Strangeness is the key: from $\bar{K}N$ to $\bar{D}_s D K$

Strangeness is the k ey: from ¯ K N to ¯ D s D K Li-Sheng Geng 1 , 2 , 3 , 4 , Ming-Zh u Liu 5 , 6 , and Jia-Ming Xie 1 , 7 1 Scho ol of Physics, Beihang University, Beijing 102206, China 2 Sino-F r ench Carb on Neutr ality R ese ar ch Center, ´ Ec ole Centr ale de P ´ ekin/Scho ol of Gener al Engine ering, Beihang University, Beijing 100191, China 3 Peng Huanwu Col lab or ative Center for R ese ar ch and Educ ation, Beihang University, Beijing 100191, China 4 Southern Center for Nucle ar-Scienc e The ory (SCNT), Institute of Mo dern Physics, Chinese A c ademy of Scienc es, Huizhou 516000, China 5 F r ontiers Scienc e Center for R ar e Isotop es, L anzhou University, L anzhou 730000, China 6 Scho ol of Nucle ar Scienc e and T e chnolo gy, L anzhou University, L anzhou 730000, China 7 Dep artment of Physics, Gr aduate Scho ol of Scienc e, The University of T okyo, T okyo 113-0033, Jap an E-mail: lisheng.geng@buaa.e du.cn (Receiv ed F ebruary 21, 2019) The k aon, the ligh test hadron con taining a strangeness quark, is v ery peculiar. It is a Nam bu- Goldstone b oson, but significan tly hea vier than the pion. As a result, its interaction with a matter particle, suc h as the nucleon or a hea vy-light meson, suc h as the D meson, is completely determined by c hiral dynamics and muc h stronger than its pion cousin. The strong attractive in teraction has brought us man y surprises and is manifested in the peculiar nature of man y particles, such as the m ysterious Λ(1405) and D ∗ s 0 (2317). These t wo particles can b e understo o d as ¯ K N and D K hadronic molecules, resp ectively . They also imply the existence of three-bo dy hadronic molecules that aw ait future disco v ery . In this talk, I review some recen t developmen ts in our understanding of hadronic interactions inv olving the k aon. KEYW ORDS: Kaon interaction, Λ(1405) , D ∗ s 0 (2317) , three-b o dy hadronic molecules, genuine three-b o dy forces . . . 1. Motiv ation: Exotic Hadrons as Hadronic Molecules One cen tral question in ph ysics addresses the fundamen tal building blo c ks of nature across scales and the rules go v e rning their in teractions. F or instance, nucleons are b ound together to form n uclei by the residual strong interaction–the nuclear force. Nuclei attract electrons via electromagnetic interactions and form atoms. Through theresidual electromagnetic inter- action, atoms attract one another, thereby giving rise to v arious molecular structures. They are the building blo c ks of our visible universe. As of to da y , the smallest strongly interacting matter particles are quarks. According to the standard mo del of particle ph ysics, they inter- act via gluons to form color-singlet hadrons. The constituent quark mo del (CQM), prop osed b y Gell-Mann and Zweig in 1964 [1–3], classifies hadrons into t wo categories: mesons as pairs of quark and antiquark ( q ¯ q ), such as the pion, the k aon, and the D meson, and baryons of three quarks ( q q q ), such as the proton and the neutron, collectively referred to as n ucleons. Beginning in 2003, such a simple but mysteriously successful picture w as challenged by the discov ery of the so-called exotic hadrons , i.e., hadrons that do not fit easily into the CQM q ¯ q or q q q states. They can b e classified into compact tetraquark states, compact p en- 1 X( 3 8 7 2 ) 2003 𝑫 𝒔𝟎 ( 𝟐𝟑𝟏𝟕 ) 2020 2019 𝑿 ( 𝟔𝟗𝟎𝟎 ) 𝑷 𝒄 ( 𝟒𝟑𝟏𝟐 ) 𝑷 𝒄 ( 𝟒𝟒𝟒𝟎 ) 𝑷 𝒄 ( 𝟒𝟒𝟓𝟎 ) 𝒁 𝒄 ( 3900 ) 2013 2007 𝒁 𝒄 (44 3 0 ) 𝑫 𝒔𝟏 ( 𝟐𝟒𝟔𝟎 ) 2005 Y( 4 2 6 0 ) 2009 𝒀 ( 3940 ) X( 3 9 1 5 ) 𝑿 ( 4140 ) 2011 X(4 2 7 4 ) 𝒁 𝒄 ( 4020 ) X( 4 1 6 0 ) 2018 𝛀 ( 2012 ) 2016 X( 5 5 6 8 ) Y ( 4 2 2 0 ) Y ( 4 3 9 0 ) Y( 4 3 6 0 ) Y( 4 6 6 0 ) 𝒁 𝒃 ( 10610 ) 𝒁 𝒃 ( 10650 ) 2015 𝑷 𝒄 ( 𝟒𝟑𝟖𝟎 ) 𝑷 𝒄 ( 𝟒𝟒𝟓𝟕 ) 𝑿 𝟎 ( 𝟐𝟖𝟔𝟔 ) 𝑿 𝟏 ( 𝟐𝟗𝟎𝟒 ) 𝒁 𝒄𝒔 ( 3985 ) 𝚵 ( 1620 ) 𝑷 𝒄𝒔 ( 𝟒𝟒 𝟓𝟗 ) 𝒁 𝒄 ( 4100 ) 2014 𝒁 𝒄 (4 2 0 0 ) 2006 𝚲 𝒄 ( 𝟐 𝟗 𝟒 𝟎 ) 2021 2022 X( 3 9 6 0 ) 𝑻 𝒄 𝒄 ( 3875 ) Y( 4 5 0 0 ) 𝑷 𝒄𝒔 ( 𝟒𝟑 𝟑𝟖 ) 𝑻 𝒄𝒔 ( 𝟐𝟗𝟎𝟎 ) 𝑷 𝒄 ( 𝟒𝟑𝟑𝟕 ) 2023 𝑿 𝟔𝟔𝟎𝟎 𝑿 ( 𝟔𝟗𝟎𝟎 ) 𝑿 𝟕𝟏𝟎𝟎 𝑿 𝟔𝟗𝟎𝟎 𝑿 ( 𝟕𝟐𝟎𝟎 ) E x otic mes ons a nd b a r y ons T etra qua rk P ent a qua rk s 𝑻 𝒄 ത 