# Investigating excited $\Omega_{c}$ states from pentaquark perspective

Ye Yan1<EMAIL_ADDRESS>Xiaohuang Hu2<EMAIL_ADDRESS>Hongxia
Huang1 hxhuang@njnu.edu.cn(Corresponding author) Jialun Ping1
jlping@njnu.edu.cn(Corresponding author) 1Department of Physics, Nanjing
Normal University, Nanjing, Jiangsu 210097, P. R. China  2Changzhou Institute
of Industry Technology, Changzhou, Jiangsu 213164, P. R. China

###### Abstract

Inspired by the recent observation of new $\Omega_{c}^{0}$ states by the LHCb
Collaboration, we explore the excited $\Omega_{c}$ states from the pentaquark
perspective in the quark delocalization color screening model. Our results
indicate that the $\Omega_{c}(3185)$ can be well interpreted as a molecular
$\Xi D$ predominated resonance state with $J^{P}=1/2^{-}$. The
$\Omega_{c}(3120)$ can also be interpreted as a molecular $\Xi_{c}^{*}\bar{K}$
state with $J^{P}=3/2^{-}$ and a new molecular state $\Xi^{*}_{c}\bar{K}^{*}$
with $J^{P}=5/2^{-}$ and a mass of 3527 MeV is predicted, which is worth
searching in the future. Other reported $\Omega_{c}$ states cannot be well
described in the framework of pentaquark systems in present work. The three-
quark excited state, or the unquenched picture may be a good explanation,
which is worth further exploration.

##  I Introduction

In the last few decades, significant experimental progress has been made in
the sector of heavy baryons. Many heavy baryons have been reported, such as
$\Lambda_{c}$ and $\Sigma_{c}$ family Knapp:1976qw ; ARGUS:1993vtm ;
CLEO:1994oxm ; ARGUS:1997snv ; E687:1993bax ; CLEO:2000mbh ; LHCb:2017jym ;
BaBar:2006itc ; Belle:2021qip ; Belle:2014fde ; Ammosov:1993pi ; CLEO:1996czm
; Belle:2004zjl ; Belle:2022hnm , $\Xi_{c}$ family LHCb:2020gge ;
ALICE:2021bli ; Belle:2016lhy ; CLEO:1998wvk ; Belle:2013htj ; Belle:2020ozq ;
LHCb:2020iby ; Belle:2020tom ; Belle:2016tai  and $\Omega_{c}$ family
Belle:2021gtf ; LHCb:2017uwr ; BaBar:2006pve ; LHCb:2021ptx ; Belle:2017ext .
These observations have stimulated extensive interest in understanding the
structures of these baryons. For one thing, verifying these heavy baryons
could deepen our understanding of the non-perturbative behavior of quantum
chromodynamics (QCD) Chen:2016qju ; Swanson:2006st ; Voloshin:2007dx ;
Chen:2016heh ; Huang:2023jec ; Esposito:2016noz ; Lebed:2016hpi ; Guo:2017jvc
. For another thing, with the appearance of heavy baryons that are difficult
to be interpreted simply as traditional three-quark baryons, the study of
multi-quark explanation has become a non-negligible subject.

Among them, the excited $\Omega_{c}$ baryons were considerably enriched by the
LHCb Collaboration in 2017 LHCb:2017uwr . Five narrow $\Omega_{c}^{0}$ states
were observed in the $\Xi_{c}^{+}K^{-}$ invariant mass spectrum, which are
$\Omega_{c}^{0}(3000)$, $\Omega_{c}^{0}(3050)$, $\Omega_{c}^{0}(3065)$,
$\Omega_{c}^{0}(3090)$ and $\Omega_{c}^{0}(3120)$. The narrow width of these
states, along with their unknown quantum numbers and structures, has attracted
broad interest in theoretical work. A classical way to describe these
$\Omega_{c}$ baryons is considering that they are conventional three-quark
excitations, and another way is treating them as multi-quark states.

On the basis of three-quark configuration, $\Omega_{c}$ states have been
studied in the framework of the Lattice QCD Padmanath:2017lng ;
Bahtiyar:2020uuj , the QCD sum rules Agaev:2017jyt ; Wang:2017zjw ;
Aliev:2017led ; Agaev:2017lip ; Wang:2017xam , the light cone QCD sum rules
Chen:2017sci ; Aliev:2018uby , the heavy hadron chiral perturbation theory
Cheng:2017ove , the Regge phenomenology Oudichhya:2021kop ; Oudichhya:2023awb
, the chiral quark model Yang:2017rpg , the constituent quark model
Wang:2017hej ; Wang:2017kfr ; Yao:2018jmc , the quark-diquark model
Wang:2017vnc ; Ali:2017wsf , the quark pair creation model Chen:2017gnu , the
${}^{3}P_{0}$ model Zhao:2017fov ; Garcia-Tecocoatzi:2022zrf , the chiral
quark-soliton model Kim:2017jpx ; Kim:2017khv , the holographic model
Liu:2017frj , the string model Sonnenschein:2017ylo , the harmonic oscillator
based model Santopinto:2018ljf , the light-front quark model Chua:2019yqh ,
the relativized potential quark model Jia:2020vek , the non-relativistic
potential model Luo:2023sra , the relativistic flux tube model Jakhad:2023mni
and other quark models Karliner:2017kfm ; Ortiz-Pacheco:2020hmj .

