Pentaquark interpretation of $\Lambda _{c}$ states in the quark model

Abstract

Motivated by the recent observation of the $\Lambda _c(2910)^+$ state in the $\Sigma _{c}(2455)^{0,++}\pi ^{\pm }$ spectrum by the Belle Collaboration, we investigate the explanation of $\Lambda _{c}$ states within the pentaquark framework using the quark delocalization color screening model (QDCSM). To check for bound states and resonance states, we utilize the real-scaling method. Additionally, we calculate the root mean square of cluster spacing to study the structure of the states and further estimate whether a state is a resonance state or not. Our numerical results show that $\Lambda _{c}(2910)$ cannot be interpreted as a molecular state, and $\Sigma _{c}(2800)$ cannot be explained as the ND molecular state with $J^P=1/2^-$. However, we find that $\Lambda _{c}(2595)$ can be interpreted as a molecular state with $J^P=\frac{1}{2}^-$, where the main component is $\Sigma _{c}\pi$. Similarly, we interpret $\Lambda _{c}(2625)$ as a molecular state with $J^P=\frac{3}{2}^-$, where the main component is $\Sigma _{c}^{*}\pi$. We also suggest that $\Lambda _{c}(2940)$ is likely to be interpreted as a molecular state with $J^P=3/2^-$, where the main component is $ND^{*}$. In addition, we predict two new states: a $J^P=3/2^-$ $\Sigma _{c}\rho$ resonance state with a mass of 3140–3142 MeV, and a $J^P=\frac{5}{2}^-$ $\Sigma _{c}^*\rho$ state with a mass about 3187–3188 MeV.

1 Introduction

Since the discovery of $\Lambda _{c}^+$ by Fermilab in 1976 [1], the charmed baryon family has been enriched through a series of experimental collaborations [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. These observations have stimulated broad interest in understanding the structures of these states among theoretical groups [28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85], particularly the excited charmed baryons such as $\Lambda _{c}(2595)$, $\Lambda _{c}(2625)$, $\Sigma _{c}(2800)$, and $\Lambda _{c}(2940)$. As these charmed baryons contain both light and heavy quarks, they provide a transitional link between light and heavy baryons, making them an important subject of study. Furthermore, the analysis of the properties of these states can deepen our understanding of the non-perturbative behavior of quantum chromodynamics (QCD).

In the last two decades, the focus of the various theoretical work and the controversial point is whether these states are excitations or multi-quark states. We can briefly review some of the theoretical work that concerning the single charmed baryons mentioned earlier. These states are studied by assuming that they are traditional three-quark excitations in the constituent quark model [28,29,30], the relativistic flux tube model [31, 32], the non-relativistic quark model [33], the relativistic quark–diquark model [34,35,36,37,38], the chiral quark model [39, 40], the QCD sum rules [41,42,43], the effective field theory [44, 45], the effective Lagrangian method [46], the Faddeev method in momentum space [47], and so on.

Besides, the multi-quark interpretations of these states were also investigated in the framework of an unitary baryon–meson coupled-channel model [48, 49], the constituent quark model [50,51,52,53], the one boson exchange model [54, 55], the QCD sum rules [56, 57], the effective field theory [44, 58], a general framework that goes beyond effective range expansion [59], the effective Lagrangian approach [60], the meson-exchange picture [61], the unitarized chiral perturbation theory [62, 63], the heavy quark spin symmetry [64], and so on.

In addition to the studies of masses and structures, the studies of the decay of these states were carried out in the effective meson Lagrangian [65,66,67], the $^3P_0$ strong decay model [68,69,70,71], and the constituent quark model [72]. Moreover, the view of these states as a mixture of three-quark and five-quark was considered in an unquenched picture [73].

For $\Lambda _{c}(2595)$, it was investigated as multi-quark state in Ref. [44], where $\Lambda _{c}(2595)$ was predicted to have a predominant molecular structure with the help the effective field theory. This is because it is either the result of the chiral $\Sigma _{c}\pi$ interaction, whose threshold is located much closer than the mass of the bare three-quark state, or because the light degrees of freedom in its inner structure are coupled to the unnatural $0^-$ quantum numbers. Meanwhile $\Lambda _{c}(2595)$ was also studied in terms of traditional three-quark state. For the low-lying $\Lambda _{c}(2595)$ baryon, the non-relativistic quark model description as the $\lambda$-mode excitation with a spin-0 diquark can explain the decay property well in Ref. [33].

As for $\Lambda _{c}(2625)$, in Ref. [64], a state with spin 3/2 that couples mostly to $D^{*}N$ was associated to the experimental found $\Lambda _{c}(2625)$ in the framework of the heavy quark spin symmetry. Meanwhile, $\Lambda _{c}(2625)$ can be explained as an excited three-quark state in Ref. [71], where $\Lambda _{c}(2625)$ was identified as a $P_{\lambda }$-wave excitation, with $J^P=3/2^-$ and $S=1/2$.

According to Ref. [56], the author investigated $\Sigma _{c}(2800)$ as the S-wave DN state with $J^P=1/2^-$ in the framework of QCD sum rules. However, the research that regards $\Sigma _{c}(2800)$ as a traditional three-quark state was carried out in Ref. [45], where the explanation of $\Sigma _{c}(2800)$ as the DN molecular state was disfavored. In Ref. [45], $\Sigma _{c}(2800)$ was more likely to be the conventional 1P charmed baryon, since its mass was well consistent with the quark model prediction.

In Ref. [50], the authors proposed a theoretical explanation of the $\Lambda _{c}(2940)$ as a molecular state in a constituent quark model that has been extensively used to describe hadron phenomenology. However, the work of Ref. [47] used the Faddeev method in momentum space and considered $\Lambda _{c}(2940)$ being the first radial excitation 2S of the $\Sigma _{c}$ with $J^P=3/2^+$.

Recently, the Belle Collaboration reported the discovery of a new structure in the $M_{\Sigma _{c}(2455)^{0,++}\pi ^{\pm }}$ spectrum, tentatively named $\Lambda _{c}(2910)^{+}$, with a significance of $4.2\sigma$ including systematic uncertainty [86]. Its mass and width were measured to be $(2913.8 \pm 5.6 \pm 3.8) \textrm{MeV} / c^{2}$ and $(51.8 \pm 20.0 \pm 18.8) \textrm{MeV}$, respectively. Several theoretical analyses have been conducted to interpret this state. In Ref. [87], the author used the light-cone QCD sum rule to suggest that the $\Lambda _{c}(2910)$ baryon is a 2P state with $J^P=1/2^-$, denoted as $\Lambda _{c}(1/2^-,2P)$. In Ref. [88], the author concluded that the newly observed $\Lambda _{c}(2910)$ can be explained as the $J^P=5/2^-$ state $\Lambda _{c}\left| J^{P}=\frac{5}{2}^{-}, 2\right\rangle _{\rho }$ in the framework of the chiral quark model. In Ref. [89], an unquenched picture was used to study $\Lambda _{c}(2910)$ by considering the $S-$wave $D^*N$ channel coupled with the bare udc core $(\Lambda _{c}(2P))$. Their results suggest that $\Lambda _{c}(2910)$ contains a significant $D^*N$ component, and the bare state can cause the $D^*N$ binding to be more compact.

