Possible charmed-strange molecular dibaryons

Abstract

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.

1 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 $\Lambda \Lambda$ 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 $\Delta \Delta (D_{03})$ was 2376 MeV, and the mass of dibaryon $N\Delta (D_{21}, D_{21})$ 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 $\Delta \Delta$ 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 $\Delta$ 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 $IJ^P=2(1^+)$ with a mass of 2140 MeV [14], which is slightly below the $N\Delta$ threshold. In our previous work [15], we studied the possible S-wave molecular states from the $N\Delta$ 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 $\eta (2225)$ [17] have mass of 2239 MeV and 2220 MeV, respectively, which are very close to a pair of baryon $\Lambda$ and antibaryon ${\bar{\Lambda }}$. Hence, they are good candidates for hidden-strange baryon $\Lambda {\bar{\Lambda }}$ molecular states. In Ref. [22, 23], two bound states with spin parities $1^-$ and $0^-$ from the $\Lambda {\bar{\Lambda }}$ interaction were assigned to the X(2239) and the $\eta (2225)$, respectively. There are also many investigations to interpret the X(2239) and the $\eta (2225)$ in other pictures [18,19,20,21].

In the heavy flavor sector, the hidden-charm and double-charm 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 systems mentioned above are all composed of two light baryons or two heavy baryons. Besides systems above, it is still a blank for the study of systems composed of a light and a heavy baryon. In fact, there exist a large amount of study about the systems with a light meson and heavy meson. For example, the experimentally observed $D^*_{s0}(2317)$, $D_{s1}(2460)$, $X_0(2900)$ are considered as the candidates of the DK molecule [33,34,35,36,37,38], $D^*K$ molecule [36,37,38,39], and ${\bar{D}}^*K^*$ molecule [38, 40,41,42,43], respectively. Hence, it is nature to investigate the systems composed of a charmed and a strange/antistrange baryon. In this work, we systematically investigate the charmed-strange interactions of a charmed baryon and a strange baryon $\Lambda _c\Lambda$, $\Lambda _c\Sigma ^{(*)}$, $\Sigma _c^{(*)}\Lambda$, and $\Sigma ^{(*)}_c\Sigma ^{(*)}$, and corresponding charmed-antistrange interactions of a charmed baryon and an antistrange baryon $\Lambda _c{\bar{\Lambda }}$, $\Lambda _c{\bar{\Sigma }}^{(*)}$, $\Sigma ^{(*)}_c{\bar{\Lambda }}$, and $\Sigma ^{(*)}_c{\bar{\Sigma }}^{(*)}$ in the qBSE approach to study the possibilities of existence of charmed-strange molecules.

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 SU(3), heavy quark, and chiral symmetries. And the qBSE approach will be introduced briefly. In Sect. 3, the results for the molecular states from the charmed-strange interactions are presented with both single and coupled channel calculations. The Sect. 4 contributes to possible molecular states from the charmed-antistrange interactions. In Sect. 5, discussion and summary are given.

2 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-boson-exchange model. The exchanges by pseudoscalar mesons ${\mathbb {P}}$, vector mesons ${\mathbb {V}}$ and scalar meson $\sigma$ will be considered. Hence, the Lagrangians depicting the couplings of charmed or strange baryons with light mesons are required.

2.1 Relevant Lagrangians

For the strange part, we consider the exchange of $\pi$, $\eta$, $\rho$, $\omega$, and $\sigma$ mesons with the strange baryons $\Lambda$, $\Sigma$ and $\Sigma ^*$ respectively. In the current work, the $\phi$ 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,

$$\begin{aligned} {\mathcal {L}}_{BBP}= & {} -\frac{g_{BBP}}{m_P}{\bar{B}}\gamma ^{5}\gamma ^{\mu }\partial _{\mu }PB, \end{aligned}$$

(1)

$$\begin{aligned} {\mathcal {L}}_{BBV}= & {} -{\bar{B}}\left[ g_{BBV}\gamma ^{\mu }-\frac{f_{BBV}}{2m_{B}}\sigma ^{\mu \nu }\partial _{\nu }\right] V_{\mu }B, \end{aligned}$$

(2)

$$\begin{aligned} {\mathcal {L}}_{B^*B^*P}= & {} \frac{g_{B^*B^*P}}{m_P}\bar{B^*}_{\mu }\gamma ^{5} \gamma ^{\nu }B^{*\mu }\partial _{\nu }P, \end{aligned}$$

