ∗ ¯ D and D ∗ s ¯ D s V = i (cid:18) i iζ g V √ (cid:19) (cid:18) i − iζ g V √ m D ∗ m D (cid:19) (cid:15) αβγδ (cid:15) α β γ δ × (cid:15) β (cid:15) ∗ α v γ − i ( g δδ − q δ q δ /m ) q − m + i(cid:15) (cid:15) α (cid:15) ∗ β v γ F ≈ ζ g V m D ∗ m D m − ∆ m ( (cid:15) × (cid:15) ∗ ) · ( (cid:15) × (cid:15) ∗ ) F = − F ζ g V m D ∗ m D m − ∆ m S · S = F F c ζ g V m D ∗ m D m − ∆ m , (C2)where S i is the spin-1 operator. Explicitly, S · S = − , − J = 0 , F c = 2 / , / − / • D ∗ ¯ D and D ∗ s ¯ D s V = i (cid:16) i √ ζ g V (cid:17) (cid:16) i √ ζ g V (cid:17) m D ∗ m D × (cid:15) µ (cid:15) ∗ µν − i ( g να − q ν q α ) /m q − m + i(cid:15) (cid:15) αβ (cid:15) β F ≈ ζ g V m D ∗ m D m − ∆ m (cid:15) µ (cid:15) ∗ µν (cid:15) β ν (cid:15) β F = F F c ζ g V m D ∗ m D m − ∆ m , (C3)where F c = − / , − J = 1 , [1] M. Gell-Mann, Phys. Lett. , 214 (1964).[2] G. Zweig, in DEVELOPMENTS IN THE QUARKTHEORY OF HADRONS. VOL. 1. 1964 - 1978 , edited by D. Lichtenberg and S. P. Rosen (1964) pp. 22–101.[3] H.-X. Chen, W. Chen, X. Liu, and S.-L. Zhu, Phys.Rept. , 1 (2016), arXiv:1601.02092 [hep-ph]. TABLE VIII. The group theory factor F , defined in Eq. (47), for the interaction of charmed-anticharmed hadron pairs withonly the light vector-meson exchanges. Here both charmed hadrons are the S -wave ground states. I is the isospin and S isthe strangeness. Positive F means attractive. For the systems with F = 0, the sub-leading exchanges of vector-charmonia alsolead to an attractive potential at threshold. System (
I, S ) Thresholds (MeV) Exchanged particles FD ( ∗ ) ¯ D ( ∗ ) (0,0) (3734 , , ρ, ω , (1,0) ρ, ω − , D ( ∗ ) s ¯ D ( ∗ ) ( ,
1) (3836 , , , − D ( ∗ ) s ¯ D ( ∗ ) s (0,0) (3937 , , φ D ( ∗ ) Λ c ( ,
0) (4154 , ω − D ( ∗ ) s Λ c (0 , −
1) (4255 , − D ( ∗ ) Ξ c (1 , −
1) (4337 , ρ, ω − , − (0 , − ρ, ω , − ¯ D ( ∗ ) s Ξ c ( , −
2) (4438 , φ − D ( ∗ ) Σ ( ∗ ) c ( ,
0) (4321 , , , ρ, ω − , − , ρ, ω , − D ( ∗ ) s Σ ( ∗ ) c (1 , −
1) (4422 , , , − D ( ∗ ) Ξ (cid:48) ( ∗ ) c (1 , −
1) (4446 , , , ρ, ω − , − (0 , − ρ, ω , − ¯ D ( ∗ ) s Ξ (cid:48) ( ∗ ) c ( , −
2) (4547 , , , φ − D ( ∗ ) Ω ( ∗ ) c ( , −
2) (4562 , , , − D ( ∗ ) s Ω ( ∗ ) c (0 , −
3) (4664 , , , φ − c ¯Λ c (0 ,
0) (4573) ω c ¯Ξ c ( ,
1) (4756) ω c ¯Ξ c (1 ,
0) (4939) ρ, ω, φ − , , , ρ, ω, φ , , c ¯Σ ( ∗ ) c (1 ,
0) (4740 , ω c ¯Ξ (cid:48) ( ∗ ) c ( ,
1) (4865 , ω c ¯Ω ( ∗ ) c (0 ,
2) (4982 , − c ¯Σ ( ∗ ) c ( , −
1) (4923 , ρ, ω − , , − ρ, ω , c ¯Ξ (cid:48) ( ∗ ) c (1 ,
0) (5048 , ρ, ω, φ − , , , ρ, ω, φ , , c ¯Ω ( ∗ ) c ( ,
1) (5165 , φ ( ∗ ) c ¯Σ ( ∗ ) c (2 ,
0) (4907 , , ρ, ω − , , ρ, ω , , ρ, ω , ( ∗ ) c ¯Ξ (cid:48) ( ∗ ) c ( ,
1) (5032 , , , ρ, ω − , , ρ, ω , ( ∗ ) c ¯Ω ( ∗ ) c (0 ,
2) (5149 , , , − (cid:48) ( ∗ ) c ¯Ξ (cid:48) ( ∗ ) c (1 ,
0) (5158 , , ρ, ω, φ − , , , ρ, ω, φ , , (cid:48) ( ∗ ) c ¯Ω ( ∗ ) c ( ,
1) (5272 , , , φ ( ∗ ) c ¯Ω ( ∗ ) c (0 ,
0) (5390 , , φ TABLE IX. The group theory factor F , defined in Eq. (47), for the interaction of charmed-anticharmed hadron pairs withonly the light vector-meson exchanges. Here one of the charmed hadrons is an s (cid:96) = 3 / System (
