Study of Singly Charmed Dibaryons in Quark Model

Abstract

We perform a systematic investigation of low-lying singly charmed dibaryon systems with $J=1$, $I=0,{\displaystyle \frac{1}{2}},1,{\displaystyle \frac{3}{2}},2,{\displaystyle \frac{5}{2}}$ and strangeness $S=-1,-2,-3,-4,-5$ in the chiral quark model. According to the analysis of effective potentials, dibaryon systems characterized by lower isospin and magnitude of strangeness exhibit stronger attractive interactions, which may enhance their tendency to form bound states. Experimental efforts may therefore prioritize the search for such configurations. The bound-state calculation results indicate that we have obtained some single-channel bound states, which are $\mathrm{\Sigma }{\mathrm{\Sigma }}_c$, $\mathrm{\Sigma }{\mathrm{\Sigma }}_c^{*}$, ${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c$, ${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c^{*}$ with $I=0,S=-1$; ${\mathrm{\Sigma }}_c\mathrm{\Delta }$, ${\mathrm{\Sigma }}_c^{*}\mathrm{\Delta }$ with $I={\displaystyle \frac{1}{2}},S=0$; $\mathrm{\Sigma }{\mathrm{\Sigma }}_c$ with $I=1,S=-1$; ${\mathrm{\Sigma }}_c\mathrm{\Delta }$ with $I={\displaystyle \frac{3}{2}},S=0$; and ${\mathrm{\Xi }}^{*}{\mathrm{\Sigma }}_c^{*}$ with $I={\displaystyle \frac{3}{2}},S=-2$. However, these states can decay through open channels. We have listed both these single-channel bound states and their corresponding decay channels in this work for experimental reference and search. In the future, we need to study the scattering processes of the open channels further to confirm whether these states are resonance states.

1. Introduction

Research on dibaryon states has been conducted for many years. The earliest established dibaryon state, dating back to 1932, is the deuteron [1]. In 1977, based on the MIT bag model, Jaffe utilized gluon exchange interactions and discovered a stabilized configuration of six quarks, thereby predicting the existence of the H-dibaryon [2]. Over the past decade, researchers have persistently pursued the search for new dibaryon states, and the discovery of the dibaryon resonance $d^{*}$ has undoubtedly inspired confidence among many scientists and physicists [3,4,5]. Beyond the H-dibaryon, the $N\mathrm{\Omega }$ configuration has emerged in recent research as a prominently studied dibaryon system involving a strange quark. In 1987, T. Goldman proposed in two different quark models that a dibaryon state with the strangeness $S=-3$ might exist [6]. Numerous studies have been dedicated to this state over many years [7,8,9,10]. Recent theoretical studies have provided substantial evidence for the existence of the $N\mathrm{\Omega }$ state [11,12]. In 2019, the STAR collaboration investigated the $N\mathrm{\Omega }$ correlation function via relativistic heavy ion collisions in $Au+Au$ collisions, providing supporting evidence for the existence of $N\mathrm{\Omega }$ [13]. Recently, the ALICE collaboration has investigated interactions between baryons containing strange quarks, specifically demonstrating the use of $N\mathrm{\Omega }$ correlation functions for precision calculations [14]. Inspired by these findings, our group further investigated the $p-\mathrm{\Omega }$ system from the perspective of the quark model, including the energy spectrum, scattering phase shifts, and correlation functions, and the results supported the existence of the $p-\mathrm{\Omega }$ bound state [15].

The experimental discovery of the doubly charmed baryon ${\mathrm{\Xi }}_{cc}$ by the LHCb Collaboration has further stimulated research into charmed baryons [16]. Moreover, the discovery of heavy baryons has also sparked significant interest among physicists in studying dibaryon states involving heavy quarks. Theoretically, the substantial mass of heavy quarks suppresses the kinetic term in the Hamiltonian, thereby enhancing the likelihood of bound-state formation. Consequently, a significant number of researchers have shifted their focus to the systematic search for charmed dibaryon states. Various theoretical approaches have focused on predicting the existence of heavy-flavor dibaryons. Some theoretical studies have employed the one-boson exchange model to investigate dibaryon states involving heavy quarks [17,18,19,20,21], N. Lee et al. performed a systematic study of the possible loosely bound states composed of two charmed baryons or a charmed baryon and an anti-charmed baryon. Their results indicate that the H-dibaryon-like state ${\mathrm{\Lambda }}_c{\mathrm{\Lambda }}_c$ does not exist, but deuteron-like states ${\mathrm{\Xi }}_c{\mathrm{\Xi }}_c$ and ${\mathrm{\Xi }}_c^{\prime }{\mathrm{\Xi }}_c^{\prime }$ may exist. In Ref. [21], Y. W. Pan et al. employed the one-boson exchange model to study the interactions between ${\mathrm{\Xi }}_{cc}^{*}$ and ${\mathrm{\Sigma }}_c^{*}$, showing that several possible three-charm deuteron-like hexaquark states may exist. Some theoretical studies employed lattice quantum chromodynamics (QCD) sum rules to investigate dibaryon states with heavy flavor. In Ref. [22], Z. G. Wang constructed color-singlet-type currents to study scalar and axialvector ${\mathrm{\Xi }}_{cc}$${\mathrm{\Sigma }}_c^{*}$ dibaryon states with QCD sum rules. Some theoretical studies have utilized lattice QCD to investigate heavy dibaryon states [23,24,25]. In Ref. [23], P. Junnarkar et al. reported the first lattice QCD study of deuteron-like dibaryons with heavy quark flavors and found that the ground-state mass of dibaryon ${\mathrm{\Omega }}_c{\mathrm{\Omega }}_{cc}$ was below the two-baryon threshold and also predicted the mass precisely. In addition, some theoretical groups investigated heavy dibaryons within the framework of the quark model [26,27,28,29,30]. The work of Ref. [28] investigated doubly heavy dibaryon systems with strangeness $S=0$ in the quark delocalization color screening model (QDCSM), and several bound systems were obtained. In Ref. [30], Y. Cui et al. performed a systematic investigation of the singly charmed deuteron-like dibaryon states ($IJ=01$), and several resonance states were obtained. $\mathrm{\Sigma }{\mathrm{\Sigma }}_c$ appears as a resonance state in the $\mathrm{\Lambda }{\mathrm{\Lambda }}_c$ and $N{\mathrm{\Xi }}_c^{*}$ scattering process, and $\mathrm{\Sigma }{\mathrm{\Sigma }}_c^{*}$ exhibits a resonance state in the $N{\mathrm{\Xi }}_c$ and $N{\mathrm{\Xi }}_c^{\prime }$ scattering phase shifts. To explore more dibaryon states, it is meaningful and necessary to systematically search for dibaryon systems from light flavor to heavy flavor with all possible quantum numbers.

This paper focuses on the study of dibaryon states containing a singly charmed quark, with strangeness $S=-1,-2,-3,-4$, and $-5$, angular momentum quantum number $J=1$, and isospin $I=0,{\displaystyle \frac{1}{2}},1,{\displaystyle \frac{3}{2}},2,{\displaystyle \frac{5}{2}}$. The structure of this paper is organized as follows: Section 2 provides a concise introduction to the quark model and the resonating group method (RGM). Section 3 presents the numerical results and corresponding discussion. The final Section 4 delivers the conclusion.

2. Quark Model and Resonating Group Method

This work is conducted within the framework of the chiral quark model (ChQM), using the resonating group method (RGM) to search for dibaryon bound states containing charm quarks. This chapter will provide a brief introduction to the model and the method.

2.1. The Chiral Quark Model

The chiral quark model is one of the most common approaches to describe hadron spectra, hadron–hadron interactions, and multiquark states [31]. In this model, the short-range interaction is primarily provided by the one-gluon exchange potential, the intermediate range attraction is generated by the meson exchange potential, while the Goldstone boson exchange potential dominates the long-range interaction. Our study focuses on dibaryons with strange and charm quarks, necessitating an extension from SU(3) to SU(4) flavor symmetry. Within this expanded framework, the standard Goldstone boson exchange potential is no longer sufficient; we have incorporated the following meson exchange potentials: D-meson exchange potential between $u/d$ and c quarks, $D_s$-meson exchange potential between s and c quarks, and ${\eta }_c$-meson exchange potential between any two quarks ($u/d$, s, or c quarks). Here, we only show the Hamiltonian and the parameters used.

$$H=\sum _{i=1}^6\left(m_i+{\displaystyle \frac{p_i^2}{2m_i}}\right)-T_c+\sum _{i<j}{V}_{ij}$$

(1)

$$V_{ij}=V^{\mathrm{CON}}\left(r_{ij}\right)+V^{\mathrm{OGE}}\left(r_{ij}\right)+V^{\sigma }\left(r_{ij}\right)+V^{\mathrm{OBE}}\left(r_{ij}\right)$$

(2)

where $m_i$ is the mass of different quarks, the kinetic energy term in the Hamiltonian is ${\displaystyle \frac{p_i^2}{2m_i}}$, and the term for the kinetic energy of the center of mass is $T_c$. The potential interaction $V_{ij}$ includes the confinement potential $V^{\mathrm{CON}}\left(r_{ij}\right)$, the one-gluon exchange potential $V^{\mathrm{OGE}}\left(r_{ij}\right)$, the $\sigma$ meson exchange potential $V^{\sigma }\left(r_{ij}\right)$, and the one-boson exchange potential $V^{\mathrm{OBE}}\left(r_{ij}\right)$. Since this study only involves calculations for the ground state, we will only present the form of the central force here:

$$V^{\mathrm{CON}}\left(r_{ij}\right)=-a_c{\lambda }_{\mathit{j}}·{\lambda }_{\mathit{j}}\left[r_{ij}^2+V_0\right]$$

(3)

Additionally, based on the asymptotic freedom property of QCD, the model introduces the one-gluon exchange potential:

$$\begin{array}{cc}\hfill V^{\mathrm{OGE}}\left(r_{ij}\right)& ={\displaystyle \frac{1}{4}}{\alpha }_{s_{q_i{q}_j}}{\lambda }_{\mathit{i}}·{\lambda }_{\mathit{j}}[{\displaystyle \frac{1}{r_{ij}}}-{\displaystyle \frac{\pi }{2}}\hfill \\ \hfill & \left({\displaystyle \frac{1}{m_i^2}}+{\displaystyle \frac{1}{m_j^2}}+{\displaystyle \frac{4{\sigma }_{\mathit{i}}·{\sigma }_{\mathit{j}}}{3m_i{m}_j}}\right)-{\displaystyle \frac{3}{4m_i{m}_j{r}_{ij}^3}}{S}_{ij}]\hfill \end{array}$$

(4)

where $S_{ij}$ is the quark tensor operator. We consider only $S-$ wave states in this work; the tensor force operator does not contribute. In the chiral quark model, we introduce the scalar $\sigma$ meson exchange potential (only between u and d quarks) to provide intermediate range attraction. Its specific expression is

$$\begin{array}{cc}\hfill V^{\sigma }\left(r_{ij}\right)& =-{\displaystyle \frac{g_{\mathrm{ch}}^2}{4\pi }}{\displaystyle \frac{{\mathrm{\Lambda }}_{\sigma }^2{m}_{\sigma }}{{\mathrm{\Lambda }}_{\sigma }^2-m_{\sigma }^2}}\left[Y\left(m_{\sigma }{r}_{ij}\right)-{\displaystyle \frac{{\mathrm{\Lambda }}_{\sigma }}{m_{\sigma }}}Y\left({\mathrm{\Lambda }}_{\sigma }{r}_{ij}\right)\right]\hfill \end{array}$$

(5)

