Strong and radiative decays of D\Xi molecular state and newly observed $\Omega_c$ states

In this work, we study strong and radiative decays of S-wave D\Xi molecular state, which is related to the \Omega^*_c states newly observed at LHCb. The coupling between the D\Xi molecular state and its constituents D and \Xi is calculated by using the compositeness condition. With the obtained coupling, the partial decay widths of the D\Xi molecular state into the \Xi_c^{+}K^{-}, \Xi^{'+}_cK^{-} and \Omega^{*}_c(2695)\gamma 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 \Omega^{*}_c(3119) or \Omega^{*}_c(3050) as D\Xi molecule while the the decay width of the \Omega^{*}_c(3000), \Omega^{*}_c(3066) and \Omega^{*}_c(3090) can not be well reproduced in the molecular state picture. The partial decay widths are also presented and helpful to further understand the internal structures of \Omega^{*}_c(3119) and \Omega^{*}_c(3050).

In this work, we study strong and radiative decays of S-wave DΞ molecular state, which is related to the Ω * c states newly observed at LHCb. The coupling between the DΞ molecular state and its constituents D and Ξ is calculated by using the compositeness condition. With the obtained coupling, the partial decay widths of the DΞ molecular state into the Ξ + c K − , Ξ ′ + c K − and Ω * c (2695)γ 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 Ω * c (3119) or Ω * c (3050) as DΞ molecule while the the decay width of the Ω * c (3000), Ω * c (3066) and Ω * c (3090) can not be well reproduced in the molecular state picture. The partial decay widths are also presented and helpful to further understand the internal structures of Ω * c (3119) and Ω * c (3050).

I. INTRODUCTION
For a long time, little is known about the charmed baryon Ω c with quantum numbers C = 1 and S = −2, which is composed of one charm quark and two strange quark in the conventional constituent quark model. Only ground state Ω * c (2695) and Ω * c (2770) are listed in the newest version of the Review of Particle Physics (PDG) [1]. Recently, five new narrow Ω * c states named Ω * c (3000), Ω * c (3050),Ω * c (3066), Ω * c (3090), and Ω * c (3119) were reported by the LHCb collaboration in the Ξ + c K − mass spectrum [2]. Though the quantum numbers of these new Ω * c states are not confirmed, it is very helpful to understand the charmed baryon spectrum.
The LHCb observation stimulated a large amount of the theoretical studies bout the new Ω * c states with different assumptions of their internal structures. Naturally, many authors try to assign these states into the conventional three-quark frames. In Refs. [3][4][5][6] the new Ω * c baryons were interpreted as 1P and 2S Ω * c baryons in the conventional quark models. The QCD sum rules were also applied to study these states in three-quark picture [7,8]. The lattice calculation was also performed and try to determine their quantum numbers [9]. In Refs. [10,11], the authors investigated the decay properties to reveal the nature of these states.
