Description of the newly observed $\Xi_c^{0}$ states as molecular states

Very recently, three new structures $\Xi_c(2923)^0$, $\Xi(2938)^0$, and $\Xi(2964)^0$ at the invariant mass spectrum of $\Lambda_c^{+}K^{-}$ observed by the LHCb Collaboration triggers a hot discussion about their inner structure. In this work, we study the strong decay mode of the newly observed $\Xi_c$ assuming that the $\Xi_c$ is a $\bar{D}\Lambda-\bar{D}\Sigma$ molecular state. With the possible quantum numbers $J^p=1/2^{\pm}$ and $3/2^{\pm}$, the partial decay widths of the $\bar{D}\Lambda-\bar{D}\Sigma$ molecular state into the $\Lambda_c^{+}K^{-}$, $\Sigma_c^{+}K^{-}$,and $\Xi_c^{+}\pi^{-}$ final states through hadronic loop are calculated with the help of the effective Lagrangians. By comparison with the LHCb observation, the current results of total decay width support the $\Xi(2923)^0$ as $\bar{D}\Lambda-\bar{D}\Sigma$ molecule while the decay width of the $\Xi_c(2938)^0$ and $\Xi(2964)^0$ can not be well reproduced in the molecular state picture. In addition, the calculated partial decay widths with $S$ wave $\bar{D}\Lambda-\bar{D}\Sigma$ molecular state picture indicate that allowed decay modes, $\Xi_c^{+}\pi^{-}$, may have the biggest branching ratios for the $\Xi_c(2923)$. The experimental measurements for this strong decay process could be a crucial test for the molecule interpretation of the $\Xi_c(2923)$.

Although the authors in Refs. [3,4,5,6] try to assign these states into the conventional three-quark frames, it is obvious that the inner structure of Ξ c (2923) 0 , Ξ(2938) 0 , and Ξ(2964) 0 are not finally determined. And another interpretation is treating them as DΛ − DΣ molecular states, because the smallest mass gaps between the newly observed Ξ c baryons and the ground Ξ c , about 450 MeV, is large enough to excite a light quark-antiquark pair to form a hadronic molecular. Indeed, it is shown in Refs. [7,8,9] that the interaction between D meson and Λ or Σ c states as molecular states baryon is strong enough to form a bound state with a mass about 2930 MeV.
The key point in this work is to explain whether the Ξ c (2923) 0 , Ξ(2938) 0 , and Ξ(2964) 0 can be considered as a molecular state. Here, we will consider the strong Ξ * c → Λ + c K − ,Σ + c K − , and Ξ + c π − decays of the Ξ c (2923) 0 , Ξ(2938) 0 , and Ξ(2964) 0 with the possible quantum numbers J p = 1/2 ± and 3/2 ± using an effective Lagrangian approach. The approach is based on the hypothesis that the Ξ * c is a hadronic molecule state of DΛ-DΣ. The coupling of the Ξ * c to the constituents is described by the effective Lagrangian. The corresponding coupling constant g Ξ * c ΛD and g Ξ * c ΣD are determined by the compositeness condition Z = 0 [10,11,12,13,14], which implies that the renormalization constant of the hadron wave function is set equal to zero. By constructing a phenomenological Lagrangian including the couplings of the bound state to its constituents and the constituents with other particles we calculated one-loop diagrams describing different decays of the molecular states.
This work is organized as follows. The theoretical formalism is explained in Sec. 2. The predicted partial decay widths are presented in Sec. 3, followed by a short summary in the last section.