(3)

$$\begin{aligned} {\mathcal {L}}_{B^*B^*V}= & {} -{\bar{B}}^*_{\tau }\left[ g_{B^*B^*V}\gamma ^{\mu }-\frac{f_{B^*B^*V}}{2m_{B^*}}\sigma ^{\mu \nu }\partial _{\nu }\right] V_{\mu } B^{*\tau }, \end{aligned}$$

(4)

$$\begin{aligned} {\mathcal {L}}_{BB^*P}= & {} \frac{g_{BB^*P}}{m_{P}}{\bar{B}}^{*\mu } \partial _{\mu }PB\;+\; \text {h.c.}, \end{aligned}$$

(5)

$$\begin{aligned} {\mathcal {L}}_{BB^*V}= & {} -i \frac{g_{BB^*V}}{m_{V}}{\bar{B}}^{*\mu }\gamma ^5\gamma ^{\nu } V_{\mu \nu }B\;+\; \text {h.c.}, \end{aligned}$$

(6)

where $V_{\mu \nu }={\partial _{\mu }}\vec {V}_{\nu }-{\partial _{\mu }}\vec {V}_{\mu }$, 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 1.

Table 1 The coupling constants in effective Lagrangians. Here, $g_{BBP}=g_{NN\pi }=0.989$, $g_{BBV}=g_{NN\rho }=3.25$, $g_{{B^*B^*}P}=\sqrt{60} g_{\Delta \Delta \pi }=13.78$, $g_{{B^*B^*}V}=\sqrt{60} g_{\Delta \Delta \rho }=59.41$, $g_{B{B^*}P}=\sqrt{20} g_{N\Delta \pi }=9.48$, $g_{B{B^*}V}=\sqrt{20} g_{N\Delta \rho }=71.69$, $\alpha _{P}=0.4$, $\alpha _{V}=1.15$, $f_{NN\rho }= g_{NN\rho }\kappa _{\rho }$, $f_{\Delta \Delta \rho }= g_{\Delta \Delta \rho }\kappa _{\rho }$ with $\kappa _{\rho }=6.1$, $f_{NN\omega }=0$ [15, 44, 47]

For the coupling of strange baryons with the scalar meson $\sigma$, the Lagrangians are [22]

$$\begin{aligned} {\mathcal {L}}_{BB\sigma }= & {} -g_{BB\sigma }{\bar{B}}\sigma B, \end{aligned}$$

(7)

$$\begin{aligned} {\mathcal {L}}_{B^*B^*\sigma }= & {} g_{B^*B^*\sigma }{\bar{B}}^{*\mu }\sigma B^*_{\mu }. \end{aligned}$$

(8)

The different choices of the mass of $\sigma$ 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 $\sigma$ mass of 500 MeV. In general, we choose the coupling constants $g_{BB\sigma }$ and $g_{B^*B^*\sigma }$ as the same value as $g_{BB\sigma }=g_{B^*B^*\sigma }=6.59$ [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,