An alternative approach to study hadron–hadron interaction and the multi-quark states is the quark delocalization color screening model (QDCSM), which was developed in the 1990s with the aim of explaining the similarities between nuclear and molecular forces [90]. The model gives a good description of NN and YN interactions and the properties of deuteron [91,92,93,94]. It is also employed to calculate the baryon–baryon and baryon–meson scattering phase shifts in the framework of the resonating group method (RGM), and the exotic hadronic states are also studied in this model. Studies also show that the NN intermediate-range attraction mechanism in the QDCSM, quark delocalization, and color screening, is equivalent to the $\sigma$-meson exchange in the chiral quark model, and the color screening is an effective description of the hidden-color channel coupling [95, 96]. So it is feasible and meaningful to extend this model to investigate the charmed baryons.

In this work, we explore the pentaquark interpretation of charmed baryons in QDCSM. Both $qqq-\bar{q}c$ and $qqc-\bar{q}q$ structures, as well as the coupling of these two structures are taken into account. Our purpose is to investigate whether $\Lambda _c(2910)$ could be explained as a pentaquark state and if the molecular structure is possible. In addition, we also want to see if any other bound or resonance pentaquarks exist or not, which can be used to interpret other charmed baryons.

This paper is organized as follows. After introduction, we briefly introduce the quark model and methods in Sect. 2. Then, the numerical results and discussions are presented in Sect. 3. Finally, the paper ends with summary in Sect. 4.

2 Theoretical framework

Herein, QDCSM is employed to investigate the properties of $qqq\bar{q}c$ systems, and the channel coupling effect is considered. In this sector, we will introduce this model and the way of constructing wave functions.

2.1 Quark delocalization color screening model (QDCSM)

The QDCSM is an extension of the native quark cluster model [97,98,99,100]. It has been developed to address multi-quark systems. The detail of QDCSM can be found in Refs. [90,91,92,93,94,95, 101, 102]. Here, we mainly present the salient features of the model. The general form of the pentaquark Hamiltonian is given by

$$\begin{aligned} H=\sum _{i=1}^5\left( m_i+\frac{{\varvec{p}}_{i}^{2}}{2m_i}\right) -T_{CM} +\sum _{j>i=1}^5 V({\varvec{r}}_{ij}) \end{aligned}$$

(1)

where $m_i$ is the quark mass, ${\varvec{p}}_{i}$ is the momentum of the quark, and $T_{CM}$ is the center-of-mass kinetic energy. The dynamics of the pentaquark system is driven by a two-body potential

$$\begin{aligned} V({\varvec{r}}_{ij})= V_{CON}({\varvec{r}}_{ij})+V_{OGE} ({\varvec{r}}_{ij})+V_{\chi }({\varvec{r}}_{ij}). \end{aligned}$$

(2)

The most relevant features of QCD at its low energy regime: color confinement $(V_{CON}),$ perturbative one-gluon exchange interaction $(V_{OGE}),$ and dynamical chiral symmetry breaking $(V_{\chi })$ have been taken into consideration.

Here, a phenomenological color screening confinement potential $(V_{CON})$ is used as

$$\begin{aligned} V_{CON}({\varvec{r}}_{ij})&= -a_{c}{\varvec{\lambda }}_{i}^{c} \cdot {\varvec{\lambda }}_{j}^{c}\left[ f({\varvec{r}}_{ij}) +V_{0}\right] , \end{aligned}$$

(3)

$$\begin{aligned} f({\varvec{r}}_{ij})&= \left\{ \begin{array}{ll} {\varvec{r}}_{i j}^{2} &{}\quad i,j \text { occur in the same cluster} \\ \frac{1-e^{-\mu _{q_{i}q_{j}} {\varvec{r}}_{i j}^{2}}}{\mu _{q_{i}q_{j}}} &{}\quad i,j \text { occur in different cluster} \end{array}\right. \end{aligned}$$

(4)

where $a_c$, $V_{0}$ and $\mu _{q_{i}q_{j}}$ are model parameters, and ${\varvec{\lambda }}^{c}$ stands for the SU(3) color Gell-Mann matrices. Among them, the color screening parameter $\mu _{q_{i}q_{j}}$ is determined by fitting the deuteron properties, nucleon–nucleon scattering phase shifts, and hyperon–nucleon scattering phase shifts, respectively, with $\mu _{qq}=0.45$ fm$^{-2}$, $\mu _{qs}=0.19$ fm$^{-2}$ and $\mu _{ss}=0.08$ fm$^{-2}$, satisfying the relation, $\mu _{qs}^{2}=\mu _{qq}\mu _{ss}$ [103]. Besides, we found that the heavier the quark, the smaller this parameter $\mu _{q_{i}q_{j}}$. When extending to the heavy quark system, the hidden-charm pentaquark system, we took $\mu _{cc}$ as a adjustable parameter from 0.01 fm$^{-2}$ to 0.001 fm$^{-2}$, and found that the results were insensitive to the value of $\mu _{cc}$ [104]. Moreover, the $P_{c}$ states were well predicted in the work of Refs. [104, 105]. So here we take $\mu _{cc}=0.01$ fm$^{-2}$ and $\mu _{qc}=0.067$ fm$^{-2}$, also satisfy the relation $\mu _{qc}^{2}=\mu _{qq}\mu _{qc}$.

In the present work, we mainly focus on the low-lying negative parity $qqq\bar{q}c$ pentaquark states of S-wave, so the spin-orbit and tensor interactions are not included. The one-gluon exchange potential $(V_{OGE}),$ which includes coulomb and color-magnetic interactions, is written as

$$\begin{aligned} V_{OGE}({\varvec{r}}_{ij})&= \frac{1}{4}\alpha _{s} {\varvec{\lambda }}_{i}^{c} \cdot {\varvec{\lambda }}_{j}^{c} \nonumber \\&\quad \cdot \left[ \frac{1}{r_{i j}}-\frac{\pi }{2} \delta \left( {\textbf{r}}_{i j}\right) \left( \frac{1}{m_{i}^{2}} +\frac{1}{m_{j}^{2}}+\frac{4 {\varvec{\sigma }}_{i} \cdot {\varvec{\sigma }}_{j}}{3 m_{i} m_{j}}\right) \right] \end{aligned}$$

(5)

where ${\varvec{\sigma }}$ is the Pauli matrices and $\alpha _{s}$ is the quark–gluon coupling constant.

However, the quark–gluon coupling constant between quark and anti-quark, which offers a consistent description of mesons from light to heavy-quark sector, is determined by the mass differences between pseudoscalar mesons (spin-parity $J^P=0^-$) and vector (spin-parity $J^P=1^-$), respectively. For example, from the model Hamiltonian, the mass difference between D and $D^*$ is determined by the color-magnetic interaction in Eq. (5), so the parameter $\alpha _s(qc)$ is determined by fitting the mass difference between D and $D^*$.