For the Goldstone boson exchange potential $V^{\mathrm{OBE}}\left(r_{ij}\right)$ arising from spontaneous chiral symmetry breaking, the specific expressions for each term are

$$\begin{array}{cc}\hfill V^{\mathrm{OBE}}\left(r_{ij}\right)& ={\nu }_{\pi }\left(r_{ij}\right)\sum _{a=1}^3{\lambda }_i^a·{\lambda }_j^a+{\nu }_K\left(r_{ij}\right)\sum _{a=4}^7{\lambda }_i^a·{\lambda }_j^a\hfill \\ \hfill & +{\nu }_{\eta }\left(r_{ij}\right)\left[\left({\lambda }_i^8·{\lambda }_j^8\right)sin{\theta }_p-\left({\lambda }_i^0·{\lambda }_j^0\right)cos{\theta }_p\right]\hfill \\ \hfill & +{\nu }_D\left(r_{ij}\right)\sum _{a=9}^{12}{\lambda }_i^a·{\lambda }_j^a+{\nu }_{D_s}\left(r_{ij}\right)\sum _{a=13}^{14}{\lambda }_i^a·{\lambda }_j^a\hfill \\ \hfill & +{\nu }_{{\eta }_c}\left(r_{ij}\right){\lambda }_i^{15}·{\lambda }_j^{15}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\nu }_{\chi }\left(r_{ij}\right)& =-{\displaystyle \frac{g_{\mathrm{ch}}^2}{4\pi }}{\displaystyle \frac{m_{\chi }^2}{12m_i{m}_j}}{\displaystyle \frac{{\mathrm{\Lambda }}_{\chi }^2}{{\mathrm{\Lambda }}_{\chi }^2-m_{\chi }^2}}{m}_{\chi }\{\hfill \\ \hfill & \left[Y\left(m_{\chi }{r}_{ij}\right)-{\displaystyle \frac{{\mathrm{\Lambda }}_{\chi }^3}{m_{\chi }^3}}Y\left({\mathrm{\Lambda }}_{\chi }{r}_{ij}\right)\right]{\sigma }_{\mathit{i}}·{\sigma }_{\mathit{j}}\hfill \\ \hfill & +\left[H\left(m_{\chi }{r}_{ij}\right)-{\displaystyle \frac{{\mathrm{\Lambda }}_{\chi }^3}{m_{\chi }^3}}H\left({\mathrm{\Lambda }}_{\chi }{r}_{ij}\right)\right]S_{ij}\}\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill S_{ij}& ={\displaystyle \frac{({\mathit{\sigma }}_i·{\mathit{r}}_{ij})({\mathit{\sigma }}_j·{\mathit{r}}_{ij})}{r_{ij}^2}}-{\displaystyle \frac{1}{3}}{\mathit{\sigma }}_i·{\mathit{\sigma }}_j\hfill \end{array}$$

(8)

where $Y\left(x\right)$ and $H\left(x\right)$ are standard Yukawa functions [31]. The model parameters are determined as follows. The mass parameters (e.g., $m_{\pi }$, $m_K$, $m_{\eta }$, $m_D$, $m_{\eta }$, $m_{D_s}$, and $m_{{\eta }_c}$) take their experimental values. The mass of $u/d$-quark ($m_{u,d}$), cutoff parameters (e.g., ${\mathrm{\Lambda }}_{\pi }$, ${\mathrm{\Lambda }}_K$, and ${\mathrm{\Lambda }}_{\eta }$), and the mixing angle ${\theta }_P$ are empirically fit [32]. The chiral coupling constant $g_{ch}$ can be obtained from the $\pi NN$ coupling constant through

$$\begin{array}{ccc}\hfill {\displaystyle \frac{g_{ch}^2}{4\pi }}& =& {\left({\displaystyle \frac{3}{5}}\right)}^2{\displaystyle \frac{g_{\pi NN}^2}{4\pi }}{\displaystyle \frac{m_{u,d}^2}{m_N^2}}.\hfill \end{array}$$

(9)

The other adjustable parameters can be determined by fitting the ground-state light baryons and singly heavy baryons, and the MINUIT program is employed for the fitting. In quark model calculations, equivalent coupling constants are used to describe the spectra of baryons and mesons. As shown in $V^{\mathrm{OGE}}$, the chromomagnetic term is related to ${\displaystyle \frac{{\alpha }_{s_{q_i{q}_j}}}{m_i{m}_j}}$, where ${\alpha }_{s_{q_i{q}_j}}$ is associated with the quark flavor and determined by the mass difference between two baryons with spins $S={\displaystyle \frac{1}{2}},S={\displaystyle \frac{3}{2}}$, respectively. The mass of heavy quarks is relatively large, which reduces the mass difference between the two baryons with spins $S={\displaystyle \frac{1}{2}},S={\displaystyle \frac{3}{2}}$. Therefore, it is necessary to increase the value of ${\alpha }_{s_{q_i{q}_j}}$ to counteract the influence of quark mass and obtain a mass difference more consistent with experimental values. Thus, a phenomenon occurs where ${\alpha }_{s_{uu}}$ is less than ${\alpha }_{s_{cc}}$. However, when considering quark mass, ${\displaystyle \frac{{\alpha }_{s_{uu}}}{m_i{m}_j}}$ remains greater than ${\displaystyle \frac{{\alpha }_{s_{cc}}}{m_i{m}_j}}$. The model parameters and the masses of the fitted baryons with errors are shown in

Table 1. Model parameters.

	$b\left(fm\right)$	$m_{u,d}$ (MeV)	$m_s$ (MeV)
ChQM	0.52088$\pm 0.57\times {10}^{-7}$	313	590$\pm 0.09$
	$m_c($MeV)	$a_c$ (MeV$f{m}^{-2})$	$V_0\left(f{m}^2\right)$
ChQM	1700$\pm 0.10$	49.560$\pm 0.11$	−1.0778$\pm 0.67\times {10}^{-3}$
	${\alpha }_{s_{qq}}$	${\alpha }_{s_{qs}}$	${\alpha }_{s_{ss}}$
ChQM	0.67249$\pm 0.14\times {10}^{-2}$	0.83964$\pm 0.13\times {10}^{-2}$	0.71678$\pm 0.17\times {10}^{-2}$
	${\alpha }_{s_{qc}}$	${\alpha }_{s_{sc}}$	${\alpha }_{s_{cc}}$
ChQM	0.59298$\pm 0.13\times {10}^{-2}$	0.60778$\pm 0.15\times {10}^{-2}$	1.0810$\pm 0.50\times {10}^{-2}$

and

Table 2. The calculated masses (in MeV) of the baryons in ChQM. Experimental values (Exp) are taken from the Particle Data Group (PDG) [33].

	N	$\mathbf{\Delta }$	$\mathbf{\Lambda }$	$\mathbf{\Sigma }$
ChQM	932.7$\pm 0.975$	1254.03$\pm 0.561$	1104.85$\pm 1.018$	1207.94$\pm 0.878$
Exp	939	1233	1116	1189
	${\mathrm{\Sigma }}^{*}$	$\mathrm{\Omega }$	$\mathrm{\Xi }$	${\mathrm{\Xi }}^{*}$
ChQM	1374.63$\pm 0.675$	1662.98$\pm 0.232$	1344.26$\pm 0.768$	1510.95$\pm 0.565$
Exp	1385	1672	1315	1530
	${\mathbf{\Lambda }}_{\mathit{c}}$	${\mathbf{\Sigma }}_{\mathit{c}}$	${\mathbf{\Sigma }}_{\mathit{c}}^{*}$	${\mathbf{\Xi }}_{\mathit{c}}$
ChQM	2224.08$\pm 0.617$	2416.80$\pm 0.342$	2449.04$\pm 0.342$	2453.80$\pm 0.534$
Exp	2286	2455	2520	2470
	${\mathrm{\Xi }}_c^{\prime }$	${\mathrm{\Xi }}_c^{*}$	${\mathrm{\Xi }}_{cc}$	${\mathrm{\Omega }}_c$
ChQM	2548.35$\pm 0.399$	2573.26$\pm 0.398$	3492.87$\pm 0.704$	2695.61$\pm 0.232$
Exp	2578	2645	3519	2695
	${\mathrm{\Omega }}_c^{*}$
ChQM	2713.10$\pm 0.232$
Exp	2700

, respectively. The energies listed in Table 1 and Table 2 are expressed in the natural unit system (ℏ = c = 1). Table 1 and Table 2 show that the uncertainties of the parameters are very small, so the error of the calculated baryon spectrum is also small.

Table 2 shows that the influence of parameter errors on baryon spectra is minimal.

2.2. The Resonating Group Method

The resonating group method (RGM) represents one of the most common approximation approaches for nuclear physics problems. When extending this method to six-quark systems, the primary approximation in RGM involves assuming unchanged internal wave functions of the constituent baryons while focusing solely on the relative motion wave function between the two baryons. This approximation effectively reduces the complex six-body problem to a simpler two-body problem. The only quantity that needs to be solved is the relative motion wave function of the two baryons.

$$\begin{array}{c}\hfill {\mathsf{\Psi }}_{6q}=\mathcal{A}{\left[{\left[{\varphi }_{B_1}\left({\mathit{\rho }}_{\mathbf{1}},{\mathit{\lambda }}_{\mathbf{1}}\right){\varphi }_{B_2}\left({\mathit{\rho }}_{\mathbf{2}},{\mathit{\lambda }}_{\mathbf{2}}\right)\right]}^{\left[\sigma \right]IS}\chi \left(\mathit{R}\right)Z\left({\mathit{R}}_{\mathit{c}}\right)\right]}^J\end{array}$$

(10)

where the symbol $\mathcal{A}$ is the antisymmetrization operator. With the $SU\left(4\right)$ extension, both the light and heavy quarks are considered as identical particles. So, the symbol $\mathcal{A}$ is written as $\mathcal{A}=1-9P_{36}$. The internal wave functions of the two three-quark clusters is ${\varphi }_{B_i}$; both ${\mathit{\rho }}_{\mathit{i}}$ and ${\mathit{\lambda }}_{\mathit{i}}$ are internal coordinates. The relative motion wave function between the two clusters is $\chi \left(\mathit{R}\right)$, in which $\mathit{R}$ refers to the relative coordinate between the two clusters. $Z\left({\mathit{R}}_{\mathit{c}}\right)$ denotes the center of mass motion wave function of the two clusters, where ${\mathit{R}}_{\mathit{c}}$ represents the center-of-mass coordinate. In Formula (10), $\sigma =\left[222\right]$ provides the total color symmetry, I represents isospin quantum number, and S refers to the spin quantum number. We assume that the three-quark cluster wave function ${\varphi }_{B_i}$ is the product of the harmonic oscillator ground-state wave function and the isospin–spin wave function ${\eta }_{I_i{S}_i}\left(B_i\right)$ and color wave function ${\chi }_{c_i}$, expressed as

$$\begin{array}{c}\hfill {\varphi }_{B_i}={\left({\displaystyle \frac{2}{3\pi b^2}}\right)}^{3/4}{\left({\displaystyle \frac{2}{4\pi b^2}}\right)}^{3/4}exp\left(-{\displaystyle \frac{{\mathit{\lambda }}_{\mathit{i}}^2}{3b^2}}-{\displaystyle \frac{{\mathit{\rho }}_{\mathit{i}}^2}{4b^2}}\right){\eta }_{I_i{S}_i}\left(B_i\right){\chi }_{c_i}\left(B_i\right)\end{array}$$

(11)

Now, we can determine the relative motion wave function by solving the Schrödinger equation. From the variational principle,

$$\begin{array}{c}\hfill \langle \delta {\mathit{\Psi }}^{″}|H-E|{\mathit{\Psi }}^{\prime }\rangle =0\end{array}$$

(12)

the Hamiltonian in the expression above is Formula (1). After performing the variation, we can obtain the RGM equation:

$$\begin{array}{c}\hfill \int H({\mathit{R}}^{″},{\mathit{R}}^{\prime })\chi \left({\mathit{R}}^{\prime }\right)d{\mathit{R}}^{\prime }=E\int N({\mathit{R}}^{″},{\mathit{R}}^{\prime })\chi \left({\mathit{R}}^{\prime }\right)d{\mathit{R}}^{\prime }\end{array}$$