On the other hand, the pentaquark interpretation of $\Omega_{c}$ states has
been investigated in the framework of the QCD sum rules Wang:2018alb ;
Wang:2021cku , the chiral quark model Yang:2017rpg ; Huang:2017dwn , the
constituent quark model An:2017lwg , the diquark-diquark-antiquark model
Anisovich:2017aqa , the heavy-quark spin symmetry model Nieves:2017jjx , the
one boson exchange model Liu:2018bkx , the vector meson exchange model
Montana:2018teh ; Montana:2017kjw , the meson-baryon interaction model
Ramos:2020bgs , the chiral quark-soliton model Praszalowicz:2022hcp , the
extended local hidden gauge approach Debastiani:2017ewu ; Debastiani:2018adr ,
the Bethe–Salpeter formalism Wang:2017smo , the effective Lagrangian approach
Huang:2018wgr  and the quasipotential Bethe-Salpeter equation approach
Zhu:2022fyb .

Very recently, two new excited states, $\Omega_{c}^{0}(3185)$ and
$\Omega_{c}^{0}(3327)$ are observed in the $\Xi_{c}^{+}K^{-}$ invariant-mass
spectrum by the LHCb Collaboration LHCb:2023zpu . Bisides, the five narrow
$\Omega_{c}$ states obtained before LHCb:2017uwr  are also confirmed. The
measured masses and widths of the two newly found states are

$\displaystyle M_{\Omega_{c}(3185)}$ $\displaystyle=3185.1\pm
1.7_{-0.9}^{+7.4}\pm 0.2\mathrm{MeV},$
$\displaystyle\Gamma_{\Omega_{c}(3185)}$ $\displaystyle=50\pm
7_{-20}^{+10}\mathrm{MeV},$ $\displaystyle M_{\Omega_{c}(3327)}$
$\displaystyle=3327.1\pm 1.2_{-1.3}^{+0.1}\pm 0.2\mathrm{MeV},$
$\displaystyle\Gamma_{\Omega_{c}(3327)}$ $\displaystyle=20\pm
5_{-1}^{+13}\mathrm{MeV}.$ (1)

So far, there have been a few studies on the two newly discovered states. In
Ref. Luo:2023sra , in the framework of the non-relativistic potential model
with Gaussian Expansion Method, the authors’ results imply that the
$\Omega_{c}^{0}(3327)$ is a good candidate of $\Omega_{c}^{0}(1D)$ state with
$J^{P}=5/2^{+}$. In Ref. Yu:2023bxn , the ${}^{3}P_{0}$ model calculation
results support assigning the observed $\Omega_{c}^{0}(3185)$ and
$\Omega_{c}^{0}(3327)$ as the $2S(3/2^{+}$) and $1D(3/2^{+}$) states,
respectively. In Ref. Wang:2023wii , using the QCD sum rules, the
$\Omega_{c}^{0}(3327)$ is assigned to be $D$-wave $\Omega_{c}$ state with
$J^{P}=1/2^{+},3/2^{+}$ or $5/2^{+}$. In Ref. Feng:2023ixl ,
$\Omega_{c}^{0}(3185)$ and $\Omega_{c}^{0}(3327)$ are studied in the effective
Lagrangian approach by assuming they are molecular states. The results support
$\Omega_{c}^{0}(3327)$ as a $J^{P}=3/2^{-}$ $D^{*}\Xi$ molecular state, and
the $\Omega_{c}^{0}(3185)$ may be a meson-baryon molecule with a big $D\Xi$
component. In Ref. Karliner:2023okv , the assignment of $\Omega_{c}^{0}(3185)$
and $\Omega_{c}^{0}(3327)$ to $2S_{1/2}$ and $2S_{3/2}$ is discussed. In Ref.
Yan:2023ttx , within a simple contact-range theory in which the couplings are
saturated by light-meson exchanges, $\Omega_{c}^{0}(3185)$ and
$\Omega_{c}^{0}(3327)$ match the masses of $J=1/2$ $\Xi D$ and $J=3/2$ $\Xi
D^{*}$, respectively. In Ref. Jakhad:2023mni , based on the quark-diquark
configuration with relativistic flux tube model, $\Omega_{c}^{0}(3185)$ and
$\Omega_{c}^{0}(3327)$ is assigned to be $\left|2S,3/2^{+}\right\rangle$ and
$\left|1D,3/2^{+}\right\rangle$. In Ref. Xin:2023gkf , via the QCD sum rules,
the numerical results favor assigning $\Omega_{c}^{0}(3185)$ as the $D\Xi$
molecular state with the $J^{P}=1/2^{-}$, assigning $\Omega_{c}^{0}(3327)$ as
the $D^{*}\Xi$ molecular state with the $J^{P}=3/2^{-}$.

