Possible open-charmed pentaquark molecule $\Omega_c(3188)$ --- the $D \Xi$ bound state --- in the Bethe-Salpeter formalism

We study the $S$-wave $D\Xi$ bound state in the Bethe-Salpeter formalism in the ladder and instantaneous approximations. With the kernel generated by the hadronic effective Lagrangian, two open-charmed bound states, which quantum numbers are $I=0$, $J^P=(\frac{1}{2})^-$ and $I=1$, $J^P=(\frac{1}{2})^-$, respectively, are predicted as new candidates of hadronic pentaquark molecules in our formalism. If existing, they could contribute to the broad 3188 eV structure near the five new narrow $\Omega_c$ states observed recently by the LHCb Collaboration.


I. INTRODUCTION
In the last decade, many important experimental progresses were made in the study of charm hadrons. Several charm baryons and their excited states have been reported and this stimulates great interest in understanding the structures of charm baryons. In these experimental observations, there are some non-conventional states, which are more complicated than the hadronic states in the classical quark model. These states can be interpreted as five-quark or meson-baryon bound states. In 2015, the LHCb Collaboration discovered two hidden-charmed pentaquark-like structures P c (4380) + and P c (4450) + [1- 3], which are considered as Σ cD * , Σ * cD , or Σ * cD * pentaquark molecule [4][5][6][7]. Besides the hidden-charmed pantaquarks, only a few open-charmed baryons are treated as candidates for pentaquark molecules which have not yet been confirmed by experiments. For example, Σ c (2800) and Λ c (2940) + have been suggested to be S-wave DN and D * N molecular states, respectively [8][9][10][11].
Very recently, the LHCb Collaboration declared that they observed five new narrow Ω c states, which are Ω c (3000) 0 , Ω c (3050) 0 , Ω c (3066) 0 , Ω c (3090) 0 , and Ω c (3119) 0 [12]. Actually, there are six structures existing in the spectrum of invariant mass of Ξ + c K − . The data also indicate the presence of a broad structure around 3188 MeV that is fitted as single resonance [shortly denoted by Ω c (3188) in our work]. In the experiment, this resonance is described as the sum of four incoherent Breit-Wigner functions to be left into the systematic uncertainties of the others five resonances [12]. However, one notes that this structure is located at the DΞ threshold (3179-3191 MeV) and therefore can also be interpreted as the DΞ bound state naturally. At the same time, since the width of corresponding peak is quite broad [12], if this structure is a real signal of resonance, there is reason to believe that it would be a new pentaquark molecule with a single charm quark. At present, there is no enough information concerning this structure, such as spin-parity, it is interesting and significative to make efforts on the theoretical side to confirm its existence and reveal its properties. In this work, we will focus on this structure and study the possible S-wave DΞ bound state.
The Bethe-Salpeter (BS) equation is a formally exact equation to describe the relativistic bound state. This technique was developed by Feynman, Bethe, and Salpeter et al. [13][14][15].
It has been applied to theoretical studies concerning heavy baryons and molecular bound states [16][17][18][19][20][21]. In previous studies, the possible bound states of KK, DK, BK, and K − p have been investigated in the BS formalism in the ladder and instantaneous approximations [17][18][19][20]. We will try to study S-wave DΞ molecular bound state with the kernel introduced by the vector meson exchange interactions in this framework. We will investigate whether this state exists or not and study its decay. We will also discuss the possibility of the Ω c (3188) structure to be the DΞ bound state.
The remainder of this paper is organized as follows. In Sect. II, we give the generalized formalism of the BS equation for the fermion-scalar system. In Sect. III, we derive the BS equation for the DΞ system in detail and present the normalization condition of the corresponding BS amplitude. In Sect. III, the decay of the DΞ bound state to Ξ + c K − is discussed. The numerical results are presented in Sect. IV. In the last section, we give a summary and some discussions.