𝒔 𝟎 ∗ ( 𝟐𝟗𝟎𝟎 ) 2024 Fig. 1. Disco v ery timeline of selected exotic hadrons. The color co de denotes their classifica- tion: “exotic baryons and mesons” denote that they hav e quantum num b ers of CQM hadrons, but ha v e p eculiar prop erties; “tetraquarks” denote mesons that hav e a minim um of four constituen t quarks/an tiquarks; “p entaquarks” denote bary ons that contain at least five constituen t quarks. taquark states, hadronic molecules, glueballs, h ybrids, or ev en kinematic cusps. An in teresting observ ation is that man y exotic hadrons, if not all, lie near tw o-b o dy hadronic thresholds, suggesting a hadronic molecular nature, such as the familiar deuteron, a weakly b ound state of a neutron and a proton with a binding energy of ab out 2.2 MeV. Fig. 1 shows a selection of these hadronic states and their discov ery time. W e will not discuss each of them and refer interested readers to the many excellent reviews [4–15]. 2. Λ(1405) as a ¯ K N Bound State and Its Two-P ole Structure The Λ(1405), disco vered in 1961 [16], can b e considered the first exotic hadronic state b ecause it is significantly ligh ter than its nucleon counterpart, N ∗ (1535), though it contains a hea vier strangeness quark. The Λ(1405) was predicted to b e a ¯ K N b ound state even b efore its exp erimental disco very [17]. Suc h a picture received strong supp ort in the chiral unitary approac hes that combine SU(3) L × SU(3) R c hiral dynamics and elastic unitarity (see Refs. [18, 19] for a more complete list of references). An unexp ected finding of the c hiral unitary approaches is that the Λ(1405) actually corresp onds to tw o dynamically generated p oles [20, 21], b et ween the π Σ and ¯ K N thresholds. Such a t wo-pole picture was recently confirmed in the unified description of meson-baryon scattering at next-to-next-to-leading order (NNLO) [22] and b y lattice QCD sim ulations [23, 24]. In Ref. [25], we show ed how the tw o-p ole structure emerges from the underlying chiral dynamics, namely , the universal W ein b erg-T omoza w a interaction, the large mass difference b et w e en the k aon and the pion, and the SU(3) flav or symmetry breaking of the ¯ K N and π Σ channels. W e also argued that giv en the universalit y of the W einberg-T omozaw a interaction, suc h a tw o-p ole structure is exp ected in other systems as well, suc h as the K 1 (1270) [26, 27] and the Ξ(1890) [28]. In Fig.2, we sho w the tra jectories of the tw o p oles of the Λ(1405). The evolution of the higher p ole is simple. As the pion mass increases, b oth its real and imaginary parts decrease. This indicates that the effectiv e ¯ K N interaction and coupling to π Σ decrease as the pion (k aon) mass increases. Note that as the pion mass increases, the tw o thresholds also increase. 2 On the other hand, the tra jectory of the lo wer p ole is more complicated and highly nontrivial. As the pion mass increases, it first b ecomes a virtual state from a resonan t state for a pion mass of ab out 200 MeV. F or a pion mass of ab out 300 MeV, it b ecomes a b ound state and remains so up to the pion mass of 500 MeV. The evolution of the low er p ole clearly demonstrates the chiral dynamics underlying the tw o-p ole structure of the Λ(1405). Suc h a highly nontrivial quark mass dep endence of the t w o poles can be deemed as k ey to deciphering the nature of the tw o-p ole structure. In a recen t w ork, we similarly studied the tra jectories of the tw o poles of the K 1 (1270) [29]. F ollowing the lattice QCD studies [23, 24], several w orks hav e examined the quark-mass dep endence of the Λ(1405) [30–32]. These studies yield largely consistent conclusions, but a question remains. That is, whether, at NLO, the higher and lo wer p oles exc hange their flav or-structure assignmen ts [30, 32]. In Ref. [33], it w as argued that one can use the hadronic deca ys of charmonia in to ¯ ΛΣ π and ¯ ΛΣ π as fla v or filters to single out the flav or o ctet and singlet p oles. In addition, it will b e interesting to c heck what happ ens at the next-to-next-to-leading order [22]. - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 - 1 0 - 8 - 6 - 4 - 2 0 2 I m z R [ M e V ] R e z R - ( m K + M N ) [ Μ e V ] K N p o l e m π = 1 3 7 m π = 4 9 7 b o u n d - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 - 1 2 0 - 9 0 - 6 0 - 3 0 0 3 0 I m z R [ M e V ] R e z R - ( m π + M Σ ) [ Μ e V ] π Σ p o l e r e s o n a n t b o u n d m π = 1 3 7 m π = 4 9 7 m π = 1 9 7 m π = 2 3 7 m π = 2 5 7 m π = 2 9 7 v i r t u a l Fig. 2. T ra jectories of the tw o p oles of Λ(1405) as functions of the pion mass m π from 137 MeV to 497 MeV. Critical masses are lab eled by solid squares, with p oints equally spaced b etw een them. T aken from Ref. [25]. 3. D ∗ s 0 (2317) as a D K Molecule and the Unique ¯ D s D K Three-Bo dy State D ∗ s 0 (2317) was first discov ered b y the BaBar Collab oration [34] and then confirmed by the CLEO [35] and Belle [36] collab orations. It is lo cated 45 MeV b elow the D K threshold and has a decay width less than 3.8 MeV. The observ ed mass and width are far from the predicted mass of 2460 MeV and the width of h undreds of MeV in the quark model [37]. Th us, D ∗ s 0 (2317) is difficult to b e interpreted as a conv entional c ¯ s state. On the other hand, due to the strongly attractive D K interaction, it can b e easily explained as a D K molecule [37–47]. It is interesting to note that in the unitarized chiral approac h, the D ∗ K interaction is the same as the D K interaction up to hea vy quark spin symmetry breaking effects. As a result, the existence of a D K molecule implies the existence of a D ∗ K molecule. In Ref. [43], fixing the next-to-leading order low-energy constants (LECs) and a subtraction constant by fitting to the lattice QCD scattering lengths [42], and then solving the Bethe-Salp eter equation, one found t w o p oles in the strangeness − 1, whic h coincide with the exp erimentally kno wn D ∗ s 0 (2317) and D s 1 (2460). In suc h a picture, one can easily understand the fact that the 3 Fig. 3. RMS radii of subsystems in the D ¯ D K (left), D ¯ D ∗ K (middle), and D D K [48] bound states with a cutoff R c = 1 . 0 fm. T ak en from Ref. [49]. mass difference b et ween D s 1 (2460) and D ∗ s 0 (2317) is almost the same as the mass difference b et w een D ∗ and D , b ecause in the molecule picture, the mass difference originates from the differen t masses of the constituents, as the in teractions b et ween the constituents are the same due to hea vy quark spin symmetry . If the D ∗ s 0 (2317) is dominantly a D K b ound state, it is natural to ask what happ ens if one adds one D / ¯ D / ¯ D ∗ in to the D K pair. Will the resulting three-b o dy systems bind? If they do, what are the binding energies and strong deca y widths? Where can future exp eriments searc h for them? Many such systems ha v e b een studied previously , such as the D D K [48], D ¯ D K [49, 50], D ¯ D ∗ K [49, 51, 52], D ∗ D K [53], and D D K K [54] systems. See Refs. [13, 55] for reviews. The binding energies of the D D K , D ¯ D K , and D ¯ D ∗ K b ound states are 67 . 1 ∼ 71 . 2, 48 . 9 +1 . 4 − 2 . 4 , and 77 . 3 +3 . 1 − 6 . 6 MeV, resp ectively . Among the three b ound states, the D ¯ D ∗ K state has the largest binding energy , b ecause the D ¯ D ∗ in teraction is attractive enough to generate X (3872) dynamically . The uncertain ties arise from the consideration of the lik ely existence of a short-range repulsion and from the use of several cutoffs (see Refs. [48, 49] for more details). In Fig. 3, w e show the ro ot mean square (RMS) radii of the tw o-b o dy subsystems of the three b ound states. Consistent with the binding energies, the D ¯ D K system is more extended, while the D ¯ D ∗ K state is more compact. One should note that the spatial distributions are more sensitiv e to the details of the tw o-b o dy p otentials, and the results sho wn in Fig. 3 are for illustrativ e purp oses only . The Belle collab oration has searc hed for the D D K state in e + e − collisions, but has obtained only upp er limits on the product of branching fractions [56]. In contrast to the op en-c harm D D K state, the hidden-c harm D ¯ D K state is more lik ely to b e observ ed in e + e − collisions. The p ossibilit y of detecting the D ¯ D K state in inclusive e + e − → c ¯ c collisions has b een studied in Ref. [57]. As indicated in Ref. [13], these three-b o dy molecules can b e pro duced in the exclusive b -flav ored meson decays, i.e., D D K in B c meson decays and D ¯ D K in B meson decays. 