Possible molecular dibaryons with csssqq quarks and their baryon–antibaryon partners

In this work, we systematically investigate the charmed–strange dibaryon systems with csssqq quarks and their baryon–antibaryon partners from the interactions Ξc(′,∗)Ξ(∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varXi ^{(',*)}_{c}\varXi ^{(*)}$$\end{document}, Ωc(∗)Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega ^{(*)}_c\varLambda $$\end{document}, Ωc(∗)Σ(∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega ^{(*)}_c\varSigma ^{(*)}$$\end{document}, ΛcΩ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda _c\varOmega $$\end{document} and Σc(∗)Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varSigma ^{(*)}_c\varOmega $$\end{document} and their baryon–antibaryon partners from interactions Ξc(′,∗)Ξ¯(∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varXi ^{(',*)}_{c}\bar{\varXi }^{(*)}$$\end{document}, Ωc(∗)Λ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega ^{(*)}_c\bar{\varLambda }$$\end{document}, Ωc(∗)Σ¯(∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega ^{(*)}_c\bar{\varSigma }^{(*)}$$\end{document}, ΛcΩ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda _c\bar{\varOmega }$$\end{document} and Σc(∗)Ω¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varSigma ^{(*)}_c\bar{\varOmega }$$\end{document}. The potential kernels are constructed with the help of effective Lagrangians under SU(3), heavy quark, and chiral symmetries to describe these interactions. To search for possible molecular states, the kernels are inserted into the quasipotential Bethe–Salpeter equation, which is solved to find poles from scattering amplitude. The results suggest that 36 and 24 bound states can be found in the baryon–baryon and baryon–antibaryon interactions, respectively. However, much large values of parameter α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} are required to produce the bound states from the baryon–antibaryon interactions, which questions the existence of these bound states. Possible coupled-channel effect are considered in the current work to estimate the couplings of the molecular states to the channels considered.


Introduction
As an important type of exotic hadrons, the dibaryons with baryon quantum number B = 2 attract much attention from the hadron physics community.In fact, one type of the exotic hadrons proposed earliest in the literature is the dibaryons predicted by Dyson and Xuong in 1964 based on the SU (6) symmetry almost at the same time of the proposal of the quark model [1].The WASA-at-COSY collaboration reported a new resonance d * (2380) with quantum number I(J P ) = 0(3 + ), a mass of about 2370 MeV, and a width of about 70 MeV in the process pp → dπ 0 π 0 at [2].Soon after the observation of the d * (2380), it is related to the dibaryon a Corresponding author: junhe@njnu.edu.cnpredicted [3,4] while there is still other interpretations, such as a triangle singularity in the last step of the reaction in a sequential single pion production process [5].More experimental and theoretical works are still required to clarify its origin.
These early proposed dibaryons are exotic hadrons in the light flavor sector.In the past decades, many candidates of exotic states in charmed sector, such as hidden-charm tetraquarks and pentaquarks, have been observed in experiment, for example, the X(3872) and Z c (3900) [6][7][8][9][10], and a series of hidden-charm pentaquarks P c [11][12][13][14].These states were observed near the thresholds of two charmed hadrons.Hence, it is natural to interpret them as the molecular states produced from interactions of a pair of charm and anticharm hadrons.Motivated by the observations of these states, theorists expect that there may exist dibaryon molecules composed of two heavy baryons.Due to large masses of the heavy baryons, the kinetic energy of a dibaryon system is reduced, which makes it easier to form a bound state.Possible hidden-charm and double-charm dibaryons were investigated in different approaches [15][16][17][18][19][20][21][22][23].These results suggest that attraction may exist between a charmed baryon and an anticharmed or charmed baryon by light meson exchanges, which favors the existence of hidden-charm dibaryon molecular states and their double-charm partners.
In addition to the above hidden-charm and double-charm states, some charmed-strange states were also observed these years, and taken as the candidates of molecular states of a charmed meson and a strange meson in the literature.As early as 2003, the BaBar collaboration reported a narrow peak D * s0 (2317) near the DK threshold [24], and later confirmed at CLEO and BELLE [25,26].The CLEO collaboration also observed another narrow peak, the D s1 (2460) near the D * K threshold [26].These states can not be well put into the conventional quark model with a charmed and an antistrange quark.Since these charmed-strange states are very c Σ( * ) , Λ c Ω and Σ ( * )  c Ω.The coupling between different channels will also be included to make a coupled-channel calculation to obtain the scattering amplitude by solving the qBSE.To achieve this aim, the potential will be constructed by the light meson exchanges.The Lagrangians are required to obtain the vertices, and will be given below.