The dynamical breaking of chiral symmetry results in the SU(3) Goldstone boson exchange interactions appear between constituent light quarks u, d and s. Hence, the chiral interaction is expressed as

$$\begin{aligned} V_{\chi }({\varvec{r}}_{ij})=V_{\pi }({\varvec{r}}_{ij}) +V_{K}({\varvec{r}}_{ij})+V_{\eta }({\varvec{r}}_{ij}). \end{aligned}$$

(6)

Among them

$$\begin{aligned} V_{\pi }\left( {\varvec{r}}_{i j}\right)&=\frac{g_{c h}^{2}}{4 \pi } \frac{m_{\pi }^{2}}{12 m_{i} m_{j}} \frac{\Lambda _{\pi }^{2}}{\Lambda _{\pi }^{2}-m_{\pi }^{2}} m_{\pi }\left[ Y\left( m_{\pi } {\varvec{r}}_{i j}\right) \right. \nonumber \\&\quad \left. -\frac{\Lambda _{\pi }^{3}}{m_{\pi }^{3}} Y \left( \Lambda _{\pi } {\varvec{r}}_{i j}\right) \right] \left( {\varvec{\sigma }}_{i} \cdot {\varvec{\sigma }}_{j}\right) \sum _{a=1}^{3}\left( {\varvec{\lambda }}_{i}^{a} \cdot {\varvec{\lambda }}_{j}^{a}\right) \end{aligned}$$

(7)

$$\begin{aligned} V_{K}\left( {\varvec{r}}_{i j}\right)&=\frac{g_{c h}^{2}}{4 \pi } \frac{m_{K}^{2}}{12 m_{i} m_{j}} \frac{\Lambda _{K}^{2}}{\Lambda _{K}^{2}-m_{K}^{2}} m_{K} \left[ Y\left( m_{K} {\varvec{r}}_{i j}\right) \right. \nonumber \\&\quad \left. -\frac{\Lambda _{K}^{3}}{m_{K}^{3}} Y \left( \Lambda _{K} {\varvec{r}}_{i j}\right) \right] \left( {\varvec{\sigma }}_{i} \cdot {\varvec{\sigma }}_{j}\right) \sum _{a=4}^{7} \left( {\varvec{\lambda }}_{i}^{a} \cdot {\varvec{\lambda }}_{j}^{a}\right) \end{aligned}$$

(8)

$$\begin{aligned} V_{\eta }\left( {\varvec{r}}_{i j}\right)&=\frac{g_{c h}^{2}}{4 \pi } \frac{m_{\eta }^{2}}{12 m_{i} m_{j}} \frac{\Lambda _{\eta }^{2}}{\Lambda _{\eta }^{2}-m_{\eta }^{2}} m_{\eta }\left[ Y\left( m_{\eta } {\varvec{r}}_{i j}\right) \right. \nonumber \\&\quad \left. -\frac{\Lambda _{\eta }^{3}}{m_{\eta }^{3}} Y\left( \Lambda _{\eta } {\varvec{r}}_{i j}\right) \right] \left( {\varvec{\sigma }}_{i} \cdot {\varvec{\sigma }}_{j}\right) \left[ \cos \theta _{p} \left( {\varvec{\lambda }}_{i}^{8} \cdot {\varvec{\lambda }}_{j}^{8}\right) \right. \nonumber \\&\quad \left. -\sin \theta _{p}\left( {\varvec{\lambda }}_{i}^{0} \cdot {\varvec{\lambda }}_{j}^{0}\right) \right] \end{aligned}$$

(9)

where $Y(x) = e^{-x}/x$ is the standard Yukawa function. The physical $\eta$ meson is considered by introducing the angle $\theta _{p}$ instead of the octet one. The ${\varvec{\lambda }}^a$ are the SU(3) flavor Gell-Mann matrices. The values of $m_\pi$, $m_K$ and $m_\eta$ are the masses of the SU(3) Goldstone bosons, which adopt the experimental values [106]. The chiral coupling constant $g_{ch}$, is determined from the $\pi N N$ coupling constant through

$$\begin{aligned} \frac{g_{c h}^{2}}{4 \pi } =\left( \frac{3}{5}\right) ^{2} \frac{g_{\pi N N}^{2}}{4 \pi } \frac{m_{u, d}^{2}}{m_{N}^{2}}. \end{aligned}$$

(10)

Since there exists no strange quark in the $qqq\bar{q}c$ systems, the K-meson exchange potential will not influence the energies of the pentaquark systems. The other symbols in the above expressions have their usual meanings.

For the purpose of testing the stability of the calculated results against parameter changes, we used two sets of parameters in this work. All the parameters shown in Table 1 are fixed by masses of the ground baryons and mesons. Table 2 shows the masses of the baryons and mesons used in this work.

Table 1 The values of the Model parameters. The masses of mesons take their experimental values

Table 2 The masses (in MeV) of the baryons and mesons used in this work. Experimental values are taken from the Particle Data Group (PDG) [106]

2.2 Resonating group method and wave functions

The resonating group method (RGM) [107, 108] and generating coordinates method [109, 110] are used to carry out a dynamical calculation. The main feature of the RGM for two-cluster systems is that it assumes two clusters are frozen inside, and only considers the relative motion between the two clusters. So, the conventional ansatz for the two-cluster wave functions is

$$\begin{aligned} \psi _{5q} = {\mathcal {A}}\left[ [\phi _{B}\phi _{M}]^{[\sigma ]IS} \otimes \chi ({\varvec{R}})\right] ^{J} \end{aligned}$$

(11)

where the symbol ${\mathcal {A}}$ is the anti-symmetrization operator, and ${\mathcal {A}} = 1-P_{14}-P_{24}-P_{34}$. $[\sigma ]=[222]$ gives the total color symmetry and all other symbols have their usual meanings. $\phi _{B}$ and $\phi _{M}$ are the $q^{3}$ and $\bar{q}q$ cluster wave functions, respectively. From the variational principle, after variation with respect to the relative motion wave function $\chi (\textbf{R}) =\sum _{L}\chi _{L}(\textbf{R})$, one obtains the RGM equation:

$$\begin{aligned} \int H(\textbf{R},\textbf{R}') \chi (\textbf{R}')d\textbf{R}' =E\int N(\textbf{R},\textbf{R}') \chi (\textbf{R}')d\textbf{R}' \end{aligned}$$

(12)

where $H(\textbf{R},\textbf{R}')$ and $N(\textbf{R},\textbf{R}')$ are Hamiltonian and norm kernels. By solving the RGM equation, we can get the energies E and the wave functions. In fact, it is not convenient to work with the RGM expressions. Then, we expand the relative motion wave function $\chi (\textbf{R})$ by using a set of gaussians with different centers

$$\begin{aligned} \chi ({\varvec{R}})&= \frac{1}{\sqrt{4 \pi }} \left( \frac{6}{5 \pi b^{2}}\right) ^{3 / 4} \sum _{i,L,M} C_{i,L} \nonumber \\&\quad \cdot \int \exp \left[ -\frac{3}{5 b^{2}}\left( {\varvec{R}} -{\varvec{S}}_{i}\right) ^{2}\right] Y_{LM} \left( \hat{{\varvec{S}}}_{i}\right) d \Omega _{{\varvec{S}}_{i}} \end{aligned}$$

(13)

where L is the orbital angular momentum between two clusters, and ${\varvec{S}}_{{\varvec{i}}},$ $i=1,2,\ldots ,n$ are the generator coordinates, which are introduced to expand the relative motion wave function. By including the center of mass motion:

$$\begin{aligned} \phi _{C} ({\varvec{R}}_{C}) = \left( \frac{5}{\pi b^{2}}\right) ^{3/4} e^{-\frac{5{\varvec{R}}^{2}_{C}}{2b^{2}}} \end{aligned}$$