(13)

where $H({\mathit{R}}^{\prime },{\mathit{R}}^{\prime })$ and $N({\mathit{R}}^{″},{\mathit{R}}^{\prime })$ are Hamiltonian and norm kernels, specifically expressed as

$$\begin{array}{c}\hfill H({\mathit{R}}^{″},{\mathit{R}}^{\prime })=\langle \mathcal{A}\left[{\varphi }_1{\varphi }_2\delta (\mathit{R}-{\mathit{R}}^{\prime \prime })\right]\left|H\right|\mathcal{A}\left[{\varphi }_1{\varphi }_2\delta (\mathit{R}-{\mathit{R}}^{\prime })\right]\rangle \\ \hfill N({\mathit{R}}^{″},{\mathit{R}}^{\prime })=\langle \mathcal{A}\left[{\varphi }_1{\varphi }_2\delta (\mathit{R}-{\mathit{R}}^{\prime \prime })\right]|\mathcal{A}\left[{\varphi }_1{\varphi }_2\delta (\mathit{R}-{\mathit{R}}^{\prime })\right]\rangle \end{array}$$

(14)

Since the operator $\mathcal{A}$ is Hermitian, satisfying $\mathcal{A}H=H\mathcal{A},{\mathcal{A}}^2=\mathcal{A}$, the above expression can be written as

$$\begin{array}{c}\hfill H({\mathit{R}}^{″},{\mathit{R}}^{\prime })=\langle \left[{\varphi }_1{\varphi }_2\delta (\mathit{R}-{\mathit{R}}^{\prime \prime })\right]\left|H\right|\mathcal{A}\left[{\varphi }_1{\varphi }_2\delta (\mathit{R}-{\mathit{R}}^{\prime })\right]\rangle \\ \hfill N({\mathit{R}}^{″},{\mathit{R}}^{\prime })=\langle \left[{\varphi }_1{\varphi }_2\delta (\mathit{R}-{\mathit{R}}^{\prime \prime })\right]|\mathcal{A}\left[{\varphi }_1{\varphi }_2\delta (\mathit{R}-{\mathit{R}}^{\prime })\right]\rangle \end{array}$$

(15)

the operator $\mathcal{A}$ can be written as $\mathcal{A}=1+{\mathcal{A}}^{\prime }$, in which ${\mathcal{A}}^{\prime }$ represents antisymmetrization operator acting between the two quark clusters. Thus, $H({\mathit{R}}^{\prime \prime },{\mathit{R}}^{\prime })$ and $N({\mathit{R}}^{\prime \prime },{\mathit{R}}^{\prime })$ can be expressed as the sum of direct terms and exchange terms:

$$\begin{array}{c}\hfill H({\mathit{R}}^{″},{\mathit{R}}^{\prime })=H^D\left({\mathit{R}}^{″},{\mathit{R}}^{\prime }\right)\delta (\mathit{R}-{\mathit{R}}^{″})+H^{EX}\left({\mathit{R}}^{″},{\mathit{R}}^{\prime }\right)\\ \hfill N({\mathit{R}}^{″},{\mathit{R}}^{\prime })=N^D\left({\mathit{R}}^{″},{\mathit{R}}^{\prime }\right)\delta (\mathit{R}-{\mathit{R}}^{″})+N^{EX}\left({\mathit{R}}^{″},{\mathit{R}}^{\prime }\right)\end{array}$$

(16)

where the superscript D represents the direct item, and the superscript $EX$ represents the exchange item. Then, integrate with respect to the coordinates ${\mathit{\rho }}_{\mathbf{1}}$, ${\mathit{\rho }}_{\mathbf{2}}$, ${\mathit{\lambda }}_{\mathbf{1}}$, ${\mathit{\lambda }}_{\mathbf{2}}$, and $\mathit{R}$ and we can obtain the formula

$$\begin{gathered} \left(\begin{array}{c}{N}^D({\mathit{R}}^{\prime \prime },{\mathit{R}}^{\prime })\\ H^D({\mathit{R}}^{″},{\mathit{R}}^{\prime })\end{array}\right)\delta ({\mathit{R}}^{″}-{\mathit{R}}^{\prime })=\int {\varphi }_1^{*}({\mathit{\rho }}_{\mathbf{1}},{\mathit{\lambda }}_{\mathbf{1}}){\varphi }_2^{*}({\mathit{\rho }}_{\mathbf{2}},{\mathit{\lambda }}_{\mathbf{2}})\delta (\mathit{R}-{\mathit{R}}^{″})\left(\begin{array}{c}1\\ H\end{array}\right) \\ \times {\varphi }_1({\mathit{\rho }}_{\mathbf{1}},{\mathit{\lambda }}_{\mathbf{1}}){\varphi }_2({\mathit{\rho }}_{\mathbf{2}},{\mathit{\lambda }}_{\mathbf{2}})\delta (\mathit{R}-{\mathit{R}}^{\prime })d{\mathit{\rho }}_{\mathbf{1}}d{\mathit{\lambda }}_{\mathbf{1}}d{\mathit{\rho }}_{\mathbf{2}}d{\mathit{\lambda }}_{\mathbf{2}}d\mathit{R} \end{gathered}$$

(17)

$$\begin{gathered} \left(\begin{array}{c}{N}^{EX}({\mathit{R}}^{\prime \prime },{\mathit{R}}^{\prime })\\ H^{EX}({\mathit{R}}^{\prime \prime },{\mathit{R}}^{\prime })\end{array}\right)=\int {\varphi }_1^{*}({\mathit{\rho }}_{\mathbf{1}},{\mathit{\lambda }}_{\mathbf{1}}){\varphi }_2^{*}({\mathit{\rho }}_{\mathbf{2}},{\mathit{\lambda }}_{\mathbf{2}})\delta (\mathit{R}-{\mathit{R}}^{″})\left(\begin{array}{c}1{\mathcal{A}}^{″}\\ H{\mathcal{A}}^{″}\end{array}\right) \\ \times {\mathit{\varphi }}_1({\mathit{\rho }}_{\mathbf{1}},{\mathit{\lambda }}_{\mathbf{1}}){\mathit{\varphi }}_2({\rho }_{\mathbf{2}},{\lambda }_{\mathbf{2}})\delta (\mathit{R}-{\mathit{R}}^{\prime })d{\mathit{\rho }}_{\mathbf{1}}d{\mathit{\lambda }}_{\mathbf{1}}d{\mathit{\rho }}_{\mathbf{2}}d{\mathit{\lambda }}_{\mathbf{2}}d\mathit{R} \end{gathered}$$

(18)

So, Formula (13) can be written as

$$\begin{array}{c}\hfill \int L({\mathit{R}}^{″},{\mathit{R}}^{\prime })\chi \left({\mathit{R}}^{\prime }\right)d{\mathit{R}}^{\prime }=0\end{array}$$

(19)

where

$$\begin{array}{cc}\hfill L({\mathit{R}}^{\prime \prime },{\mathit{R}}^{\prime })=& H({\mathit{R}}^{″},{\mathit{R}}^{\prime })-EN({\mathit{R}}^{″},{\mathit{R}}^{\prime })\hfill \\ \hfill =& \left[-{\displaystyle \frac{{\nabla }_{{\mathit{R}}^{\prime }}^2}{2\mu }}+V_{rel}^D\left({\mathit{R}}^{\prime }\right)-E_{rel}\right]\delta \left({\mathit{R}}^{″}-{\mathit{R}}^{\prime }\right)\hfill \\ \hfill & +H^{EX}\left({\mathit{R}}^{″},{\mathit{R}}^{\prime }\right)-E{N}^{EX}\left({\mathit{R}}^{″},{\mathit{R}}^{\prime }\right)\hfill \end{array}$$

(20)

In the above equation, the approximate mass for the two-quark system is defined as $\mu$. $E_{rel}=E-E_{int}$ denotes the relative motion energy, and $V_{rel}^D$ represents the direct term in the interaction potential.

Formula (19) is a differential–integral equation that is difficult to solve. Therefore, we will not directly solve this equation here but will expand the relative motion wave function $\chi \left(\mathit{R}\right)$ using a series of known Gaussian wave functions with different generating coordinates ${\mathit{S}}_i$ ($i=1,2,\cdots n$):

$$\begin{array}{cc}\hfill \chi \left(\mathit{R}\right)& ={\displaystyle \frac{1}{\sqrt{4\pi }}}{\left({\displaystyle \frac{3}{2\pi b^2}}\right)}^{3/4}\sum _{i=1}^n{C}_i\hfill \\ \hfill & \times \int exp\left[-{\displaystyle \frac{3}{4b^2}}{\left(\mathit{R}-{\mathit{S}}_i\right)}^2\right]Y_{LM}\left({\widehat{\mathit{S}}}_i\right)d{\widehat{\mathit{S}}}_i\hfill \end{array}$$

(21)

where L denotes the orbital angular momentum between the two clusters. As all systems studied in this work are S-waves, L = 0. The center of mass motion wave function is

$$\begin{array}{c}\hfill Z\left({\mathit{R}}_c\right)={\left({\displaystyle \frac{6}{\pi b^2}}\right)}^{3/4}{e}^{-{\displaystyle \frac{3{\mathit{R}}_c^2}{b^2}}}\end{array}$$

(22)

and, for the orbital wave functions, ${\varphi }_{\alpha }\left({\mathit{S}}_i\right)$ and ${\varphi }_{\beta }(-{\mathit{S}}_i)$ represent single-particle orbital wave functions centered at different reference points:

$$\begin{array}{cc}\hfill {\varphi }_{\alpha }\left({\mathit{S}}_i\right)& ={\left({\displaystyle \frac{1}{\pi b^2}}\right)}^{{\displaystyle \frac{3}{4}}}{e}^{-{\displaystyle \frac{{(r_{\alpha }-{\mathit{S}}_i/2)}^2}{2b^2}}},\hfill \\ \hfill {\varphi }_{\beta }(-{\mathit{S}}_i)& ={\left({\displaystyle \frac{1}{\pi b^2}}\right)}^{{\displaystyle \frac{3}{4}}}{e}^{-{\displaystyle \frac{{(r_{\beta }+{\mathit{S}}_i/2)}^2}{2b^2}}}\hfill \end{array}$$

(23)

Formula (10) can be rewritten by substituting Formulas (21)–(23):

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_{6q}& =\sum _{i=1}^n{C}_i{\psi }_i\hfill \\ \hfill & =\mathcal{A}\sum _{i=1}^n{C}_i\int {\displaystyle \frac{d{\widehat{\mathit{S}}}_i}{\sqrt{4\pi }}}\prod _{\alpha =1}^3{\varphi }_{\alpha }\left({\mathit{S}}_i\right)\prod _{\beta =4}^6{\varphi }_{\beta }(-{\mathit{S}}_i)\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & \times {\left[{\left[{\eta }_{I_1{S}_1}\left(B_1\right){\eta }_{I_2{S}_2}\left(B_2\right)\right]}^{IS}{Y}_{LM}\left({\widehat{\mathit{S}}}_i\right)\right]}^J\hfill \\ \hfill & \times {\left[{\chi }_c\left(B_1\right){\chi }_c\left(B_2\right)\right]}^{\left[\sigma \right]},\hfill \end{array}$$

(25)

Finally, substituting Formula (21) into Formula (13) leads to the algebraic form of the RGM equation:

$$\begin{array}{c}\hfill \sum _j{C}_j{H}_{i,j}=E\sum _j{C}_j{N}_{i,j}.\end{array}$$

(26)

where $H_{i,j}$ represents the Hamiltonian matrix elements ($H_{ij}=\langle {\psi }_i\left|H\right|{\psi }_j\rangle$), and $N_{i,j}$ is the overlap ($N_{ij}=\langle {\psi }_i|{\psi }_j\rangle$). By solving Equation (26), the energy and wave function of the six-quark system can be obtained. More details on the resonating group method can be found in Ref. [34].