I, S ) Thresholds (MeV) Exchanged particles FD ( ∗ ) ¯ D , (0,0) (4289 , , , ρ, ω , (1,0) ρ, ω − , D ( ∗ ) ¯ D s ,s ( , −
1) (4390 , , , − D ( ∗ ) s ¯ D , ( ,
1) (4402 , , , − D ( ∗ ) s ¯ D s ,s (0,0) (4503 , , , φ D , ¯ D , (0,0) (4844 , , ρ, ω , (1,0) ρ, ω − , D s ,s ¯ D , ( ,
1) (4957 , , , − D s ,s ¯ D s ,s (0,0) (5070 , , φ c ¯ D , ( ,
0) (4708 , ω − c ¯ D s ,s (0 , −
1) (4822 , − c ¯ D , (1 , −
1) (4891 , ρ, ω − , − (0 , − ρ, ω , − Ξ c ¯ D s ,s ( , −
2) (5005 , φ − ( ∗ ) c ¯ D , ( ,
0) (4876 , , , ρ, ω − , − , ρ, ω , − ( ∗ ) c ¯ D s ,s (1 , −
1) (4989 , , , − (cid:48) ( ∗ ) c ¯ D , (1 , −
1) (5001 , , , ρ, ω − , − (0 , − ρ, ω , − Ξ (cid:48) ( ∗ ) c ¯ D s ,s ( , −
2) (5114 , , , φ − ( ∗ ) c ¯ D , ( , −
2) (5117 , , , − ( ∗ ) c ¯ D s ,s (0 , −
3) (5230 , , , φ − [4] A. Hosaka, T. Iijima, K. Miyabayashi, Y. Sakai, andS. Yasui, PTEP , 062C01 (2016), arXiv:1603.09229[hep-ph].[5] J.-M. Richard, Few Body Syst. , 1185 (2016),arXiv:1606.08593 [hep-ph].[6] R. F. Lebed, R. E. Mitchell, and E. S. Swanson, Prog.Part. Nucl. Phys. , 143 (2017), arXiv:1610.04528[hep-ph].[7] A. Esposito, A. Pilloni, and A. Polosa, Phys. Rept. , 1 (2017), arXiv:1611.07920 [hep-ph].[8] F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang,Q. Zhao, and B.-S. Zou, Rev. Mod. Phys. , 015004(2018), arXiv:1705.00141 [hep-ph].[9] A. Ali, J. S. Lange, and S. Stone, Prog. Part. Nucl.Phys. , 123 (2017), arXiv:1706.00610 [hep-ph].[10] S. L. Olsen, T. Skwarnicki, and D. Zieminska, Rev.Mod. Phys. , 015003 (2018), arXiv:1708.04012 [hep-ph].[11] W. Altmannshofer et al. (Belle-II), PTEP ,123C01 (2019), [Erratum: PTEP 2020, 029201 (2020)],arXiv:1808.10567 [hep-ex].[12] Y. S. Kalashnikova and A. Nefediev, Phys. Usp. , 568(2019), arXiv:1811.01324 [hep-ph].[13] A. Cerri et al. , CERN Yellow Rep. Monogr. , 867(2019), arXiv:1812.07638 [hep-ph]. [14] Y.-R. Liu, H.-X. Chen, W. Chen, X. Liu, and S.-L. Zhu, Prog. Part. Nucl. Phys. , 237 (2019),arXiv:1903.11976 [hep-ph].[15] N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev,C.-P. Shen, C. E. Thomas, A. Vairo, and C.-Z. Yuan,Phys. Rept. , 1 (2020), arXiv:1907.07583 [hep-ex].[16] F.-K. Guo, X.-H. Liu, and S. Sakai, Prog. Part. Nucl.Phys. , 103757 (2020), arXiv:1912.07030 [hep-ph].[17] G. Yang, J. Ping, and J. Segovia, Symmetry , 1869(2020), arXiv:2009.00238 [hep-ph].[18] P. G. Ortega and D. R. Entem, (2020),arXiv:2012.10105 [hep-ph].[19] S. Choi et al. (Belle), Phys. Rev. Lett. , 262001(2003), arXiv:hep-ex/0309032.[20] M. Ablikim et al. (BESIII), Phys. Rev. Lett. ,252001 (2013), arXiv:1303.5949 [hep-ex].[21] Z. Liu et al. (Belle), Phys. Rev. Lett. , 252002(2013), [Erratum: Phys. Rev. Lett. 111, 019901 (2013)],arXiv:1304.0121 [hep-ex].[22] M. Ablikim et al. (BESIII), Phys. Rev. Lett. ,022001 (2014), arXiv:1310.1163 [hep-ex].[23] M. Ablikim et al. (BESIII), Phys. Rev. Lett. ,132001 (2014), arXiv:1308.2760 [hep-ex].