$\bar{K}\Lambda$ molecular explanation to the newly observed $\Xi(1620)^0$

The newly observed $\Xi(1620)^0$ by the Belle Collaboration inspires our interest in performing a systematic study on the interaction of an anti-strange meson $(\bar{K}^{(*)})$ with a strange or doubly strange ground octet baryon $\mathcal{B}$ ($\Lambda$, $\Sigma$, and $\Xi$), where the spin-orbit force and the recoil correction are considered in the adopted one-boson-exchange model. Our results indicate that $\Xi(1620)^0$ can be explained as a $\bar{K}\Lambda$ molecular state with $I(J^P)=1/2(1/2^-)$ and the intermediate force from $\sigma$ exchange plays an important role. Additionally, we also predict several other possible molecular candidates, i.e., the $\bar{K}\Sigma$ molecular state with $I(J^P)=1/2(1/2^-)$ and the triply strange $\bar{K}\Xi$ molecular state with $I(J^P)=0(1/2^-)$.

The newly observed Ξ(1620) 0 by the Belle Collaboration inspires our interest in performing a systematic study on the interaction of an anti-strange meson (K ( * ) ) with a strange or doubly strange ground octet baryon B (Λ, Σ, and Ξ), where the spin-orbit force and the recoil correction are considered in the adopted one-bosonexchange model. Our results indicate that Ξ(1620) 0 can be explained as aKΛ molecular state with I(J P ) = 1/2(1/2 − ) and the intermediate force from σ exchange plays an important role. Additionally, we also predict several other possible molecular candidates, i.e., theKΣ molecular state with I(J P ) = 1/2(1/2 − ) and the triply strangeKΞ molecular state with I(J P ) = 0(1/2 − ).
Since M. Gell-Mann [6] and G. Zweig [7] firstly proposed the existence of the exotic states in their pioneer work of quark model, great theoretical and experimental efforts were made on searching for exotic hadronic matter. The studies of exotic hadronic matter can deepen our understanding of the nonperturbative behavior of QCD. As an important configuration of exotic hadronic matter, hadronic molecules have received extensive attentions in the past decade [8][9][10]. In particular, the updated analysis from the LHCb Collaboration indicated the observation of three near threshold hidden-charm pentaquarks, P c (4312), P c (4440), and P c (4457) [11], which provides a strong evidence for the existence of hidden-charm meson-baryon configuration molecular states [12][13][14][15][16][17][18][19].
In fact, there is a long-term discussion on the meson-baryon molecules in the light flavor sector. Λ(1405) is a typical example which was assigned as aKN molecular candidate with I(J P ) = 0(1/2 − ), since its mass is close to theKN threshold but far away from the prediction in quark model (see the review articles [20,21] for details).
Due to these similarities between Ξ(1620) and Λ(1405), it is interesting to study whether the Ξ(1620) can be the doubly strange molecular partner of the Λ(1405). In Refs. [22,23], the Ξ(1620) was interpreted as a J P = 1/2 − resonance dynamically generated in chiral unitary approach. By introducing the vector exchange interaction, the Bethe-Salpeter equation approach [24] was applied to identify the Ξ(1620) as aKΛ or ā KΣ molecular state.
In this work, we will discuss the Ξ(1620) as a moelcular state in the framework of One-Boson-Exchange (OBE) model. In general, the spin-orbit force and the recoil correction are very important for hadron-hadron interactions in the light fla-vor sector, which will be also included in the following calculations. As a byproduct, theK ( * ) B (B = Λ/Σ/Ξ) systems will also be investigated. Here, we predict the existence of other possible molecular candidates with strange numbers S = −2 and −3.
This paper is organized as follows. In Sec. II, we present the deduction of the OBE effective potentials. In Sec. III, the corresponding numerical results and discussion forK ( * ) B systems are given. This paper will end with a summary in Sec. IV.