(14)

the ansatz Eq. (11) can be rewritten as

$$\begin{aligned} \psi _{5 q}&= {\mathcal {A}} \sum _{i,L} C_{i,L} \int \frac{d \Omega _{{\varvec{S}}_{i}}}{\sqrt{4 \pi }} \prod _{\alpha =1}^{3} \phi _{\alpha }\left( {\varvec{S}}_{i}\right) \prod _{\beta =4}^{5} \phi _{\beta }\left( -{\varvec{S}}_{i}\right) \nonumber \\&\quad \cdot \left[ \left[ \chi _{I_{1} S_{1}}\left( B\right) \chi _{I_{2} S_{2}} \left( M\right) \right] ^{I S} Y_{LM} \left( \hat{{\varvec{S}}}_{i}\right) \right] ^{J} \nonumber \\&\quad \cdot \left[ \chi _{c}\left( B\right) \chi _{c} \left( M\right) \right] ^{[\sigma ]} \end{aligned}$$

(15)

where $\chi _{I_{1}S_{1}}$ and $\chi _{I_{2}S_{2}}$ are the product of the flavor and spin wave functions, and $\chi _{c}$ is the color wave function. These will be shown in detail later. $\phi _{\alpha }({\varvec{S}}_{i})$ and $\phi _{\beta }(-{\varvec{S}}_{i})$ are the single-particle orbital wave functions with different reference centers:

$$\begin{aligned} \phi _{\alpha }\left( {\varvec{S}}_{i}\right)&= \left( \frac{1}{\pi b^{2}}\right) ^{3 / 4} e^{-\frac{1}{2 b^{2}}\left( r_{\alpha }-\frac{2}{5} {\varvec{S}}_{i}\right) ^{2}}, \nonumber \\ \phi _{\beta }\left( \varvec{-S}_{i}\right)&= \left( \frac{1}{\pi b^{2}}\right) ^{3 / 4} e^{-\frac{1}{2 b^{2}}\left( r_{\beta }+\frac{3}{5} {\varvec{S}}_{i}\right) ^{2}}. \end{aligned}$$

(16)

With the reformulated ansatz Eq. (15), the RGM Eq. (12) becomes an algebraic eigenvalue equation:

$$\begin{aligned} \sum _{j} C_{j}H_{i,j}= E \sum _{j} C_{j}N_{i,j} \end{aligned}$$

(17)

where $H_{i,j}$ and $N_{i,j}$ are the Hamiltonian matrix elements and overlaps, respectively. By solving the generalized eigen problem, we can obtain the energy and the corresponding wave functions of the pentaquark systems.

In QDCSM, in order to take into account the mutual distortion or the internal excitations of nucleons in the course of interaction, quark delocalization was introduced to enlarge the model variational space. It is realized by specifying the single particle orbital wave function of QDCSM as a linear combination of left and right Gaussians. The single particle orbital wave functions used in the ordinary quark cluster model are

$$\begin{aligned} \psi _{\alpha }({\varvec{S}}_{{\varvec{i}}},\epsilon )= & {} \left( \phi _{\alpha }({\varvec{S}}_{{\varvec{i}}}) + \epsilon \phi _{\alpha }(-{\varvec{S}}_{{\varvec{i}}})\right) / N(\epsilon ), \nonumber \\ \psi _{\beta }(-{\varvec{S}}_{{\varvec{i}}},\epsilon )= & {} \left( \phi _{\beta }(-{\varvec{S}}_{{\varvec{i}}}) + \epsilon \phi _{\beta }({\varvec{S}}_{{\varvec{i}}})\right) / N(\epsilon ), \nonumber \\ N(S_{i},\epsilon )= & {} \sqrt{1+\epsilon ^2+2\epsilon e^{-S_i^2/4b^2}}. \end{aligned}$$

(18)

It is important to note that, the mixing parameter $\epsilon$ is not an adjustable parameter, but is instead determined by the dynamics of the multi-quark system itself. In this way, the multi-quark system could choose its favorable configuration in the interacting process. The intermediate-range attraction and the structure of the formed state can be revealed by studying the variation of delocalization parameter $\epsilon$. If the value of the parameter $\epsilon$ is still large at large distance between two cluster, which indicates that the quarks are willing to run between two clusters, then the state is inappropriate to be explained as a molecular state. Conversely, if the value of the parameter $\epsilon$ is close to zero at large distance between two cluster, then the formed state is likely to be a molecular state. This mechanism has been used to explain the cross-over transition between hadron phase and quark–gluon plasma phase [111].

For the spin wave function, we first construct the spin wave functions of the $q^{3}$ and $\bar{q}q$ clusters with SU(2) algebra, and then the total spin wave function of the pentaquark system is obtained by coupling the spin wave functions of two clusters together. The spin wave functions of the $q^{3}$ and $\bar{q}q$ clusters are Eq. (19) and Eq. (20), respectively

$$\begin{aligned} \chi _{\frac{3}{2}, \frac{3}{2}}^{\sigma }(3)&= \alpha \alpha \alpha \nonumber \\ \chi _{\frac{3}{2}, \frac{1}{2}}^{\sigma }(3)&= \frac{1}{\sqrt{3}}(\alpha \alpha \beta +\alpha \beta \alpha +\beta \alpha \alpha )\nonumber \\ \chi _{\frac{3}{2},-\frac{1}{2}}^{\sigma }(3)&= \frac{1}{\sqrt{3}}(\alpha \beta \beta +\beta \alpha \beta +\beta \beta \alpha ) \nonumber \\ \chi _{\frac{1}{2}, \frac{1}{2}}^{\sigma 1}(3)&= \sqrt{\frac{1}{6}}(2 \alpha \alpha \beta -\alpha \beta \alpha -\beta \alpha \alpha ) \end{aligned}$$

(19)

$$\begin{aligned} \chi _{\frac{1}{2}, \frac{1}{2}}^{\sigma 2}(3)&= \sqrt{\frac{1}{2}}(\alpha \beta \alpha -\beta \alpha \alpha ) \nonumber \\ \chi _{\frac{1}{2},-\frac{1}{2}}^{\sigma 1}(3)&= \sqrt{\frac{1}{6}}(\alpha \beta \beta +\beta \alpha \beta -2 \beta \beta \alpha ) \nonumber \\ \chi _{\frac{1}{2},-\frac{1}{2}}^{\sigma 2}(3)&= \sqrt{\frac{1}{2}}(\alpha \beta \beta -\beta \alpha \beta ) \nonumber \\ \chi _{1,1}^{\sigma }(2)&= \alpha \alpha \nonumber \\ \chi _{1,0}^{\sigma }(2)&=\frac{1}{\sqrt{2}}(\alpha \beta +\beta \alpha )\nonumber \\ \chi _{1,-1}^{\sigma }(2)&= \beta \beta \nonumber \\ \chi _{0,0}^{\sigma }(2)&= \frac{1}{\sqrt{2}} (\alpha \beta -\beta \alpha ). \end{aligned}$$

(20)