$$\begin{aligned} {{{\mathcal {L}}}}_{BB{\mathbb {P}}}= & {} -\frac{3g_1}{4f_\pi \sqrt{m_{{\bar{B}}}m_{B}}}~\epsilon ^{\mu \nu \lambda \kappa }\partial ^\nu {\mathbb {P}}~ \sum _{i=0,1}{\bar{B}}_{i\mu } \overleftrightarrow {\partial }_\kappa B_{j\lambda },\nonumber \\ {{{\mathcal {L}}}}_{BB{\mathbb {V}}}= & {} -i\frac{\beta _S g_V}{2\sqrt{2m_{{\bar{B}}}m_{B}}}{\mathbb {V}}^\nu \sum _{i=0,1}{\bar{B}}_i^\mu \overleftrightarrow {\partial }_\nu B_{j\mu }\nonumber \\&-i\frac{\lambda _S g_V}{\sqrt{2}}(\partial _\mu {\mathbb {V}}_\nu -\partial _\nu {\mathbb {V}}_\mu ) \sum _{i=0,1}{\bar{B}}_i^\mu B_j^\nu ,\nonumber \\ {{{\mathcal {L}}}}_{BB\sigma }= & {} \ell _S\sigma \sum _{i=0,1}{\bar{B}}_i^\mu B_{j\mu },\nonumber \\ {{{\mathcal {L}}}}_{B_{{\bar{3}}}B_{{\bar{3}}}{\mathbb {V}}}= & {} -i\frac{g_V\beta _B}{2\sqrt{2m_{{\bar{B}}_{{\bar{3}}}}m_{B_{{\bar{3}}}}} }{\mathbb {V}}^\mu {\bar{B}}_{{\bar{3}}}\overleftrightarrow {\partial }_\mu B_{{\bar{3}}},\nonumber \\ {{{\mathcal {L}}}}_{B_{{\bar{3}}}B_{{\bar{3}}}\sigma }= & {} \ell _B \sigma {\bar{B}}_{{\bar{3}}}B_{{\bar{3}}},\nonumber \\ {{{\mathcal {L}}}}_{BB_{{\bar{3}}}{\mathbb {P}}}= & {} -i\frac{g_4}{f_\pi } \sum _i{\bar{B}}_i^\mu \partial _\mu {\mathbb {P}} B_{{\bar{3}}}+\mathrm{H.c.},\nonumber \\ {{{\mathcal {L}}}}_{BB_{{\bar{3}}}{\mathbb {V}}}= & {} \frac{g_{\mathbb {V}}\lambda _I}{\sqrt{2 m_{{\bar{B}}}m_{B_{{\bar{3}}}}}}\epsilon ^{\mu \nu \lambda \kappa } \partial _\lambda {\mathbb {V}}_\kappa \sum _i{\bar{B}}_{i\nu } \overleftrightarrow {\partial }_\mu B_{{\bar{3}}}+\mathrm{H.c.},\nonumber \\ \end{aligned}$$

(9)

where $S^{\mu }_{ab}$ is composed of Dirac spinor operators as,

$$\begin{aligned} S^{ab}_{\mu }= & {} -\sqrt{\frac{1}{3}}(\gamma _{\mu }+v_{\mu }) \gamma ^{5}B^{ab}+B^{*ab}_{\mu }\equiv { B}^{ab}_{0\mu }+B^{ab}_{1\mu },\nonumber \\ {\bar{S}}^{ab}_{\mu }= & {} \sqrt{\frac{1}{3}}{\bar{B}}^{ab} \gamma ^{5}(\gamma _{\mu }+v_{\mu })+{\bar{B}}^{*ab}_{\mu }\equiv {{\bar{B}}}^{ab}_{0\mu }+{\bar{B}}^{ab}_{1\mu }, \end{aligned}$$

(10)

and the charmed baryon matrices are defined as,

$$\begin{aligned}&B_{{\bar{3}}}=\left( \begin{array}{ccc} 0&{}\quad \Lambda ^+_c&{}\quad \Xi _c^+\\ -\Lambda _c^+&{}\quad 0&{}\quad \Xi _c^0\\ -\Xi ^+_c&{}\quad -\Xi _c^0&{}\quad 0 \end{array}\right) ,\quad \nonumber \\&B=\left( \begin{array}{ccc} \Sigma _c^{++}&{}\quad \frac{1}{\sqrt{2}}\Sigma ^+_c&{}\quad \frac{1}{\sqrt{2}}\Xi '^+_c\\ \frac{1}{\sqrt{2}}\Sigma ^+_c&{}\quad \Sigma _c^0&{}\quad \frac{1}{\sqrt{2}}\Xi '^0_c\\ \frac{1}{\sqrt{2}}\Xi '^+_c&{}\quad \frac{1}{\sqrt{2}}\Xi '^0_c&{}\quad \Omega ^0_c \end{array}\right) , \nonumber \\&B^*=\left( \begin{array}{ccc} \Sigma _c^{*++}&{}\quad \frac{1}{\sqrt{2}}\Sigma ^{*+}_c&{}\quad \frac{1}{\sqrt{2}}\Xi ^{*+}_c\\ \frac{1}{\sqrt{2}}\Sigma ^{*+}_c&{}\quad \Sigma _c^{*0}&{}\quad \frac{1}{\sqrt{2}}\Xi ^{*0}_c\\ \frac{1}{\sqrt{2}}\Xi ^{*+}_c&{}\quad \frac{1}{\sqrt{2}}\Xi ^{*0}_c&{}\quad \Omega ^{*0}_c \end{array}\right) . \end{aligned}$$

(11)