3. Results and Discussion

In this work, we systematically investigate S-wave singly charmed ($C=1$) dibaryon states with the angular momentum $J=1$, isospin $I=0,{\displaystyle \frac{1}{2}},1,{\displaystyle \frac{3}{2}},2,{\displaystyle \frac{5}{2}}$, and strangeness $S=-1,-2,-3,-4,-5$. In the following discussion, we will first introduce the construction of the wave function, study the interaction between two baryons by calculating the effective potentials, and finally perform bound-state calculations on the system to search for possible dibaryon states.

3.1. Wave Function Construction

The total wave function of the six-quark system consists of spatial, flavor, spin, and color components. The detailed construction of the spatial wave function is presented in Section 2.2. Here, we introduce the wave functions of baryons and dibaryons separately. Due to the numerous channels involved in the system, we will use $N{\mathrm{\Xi }}_c$ as an example for detailed elaboration.

In order to construct the wave function of the six-quark system, we must first construct the baryon wave functions. For the $N{\mathrm{\Xi }}_c$ channel, the involved baryons are $p\left(uud\right),n\left(udd\right),{\mathrm{\Xi }}_c^0\left(dsc\right),{\mathrm{\Xi }}_c^{+}\left(usc\right)$. The spin, flavor, and color wave functions of these four baryons will be presented in detail below.

(1) Baryon spin wave functions: The system involves a total of 8 distinct spin wave functions, as detailed in Ref. [30]. The spin wave functions used in this channel are listed below:

$$\begin{array}{cc}\hfill {\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{\sigma 1}=& {\displaystyle \frac{1}{\sqrt{6}}}\left(2\alpha \alpha \beta -\alpha \beta \alpha -\beta \alpha \alpha \right)\hfill \\ \hfill {\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{\sigma 2}=& {\displaystyle \frac{1}{\sqrt{2}}}\left(\alpha \beta \alpha -\beta \alpha \alpha \right)\hfill \end{array}$$

where the symbol $\alpha$ represents the spin-up state; the symbol $\beta$ represents the spin-down state. We use ${\chi }_{S,S_z}^{\sigma }$ to denote the baryon spin wave function, where S and $S_z$ represent the spin quantum number and its third component, respectively. For wave functions with identical quantum numbers but different symmetries, we distinguish them using different superscripts. For example: ${\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{\sigma 1}$ and ${\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{\sigma 2}$ represent symmetric and antisymmetric wave functions with spin quantum number ${\displaystyle \frac{1}{2}}$.

(2) Baryon flavor wave functions: The system involves a total of 36 distinct flavor wave function configurations, as detailed in Ref. [30]. The flavor wave functions for these four baryons are, respectively, expressed as

$$\begin{array}{cc}\hfill {\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{f1}& =\sqrt{{\displaystyle \frac{1}{6}}}(2uud-udu-duu)\hfill \\ \hfill {\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{f2}& =\sqrt{{\displaystyle \frac{1}{2}}}(udu-duu)\hfill \\ \hfill {\chi }_{{\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{2}}}^{f3}& =\sqrt{{\displaystyle \frac{1}{6}}}(udd+dud-2ddu)\hfill \\ \hfill {\chi }_{{\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{2}}}^{f4}& =\sqrt{{\displaystyle \frac{1}{2}}}(udd-dud)\hfill \\ \hfill {\chi }_{{\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{2}}}^{f1}& ={\displaystyle \frac{1}{2}}(dcs+cds-csd-scd)\hfill \\ \hfill {\chi }_{{\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{2}}}^{f2}& =\sqrt{{\displaystyle \frac{1}{12}}}(2dsc-2sdc+csd+dcs-cds-scd)\hfill \\ \hfill {\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{f3}& ={\displaystyle \frac{1}{2}}(ucs+cus-csu-scu)\hfill \\ \hfill {\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{f4}& =\sqrt{{\displaystyle \frac{1}{12}}}(2usc-2suc+csu+ucs-cus-scu)\hfill \end{array}$$

Similar to the spin wave function, we use ${\chi }_{I,I_z}^f$ to denote the baryon flavor wave function, where I and $I_z$ represent the isospin quantum number and its third component, respectively. Here, both light quarks and heavy quarks are treated as identical particles with $SU\left(4\right)$ symmetry.

(3) Baryon color wave function: Each quark carries one of three color charges: red (r), green (g), or blue (b). As is well known, all physically observed hadronic states must be color singlets (colorless), which implies that the color wave function must be fully antisymmetric. The color wave function for a baryon cluster can be expressed as

$${\chi }^c=\sqrt{{\displaystyle \frac{1}{6}}}\left(rgb-rbg+gbr-grb+brg-bgr\right)$$

Since the spin, flavor, and color degrees of freedom are independent, we directly multiply the three parts of the wave functions to obtain the total wave function of the baryon, denoted as ${\varphi }_{I_z,S_z}^B$, where $I_z$ is the third component of isospin, $S_z$ represents the third component of spin, and B denotes the baryon. The system involves a total of 80 distinct baryon wave functions, as detailed in Ref. [30]. The wave functions of the four baryons can be expressed as

$$\begin{array}{cc}\hfill {\varphi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^p=& \sqrt{{\displaystyle \frac{1}{2}}}\left({\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{f1}{\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{\sigma 1}+{\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{f2}{\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{\sigma 2}\right){\chi }^c\hfill \\ \hfill {\varphi }_{-{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{{\mathrm{\Xi }}_c}=& \sqrt{{\displaystyle \frac{1}{2}}}\left({\chi }_{{\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{2}}}^{f1}{\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{\sigma 1}+{\chi }_{{\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{2}}}^{f2}{\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{\sigma 2}\right){\chi }^c\hfill \\ \hfill {\varphi }_{-{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^n=& \sqrt{{\displaystyle \frac{1}{2}}}\left({\chi }_{{\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{2}}}^{f3}{\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{\sigma 1}+{\chi }_{{\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{2}}}^{f4}{\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{\sigma 2}\right){\chi }^c\hfill \\ \hfill {\varphi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{{\mathrm{\Xi }}_c}=& \sqrt{{\displaystyle \frac{1}{2}}}\left({\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{f3}{\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{\sigma 1}+{\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{f4}{\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{\sigma 2}\right){\chi }^c\hfill \end{array}$$

By substituting the wave functions of the flavor, spin, and color components according to the given quantum numbers of the system, the total flavor–spin–color wave function of the dibaryon system can be obtained. The wave function of the dibaryon is then constructed by coupling the two baryon wave functions using Clebsch–Gordan coefficients according to the total quantum numbers. Therefore, the wave function of $N{\mathrm{\Xi }}_c$ is expressed as

$$\left|N{\mathrm{\Xi }}_c\right.〉=\sqrt{{\displaystyle \frac{1}{2}}}\left[{\varphi }_{\frac{1}{2},\frac{1}{2}}^p{\varphi }_{-\frac{1}{2},\frac{1}{2}}^{{\mathrm{\Xi }}_c}-{\varphi }_{-\frac{1}{2},\frac{1}{2}}^n{\varphi }_{\frac{1}{2},\frac{1}{2}}^{{\mathrm{\Xi }}_c}\right]$$

All possible channels are listed in

Table 3. The channels involved in the calculation.

$\mathit{I}$	S	Channels
0	−1	$\mathrm{\Lambda }{\mathrm{\Lambda }}_c$$N{\mathrm{\Xi }}_c$$N{\mathrm{\Xi }}_c^{\prime }$$N{\mathrm{\Xi }}_c^{*}$$\mathrm{\Sigma }{\mathrm{\Sigma }}_c$$\mathrm{\Sigma }{\mathrm{\Sigma }}_c^{*}$${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c$${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c^{*}$
	−3	$\mathrm{\Lambda }{\mathrm{\Omega }}_c$$\mathrm{\Lambda }{\mathrm{\Omega }}_c^{*}$${\mathrm{\Lambda }}_c\mathrm{\Omega }$$\mathrm{\Xi }{\mathrm{\Xi }}_c$$\mathrm{\Xi }{\mathrm{\Xi }}_c^{\prime }$$\mathrm{\Xi }{\mathrm{\Xi }}_c^{*}$${\mathrm{\Xi }}^{*}{\mathrm{\Xi }}_c$${\mathrm{\Xi }}^{*}{\mathrm{\Xi }}_c^{*}$${\mathrm{\Xi }}^{*}{\mathrm{\Xi }}_c^{\prime }$
	−5	$\mathrm{\Omega }{\mathrm{\Omega }}_c$$\mathrm{\Omega }{\mathrm{\Omega }}_c^{*}$
${\displaystyle \frac{1}{2}}$	0	$N{\mathrm{\Sigma }}_c$$N{\mathrm{\Sigma }}_c^{*}$${\mathrm{\Sigma }}_c\mathrm{\Delta }$${\mathrm{\Sigma }}_c^{*}\mathrm{\Delta }$
	−2	${\mathrm{\Sigma }}_c\mathrm{\Xi }$$\mathrm{\Sigma }{\mathrm{\Xi }}_c$$\mathrm{\Sigma }{\mathrm{\Xi }}_c^{\prime }$${\mathrm{\Sigma }}_c{\mathrm{\Xi }}^{*}$$\mathrm{\Sigma }{\mathrm{\Xi }}_c^{*}$$\mathrm{\Xi }{\mathrm{\Sigma }}_c^{*}$${\mathrm{\Xi }}_c{\mathrm{\Sigma }}^{*}$${\mathrm{\Xi }}_c^{\prime }{\mathrm{\Sigma }}^{*}$${\mathrm{\Xi }}^{*}{\mathrm{\Sigma }}_c^{*}$${\mathrm{\Xi }}_c^{*}{\mathrm{\Sigma }}^{*}$
1	−1	$\mathrm{\Sigma }{\mathrm{\Sigma }}_c$$N{\mathrm{\Xi }}_c$$N{\mathrm{\Xi }}_c^{\prime }$$\mathrm{\Lambda }{\mathrm{\Sigma }}_c$${\mathrm{\Lambda }}_c\mathrm{\Sigma }$$\mathrm{\Lambda }{\mathrm{\Sigma }}_c^{*}$${\mathrm{\Lambda }}_c{\mathrm{\Sigma }}^{*}$
		$N{\mathrm{\Xi }}_c^{*}$${\mathrm{\Xi }}_c\mathrm{\Delta }$${\mathrm{\Xi }}_c^{\prime }\mathrm{\Delta }$$\mathrm{\Sigma }{\mathrm{\Sigma }}_c^{*}$${\mathrm{\Sigma }}_c{\mathrm{\Sigma }}^{*}$${\mathrm{\Xi }}_c^{*}\mathrm{\Delta }$${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c^{*}$
	−3	$\mathrm{\Sigma }{\mathrm{\Omega }}_c$$\mathrm{\Xi }{\mathrm{\Xi }}_c$${\mathrm{\Sigma }}_c\mathrm{\Omega }$$\mathrm{\Sigma }{\mathrm{\Omega }}_c^{*}$${\mathrm{\Xi }}_c^{\prime }{\mathrm{\Xi }}^{*}$
		${\mathrm{\Xi }}_c{\mathrm{\Xi }}^{*}$$\mathrm{\Xi }{\mathrm{\Xi }}_c^{*}$${\mathrm{\Sigma }}^{*}{\mathrm{\Omega }}_c$$\mathrm{\Omega }{\mathrm{\Sigma }}_c^{*}$${\mathrm{\Sigma }}^{*}{\mathrm{\Omega }}_c^{*}$${\mathrm{\Xi }}^{*}{\mathrm{\Xi }}_c^{*}$
${\displaystyle \frac{3}{2}}$	0	${\mathrm{\Sigma }}_{c}N$${\mathrm{\Sigma }}_c^{*}N$${\mathrm{\Sigma }}_c\mathrm{\Delta }$${\mathrm{\Sigma }}_c^{*}\mathrm{\Delta }$
	−2	${\mathrm{\Sigma }}_c\mathrm{\Xi }$$\mathrm{\Sigma }{\mathrm{\Xi }}_c$$\mathrm{\Sigma }{\mathrm{\Xi }}_c^{\prime }$${\mathrm{\Sigma }}_c{\mathrm{\Xi }}^{*}$$\mathrm{\Sigma }{\mathrm{\Xi }}_c^{*}$$\mathrm{\Xi }{\mathrm{\Sigma }}_c^{*}$${\mathrm{\Xi }}_c{\mathrm{\Sigma }}^{*}$${\mathrm{\Xi }}_c^{\prime }{\mathrm{\Sigma }}^{*}$${\mathrm{\Xi }}^{*}{\mathrm{\Sigma }}_c^{*}$${\mathrm{\Xi }}_c^{*}{\mathrm{\Sigma }}^{*}$
2	−1	${\mathrm{\Xi }}_c\mathrm{\Delta }$${\mathrm{\Xi }}_c^{\prime }\mathrm{\Delta }$$\mathrm{\Sigma }{\mathrm{\Sigma }}_c^{*}$${\mathrm{\Sigma }}_c{\mathrm{\Sigma }}^{*}$$\mathrm{\Sigma }{\mathrm{\Sigma }}_c$${\mathrm{\Xi }}_c^{*}\mathrm{\Delta }$${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c^{*}$
${\displaystyle \frac{5}{2}}$	0	${\mathrm{\Sigma }}_c\mathrm{\Delta }$${\mathrm{\Sigma }}_c^{*}\mathrm{\Delta }$

.