In addition to the above theoretical methods, quark delocalization color
screening model (QDCSM) is a reliable approach, which was developed in the
1990s with the aim of explaining the similarities between nuclear and
molecular forces Wu:1996fm . The model gives a good description of $NN$ and
$YN$ interactions and the properties of deuteron Ping:2000dx ; Ping:1998si ;
Wu:1998wu ; Pang:2001xx . It is also employed to calculate the baryon-baryon
and baryon-meson scattering phase shifts, and the exotic hadronic states are
also studied in this model. Studies show that color screening is an effective
description of the hidden-color channel coupling ChenLZ ; Huang:2011kf . So it
is feasible and meaningful to extend this model to investigate the pentaquark
interpretation of excited $\Omega_{c}$ states.

In this work, we systematically investigate the $ssc\bar{q}q$ systems in order
to find out if there are $\Omega_{c}$ states that are possible to be
interpreted as pentaquark states. The five-body system is calculated by means
of the resonating group method to search for bound states. The strong decay
channels of the $ssc\bar{q}q$ systems are investigated to determine the
resonance states, based on the conservation of the quantum numbers and the
limit of phase space. In order to ensure the reliability and stability of the
calculation results, the parameters used in this work are the same as those
used in the previous work Yan:2022nxp .

This paper is organized as follows. After introduction, the details of QDCSM
are presented in section II. The calculation of the bound state and scattering
phase shift is presented in Section III, along with the discussion and
analysis of the results. Finally, the paper ends with summary in Section IV.

##  II QUARK DELOCALIZATION COLOR SCREENING MODEL (QDCSM)

Herein, the QDCSM is employed to investigate the properties of $ssc\bar{q}q$
systems. The QDCSM is an extension of the native quark cluster model
DeRujula:1975qlm ; Isgur:1978xj ; Isgur:1978wd ; Isgur:1979be . It has been
developed to address multi-quark systems. The detail of the QDCSM can be found
in Refs. Wu:1996fm ; Huang:2011kf ; ChenLZ ; Ping:1998si ; Wu:1998wu ;
Pang:2001xx ; Ping:2000cb ; Ping:2000dx ; Ping:2008tp . In this sector, we
mainly introduce the salient features of this model. The general form of the
pentaquark Hamiltonian is given by

$\displaystyle H=$
$\displaystyle\sum_{i=1}^{5}\left(m_{i}+\frac{\boldsymbol{p}_{i}^{2}}{2m_{i}}\right)-T_{CM}+\sum_{j>i=1}^{5}V(\boldsymbol{r}_{ij}),$
(2)

where $m_{i}$ is the quark mass, $\boldsymbol{p}_{i}$ 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

$\displaystyle V(\boldsymbol{r}_{ij})=$ $\displaystyle
V_{CON}(\boldsymbol{r}_{ij})+V_{OGE}(\boldsymbol{r}_{ij})+V_{\chi}(\boldsymbol{r}_{ij}).$
(3)

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

$\displaystyle V_{CON}(\boldsymbol{r}_{ij})=$ $\displaystyle-
a_{c}\boldsymbol{\lambda}_{i}^{c}\cdot\boldsymbol{\lambda}_{j}^{c}\left[f(\boldsymbol{r}_{ij})+V_{0}\right],$
(4) $\displaystyle f(\boldsymbol{r}_{ij})=$
$\displaystyle\left\\{\begin{array}[]{l}\boldsymbol{r}_{ij}^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}i,j~{}\text{occur
in the same cluster }\\\
\frac{1-e^{-\mu_{q_{i}q_{j}}\boldsymbol{r}_{ij}^{2}}}{\mu_{q_{i}q_{j}}},~{}~{}~{}i,j~{}\text{occur
in different cluster }\end{array}\right.$ (7)

where $a_{c}$, $V_{0}$ and $\mu_{q_{i}q_{j}}$ are model parameters, and
$\boldsymbol{\lambda}^{c}$ 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, $\mu_{qs}^{2}=\mu_{qq}\mu_{ss}$ ChenM . 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 $\mu_{cc}$ HuangPc1 . Moreover, the $P_{c}$ states were well
predicted in the work of Refs. HuangPc1 ; HuangPc2 . 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
$ssc\bar{q}q$ 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 chromomagnetic interactions, is written as

$\displaystyle V_{OGE}(\boldsymbol{r}_{ij})=$
$\displaystyle\frac{1}{4}\alpha_{s_{q_{i}q_{j}}}\boldsymbol{\lambda}_{i}^{c}\cdot\boldsymbol{\lambda}_{j}^{c}$
(8)
$\displaystyle\cdot\left[\frac{1}{r_{ij}}-\frac{\pi}{2}\delta\left(\mathbf{r}_{ij}\right)\left(\frac{1}{m_{i}^{2}}+\frac{1}{m_{j}^{2}}+\frac{4\boldsymbol{\sigma}_{i}\cdot\boldsymbol{\sigma}_{j}}{3m_{i}m_{j}}\right)\right],$

where $\boldsymbol{\sigma}$ is the Pauli matrices and
$\alpha_{s_{q_{i}q_{j}}}$ 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 chromomagnetic interaction in Eq. (II), 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

