Possible charmed-strange molecular dibaryons

In this work, we systematically investigate the dibaryons with charm number $C$=1 and strangeness number $S$=$\pm$ 1 from the interactions of a charmed baryon and a strange baryon $\Lambda_c\Lambda$, $\Lambda_c\Sigma^{(*)}$, $\Sigma_c^{(*)}\Lambda$, and $\Sigma^{(*)}_c\Sigma^{(*)}$, and corresponding interactions of a charmed baryon and an antistrange baryon $\Lambda_c\bar{\Lambda}$, $\Lambda_c\bar{\Sigma}^{(*)}$, $\Sigma^{(*)}_c\bar{\Lambda}$, and $\Sigma^{(*)}_c\bar{\Sigma}^{(*)}$. With the help of the effective Lagrangians with $SU(3)$, heavy quark, and chiral symmetries, the potentials of the interactions considered are constructed by light meson exchanges. To search for the possible molecules, the quasipotential Bethe-Salpeter equation with the interaction potential kernel is solved to find poles from scattering amplitude. The results suggest that attractions widely exist in charmed-strange system with $C$=1 and $S$=$-1$. The $S$-wave bound states can be produced from most of the channels. Few bound states are also produced from the charmed-antistrange interactions. Couple-channel effect are considered in the current work to discuss the couplings of the molecular states to the channels considered. More experimental research for these charmed-strange dibaryons are suggested.


I. INTRODUCTION
In the last two decades, a growing number of exotic particles have been discovered in experiment. It is hard to put these exotic particles into the conventional quark model and their inner structures are still under debate. It implies other interpretations such as molecular states, compact multiquarks, hybrids and some nonresonant state interpretations. Inspired by the observation that many exotic particles were observed near the thresholds of two hadrons, the molecular state picture, which is a shallow bound state of two or more hadrons, naturally becomes a popular interpretation of the exotic particles.
All the time, it is a big challenge to find possible dibaryon molecules, which is an important part of the spectrum of the hadronic molecular states. The well-known deuteron can be seen as a dibaryon molecular state composed of two nucleons. Jaffe suggested another famous dibaryon, H dibaryon, which is a bound state of the ΛΛ system with [uuddss] configuration [1]. Actually, the idea of possible dibaryon molecules can go back to 1964. Dyson and Xuong predicted dibaryon states based on the SU(6) symmetry [2]. In their prediction, the mass of dibaryon ∆∆(D 03 ) was 2376 MeV, and the mass of dibaryon N∆(D 21 , D 21 ) was 2176 MeV. The dibaryon D 03 predicted has a mass surprisingly close to the later discovery of d * (2380) at WASA [3]. The resonant structure was studied in many theoretical methods [4][5][6][7][8][9][10], including studies of assigning it as a molecular state from the ∆∆ interaction [7]. However, such an assumption leads to a binding energy of about 80 MeV, which tends to assign it as a compact hexaquark rather than a bound state of two ∆ baryons. The peak structure was suggested to be a triangle singularity in the last step of the reaction [11,12], which can be traced back to early work in Ref. [13]. The WASA-at-COSY Collaboration reported a hint of an isotensor dibaryon with quantum numbers I J P = 2(1 + ) with a mass of 2140 MeV [14], * Corresponding author: junhe@njnu.edu.cn which is slightly below the N∆ threshold. In our previous work [15], we studied the possible S-wave molecular states from the N∆ interaction within the quasipotential Bethe-Salpeter equation (qBSE) approach. The results suggest a D 21 bound state can be produced from the interaction and there may exist two more possible D 12 and D 22 states with smaller binding energies. More theoretical and experimental works are required to clarify existence of these states and the origin of the experimentally observed states.
There also exist some studies about the molecular state with a baryon and an antibaryon. Such molecular states can carry quantum numbers as a meson. For example, the X(2239) [16] and η(2225) [17] have mass of 2239 MeV and 2220 MeV, respectively, which are very close to a pair of baryon Λ and antibaryonΛ. Hence, they are good candidates for hidden-strange baryon ΛΛ molecular states. In Ref. [22,23], two bound states with spin parities 1 − and 0 − from the ΛΛ interaction were assigned to the X(2239) and the η(2225), respectively. There are also many investigations to interpret the X(2239) and the η(2225) in other picutres [18][19][20][21].
In the heavy flavor sector, the hidden-charm and doublecharm dibaryons also enter the view of the community of exotic hadrons even though there are not many relevant experimental results right now. It is not difficult to understand that heavy quark systems are more likely to have attraction and form bound states in the molecular state picture due to the reduction of the kinetic of the systems resulting from the large masses of the heavy quark systems. Up to now, some theoretical investigations have been performed to look for possible bound states composed of two heavy baryons or a pair of heavy baryon and antibaryon within the constituent quark model [24][25][26], one-boson-exchange model [27][28][29], a quark level effective potential [30], a chromo-magnetic interaction model [31]. In Ref. [32], we performed a systematic study of possible molecular states composed of two charmed baryons. The results suggest that strong attractions exist in both hidden-charm and double-charm systems and bound states can be produced in most of the systems. All these theoretical results support the existence of bound states of heavy flavor dibaryons.
The work is organized as follows. After introduction, the potential kernels of charmed-strange baryon systems are presented, which is obtained with the help of the effective Lagrangians with S U(3), heavy quark, and chiral symmetries. And the qBSE approach will be introduced briefly. In Section III, the results for the molecular states from the charmed-strange interactions are presented with both single and coupled channel calculations. The Section IV contributes to posbbile molecular states from the charmed-antistrange interactions. In Section V, discussion and summary are given.