II. THE BETHE-SALPETER EQUATION FOR THE FERMION-SCALAR SYS-TEM
In this section, we will present the BS equation for the fermion-scalar system. We assume that the bound state exists in a fermion-scalar system and its mass is M. The BS amplitude can be defined as [15][16][17][18][19][20] with ψ(x 1 ) and φ(x 2 ) being field operators of the fermion and scalar particles , respectively, and P being the momentum of the system. In momentum space, the Bethe-Salpeter amplitude, χ P (p), is related to χ(x 1 , x 2 , P ) through the following equation [15]: where p and x(= x 1 − x 2 ) are the relative momentum and the relative coordinate of two constituents, respectively, and X is the center of mass coordinate which is defined as X = λ 1 x 1 +λ 2 x 2 , where λ 1 = m 1 m 1 +m 2 , λ 2 = m 2 m 1 +m 2 , with m 1 and m 2 being the masses of the fermion and the scalar constituent, respectively. The momentum of the fermion is p 1 = λ 1 P + p and that of the scalar particle is p 2 = λ 2 P − p. The derivation of the BS formalism for the two fermion system can be found in the textbook [15]. In the same way, one can prove that the form of the BS equation is still valid for the fermion and scalar object system. The BS amplitude in our case satisfies the follow homogeneous integral equation [15][16][17][18][19][20]: where s F and s S are propagators of the fermion and the scalar particle, respectively, and K(P, p, q) is the interaction kernel which can be described by the sum of all the irreducible graphs which cannot be split into two pieces by cutting two particle lines as defined in Ref. [15]. For convenience, we also define the relative longitudinal momentum p l (= v · p) and the transverse momentum with The isoscalar bound state can be written as while the isovector one is where the subscripts (I, I 3 ) denote the isospin and the third component of the isospin.
Since the interaction between the two constituents is dominated by strong interaction, the BS amplitude depends only on the isospin I. Therefore, the BS amplitude of the DΞ system can be defined as where the isospin coefficients C ij (I, for the isoscalar state and for the isovector state.
Considering the isospin structure, the BS equation for the DΞ system can be written as where i(j) and l(k) refer to the components of the D(Ξ) field doublets, we neglect the effect of isospin violation and take m Ξ = 1 2 (m Ξ 0 + m Ξ − ) and m D = 1 2 (m D + + m D 0 ). Explicitly, we give the isoscalar case as an example: B. The Bethe-Salpeter equation for the DΞ bound state In general, considering v /u(v, s) = u(v, s) , χ P (p) can be written as [22][23][24] where u(v, s) is the spinor of the bound state with helicity s and g i (i = 1, 2 · · · 5) are Lorentzscalar functions. According to our previous works, the momentum transfer between the two constituents in a hadronic bound state is in the order of 0.1 GeV [17][18][19][20] and this quantity is quite smaller than the mass of D. Thus, we can apply the heavy quark symmetry and have v /χ P (p) = χ P (p). With the constraints imposed by parity and Lorentz transformations, it is easy to prove that χ P (p) can be simplified as [22,23] in which f (p) is a Lorentz-scalar function of p.
In this work, we describe the DΞ interaction by one-particle exchange diagrams as shown in Fig 1. In the chiral limit and the heavy quark limit, the effective Lagrangian of the interacting vertices involved are [19,25] where g DDV and g ΞΞV are the coupling constants, and V refers to the fields of the vector mesons and have the following form: where the φ meson exchange process is neglected because of the OZI suppressed. In order to include the finite-size effects of these hadrons, we also introduce a form factor at each interacting vertex of hadrons. Following Refs. [17][18][19], we take the monopole form: where m V is the mass of the exchanged meson, k is the momentum transfer carried by m V , and Λ is a cutoff parameter.
The BS equation can be treated in the so-called ladder approximation [15]. In this approximation, K(P, p, q) is replaced by its lowest order form. Using these interaction vertices and the form factor, one can get the following kernel: where we have used the covariant instantaneous approximation [16][17][18][19][20], p l = q l , and c ρ(ω) = 1 2 (− 1 2 ) and 3 2 ( 1 2 ) corresponding to isoscalar and isovector mesons, respectively. In the chiral limit and the heavy quark limit, the propagator of the D meson can be expressed at the leading order of the 1/m D expansion as follows [26]: .
The propagator of Ξ has the form [20,23] Then, we substitute Eqs.
In the bound state rest frame, one has p t = (0, −p t ) and |p t | = −|p t |. Performing the integration over p l on both sides with the residue theorem, we havẽ where we have defined the functionf (p t ) = dp l 2π f (p l , p t ). One may note that this equation involves the integration of q t and it looks like a divergent integration since q t varies from 0 to +∞. However, the Lorentz-scalar function,f (p t ), decreases to zero rapidly at the large momentum transfer and thus there is no divergence in practice [17][18][19][20]. Since we study the ground state of the DΞ bound state, the BS amplitude is in fact rotationally invariant and depends only on the norm of the three momentum, |p t |. Then, the BS equation becomes a one-dimensional integral equation.

C. Normalization condition
In Eq. (20), we leave the normalization undetermined. Following Ref. [15], the normalization condition for the BS equation can be written as [I(P, p, q) + K(P, p, q)] χ P (q) = 1, (P 0 = E P ), (21) where I(P, p, q) is the inverse of the four-point propagator In the DΞ bound state rest frame, the normalization condition can be written in the following form: [I(P, p, q) + K(P, p, q)]χ P (q) According to Eqs. (19) and (20), we have Then, One can recast the normalization condition for the BS amplitude into the form We note that the unit off (p t ) is GeV −5/2 .
Feynman diagrams for the DΞ bound states decay into the Ξ + c K − final state.