3.2. The Effective Potential

In particle physics, the effective potential refers to a simplified model potential that describes interparticle interactions by integrating out microscopic degrees of freedom and reducing complex many-body interactions to an equivalent particle- or two-body potential function [35]. In this work, the effective potential between the two baryons can be expressed in the following form:

$$\begin{array}{c}\hfill V\left(S_i\right)=E\left(S_i\right)-E\left(\infty \right)\end{array}$$

(27)

The symbol $S_i$ denotes the distance between the two baryon clusters, $E\left(\infty \right)$ represents the interaction energy at a sufficiently large separation distance, and $E\left(S_i\right)$ is explicitly given by

$$E\left(S_i\right)={\displaystyle \frac{\langle {\mathsf{\Psi }}_{6q}\left(S_i\right)\left|H\right|{\mathsf{\Psi }}_{6q}\left(S_i\right)\rangle }{\langle {\mathsf{\Psi }}_{6q}\left(S_i\right)|{\mathsf{\Psi }}_{6q}\left(S_i\right)\rangle }}$$

(28)

${\mathsf{\Psi }}_{6q}\left(S_i\right)$ represents the specific wave function of a dibaryon state; $\langle {\mathsf{\Psi }}_{6q}\left(S_i\right)\left|H\right|{\mathsf{\Psi }}_{6q}\left(S_i\right)\rangle$ and $\langle {\mathsf{\Psi }}_{6q}\left(S_i\right)|{\mathsf{\Psi }}_{6q}\left(S_i\right)\rangle$ are Hamiltonian matrix and the overlap of the states. The overlap of the states is always positive, so the sign variation of the effective potential cannot be caused by the scalar product between wave functions. The effective potentials for systems with different quantum numbers and varying strange quarks are presented in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11. Although most of them have attractive and repulsive regions, as long as there is a negative minimum potential between two hadrons, it indicates the existence of an attractive potential between these two hadrons, which may form a bound state.

For the $I=0,S=-1$ system, the effective potentials of different channels are shown in Figure 1. Among these eight channels, only the $N{\mathrm{\Xi }}_c^{*}$ channel exhibits a purely repulsive interaction, while the other seven channels all exhibit attractive potentials. Additionally, the effective potential of $\mathrm{\Sigma }{\mathrm{\Sigma }}_c$, $\mathrm{\Sigma }{\mathrm{\Sigma }}_c^{*}$, ${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c$, and ${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c^{*}$ is deeper than that of the other three channels, indicating that these four channels are more prone to forming bound states. The attractive potential region of $\mathrm{\Sigma }{\mathrm{\Sigma }}_c$ is 0.5–2 fm, and the deepest attractive potential has a value of about 100 MeV at a distance of around 1 fm between the two clusters. Meanwhile, the repulsive potential region of $\mathrm{\Sigma }{\mathrm{\Sigma }}_c$ is 0–0.5 fm.

For the $I=0,S=-3$ system, the effective potentials of different channels are shown in Figure 2. $\mathrm{\Lambda }{\mathrm{\Omega }}_c^{*}$, $\mathrm{\Xi }{\mathrm{\Xi }}_c$, $\mathrm{\Xi }{\mathrm{\Xi }}_c^{*}$, ${\mathrm{\Xi }}^{*}{\mathrm{\Xi }}_c^{\prime }$ and ${\mathrm{\Xi }}^{*}{\mathrm{\Xi }}_c^{*}$ channels exhibit attractive potentials, while the other four channels have purely repulsive potentials. Also, the effective potentials of $\mathrm{\Xi }{\mathrm{\Xi }}_c$ and $\mathrm{\Xi }{\mathrm{\Xi }}_c^{*}$ are deeper than that of the other three channels, indicating that these two channels are more prone to forming bound states. Meanwhile, the attractive potential region of $\mathrm{\Xi }{\mathrm{\Xi }}_c$ is 0.6–2 fm, and the repulsive potential region of this state is 0–0.6 fm. The deepest attractive potential is less than 50 MeV at a distance of around 1 fm between the two clusters.

For the $I=0,S=-5$ system, the effective potentials of different channels are shown in Figure 3. We can identify two distinct channels in this system. Both $\mathrm{\Omega }{\mathrm{\Omega }}_c$ and $\mathrm{\Omega }{\mathrm{\Omega }}_c^{*}$ states exhibit a weakly attractive potential. Consequently, we conclude that neither channel is likely to form bound states.

For $I={\displaystyle \frac{1}{2}},S=0$ system, the effective potentials of different channels are shown in Figure 4. It can be observed that this system exhibits four distinct channels, among which only the $N{\mathrm{\Sigma }}_c^{*}$ channel has purely repulsive interactions, while the other three channels exhibit attractive interactions. Furthermore, the effective potentials of both the ${\mathrm{\Sigma }}_c\mathrm{\Delta }$ and ${\mathrm{\Sigma }}_c^{*}\mathrm{\Delta }$ are deeper than the $N{\mathrm{\Sigma }}_c$ state. In comparison, the ${\mathrm{\Sigma }}_c\mathrm{\Delta }$ and ${\mathrm{\Sigma }}_c^{*}\mathrm{\Delta }$ are more likely to form bound states. Meanwhile, the attractive potential region of ${\mathrm{\Sigma }}_c^{*}\mathrm{\Delta }$ is 0–2 fm, and the deepest attractive potential is more than 100 MeV at a distance of around 0.6 fm between the two clusters.

For the $I={\displaystyle \frac{1}{2}},S=-2$ system, the effective potentials of different channels are shown in Figure 5. Among these ten channels, only the $\mathrm{\Sigma }{\mathrm{\Xi }}_c^{*}$ channel has a purely repulsive effective potential, while the other nine channels all exhibit attractive interactions. Furthermore, within these nine attractive channels, we note that four specific channels, ${\mathrm{\Sigma }}_c\mathrm{\Xi }$, $\mathrm{\Sigma }{\mathrm{\Xi }}_c$, $\mathrm{\Xi }{\mathrm{\Sigma }}_c^{*}$, and ${\mathrm{\Xi }}_c^{\prime }{\mathrm{\Sigma }}^{*}$, exhibit significantly deeper effective potentials, indicating their greater propensity to form bound states. Meanwhile, the attractive potential region of ${\mathrm{\Sigma }}_c\mathrm{\Xi }$, $\mathrm{\Sigma }{\mathrm{\Xi }}_c$, $\mathrm{\Xi }{\mathrm{\Sigma }}_c^{*}$, and ${\mathrm{\Xi }}_c^{\prime }{\mathrm{\Sigma }}^{*}$ is 0.6–2 fm, and the deepest attractive potential is more than 25 MeV at a distance of around 1 fm between the two clusters. The repulsive potential region of these states is 0–0.6 fm.

For the $I=1,S=-1$ system, the effective potentials of different channels are shown in Figure 6. Among these fourteen channels, a total of seven channels exhibit attractive interactions, specifically $\mathrm{\Sigma }{\mathrm{\Sigma }}_c,{\mathrm{\Xi }}_c\mathrm{\Delta },{\mathrm{\Xi }}_c^{\prime }\mathrm{\Delta },\mathrm{\Sigma }{\mathrm{\Sigma }}_c^{*},{\mathrm{\Sigma }}_c{\mathrm{\Sigma }}^{*},{\mathrm{\Xi }}_c^{*}\mathrm{\Delta },{\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c^{*}$, while all other channels have purely repulsive behavior. Furthermore, the effective potential of the $\mathrm{\Sigma }{\mathrm{\Sigma }}_c$ state is deeper than the seven other channels. We therefore conclude that the $\mathrm{\Sigma }{\mathrm{\Sigma }}_c$ state is more likely to form bound states. Meanwhile, the attractive potential region of $\mathrm{\Sigma }{\mathrm{\Sigma }}_c$ is 0.5–2 fm, and the deepest attractive potential is close to 25 MeV at a distance of around 0.9 fm between the two clusters. The repulsive potential region of this state is 0–0.5 fm.

For the $I=1,S=-3$ system, the effective potentials of different channels are shown in Figure 7. Only the ${\mathrm{\Xi }}_c^{\prime }{\mathrm{\Xi }}^{*}$ state exhibits a very weakly attractive effective potential, while all other channels have purely repulsive interactions. Consequently, bound-state formation is unlikely in this system.

For the $I={\displaystyle \frac{3}{2}},S=0$ system, the effective potentials of different channels are shown in Figure 8. The $N{\mathrm{\Sigma }}_c$ channel has a purely repulsive potential. The remaining three channels, $N{\mathrm{\Sigma }}_c^{*}$, ${\mathrm{\Sigma }}_c\mathrm{\Delta }$, and ${\mathrm{\Sigma }}_c^{*}\mathrm{\Delta }$, exhibit attractive potentials, with the ${\mathrm{\Sigma }}_c\mathrm{\Delta }$ channel showing deeper potential compared to the other two, indicating a greater propensity for bound-state formation. Meanwhile, the attractive potential region of ${\mathrm{\Sigma }}_c\mathrm{\Delta }$ is 0.6–2 fm, and the deepest attractive potential is more than 25 MeV at a distance of around 1 fm between the two clusters. The repulsive potential region of this state is 0–0.6 fm.

For the $I={\displaystyle \frac{3}{2}},S=-2$ system, the effective potentials of different channels are shown in Figure 9. Five states, $\mathrm{\Sigma }{\mathrm{\Xi }}_c$, ${\mathrm{\Sigma }}_c{\mathrm{\Xi }}^{*}$, ${\mathrm{\Xi }}_c^{\prime }{\mathrm{\Sigma }}^{*}$${\mathrm{\Xi }}^{*}{\mathrm{\Sigma }}_c^{*}$, and ${\mathrm{\Xi }}_c^{*}{\mathrm{\Sigma }}^{*}$, exhibit very weakly attractive potentials, while other states exhibit purely repulsive behavior. Although these five states exhibit attractive potentials, the potentials are too shallow to support bound-state formation under these conditions.

For the $I=2,S=-1$ system, the effective potentials of different channels are shown in Figure 10. This system contains a total of seven channels, among which the ${\mathrm{\Xi }}_c\mathrm{\Delta }$ and $\mathrm{\Sigma }{\mathrm{\Sigma }}_c$ channels exhibit purely repulsive interactions, while the remaining channels exhibit weakly attractive potentials. Although these channels exhibit attractive effective potentials, the interaction depths are exceptionally shallow. Consequently, we conclude that no bound states can exist in this system under the given conditions.