II. THEORETICAL FRAME
To study the charmed-strange systems considered in the current work and the couplings between different channels, the potential will be constructed within the one-bosonexchange model. The exchanges by peseudoscalar mesons P, vector mesons V and scalar meson σ will be considered. Hence, the Lagrangians depicting the couplings of charmed or strange baryons with light mesons are required.

A. Relevant Lagrangians
For the strange part, we consider the exchange of π, η, ρ, ω, and σ mesons with the strange baryons Λ, Σ and Σ * respectively. In the current work, the φ exchange does not contribute due to the suppression by the OZI rule. For the former four mesons, the interaction can be described by the effective Lagrangians with SU(3) and chiral symmetries [44,45]. The explicit forms can be written as, where V µν = ∂ µ V ν − ∂ µ V µ , and the coupling constants can be determined by the SU(3) symmetry [15,44,46,47]. The SU(3) relations and the explicit values of coupling constants are listed in Table I. For the coupling of strange baryons with the scalar meson σ, the Lagrangians are [22] L BBσ = −g BBσB σB, The different choices of the mass of σ meson from 400 to 550 MeV affects the result a little, which can be smeared by a small variation of the cutoff. In this work, we adopt a σ mass of 500 MeV. In general, we choose the coupling constants g BBσ and g B * B * σ as the same value as g BBσ = g B * B * σ = 6.59 [22]. For the charmed part, the Lagrangians for the couplings between the charmed baryons and exchanged mesons can be constructed under the heavy quark and chiral symmetry [48][49][50][51]. The explicit forms of the Lagrangians can be written as, (9) where S µ ab is composed of Dirac spinor operators as, and the charmed baryon matrices are defined as, The P and V are the pseudoscalar and vector matrices as, The coupling constants in the above Lagrangians are listed in Table II, which are cited from the literatures [52][53][54][55]. The parameters and coupling constants. The λ, λ S ,I and f π are in the unit of GeV −1 . Others are in the unit of 1.