$\displaystyle V_{\chi}(\boldsymbol{r}_{ij})=$ $\displaystyle
V_{\pi}(\boldsymbol{r}_{ij})+V_{K}(\boldsymbol{r}_{ij})+V_{\eta}(\boldsymbol{r}_{ij}).$
(9)

Among them

$\displaystyle V_{\pi}\left(\boldsymbol{r}_{ij}\right)=$
$\displaystyle\frac{g_{ch}^{2}}{4\pi}\frac{m_{\pi}^{2}}{12m_{i}m_{j}}\frac{\Lambda_{\pi}^{2}}{\Lambda_{\pi}^{2}-m_{\pi}^{2}}m_{\pi}\left[Y\left(m_{\pi}\boldsymbol{r}_{ij}\right)\right.$
$\displaystyle\left.-\frac{\Lambda_{\pi}^{3}}{m_{\pi}^{3}}Y\left(\Lambda_{\pi}\boldsymbol{r}_{ij}\right)\right]\left(\boldsymbol{\sigma}_{i}\cdot\boldsymbol{\sigma}_{j}\right)\sum_{a=1}^{3}\left(\boldsymbol{\lambda}_{i}^{a}\cdot\boldsymbol{\lambda}_{j}^{a}\right),$
(10) $\displaystyle V_{K}\left(\boldsymbol{r}_{ij}\right)=$
$\displaystyle\frac{g_{ch}^{2}}{4\pi}\frac{m_{K}^{2}}{12m_{i}m_{j}}\frac{\Lambda_{K}^{2}}{\Lambda_{K}^{2}-m_{K}^{2}}m_{K}\left[Y\left(m_{K}\boldsymbol{r}_{ij}\right)\right.$
$\displaystyle\left.-\frac{\Lambda_{K}^{3}}{m_{K}^{3}}Y\left(\Lambda_{K}\boldsymbol{r}_{ij}\right)\right]\left(\boldsymbol{\sigma}_{i}\cdot\boldsymbol{\sigma}_{j}\right)\sum_{a=4}^{7}\left(\boldsymbol{\lambda}_{i}^{a}\cdot\boldsymbol{\lambda}_{j}^{a}\right),$
(11) $\displaystyle V_{\eta}\left(\boldsymbol{r}_{ij}\right)=$
$\displaystyle\frac{g_{ch}^{2}}{4\pi}\frac{m_{\eta}^{2}}{12m_{i}m_{j}}\frac{\Lambda_{\eta}^{2}}{\Lambda_{\eta}^{2}-m_{\eta}^{2}}m_{\eta}\left[Y\left(m_{\eta}\boldsymbol{r}_{ij}\right)\right.$
$\displaystyle\left.-\frac{\Lambda_{\eta}^{3}}{m_{\eta}^{3}}Y\left(\Lambda_{\eta}\boldsymbol{r}_{ij}\right)\right]\left(\boldsymbol{\sigma}_{i}\cdot\boldsymbol{\sigma}_{j}\right)\left[\cos\theta_{p}\left(\boldsymbol{\lambda}_{i}^{8}\cdot\boldsymbol{\lambda}_{j}^{8}\right)\right.$
$\displaystyle\left.-\sin\theta_{p}\left(\boldsymbol{\lambda}_{i}^{0}\cdot\boldsymbol{\lambda}_{j}^{0}\right)\right],$
(12)

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 $\boldsymbol{\lambda}^{a}$ 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
ParticleDataGroup:2020ssz . The chiral coupling constant $g_{ch}$, is
determined from the $\pi NN$ coupling constant through

$\displaystyle\frac{g_{ch}^{2}}{4\pi}$
$\displaystyle=\left(\frac{3}{5}\right)^{2}\frac{g_{\pi
NN}^{2}}{4\pi}\frac{m_{u,d}^{2}}{m_{N}^{2}}.$ (13)

Assuming that flavor SU(3) is an exact symmetry, it will only be broken by the
different mass of the strange quark. The other symbols in the above
expressions have their usual meanings. 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.

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

$\displaystyle\psi_{\alpha}(\boldsymbol{S_{i}},\epsilon)$ $\displaystyle=$
$\displaystyle\left(\phi_{\alpha}(\boldsymbol{S_{i}})+\epsilon\phi_{\alpha}(-\boldsymbol{S_{i}})\right)/N(\epsilon),$
$\displaystyle\psi_{\beta}(-\boldsymbol{S_{i}},\epsilon)$ $\displaystyle=$
$\displaystyle\left(\phi_{\beta}(-\boldsymbol{S_{i}})+\epsilon\phi_{\beta}(\boldsymbol{S_{i}})\right)/N(\epsilon),$
$\displaystyle N(S_{i},\epsilon)$ $\displaystyle=$
$\displaystyle\sqrt{1+\epsilon^{2}+2\epsilon e^{-S_{i}^{2}/4b^{2}}}.$ (14)

It is worth noting that the mixing parameter $\epsilon$ is not an adjusted one
but determined variationally by the dynamics of the multi-quark system itself.
In this way, the multi-quark system chooses its favorable configuration in the
interacting process. This mechanism has been used to explain the cross-over
transition between hadron phase and quark-gluon plasma phase Xu .