[24] M. Ablikim et al. (BESIII), Phys. Rev. Lett. ,242001 (2013), arXiv:1309.1896 [hep-ex]. [25] A. Bondar et al. (Belle), Phys. Rev. Lett. , 122001(2012), arXiv:1110.2251 [hep-ex].[26] A. Garmash et al. (Belle), Phys. Rev. Lett. , 212001(2016), arXiv:1512.07419 [hep-ex].[27] M. Ablikim et al. (BESIII), (2020), arXiv:2011.07855[hep-ex].[28] R. Aaij et al. (LHCb), Phys. Rev. Lett. , 222001(2019), arXiv:1904.03947 [hep-ex].[29] G. Ecker, J. Gasser, A. Pich, and E. de Rafael, Nucl.Phys. B , 311 (1989).[30] E. Epelbaum, U.-G. Meißner, W. Gl¨ockle, and C. El-ster, Phys. Rev. C , 044001 (2002), arXiv:nucl-th/0106007.[31] F.-Z. Peng, M.-Z. Liu, M. S´anchez S´anchez, andM. Pavon Valderrama, Phys. Rev. D , 114020(2020), arXiv:2004.05658 [hep-ph].[32] X. Liu, Y.-R. Liu, W.-Z. Deng, and S.-L. Zhu, Phys.Rev. D , 034003 (2008), arXiv:0711.0494 [hep-ph].[33] G.-J. Ding, Eur. Phys. J. C , 297 (2009),arXiv:0904.1782 [hep-ph].[34] J.-J. Wu, R. Molina, E. Oset, and B. S. Zou, Phys. Rev.Lett. , 232001 (2010), arXiv:1007.0573 [nucl-th].[35] J.-J. Wu, R. Molina, E. Oset, and B. S. Zou, Phys.Rev. C , 015202 (2011), arXiv:1011.2399 [nucl-th].[36] Z.-C. Yang, Z.-F. Sun, J. He, X. Liu, and S.-L. Zhu,Chin. Phys. C , 6 (2012), arXiv:1105.2901 [hep-ph].[37] M. Pavon Valderrama, Eur. Phys. J. A , 109 (2020),arXiv:1906.06491 [hep-ph].[38] X.-K. Dong, F.-K. Guo, and B.-S. Zou, (2020),arXiv:2011.14517 [hep-ph].[39] N. Isgur and M. B. Wise, Phys. Lett. B , 527 (1990).[40] N. Isgur and M. B. Wise, Phys. Lett. B , 113 (1989).[41] M. B. Wise, Phys. Rev. D , 2188 (1992).[42] R. Casalbuoni, A. Deandrea, N. Di Bartolomeo,R. Gatto, F. Feruglio, and G. Nardulli, Phys. Lett.B , 371 (1992), arXiv:hep-ph/9209248.[43] R. Casalbuoni, A. Deandrea, N. Di Bartolomeo,R. Gatto, F. Feruglio, and G. Nardulli, Phys. Rept. , 145 (1997), arXiv:hep-ph/9605342 [hep-ph].[44] B. Grinstein, E. E. Jenkins, A. V. Manohar, M. J. Sav-age, and M. B. Wise, Nucl. Phys. B , 369 (1992),arXiv:hep-ph/9204207.[45] A. F. Falk, Nucl. Phys. B , 79 (1992).[46] A. F. Falk and M. E. Luke, Phys. Lett. B , 119(1992), arXiv:hep-ph/9206241.[47] P. Zyla et al. (Particle Data Group), PTEP ,083C01 (2020).[48] A. Filin, A. Romanov, V. Baru, C. Hanhart,Y. Kalashnikova, A. Kudryavtsev, U.-G. Meißner, andA. Nefediev, Phys. Rev. Lett. , 019101 (2010),arXiv:1004.4789 [hep-ph].[49] F.-K. Guo and U.-G. Meißner, Phys. Rev. D , 014013(2011), arXiv:1102.3536 [hep-ph].[50] M.-L. Du, F.-K. Guo, and U.-G. Meißner, Phys. Rev.D , 114002 (2019), arXiv:1903.08516 [hep-ph].[51] M.-L. Du, F.-K. Guo, C. Hanhart, B. Kubis, and U.-G.Meißner, (2020), arXiv:2012.04599 [hep-ph].[52] R. Aaij et al. (LHCb), Phys. Rev. D , 072001 (2016),arXiv:1608.01289 [hep-ex].[53] T. Barnes, F. Close, and H. Lipkin, Phys. Rev. D ,054006 (2003), arXiv:hep-ph/0305025.[54] E. van Beveren and G. Rupp, Phys. Rev. Lett. ,012003 (2003), arXiv:hep-ph/0305035.[55] E. Kolomeitsev and M. Lutz, Phys. Lett. B , 39 (2004), arXiv:hep-ph/0307133.[56] Y.-Q. Chen and X.-Q. Li, Phys. Rev. Lett. , 232001(2004), arXiv:hep-ph/0407062.[57] F.-K. Guo, P.-N. Shen, H.