For the $I={\displaystyle \frac{5}{2}},S=0$ system, the effective potentials of different channels are shown in Figure 11. We can observe that this system consists of two channels, both exhibiting attractive potentials. The potential well of the ${\mathrm{\Sigma }}_c\mathrm{\Delta }$ channel is deeper than that of the ${\mathrm{\Sigma }}_c^{*}\mathrm{\Delta }$ channel. However, the attraction in these two channels is too weak to support a bound state.

From the study of the effective potentials above, it can be observed that some states exhibit deeply attractive interactions, some exhibit weakly attractive interactions, while others are purely repulsive. These findings provide valuable insights for the subsequent research on bound states and resonance states.

3.3. Bound-State Calculation

To investigate possible bound states, we conduct systematic bound-state calculations. We calculate the energies of each single channel. Meanwhile, we also consider the effects of channel coupling and perform coupled-channel calculations. The computational results are listed in

Table 4. The energy (in MeV) of different channels of the singly charmed dibaryons with $I=0,S=-1$.

Channels	${\mathit{E}}_{\mathbf{th}}$	${\mathit{E}}_{\mathbf{sc}}$	${\mathit{B}}_{\mathbf{sc}}$	${\mathit{E}}_{\mathbf{cc}}$	${\mathit{B}}_{\mathbf{cc}}$
$\mathrm{\Lambda }{\mathrm{\Lambda }}_c$	3328.93$\pm 1.635$	3337.16$\pm 1.163$	ub
$N{\mathrm{\Xi }}_c$	3386.50$\pm 1.509$	3394.06$\pm 1.045$	ub
$N{\mathrm{\Xi }}_c^{\prime }$	3481.05$\pm 1.374$	3486.15$\pm 0.909$	ub
$N{\mathrm{\Xi }}_c^{*}$	3505.96$\pm 1.373$	3515.29$\pm 0.909$	ub	3335.76$\pm 1.176$	ub
$\mathrm{\Sigma }{\mathrm{\Sigma }}_c$	3624.74$\pm 1.220$	3602.88$\pm 0.779$	−21.86$\pm 1.447$
$\mathrm{\Sigma }{\mathrm{\Sigma }}_c^{*}$	3656.98$\pm 1.220$	3632.82$\pm 0.822$	−24.16$\pm 1.471$
${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c$	3791.64$\pm 1.017$	3759.45$\pm 0.640$	−32.19$\pm 1.201$
${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c^{*}$	3823.67$\pm 1.017$	3798.47$\pm 0.653$	−25.20$\pm 1.208$

,

Table 5. The energy (in MeV) of different channels of the singly charmed dibaryons with $I=0,S=-3$.

Channels	${\mathit{E}}_{\mathbf{th}}$	${\mathit{E}}_{\mathbf{sc}}$	${\mathit{B}}_{\mathbf{sc}}$	${\mathit{E}}_{\mathbf{cc}}$	${\mathit{B}}_{\mathbf{cc}}$
$\mathrm{\Lambda }{\mathrm{\Omega }}_c$	3800.46$\pm 1.250$	3808.17$\pm 0.786$	ub
$\mathrm{\Lambda }{\mathrm{\Omega }}_c^{*}$	3817.95$\pm 1.250$	3857.99$\pm 0.751$	ub
${\mathrm{\Lambda }}_c\mathrm{\Omega }$	3887.06$\pm 0.849$	3893.80$\pm 0.386$	ub
$\mathrm{\Xi }{\mathrm{\Xi }}_c$	3798.06$\pm 1.302$	3801.09$\pm 0.834$	ub	3801.09$\pm 0.834$	ub
$\mathrm{\Xi }{\mathrm{\Xi }}_c^{\prime }$	3892.61$\pm 1.167$	3899.32$\pm 0.703$	ub
$\mathrm{\Xi }{\mathrm{\Xi }}_c^{*}$	3917.52$\pm 1.166$	3921.04$\pm 0.709$	ub
${\mathrm{\Xi }}^{*}{\mathrm{\Xi }}_c$	3964.75$\pm 0.963$	3971.04$\pm 0.635$	ub
${\mathrm{\Xi }}^{*}{\mathrm{\Xi }}_c^{*}$	4084.21$\pm 0.963$	4089.45$\pm 0.506$	ub
${\mathrm{\Xi }}^{*}{\mathrm{\Xi }}_c^{\prime }$	4059.30$\pm 0.964$	4061.86$\pm 0.503$	ub

,

Table 6. The energy (in MeV) of different channels of the singly charmed dibaryons with $I=0,S=-5$.

Channels	${\mathit{E}}_{\mathbf{th}}$	${\mathit{E}}_{\mathbf{sc}}$	${\mathit{B}}_{\mathbf{sc}}$	${\mathit{E}}_{\mathbf{cc}}$	${\mathit{B}}_{\mathbf{cc}}$
$\mathrm{\Omega }{\mathrm{\Omega }}_c$	4358.59$\pm 0.464$	4364.18$\pm 1.016$	ub	4362.77$\pm 1.133$	ub
$\mathrm{\Omega }{\mathrm{\Omega }}_c^{*}$	4376.08$\pm 0.464$	4381.11$\pm 1.134$	ub

,

Table 7. The energy (in MeV) of different channels of the singly charmed dibaryons with $I={\displaystyle \frac{1}{2}},S=0$.

Channels	${\mathit{E}}_{\mathbf{th}}$	${\mathit{E}}_{\mathbf{sc}}$	${\mathit{B}}_{\mathbf{sc}}$	${\mathit{E}}_{\mathbf{cc}}$	${\mathit{B}}_{\mathbf{cc}}$
$N{\mathrm{\Sigma }}_c$	3349.50$\pm 1.317$	3353.24$\pm 0.849$	ub
$N{\mathrm{\Sigma }}_c^{*}$	3381.74$\pm 1.317$	3392.19$\pm$0.853	ub	3353.24$\pm 0.849$	ub
${\mathrm{\Sigma }}_c\mathrm{\Delta }$	3670.83$\pm 0.903$	3647.01$\pm 0.474$	−23.83$\pm 1.019$
${\mathrm{\Sigma }}_c^{*}\mathrm{\Delta }$	3703.07$\pm 0.903$	3659.46$\pm 0.556$	−43.61$\pm 1.060$

,

Table 8. The energy (in MeV) of different channels of the singly charmed dibaryons with $I={\displaystyle \frac{1}{2}},S=-2$.

Channels	${\mathit{E}}_{\mathbf{th}}$	${\mathit{E}}_{\mathbf{sc}}$	${\mathit{B}}_{\mathbf{sc}}$	${\mathit{E}}_{\mathbf{cc}}$	${\mathit{B}}_{\mathbf{cc}}$
${\mathrm{\Sigma }}_c\mathrm{\Xi }$	3761.06$\pm 1.110$	3763.79$\pm 0.651$	ub
$\mathrm{\Sigma }{\mathrm{\Xi }}_c$	3661.74$\pm 1.412$	3664.94$\pm 0.956$	ub
$\mathrm{\Sigma }{\mathrm{\Xi }}_c^{\prime }$	3756.29$\pm 1.277$	3760.88$\pm 0.813$	ub
${\mathrm{\Sigma }}_c{\mathrm{\Xi }}^{*}$	3927.75$\pm 0.907$	3931.17$\pm 1.339$	ub
$\mathrm{\Sigma }{\mathrm{\Xi }}_c^{*}$	3781.20$\pm 1.267$	3788.38$\pm 0.812$	ub	3663.83$\pm 0.955$	ub
$\mathrm{\Xi }{\mathrm{\Sigma }}_c^{*}$	3793.30$\pm 1.110$	3796.45$\pm 0.649$	ub
${\mathrm{\Xi }}_c{\mathrm{\Sigma }}^{*}$	3828.43$\pm 1.209$	3833.93$\pm 0.745$	ub
${\mathrm{\Xi }}_c^{\prime }{\mathrm{\Sigma }}^{*}$	3922.98$\pm 1.074$	3925.13$\pm 0.614$	ub
${\mathrm{\Xi }}^{*}{\mathrm{\Sigma }}_c^{*}$	3959.99$\pm 0.907$	3964.89$\pm 0.444$	ub
${\mathrm{\Xi }}_c^{*}{\mathrm{\Sigma }}^{*}$	3947.89$\pm 1.073$	3951.90$\pm 0.616$	ub

,

Table 9. The energy (in MeV) of different channels of the singly charmed dibaryons with $I=1,S=-1$.

Channels	${\mathit{E}}_{\mathbf{th}}$	${\mathit{E}}_{\mathbf{sc}}$	${\mathit{B}}_{\mathbf{sc}}$	${\mathit{E}}_{\mathbf{cc}}$	${\mathit{B}}_{\mathbf{cc}}$
$\mathrm{\Sigma }{\mathrm{\Sigma }}_c$	3624.74$\pm 1.220$	3623.86$\pm 0.762$	−0.88$\pm 1.438$
$N{\mathrm{\Xi }}_c$	3386.50$\pm 1.509$	3395.74$\pm 1.045$	ub
$N{\mathrm{\Xi }}_c^{\prime }$	3481.05$\pm 1.374$	3489.93$\pm 0.910$	ub
$\mathrm{\Lambda }{\mathrm{\Sigma }}_c$	3521.65$\pm 1.352$	3529.89$\pm 0.896$	ub
${\mathrm{\Lambda }}_c\mathrm{\Sigma }$	3432.02$\pm 1.495$	3441.02$\pm 1.032$	ub
$\mathrm{\Lambda }{\mathrm{\Sigma }}_c^{*}$	3553.89$\pm 1.352$	3562.78$\pm 0.896$	ub
${\mathrm{\Lambda }}_c{\mathrm{\Sigma }}^{*}$	3598.71$\pm 1.292$	3607.14$\pm 0.828$	ub	3395.74$\pm 1.045$	ub
$N{\mathrm{\Xi }}_c^{*}$	3505.96$\pm 1.373$	3515.30$\pm 0.909$	ub
${\mathrm{\Xi }}_c\mathrm{\Delta }$	3707.83$\pm 1.095$	3710.28$\pm 0.635$	ub
${\mathrm{\Xi }}_c^{\prime }\mathrm{\Delta }$	3802.38$\pm 0.960$	3804.44$\pm 0.504$	ub
$\mathrm{\Sigma }{\mathrm{\Sigma }}_c^{*}$	3656.98$\pm 1.220$	3661.94$\pm 0.755$	ub
${\mathrm{\Sigma }}_c{\mathrm{\Sigma }}^{*}$	3791.64$\pm 1.017$	3794.29$\pm 0.553$	ub
${\mathrm{\Xi }}_c^{*}\mathrm{\Delta }$	3827.29$\pm 0.959$	3829.58$\pm 0.506$	ub
${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c^{*}$	3823.67$\pm 1.017$	3827.95$\pm 0.553$	ub

,

Table 10. The energy (in MeV) of different channels of the singly charmed dibaryons with $I=1,S=-3$.