B. Potential kernel of interactions
With the above Lagrangians for the vertices, the potential kernel can be constructed in the one-boson-exchange model with the help of the standard Feynman rule as in Refs. [56,57]. The propagators of the exchanged light mesons are defined as, where the form factor f i (q 2 ) is adopted to compensate the off-shell effect of exchanged meson, which is in form of e −(m 2 e −q 2 ) 2 /Λ 4 e with m e and q being the mass and momentum of the exchanged light mesons, respectively.
In this work, we do not give the explicit form of the potential due to the large number of channels to be considered. Instead, we input the vertices Γ and the above propagators P into the code directly and the potential can be constructed with the help of the standard Feynman rule as [56], The I P,V,σ is the flavor factors of the certain meson exchange as listed in Table III. The interaction of charmed-antistrange interactions will be rewritten to the charmed-strange interactions by the well-known G-parity rule [58,59], The G parities of the exchanged mesons i are left as a ζ i factor. Since π and ω mesons carry odd G parity, ζ π , and ζ ω should equal −1, and others equal 1.
The Bethe-Salpeter equation is widely used to treat two body scattering. The potentials obtained above can be taken as Bethe-Salpeter equation under the ladder approximation, which describes the interaction well. In order to reduce the 4-dimensional Bethe-Salpeter equation to a 3-dimensional equation, we adopt the covariant spectator approximation, which keeps the unitary and covariance of the equation [60]. In such treatment, one of the constituent particles, usually heavier one, is put on shell, which leads to a reduced propagator for two constituent particles in the center-of-mass frame as [57,61], As required by the spectator approximation, the heavier particle (h represents the charmed baryons) satisfies p ′′0 l for the lighter particle (remarked as l) is then W − E h (p ′′ ). Here and hereafter, the value of the momentum in center-of-mass frame are defiend as p = |p|.
After the covariant spectator approximation, the 3dimensional Bethe-Saltpeter equation can be reduced to a 1-dimensional equation with fixed spin-parity J P by partial wave decomposition [57], (16) where the sum extends only over nonnegative helicity λ ′′ . The partial wave potential in 1-dimensional equation is defined with the potential of the interaction obtained in the above as where η = PP 1 P 2 (−1) J−J 1 −J 2 with P and J being parity and spin for the system. The initial and final relative momenta are chosen as p = (0, 0, p) and A regularization is usually introduced to avoid divergence, when we treat an integral equation. In the qBSE approach, we usually adopt an exponential regularization by introducing a form factor into the propagator as f (q 2 ) = e −(k 2 l −m 2 l ) 2 /Λ 4 r , where k l and m l are the momentum and mass of the lighter baryon. In the current work, the relation of the cutoff Λ r = m + α r 0.22 GeV with m being the mass of the exchanged meson is also introduced into the regularization form factor as in those for the exchanged mesons. Cutoffs Λ e and Λ r play analogous roles in the calculation of the binding energy. For simplification, we set Λ e = Λ r in the calculations.
The partial-wave qBSE is a one-dimensional integral equation, which can be solved by discretizing the momenta with the Gauss quadrature. It leads to a matrix equation of a form M = V + VGM [57]. The molecular state corresponds to the pole of the amplitude, which can be obtained by varying z to satisfy |1 − V(z)G(z)| = 0 where z = E R − iΓ/2 being the exact position of the bound state.

III. MOLECULAR STATES FROM CHARMED-STRANGE INTERACTIONS
First we consider the charmed-strange interactions with C=1, S =−1. Only S -wave states are considered in singlechannel calculation. The results for scalar, vector and tensor isospins are presented in the followings.

A. Isoscalar charmed-strange molecular states
In the current model, we have only one free parameter α. In the following, we vary the free parameter in a range of 0-5 to find the bound states with binding energy smaller than 30 MeV. The single-channel results of charmed-strange interaction with isospin I = 0 are illustrated in Fig. 1. For the isoscalar charmed-strange interaction, we consider twelve channels, Λ c Λ with spin parities J P = (0, 1) + , Σ c Σ with (0, 1) + , Σ c Σ * and Σ * c Σ with (1, 2) + , and Σ * c Σ * with (0, 1, 2, 3) + . As shown in Fig. 1, the isoscalar single-channel calculation suggests that except Σ c Σ interaction with 0 + , all other eleven isoscalar channels can produce bound state. Their binding energies increase with increaseing the free parameter α. The bound states from the interactions Λ c Λ appear at α value of about 0, and increase rapidly to 30 MeV at α value of about 1. The bound states from the Σ c Σ interaction with 1 + and the Σ * c Σ interaction with 2 + also appear at an α of about 1, and increase relatively slowly to 30 MeV at α value of about 3. The attraction of the Σ * c Σ interaction with 1 + is weak, and produce a bound state at an α value of about 3.5. The variation tendencies of the binding energies of two states with different spin parities from the Λ c Λ interaction are analogous.
The channels with the same quantum numbers can couple to each other, which will make the poles move in the complex energy plane. In Table IV,  The results are similar to those from the single-channel calculations. For the states above the lowest threshold, with the coupled-channel effect, the pole of a bound state will deviate from the real axis and acquire an imaginary part, which corresponds to the width as Γ = −2Imz. The results suggest small width produced from the couplings with the channels considered.
The two-channel calculations are also performed to show the strength of the coupling between the molecular states and the corresponding decay channels. Larger variation of the mass and value of width reflect stronger couplings. In the fourth to seventh columns, the results for the couplings to labeled channels are presented. For the state with (0, 1) + near Σ * c Σ * thresholds, the strong couplings to the Σ c Σ channel can be found based on the mass and width, while the states with (2, 3) + couples strongly to the Σ c Σ * channel. No obvious strongly coupled channel can be found for the states near the Σ c Σ * , Σ * c Σ and Σ c Σ thresholds. Since the Λ c Λ interaction has the lowest threshold, no width will be acquired from the coupled-channel calculation, and the results are not presented due to absence of the decay channels.