It is quite rare to observe five states in one observation simultaneously. So many states observed also make it difficult to put all states into the conventional quark model. Hence, after the observation at LHCb, the newly observed Ω * c was immediately interpreted as exotic state beyond three-quark picture, i. e., the pentaquark state. The largest mass gaps between * huangy2017@buaa.edu.cn † Corresponding author: junhe@njnu.edu.cn ‡ Corresponding author: lisheng.geng@buaa.edu.cn the newly observed Ω * c baryons and the ground Ω * c baryon are about 400 MeV, which is large enough to excite a light quark-antiquark pair. Indeed, in Ref. [12], pentaquark-like Ω * c baryons were studied in the constituent quark model and associated to some of the LHCb Ω * c baryons. In Ref. [13], it was found that four sscqq states with J P = 1/2 − or J P = 3/2 − have masses close to the newly observed Ω * c states. In the chiral quark-soliton model, pentaquark-like structures were suggested for the Ω * c (3050) and Ω * c (3119) [14,15]. Since the Ξ ′ cK and ΞD thresholds fall in the mass region of the LHCb observed Ω * c states, hadronic molecule interpretations can not be excluded. In Ref. [16], the Ω * c (3050) and Ω * c (3090) were regarded as meson-baryon molecules and with a similar method, the Ω * c (3119) was also proposed to be a hadronic molecule [17]. Moreover, the Ω * c (3000), Ω * (3050) ,and Ω * (3090) or Ω * (3119) can all be explained as mesonbaryon molecular state in Ref. [18]. With the one-gluonexchange and the Goldstone-boson-exchange in addition to the color confinement, the authors in Ref. [19] suggested that only Ω * c (3119) can be explained as an S -wave resonance state of ΞD with J P = 1/2 − , which decays mainly through S wave into Ξ c K and Ξ ′ c K. Until now, the nature of the observed Ω * c baryons remains unclear. In addition to their masses, decay property also serves as an important way to unveil the nature of hadrons. In Ref. [10] the authors studied the decay patterns of the Ω * c baryons in a chiral quark model in three-quark picture and suggested that most of the low-lying Ω * c baryons have masses in the vicinity of the Ξ + c K − and Ξ ′ + c K − thresholds, to which the strong decay will almost saturate their total decay widths. However, the decays of the Ω * c baryons, which are helpful to understand their internal structures, have not been studied in the molecular state picture.
In Refs. [20][21][22][23][24], the decays of hadronic molecular states have been studied by calculating the hadronic loop with the assumption that a molecular state prefers to decay into its two constituents. The technique for evaluating composite hadron systems has been widely used to study hadronic molecular state, where the compositeness condition, corresponding to Z = 0, has been employed to extract the coupling of a molecular state to its constituents [25][26][27][28]. In this work, we will calculate the radiative and strong decay pattern of S-wave DΞ molecular state within the effective Lagrangians approach, and find the relation between the DΞ molecular state and the Ω * c states by comparing with the LHCb observation, This paper is organized as follows. The theoretical formalism is explained in Sec. II. The predicted partial decay widths are presented in Sec. III. Finally, we give discussion and summary in the last section.

II. FORMALISM AND INGREDIENTS
In the molecule scenario, the interaction between the state Ω * c and its components ΞD is mainly via S -wave and the simplest Feynman diagrams are shown in Figs. 1. For the Ω * c ΞD coupling, following Refs. [25,26], we take the Lagrangian densities as where In the Lagrangian, an effective correlation function Φ(y 2 ) is introduced to reflect the distribution of two constituents, Ξ and D, in the hadronic molecule Ω * c state. It also play a role to avoid the Feynman diagrams ultraviolet divergence, which requires that its Fourier transform should vanish quickly in the ultraviolet region in the Euclidean space. Since only S wave is considered in current work, we adopt an exponential form Φ(−p 2 E ) exp(−p 2 E /χ 2 ) with p E being the Euclidean Jacobi momentum as used in Refs. [25,26]. The χ is a free size parameter characterizing the distribution of the two components in the molecule and we adopt the χ = 1 that is often used in Refs. [20][21][22][24][25][26][27][28] .
The only undetermined parameter is the coupling between molecular state and two constituents, g Ω * c ΞD , which strength is a key factor to the value of the decay width on which we focus in the current work. Following Refs. [29][30][31], we will adopt the compositeness condition to calculate the coupling of the hadronic molecule Ω * c and its consituents Ξ and D. This condition requires that the renormalization constant of the hadronic molecular wave function is equal to zero, 1 − c being the self-energy of the hadronic molecule Ω * c . Such relation connects the binding energy and the coupling strength of bound state and its constituents. Now that the masses of Ω * c baryons have been observed in experiment, the couplings can be determined with such relation. The Feynman diagram describing the self-energy of the Ω * c states is presented in Fig. 1. With the help of the effective Lagrangian in Eq. (1), we can obtian the self energy of the Ω * c as While k 1 , m Ξ and m D are the four-momenta, mass of the Ξ and mass of D, respectively.