3.1 Unique F e atur es of the ¯ D s D K Thr e e-Bo dy System The ¯ D s D K system is unique b ecause of the following reasons. First, there exists a C - parit y-dep enden t interaction. Second, it has exotic quantum num b ers, J P C = 0 −− , implying no mixing with conv entional q ¯ q or qq q states. Third, there are no b ound tw o-b o dy subsystems, indicating a gen uine three-b o dy b ound state, as shown in Fig. 4. The three-b o dy wa v e function for the ¯ D s D K system of go o d C parit y reads Ψ C = 1 √ 2 (Ψ ¯ D s DK + C Ψ ′ D s ¯ D ¯ K ) , (1) where C = ± 1 represents the C -parity eigen v alue, and Ψ (Ψ ′ ) denotes the wa v e function of the 4 Fig. 4. The predicted masses of 0 −− ¯ D s D K three-bo dy molecular state and 0 − + ¯ D s D K three- b o dy molecular state compared with the 0 − + ¯ D s D ∗ s 0 molecular state and their thresholds. T aken from Ref. [58]. ¯ D s D K ( D s ¯ D ¯ K ) system. The wa ve function Ψ C can b e obtained by solving the Schr¨ odinger equation with the Hamiltonian H = T + T ′ + V + V ′ + V C , where T ( T ′ ) and V ( V ′ ) are the kinetic energy term and the hadron-hadron p otentials of Ψ (Ψ ′ ), resp ectively . Since the strong interaction is inv ariant under charge conjugation, the Schr¨ odinger equation of Ψ C can b e simplified as ⟨ Ψ | ( T + V DK + V ¯ D s K + V ¯ D s D + V C − E ) | Ψ ⟩ = 0 , (2) whic h can b e solved using the Gaussian Expansion Metho d (GEM) [59]. Here w e employ the contact-range effective field theory (EFT) to construct the D K , ¯ D s K , and D ¯ D s p oten tials [60, 61]. A Gaussian-shap ed p otential in coordinate space can parameterize the con tact-range p otential in momentum space V ( r ) = C a e ( r/R c )2 π 3 / 2 R 3 c , (3) where R c is a cut-off radius of the order of a typical hadronic size. The parameter C a c har- acterizes the strength of tw o-b o dy p oten tials, which are determined b y fitting the masses of molecular candidates as well as symmetries [40, 42, 43, 62, 63]. W e assume the D ∗ s 0 (2317) as a mixture of a D K − D s η molecular state and a c ¯ s bare state rather than a pure D K molecule. The uncertaint y in the extracted D K p oten tial is estimated by v arying the molecular comp os- iteness of the D ∗ s 0 (2317) from 50% to 100%. Finally , one can obtain the follo wing appro ximate ratio for the contact p oten tials for a cutoff Λ = 1 GeV: C DK a : C ¯ D s K a : C ¯ D s D a = 1 : 0 . 5 : 0 . 1. The p otential V C , dep endent on the C -parity , is a three-b o dy interaction correlating the w av e functions Ψ and Ψ ′ . Here, since the D K p oten tial can form a b ound state D ∗ s 0 , w e use an effectiv e t wo-bo dy ¯ D s D ∗ s 0 p oten tial utilizing the one-b oson-exchange (OBE) mo del [64, 65]. 5 T able I. Binding energy (in units of MeV), weigh ts of Jacobi channels, ro ot mean square radii (in units of fm), and expectation v alues of the Hamiltonian (in units of MeV) of the 0 −− ¯ D s D K molecule. T aken from Ref. [58] Scenarios B.E.(0 −− ) P ¯ D s K − D P DK − ¯ D s P ¯ D s D − K α = 1 22 +23 − 14 11 +1 − 1 % 78 − 1 +2 % 11 +0 − 1 % α = 2 20 +22 − 13 10 +1 − 1 % 80 − 1 +2 % 10 +0 − 1 % Scenarios r ¯ D s K r DK r ¯ D s D ⟨ T ⟩ α = 1 1 . 6 − 0 . 4 +0 . 9 1 . 2 − 0 . 3 +0 . 5 1 . 4 − 0 . 3 +0 . 8 177 +81 − 74 α = 2 1 . 7 − 0 . 4 +1 . 4 1 . 2 − 0 . 3 +0 . 7 1 . 6 − 0 . 5 +1 . 3 169 +82 − 78 Scenarios ⟨ V D s ¯ K ⟩ ⟨ V DK ⟩ ⟨ V D s ¯ D ⟩ ⟨ V C = − D s ¯ D ∗ s 0 ⟩ α = 1 − 40 − 25 +20 − 147 − 73 +62 − 14 − 7 +6 2 +0 − 1 α = 2 − 37 − 25 +22 − 143 − 73 +64 − 13 − 6 +7 3 +2 − 1 The η exchange p otential for the ¯ D s D ∗ s 0 system is V C = ± = ∓ 2 3 k 2 f 2 π q 2 0 ( e − mr − e − Λ r 4 π r − Λ 2 − m 2 8 π Λ e − Λ r ) , (4) where q 0 = m D ∗ s 0 − m D s , k = 0 . 