FORMALISM AND INGREDIENTS
In the molecular scenario, the details of the calculations for Ξ 0 c → Λ + c K − , Ξ 0 c → Σ + c K − , and Ξ 0 c → Ξ + c π − are presented for Ξ c state with two different total angular momentum J possibilities. The molecular structure of the Ξ * c baryon with quantum numbers J p = 1/2 ± is described by the Lagrangian [14] while for the choice J p = 3/2 ± the Lagrangian contains a derivative where Γ is the corresponding Dirac matrix related to the spin-parity of the Ξ * c . In particular, for J p = 1/2 + , 3/2 − we have Γ = γ 5 while for J p = 1/2 − , 3/2 + the Dirac structure Γ = 1. In the Lagrangian, an effective correlation function Φ(y 2 ) is introduced to reflect the distribution of two constituents in the hadronic molecular Ξ * c state. The introduced correlation function also makes the Feynman diagrams finite in the ultraviolet region of Euclidean space, which indicates that the Fourier transformation of the correlation function should drop fast enough in the ultraviolet region. Here we choose the Fourier transformation of the correlation in the Gaussian form [10,11,12,13,14], with α being the size parameter which characterize the distribution of components inside the molecule. Though the value of α could not be determined from first principles, it is usually chosen to be about 1 GeV in the literature [10,11,12,13,14]. In this work, we set Λ = 1.0 GeV. With the help of the effective Lagrangian in Eq.( 1) and Eq.( 2), we can obtain the self energy of the where k 2 0 = m 2 Ξ * c with k 0 , m * Ξc denoting the four momenta and mass of the Ξ * c , respectively, k 1 , m D , and m Λ , m Σ are the four-momenta, the mass of the D meson, and the mass of the Λ baryon, the mass of the Σ baryon, respectively. The coupling constant g Ξ * c ΛD and g Ξ * c ΣD is determined by the compositeness condition [15,16]. It implies that the renormalization constant of the hadron wave function is set equal to zero with where the x AB is the probability to find the Ξ * c in the hadronic state AB with the normalization x DΛ + x DΣ = 1.0. And the Σ Fig. 1. Self-energy of the Ξ * c states.. Fig. (2) shows the hadronic decay of the ΛD − ΣD molecular state into the Λ + c K − ,Ξ + c π − , and Σ + c K − final states occuring by exchanging nucleon, D * s meson, and D * meson. To compute the amplitudes of the diagrams shown in Fig.( 2), we need the effective Lagrangian densities for the relevant interaction vertices. In Refs. [17,18], coupling of the vector meson to charm baryons are described from effective Lagrangians, which are obtained using the hidden gauge formalism and assuming SU(4) symmetry: where the coupling constant g = 6.6 is from Ref. [17]. The symbol V µ represents the vector fields of the 16-plet of the ρ, given by and B is the tensor of baryons belonging to the 20-plet of p where the indices i, j, k of B ijk denote the quark content of the baryon fields with the identification 1 ↔ u, 2 ↔ d, 3 ↔ s,and 4 ↔ c.
To evaluate the diagrams in Fig.( 2), in addition to the Lagrangian in Eq.( 1), Eq.( 2), and Eq.( 8), the following effective Lagrangians, responsible for vector mesons and pseudoscalar mesons interactions are needed as well [17] where P is the SU (4) pseudoscalar meson matrices, and ... in the trace over the SU (4) matrices. The meson matrices are [17] The coupling g h is fixed from the strong decay width of K * → Kπ. With the help of Eq. (12), the two-body decay width Γ (K * + → K 0 π + ) is related to g h as where P πK * is the three-momentum of the π in the rest frame of the K * . Using the experimental strong decay width(Γ K * + = 50.3 ± 0.8 MeV) and the masses of the particles needed in the present work are listed in Table. (1) we obtain g = 4.64 [1]. The vertexes for the meson-baryon interaction are needed and the form in the SU (3) sector is given by the chiral Lagrangian [19] where F = 0.51,D = 0.75 [19] and at lowest order in the pseudoscalar field u µ = − √ 2∂ µ φ/f , with f = 93 MeV. And B and φ is now the SU (3) matrix of the baryon octet and meson, respectively.
With the above vertices, the amplitudes of the triangle diagrams of Fig.( 2), evaluated in the center of mass frame of final states, are Once the amplitudes are determined, the corresponding partial decay widths can be obtained, which read, where J is the total angular momentum of the Ξ * b state, the |p 1 | is the three-momenta of the decay products in the center of mass frame, the overline indicates the sum over the polarization vectors of the final hadrons, and M B denotes the decay channel of M B, i.e.,Λ cK , Ξ c π, Σ cK . In this work, the main ideal is to explain whether the Ξ c (2923) 0 , Ξ(2938) 0 , and Ξ(2964) 0 can be considered as a DΛ − DΣ molecular state by computing allowed twobody strong decay modes. To compute the partial decay widths of the Ξ * c , we first need the information of the coupling constants related to the molecular state and its components. In Fig. 3 we show the results for the coupling constant g Ξ * c ΛD and g Ξ * c ΣD of the various Ξ * c states for different spin-parity assignments and for a variation of the size parameter x DΛ in a range of 0.0 − 1.0. In the discussed x DΛ range, the coupling constant g Ξ * c ΣD decreases, while the coupling constant g Ξ * c ΛD increases. The results in Fig. 3 also show that the coupling constants increases (or decreases) sharply with the increase of the x DΛ for the J p = 3/2 − case, where the Ξ * c is a D-wave DΛ − DΣ molecular state. For the case of J p = 1/2 − , the coupling constants increases (or decreases) but relatively slowly compared with that coupling constants in the case of J p = 3/2 − . We show the dependence of the total decay width on the x DΛ in Fig. 4. The total decay widths increase with x DΛ for the J P = 1/2 + and J P = 3/2 ± assignments. For the J P = 1/2 − assignments, we found that the line shape of the the total decay widths are huge different and the total decay widths first increases, then decreases but very slowly. A possible explanation for this may be that for an S−wave loosely bound state the effective coupling strength of the bound state to its components is more sensitive to the inner structure than the effective coupling strength of another possible molecular state, such as P − wave molecular state, relative to their inner component. This is why people often focus on the bound state from Swave interaction and assume the P − and D-wave bound state should be difficult to form from hadron-hadron interaction [23].

RESULTS AND DISCUSSIONS
From the Fig. 4, one find that the predicted total decay widths for the Ξ c (2964) state and Ξ c (2938) state in the four spin-parity assignments are all smaller than the experimental total width. Such results disfavor the assignment of these two states as DΛ − DΣ molecular state. For the Ξ c (2923) state, the predicted total decay width is much smaller than the experimental total width in the case of J P = 1/2 + , which disfavors such a spin-parity assignment for the Ξ c (2923) in the DΛ − DΣ molecular picture. For the case of J P = 3/2 + , Since the estimated total decay widths is much smaller than the experimental total width, this case can be completely excluded as well. The J P = 3/2 − case is also disfavored due to the smallest width predicted. Hence, only the Ξ c (2923) can be considered as the molecular states composed of DΛ and DΣ components by comparison with the total decay width experimentally measured. The results in Fig. 4 also show that the total decay width of the Ξ c (2923) can not be reproduced when only consider Ξ c (2923) as pure DΛ or pure DΣ molecular state.