-C. Chiang, R.-G. Ping, andB.-S. Zou, Phys. Lett. B , 278 (2006), arXiv:hep-ph/0603072.[58] F.-K. Guo, P.-N. Shen, and H.-C. Chiang, Phys. Lett.B , 133 (2007), arXiv:hep-ph/0610008.[59] F.-K. Guo, EPJ Web Conf. , 02001 (2019).[60] L. Ma, Q. Wang, and U.-G. Meißner, Chin. Phys. C , 014102 (2019), arXiv:1711.06143 [hep-ph].[61] A. Mart´ınez Torres, K. Khemchandani, and L.-S. Geng,Phys. Rev. D , 076017 (2019), arXiv:1809.01059 [hep-ph].[62] T.-W. Wu, M.-Z. Liu, L.-S. Geng, E. Hiyama, andM. Pavon Valderrama, Phys. Rev. D , 034029(2019), arXiv:1906.11995 [hep-ph].[63] T.-W. Wu, M.-Z. Liu, and L.-S. Geng, (2020),arXiv:2012.01134 [hep-ph].[64] M. Bando, T. Kugo, S. Uehara, K. Yamawaki, andT. Yanagida, Phys. Rev. Lett. , 1215 (1985).[65] M. Bando, T. Kugo, and K. Yamawaki, Phys. Rept. , 217 (1988).[66] U.-G. Meißner, Phys. Rept. , 213 (1988).[67] R. Casalbuoni, A. Deandrea, N. Di Bartolomeo,R. Gatto, F. Feruglio, and G. Nardulli, Phys. Lett.B , 139 (1993), arXiv:hep-ph/9211248.[68] C. Isola, M. Ladisa, G. Nardulli, and P. Santorelli,Phys. Rev. D , 114001 (2003), arXiv:hep-ph/0307367.[69] X.-K. Dong, Y.-H. Lin, and B.-S. Zou, Phys. Rev. D , 076003 (2020), arXiv:1910.14455 [hep-ph].[70] T.-M. Yan, H.-Y. Cheng, C.-Y. Cheung, G.-L. Lin,Y. Lin, and H.-L. Yu, Phys. Rev. D , 1148 (1992),[Erratum: Phys. Rev. D 55, 5851 (1997)].[71] W. Rarita and J. Schwinger, Phys. Rev. , 61 (1941).[72] Y.-R. Liu and M. Oka, Phys. Rev. D , 014015 (2012),arXiv:1103.4624 [hep-ph].[73] R. Machleidt, K. Holinde, and C. Elster, Phys. Rept. , 1 (1987).[74] R. Chen, Z.-F. Sun, X. Liu, and S.-L. Zhu, Phys. Rev.D , 011502 (2019), arXiv:1903.11013 [hep-ph].[75] J. Oller and E. Oset, Nucl. Phys. A , 438 (1997),[Erratum: Nucl. Phys. A 652, 407 (1999)], arXiv:hep-ph/9702314.[76] J. F. Donoghue, C. Ramirez, and G. Valencia, Phys.Rev. D , 1947 (1989).[77] Z.-w. Lin and C. Ko, Phys. Rev. C , 034903 (2000),arXiv:nucl-th/9912046.[78] M. Albaladejo, F.-K. Guo, C. Hidalgo-Duque,and J. Nieves, Phys. Lett. B , 337 (2016),arXiv:1512.03638 [hep-ph].[79] A. Pilloni, C. Fern´andez-Ram´ırez, A. Jackura, V. Math-ieu, M. Mikhasenko, J. Nys, and A. Szczepa-niak (JPAC), Phys. Lett. B , 200 (2017),arXiv:1612.06490 [hep-ph].[80] Q.-R. Gong, J.-L. Pang, Y.-F. Wang, and H.-Q. Zheng,Eur. Phys. J. C , 276 (2018), arXiv:1612.08159 [hep-ph].[81] M. Veltman, Diagrammatica: The Path to Feynmanrules (Cambridge University Press, 2012).[82] M. Aghasyan et al. (COMPASS), Phys. Lett. B ,334 (2018), arXiv:1707.01796 [hep-ex].[83] R. Aaij et al. (LHCb), (2020), arXiv:2012.10380 [hep-ex]. [84] N. A. T¨ornqvist, (2003), arXiv:hep-ph/0308277.[85] C.-Y. Wong, Phys. Rev. C , 055202 (2004),arXiv:hep-ph/0311088.[86] E. S. Swanson, Phys. Lett. B , 189 (2004),arXiv:hep-ph/0311229.[87] N. A. T¨ornqvist, Phys. Lett. B , 209 (2004),arXiv:hep-ph/0402237.[88] N. A. T¨ornqvist, Z. Phys. C61 , 525 (1994), arXiv:hep-ph/9310247 [hep-ph].[89] D. Gamermann, E. Oset, D. Strottman, and M. Vi-cente Vacas, Phys. Rev. D , 074016 (2007), arXiv:hep-ph/0612179.