56, and f π = 130 MeV [64]. In T able I, the 0 −− ¯ D s D K system is predicted to b e a b ound state with a binding energy of 21 +24 − 14 MeV. Our results indicate that when the D K − D s η molecular comp onent of the D ∗ s 0 (2317) ranges from 50% to 100%, the ¯ D s D K system alwa ys remains b ound. Such a three-bo dy molecule is denoted by X (4310). The w eight of each Jacobi channel Φ i is calculated by P i = ⟨ Ψ | ˆ P G i | Ψ ⟩ , with | Ψ ⟩ = P i | Φ i ⟩ and the generalized pro jection op erator ˆ P G i = P j | Φ i ⟩ S − 1 ij ⟨ Φ j | suited for a non-orthogonal basis [66], where S − 1 ij is the element of the inv erse of the ov erlap S -matrix of Jacobi channels with S ij = ⟨ Φ i | Φ j ⟩ ( i, j = 1 − 3). In terestingly , the weigh ts of Jacobi channels i = 1 − 3 in Fig. 3 are stable, which are ab out 10%, 80%, and 10% of ¯ D s K − D , D K − ¯ D s , and ¯ D s D − K , resp ectively , v arying only 1 ∼ 2 p ercen t. This indicates the 0 −− three-b o dy b ound state is dominated b y the ( D K ) − ¯ D s c hannel, w eakly dep endent on the molecular comp onent of the D ∗ s 0 (2317). The ro ot mean square radii and exp ectation v alues of the Hamiltonian of the 0 −− ¯ D s D K bound state are also presented in T able I, thereby providing a more in tuitive demonstration of the spatial exten t of the predicted molecular state and the relative con tributions of individual interaction terms in the Hamiltonian. T o facilitate p oten tial observ ation of the predicted state X (4310), w e ha v e in vestigated its deca y properties and pro duction mechanisms through triangle diagrams, fo cusing particularly on B decays [67, 68]. W e assume that the B meson firstly deca ys into a ¯ D ∗ meson and a D ∗ s 0 hadronic molecule, and then the ¯ D ∗ meson scatters into a ¯ D s and a K meson. Finally , the ¯ D s D K molecule is dynamically generated by the subsystem ¯ D s D ∗ s 0 . Our results indicate that the X (4310) dominantly decays into ¯ D ∗ D . Using a similar approach, w e estimated the branc hing fraction of the decay B → K X (4310) is approximately 10 − 6 . In the isospin limit, w e estimate the branc hing fractions of the deca ys B + → [ X (4310) → D ∗± D ∓ ] K + to b e 5 × 10 − 7 . Referring to the branching fractions B ( B + → D ∗± D ∓ K + ) = 6 × 10 − 4 [69, 70], we estimate the ratios of B [ B + → ( X (4310) → D ∗± D ∓ ) K + ] / B ( B + → D ∗± D ∓ K + ) ∼ 10 − 3 . The ev ent num b er of the decays B + → D ∗± D ∓ K + of the LHCb Collab oration corresp onding to an in tegrated luminosit y of 9 fb − 1 is around 2 × 10 3 [71]. One can exp ect that the even t num b er of the decays B + → [ X (4310) → D ∗± D ∓ ] K + w ould reach at least 10 and 10 2 corresp onding to the in tegrated luminosit y of 50 fb − 1 and 350 fb − 1 . Therefore, w e strongly recommend exp erimen tal searches for the 0 −− ¯ D s D K mole cule in the decay c hannels B + → D ∗± D ∓ K + . 6 4. Summary and Outlo ok W e hav e pro vided a concise review on Λ(1405), D ∗ s 0 (2317), and the p ertinen t ¯ K N and D K in teractions. V arious studies show that b oth Λ(1405) and D ∗ s 0 (2317) can b e largely understo o d as hadronic molecules of ¯ K N and D K . They can b e dynamically generated via c hiral dynamics. W e argued that Λ(1405)’s tw o-p ole structure is a univ ersal feature of chiral dynamics, confirmed by the N 2 LO calculations, lattice QCD, and exp erimental data. Based on the molecular picture of the D ∗ s 0 (2317), we predicted the existence of a unique 0 −− ¯ D s D K three-b o dy b ound state X (4310) and studied its pro duction rates in B deca ys. Interestingly , the charge-parit y-dep endent in teraction implies that such systems provide a unique platform for understanding genuine three-b o dy interactions, whic h hav e remained elusive in nuclear ph ysics for decades. In a recent study [72], w e sho wed that three-b o dy forces play a more imp ortan t role in determining whether the ¯ D ∗ D η system is b ound. Lo oking ahead, we hop e that some of the predictions, particularly the existence of three- b o dy hadronic molecules, can inspire more theoretical and exp erimental studies. 