IV. THE DECAY WIDTH OF THE DΞ BOUND STATE
In this section, we will proceed to calculate the decay of the DΞ bound state through BS technique. In the experiments, the LHCb collaboration observed the new Ω 0 c resonances in the Ξ + c K − spectroscopy [12]. In order to compare with the experimental results, we will study the process of the DΞ bound state decay into above final state. The interaction is described via the vector meson exchange (the D * s meson) as shown in Fig. 2. The D 0 K − D * s coupling vertex is given below in the heavy quark limit and the chiral limit [27][28][29][30] The effective Lagrangian for the Ξ 0 Ξ + c D * s vertex is where SU(3) flavor symmetry is extended to most general SU(3) symmetry including the charm quark [31]. Considering the charm quark, the Ξ 0 and Ξ + c are collected into SU(3) multiplet fields 8 F and 6 F , respectively, and the D * s meson is belong to SU(3) multiplet fields 3 F . The coupling strength among baryon 8 F -plet, 6 F -plet and meson3 F -plet is where g is a universal coupling constant in the SU(3) symmetry without the charm quark.
One may doubt the validity of this relation with the limit of a light charm quark mass.
We define p a [= (E a , −p a )] and p b [= (E b , −p b )] to be the momenta of Ξ + c and K − , respectively. p ′ (= λ ′ 2 p a − λ ′ 1 p b ) is defined as the relative momentum between Ξ + c and with m a and m b being the masses of Ξ + c and K − , respectively. According to the kinematics of the two-body decay, in the rest frame of the bound state one has to solve the homogeneous BS equation. One solution corresponds to a possible bound state.
Since the BS amplitude for the ground state is in fact rotationally invariant,f depends only on |p t |. Generally, |p t | varies from 0 to +∞ andf would decrease to zero when |p t | → +∞.
We replace |p t | by the variable, t: where ǫ is a small parameter and is introduced to avoid divergence in numerical calculations and t varies from -1 to 1. We then discretize Eq. (20) into n pieces (n is large enough) through the Gauss quadrature rule. The BS amplitude can be written as n-dimension vectors, f (n) .

The coupled integral equation becomes a matrix equation f
to the coefficients in Eq. (20)]. One can obtain the numerical results of the BS amplitude by solving the eigenvalue equation obtained from the above matrix equation.
In our calculation, we take the mass of the mesons and baryons from the Review of PDG [32]. The cutoff parameter Λ contains the information of the non-point interaction among the hadrons. Although Λ depends on the specific process and cannot be exactly determined, it should be the typical hadronic scale, which is about 1 GeV. In Ref. [25] one takes Λ πN N = 1.3 GeV and Λ ρN N = 1.4 GeV. For the mesons with heavy quarks, the value of Λ can be as large as 3 GeV [33]. Λ is varied in the ranges 0.8-4.5 GeV [18] and 1-4 GeV [19] when studing the DK bound state. In the present paper, we vary Λ from 1 to 5 GeV.
Besides, there are several coupling constants in our calculations. Following Ref. [26], we The coupling constants for the ΞΞV vertex in Fig. 1 and the D 0 K − D * s vertex in Fig. 2 are obtained in the framework of light-cone QCD sum rules, g ΞΞV = 1.5 [34], and g D 0 K − D * s = 1.84 [35]. As for the coupling constant g Ξ 0 Ξ + c D * s , we estimate its value by extending to most general SU(3) symmetry including the charm quark as mentioned in Sect. IV. We take g Ξ 0 Ξ + c D * s = 2 √ 6 g3 86 (≃ 7.6) with g ≃ 6.6 [31]. According to the LHCb collaboration experiments, the peak and width of Ω c (3188) is 3188 ± 5 ± 13 and 60 ± 15 ± 11 MeV, respectively [12]. We can see the two DΞ bound states in our calculations are located in the range of Ω c (3188) and its decay width is smaller than that of Ω c (3188). That is to say, the DΞ bound states are likely to exist and could contribute to the observed Ω c (3188). However, Ω c (3188) might also be produced by a superposition of others resonances. We still need more experimental data to support our results. If Ω c (3188) corresponds to the S-wave DΞ bound states, its the quantum number should be I(J P ) = 0( 1 2 − ) or 1( 1 2 − ) and one can verify this resonance by its quantum numbers. This is important to ascertain the structure of Ω c (3188).

VI. SUMMARY AND DISCUSSION
In this paper, we studied the possible S-wave molecular bound states of the DΞ system in the BS formalism. Considering the interaction kernel based on vector meson exchange diagrams, we established the BS equation for the DΞ system in the ladder and instantaneous approximations. Then, we discretized the integral equation and solved the eigenvalue equation numerically. We confirmed the existence of the S-wave isoscalar and isovector DΞ bound states in this formalism and obtained their BS amplitudes. We also calculated the decay widths of the bound states by using the BS amplitudes. According to our results, the DΞ bound states are compatible with the new observed Ω c (3188) structure. Or, at least, we can say safely that these two bound states could contribute to the Ω c (3188) peak.
In this work, we predicted the existence of the DΞ bound states and discussed the possibility of Ω c (3188) to be the DΞ bound states. Unfortunately, there are several parameters not being well determinated in the calculations. These uncertainties reduced the predictability of our model. In order to describe non-pointlike effects, we introduced a cutoff parameter Λ. This cutoff parameter is dominated by nonperturbative QCD and cannot be determined at present, which leads to large uncertainties. Meanwhile, the mass and decay width of the DΞ bound states have a strong dependency on this cutoff parameter.
Thus, it is difficult for us to determine their masses and widths much more accurately at present. Further experimental data are expected to verify Ω c (3188) to be the DΞ bound states with its quantum numbers determined.