Possible molecular states from interactions of charmed baryons

In this work, we perform a systematic study of possible molecular states composed of two charmed baryons including hidden-charm systems $\Lambda_c\bar{\Lambda}_c$, $\Sigma_c^{(*)}\bar{\Sigma}_c^{(*)}$, and $\Lambda_c\bar{\Sigma}_c^{(*)}$, and corresponding double-charm systems $\Lambda_c\Lambda_c$, $\Sigma_c^{(*)}\Sigma_c^{(*)}$, and $\Lambda_c\Sigma_c^{(*)}$. With the help of the heavy quark chiral effective Lagrangians, the interactions are described with $\pi$, $\rho$, $\eta$, $\omega$, $\phi$, and $\sigma$ exchanges. The potential kernels are constructed, and inserted into the quasipotential Bethe-Salpeter equation. The bound states from the interactions considered is studied by searching for the poles of the scattering amplitude. The results suggest that strong attractions exist in both hidden-charm and double-charm systems considered in the current work, and bound states can be produced in most of the systems. More experiment studies about these molecular states are suggested though the nucleon-nucleon collison at LHC and nucleon-antinucleon collison at $\rm \bar{P}ANDA$.


I. INTRODUCTION
With the development of the experimental technology, a large amount of data accumulated in experiment provide opportunity to the study of the hadron spectrum. In the recent years, more and more hadrons have been observed in experiment [1]. Many of these new observed hadrons cannot be put into the conventional quark model, which is the basic frame to understand the hadron spectrum [2,3]. A growing number of efforts have been paid to explain their origin and internal structure. An obvious observation is that many newly observed particles are close to the threshold of two hadrons, so a popular picture to understand these exotic hadrons is the molecular state, which is a loosely bound state of hadrons. The XYZ particles, such as X(3872), Z c (3900) and Z b (10610) and Z c (10650), were widely assigned as molecular states in the literature [4][5][6][7][8]. Particularly, the observed hidden-charm pentaquarks provide a wonderful spectrum of molecular states composed of an anticharmed meson and a charmed baryon [9][10][11][12][13][14][15]. Such picture is enhanced by the recent observed strange hidden-charm pentaquarks [16][17][18][19]. However, though the well-known deuteron and the dibaryon with nucleon, ∆, and Λ baryon were predicted and studied in both theory and experiment very far before the XYZ particle and pentaquarks, few predicted molecular states of two baryons are observed in experiment [1,20]. Some theoretical studies have been performed to discuss the possibility of existence of molecular states composed of two baryons beyond nucleon, ∆, and Λ baryon [21][22][23][24][25][26].
Most of the molecular state candidates observed in the past two decades are in the hidden-charm sector. Hence, it is natural to expect the molecular state composed of a charmed baryon and an anticharmed baryon. In recent years, the structures near the Λ cΛc threshold has attracted much attentions. A charmoniumlike Y(4630) with quantum numbers J PC = 1 −− was observed at Belle [27]. After the experimental discovery of Y(4630), many theoretical works have performed to understand its origin, such as conventional charmonium state [28,29] and compact multiquark state [30][31][32][33]. Due to * Corresponding author: junhe@njnu.edu.cn the closeness of the mass of Y(4630) and the Λ cΛc threshold, the relation between Y(4630) and the threshold effect was studies in Ref. [24]. In Ref. [35], the mechanism of Y(4630) enhancement in Λ cΛc electroproduction was also studied. The Λ cΛc molecular state also attracts much attention [21,[33][34][35]. Theoretical calculations suggest strong attraction between a Λ c baryon and anΛ c baryon by σ and ω exchanges, which favors the existence of a Λ cΛc molecular state [21,35]. In our previous work, the Λ cΛc molecular state can be produced from the interaction, but it is difficult to be used to interpret the Y(4630) [25]. The studies of more molecular states with a charmed baryon and an anticharmed baryon are also helpful to understand this exotic structure. In the current work, the interactions Λ cΛc , Σ ( * ) cΣ ( * ) c , and Λ cΣ ( * ) c will be studied in a quasipotential Bethe-Salpeter equation (qBSE) approach.
This article is organized as follows. After the Introduction, Section II shows the details of dynamics of the charmed baryons interactions, reduction of potential kernel and a brief introduction of the qBSE. In Section III, the numerical results are given. Finally, summary and discussion are given in Section IV.

II. THEORETICAL FRAME
To study the interactions of charmed baryons, we need to construct the potential kernel, which is performed by introducing the exchanges of peseudoscalar P, vector V and scalar σ mesons. The Lagrangians depicting the couplings of light mesons and baryons are required and will be presented below.