5. Ac knowledgmen ts This work is partly supp orted b y the National Key R&D Program of China under Grant No. 2023YF A1606703 and the National Natural Science F oundation of China under Grant No. W2543006 and No. 12435007. M.Z.L ackno wledges supp ort from the National Natural Science F oundation of China under Grant No. 12575086. References [1] Murra y Gell-Mann. A Schematic Mo del of Baryons and Mesons. Phys. L ett. , 8:214–215, 1964. [2] G. Zweig. A n SU(3) mo del for str ong inter action symmetry and its br e aking. V ersion 2 , pages 22–101. 2 1964. [3] G. Zw eig. An SU(3) model for strong in teraction symmetry and its breaking. V ersion 1. CERN- TH-401 , 1 1964. [4] Hua-Xing Chen, W ei Chen, Xiang Liu, and Shi-Lin Zh u. The hidden-c harm p entaquark and tetraquark states. Phys. R ept. , 639:1–121, 2016. [5] Ric hard F. Leb ed, Ryan E. Mitc hell, and Eric S. Swanson. Heavy-Quark QCD Exotica. Pr o g. Part. Nucl. Phys. , 93:143–194, 2017. [6] Eulogio Oset et al. W eak decays of hea vy hadrons into dynamically generated resonances. Int. J. Mo d. Phys. E , 25:1630001, 2016. [7] A. Esp osito, A. Pilloni, and A. D. Polosa. Multiquark Resonances. Phys. R ept. , 668:1–97, 2017. [8] Y ubing Dong, Amand F aessler, and V alery E. Lyub o vitskij. Description of heavy exotic resonances as molecular states using phenomenological Lagrangians. Pr o g. Part. Nucl. Phys. , 94:282–310, 2017. [9] F eng-Kun Guo, Christoph Hanhart, Ulf-G. Meißner, Qian W ang, Qiang Zhao, and Bing-Song Zou. Hadronic molecules. R ev. Mo d. Phys. , 90(1):015004, 2018. [10] Ahmed Ali, Jens S¨ oren Lange, and Sheldon Stone. Exotics: Hea vy Pen taquarks and T etraquarks. Pr o g. Part. Nucl. Phys. , 97:123–198, 2017. [11] Marek Karliner, Jonathan L. Rosner, and T omasz Skwarnic ki. Multiquark States. A nn. R ev. Nucl. Part. Sci. , 68:17–44, 2018. [12] F eng-Kun Guo, Xiao-Hai Liu, and Sh untaro Sak ai. Threshold cusps and triangle singularities in hadronic reactions. Pr o g. Part. Nucl. Phys. , 112:103757, 2020. [13] Ming-Zh u Liu, Y a-W en Pan, Zhi-W ei Liu, Tian-W ei W u, Jun-Xu Lu, and Li-Sheng Geng. Three w a ys to decipher the nature of exotic hadrons: Multiplets, three-b o dy hadronic molecules, and correlation functions. Phys. R ept. , 1108:1–108, 2025. [14] Zhi-Gang W ang. Review of the QCD sum rules for exotic states. F r ont. Phys. (Beijing) , 21(1):016300, 2026. 7 [15] Mic hael D¨ oring, Johann Haidenbauer, Maxim Mai, and T oru Sato. Dynamical coupled-channel mo dels for hadron dynamics. Pr o g. Part. Nucl. Phys. , 146:104213, 2026. [16] Margaret H. Alston et al. Study of Resonances of the Sigma-pi System. Phys. R ev. L ett. , 6:698–702, 1961. [17] R. H. Dalitz and S. F. T uan. A p ossible resonan t state in pion-hyperon scattering. Phys. R ev. L ett. , 2:425–428, 1959. [18] J. A. Oller, E. Oset, and A. Ramos. Chiral unitary approac h to meson meson and meson - bary on in teractions and nuclear applications. Pr o g. Part. Nucl. Phys. , 45:157–242, 2000. [19] J. A. Oller. Coupled-channel approach in hadron–hadron scattering. Pr o g. Part. Nucl. Phys. , 110:103728, 2020. [20] J.A. Oller and Ulf G. Meissner. Chiral dynamics in the presence of b ound states: Kaon nucleon in teractions revisited. Phys. L ett. B , 500:263–272, 2001. [21] D. Jido, J. A. Oller, E. Oset, A. Ramos, and U. G. Meissner. Chiral dynamics of the tw o Lam b da(1405) states. Nucl. Phys. , A725:181–200, 2003. [22] Jun-Xu Lu, Li-Sheng Geng, Michael Do ering, and Maxim Mai. Cross-Channel Constraints on Resonan t Antik aon-Nucleon Scattering. Phys. R ev. L ett. , 130(7):071902, 2023. [23] John Bulav a et al. Two-P ole Nature of the Λ(1405) resonance from Lattice QCD. Phys. R ev. L ett. , 132(5):051901, 2024. [24] John Bulav a et al. Lattice QCD study of π Σ- ¯ K N scattering and the Λ(1405) resonance. Phys. R ev. D , 109(1):014511, 2024. [25] Jia-Ming Xie, Jun-Xu Lu, Li-Sheng Geng, and Bing-Song Zou. Tw o-p ole structures as a univ ersal phenomenon dictated b y coupled-c hannel chiral dynamics. Phys. R ev. D , 108(11):L111502, 2023. [26] L. Ro ca, E. Oset, and J. Singh. Low lying axial-vector mesons as dynamically generated reso- nances. Phys. R ev. D , 72:014002, 2005. [27] L. S. Geng, E. Oset, L. Roca, and J. A. Oller. Clues for the existence of t wo K 1 (1270) resonances. Phys. R ev. D , 75:014017, 2007. [28] R. Molina, W ei-Hong Liang, Chu-W en Xiao, Zhi-F eng Sun, and E. Oset. Tw o states for the Ξ(1820) resonance. Phys. L ett. B , 856:138872, 2024. [29] Jia-Ming Xie, Zhi-W ei Liu, Jun-Xu Lu, Haozhao Liang, Raquel Molina, and Li-Sheng Geng. Chiral Evolution and F emtoscopic Signatures of the K 1 (1270) Resonance. [hep-ph], 2025. [30] Zejian Zh uang, Raquel Molina, Jun-Xu Lu, and Li-Sheng Geng. Pole tra jectories of the Λ(1405) help establish its dynamical nature. Sci. Bul l. , 70:1953–1961, 2025. [31] Xiu-Lei Ren. Ligh t-quark mass dep endence of the Λ(1405) resonance. Phys. L ett. B , 855:138802, 2024. [32] F eng-Kun Guo, Y uki Kamiya, Maxim Mai, and Ulf-G. Meißner. New insights in to the nature of the Λ(1380) and Λ(1405) resonances aw ay from the SU(3) limit. Phys. L ett. B , 846:138264, 2023. [33] Xiao-Hai Liu, Ying-Bo He, Li-Sheng Geng, F eng-Kun Guo, and Ju-Jun Xie. Iden tifying the t w o- p ole structure of Λ(1405). arXiv:2407.13486 [hep-ph], 2024. [34] B. Aub ert et al. Observ ation of a narrow meson decaying to D + s π 0 at a mass of 2.32-GeV/c 2 . Phys. R ev. L ett. , 90:242001, 2003. [35] D. Besson et al. Observ ation of a narrow resonance of mass 2.46-GeV/c**2 deca ying to D*+(s) pi0 and confirmation of the D*(sJ)(2317) state. Phys. R ev. , D68:032002, 2003. [Erratum: Phys. Rev.D75,119908(2007)]. [36] P . Kroko vny et al. Observ ation of the D(sJ)(2317) and D(sJ)(2457) in B deca ys. Phys. R ev. L ett. , 91:262002, 2003. [37] T. Barnes, F. E. Close, and H. J. Lipkin. Implications of a DK molecule at 2.32-GeV. Phys. R ev. D , 68:054006, 2003. [38] E. E. Kolomeitsev and M. F. M. Lutz. On Heavy light meson resonances and chiral symmetry. Phys. L ett. B , 582:39–48, 2004. [39] J. Hofmann and M. F. M. Lutz. Op en c harm meson resonances with negativ e strangeness. Nucl. Phys. A , 733:142–152, 2004. [40] F eng-Kun Guo, P eng-Nian Shen, Huan-Ching Chiang, Rong-Gang Ping, and Bing-Song Zou. Dy- namically generated 0+ heavy mesons in a hea vy chiral unitary approac h. Phys. L ett. , B641:278– 285, 2006. 8 [41] D. Gamermann, E. Oset, D. Strottman, and M. J. Vicente V acas. Dynamically generated op en and hidden charm meson systems. Phys. R ev. , D76:074016, 2007. [42] Liuming Liu, Kostas Orginos, F eng-Kun Guo, Christoph Hanhart, and Ulf-G. Meissner. Interac- tions of charmed mesons with ligh t pseudoscalar mesons from lattice QCD and implications on the nature of the D ∗ s 0 (2317). Phys. R ev. , D87(1):014508, 2013. [43] M. Altenbuc hinger, L. S. Geng, and W. W eise. Scattering lengths of Nambu-Goldstone b osons off D mesons and dynamically generated heavy-ligh t mesons. Phys. R ev. D , 89(1):014026, 2014. [44] Daniel Mohler, C. B. Lang, Luk a Lesk ov ec, Sasa Prelovsek, and R. M. W oloshyn. D ∗ s 0 (2317) Meson and D -Meson-Kaon Scattering from Lattice QCD. Phys. R ev. L ett. , 111(22):222001, 2013. [45] C. B. Lang, Luk a Lesko vec, Daniel Mohler, Sasa Prelo vsek, and R. M. W olosh yn. Ds mesons with DK and D*K scattering near threshold. Phys. R ev. D , 90(3):034510, 2014. [46] Gunnar S. Bali, Sara Collins, Antonio Cox, and Andreas Sch¨ afer. Masses and decay constants of the D ∗ s 0 (2317) and D s 1 (2460) from N f = 2 lattice QCD close to the physical p oint. Phys. R ev. D , 96(7):074501, 2017. [47] Zhi Y ang, Guang-Juan W ang, Jia-Jun W u, Mak oto Ok a, and Shi-Lin Zhu. Nov el Coupled Channel F ramework Connecting the Quark Mo del and Lattice QCD for the Near-threshold Ds States. Phys. R ev. L ett. , 128(11):112001, 2022. [48] Tian-W ei W u, Ming-Zhu Liu, Li-Sheng Geng, Emiko Hiyama, and Manuel P a von V alderrama. D K , DD K , and D D D K molecules–understanding the nature of the D ∗ s 0 (2317). Phys. R ev. D , 100(3):034029, 2019. [49] Tian-W ei W u, Ming-Zhu Liu, and Li-Sheng Geng. Excited K meson, K c (4180) , with hidden c harm as a D ¯ D K bound state. Phys. R ev. D , 103(3):L031501, 2021. [50] Xiang W ei, Qing-Hua Shen, and Ju-Jun Xie. F addeev fixed-cen ter approximation