Relevant Lagrangians
For the couplings of strange baryons with light mesons, we consider the exchange of pseudoscalar mesons P (π, η, ρ), vector mesons V (ω, φ, K, K * ), and σ mesons.For the former seven mesons, the vertices can be described by the effective Lagrangians with SU(3) and chiral symmetries [48,49].The explicit the effective Lagrangians reads, where m p,V is the mass of the pseudoscalar or vector meson.B ( * ) is the field of the strange baryon.
The coupling constants can be determined by the SU(3) symmetry [48,[50][51][52] with the coupling constants for the nucleon and ∆.The SU(3) relations and the explicit values of coupling constants are calculated and listed in Table 1.
For the couplings of strange baryons with the scalar meson σ, the Lagrangians read [53] 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 the calculation.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 [53].
For the couplings of charmed baryons with light mesons, the Lagrangians can be constructed under the heavy quark and chiral symmetries [54][55][56][57].The explicit forms of the Lagrangians can be written as, 3 (4α P − 1)g BBP 0.34 g ΞΣK −g BBP −0.98 g BB * V −6.54 where m B,B, B3 ,B 3 is the mass of the charmed baryon.S µ ab is composed of the Dirac spinor operators, and the charmed baryon matrices are defined as, The P and V are the pseudoscalar and vector matrices as, The parameters in the above Lagrangians are listed in Table 2, which are cited from the literature [58][59][60][61].
Table 2 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.

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.[62,63].The propagators of the exchanged light mesons are defined as, where the form factor f i (q 2 ) is adopted to reflect 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 mesons, respectively.In this work, we still 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 obtained from the Lagrangians and the above propagators P into the code directly.The dibaryon systems potential can be constructed with the help of the standard Feynman rule as [62], (13) where I P,V,σ is the flavor factors of the certain meson exchange, which are listed in Table 3.The interaction of their baryon-antibaryon partners interactions will be rewritten to the charmed-strange interactions by the well-known Gparity rule V = i ζ i V i [64,65].The G parities of the exchanged mesons i are left as a ζ i factor.Since π, ω and φ mesons carry odd G parity, the ζ π , ζ ω and ζ φ should equal −1, and others equal 1.

The qBSE approach
The Bethe-Salpeter equation is a 4-dimensional relativistic integral equation, which can be used to treat two body scattering.In order to reduce the 4-dimensional Bethe-Salpeter equation to a 3-dimensional integral equation, we adopt the covariant spectator approximation, which keeps the unitary and covariance of the equation [66].In such treatment, one of the constituent particles, usually the heavier one, is put on shell, which leads to a reduced propagator for two constituent particles in the center-of-mass frame as [63,67], Table 3 The flavor factors I e for charmed-strange interactions.The values for charmed-antistrange interactions can be obtained by Gparity rule from these of charmed-strange interactions.The I σ should be 0 for coupling between different channels.
As required by the spectator approximation adopted in the curren work, the heavier particle (h represents the charmed baryons) satisfies . The 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 is defined as p = | p|.
Then the 3-dimensional Bethe-Saltpeter equation can be reduced to a 1-dimensional integral equation with fixed spinparity J P by partial wave decomposition [63], 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 p ′ = (p ′ sin θ, 0, p ′ cos θ).The d J λλ ′ (θ) is the Wigner d-matrix.Here, 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 where k l and m l are the momentum and mass of the lighter one of and 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.The cutoff 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 [63].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.