The ${\mathbb {P}}$ and ${\mathbb {V}}$ are the pseudoscalar and vector matrices as,

$$\begin{aligned}&{{\mathbb {P}}}=\left( \begin{array}{ccc} \frac{\sqrt{3}\pi ^0+\eta }{\sqrt{6}}&{}\quad \pi ^+&{}K^+\\ \pi ^-&{}\quad \frac{-\sqrt{3}\pi ^0+\eta }{\sqrt{6}}&{}\quad K^0\\ K^-&{}\quad {\bar{K}}^0&{}\quad -\frac{2\eta }{\sqrt{6}} \end{array}\right) ,\\&{\mathbb {V}}=\left( \begin{array}{ccc} \frac{\rho ^0+\omega }{\sqrt{2}}&{}\quad \rho ^{+}&{}\quad K^{*+}\\ \rho ^{-}&{}\quad \frac{-\rho ^{0}+\omega }{\sqrt{2}}&{}\quad K^{*0}\\ K^{*-}&{}\quad {\bar{K}}^{*0}&{}\quad \phi \end{array}\right) . \end{aligned}$$

The coupling constants in the above Lagrangians are listed in Table 2, which are cited from the literatures [52,53,54,55].

Table 2 The parameters and coupling constants. The $\lambda$, $\lambda _{S,I}$ and $f_\pi$ are in the unit of GeV$^{-1}$. Others are in the unit of 1

2.2 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,

$$\begin{aligned} P_\mathbb {P,\sigma }(q^2)= & {} \frac{i}{q^2-m_\mathbb {P,\sigma }^2}~f_i(q^2),\nonumber \\ P^{\mu \nu }_{\mathbb {V}}(q^2)= & {} i\frac{-g^{\mu \nu }+q^\mu q^\nu /m^2_{{\mathbb {V}}}}{q^2-m_{\mathbb {V}}^2}~f_i(q^2), \end{aligned}$$

(12)

where the form factor $f_i(q^2)$ is adopted to compensate the off-shell effect of exchanged meson, which is in form of $e^{-(m_e^2-q^2)^2/\Lambda _e^4}$ 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 $\Gamma$ 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],

$$\begin{aligned} {{{\mathcal {V}}}}_{{\mathbb {P}},\sigma }=I_{{\mathbb {P}},\sigma }\Gamma _1\Gamma _2 P_{{\mathbb {P}},\sigma }(q^2),\quad \mathcal{V}_{{\mathbb {V}}}=I_{{\mathbb {V}}}\Gamma _{1\mu }\Gamma _{2\nu } P^{\mu \nu }_{{\mathbb {V}}}(q^2).\nonumber \\ \end{aligned}$$

(13)

The $I_{{\mathbb {P}},{\mathbb {V}},\sigma }$ is the flavor factors of the certain meson exchange as listed in Table 3. The interaction of charmed-antistrange interactions will be rewritten to the charmed-strange interactions by the well-known G-parity rule [58, 59],

$$\begin{aligned} V= & {} \sum _{i}{\zeta _{i}V_{i}}=-V_{\pi }+V_{\eta }+V_{\rho }-V_{\omega }+V_{\sigma }. \end{aligned}$$

(14)

The G parities of the exchanged mesons i are left as a $\zeta _{i}$ factor. Since $\pi$ and $\omega$ mesons carry odd G parity, $\zeta _{\pi }$, and $\zeta _{\omega }$ should equal $-1$, and others equal 1.

Table 3 The flavor factors $I_e$ for charmed-strange interactions. The values for charmed-antistrange interactions can be obtained by Eq. (14) from these of charmed-strange interactions. The $I_\sigma$ should be 0 for coupling between different channels

2.3 The qBSE approach

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],

$$\begin{aligned} G_0= & {} \frac{\delta ^+(p''^{0}_h-E_h(\mathrm{p}''))}{2E_h(\mathrm{p''})[(W-E_h(\mathrm{p}''))^2-E_l^{2}(\mathrm{p}'')]}. \end{aligned}$$

(15)

As required by the spectator approximation, the heavier particle (h represents the charmed baryons) satisfies $p''^0_h=E_{h}(\mathrm{p}'')=(m_{h}^{~2}+\mathrm p''^2)^{1/2}$. The $p''^0_l$ for the lighter particle (remarked as l) is then $W-E_{h}(\mathrm{p}'')$. Here and hereafter, the value of the momentum in center-of-mass frame are defined as $\mathrm{p}=|{\varvec{p}}|$.