In addition, the dynamical calculation is carried out using the resonating
group method and the generating coordinates method. The details of the two
methods can be seen in Appendix A, and the way of constructing wave functions
are presented in Appendix B.

Table 1: Model parameters used in this work: $m_{\pi}=0.7$ fm-1, $m_{K}=2.51$ fm-1, $m_{\eta}=2.77$ fm-1, $\Lambda_{\pi}=4.2$ fm-1, $\Lambda_{K}=5.2$ fm-1, $\Lambda_{\eta}=5.2$ fm-1, $g_{ch}^{2}/(4\pi)$=0.54. $b$ |  $m_{q}$ |  $m_{c}$ |  $V_{0_{qq}}$ |  $V_{0_{q\bar{q}}}$ |  $a_{c}$
---|---|---|---|---|---
(fm) | (MeV) | (MeV) | (fm-2) | (fm-2) |  (MeV fm-2)
0.518 | 313 | 1788 | -1.288 | -0.743 | 58.03
$\alpha_{s_{qs}}$ | $\alpha_{s_{qc}}$ | $\alpha_{s_{sc}}$ | $\alpha_{s_{q\bar{q}}}$ | $\alpha_{s_{s\bar{q}}}$ | $\alpha_{s_{c\bar{q}}}$
0.524 | 0.467 | 0.351 | 1.491 | 1.423 | 1.200
Table 2: The masses (in MeV) of the baryons and mesons. Experimental values are taken from the Particle Data Group (PDG) ParticleDataGroup:2020ssz . Hadron |  $I(J^{P})$ |  $M_{Exp}$ |  $M_{Theo}$
---|---|---|---
$N$ | $1/2(1/2^{+})$ | 939 | 939
$\Delta$ | $3/2(3/2^{+})$ | 1232 | 1232
$\Sigma_{c}$ | $1(1/2^{+})$ | 2455 | 2465
$\Sigma^{*}_{c}$ | $1(3/2^{+})$ | 2490 | 2518
$\Lambda_{c}$ | $0(1/2^{+})$ | 2286 | 2286
$\Xi$ | $1/2(1/2^{+})$ | 1318 | 1375
$\Xi^{*}$ | $1/2(3/2^{+})$ | 1536 | 1496
$\Xi_{c}$ | $1/2(1/2^{+})$ | 2467 | 2551
$\Xi_{c}^{\prime}$ | $1/2(1/2^{+})$ | 2577 | 2621
$\Xi_{c}^{*}$ | $1/2(3/2^{+})$ | 2645 | 2638
$\Omega_{c}$ | $0(1/2^{+})$ | 2695 | 2785
$\Omega_{c}^{*}$ | $0(3/2^{+})$ | 2766 | 2796
$\pi$ | $1(0^{-})$ | 139 | 139
$\rho$ | $1(1^{-})$ | 770 | 770
$\omega$ | $0(1^{-})$ | 782 | 722
$\bar{K}$ | $1/2(0^{-})$ | 495 | 495
$\bar{K}^{*}$ | $1/2(1^{-})$ | 892 | 814
$D$ | $1/2(0^{-})$ | 1869 | 1869
$D^{*}$ | $1/2(1^{-})$ | 2007 | 1952

##  III The results and discussions

In this work, we investigate the $S-$wave $ssc\bar{q}q$ pentaquark systems in
the framework of QDCSM with resonating group method. The quantum numbers of
the pentaquark system are $I=0$, $J^{P}=1/2^{-},3/2^{-}$ and $5/2^{-}$. Three
structures $qss-\bar{q}c$, $qsc-\bar{q}s$ and $ssc-\bar{q}q$, as well as the
coupling of these structures are taken into consideration. To find out if
there exists any bound state, we carry out a dynamic bound-state calculation.
The scattering process is also studied to obtain the genuine resonance state.
The introduction of the bound state calculation and scattering process can be
seen in Appendix A. Moreover, the calculation of root mean square (RMS) of
cluster spacing is helpful to explore the structure of the bound state or
resonance state on the one hand, and to further estimate whether the observed
states are resonance state or scattering state on the other hand.

The single-channel results of different systems are listed in Tables 3, 5 and
7, respectively. The first column headed with Structure incluedes
$qss-\bar{q}c$, $qsc-\bar{q}s$ and $ssc-\bar{q}q$ three kinds. The second and
third columns, headed with $\chi^{f_{i}}$ and $\chi^{\sigma_{j}}$, denote the
way how wave functions constructed, which can be seen in Appendix B. The forth
column headed with Channel gives the physical channels involved in the present
work. The fifth column headed with $E_{th}^{Theo}$ refers to the theoretical
value of non-interacting baryon-meson threshold. The sixth 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. Finally, the experimental thresholds $E_{th}^{Exp}$ (the sum
of the experimental masses of the corresponding baryon and meson) along with
corrected energies $E^{\prime}=E_{th}^{Exp}+E_{B}$ are given in last two
columns.