For pentaquark system, the total spin quantum number can be 1/2, 3/2 or 5/2. Considering that the Hamiltonian does not contain an interaction that can distinguish the third component of the spin quantum number, so the wave function of each spin quantum number can be written as follows

$$\begin{aligned} \chi _{\frac{1}{2}, \frac{1}{2}}^{\sigma 1}(5)&= \chi _{\frac{1}{2}, \frac{1}{2}}^{\sigma }(3) \chi _{0,0}^{\sigma }(2) \nonumber \\ \chi _{\frac{1}{2}, \frac{1}{2}}^{\sigma 2}(5)&= -\sqrt{\frac{2}{3}} \chi _{\frac{1}{2}, -\frac{1}{2}}^{\sigma }(3) \chi _{1,1}^{\sigma }(2) +\sqrt{\frac{1}{3}} \chi _{\frac{1}{2}, \frac{1}{2}}^{\sigma }(3) \chi _{1,0}^{\sigma }(2) \nonumber \\ \chi _{\frac{1}{2}, \frac{1}{2}}^{\sigma 3}(5)&= \sqrt{\frac{1}{6}} \chi _{\frac{3}{2}, -\frac{1}{2}}^{\sigma }(3) \chi _{1,1}^{\sigma }(2) -\sqrt{\frac{1}{3}} \chi _{\frac{3}{2}, \frac{1}{2}}^{\sigma }(3) \chi _{1,0}^{\sigma }(2)\nonumber \\&\quad +\sqrt{\frac{1}{2}} \chi _{\frac{3}{2}, \frac{3}{2}}^{\sigma }(3) \chi _{1,-1}^{\sigma }(2) \nonumber \\ \chi _{\frac{3}{2}, \frac{3}{2}}^{\sigma 4}(5)&= \chi _{\frac{1}{2}, \frac{1}{2}}^{\sigma }(3) \chi _{1,1}^{\sigma }(2) \nonumber \\ \chi _{\frac{3}{2}, \frac{3}{2}}^{\sigma 5}(5)&= \chi _{\frac{3}{2}, \frac{3}{2}}^{\sigma }(3) \chi _{0,0}^{\sigma }(2) \nonumber \\ \chi _{\frac{3}{2}, \frac{3}{2}}^{\sigma 6}(5)&= \sqrt{\frac{3}{5}} \chi _{\frac{3}{2}, \frac{3}{2}}^{\sigma }(3) \chi _{1,0}^{\sigma }(2) -\sqrt{\frac{2}{5}} \chi _{\frac{3}{2}, \frac{1}{2}}^{\sigma }(3) \chi _{1,1}^{\sigma }(2)\nonumber \\ \chi _{\frac{5}{2}, \frac{5}{2}}^{\sigma 7}(5)&= \chi _{\frac{3}{2}, \frac{3}{2}}^{\sigma }(3) \chi _{1,1}^{\sigma }(2). \end{aligned}$$

(21)

Similar to constructing spin wave functions, we first write down the flavor wave functions of the $q^{3}$ clusters, which are

$$\begin{aligned} \chi _{\frac{3}{2}, \frac{3}{2}}^{f}(3)&= u u u \nonumber \\ \chi _{\frac{3}{2}, \frac{1}{2}}^{f}(3)&= \sqrt{\frac{1}{3}} (u u d+u d u+d u u) \nonumber \\ \chi _{1,1}^{f 1}(3)&= \frac{1}{\sqrt{6}} (2 u u c -u c u- c u u) \nonumber \\ \chi _{1,1}^{f 2}(3)&= \frac{1}{\sqrt{2}}(ucu -cuu) \nonumber \\ \chi _{1,1}^{f 3}(3)&= \frac{1}{\sqrt{3}}(uuc+ucu+cuu) \nonumber \\ \chi _{1,0}^{f 1}(3)&= \frac{1}{\sqrt{12}} (2 u d c+2 d u c-c d u-u c d-c u d-d c u) \nonumber \\ \chi _{1,0}^{f 2}(3)&= \frac{1}{\sqrt{4}} (-c d u+u c d-c u d+d c u) \nonumber \\ \chi _{1,0}^{f 3}(3)&= \frac{1}{\sqrt{6}} (udc+duc+ucd+cud+dcu+cdu) \nonumber \\ \chi _{1,-1}^{f 1}(3)&= \frac{1}{\sqrt{6}}(2 d d c -d c d- c d d) \nonumber \\ \chi _{1,-1}^{f 2}(3)&= \frac{1}{\sqrt{2}}(dcd -cdd) \nonumber \\ \chi _{1,-1}^{f 3}(3)&= \frac{1}{\sqrt{3}}(ddc+dcd+cdd) \nonumber \\ \chi _{\frac{1}{2}, \frac{1}{2}}^{f 1}(3)&= \sqrt{\frac{1}{6}}(2 u u d-u d u-d u u) \nonumber \\ \chi _{\frac{1}{2}, \frac{1}{2}}^{f 2}(3)&= \sqrt{\frac{1}{2}}(u d u-d u u) \nonumber \\ \chi _{\frac{1}{2},-\frac{1}{2}}^{f 1}(3)&= \sqrt{\frac{1}{6}}(u d d+d u d-2 d d u) \nonumber \\ \chi _{\frac{1}{2},-\frac{1}{2}}^{f 2}(3)&= \sqrt{\frac{1}{2}}(u d d-d u d). \end{aligned}$$

(22)

Then, the flavor wave functions of $\bar{q}q$ clusters are

$$\begin{aligned} \chi _{1,1}^{f}(2)&= \bar{d} u \nonumber \\ \chi _{1,0}^{f}(2)&= \sqrt{\frac{1}{2}} (\bar{d} d-\bar{u} u) \nonumber \\ \chi _{1,-1}^{f}(2)&= -\bar{u} d \nonumber \\ \chi _{\frac{1}{2}, \frac{1}{2}}^{f}(2)&= \bar{d} c \nonumber \\ \chi _{\frac{1}{2},-\frac{1}{2}}^{f}(2)&= -\bar{u} c. \end{aligned}$$

(23)

As for the flavor degree of freedom, the isospin of pentaquark systems we investigated in this work is $I=0$. The flavor wave functions of pentaquark systems can be expressed as

$$\begin{aligned} \chi _{0,0}^{f 1}(5)&= \chi _{0,0}^{f}(3) \chi _{0,0}^{f}(2) \nonumber \\ \chi _{0,0}^{f 2}(5)&= \sqrt{\frac{1}{2}} \chi _{\frac{1}{2}, \frac{1}{2}}^{f }(3) \chi _{\frac{1}{2},-\frac{1}{2}}^{f}(2) -\sqrt{\frac{1}{2}} \chi _{\frac{1}{2},-\frac{1}{2}}^{f }(3) \chi _{\frac{1}{2}, \frac{1}{2}}^{f}(2) \nonumber \\ \chi _{0,0}^{f 3}(5)&= \sqrt{\frac{1}{3}} \chi _{1,1}^{f}(3) \chi _{1,-1}^{f}(2)-\sqrt{\frac{1}{3}} \chi _{1,0}^{f}(3) \chi _{1,0}^{f}(2) \nonumber \\&\quad +\sqrt{\frac{1}{3}} \chi _{1,-1}^{f}(3) \chi _{1,1}^{f}(2). \end{aligned}$$

(24)

For the color-singlet channel (two clusters are color-singlet), the color wave function can be obtained by $1 \otimes 1$

$$\begin{aligned} \chi ^{c}&= \frac{1}{\sqrt{6}}(r g b-r b g +g b r-g r b+b r g-b g r)\nonumber \\&\quad \cdot \frac{1}{\sqrt{3}}(\bar{r}r+\bar{g}g+\bar{b}b). \end{aligned}$$

(25)

Finally, we can acquire the total wave functions by combining the wave functions of the orbital, spin, flavor and color parts together according to the quantum numbers of the pentaquark systems.