After the covariant spectator approximation, the 3-dimensional Bethe-Saltpeter equation can be reduced to a 1-dimensional equation with fixed spin-parity $J^P$ by partial wave decomposition [57],

$$\begin{aligned} i{{{\mathcal {M}}}}^{J^P}_{\lambda '\lambda }(\mathrm{p}',\mathrm{p})= & {} i\mathcal{V}^{J^P}_{\lambda ',\lambda }(\mathrm{p}',\mathrm{p})+\sum _{\lambda ''}\int \frac{\mathrm{p}''^2d\mathrm{p}''}{(2\pi )^3}\nonumber \\&\cdot i{{{\mathcal {V}}}}^{J^P}_{\lambda '\lambda ''}(\mathrm{p}',\mathrm{p}'') G_0(\mathrm{p}'')i{{{\mathcal {M}}}}^{J^P}_{\lambda ''\lambda }(\mathrm{p}'',\mathrm{p}),\quad \quad \end{aligned}$$

(16)

where the sum extends only over nonnegative helicity $\lambda ''$. The partial wave potential in 1-dimensional equation is defined with the potential of the interaction obtained in the above as

$$\begin{aligned} {{{\mathcal {V}}}}_{\lambda '\lambda }^{J^P}(\mathrm{p}',\mathrm{p})= & {} 2\pi \int d\cos \theta ~[d^{J}_{\lambda \lambda '}(\theta ) {{{\mathcal {V}}}}_{\lambda '\lambda }({\varvec{p}}',{\varvec{p}})\nonumber \\&+\eta d^{J}_{-\lambda \lambda '}(\theta ) \mathcal{V}_{\lambda '-\lambda }({\varvec{p}}',{\varvec{p}})], \end{aligned}$$

(17)

where $\eta =PP_1P_2(-1)^{J-J_1-J_2}$ with P and J being parity and spin for the system. The initial and final relative momenta are chosen as ${\varvec{p}}=(0,0,\mathrm{p})$ and ${\varvec{p}}'=(\mathrm{p}'\sin \theta ,0,\mathrm{p}'\cos \theta )$. The $d^J_{\lambda \lambda '}(\theta )$ is the Wigner d-matrix. 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 $f(q^2)=e^{-(k_l^2-m_l^2)^2/\Lambda _r^4}$, where $k_l$ and $m_l$ are the momentum and mass of the lighter baryon. In the current work, the relation of the cutoff $\Lambda _r= m + \alpha _r$ 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 $\Lambda _e$ and $\Lambda _r$ play analogous roles in the calculation of the binding energy. For simplification, we set $\Lambda _e=\Lambda _r=m+\alpha ~0.22$ GeV 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 $z=E_R-i\Gamma /2$ being the exact position of the bound state.

3 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 single-channel calculation. The results for scalar, vector and tensor isospins are presented in the followings.

3.1 Isoscalar charmed-strange molecular states

In the current model, we have only one free parameter $\alpha$. 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.

Fig. 1

Binding energies of isoscalar bound states from the charmed-strange interactions with the variation of $\alpha$ in single-channel calculation

For the isoscalar charmed-strange interaction, we consider twelve channels, $\Lambda _c\Lambda$ with spin parities $J^P=(0,1)^+$, $\Sigma _c\Sigma$ with $(0,1)^+$, $\Sigma _c\Sigma ^*$ and $\Sigma ^*_c\Sigma$ with $(1,2)^+$, and $\Sigma ^*_c\Sigma ^*$ with $(0,1,2,3)^+$. As shown in Fig. 1, the isoscalar single-channel calculation suggests that except $\Sigma _c\Sigma$ interaction with $0^+$, all other eleven isoscalar channels can produce bound state. Their binding energies increase with increasing the free parameter $\alpha$. The bound states from the interactions $\Lambda _c\Lambda$ appear at $\alpha$ value of about 0, and increase rapidly to 30 MeV at $\alpha$ value of about 1. The bound states from the $\Sigma _c\Sigma$ interaction with $1^+$ and the $\Sigma ^*_c\Sigma$ interaction with $2^+$ also appear at an $\alpha$ of about 1, and increase relatively slowly to 30 MeV at $\alpha$ value of about 3. The attraction of the $\Sigma ^*_c\Sigma$ interaction with $1^+$ is weak, and produce a bound state at an $\alpha$ value of about 3.5. The variation tendencies of the binding energies of two states with different spin parities from the $\Lambda _c\Lambda$ 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 4, we present the coupled-channel results of isoscalar charmed-strange interactions. The results of pole under the corresponding threshold with different $\alpha$ is given in the second and third columns with full coupled-channel interaction. To compare with the single-channel results, we present the position as $M_{th}-z$ instead of the position z of the pole, with the $M_{th}$ being the nearest threshold. 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 $\Gamma =-2 \mathrm{Im} z$. The results suggest small width produced from the couplings with the channels considered.