B. Isovector charmed-strange molecular states
For isovector charmed-strange interaction states, we consider eighteen channels, Λ c Σ , Σ c Λ and Σ c Σ with (0, 1) + , Σ * c Λ, Λ c Σ * , Σ * c Σ and Σ c Σ * with (1, 2) + , and Σ * c Σ * with (0, 1, 2, 3) + . The results are shown in the following Fig. 2. From Fig. 2, the results suggest that bound states are produced from all eighteen channels and all bound states are produced at a value of α less than 1. It is still worth mentioning that at a α value of about 4 or larger, the repulsion from the ω meson and η meson exchanges increase faster than attractions of other mesons, which make the states with 0 + and 1 + of Σ * c Σ * shallower. Also, the variation tendencies of the binding energies of states with different spin parities from Λ c Σ ( * ) or Σ ( * ) Λ interaction are still analogous. One can generally find that the binding energies of states with the smaller spin increase more rapidly for the isovector bound states.  Table IV.
In Table V, we present the coupled-channel results of isovector charmed-strange interactions. There are four isovector states near the Σ * c Σ * threshold. The width for the 0 + state is mainly from the Σ c Λ channel. For the state with 1 + , large couplings can be found in the channels Λ c Σ * and Σ c Σ * . The state with 2 + strongly couples to the channels Λ c Σ * and Σ * c Λ. The states with 3 + has strong couplings to the channels Σ * c Λ, Λ c Σ * , and Σ c Λ. Near the Σ c Σ * threshold, there exist two poles with spin parities 1 + and 2 + . For the state with 1 + , the Λ c Σ * is found to be its dominant decay channel while no obvious dominant channel can be found for the state with 2 + . For two states near the Σ * c Σ threshold, there is also no obvious dominant channel for the 1 + states while the 2 + state has the strongest coupling to channel Λ c Σ * . For two Λ c Σ * states, the dominant channel shifts to the Σ * c Λ channel. For other states, the widths are very small partly due to few channels below their masses.

C. Isotensor molecular states
In Fig. 3, the isotensor bound states from the charmedstrange interactions in a single-channel calculation are presented. Channels considered include Σ c Σ with (0, 1) + , Σ * c Σ and Σ c Σ * with (1, 2) + , and Σ * c Σ * with (0, 1, 2, 3) + . In the considered range of parameter α, bound states are produced from all channels. However, two Σ c Σ and Σ * c Σ bound states with (0, 1) + appears at α values of about 0, and increase rapidly to 30 MeV at α values less than 1 while the states from the interactions Σ c Σ * and Σ * c Σ * appears at an α values of 1.5 or larger and reach 30 MeV at α values about 3. Such results disfavor the coexistence of the these states. In Table. VI, we present the coupled-channel results of isotensor charmed-strange interactions. Though the overall coupled-channel results with all four channels, the mass of the Σ * c Σ * state with the 3 + state becomes obviously smaller, which leads to a larger parameter α to produce the state, about 2.5, compared with the single-channel calculation in Fig. 3, which is mainly from coupling to the Σ * c Σ channel. From the twochannel calculations listed in the fourth to sixth columns, the isotensor Σ * c Σ * states with (0, 1) + have the strongest coupling to the Σ c Σ channel, while states with (2, 3) + states prefer the Σ c Σ * channel. For the Σ c Σ * states with (1, 2) + , the Σ * c Σ channel is dominant to produce their total width. The coupling effect has no effect on the width for the Σ * c Σ states with (1, 2) + , while the degeneration in mass disappears for the two states.  Table IV.

IV. MOLECULAR STATES FROM CHARMED-ANTISTRANGE INTERACTIONS
Now we turn to the charmed-antistrange systems with C=1 and S =1 by replacing the strange baryons with their antibaryons in the corresponding systems discussed above by the G parity rule. We still consider S -wave states with scalar, vector and tensor isospins in single-channel calculation. The coupled-channel calculation with all channels and two channels will be performed, and the isoscalar, isovector and isotensor results will be shown in the followings.