According to the normalization conditions, the coupling constants is given by where the η and β will be integrated out, and the α is a free parameter, which will be discussed later.
Considering the quantum numbers and phase space, the strong decay modes of Ω cK 0 ). The sum of the two parts is the total decay width of the Ω * c →KΞ ( ′ )+ c . In the hadronic molecule picture, Ω * c can decay into Ξ + c K − , Ξ ′ + c K − and γΩ * c (2695) by rearranging the quarks in its components. At the hadron level, Ω * c is treated as a bound state of ΞD and the decay Ω * occurs by exchanging a proper strange meson and hyperon as shown in Fig. 2. In the present work, we estimate these triangle diagrams in an effective Lagrangian approach. Besides the Lagrangian in Eq. 1, the effective Lagrangians of relevant interaction vertices are also needed [32][33][34][35][36].
where the m Λ , m Σ , and m Ξ ( ′ ) c are the masses of the particle Λ, Σ, and Ξ ( ′ ) c , respectively. The coupling constant g KD * s D = 1.84 is estimated in the framework of light-cone QCD sum rules [37] and g = 6.6 is taken from [32,35]. The τ is the Pauli matrix, Σ represents the Σ triplets, andD andK are the doublets of charmed and K mesons.
The couplings for the different charge states are related by isospin symmetry: One can estimate the couplings constants from SU(4) symmetry and phenomenological constraints [38] g where g NNπ = 13.26 [36], and α NNπ = 0.64 [38]. The numerical values of the couplings constant are listed in Table I. The involved interaction related to the photon field and the charmed mesons is [39].
where the field-strength tensors are defined as F µν = ∂ µ A ν − ∂ ν A µ , D * αβ = ∂ α D * β − ∂ β D * α , and e = √ 4π/137. According to the Lagrangian and the radiative decay width of Γ D * 0 →D 0 γ = 26 KeV that was deduced from the data on strong and radiative decays of D * meson by theoretical predictions [25,40], the coupling constant g D * 0 D 0 γ can be determined as where m D * 0 = 2.007 GeV, m D 0 = 1.865 GeV. Similar, the coupling constant g D * + D + γ = −0.5GeV −1 is estimated from the partial decay width of Γ D * + →D + γ =1.334 KeV [1] with m D * ± =2.010 GeV. The minus sign is adopted according to the lattice QCD and QCD sum rule calculations [41,42]. In evaluating the amplitudes which are shown in Figs. 2, we need to include the form factors because hadrons are not pointlike particles. We adopt here the monopole-type form factor F B (q 2 ) that was used in many previous works [26,43], with M being the mass of the exchanged meson and baryon. The cutoff Λ = M + λΛ QCD with Λ QCD = 220 MeV is taken from Refs. [44,45]. The parameter λ reflects the nonperturbative property of QCD at the low-energy scale, which will be taken as a parameter and discussed later. Putting all pieces together, we obtain the amplitudes for and Ω * c (2695)γ which correspond to the diagrams in Fig. 2, which reads The corresponding partial decay widths then read where J is the total angular momentum of the initial Ω * c state, the overline indicates the sum over the polarization vectors of final hadrons. Here | p K/γ 1 | is the 3-momenta of the decay products in the center of mass frame.