Table 4 The masses and widths of isoscalar charmed-strange molecular states at different values of $\alpha$. 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 decay channel does not exist. The imaginary part shown as “0.0” means too small value under the current precision chosen

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 $\Sigma _c^*\Sigma {^*}$ thresholds, the strong couplings to the $\Sigma _c\Sigma$ channel can be found based on the mass and width, while the states with $(2,3)^+$ couples strongly to the $\Sigma _c\Sigma ^*$ channel. No obvious strongly coupled channel can be found for the states near the $\Sigma _c\Sigma ^*$, $\Sigma _c^*\Sigma$ and $\Sigma _c\Sigma$ thresholds. Since the $\Lambda _c\Lambda$ 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.

3.2 Isovector charmed-strange molecular states

For isovector charmed-strange interaction states, we consider eighteen channels, $\Lambda _c\Sigma$ , $\Sigma _c\Lambda$ and $\Sigma _c\Sigma$ with $(0,1)^+$, $\Sigma ^*_c\Lambda$, $\Lambda _c\Sigma ^*$, $\Sigma ^*_c\Sigma$ and $\Sigma _c\Sigma ^*$ with $(1,2)^+$, and $\Sigma ^*_c\Sigma ^*$ with $(0,1,2,3)^+$. The results are shown in the following Fig. 2.

Fig. 2

Binding energies of isovector bound states from the charmed-strange interactions with the variation of $\alpha$ in single-channel calculation

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 $\alpha$ less than 1. It is still worth mentioning that at a $\alpha$ value of about 4 or larger, the repulsion from the $\omega$ meson and $\eta$ meson exchanges increase faster than attractions of other mesons, which make the states with $0^+$ and $1^+$ of $\Sigma ^*_c\Sigma ^*$ shallower. Also, the variation tendencies of the binding energies of states with different spin parities from $\Lambda _c\Sigma ^{(*)}$ or $\Sigma ^{(*)}\Lambda$ 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 5 The masses and widths of isovector charmed-strange molecular states at different values of $\alpha$. Other notations are the same as Table 4

In Table 5, we present the coupled-channel results of isovector charmed-strange interactions. There are four isovector states near the $\Sigma ^*_c\Sigma ^*$ threshold. The width for the $0^+$ state is mainly from the $\Sigma _c\Lambda$ channel. For the state with $1^+$, large couplings can be found in the channels $\Lambda _c\Sigma ^*$ and $\Sigma _c\Sigma ^*$. The state with $2^+$ strongly couples to the channels $\Lambda _c\Sigma ^*$ and $\Sigma _c^*\Lambda$. The states with $3^+$ has strong couplings to the channels $\Sigma ^*_c\Lambda$, $\Lambda _c\Sigma ^*$, and $\Sigma _c\Lambda$. Near the $\Sigma _c\Sigma ^*$ threshold, there exist two poles with spin parities $1^+$ and $2^+$. For the state with $1^+$, the $\Lambda _c\Sigma ^*$ 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 $\Sigma ^*_c\Sigma$ threshold, there is also no obvious dominant channel for the $1^+$ states while the $2^+$ state has the strongest coupling to channel $\Lambda _c\Sigma ^*$. For two $\Lambda _c\Sigma ^*$ states, the dominant channel shifts to the $\Sigma ^*_c\Lambda$ channel. For other states, the widths are very small partly due to few channels below their masses.

3.3 Isotensor molecular states

In Fig. 3, the isotensor bound states from the charmed-strange interactions in a single-channel calculation are presented. Channels considered include $\Sigma _c\Sigma$ with $(0,1)^+$, $\Sigma ^*_c\Sigma$ and $\Sigma _c\Sigma ^*$ with $(1,2)^+$, and $\Sigma ^*_c\Sigma ^*$ with $(0,1,2,3)^+$. In the considered range of parameter $\alpha$, bound states are produced from all channels. However, two $\Sigma _c\Sigma$ and $\Sigma ^*_c\Sigma$ bound states with $(0,1)^+$ appears at $\alpha$ values of about 0, and increase rapidly to 30 MeV at $\alpha$ values less than 1 while the states from the interactions $\Sigma _c\Sigma ^*$ and $\Sigma ^*_c\Sigma ^*$ appears at an $\alpha$ values of 1.5 or larger and reach 30 MeV at $\alpha$ values about 3. Such results disfavor the coexistence of the these states.