2.3 Stabilization method

To provide the necessary information for experiments to search for exotic hadron states, the coupling calculation between the bound channels and open channels is indispensable. As we know that some channels are bound because of the strong attractions of the system. However, these states will decay to the corresponding open channels by coupling to open channels and become resonance states. Besides, some states will become scattering state by the effect of coupling to both the open and closed channels. The stabilization method, which is also called the real-scaling method, is one of the effective ways to look for the genuine resonance states. This method was proved to be a reliable and mature approach for estimating the energies of long-lived metastable states of electron-atom, electron-molecule, and atom-diatom complexes [112]. It was firstly applied to quark model by Emiko Hiyama [114] to search for $P_c$ states.

Fig. 1

The shape of the resonance state in real-scaling method

Table 3 The energies of the $qqq\bar{q}c$ pentaquark system with quantum numbers $J^P=\frac{1}{2}^-$ (unit: MeV)

In this approach, the distance between baryon and meson clusters is labeled as $S_i$, and the largest one is $S_m$. We scale the finite volume for expansion by increasing $S_m$ continuously. The continuum states (scattering states) will fall down to the corresponding thresholds, while the genuine resonance states will survive after coupling to the scattering states and keep stable with increasing $S_m$. In this way, there will be two cases of the genuine resonance states, which are shown in Fig. 1. (a) If the energy of a scattering state is far away from that of the resonance state which means there is a weak coupling (or no coupling) between the resonance states and the scattering states, the shape of the resonance state is a stable straight line. (b) If the energy of a scattering state is close to the resonance state, there will be a strong coupling between the resonance state and the scattering state, which would appear as an avoid-crossing structure between two declining lines. The avoid-crossing structure will reproduce stably with the continual increase of $S_m$. In this case, one can obtain the possible resonance state by the corresponding avoid-crossing structure. Furthermore, by analyzing the components of avoid-crossing structures, we can determine the formation mechanism of the resonance state. The main component of the avoid-crossing structure is usually bound in single-channel calculation and the avoid-crossing structure is below the threshold of the main component, which can account for the mechanism of the resonance state. Otherwise, if the main component of the avoid-crossing structure is unbound in the single-channel calculation, then it is usually a false resonance state. By this mean, we can distinguish resonance states from scattering states. More details can be found in Refs. [113,114,115,116].

3 The results and discussions

In this work, we investigate the S-wave $qqq\bar{q}c$ pentaquark systems in the framework of QDCSM using two sets of parameters. The quantum numbers of these systems are $I=0$, $J^P = 1/2^-, 3/2^-$ and $5/2^-$. Two structures $qqq-\bar{q}c$ and $qqc-\bar{q}q$, as well as the coupling of these two structures are taken into consideration. To find out if there exists any bound state, we carry out a dynamic bound-state calculation. The real-scaling method is employed to investigate the genuine resonance states. Moreover, the calculation of root mean square (RMS) is helped to explore the structure of the bound states or resonance states on the one hand, and to further estimate whether the observed states are resonance states or scattering states on the other hand.

The numerical results of different systems are listed in Tables 3, 5 and 7, respectively. The first column headed with Parameter represents the results of two sets of parameters. The second column headed with Structure includes $qqq-\bar{q}c$ and $qqc-\bar{q}q$ two kinds. The third and forth columns, headed with $\chi ^{f_i}$ and $\chi ^{\sigma _{j}}$, denote the way how wave functions constructed, which can be seen in Sect. 2.2. The fifth column headed with Channel gives the physical channels involved in the present work. The sixth column headed with $E_{th}(Theo.)$ refers to the theoretical value of non-interacting baryon–meson threshold. The seventh column headed with $E_{sc}$ shows the energy of each single channel. The values of binding energies $(E_B= E_{sc} -E_{th}(Theo.))$ are listed in the eighth column only if $E_B<0$ MeV. $E'_{sc}$ is the corrected energy after the mass correction, which will be introduced in detail later. $E_{ccs}$ is the energy of the channel coupling of single spatial structure, while $E_{cct}$ takes two spatial structures into account in channel coupling. $E'_{ccs}$ and $E'_{cct}$ are the corrected energies, corresponding to different coupling modes. In addition, the proportion of each channel in the channel coupling calculation is shown at the bottom of Tables 3, 5 and 7. With the help of the proportion of each channel, we can further study the influence of the channel coupling.

Fig. 2

Energy spectrum of $J^P=\frac{1}{2}^-$ system

To reduce the theoretical errors, we can shift the mass to $E'_{sc}=M_{1}(Exp.)+M_{2}(Exp.)+E_{B}$, where the experimental values of a baryon $M_{1}(Exp.)$ and a meson $M_{2}(Exp.)$ are used. The above formula is used to handle single channel mass correction. When we deal with the mass correction of the coupled channels, a modified formula will be used. $E^{\prime }_{cc}=E_{cc}+\sum _{i} p_{i}\left[ E_{\textrm{th }}^{i}(\text {Exp.})-E_{\textrm{th }}^{i}(\text {Theo.})\right]$, where $p_{i}$ is the proportion of various physical channels.

3.1 $J^P=\frac{1}{2}^-$ sector

The energies of the systems with $J^P=1/2^-$ are listed in Table 3, including two spatial structures $qqq-\bar{q}c$ and $qqc-\bar{q}q$. Both single channel and channel coupling results are presented. First of all, the most intuitive analysis can be based on the results of single-channel calculations. Obviously, the energies of ND, $ND^*$, $\Lambda _{c}\omega$ and $\Sigma _{c}\pi$ channels are above the corresponding theoretical threshold, which means that none of these channels is bound. The energies of $\Sigma _{c}\rho$ and $\Sigma _{c}^*\rho$ channels are below the corresponding theoretical threshold, with the binding energies $-3$ to $-6$ MeV and $-13$ to $-15$ MeV, respectively. However, the channels of the system are influenced by each other, so it is unavoidable to take into account the channel coupling effect.

To give a better understanding of channel coupling, we couple the channels with the same spatial structure, as well as all the channels with two spatial structures. From Table 3, we can see that for the $qqq-\bar{q}c$ structure, a bound state with the corrected mass 2809–2810 MeV is obtained, which is close to the charmed baryon $\Sigma _{c}(2800)$. Similar conclusions can be found in Refs. [52, 53, 57, 58, 65]. However, this state can decay to the $\Sigma _{c}\pi$ channel, which has a lower energy. For the $qqc-\bar{q}q$ structure, the lowest energy of this system is 2614–2615 MeV, which is 8–9 MeV lower than the lowest energy channel $\Sigma _{c}\pi$. So a bound state with the main component $\Sigma _{c}\pi$ is obtained here. However, it is unclear whether the bound single channels $\Sigma _{c}\rho$ and $\Sigma _{c}^*\rho$ become resonance states or not. So the channel coupling of all channels is needed and the real-scaling method is employed to explore the resonance states. By solving the Schrodinger equation with channel coupling, we can obtain a series of eigenvalues theoretically. All these eigenvalues are used to draw for real-scaling method, although some extreme high energies are not visible in the images that finally presented. On the other hand, only the lowest energy is presented in Table 3, because whether the system can form a bound state depends on whether the lowest energy is below the lowest threshold.