Channels	${\mathit{E}}_{\mathbf{th}}$	${\mathit{E}}_{\mathbf{sc}}$	${\mathit{B}}_{\mathbf{sc}}$	${\mathit{E}}_{\mathbf{cc}}$	${\mathit{B}}_{\mathbf{cc}}$
$\mathrm{\Sigma }{\mathrm{\Omega }}_c$	3903.55$\pm 1.110$	3910.56$\pm 0.647$	ub
$\mathrm{\Xi }{\mathrm{\Xi }}_c$	3798.06$\pm 1.302$	3805.30$\pm 0.838$	ub
${\mathrm{\Sigma }}_c\mathrm{\Omega }$	4079.78$\pm 0.574$	4085.53$\pm 0.504$	ub
$\mathrm{\Sigma }{\mathrm{\Omega }}_c^{*}$	3921.04$\pm 1.110$	3928.33$\pm 0.646$	ub
${\mathrm{\Xi }}_c^{\prime }{\mathrm{\Xi }}^{*}$	4059.30$\pm 0.964$	4063.89$\pm 0.499$	ub
${\mathrm{\Xi }}_c{\mathrm{\Xi }}^{*}$	3964.75$\pm 1.099$	3972.15$\pm 0.636$	ub	3805.26$\pm 1.173$	ub
$\mathrm{\Xi }{\mathrm{\Xi }}_c^{*}$	3917.52$\pm 1.110$	3924.83$\pm 0.702$	ub
${\mathrm{\Sigma }}^{*}{\mathrm{\Omega }}_c$	4070.24$\pm 0.907$	4076.09$\pm 0.444$	ub
${\mathrm{\Sigma }}_c^{*}\mathrm{\Omega }$	4112.02$\pm 0.574$	4118.34$\pm 0.110$	ub
${\mathrm{\Sigma }}^{*}{\mathrm{\Omega }}_c^{*}$	4087.73$\pm 0.907$	4093.79$\pm 0.444$	ub
${\mathrm{\Xi }}^{*}{\mathrm{\Xi }}_c^{*}$	4084.21$\pm 0.963$	4090.34$\pm 0.499$	ub

,

Table 11. The energy (in MeV) of different channels of the singly charmed dibaryons with $I={\displaystyle \frac{3}{2}},S=0$.

Channels	${\mathit{E}}_{\mathbf{th}}$	${\mathit{E}}_{\mathbf{sc}}$	${\mathit{B}}_{\mathbf{sc}}$	${\mathit{E}}_{\mathbf{cc}}$	${\mathit{B}}_{\mathbf{cc}}$
$N{\mathrm{\Sigma }}_c$	3349.50$\pm 1.317$	3359.29$\pm 0.853$	ub
$N{\mathrm{\Sigma }}_c^{*}$	3381.74$\pm 1.317$	3389.20$\pm 0.850$	ub	3359.28$\pm 0.853$	ub
${\mathrm{\Sigma }}_c\mathrm{\Delta }$	3670.83$\pm 0.903$	3667.72$\pm 0.443$	−3.11$\pm 1.005$
${\mathrm{\Sigma }}_c^{*}\mathrm{\Delta }$	3703.07$\pm 0.903$	3707.86$\pm 0.436$	ub

,

Table 12. The energy (in MeV) of different channels of the singly charmed dibaryons with $I={\displaystyle \frac{3}{2}},S=-2$.

Channels	${\mathit{E}}_{\mathbf{th}}$	${\mathit{E}}_{\mathbf{sc}}$	${\mathit{B}}_{\mathbf{sc}}$	${\mathit{E}}_{\mathbf{cc}}$	${\mathit{B}}_{\mathbf{cc}}$
${\mathrm{\Sigma }}_c\mathrm{\Xi }$	3761.06$\pm 1.110$	3768.49$\pm 0.646$	ub
$\mathrm{\Sigma }{\mathrm{\Xi }}_c$	3661.74$\pm 1.412$	3669.34$\pm 0.949$	ub
$\mathrm{\Sigma }{\mathrm{\Xi }}_c^{\prime }$	3756.29$\pm 1.277$	3764.25$\pm 0.813$	ub
${\mathrm{\Sigma }}_c{\mathrm{\Xi }}^{*}$	3927.75$\pm 0.907$	3933.88$\pm 0.442$	ub
$\mathrm{\Sigma }{\mathrm{\Xi }}_c^{*}$	3781.20$\pm 1.267$	3788.98$\pm 0.812$	ub	3668.94$\pm 0.948$	ub
$\mathrm{\Xi }{\mathrm{\Sigma }}_c^{*}$	3793.30$\pm 1.110$	3801.80$\pm 0.646$	ub
${\mathrm{\Xi }}_c{\mathrm{\Sigma }}^{*}$	3828.43$\pm 1.209$	3836.46$\pm 0.745$	ub
${\mathrm{\Xi }}_c^{\prime }{\mathrm{\Sigma }}^{*}$	3922.98$\pm 1.074$	3928.37$\pm 0.610$	ub
${\mathrm{\Xi }}^{*}{\mathrm{\Sigma }}_c^{*}$	3959.99$\pm 0.907$	3954.42$\pm 0.609$	−5.57$\pm 1.092$
${\mathrm{\Xi }}_c^{*}{\mathrm{\Sigma }}^{*}$	3947.89$\pm 1.073$	3966.57$\pm 0.442$	ub

,

Table 13. The energy (in MeV) of different channels of the singly charmed dibaryons with $I=2,S=-1$.

Channels	${\mathit{E}}_{\mathbf{th}}$	${\mathit{E}}_{\mathbf{sc}}$	${\mathit{B}}_{\mathbf{sc}}$	${\mathit{E}}_{\mathbf{cc}}$	${\mathit{B}}_{\mathbf{cc}}$
${\mathrm{\Xi }}_c\mathrm{\Delta }$	3707.83$\pm 1.095$	3716.40$\pm 0.632$	ub
${\mathrm{\Xi }}_c^{\prime }\mathrm{\Delta }$	3802.38$\pm 0.960$	3808.93$\pm 0.495$	ub
$\mathrm{\Sigma }{\mathrm{\Sigma }}_c^{*}$	3656.98$\pm 1.220$	3664.12$\pm 0.754$	ub
${\mathrm{\Sigma }}_c{\mathrm{\Sigma }}^{*}$	3791.43$\pm 1.017$	3797.31$\pm 0.541$	ub	3633.55$\pm 0.756$	ub
$\mathrm{\Sigma }{\mathrm{\Sigma }}_c$	3624.74$\pm 1.220$	3633.55$\pm 0.756$	ub
${\mathrm{\Xi }}_c^{*}\mathrm{\Delta }$	3827.29$\pm 0.959$	3834.01$\pm 0.501$	ub
${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c^{*}$	3823.67$\pm 1.017$	3830.44$\pm 0.551$	ub

and

Table 14. The energy (in MeV) of different channels of the singly charmed dibaryons with $I={\displaystyle \frac{5}{2}},S=0$.

Channels	${\mathit{E}}_{\mathbf{th}}$	${\mathit{E}}_{\mathbf{sc}}$	${\mathit{B}}_{\mathbf{sc}}$	${\mathit{E}}_{\mathbf{cc}}$	${\mathit{B}}_{\mathbf{cc}}$
${\mathrm{\Sigma }}_c\mathrm{\Delta }$	3670.83$\pm 0.903$	3674.72$\pm 0.434$	ub	3674.72$\pm 0.434$	ub
${\mathrm{\Sigma }}_c^{*}\mathrm{\Delta }$	3703.07$\pm 0.903$	3709.99$\pm 0.437$	ub

. In these tables, the first column indicates the particle species; the second column $E_{th}$ shows the corresponding theoretical thresholds for each state; the third column $E_{sc}$ presents the single-channel energy; the fourth column $B_{sc}$ displays the single-channel binding energy (where $B_{sc}=E_{sc}-E_{th}$). The fifth $E_{cc}$ column provides the lowest energy of the system after channel coupling, and the final column $B_{cc}$ (where $B_{cc}=E_{cc}-E_{th}^{min}$) provides the binding energy from channel coupling. For states where no bound state exists, we denote them with ’ub’ (unbound). The calculation results show that the error values of energy in each system caused by parameter errors are very small. Therefore, the results of the these dibaryon systems are not sensitive to parameters.

For the $I=0,S=-1$ system, the energies of different channels are shown in Table 4. Within this system, the single-channel energy calculations demonstrate that four channels, $\mathrm{\Sigma }{\mathrm{\Sigma }}_c,\mathrm{\Sigma }{\mathrm{\Sigma }}_c^{*},{\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c$, and ${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c^{*}$, can form bound states with the binding energies of −21.86$\pm 1.447$ MeV, −24.16$\pm 1.471$ MeV, −32.19$\pm 1.201$ MeV, and −25.20$\pm 1.208$ MeV, respectively. The remaining four channels exhibit energies above the theoretical threshold, which indicates that no bound states are formed in these channels. These results are consistent with the predictions from effective potential analysis. The effective potentials of the four channels ($\mathrm{\Sigma }{\mathrm{\Sigma }}_c,\mathrm{\Sigma }{\mathrm{\Sigma }}_c^{*},{\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c$, and ${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c^{*}$) in Figure 1 are relatively deep, approaching 100 MeV, which indicates that the attraction in these four channels is strong enough to form bound states. Although the effective potentials in the three channels ($\mathrm{\Lambda }{\mathrm{\Lambda }}_c,N{\mathrm{\Xi }}_c,N{\mathrm{\Xi }}_c^{\prime }$) are attractive, the attraction is too weak to form bound states. The remaining channel ($N{\mathrm{\Xi }}_c^{*}$) exhibits a purely repulsive effective potential, which prevents the bound-state formation. Coupled-channel energy calculations further reveal that the system ground-state energy remains higher than the minimum theoretical threshold $\left(\mathrm{\Lambda }{\mathrm{\Lambda }}_c\right)$. We therefore conclude that no bound states below the lowest threshold exist in this system. However, there may exist resonance states. For higher-energy single-channel bound states, they may decay to open channels ($\mathrm{\Lambda }{\mathrm{\Lambda }}_c,N{\mathrm{\Xi }}_c,N{\mathrm{\Xi }}_c^{\prime }$, and $N{\mathrm{\Xi }}_c^{*}$), and their existence as resonance states may be verified through scattering processes, which will be investigated in the future.

For the $I=0,S=-3$ system, the energies of different channels are shown in Table 5. We observe that no bound states are formed in these channels. All single-channel calculation results lie above their theoretical thresholds, respectively. After performing coupled-channel calculations, the system ground-state energy 3801.09$\pm 0.834$ MeV remains above the minimum threshold 3798.06$\pm 1.302$ MeV ($\mathrm{\Xi }{\mathrm{\Xi }}_c$), again indicating that no bound states exist. The results are physically reasonable, as clearly demonstrated in Figure 2. The effective attraction in these five channels ($\mathrm{\Lambda }{\mathrm{\Omega }}_c^{*}$, $\mathrm{\Xi }{\mathrm{\Xi }}_c$, $\mathrm{\Xi }{\mathrm{\Xi }}_c^{*}$, ${\mathrm{\Xi }}^{*}{\mathrm{\Xi }}_c^{\prime }$, and ${\mathrm{\Xi }}^{*}{\mathrm{\Xi }}_c^{*}$) is insufficient to form any bound states, while the remaining channels exhibit purely repulsive effective potentials.

For the $I=0,S=-5$ system, the energies of different channels are shown in Table 6. We observe that all single-channel energies remain above their theoretical thresholds, respectively. Coupled-channel calculations further demonstrate that the system ground-state energy of 4362.77$\pm 1.133$ MeV exceeds the lowest 4358.59$\pm 0.464$ MeV threshold ($\mathrm{\Omega }{\mathrm{\Omega }}_c$), confirming the absence of bound states even with channel coupling effects.

For the $I={\displaystyle \frac{1}{2}},S=0$ system, the energies of different channels are shown in Table 7. We observe that, among these four channels, only the ${\mathrm{\Sigma }}_c\mathrm{\Delta }$ and ${\mathrm{\Sigma }}_c^{*}\mathrm{\Delta }$ channels form bound states, with binding energies of −23.83$\pm 1.019$ MeV and -43.61$\pm 1.060$ MeV, respectively. The other two channels exhibit energies above their theoretical thresholds. Consequently, no bound states exist, which is fully consistent with our analysis of the effective potentials. After performing four-channel coupling calculations, the total energy of 3353.24$\pm 0.849$ MeV remains above the lowest single-channel threshold of 3349.50$\pm 1.317$ MeV ($N{\mathrm{\Sigma }}_c$); no bound states below the lowest threshold exist in this system. However, after these channels couple, they may become resonance states, with the corresponding decay channels being $N{\mathrm{\Sigma }}_c$ and $N{\mathrm{\Sigma }}_c^{*}$. To investigate whether or not these states are resonance states, we will continue to study the scattering phase shifts of the open channels in the future.