[90] D. Gamermann and E. Oset, Eur. Phys. J. A , 119(2007), arXiv:0704.2314 [hep-ph].[91] Z.-G. Wang, (2020), arXiv:2012.11869 [hep-ph].[92] M. T. AlFiky, F. Gabbiani, and A. A. Petrov, Phys.Lett. B , 238 (2006), arXiv:hep-ph/0506141.[93] J. Nieves and M. Valderrama, Phys. Rev. D , 056004(2012), arXiv:1204.2790 [hep-ph].[94] C. Hidalgo-Duque, J. Nieves, and M. Valderrama, Phys.Rev. D , 076006 (2013), arXiv:1210.5431 [hep-ph].[95] F.-K. Guo, C. Hidalgo-Duque, J. Nieves, andM. Pavon Valderrama, Phys. Rev. D , 054007 (2013),arXiv:1303.6608 [hep-ph].[96] C. Hidalgo-Duque, J. Nieves, A. Ozpineci, and V. Za-miralov, Phys. Lett. B , 432 (2013), arXiv:1305.4487[hep-ph].[97] V. Baru, E. Epelbaum, A. Filin, C. Hanhart, U.-G.Meißner, and A. Nefediev, Phys. Lett. B , 20 (2016),arXiv:1605.09649 [hep-ph].[98] Y.-J. Zhang, H.-C. Chiang, P.-N. Shen, and B.-S. Zou,Phys. Rev. D , 014013 (2006), arXiv:hep-ph/0604271.[99] X. Liu, Z.-G. Luo, Y.-R. Liu, and S.-L. Zhu, Eur. Phys.J. C , 411 (2009), arXiv:0808.0073 [hep-ph].[100] S. Prelovsek, S. Collins, D. Mohler, M. Padmanath, andS. Piemonte, (2020), arXiv:2011.02542 [hep-lat].[101] D. Gamermann and E. Oset, Eur. Phys. J. A , 189(2008), arXiv:0712.1758 [hep-ph].[102] L. Dai, G. Toledo, and E. Oset, Eur. Phys. J. C ,510 (2020), arXiv:2004.05204 [hep-ph].[103] E. Wang, H.-S. Li, W.-H. Liang, and E. Oset, (2020),arXiv:2010.15431 [hep-ph].[104] S. Uehara et al. (Belle), Phys. Rev. Lett. , 082003(2006), arXiv:hep-ex/0512035.[105] P. Pakhlov et al. (Belle), Phys. Rev. Lett. , 202001(2008), arXiv:0708.3812 [hep-ex].[106] B. Aubert et al. (BaBar), Phys. Rev. D , 092003(2010), arXiv:1002.0281 [hep-ex].[107] M. Albaladejo, F.-K. Guo, C. Hidalgo-Duque, J. Nieves,and M. Pavon Valderrama, Eur. Phys. J. C , 547(2015), arXiv:1504.00861 [hep-ph].[108] E. Cincioglu, J. Nieves, A. Ozpineci, and A. Yilmazer,Eur. Phys. J. C , 576 (2016), arXiv:1606.03239 [hep-ph].[109] I. Hammer, C. Hanhart, and A. Nefediev, Eur. Phys.J. A , 330 (2016), arXiv:1607.06971 [hep-ph].[110] P. G. Ortega, J. Segovia, D. R. Entem, andF. Fern´andez, Phys. Lett. B , 1 (2018),arXiv:1706.02639 [hep-ph].[111] Y. Kalashnikova, Phys. Rev. D , 034010 (2005),arXiv:hep-ph/0506270.[112] Z.-Y. Zhou and Z. Xiao, Phys. Rev. D , 054031(2017), [Erratum: Phys. Rev. D 96, 099905 (2017)],arXiv:1704.04438 [hep-ph]. [113] E. Cincioglu, A. Ozpineci, and D. Y. Yilmaz, (2020),arXiv:2012.14013 [hep-ph].[114] Q. Wang, C. Hanhart, and Q. Zhao, Phys. Rev. Lett. , 132003 (2013), arXiv:1303.6355 [hep-ph].[115] F. Aceti, M. Bayar, E. Oset, A. Mart´ınez Tor-res, K. Khemchandani, J. M. Dias, F. Navarra,and M. Nielsen, Phys. Rev. D , 016003 (2014),arXiv:1401.8216 [hep-ph].[116] J. He and D.-Y. Chen, Eur. Phys. J. C , 94 (2018),arXiv:1712.05653 [hep-ph].[117] P. G. Ortega, J. Segovia, D. R. Entem, andF. Fern´andez, Eur. Phys. J. C , 78 (2019),arXiv:1808.00914 [hep-ph].[118] M. Albaladejo, P. Fernandez-Soler, and J. Nieves, Eur.Phys. J. C , 573 (2016), arXiv:1606.03008 [hep-ph].