III. RESULTS
Regarding the five new Ω * c as ΞD hadronic molecules, the coupling constants g Ω * c ΞD can be estimated from the compositeness condition. As shown in Eq. (3), the coupling constant is dependent on the parameter α. In Fig. 3, we show the dependence of the coupling constants g Ω * c ΞD on the cutoff parameter α. The coupling constant g Ω * c ΞD decreases with the increase of α. Taking the Ω * c (3119) as an example, the value of the coupling constant g Ω * c (3119)ΞD is not very sensitive to the model parameter α when varying cutoff parameter λ from 0.7 GeV to 1.3 GeV (not sensitive to λ also). Fixing the α at certain value, such as 1.00 GeV, the coupling constants decrease with increase of the mass m Ω * c . According to the studies of the XYZ resonances and the deuteron [40,46], a typical value of α ∼1 GeV is often employed. Thus, in this work we take α = 1.0 and the corresponding coupling constants are listed in Table. II, which are used to calculate the decay processes of Fig. 2.  Once the coupling constants of the molecular Ω * c baryons and ΞD are determined, the partial decay widths of the Ω 2695)γ, and the total decay width are only dependent on the parameter λ in the cutoff. Though the value of λ could not be determined in first principles, it is usually chosen as about 1 in the literature. In Ref. [26], by comparing the sum of the partial decay modes of the η(2225) and φ(2170) with the total width, the parameter λ was constrained as λ = 0.91 − 1.00. In addition, the experimental branching ratios of ψ(4040) → J/ψη and ψ(4160) → J/ψη can be well explained with λ = 0.53 − 1.20 [45]. Larger range of 0.5 to 5 can be found in Refs. [47][48][49][50]. Considering the values adopted in above literatures, we adopt a parameter λ in the a range of 0.91 ≤ λ ≤ 1.0 because this range is determined from the experimental data of branching ratios within the same theoretical framework adopted in current work in Ref. [26]. The numerical results are presented in Fig. 4 with the variation of λ from 0.90 to 1.0. In the discussed range, the partial decay widths increase with λ, and the Ω * c states mainly decay into Ξ cK and the partial width into Ξ cK is much larger than those into Ξ ′ cK and, of course, γΩ * (2695). The total width of Ω * c (3119) and Ω * c (3050) can be well reproduced in the λ range considered here. If we increase λ to higher values, the total widths of all five Ω * c baryons can not be reproduced until a much larger λ value of about 2 adopted. Hence, it is reasonable to adopt a λ of about 1 in the current work. , Ω * c (3000), and Ω * c (3065), their total decay widths are much smaller than the experimental total width. Such results disfavor the assignment of these three states as DΞ molecular state. Hence, only the Ω * c (3119) or Ω * c (3050) states can be considered as S −wave ΞD molecules. Hence, we only list the decay widths of Ω * c (3119) and Ω * c (3050) with λ = 0.91 − 1.00 in Table. III. For comparisons, we show the results in the constituent quark model as well [10]. The decay width Γ Ξ cK is close to that in the constituent quark model if we assign the S -wave ΞD bound state as Ω * c (3050). Assuming this channel is dominant decay channel, the total decay width under such assignment is also consistent with that in constituent quark model and the experimental value. Under assignment as Ω * c (3119), the total widths decay width Γ Ξ cK is lager than that in the constituent quark model while Γ Ξ ′ cK is smaller, which leads to a comparable total decay width to those in the constituent quark model and in experiment. Now we turn to the radiative decay Ω * c → Ω * c (2695)γ. The individual contributions of the D * 0 and D * − exchange and total decay width with varying λ from 0.90-1.00 for the Ω * c → Ω * c (2695)γ are presented in Fig. 6 and Fig. 4, respectively. Our study shows that the partial width of the Ω * c → Ω * c (2695)γ is rather small and the D * 0 exchange plays a dominant role, weakly increasing with the λ increasing. In the considered parameter region, the partial widths for the Ω * c → Ω * c (2695)γ are predicted and listed in Table. III, compared with the results in conventional charmed baryons scheme [10]. The partial widths of Ω * c (3119) → Ω * c (2695)γ and Ω * c (3050) → Ω * c (2695)γ in Ref. [10] were 1.2 and 2.9/1.0×10 −3 MeV, re- ,Ω * c (2695)γ, and the total decay width Γ total with λ = 0.91 − 1.00 that is introduced by the form factor, in comparison with the results in the constituent quark model [10]. The total width obtained from the LHCb experiments [2]. All masses and widths are in units of MeV. The two values of decay width for the Ω * c (3119) in Ref [10] are for the assignments |2 2 S ΛΛ 1/2 + or |2 4 S ΛΛ 3/2 + , respectively.