For the $I={\displaystyle \frac{1}{2}},S=-2$ system, the energies of different channels are shown in Table 8. The tabulated results demonstrate that all single-channel ground-state energies exceed their theoretical thresholds, respectively. Even after accounting for channel coupling effects, the system’s lowest energy level of 3663.83$\pm 0.955$ MeV remains above the minimum single-channel threshold of 3661.74$\pm 1.412$ MeV $\left(\mathrm{\Sigma }{\mathrm{\Xi }}_c\right)$. Therefore, we conclusively rule out the existence of bound states in this channel.

For the $I=1,S=-1$ system, the energies of different channels are shown in Table 9. Within this system, the single-channel calculations reveal that only the $\mathrm{\Sigma }{\mathrm{\Sigma }}_c$ forms a shallow bound state with a binding energy of −0.88$\pm 1.438$ MeV. All other single-channel energies lie above their theoretical thresholds, indicating the absence of bound states in these channels. Furthermore, our coupled-channel calculation energy is higher than the lowest threshold value $\left(N{\mathrm{\Xi }}_c\right)$. Therefore, we conclude that no bound states exist below the lowest threshold in this system. However, this state may decay into the $N{\mathrm{\Xi }}_c$, $N{\mathrm{\Xi }}_c^{\prime }$, $\mathrm{\Lambda }{\mathrm{\Sigma }}_c$, ${\mathrm{\Lambda }}_c\mathrm{\Sigma }$, $\mathrm{\Lambda }{\mathrm{\Sigma }}_c^{*}$, ${\mathrm{\Lambda }}_c{\mathrm{\Sigma }}^{*}$, and $N{\mathrm{\Xi }}_c^{*}$ channels and form a resonance. We will investigate this further by studying the open-channel scattering phase shifts.

For the $I=1,S=-3$ system, the energies of different channels are shown in Table 10. All single-channel energies exceed their theoretical thresholds, demonstrating the absence of bound states in the single-channel approximation. After channel coupling, the computed energy, 3805.26$\pm 1.173$ MeV, is higher than the lowest threshold, 3798.06$\pm 1.302$ MeV $\left(\mathrm{\Xi }{\mathrm{\Xi }}_c\right)$, confirming the absence of bound states in these channels. These numerical results align consistently with our effective potential analysis, demonstrating purely repulsive interactions in some channels, as shown in Figure 7, with the exception of the ${\mathrm{\Xi }}_c^{\prime }{\mathrm{\Xi }}^{*}$ state.

For the $I={\displaystyle \frac{3}{2}},S=0$ system, the energies of different channels are shown in Table 11. Within this system, single-channel calculations indicate that only the ${\mathrm{\Sigma }}_c\mathrm{\Delta }$ state forms a bound state with a binding energy of −3.11$\pm 1.005$ MeV. All other channels yield energies above their theoretical thresholds, respectively, confirming that no bound states exist in these channels. When channel coupling is considered, the system ground-state energy 3359.28$\pm 0.853$ MeV remains above the fundamental theoretical threshold 3349.50$\pm 1.317$ MeV $\left(N{\mathrm{\Sigma }}_c\right)$. Therefore, we conclude that in this state there are no bound states below the lowest threshold. Yet, this state may decay into the $N{\mathrm{\Sigma }}_c$, $N{\mathrm{\Sigma }}_c^{*}$ channel, giving rise to a resonance. We will pursue this question through the scattering phase shifts in the open-channel analysis.

For the $I={\displaystyle \frac{3}{2}},S=-2$ system, the energies of different channels are shown in Table 12. The single-channel calculations for this state reveal that only the ${\mathrm{\Xi }}^{*}{\mathrm{\Sigma }}_c^{*}$ state constitutes a bound state with a binding energy of −5.57$\pm 1.092$ MeV. For all other states, the single-channel calculation results exceed their theoretical thresholds due to the purely repulsive nature of their effective potentials, making bound-state formation unlikely. Furthermore, the channel coupling calculations demonstrate that the system’s lowest energy level 3668.94$\pm 0.948$ MeV remains above the minimum theoretical threshold 3661.74$\pm 1.412$ MeV of the system $\left(\mathrm{\Sigma }{\mathrm{\Xi }}_c\right)$. Hence, we find that, within this configuration, no bound states exist whose energies fall beneath the minimal threshold. Nevertheless, this state could potentially decay through the remaining seven channels, giving rise to resonance. We plan to examine this possibility through detailed analysis of open-channel scattering phase shifts in subsequent research.

For the $I=2,S=-1$ system, the energies of different channels are shown in Table 13. In this state, all seven channels exhibit single-channel energies above their theoretical thresholds, which implies that no bound states are present in these channels. Coupled-channel calculations yield energies of 3633.55$\pm 0.756$ MeV, which exceed the system minimum threshold energy 3624.74$\pm 1.220$ MeV $\left(\mathrm{\Sigma }{\mathrm{\Sigma }}_c\right)$, further confirming the absence of bound states. These results are fully consistent with the repulsive effective potentials presented in Figure 10.

For the $I={\displaystyle \frac{5}{2}},S=0$ system, the energies of different channels are shown in Table 14. Table 14 clearly demonstrates that the single-channel energies of both channels exceed their theoretical thresholds, ruling out the possibility of bound states. Even after considering channel coupling effects, the system ground-state energy 3674.72$\pm 0.434$ MeV remains above the fundamental theoretical threshold 3670.83$\pm 0.903$ MeV (${\mathrm{\Sigma }}_c\mathrm{\Delta }$), confirming the absence of bound states. This finding contradicts our initial assessment based on the effective potentials presented in Figure 11.

4. Summary

In this work, we systematically investigate the existence of S-wave singly charmed dibaryon states within the framework of the chiral quark model. We begin by constructing dibaryon wave functions and then calculate the effective potentials to analyze the interactions between two baryons of the singly charmed dibaryon systems, which can provide preliminary indications of possible bound-state formation. To further verify the existence of bound states, we perform both single-channel and coupled-channel energy calculations.

From the results of the effective potentials for different systems, we find that, for attractive interactions, states with the same isospin exhibit weaker attraction as the number of strange quarks increases. For example, in the isospin $I=0$ sector, the deepest attractive potential reaches about 100 MeV for the channel ${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c$ with strangeness $S=-1$, while the deepest attraction is less than 10 MeV for the channel $\mathrm{\Omega }{\mathrm{\Omega }}_c$ with $S=-5$. We also find that, for the fixed number of strange quarks, the attraction strengthens as isospin decreases. For instance, for systems with $S=-1$, the deepest attractive potential reaches 100 MeV for the channel ${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c$ with $I=0$ but drops below 5 MeV for the channel ${\mathrm{\Sigma }}_c{\mathrm{\Sigma }}^{*}$ with $I=2$. The patterns of these changes in effective potentials indicate that, for dibaryon systems with smaller isospin quantum numbers and fewer strangeness numbers, the attraction between the two baryons is stronger, potentially making them more likely to form bound states. Therefore, experiments could initially focus on searching for these states.

From the results of the bound-state calculation, we find that there is no bound state below the lowest threshold in the present systems. Nevertheless, single-channel bound states do appear, and each of them can decay via open channels. Their nature can be probed further through the scattering process of the open channels. All the potentially existing bound states are compiled in

Table 15. Quasi-stable states under distinct quantum numbers and their possible two-body decay channels.

I	S	Quasi-Stable States	Decay Channels
0	−1	$\mathrm{\Sigma }{\mathrm{\Sigma }}_c$$\mathrm{\Sigma }{\mathrm{\Sigma }}_c^{*}$${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c$${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c^{*}$	$\mathrm{\Lambda }{\mathrm{\Lambda }}_c,N{\mathrm{\Xi }}_c,N{\mathrm{\Xi }}_c^{\prime },$$N{\mathrm{\Xi }}_c^{*}$
${\displaystyle \frac{1}{2} }$	0	${\mathrm{\Sigma }}_c\mathrm{\Delta }$${\mathrm{\Sigma }}_c^{*}\mathrm{\Delta }$	$N{\mathrm{\Sigma }}_c$$N{\mathrm{\Sigma }}_c^{*}$
1	−1	$\mathrm{\Sigma }{\mathrm{\Sigma }}_c$	$N{\mathrm{\Xi }}_c$$N{\mathrm{\Xi }}_c^{\prime }$$\mathrm{\Lambda }{\mathrm{\Sigma }}_c$${\mathrm{\Lambda }}_c\mathrm{\Sigma }$
			$\mathrm{\Lambda }{\mathrm{\Sigma }}_c^{*}$${\mathrm{\Lambda }}_c{\mathrm{\Sigma }}^{*}$$N{\mathrm{\Xi }}_c^{*}$
${\displaystyle \frac{3}{2} }$	0	${\mathrm{\Sigma }}_c\mathrm{\Delta }$	$N{\mathrm{\Sigma }}_c,$$N{\mathrm{\Sigma }}_c^{*}$
${\displaystyle \frac{3}{2} }$	−2	${\mathrm{\Xi }}^{*}{\mathrm{\Sigma }}_c^{*}$	${\mathrm{\Sigma }}_c\mathrm{\Xi }$$\mathrm{\Sigma }{\mathrm{\Xi }}_c$$\mathrm{\Sigma }{\mathrm{\Xi }}_c^{\prime }$${\mathrm{\Sigma }}_c{\mathrm{\Xi }}^{*}$$\mathrm{\Sigma }{\mathrm{\Xi }}_c^{*}$
			${\mathrm{\Xi }}^{*}{\mathrm{\Sigma }}_c^{*}$${\mathrm{\Xi }}_c{\mathrm{\Sigma }}^{*}$${\mathrm{\Xi }}_c^{\prime }{\mathrm{\Sigma }}^{*}$${\mathrm{\Xi }}_c^{*}{\mathrm{\Sigma }}^{*}$

, together with their possible two-body decay channels. The results show that the interaction potentials of the $\mathrm{\Sigma }{\mathrm{\Sigma }}_c$, $\mathrm{\Sigma }{\mathrm{\Sigma }}_c^{*}$, ${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c$, ${\mathrm{\Sigma }}^{*}{\mathrm{\Sigma }}_c^{*}$, ${\mathrm{\Sigma }}_c\mathrm{\Delta }$, and ${\mathrm{\Sigma }}_c^{*}\mathrm{\Delta }$ states are rather deep and strongly attractive, giving rise to a pronounced single-channel bound state. Such states are likely to appear as resonances, and they will therefore be primary targets in our future scattering analyses as well as worthwhile seeking states for experimental investigation.

While this work has systematically investigated the existence of singly charmed dibaryon states, there remain several directions for future research. First, our current study has focused exclusively on bound states and has not addressed possible resonance states; studying the scattering process is necessary for confirming the existence of resonance states. Second, the present work only studies ground-state configurations; dibaryon systems with higher partial waves (P-waves, D-waves, etc.) are also worth studying. From the study of dibaryon systems composed of light quarks, it can be seen that the possible dibaryons are mainly in the ground state, such as $d^{*}$. Therefore, it is unknown whether dibaryons composed of heavy quarks exist in a highly excited state, and specific computational research is needed. However, for $S-$wave bound states, they can decay into $D-$wave open channels, which will affect the decay width of the resonance states. Due to the fact that $S-$ and $D-$ waves are coupled through tensor forces, this effect will not be too significant, but it still needs to be judged through specific calculations, which is work that will be conducted in the future. Such extensions will provide a more complete understanding of the exotic hadron spectrum.