[119] S. Prelovsek, C. Lang, L. Leskovec, and D. Mohler,Phys. Rev. D , 014504 (2015), arXiv:1405.7623 [hep-lat].[120] I. Matuschek, V. Baru, F.-K. Guo, and C. Hanhart,(2020), arXiv:2007.05329 [hep-ph].[121] M. Cleven, F.-K. Guo, C. Hanhart, and U.-G. Meissner,Eur. Phys. J. A , 120 (2011), arXiv:1107.0254 [hep-ph].[122] C. Hanhart, Y. S. Kalashnikova, P. Matuschek,R. Mizuk, A. Nefediev, and Q. Wang, Phys. Rev. Lett. , 202001 (2015), arXiv:1507.00382 [hep-ph].[123] F.-K. Guo, C. Hanhart, Y. S. Kalashnikova, P. Ma-tuschek, R. Mizuk, A. Nefediev, Q. Wang, andJ. L. Wynen, Phys. Rev. D , 074031 (2016),arXiv:1602.00940 [hep-ph].[124] Q. Wang, V. Baru, A. Filin, C. Hanhart, A. Nefediev,and J.-L. Wynen, Phys. Rev. D , 074023 (2018),arXiv:1805.07453 [hep-ph].[125] V. Baru, E. Epelbaum, A. Filin, C. Hanhart, R. Mizuk,A. Nefediev, and S. Ropertz, (2020), arXiv:2012.05034[hep-ph].[126] G.-J. Wang, X.-H. Liu, L. Ma, X. Liu, X.-L. Chen, W.-Z. Deng, and S.-L. Zhu, Eur. Phys. J. C , 567 (2019),arXiv:1811.10339 [hep-ph].[127] L.-Y. Xiao, G.-J. Wang, and S.-L. Zhu, Phys. Rev. D , 054001 (2020), arXiv:1912.12781 [hep-ph].[128] R. Aaij et al. (LHCb), Phys. Rev. Lett. , 242001(2020), arXiv:2009.00025 [hep-ex].[129] R. Aaij et al. (LHCb), Phys. Rev. D , 112003 (2020),arXiv:2009.00026 [hep-ex].[130] L. Meng, B. Wang, and S.-L. Zhu, (2020),arXiv:2012.09813 [hep-ph].[131] R. Aaij et al. (LHCb), Phys. Rev. Lett. , 022003(2017), arXiv:1606.07895 [hep-ex].[132] T. Aaltonen et al. (CDF), Phys. Rev. Lett. , 242002(2009), arXiv:0903.2229 [hep-ex].[133] X. Liu and S.-L. Zhu, Phys. Rev. D , 017502(2009), [Erratum: Phys. Rev. D 85, 019902 (2012)],arXiv:0903.2529 [hep-ph].[134] T. Branz, T. Gutsche, and V. E. Lyubovitskij, Phys.Rev. D , 054019 (2009), arXiv:0903.5424 [hep-ph].[135] R. M. Albuquerque, M. E. Bracco, and M. Nielsen,Phys. Lett. B , 186 (2009), arXiv:0903.5540 [hep-ph].[136] J.-R. Zhang and M.-Q. Huang, J. Phys. G , 025005(2010), arXiv:0905.4178 [hep-ph].[137] X. Chen, X. L¨u, R. Shi, and X. Guo, (2015),arXiv:1512.06483 [hep-ph].[138] M. Karliner and J. L. Rosner, Nucl. Phys. A , 365 (2016), arXiv:1601.00565 [hep-ph].[139] R. Aaij et al. (LHCb), Phys. Rev. D , 012002 (2017),arXiv:1606.07898 [hep-ex].[140] E. Wang, J.-J. Xie, L.-S. Geng, and E. Oset, Phys. Rev.D , 014017 (2018), arXiv:1710.02061 [hep-ph].[141] J.-Z. Wang, D.-Y. Chen, X. Liu, and T. Matsuki,(2020), arXiv:2011.08501 [hep-ph].[142] B.-D. Wan and C.-F. Qiao, (2020), arXiv:2011.08747[hep-ph].[143] J.-Z. Wang, Q.-S. Zhou, X. Liu, and T. Matsuki,(2020), arXiv:2011.08628 [hep-ph].[144] L. Meng, B. Wang, and S.-L. Zhu, Phys. Rev. D ,111502 (2020), arXiv:2011.08656 [hep-ph].[145] Z. Yang, X. Cao, F.-K. Guo, J. Nieves, andM. Pavon Valderrama, (2020), arXiv:2011.08725 [hep-ph].[146] R. Chen and Q. Huang, (2020), arXiv:2011.09156 [hep-ph].[147] M.-C. Du, Q. Wang, and Q. Zhao, (2020),arXiv:2011.09225 [hep-ph].[148] X. Cao, J.-P. Dai, and Z. Yang, (2020),arXiv:2011.09244 [hep-ph].[149] Z.-F. Sun and C.-W. Xiao, (2020), arXiv:2011.09404[hep-ph].[150] Q.