A. Relevant Lagrangians
The Lagrangians for the couplings between charmed baryon and light mesons are constructed under the heavy quark limit and chiral symmetry as [26,62,63], where S µ ab is composed of the Dirac spinor operators, and the bottomed baryon matrices are defined as The explicit forms of the Lagrangians can be written as, The V and P are the vector and pseudoscalar matrices as The masses of particles involved in the calculation are chosen as suggested central values in the Review of Particle Physics (PDG) [1]. The mass of broad σ meson is chosen as 500 MeV.
The coupling constants involved are listed in Table I.
First, we should construct flavor wave functions with definite isospin under S U(3) symmetry. In this paper, we take the following charge conjugation conventions for two-baryon system as [23], where J and J i are the spins of system |B 1B2 and |B i , respectively, and c i is defined by C|B i = c iBi . For the isovector state, the C parity cannot be defined, so we will use the G parity instead as G = (−1) I C with C = c. Following the method in Ref. [66], we input vertices Γ and propagators P into the code directly. The potential can be written as In this work, both hidden-charm and double-charm systems will be considered in the calculation. The well-known Gparity rule will be adopted to write the interaction of a charmed and an anticharmed baryon from the interaction of two charmed baryons. By inserting the G −1 G operator into the potential, the G-parity rule can be obtained easily as [21,22,67,68], The G parity of the exchanged meson is left as a ζ i factor for i meson.
The propagators are defined as usual as where the form factor f (q 2 ) is adopted to compensate the offshell effect of exchanged meson as f (q 2 ) = e −(m 2 e −q 2 ) 2 /Λ 2 e with m e and q being the m P,V,σ and the momentum of the exchanged meson. The I i is the flavor factor for certain meson exchange i of certain interaction, and the explicit values are listed in Table II.
where the sum extends only over non-negative helicity λ ′′ . The G 0 (p ′′ ) is reduced from the 4-dimensional propagator G(p ′′ ) under quasipotential approximation with one of two baryons on-shell as where p ′′ l and m l are the momentum and mass of light hadron, respectively. As required by the spectator approximation, the heavier particle is on shell, which satisfies p ′′0 h = E h (p ′′ ) = m 2 h + p ′′2 . The p ′′0 l for the lighter particle is then W − E h (p ′′ ). Here and hereafter, a definition p = |p| will be adopted.
The partial wave potential is defined with the potential of interaction obtained in the above in Eq. (7) as where η = PP 1 P 2 (−1) J−J 1 −J 2 with P and J being parity and spin for system. The initial and final relative momenta are chosen as p = (0, 0, p) and p ′ = (p ′ sin θ, 0, p ′ cos θ).
The d J λλ ′ (θ) is the Wigner d-matrix. We also adopt an exponential regularization by introducing a form factor into the propagator as with Λ r being a cutoff [71].

III. NUMERICAL RESULTS
With the preparation above, numerical calculation can be performed to study the molecular states from the interac- After transformation of the one dimensional integral qBSE into a matrix equation, the scattering amplitude can be obtained, and the molecular states can be searched for as the poles of the amplitude. The parameters of the Lagrangians in the current work are chosen as the same as those in our previous study of the hidden-charm pentaquarks [15,66]. The only free parameters are cutoffs Λ e and Λ r , which are rewritten as a form of Λ r = Λ e = m + α 0.22 GeV with m being the mass of the exchanged meson, which is also introduced into the regularization form factor to suppress large momentum, i. e., the short-range contribution of the exchange as warned in Ref [73]. Hence, in the current work, only one parameter α is involved.