II. INTERACTIONS
In the local hidden gauge approach [25,26], the effective Lagrangians depicting the interaction of vector mesons, vector meson with pseudoscalar mesons can be constructed as In the above Lagrangians, g = 12 2 √ 2 and G = 3g 2 4π 2 f π with f π = 93 MeV were given in Ref. [25]. In Ref. [26], the lowest order baryon meson Lagrangians are expressed as where D = 0.75 and F = 0.51 [26]. Here, matrixes for vector mesons, pseudoscalar mesons, and light baryons in SU(3) octet are respectively written as In the OBE model, the intermediate-range interaction for theK ( * ) B systems is provided by σ exchange process, which corresponds to the Lagrangians below Here, m V and m P denote the masses of vector and pseudoscalar mesons, respectively. In quark model, the coupling constants in Eqs. (2.6)-(2.7) have the relation of g ′ σ = g σNN = 3 2 g σ . In Ref. [27], g 2 σNN 4π = 5.69 was determined. With the Lagrangians given in Eqs. (2.1)-(2.7), we can derive the scattering amplitude for theK ( * ) B →K ( * ) B process in t−channel. In Fig. 2, we present the corresponding Feynman diagram and the four momentum for the initial and the final states. The OBE effective potential V h 1 h 2 →h 3 h 4 E ( q) can be related to the scattering amplitude for the process h 1 h 2 → h 3 h 4 via the Breit approximation, i.e., is the scattering amplitude. M i and M f denote the masses of initial and final states, respectively. And then, we get the OBE effective potentials for theKB ( * ) systems Here, V σ and V V are σ exchange and vector exchange potentials for theKB →KB processes, respectively. While in thē K * B →K * B processes, σ exchange, vector exchange, and pseudoscalar exchange potentials are respectively denoted by V ′ σ , V ′ V , and V P . q = p ′ − p and k = 1/2( p + p ′ ). Additionally, m σ , m P , and m V denote the masses of exchanged scalar meson (σ), pseudoscalar mesons (π, η) and vector mesons (ρ, ω, φ), respectively. After performing the Fourier transformation, we may extract the effective potentials in the coordinate space, i.e., Here, the form factor F (q 2 , m 2 is introduced in every interactive vertex, which can reflect the finite size effect of the discussed hadrons and compensate the off-shell effects of the exchanged mesons. Λ, m E , and q are the cutoff, mass and four momentum of the exchanged mesons, respectively. According to the experience from deuteron [28,29], the cutoff Λ is taken around 1.0 GeV, which is often regarded as a typical cutoff value for a loosely bound hadronic molecular state. Since the S − D wave mixing effect is considered, the spinorbital wave functions for theK ( * ) B systems with quantum numbers J P can be written as The expansions of the spin-orbital wave functions Here, C J,M S m S ,Lm L and C S ,m s 1 2 λ,1λ ′ are the Clebsch-Gordan coefficients. χ 1 2 λ and Y L,m L denote the spin wave function and the spherical harmonic function, respectively. ǫ λ ′ is the polarization vector of a vector meson in the laboratory frame [30], with the explicit expression where p = p 0 , p is the four-momentum in the laboratory frame and m denotes the mass of vector meson.
The detailed Fourier transformations for different types of effective potentials are expressed as [31] FT Here, the function Y (Λ, m, r) is defined as In the above effective potentials, we also introduce several spin-spin interaction operators D 1 , D 2 , spin-orbital operators E 1 , E 2 , E 3 , and tensor operators F 1 , F 2 . The explicit form of these operators are where S ( r, x, y) is the tensor force operator S ( r, x, y) = 3( r · x)( r · y) − x · y, with r = r/| r|. In Table I, we present the numerical matrices for these operators.
With the above preparation, we obtain the total effective potentials for theK ( * ) B systems

H(I) and G(I) are the isospin factors
The flavor wave functions for theK ( * ) B systems are collected in Table II.

III. NUMERICAL RESULTS
After obtaining the effective potentials and solving the Schrödinger equations, we firstly study whether the newly observed Ξ(1620) can be assigned as aKΛ molecular state with I(J P ) = 1/2(1/2 − ). In addition, other possible doubly strange and triply strangeK ( * ) B molecular candidates will be predicted.
A.KB molecules and the Ξ(1620) For theKΛ system, there does not exist the π/η/ρ exchange process due to the spin-parity conservation. As shown in Fig.  3 (a), we present the OBE potentials for theKΛ system with I(J P ) = 1/2(1/2 − ) which depends on r. We need to emphasize that we ignore the the contribution from the recoil correction. We can see that the dominant σ exchange and ω exchange interactions are both attractive, while the φ exchange is weakly repulsive.
In Fig. 3 (b), the recoil correction is considered which corresponds to the k 2 /m 2 terms. The recoil correction only has obvious contribution in the short distance. Comparing Fig.  3 (b) with Fig. 3 (a), we may see that the recoil correction significantly changes the line shape of φ and ω exchange potentials at r ≤ 0.5 fm. As shown in Fig. 4, when the cutoff is taken as 1.26 GeV, we obtain aKN[0(1/2 − )] molecular state with the binding energy E = −30.9 MeV and the root-mean-square radius r rms = 1.31 fm. This molecular state can correspond to the observed Λ(1405).
We also check the results when only considering S -wave contribution in the potentials. And we find that the above conclusions keep the same, as the D-wave contribution is negligible compared with the S -wave contribution.

IV. SUMMARY
Searching for exotic hadronic matter is an interesting research issue for hadron physics. Especially, with more and more observations of charmonium-like XYZ states and P c states in the past years, the candidates of hidden-charm tetraquark and pentaquark have been provided, which also stimulated extensive discussions of different hadronic configurations [8][9][10][32][33][34][35][36][37][38]. Among them, hadronic molecular state is very popular to apply to explain these novel phenomena. Recently, the LHCb's observation of three P c states again gave strong evidence of hadronic molecular states composed of an anti-charmed meson and a charmed baryon.
Besides the heavy flavor sector, theorists and experimentalists also paid more attentions to the light flavor sector. For example, the Λ(1405) as aKN molecule with I(J P ) = 0(1/2 − ) have been proposed [20,21]. Recently, Belle reported the observation of Ξ(1620) [1] in the Ξ + c → Ξ − π + π + process. If comparing the properties of the Ξ(1620) and the Λ(1405), we may find their similarities, which inspires our interest to exam the possibility of the newly observed Ξ(1620) as theKΛ molecular state.
In this work, we perform a systematical study on theK ( * ) B interactions within the framework of the one-boson-exchange model, where B stands for the strange or doubly strange ground octet baryons. Here, the S − D wave mixing effect, the spin-orbit potential, and the recoil correction are taken into account. By reproducing the mass of Λ(1405) under theKN[0(1/2) − ] molecular picture, the parameter Λ = 1.26 GeV can be fixed, which is directly applied to obtain the corresponding bound state solution for theKΛ molecular state. Our result shows that the newly observed Ξ(1620) as theKΛ molecular state with I(J P ) = 1/2(1/2 − ) can be supported in our theoretical framework.
Experimental search for these predicted states will be an interesting research topic. More theoretical efforts should be paid in the near future. With the running of Belle II at Super KEKB, we have reason to believe that more evidence of light flavor molecular states will be revealed, which will provide more abundant information of exotic hadronic matter. It will be an effective way to deepen our understanding to the nonperturbative behavior of QCD.