We calculate the energy eigenvalues of the $qqq\bar{q}c$ system by taking the value of $S_m$ from 6.0 to 12.0 fm, to see if there is any stable state. The results of the two sets of parameters are consistent with each other. To save space, we only show the results of PARAM I here. The stabilization plots of the energies of the $qqq\bar{q}c$ system with quantum numbers $J^P=1/2^-$ are shown in Fig. 2. The continuum states fall off towards their respective thresholds, which are marked with red lines. When the distance between baryon and meson clusters is small, the threshold structure is not obvious enough at close distance. This phenomenon is caused by a lack of computing space and this situation will improve with the increase of $S_m$. And for genuine resonance states, which appear as avoid-crossing structure will be marked with blue lines. Bound states below the lowest threshold of the systems are also marked with blue lines. In addition, the main component of the avoid-crossing structures is usually bound in single-channel calculations, which can account for the mechanism of the resonance state.

The RMS can also be used to further estimate whether the observed states are resonance states or scattering states. It is worth noting that, the scattering states have no real RMS since the relative motion wave functions of the scattered states are non-integrable in the infinite space. If we calculate the RMS of scattering states in a limited space, we can only obtain a value that increases with the expansion of computing space. Although the wave function of a resonance state is also non-integrable, we can calculate the RMS of the main component of the resonance state, which is usually bound in single-channel calculation and its wave function is integrable. In this way, we can calculate the RMS of various states to identify the nature of these states by keep expanding the computing space. Besides, the structure of a multi-quark system can also be estimated by calculating the RMS. The results of RMS of the single channel and channel-coupling are listed in the Table 4.

Table 4 The RMS of the $qqq\bar{q}c$ pentaquark system with quantum numbers $J^P=\frac{1}{2}^-$ (unit: fm)

Fig. 3

The delocalization parameter $\epsilon$ of different states

After taking into account the full channel coupling, the lowest energy of the $J^P=1/2^-$ system is pushed down to 2598–2599 MeV, and the corrected mass is 2575 MeV, as shown in Table 3. After calculating the composition, we find that the channel of $\Sigma _{c}\pi$ is the main component with the proportion of 90.0%, while the channel of ND and the rest channels are the minor components, accounting for 8.1% and 1.9%, respectively. From Fig. 2 we can see that the energy of this state is very stable with the increase of $S_m$, which confirms that it is a bound state. Besides, the RMS of this state is stable with the increase of the computing space, which further confirms the bound state conclusion. The value of the RMS of this state is 1.4 fm, indicating that the two clusters are not too close to each other. As discussed in Sect. 2.2, the delocalization parameter $\epsilon$ can be used to reveal the internal structure of multi-quark state. The variation of delocalization parameter $\epsilon$ of different states are shown is Fig. 3. To be clear, the variation of delocalization parameter $\epsilon$ for $\Sigma _{c}\pi$ with $J^P=1/2^-$ and $\Sigma _{c}^{*}\pi$ with $J^P=3/2^-$ are similar and that causes these two lines to overlap. The delocalization parameter $\epsilon$ of $\Sigma _{c}\pi$ with $J^P=1/2^-$ is quite small around 1.4 fm, which means that the quarks are not likely to run between baryon and meson clusters. All these properties show that the bound state with $J^P=1/2^-$ is inclined to be a molecular state, and its mass is close the $\Lambda _{c}(2595)$. Since the $\Lambda _{c}(2595)$ is located very close to the $\Sigma _{c}\pi$ threshold, this observation leads us naturally to consider a predominant baryon–meson structure of this lowest-lying odd parity charmed baryon. Here, we prefer to interpreted $\Lambda _{c}(2595)$ as the molecular state with the main component of $\Sigma _{c}\pi$, and the quantum number is $J^P=1/2^-$. The similar conclusions can be seen in Refs. [44, 52, 62].

As mentioned above, a quasi-bound state is obtained by coupling ND and $ND^*$ channels. From Fig. 2 we can see that the avoid-crossing structures appear at the place around the threshold of ND. However, it is difficult to estimate if there is a resonance state, because the avoid-crossing structure is too close to the threshold of ND. We calculate the RMS of the single ND and $ND^*$ channel, as well as the case by channel-coupling (labeled as $E_{ccs}(2776)$). We find that the values of RMS are very large and they are not stable with the increase of the computing space, which indicates that they are scattering states. So, we conclude that the avoid-crossing structure around the threshold of ND or $ND^*$ is not a resonance state. It is because that the decay rate of different channels to open channels is different, which leads to the different slope of the dots representing energies in the real-scaling figure and forms the avoid-crossing structure. So, we cannot explain $\Sigma _{c}(2800)$ as the ND molecular state with $J^P=1/2^-$ in present work.

Table 5 The energies of the $qqq\bar{q}c$ pentaquark system with quantum numbers $J^P=\frac{3}{2}^-$ (unit: MeV)

It is worth noting that in the middle of Fig. 2, there exists an avoid-crossing structure around 2905 MeV, the mass of which is close to the newly observed $\Lambda _{c}(2910)$. We’re interested in whether it a resonance state and can be used to explain the $\Lambda _{c}(2910)$. At the bottom of Table 4, we present the proportion of each channel of this E(2905) and its main components are $ND^*$ and ND. Although E(2905) is above the thresholds of $ND^*$ and ND, it is still possible to be a color-structure resonance state because the effect of the hidden-color channel-coupling is included in the QDCSM. However, the value of RMS of this state is very large and it is variational with the increase of the computing space, which indicates that it is a scattering state. So it cannot be used to explain the $\Lambda _{c}(2910)$ in this work.

For the $\Sigma _{c}\rho$ and $\Sigma _{c}^*\rho$, the single channel calculation shows that both of them are bound states, and the value of RMS of each channel is also consistent with this conclusion. However, after the full channel-coupling, we cannot find any avoid-crossing structure below the threshold of $\Sigma _{c}\rho$ or $\Sigma _{c}^*\rho$. It is reasonable. There are several channels below $\Sigma _{c}\rho$ and $\Sigma _{c}^*\rho$, which will push the energy of these two states above the thresholds. So, these two bound states disappear after taking into account the effect of channel-coupling.

3.2 $J^P=\frac{3}{2}^-$ sector

The energies of $qqq\bar{q}c$ pentaquark system with quantum numbers $J^P=\frac{3}{2}^-$ are listed in the Table 5. For the $qqq-\bar{q}c$ spatial structure, the $ND^*$ channel is bound in the single channel calculation. At the same time, for the $qqc-\bar{q}q$ structure, the single channel calculation shows that both the $\Sigma _{c}\rho$ and $\Sigma _{c}^{*}\rho$ are bound states, with the binding energy of $-77$ to $-79$ MeV and $-14$ to $-16$ MeV, respectively, while the $\Lambda _{c}\omega$ and $\Sigma _{c}^{*}\pi$ channels are unbound. After coupling all possible channels, a bound state is obtained, whose energy is 2624–2628 MeV (2635–2636 MeV after mass correction). The proportion of each channel is $\Sigma _{c}^*\pi :~95.6\%$, $\Lambda _{c}\omega :~2.9\%$ and the rests: 1.5%, which means that the main component of this bound state is $\Sigma _{c}^*\pi$.