Single-channel results
With previous information, the explicit numerical calculations will be performed on the systems mentioned above.In the current model, we have the only one free parameter α.In the following, we vary the free parameter in a range of 0-5 to find the S-wave bound states with binding energy smaller than 30 MeV.In this work, we consider all possible channels with csssqq quarks, that is, c Ω and their baryon-antibaryon partners can not be considered in single-channel calculations due to the lack of exchanges of light mesons in the one-boson-exchange model considered in the current work.However, these channels will be considered in the later couple-channel calculations.Based on the quark configurations in different hadron clusters, these single-channel interactions can be divided into two categories: the Ξ ( ′ , * ) c Ξ ( ′ , * ) and Ω ( * ) c Λ or Ω ( * ) c Σ ( * ) and their baryon-antibaryon partners.Ξ( * ) state with 1 − .Again, one can still find that the states with the larger spin are easy to be produced for the isoscalar interactions, while the states with the smaller spin are easy to be produced for the isovector interactions.Still, these baryonantibaryon states are produced at α values around or more than 3, which makes their coexistence less possible.In the following Fig. 3, we present the results of the Ξ ( * ) c Ξ ( * ) and Ξ * c Ξ( * ) systems, in which the charmed baryon belongs to the multiplet B * 6 .The results suggest that bound states can be produced from eighteen interactions.For the baryon-baryon systems, the bound states can be produced from all channels, and appear at α values below 1.5.The curves of two isoscalar Ξ * c Ξ states with (0, 1) + are separated obviously, and their binding energies reach 5 MeV relative slowly at α values about 4.5 and 2, respectively.Besides the two states, other ten states increase with the parameter α to 30 MeV relatively rapidly at α values of about 2.5.Meanwhile, the interaction with the smaller spins have stronger attractions, which is reflected by the binding energies increasing faster with the variation of parameter.For their baryonanibaryon partners, two isoscalar states from the Ξ * c Ξ interaction with 2 − and interaction Ξ * c Ξ * with 3 − , as well as four isovector states from the Ξ * c Ξ interaction with (1, 2) − and the Ξ * c Ξ * interaction with (0, 1) − , can be produced at the cutoff over 2.5.For the systems composed of [css][sqq] and [css][ s q q], there exist interactions Ω ( * ) c Λ, Ω ( * ) c Σ ( * ) and their baryonantibaryon partners, interactions Ω ( * ) c Λ and Ω ( * ) c Σ( * ) .In Fig. 4, we first give the results about the interactions Ω c Λ, Ω c Σ ( * ) , Ω c Λ and Ω c Σ( * ) , in which the charmed baryons belong the multiplet B 6 .Only seven states are produced from those interactions.For the Ω c Λ interaction and its baryonantibaryon partner Ω c Λ with isospin I = 0, only the states that spin J = 1 can be produced at the cutoff about 4.0 and 3.0, respectively.There is no bound state produced from the isovector interaction Ω c Σ with (0, 1) + in the considered range of the parameter α.Two bound states from the Ω c Σ interaction with (0, 1) − appear at α values of about 3.0 and 3.6, respectively.Two bound states from the isovector Ω c Σ * interaction with (1, 2) + appear at α value of about 3.0 and 1.5, respectively, while only an isovector Ω c Σ * state with 1 − can be produced at α value of about 4.8.The states from the baryon-antibaryon interactions are still less likely to coexistence due to the large values of parameter α required to produce the bound states.Σ * are found attractive, and four states with spin parities (0, 1, 2, 3) + and two states with (0, 1) − are produced, respectively.The Ω * c Σ * states with 0 + appear at α value of about 3.5, while the (1, 2, 3) + states all appear at cutoff about 2.0.The two Ω * c Σ * with (0, 1) − is produced at cutoff about 4.6.