to the D ¯ D K system and the hidden charm K c ¯ c (4180) state. Eur. Phys. J. C , 82(8):718, 2022. [51] Li Ma, Qian W ang, and Ulf-G. Meißner. Double heavy tri-hadron b ound state via delo calized π b ond. Chin. Phys. C , 43(1):014102, 2019. [52] Xiu-Lei Ren, Brenda B. Malabarba, Li-Sheng Geng, K. P . Khemchandani, and A. Mart ´ ınez T orres. K ∗ mesons with hidden charm arising from K X (3872) and K Z c (3900) dynamics. Phys. L ett. B , 785:112–117, 2018. [53] Y ue T an, Xuejie Liu, Xiaoyun Chen, Y ouc hang Y ang, Hongxia Huang, and Jialun Ping. Dynam- ical study of D*DK and D*DD ¯ systems at quark level. Phys. R ev. D , 110(1):016005, 2024. [54] Y a-W en P an, Ming-Zhu Liu, Jun-Xu Lu, and Li-Sheng Geng. Systematic studies of DDKK and DD ¯ KK ¯ four-hadron molecules. Phys. R ev. D , 109(5):054026, 2024. [55] Tian-W ei W u, Y a-W en Pan, Ming-Zh u Liu, and Li-Sheng Geng. Multi-hadron molecules: status and prosp ect. Sci. Bul l. , 67:1735–1738, 2022. [56] Y. Li et al. Search for a doubly-charged D D K b ound state in Υ(1 S, 2 S ) inclusiv e decays and via direct production in e + e − collisions at √ s = 10.520, 10.580, and 10.867 GeV. Phys. R ev. D , 102(11):112001, 2020. [57] Tian-Chen W u and Li-Sheng Geng. Theoretical inv estigation of the molecular nature of Ds0*(2317) and Ds1(2460) and the p ossibility of observing the DD ¯ K b ound state Kcc ¯ (4180) in inclusive e+e- → cc ¯ collisions. Phys. R ev. D , 108(1):014015, 2023. [58] Tian-W ei W u, Ming-Zhu Liu, and Li-Sheng Geng. Implication of the Existence of J P C = 0 −− ¯ D s D K Bound State on the Nature of D ∗ s 0 (2317), and a New Configuration of Exotic State. Phys. R ev. L ett. , 135(3):031902, 2025. [59] E. Hiy ama, Y. Kino, and M. Kamim ura. Gaussian expansion method for few-bo dy systems. Pr o g. Part. Nucl. Phys. , 51:223–307, 2003. [60] C. Hidalgo-Duque, J. Nieves, and M. Pa von V alderrama. Ligh t flav or and heavy quark spin symmetry in heavy meson molecules. Phys. R ev. D , 87(7):076006, 2013. [61] Ming-Zh u Liu, Xi-Zhe Ling, and Li-Sheng Geng. Pro ductions of Ds0*(2317) and Ds1(2460) in B(s) and Λb(Ξb) decays. Phys. R ev. D , 109(5):056014, 2024. [62] F eng-Kun Guo, Christoph Hanhart, and Ulf-G. Meissner. In teractions b etw een heavy mesons and Goldstone b osons from chiral dynamics. Eur. Phys. J. A , 40:171–179, 2009. [63] T eng Ji, Xiang-Kun Dong, Miguel Albaladejo, Meng-Lin Du, F eng-Kun Guo, Juan Nieves, and Bing-Song Zou. Understanding the 0 ++ and 2 ++ c harmonium(-lik e) states near 3.9 GeV. Sci.Bul l. 68:688–697, 2022. 9 [64] Xiang Liu, Zhi-Gang Luo, and Shi-Lin Zhu. Nov el charmonium-lik e structures in the J /ψ ϕ and J /ψ ω in v ariant mass spectra. Phys. L ett. B , 699:341–344, 2011. [Erratum: Phys.Lett.B 707, 577 (2012)]. [65] Lei-Lei Shen, Xiao-Lin Chen, Zhi-Gang Luo, Peng-Zhi Huang, Shi-Lin Zhu, Peng-F ei Y u, and Xiang Liu. The Molecular systems comp osed of the charmed mesons in the H ¯ S + h.c. doublet. Eur. Phys. J. , C70:183–217, 2010. [66] M. Soriano and J. J. Palacios. Theory of pro jections with nonorthogonal basis sets: P artitioning tec hniques and effective hamiltonians. Phys. R ev. B , 90(7):075128, 2014. [67] Hai-Y ang Cheng, Ch un-Khiang Chua, and Amarjit Soni. Final state interactions in hadronic B deca ys. Phys. R ev. D , 71:014030, 2005. [68] Amand F aessler, Thomas Gutsche, V alery E. Lyub o vitskij, and Y ong-Liang Ma. Strong and radiativ e deca ys of the D(s0)*(2317) meson in the DK-molecule picture. Phys. R ev. D , 76:014005, 2007. [69] P . del Amo Sanc hez et al. Measuremen t of the B — > D-bar(*)D(*)K branching fractions. Phys. R ev. D , 83:032004, 2011. [70] Ro el Aaij et al. Measuremen t of branching fraction ratios for B + → D ∗ + D − K + , B + → D ∗− D + K + , and B 0 → D ∗− D 0 K + deca ys. JHEP , 12:139, 2020. [71] Ro el Aaij et al. Observ ation of New Charmonium or Charmoniumlik e States in B+ → D* ± D ∓ K+ Deca ys. Phys. R ev. L ett. , 133(13):131902, 2024. [72] Y a-W en P an, Ming-Zh u Liu, and Li-Sheng Geng. Probing the three-b o dy force in hadronic systems with sp ecific c harge parity. arXiv:2512.01468 [nucl-th], 2025. 10