Since the results of the two sets of parameters are consistent, the following discussion is mainly based on PARAM I. Figure 4 and Table 6 show the stabilization plots of the energies and the RMS of the $qqq\bar{q}c$ system with $J^P=3/2^-$, respectively. It is obvious in Fig. 4 that there is a stable state under the lowest threshold, which is marked by the blue line. The value of RMS of this state is stable with the increase of the computing space and it is 1.4 fm, indicating that the two clusters are not too close to each other. Meanwhile, from Fig. 3, the delocalization parameter $\epsilon$ of $\Sigma _{c}^{*}\pi$ with $J^P=3/2^-$ is quite small around it’s RMS. All these properties show that the bound state with $J^P=3/2^-$ tends to be a molecular state, and its mass is close the $\Lambda _{c}(2625)$. So it is possible to interpreted the $\Lambda _{c}(2625)$ as a molecular state with $J^P=3/2^-$ dominated by $\Sigma _{c}^{*}\pi$ channel. The similar explanation could be found in Refs. [48, 49, 52].

In Fig. 4, there are five red lines, which represent the thresholds for each single channel. A little below the threshold line of $ND^*$, we can see that stable avoid-crossing structures repeated periodically there. After calculating the composition, we find that the main component is the $ND^{*}$ channel with the proportion of 66.5%, while the proportion of the $\Sigma _{c}^{*}\pi$ channel is 25.1% and the one of rest channels is 8.4%. The theoretical energy of this structure is 2849 MeV, lower than the threshold of $ND^*$. So these avoid-crossing structures may represent a resonance state. By using the proportion of each channel, the corrected mass 2933 MeV is obtained for this state. Besides, the calculation of the RMS of this state shows that it is stable with the increase of the computing space and the RMS is 1.9 fm, which indicates that it is a resonance state with the molecular structure. The delocalization parameter $\epsilon$ of $ND^*$ from Fig. 3 also confirms the molecular structure of $ND^*$ state. Clearly, the corrected mass of this resonance state is close to the $\Lambda _{c}(2940)$. So the $\Lambda _{c}(2940)$ is likely to be interpreted as a molecular state with $J^P=3/2^-$, and the main component is $ND^{*}$. This conclusion is consistent with the work of Refs. [50, 51, 53,54,55,56,57, 60, 66, 67].

Particularly, there is another repeated avoid-crossing structure below the threshold of $\Sigma _{c}\rho$. The calculated mass of this state is 3160 MeV, and the main component is the $\Sigma _{c}\rho$ channel with the proportion of about 72%. The corrected mass is 3140 MeV, and the RMS of this state is 1.4 fm. All these properties show that it is also a resonance state, which is worth searching in future work. Besides, the delocalization parameter $\epsilon$ of this state around 1.4 fm is not very small, as shown in Fig. 3. This indicates that quarks are likely to run between $\Sigma _{c}$ and $\rho$ clusters, so $\Sigma _{c}\rho$ with $J^P=3/2^-$ is improper to be a molecular state.

As for PARAM II, the results of real-scaling method and RMS are consistent with that of PARAM I, and two resonance states are obtained. the corrected mass of the $ND^*$ and the $\Sigma _{c}\rho$ resonance states are 2933 MeV and 3142 MeV, respectively.

Fig. 4

Energy spectrum of $J^P=\frac{3}{2}^-$ system

Table 6 The RMS of the $qqq\bar{q}c$ pentaquark system with quantum numbers $J^P=\frac{3}{2}^-$ (unit: fm)

3.3 $J^P=\frac{5}{2}^-$ sector

Table 7 The energies of the $qqq\bar{q}c$ pentaquark system with quantum numbers $J^P=\frac{5}{2}^-$ (unit: MeV)

For the $qqq\bar{q}c$ system with $J^P=\frac{5}{2}^-$, since only S-wave channels are considered in present work, there is only one channel $\Sigma _{c}^*\rho$, which is presented in Table 7. The bound-state calculation shows that it is a deeply bound state, with the binding energy of $-105$ to $-106$ MeV. The corrected mass of this state is 3187–3188 MeV and the value of RMS of this state is 1.4 fm. According to the RMS of this state, $\Sigma _{c}^*$ and $\rho$ are not very close to each other. However, considering the deep binding energy of the formed state, it is thoughtless to interpret this state directly as a molecular state. Around 1.4 fm, delocalization parameter $\epsilon$ of this state is still large, as shown in Fig. 3. This indicates a strong attraction between $\Sigma _{c}^*$ and $\rho$ despite the distance between two hadrons. On the basis of the above discussion, the $\Sigma _{c}^*\rho$ state is not suitable to be a molecular state. Although it can decay to some D-wave channels, like ND, $ND^{*}$, $\Lambda _{c}\omega$, $\Sigma _{c}\rho$, and so on, it is still possible to be a resonance, which is worthy of experimental search and research.

4 Summary

In this work, we try to explain some charmed baryons within the pentaquark framework. The S-wave pentaquark systems $qqq\bar{q}c$ with $I = 0,$ $J^P = \frac{1}{2}^-,~\frac{3}{2}^-\ \text {and}\ \frac{5}{2}^-$ are systematically investigated in the QDCSM. The dynamic bound state calculation is carried out to search for any bound state in the $qqq\bar{q}c$ systems. Both the single channel and the channel coupling calculation are performed to explore the effect of the multi-channel coupling. Meanwhile, the real-scaling method is employed to examine the existence of the resonance states and the bound states. We also calculate the RMS of cluster spacing to study the structure of the states and estimate if the state is resonance state or not.

The numerical results show that the effect of the channel coupling is important for forming a bound state and deepening the bondage to some extent. We can draw the following conclusions: (1) Three bound states are obtained in present work, among which $\Lambda _{c}(2595)$ can be interpreted as the molecular state with $J^P=\frac{1}{2}^-$ and the main component is $\Sigma _{c}\pi$, $\Lambda _{c}(2625)$ can be interpreted as the molecular state with $J^P=\frac{3}{2}^-$ and the main component is $\Sigma _{c}^{*}\pi$. Besides, the $\Sigma _{c}^*\rho$ with $J^P=\frac{5}{2}^-$ is predicted to be a deeply bound state with the mass about 3187–3188 MeV. (2) In present work, $\Lambda _{c}(2910)$ cannot be interpreted as a molecular state, and $\Sigma _{c}(2800)$ cannot be explained as the ND molecular state with $J^P=1/2^-$. (3) Two resonance states are obtained, in which the $\Lambda _{c}(2940)$ is likely to be interpreted as a molecular state with $J^P=3/2^-$, and the main component is $ND^{*}$. Besides, a new resonance state $\Sigma _{c}\rho$ with $J^P=3/2^-$ is predicated, whose mass is 3140–3142 MeV. All these charmed states are worth searching in future work.

In describing the multi-quark system, the channel coupling effect has to be taken into account, especially for the resonance state, where the coupling to the open channels will shift the mass of the resonance state, or destroy it. The real-scaling method may be an effective method to pick up the genuine resonance states from the states with discrete energies. Besides, from the above discussion of the charmed baryons, we would like to note that there exist different points of view to the structure of these states. To explore the structure of exotic hadrons, the unquenched quark model may be another critical approach.