As for coupled-channel, the results are listed in the Tables 4, 6 and 8,
respectively. The first column represents the structures involved in the
channel coupling and the second column is the theoretical value of the lowest
threshold. The third column headed with $E_{cc}$ shows the energy of coupled-
channel. The definitions of $E_{B}$, $E_{th}^{Exp}$ and $E^{\prime}$ in
coupled-channel calculation are similar to their definitions in single-channel
calculation.

###  III.1 $J^{P}=\frac{1}{2}^{-}$ sector

First of all, an intuitive analysis can be based on the results of single-
channel calculation, which is shown in the Table 3. Except for the
$\Omega_{c}\omega$ and $\Omega_{c}^{*}\omega$, the energies of other single-
channels are all higher than the corresponding thresholds. The binding
energies of the $\Omega_{c}\omega$ and $\Omega_{c}^{*}\omega$ are -3 MeV and
-4 MeV, respectively. Since that different channels of the system are
influenced by each other, so it is unavoidable to take into account the
channel coupling effect.

Table 3: The single-channel energies of the $ssc\bar{q}q$ pentaquark system with $J^{P}=\frac{1}{2}^{-}$ (unit: MeV). Structure |  $\chi^{f_{i}}$ |  $\chi^{\sigma_{j}}$ |  Channel |  $E_{th}^{Theo}$ |  $E_{sc}$ |  $E_{B}$ |  $E_{th}^{Exp}$ |  $E^{\prime}$
---|---|---|---|---|---|---|---|---
$qss-\bar{q}c$ | $i=2$ | $j=1$ | $\Xi D$ | 3235 | 3238 | ub | 3187 | 3190
| $i=2$ | $j=2$ | $\Xi D^{*}$ | 3319 | 3321 | ub | 3325 | 3327
| $i=2$ | $j=3$ | $\Xi^{*}D^{*}$ | 3441 | 3447 | ub | 3543 | 3549
$qsc-\bar{q}s$ | $i=2$ | $j=1$ | $\Xi^{\prime}_{c}\bar{K}$ | 3130 | 3137 | ub | 3072 | 3079
| $i=2$ | $j=1$ | $\Xi_{c}\bar{K}$ | 3060 | 3066 | ub | 2962 | 2968
| $i=2$ | $j=2$ | $\Xi^{\prime}_{c}\bar{K}^{*}$ | 3449 | 3454 | ub | 3469 | 3574
| $i=2$ | $j=2$ | $\Xi_{c}\bar{K}^{*}$ | 3379 | 3386 | ub | 3359 | 3366
| $i=2$ | $j=3$ | $\Xi^{*}_{c}\bar{K}^{*}$ | 3466 | 3472 | ub | 3537 | 3543
$ssc-\bar{q}q$ | $i=1$ | $j=2$ | $\Omega_{c}\omega$ | 3548 | 3545 | -3 | 3477 | 3474
| $i=1$ | $j=3$ | $\Omega_{c}^{*}\omega$ | 3558 | 3554 | -4 | 3548 | 3544

In order to better understand channel coupling, we first couple the channels
with the same spatial structure, and then coupled the channels with different
spatial structures. By solving the Schrodinger equation with channel coupling,
we can obtain a series of eigenvalues theoretically. Only the lowest energy is
presented in Table 4, because whether the system can form a bound state
depends on whether the lowest energy is below the lowest threshold. After
coupling the channels with the same spatial structure, the loweset energiges
of $qss-\bar{q}c$ and $qsc-\bar{q}s$ systems are still higher than their
respective thresholds. Besides, the $ssc-\bar{q}q$ system forms a bound state
with the binding energy of 3 MeV.

Further, we couple the channels with two different spatial structures,
$qss-\bar{q}c$ and $ssc-\bar{q}q$, and then add the third spatial structure
$ssc-\bar{q}q$ into the coupling. The result shows that the coupling of
$qss-\bar{q}c$ and $ssc-\bar{q}q$ depresses the lowest energy and makes it 5
MeV below the threshold $\Xi D$. After an overall coupling of all channels,
the lowest energy of the system is still higher than the lowest threshold of
channel $\Xi_{c}\bar{K}$, indicating that the $J^{P}=1/2^{-}$ $ssc\bar{q}q$
pentaquark system does not form a genuine bound state.

Table 4: The coupled-channel energies of the $ssc\bar{q}q$ pentaquark system with $J^{P}=\frac{1}{2}^{-}$ (unit: MeV). Coupled-structure |  $E_{th}^{Theo}$ (Channel) |  $E_{cc}$ |  $E_{B}$ |  $E_{th}^{Exp}$ |  $E^{\prime}$
---|---|---|---|---|---
$qss-\bar{q}c$ | 3235 ($\Xi D$) | 3237 | ub | 3187 | 3190
$qsc-\bar{q}s$ | 3060 ($\Xi_{c}\bar{K}$) | 3065 | ub | 2962 | 2967
$ssc-\bar{q}q$ | 3548 ($\Omega_{c}\omega$) | 3545 | -3 | 3477 | 3474
$qss-\bar{q}c,~{}ssc-\bar{q}q$ | 3235 ($\Xi D$) | 3230 | -5 | 3187 | 3192
$qss-\bar{q}c,~{}qsc-\bar{q}s,~{}ssc-\bar{q}q$ | 3060 ($\Xi_{c}\bar{K}$) | 3064 | ub | 2962 | 2966

According to the results above, some quasi-bound states are obtained in the
single-channel calculation and structure coupling. By coupling to open
channels, these states can decay to the corresponding open channels and may
become resonance states. Yet it is not excluded that these states become
scattered states under the coupling effect of open channels and closed
channels. So to determine whether resonance states would exist, we continue to
study the scattering phase shifts of possible open channels in the QDCSM. The
resonance masses and the decay widths of the resonance states are also
calculated. The current calculation applies only to the decay of $S-$wave open
channels.