-N. Wang, W. Chen, and H.-X. Chen, (2020),arXiv:2011.10495 [hep-ph].[151] B. Wang, L. Meng, and S.-L. Zhu, (2020),arXiv:2011.10922 [hep-ph].[152] Z.-G. Wang, (2020), arXiv:2011.10959 [hep-ph].[153] K. Azizi and N. Er, (2020), arXiv:2011.11488 [hep-ph].[154] X. Jin, X. Liu, Y. Xue, H. Huang, and J. Ping, (2020),arXiv:2011.12230 [hep-ph].[155] Y. Simonov, (2020), arXiv:2011.12326 [hep-ph].[156] J. S¨ung¨u, A. T¨urkan, H. Sundu, and E. V. Veliev,(2020), arXiv:2011.13013 [hep-ph].[157] N. Ikeno, R. Molina, and E. Oset, (2020),arXiv:2011.13425 [hep-ph].[158] Y.-J. Xu, Y.-L. Liu, C.-Y. Cui, and M.-Q. Huang,(2020), arXiv:2011.14313 [hep-ph].[159] B. Aubert et al. (BaBar), Phys. Rev. Lett. , 142001(2005), arXiv:hep-ex/0506081.[160] Q. He et al. (CLEO), Phys. Rev. D , 091104 (2006),arXiv:hep-ex/0611021.[161] C. Yuan et al. (Belle), Phys. Rev. Lett. , 182004(2007), arXiv:0707.2541 [hep-ex].[162] M. Ablikim et al. (BESIII), Phys. Rev. D93 , 011102(2016), arXiv:1511.08564 [hep-ex].[163] M. Ablikim et al. (BESIII), Phys. Rev. Lett. ,092002 (2017), arXiv:1610.07044 [hep-ex].[164] M. Ablikim et al. (BESIII), Phys. Rev. Lett. ,092001 (2017), arXiv:1611.01317 [hep-ex].[165] M. Ablikim et al. (BESIII), Phys. Rev. Lett. ,102002 (2019), arXiv:1808.02847 [hep-ex].[166] X. Gao, C. Shen, and C. Yuan, Phys. Rev. D , 092007(2017), arXiv:1703.10351 [hep-ex].[167] W. Qin, S.-R. Xue, and Q. Zhao, Phys. Rev. D ,054035 (2016), arXiv:1605.02407 [hep-ph].[168] Y.-H. Chen, L.-Y. Dai, F.-K. Guo, and B. Kubis, Phys.Rev. D , 074016 (2019), arXiv:1902.10957 [hep-ph].[169] Y. Lu, M. N. Anwar, and B.-S. Zou, Phys. Rev. D ,114022 (2017), arXiv:1705.00449 [hep-ph].[170] C.-Z. Yuan, “The Y states and other vectors in e + e − annihilations,” https://indico.ihep.ac.cn/event/11793/session/7/contribution/5/material/slides/ 0.pdf , talk given at the 5th Hadron Physics OnlineForum (HAPOF), July 22, 2020.[171] Q. Wang, M. Cleven, F.-K. Guo, C. Hanhart, U.-G.Meißner, X.-G. Wu, and Q. Zhao, Phys. Rev. D ,034001 (2014), arXiv:1309.4303 [hep-ph].[172] L. Ma, X.-H. Liu, X. Liu, and S.-L. Zhu, Phys. Rev. D , 034032 (2015), arXiv:1406.6879 [hep-ph].[173] M. Cleven, F.-K. Guo, C. Hanhart, Q. Wang,and Q. Zhao, Phys. Rev. D , 014005 (2015),arXiv:1505.01771 [hep-ph].[174] C. Hanhart and E. Klempt, Int. J. Mod. Phys. A ,2050019 (2020), arXiv:1906.11971 [hep-ph].[175] M. Cleven, Q. Wang, F.-K. Guo, C. Hanhart, U.-G.Meißner, and Q. Zhao, Phys. Rev. D , 074039 (2014),arXiv:1310.2190 [hep-ph].[176] K. Olschewsky, “Heavy hadronic molecules with nega-tive parity: The vector states,” (2018), master thesis,Bonn University.[177] J. Lees et al. (BaBar), Phys. Rev. D , 111103 (2014),arXiv:1211.6271 [hep-ex].[178] X. Wang et al. (Belle), Phys. Rev. D , 112007 (2015),arXiv:1410.7641 [hep-ex].[179] F.-K. Guo, C. Hanhart, and U.-G. Meißner, Phys. Lett.B , 26 (2008), arXiv:0803.1392 [hep-ph].[180] G. Cotugno, R. Faccini, A. Polosa, and C. Sabelli, Phys.Rev. Lett. , 132005 (2010), arXiv:0911.2178 [hep-ph].[181] F.-K. Guo, J. Haidenbauer, C. Hanhart, andU.