Fig. 3

Binding energies of isotensor bound states from the charmed-strange interactions with the variation of $\alpha$ in single-channel calculation

In Table 6, 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 $\Sigma ^*_c\Sigma ^*$ state with the $3^+$ state becomes obviously smaller, which leads to a larger parameter $\alpha$ to produce the state, about 2.5, compared with the single-channel calculation in Fig. 3, which is mainly from coupling to the $\Sigma ^*_c\Sigma$ channel. From the two-channel calculations listed in the fourth to sixth columns, the isotensor $\Sigma ^*_c\Sigma ^*$ states with $(0,1)^+$ have the strongest coupling to the $\Sigma _c\Sigma$ channel, while states with $(2,3)^+$ states prefer the $\Sigma _c\Sigma ^*$ channel. For the $\Sigma _c\Sigma ^*$ states with $(1,2)^+$, the $\Sigma ^*_c\Sigma$ channel is dominant to produce their total width. The coupling effect has no effect on the width for the $\Sigma ^*_c\Sigma$ states with $(1,2)^+$, while the degeneration in mass disappears for the two states.

Table 6 The masses and widths of isotensor charmed-strange molecular states at different values of $\alpha$. Other notations are the same as Table 4

4 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.

4.1 Isoscalar molecular states

For the isoscalar charmed-antistrange interaction, there also exist twelve channels, $\Lambda _c{\bar{\Lambda }}$ with spin parities $J^P=(0,1)^-$, $\Sigma _c{\bar{\Sigma }}$ with $(0,1)^-$, $\Sigma _c{\bar{\Sigma }}^*$ and $\Sigma ^*_c{\bar{\Sigma }}$ with $(1,2)^-$, and $\Sigma ^*_c{\bar{\Sigma }}^*$ 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.

Fig. 4

Binding energies of isoscalar bound states from the charmed-antistrange interactions with the variation of $\alpha$ in single-channel calculation

The $\Sigma _c\bar{\Sigma }^*$ and $\Sigma ^*_c\bar{\Sigma }^*$ interactions are found attractive and produce two and three bound states with spin parities $(1,2)^-$ and $(1,2,3)^-$, respectively. The $\Sigma _c\bar{\Sigma }^*$ states with $(1,2)^-$ appear at $\alpha$ values about 0.5 and 1.5 and the $\Sigma ^*_c\bar{\Sigma }^*$ states with $(1,2,3)^-$ appear at $\alpha$ values of about 0.5, 1.0 and 1.5, respectively. At the same time, the larger isospin quantum number, easier it is to attract for the states. The phenomenon is the same as what we draw in the above isovector charmed-strange sector. There is no bound states for the channels $\Lambda _c{\bar{\Lambda }}$, $\Sigma _c{\bar{\Sigma }}$ and $\Sigma ^*_c{\bar{\Sigma }}$. It is mainly due to the different signs of flavor factors for the $\pi$ and $\omega$ meson exchanges in Table 3 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 7. The mass of bound states $\Sigma ^*_c{\bar{\Sigma }}^*$ 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 $\Sigma _c{\bar{\Sigma }}^*$ channel. For the states with $(2,3)^-$, considerable large couplings can be found in the channel $\Sigma _c^*{\bar{\Sigma }}$. For two states with $(1,2)^-$ near the $\Sigma _c{\bar{\Sigma }}^*$ threshold, the strongest coupling channel is the $\Sigma _c{\bar{\Sigma }}$ channel while large couplings are also found in the channels $\Sigma _c^*{\bar{\Sigma }}$ and $\Lambda _c{\bar{\Lambda }}$ for state with $1^-$.