First, in order to determine whether $\Xi D$ forms a resonance state, we study
the scattering process of open channels $\Xi_{c}\bar{K}$ and
$\Xi_{c}^{\prime}\bar{K}$, because the two channels have lower thresholds than
the energy of the $\Xi D$ state. The phase shifts of $\Xi_{c}\bar{K}$ and
$\Xi_{c}^{\prime}\bar{K}$ are shown in Fig. 1 and Fig. 2, respectively. It is
obvious that both phase shifts show a sharp increase around the corresponding
resonance mass, which indicates that the $\Xi D$ state becomes a resonance
state in both $\Xi_{c}\bar{K}$ and $\Xi_{c}^{\prime}\bar{K}$ scattering
process. The resonance mass, corrected mass and the decay width are summarized
as follows:

$\displaystyle\text{In}~{}\Xi_{c}\bar{K}~{}\text{channel}:~{}~{}~{}~{}M_{res}^{Theo}$
$\displaystyle=3230~{}\mathrm{MeV},$ $\displaystyle M_{res}^{\prime}$
$\displaystyle=3182~{}\mathrm{MeV},$ $\displaystyle\Gamma_{res}$
$\displaystyle=8.4~{}\mathrm{MeV},$
$\displaystyle\text{In}~{}\Xi_{c}^{\prime}\bar{K}~{}\text{channel}:~{}~{}~{}~{}M_{res}^{Theo}$
$\displaystyle=3221~{}\mathrm{MeV},$ $\displaystyle M_{res}^{\prime}$
$\displaystyle=3174~{}\mathrm{MeV},$ $\displaystyle\Gamma_{res}$
$\displaystyle=33.6~{}\mathrm{MeV}.$

Figure 1:  The phase shift of the open channel $\Xi_{c}\bar{K}$ with
$J^{P}=\frac{1}{2}^{-}$. Figure 2:  The phase shift of the open channel
$\Xi_{c}^{\prime}\bar{K}$ with $J^{P}=\frac{1}{2}^{-}$.

Thus, a resonance state dominated by $\Xi D$ with $J^{P}=1/2^{-}$ in the decay
channel $\Xi_{c}\bar{K}$ and $\Xi_{c}^{\prime}\bar{K}$, with the corrected
resonance mass 3174$\sim$3182 MeV and decay width 42 MeV, is confirmed. This
is consistent with the newly reported $\Omega_{c}(3185)$, the mass and decay
width of which are $3185.1\pm 1.7_{-0.9}^{+7.4}\pm 0.2$ MeV and $50\pm
7_{-20}^{+10}$ MeV, respectively. Therefore, in our quark model calculation,
the $\Omega_{c}(3185)$ can be well interpreted as a $\Xi D$ resonance state
with $J^{P}=1/2^{-}$. In addition, one may be curious about the cusps around
50 MeV and 250 MeV in the Fig. 1. They are caused by the thresholds of the
corresponding single channels and the similar situation can also be found in
Fig. 2.

In order to investigate the structure of this $\Xi D$ resonance, we further
calculate its RMS. It is worth noting that, the scattering state has 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 a scattering
state 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, whose 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. According to the numerical result, the RMS
of the resonance state $\Xi D$ is 1.9 fm, indicating that it is likely to be a
molecular state.

$\Omega_{c}\omega$, $\Omega_{c}^{*}\omega$ and their coupling also form quasi-
bound states in the previous calculations. The energies of the
$\Omega_{c}\omega$ and $\Omega_{c}^{*}\omega$ single channels are about
485$\sim$495 MeV above the threshold of $\Xi_{c}\bar{K}$ and 415$\sim$425 MeV
above the threshold of $\Xi_{c}^{\prime}\bar{K}$. However, in Fig. 1 and Fig.
2, the sharp increase structure of phase shift representing the resonance
state, does not appear around the energies of $\Omega_{c}\omega$ and
$\Omega_{c}^{*}\omega$ channels. Besides, it is still possible that
$\Omega_{c}\omega$ and $\Omega_{c}^{*}\omega$ decay to other open channels
than $\Xi_{c}\bar{K}$ or $\Xi_{c}^{\prime}\bar{K}$ channels.