-G. Meißner, Phys. Rev. D , 094008 (2010),arXiv:1005.2055 [hep-ph].[182] G. Pakhlova et al. (Belle), Phys. Rev. Lett. , 172001(2008), arXiv:0807.4458 [hep-ex].[183] M. Ablikim et al. (BESIII), Phys. Rev. Lett. ,132001 (2018), arXiv:1710.00150 [hep-ex].[184] S. Jia et al. (Belle), Phys. Rev. D , 111103 (2019),arXiv:1911.00671 [hep-ex].[185] S. Jia et al. (Belle), Phys. Rev. D , 091101 (2020),arXiv:2004.02404 [hep-ex].[186] J. He, Y. Liu, J.-T. Zhu, and D.-Y. Chen, Eur. Phys.J. C , 246 (2020), arXiv:1912.08420 [hep-ph].[187] M. Ablikim et al. (BESIII), Chin. Phys. C , 040001(2020), arXiv:1912.05983 [hep-ex].[188] A. Sommerfeld, Annalen der Physik , 257 (1931).[189] L.-Y. Dai, J. Haidenbauer, and U.-G. Meißner, Phys.Rev. D , 116001 (2017), arXiv:1710.03142 [hep-ph].[190] A. Y. Barniakov (Super Charm-Tau Factory), PoS Lep-tonPhoton2019 , 062 (2019).[191] H.-P. Peng, Y.-H. Zheng, and X.-R. Zhou, Physics ,513 (2020).[192] J.-J. Wu, L. Zhao, and B. Zou, Phys. Lett. B , 70(2012), arXiv:1011.5743 [hep-ph].[193] W. Wang, F. Huang, Z. Zhang, and B. Zou, Phys. Rev.C , 015203 (2011), arXiv:1101.0453 [nucl-th].[194] J.-J. Wu, T.-S. Lee, and B. Zou, Phys. Rev. C ,044002 (2012), arXiv:1202.1036 [nucl-th].[195] C. Xiao, J. Nieves, and E. Oset, Phys. Rev. D ,056012 (2013), arXiv:1304.5368 [hep-ph].[196] M. Karliner and J. L. Rosner, Phys. Rev. Lett. ,122001 (2015), arXiv:1506.06386 [hep-ph].[197] R. Aaij et al. (LHCb), Phys. Rev. Lett. , 072001(2015), arXiv:1507.03414 [hep-ex].[198] M.-Z. Liu, Y.-W. Pan, F.-Z. Peng, M. S´anchez S´anchez,L.-S. Geng, A. Hosaka, and M. Pavon Valderrama,Phys. Rev. Lett. , 242001 (2019), arXiv:1903.11560 [hep-ph].[199] C. Xiao, J. Nieves, and E. Oset, Phys. Rev. D ,014021 (2019), arXiv:1904.01296 [hep-ph].[200] M.-L. Du, V. Baru, F.-K. Guo, C. Hanhart, U.-G.Meißner, J. A. Oller, and Q. Wang, Phys. Rev. Lett. , 072001 (2020), arXiv:1910.11846 [hep-ph].[201] M.-Z. Liu, F.-Z. Peng, M. S´anchez S´anchez, andM. Pavon Valderrama, Phys. Rev. D , 114030 (2018),arXiv:1811.03992 [hep-ph].[202] S. Sakai, H.-J. Jing, and F.-K. Guo, Phys. Rev. D ,074007 (2019), arXiv:1907.03414 [hep-ph].[203] J. Hofmann and M. Lutz, Nucl. Phys. A , 90 (2005),arXiv:hep-ph/0507071.[204] R. Chen, J. He, and X. Liu, Chin. Phys. C , 103105(2017), arXiv:1609.03235 [hep-ph].[205] V. Anisovich, M. Matveev, J. Nyiri, A. Sarantsev, andA. Semenova, Int. J. Mod. Phys. A , 1550190 (2015),arXiv:1509.04898 [hep-ph].[206] Z.-G. Wang, Eur. Phys. J. C , 142 (2016),arXiv:1509.06436 [hep-ph].[207] A. Feijoo, V. Magas, A. Ramos, and E. Oset, Eur. Phys.J. C , 446 (2016), arXiv:1512.08152 [hep-ph].[208] J.-X. Lu, E. Wang, J.-J. Xie, L.-S. Geng, and E. Oset,Phys. Rev. D , 094009 (2016), arXiv:1601.00075 [hep- ph].[209] C. Xiao, J. Nieves, and E. Oset, Phys. Lett. B ,135051 (2019), arXiv:1906.09010 [hep-ph].[210] H.-X. Chen, L.-S. Geng, W.-H. Liang, E. Oset, E. Wang,and J.-J. Xie, Phys. Rev. C , 065203 (2016),arXiv:1510.01803 [hep-ph].[211] B. Wang, L. Meng, and S.-L. Zhu, Phys. Rev. D101