Table 7 The masses and widths of isoscalar charmed-antistrange molecular states at different values of $\alpha$. Other notations are the same as Table 4

4.2 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 $\Lambda _c{\bar{\Sigma }}^*$ states with $(1,2)^-$ are almost degenerate, which all appear at $\alpha$ values of about 2.0, and its binding energy increase to 30 MeV at $\alpha$ values of about 3. The bound states from the $\Sigma _c{\bar{\Sigma }}^*$ interaction with $(1,2)^-$ appears at $\alpha$ values of about 0.5 and 2, respectively. The three bound state from the $\Sigma ^*_c{\bar{\Sigma }}^*$ interaction with $(1,2,3)^-$ appears at $\alpha$ values about 1, 1.5, 2, respectively. These states are well distinguished, which may make their coexistence less possible.

Fig. 5

Binding energies of isovector bound states from the charmed-antistrange interactions with the variation of $\alpha$ in single-channel calculation

The coupled-channel results are presented in Table 8. In the isovector case, the widths of the $\Sigma ^*_c{\bar{\Sigma }}^*$ state with $(1,2,3)^-$ increase very rapidly to about 7 MeV or larger with the increase of the parameter $\alpha$. The state with $1^-$ has strong coupling to the channels $\Sigma _c{\bar{\Sigma }}^*$, $\Sigma ^*_c{\bar{\Lambda }}$, and $\Lambda _c{\bar{\Sigma }}$. The state with $2^-$ couples to channels $\Sigma _c{\bar{\Sigma }}^*$, $\Sigma ^*_c{\bar{\Sigma }}$, and $\Sigma ^*_c{\bar{\Lambda }}$. For the states with $3^-$, except the channels $\Sigma _c{\bar{\Lambda }}$ and $\Lambda _c{\bar{\Sigma }}$, other channels have strong couplings. For the two $\Lambda _c{\bar{\Sigma }}^*$ states with $(1,2)^-$, the $\Lambda _c{\bar{\Sigma }}$ channel is the dominant channel to produce their total widths.

Table 8 The masses and widths of isovector charmed-antistrange molecular states at different values of $\alpha$. Other notations are the same as Table 4

4.3 Isotensor molecular states

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 $\Sigma _c{\bar{\Sigma }}$ state with $0^-$ can be produced at an $\alpha$ value of about 3.0. The $\Sigma _c{\bar{\Sigma }}^*$ state with $1^-$ and the $\Sigma ^*_c{\bar{\Sigma }}^*$ states with $(0,1)^-$ can be produced at an $\alpha$ value of about 2.5, while the $\Sigma _c{\bar{\Sigma }}^*$ with $1^-$ appears at an $\alpha$ value of about 4.5. Generally speaking, the $\alpha$ 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.

Fig. 6

Binding energies of isotensor bound states from the charmed-antistrange interactions with the variation of $\alpha$ in single-channel calculation

The coupled channel results are presented in Table 9. It can be seen that large $\alpha$ values are still required to produce the poles. For the two $\Sigma ^*_c{\bar{\Sigma }}{^*}$ states with $(0,1)^-$, the dominant channel is the $\Sigma _c{\bar{\Sigma }}^*$ channel. The $\Sigma _c{\bar{\Sigma }}^*$ state with $1^-$ strongly couples to the $\Sigma _c^*{\bar{\Sigma }}$ channel. Though there exists only one channel $\Sigma _c{\bar{\Sigma }}$ below the $\Sigma _c^*{\bar{\Sigma }}$ threshold, the $\Sigma _c^*{\bar{\Sigma }}$ state with $1^-$ acquire considerable large width.

Table 9 The masses and widths of isotensor charmed-antistrange molecular states at different values of $\alpha$. Other notations are the same as Table 4

5 Summary and discussion

In this work, we systematically study the charmed-strange molecular states with quantum numbers $C=1$ and $S=\pm 1$, in a qBSE approach together with the one-boson-exchange model. The potential kernels are constructed with the help of the effective Lagrangians with SU(3), chiral and heavy quark symmetries. With the exchange potential obtained, the S-wave 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 $\Lambda _c\Lambda$, $\Lambda _c\Sigma ^{(*)}$, $\Sigma _c^{(*)}\Lambda$, and $\Sigma ^{(*)}_c\Sigma ^{(*)}$, 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 $\alpha$ 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 single-channel 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 charmed-strange sector. However, only 17 states can be produced. Besides, the attractions of many states are weak as suggested by the $\alpha$ values to produce these states. Among 17 states produced, only 4 states can be produced with an $\alpha$ 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 charm-strange meson $D_s$. It also makes it easy to be found in experiment. Hence, we suggest the experimental research for such states.