A. Interactions Λ cΛc and Λ c Λ c
In the current work, only S-wave states will be considered. For the two interactions considered, the results with spins S =1 and 0 are shown in Fig. 1. The results suggest bound states are produced from all four channels. The states with spins 1 and 0 have almost the same binding energy, which is consistent with the results in Ref. [21]. The two bound states from the Λ cΛc interaction appear even with an α value below 0, which are smaller than two states for the double-charm Λ c Λ c interaction, which indicts that the Λ cΛc interaction is more attractive than the Λ c Λ c interaction due to different contributions from the meson exchanges. Since the Λ baryon is isoscalar, the interactions Λ cΛc and Λ c Λ c arises from the σ and ω exchanges. In the Λ cΛc interaction, both σ and ω exchanges provide attraction. However, in the Λ c Λ c interaction, the ω exchange is repulsive, which reduces the attraction. Different from the isoscalar Λ c baryon, the Σ ( * ) c baryon is an isovector particle. Hence, more channels will be involved in certain interaction. In Fig. 2, the bound states from the Σ cΣc interaction and their double-charm partners are presented. Here, the siospin I can be 0, 1, or 2, and the spin S =0 or 1, which leads to six channels for each interaction. As shown in Fig. 2, bound states are produced in all channels, but with different behaviors with the variation of parameter α. For the isoscalar hidden-charm Σ cΣc system, the bound states are produced at an α value below 0, and the binding energies increase rapidly with the increase of α value. As shown in Table II, the strong attraction is from the ρ exchange with a large flavor factor −1. The corresponding double-charm partners appear at larger α value, which means that it is less attractive than the hidden-charm case due to the different signs for π and ω exchanges. The binding energies for states with different spins are almost the same. For the states with I = 1, the binding energies at an α value of 0 are smaller than those with I = 0. As shown in Table II, the flavor factors for ρ and π exchanges are half of those for I = 0, which leads to less attraction. For the states with I = 2, the attraction becomes weaker due to reversing the signs of the ρ and π exchanges. The hidden-charm states are produced at a small α value, and binding energies increase to a value larger than 30 GeV very quickly at an α value of about 0.7. However, the binding energies of their double-charm partners appear at α value of about 0.2, and increase relatively slowly. The binding energies of the states produced from the Σ * spins S =0, 1, 2, and 3, due to the flavor factors are the same as those for the Σ cΣc system, the results are similar to the results in Fig. 2. For the hidden-charm system with I = 0, there are three states with spins J = 1, 2, and 3 producing at an α values of about 0. As in the case of Σ cΣc , the attractions for the corresponding double-charm systems are weaker than the hidden-charm systems. The hidden-charm bound states with I = 1 appear at an α value of about 0, and the binding energies increase to 30 MeV at α value about 0.7. The hidden-charm states with I = 2 appear at an α value little larger than 0 while their double-charm partners appear at α value of 0.5 or larger. Generally speaking, the attractions of Σ * cΣ * c /Σ * c Σ * c interaction are a little weaker than the case of Σ cΣc /Σ c Σ c interaction.
The results for the Σ cΣ * c and Σ c Σ * c interactions are presented in Fig 4. For the hidden-charm states, there are two G parities, G = ±1, which do not involve in the double-charm sector. For the hidden-charm systems with I=0, the bound states appear at α value a little below 0, and increase with the increase of the parameter α to 30 MeV at α value of about 1. For their double-charm partners, the bound states appear at an α value about 0, and the binding energies increase more slowly than the hidden-charm states. In the case with I=1, the states appear at an α value of about 0, and increase to 30 MeV at an α value about 1.2. The hidden-charm states with I = 2 appear at α value of about 0, which is smaller than these for the double-charm states, about 0.5.  Fig. 5. Here, the S-wave states with spin J = 0 and 1 are considered. Since the Λ c /Λ c baryon is isoscalar, the isospin only can be 1, and the G parity will involve in the hidden charm sector. Due to the same flavor factors, the results of systems with Σ c and Σ * c are similar. The hidden-charm states are first produced at an α value a little below 0, and the binding energies increase to 30 GeV at an α value about 1. The double-charm states appear at an α value a litter larger and the binding energies increase slowly, reach 30 MeV at an α value of about 2.

IV. SUMMARY
In the current work, the study of the molecular states from interactions of charmed baryons is performed. The hidden-charm systems Λ cΛc , Σ ( * ) cΣ ( * ) c , and Λ c Σ ( * ) c , as well as their double-charm partners, are considered in the calculation. With the help of the Lagrangians in heavy quark limit and with chiral symmetry. The potential kernels are constructed in a one-boson-exchange model, and inserted into the qBSE to search the bound states.
The calculation suggests that the attractions widely exist in the systems of two charmed baryons. For the Λ cΛc interaction, the bound states are produced with spin parities J P =0 − and 1 − , and their double-charm partner can be produced with a binding energies smaller than 30 MeV in a larger range of the parameter α. Due to the same favor factors for the Σ cΣc , Σ * cΣ * c , and Σ cΣ * c interactions, the binding energies for these three interactions behave in a similar manner. The most strong attraction can be found in the case with I = 0 for both hiddencharm and doubly-charm cases due to the large ρ exchange as suggested by its flavor factor, which is consistent with the results in Ref. [23,61]. For the interactions Λ cΣ ( * ) c and Λ c Σ ( * ) c , all bound states produced are relatively stable, has a binding energy below 30 MeV in a large range of α value. Generally speak, the interactions of two charmed baryons are attractive, and many bound states are produced. However, only a few candidates, such as Y(4630), were reported in experiment. More experiment studies about these states are suggested though the processes including the nucleon-nucleon collision at LHC and nucleon-antinucleon collision atPANDA.