Coupled-channel results
In the previous single-channel calculations, many bound states are produced from the considered interactions within allowed range of parameter α.To estimate the strength of the coupling between a molecular state and the corresponding decay channels, we will consider the couple-channel effects.In the coupled-channel calculations, the channels with the same quark components and the same quantum numbers can couple to each other, which will make the pole of the bound state deviate from the real axis to the complex energy plane and acquire an imaginary part.The imaginary part corresponds to the state of the width as Γ = 2Imz.Here, we present the coupled-channel results of the position of bound state as M th − z instead of the origin position z of the pole, with the M th being the nearest threshold.In the above singlechannel calculations, much larger α values are required to produce the bound states from the baryon-antibaryon interactions, which suggests that the possibility of the existence of these states are very low.Hence, in the following coupledchannel calculations, we only consider the baryon-baryon interactions.In the Table.4, we present the coupled-channel results of the isoscalar baryon-baryon interactions, which involve all possible couplings between the channels Ξ ( ′ , * ) c Ξ ( * ) , Λ c Ω and Ω ( * ) c Λ.The poles of full coupled-channel interaction under the corresponding threshold with different α are given in the second and third columns.
Glancing over the coupled-channel results of channels 4, we can find that the real parts of most poles from the coupled-channel calculation are similar to those from the single-channel calculations, and the small widths are acquired from the couplings with the channels considered.However, it has a great impact on the Ξ * c Ξ channel after including the full coupled-channel interactions as suggested by the variation in the mass and width.Compared with single-channel calculations, the masses change significantly, and the widths are much larger.Two-channel calculations are also performed, and the results are presented in the fourth to eleventh columns.For the states near the Ξ ( ′ , * ) c Ξ ( * ) threshold with (0, 1, 2, 3) + , relatively obvious twochannel couplings can be found in the Ξ ′ c Ξ channel.For the states near the Ξ ′ c Ξ * threshold with (1, 2) + , the main twochannel couplings can be found in the Ξ ′ c Ξ channel.For two states near the Ξ c Ξ * threshold with (1, 2) + , the widths from two-channel couplings are both less than 1.0 MeV.For the states near the Ξ * c Ξ threshold with (1, 2) + , the main decay channel are Ω * c Λ, which leads to a width of about a dozen of MeVs and large increase of binding energy.Similarly, the states near the Ξ ′ c Ξ threshold with (0, 1) + have considerable large couplings with the Ω c Λ channel, which leads to obvious increase of mass.For the state near the Ω * c Λ threshold with 1 + , the Ξ c Ξ channel is the dominant channel to produce their total widths.Since the Ω c Λ channel has the second lowest threshold, it can only couple to the Ξ c Ξ channel so that the only two-channel coupling width came from the Ξ c Ξ channel.
The coupled-channel results of isovector baryon-baryon interactions are presented in Table 5.For the isovector states near the Ξ * c Ξ * threshold with (0, 1, 2) + , large couplings can be found in the Ω * c Σ * channel and their binding energies also decrease a little compared with the single-channel results after including the two-channel couplings.Among the states near the Ω * c Σ * threshold with (0, 1, 2, 3) + , there exist some differences between different two-channel couplings.After including the two-channel couplings between the channel Ω * c Σ * and the channels Ξ c Σ( * ) , Λ c Ω and Σ ( * ) c Ω are considered, which leads to 84 channels with different spin parities.
The single-channel calculations suggest that 36 and 24 bound states can be produced from the baryon-baryon and baryon-antibaryon interactions, respectively.Most bound states from baryon-antibaryon interactions are produced at much larger values of parameter α, which suggests that these bound states are less possible to be found in future experiments than corresponding dibaryon states.Such results are consistent with our previous results [47] that fewer states can be produced in the charmed-antistrange interaction than charmed-strange interactions.
Furthermore, the coupling effects on the produced bound states in the single-channel calculations are studied.Since the states from the baryon-antibaryon interactions are less possible to exist, we do not consider these interactions in coupled-channel calculations.For the isoscalar interactions, the coupled-channel calculations hardly change the conclusion from the single-channel calculations, which means that Table 4 The masses and widths of isoscalar baryon-baryon molecular states at different values of α.The "CC" means full coupled-channel calculation.The values of the complex position means mass of corresponding threshold subtracted by the position of a pole, M th − z, in the unit of MeV.The two short line "−−" means the coupling does not exist.The imaginary part shown as "0.0" means too small value under the current precision chosen.
Table 5 The masses and widths of isovector charmed-strange molecular states at different values of α.Other notations are the same as Table 4. the coupled-channel effects are not very significant.However, for the isovector interactions, the coupled-channel effects have obvious effects, which usually cause great variations of binding energy together with considerable widths.Compared with our previous coupled-channel calculations in Refs.[68,69], the coupled-channel effect has obvious large influence on both the real part and imaginary part of poles.It may be related to the constituent hadrons considered in the current work.The systems studied in the current work are composed of a light hadron and a charmed hadron.
Compared with the double-charmed or double-bottom systems, the systems containing light hadrons are usually more unstable.
Generally speaking, the charmed-strange dibaryon systems with csssqq quarks are usually attractive enough to produce bound states, while their baryon-antibaryon partners are less or hardly attractive.Both theoretical and experimental studies are suggested to give more valuable information.

3. 1
Molecular states from interactions Ξ ( ′ , * ) c Ξ ( ′ , * ) and Ξ ( ′ , * ) c Ξ( ′ , * ) First, we consider the interactions Ξ ( ′ , * ) c Ξ ( * ) and Ξ ( ′ , * ) c Ξ( * ) with quark configurations as [csq][ssq] and [csq][ s s q], respectively.The single-channel results for the interactions Ξ c Ξ ( * ) and Ξ c Ξ( * ) , in which the charmed baryon belongs to the multiplet B3, are illustrated in Fig 1.The results suggest that fourteen interactions produce bound states in considered range of parameter α.All eight bound states from the Ξ * c Ξ ( * ) interaction can appear at α values less than 1.The binding energies of the isovector Ξ c Ξ states with (0, 1) + and the isoscalar and isovector Ξ c Ξ * states with (1, 2) + both increase rapidly to 30 MeV at α values of about 1.5, which indicates the strong attraction.However, the binding energies of isoscalar bound states from the Ξ c Ξ interaction with (1, 2) + increase slowly to 20 MeV at α values of about 5. The variation tendencies of the binding energies of the Ξ c Ξ ( * ) states with different spin parities are analogous.Almost all bound states from baryon-antibaryon interactions appear at the α values more than 3 and the isovector Ξ c Ξ * interaction with (1, 2) − can no produce bound state.It suggests that the possibility of the existence of these baryon-antibaryon bound states is relatively low.