A. Isoscalar molecular states
For the isoscalar charmed-antistrange interaction, there also exist twelve channels, Λ cΛ with spin parities J P = (0, 1) − , Σ cΣ with (0, 1) − , Σ cΣ * and Σ * cΣ with (1, 2) − , and Σ * cΣ * with (0, 1, 2, 3) − . Compared with the charmed-strange interactions, where eleven bound states are produced, only five states were produced here from the twelve channels as shown in Fig. 4. There is no bound states for the channels Λ cΛ , Σ cΣ and Σ * cΣ . It is mainly due to the different signs of flavor factors for the π and ω meson exchanges in Table. III according to the G-parity rule [58,59], which means that the attraction and repulsion are opposite for these two exchanges.
The coupled-channel results are presented in Table VII. The mass of bound states Σ * cΣ * with (1, 2, 3) − decrease obviously after including all channels. And these states acquire widths of several to tens MeV with binding energies of several MeV, which is mainly from the Σ cΣ * channel. For the states with (2, 3) − , considerable large couplings can be found in the channel Σ * cΣ . For two states with (1, 2) − near the Σ cΣ * threshold, the strongest coupling channel is the Σ cΣ channel while large couplings are also found in the channels Σ * cΣ and Λ cΛ for state with 1 − .

B. Isovector molecular states
In Fig. 5, the binding energies of all bound states produced from isovector charmed-antistrange interactions are presented. The single-channel calculation suggests that only seven bound states are produced from eighteen channels considered. The Λ cΣ * states with (1, 2) − are almost degenerate, which all appear at α values of about 2.0, and its binding energy increase to 30 MeV at α values of about 3. The bound states from the Σ cΣ * interaction with (1, 2) − appears at α values of about 0.5 and 2, respectively. The three  The coupled-channel results are presented in Table VIII. In the isovector case, the widths of the Σ * cΣ * state with (1, 2, 3) − increase very rapidly to about 7 MeV or larger with the increase of the parameter α. The state with 1 − has strong coupling to the channels Σ cΣ * , Σ * cΛ , and Λ cΣ . The state with 2 − couples to channels Σ cΣ * , Σ * cΣ , and Σ * cΛ . For the states with 3 − , except the channels Σ cΛ and Λ cΣ , other channels have strong couplings. For the two Λ cΣ * states with (1, 2) − , the Λ cΣ channel is the dominant channel to produce their total widths. In Fig. 6, the binding energies of all bound states produced from isotensor charmed-antistrange interactions are presented. There are only five bound states can be produced from ten channels considered within the reasonable range of cutoffs. The Σ cΣ state with 0 − can be produced at an α value of about 3.0. The Σ cΣ * state with 1 − and the Σ * cΣ * states with (0, 1) − can be produced at an α value of about 2.5, while the Σ cΣ * with 1 − appears at an α value of about 4.5. Generally speaking, the α value to produce these states are considerable larger than these to produce most of the bound states in the above, which suggest weak attraction in these channels. The possibility of existence of such states is also very low. The coupled channel results are presented in Table IX. It can be seen that large α values are still required to produce the poles. For the two Σ * cΣ * states with (0, 1) − , the dominant channel is the Σ cΣ * channel. The Σ cΣ * state with 1 − strongly couples to the Σ * cΣ channel. Though there exists only one channel Σ cΣ below the Σ * cΣ threshold, the Σ * cΣ state with 1 − acquire considerable large width.

V. SUMMARY AND DISCUSSION
In this work, we systematically study the charmed-strange molecular states with quantum numbers C=1 and S=±1, in a qBSE approach together with the one-boson-exchange model. The potential kernels are constructed with the help of the effective Lagrangians with S U(3), chiral and heavy quark symmetries. With the exchange potential obtained, the Swave bound states are searched for as the pole of the scattering amplitudes.
The attractions widely exist in the charmed-strange interactions. In the current work, we consider all S -wave interactions of a charmed baryon and a strange baryon Λ c Λ, Λ c Σ ( * ) , Σ ( * ) c Λ, and Σ ( * ) c Σ ( * ) , which results in 40 channels with different spin parities. Among these channels, 39 bound states are produced in the range of the parameter considered in the current work. Among these bound states, 32 states can be produced from the interactions in a range of α value from 0 to 1. For channels that have two or three different isospins, the bound state with smaller isospin, the binding energies tend to larger than the state with larger isospin. The coupled-channel calculations do not change the conclusion from the singlechannel calculations, and the possible strong couplings are also suggested.
Compared with the charmed-strange interactions, fewer states can be produced in the charmed-antistrange interactions. There are also 40 S -wave channels as in the charmedstrange sector. However, only 17 states can be produced. Besides, the attractions of many states are weak as suggested by the α values to produce these states. Among 17 states produced, only 4 states can be produced with an α value smaller than 1. For the states from charmed-antistrange interactions, the quark and antiquark in the systems can be annihilated, which results in quantum numbers as a charmstrange meson D s . It also makes it easy to be found in experiment. Hence, we suggest the experimental research for such states