Therefore, we also calculate the phase shifts of other different open channels
with channel coupling, which are shown in Fig. 3. The ranges of incident
energy for different open channels are determined to fit the energy of
$\Omega_{c}^{*}\omega$, which is the highest energy of the system. As a
result, the ranges of incident energy for different open channels are not the
same in Fig. 3. After considering the different decay channels, no resonance
state of $\Omega_{c}\omega$ or $\Omega_{c}^{*}\omega$ is found. This can be
explained by the effect of channel coupling, which should be fully considered.
As listed in the Table 4, the energy of $qss-\bar{q}c$ structure coupling is
above the corresponding threshold $\Xi D$. Nevertheless, the energy of
$qss-\bar{q}c$ structure is pushed below the threshold $\Xi D$, after being
coupled to the $ssc-\bar{q}q$ structure. As a result, the energy of the
previous quasi-bound state $\Omega_{c}\omega$ is pushed above the
corresponding threshold by a reaction, which causes the narrow resonance state
to disappear after coupling with the $qss-\bar{q}c$ structure. The same thing
happens in the phase shift of the open channel $\Xi_{c}^{\prime}\bar{K}$.
Therefore, the narrow resonance state we just discussed is not a genuine
resonance state.

Figure 3:  The phase shifts of different open channels with
$J^{P}=\frac{1}{2}^{-}$.

###  III.2 $J^{P}=\frac{3}{2}^{-}$ sector

The single-channel energies of $ssc\bar{q}q$ system with $J^{P}=3/2^{-}$ are
listed in the Table 5. Two bound states are obtained in the $\Omega_{c}\omega$
and $\Omega^{*}_{c}\omega$ channel, while the energies of other channels are
all above the corresponding thresholds. The binding energies of the
$\Omega_{c}\omega$ and $\Omega^{*}_{c}\omega$ state are -2 MeV and -4 MeV,
respectively.

Table 5: The single-channel energies of the $ssc\bar{q}q$ pentaquark system with $J^{P}=\frac{3}{2}^{-}$ (unit: MeV). Structure |  $\chi^{f_{i}}$ |  $\chi^{\sigma_{j}}$ |  Channel |  $E_{th}^{Theo}$ |  $E_{sc}$ |  $E_{B}$ |  $E_{th}^{Exp}$ |  $E^{\prime}$
---|---|---|---|---|---|---|---|---
$qss-\bar{q}c$ | $i=2$ | $j=4$ | $\Xi D^{*}$ | 3319 | 3323 | ub | 3325 | 3329
| $i=2$ | $j=5$ | $\Xi^{*}D$ | 3357 | 3362 | ub | 3405 | 3410
| $i=2$ | $j=6$ | $\Xi^{*}D^{*}$ | 3441 | 3446 | ub | 3543 | 3548
$qsc-\bar{q}s$ | $i=2$ | $j=4$ | $\Xi^{\prime}_{c}\bar{K}^{*}$ | 3449 | 3457 | ub | 3469 | 3477
| $i=2$ | $j=4$ | $\Xi_{c}\bar{K}^{*}$ | 3379 | 3386 | ub | 3359 | 3366
| $i=2$ | $j=5$ | $\Xi_{c}^{*}\bar{K}$ | 3147 | 3153 | ub | 3140 | 3146
| $i=2$ | $j=6$ | $\Xi^{*}_{c}\bar{K}^{*}$ | 3466 | 3472 | ub | 3537 | 3543
$ssc-\bar{q}q$ | $i=1$ | $j=4$ | $\Omega_{c}\omega$ | 3548 | 3546 | -2 | 3477 | 3475
| $i=1$ | $j=6$ | $\Omega_{c}^{*}\omega$ | 3558 | 3554 | -4 | 3548 | 3544

For the $ssc\bar{q}q$ system with $J^{P}=3/2^{-}$, channel coupling of various
structures is also considered, which is listed in Table 6. Similar to the
previous section with $J^{P}=1/2^{-}$, we first carry out the channel coupling
with the same spatial structure. Single structure coupling $qss-\bar{q}c$ and
$qsc-\bar{q}s$ are all unbound according to the numerical results. In
addition, the $ssc-\bar{q}q$ structure coupling slightly depresses the energy
of $\Omega_{c}\omega$, although it is not numerically significant.

Table 6: The coupled-channel energies of the $ssc\bar{q}q$ pentaquark system with $J^{P}=\frac{3}{2}^{-}$ (unit: MeV). Coupled-structure |  $E_{th}^{Theo}$ (Channel) |  $E_{cc}$ |  $E_{B}$ |  $E_{th}^{Exp}$ |  $E^{\prime}$
---|---|---|---|---|---
$qss-\bar{q}c$ | 3319 ($\Xi D^{*}$) | 3322 | ub | 3325 | 3328
$qsc-\bar{q}s$ | 3147 ($\Xi_{c}^{*}\bar{K}$) | 3150 | ub | 3140 | 3143
$ssc-\bar{q}q$ | 3548 ($\Omega_{c}\omega$) | 3546 | -2 | 3477 | 3475
$qss-\bar{q}c,~{}ssc-\bar{q}q$ | 3319 ($\Xi D^{*}$) | 3321 | ub | 3325 | 3327
$qss-\bar{q}c,~{}qsc-\bar{q}s,~{}ssc-\bar{q}q$ | 3147 ($\Xi_{c}^{*}\bar{K}$) | 3145 | -2 | 3140 | 3138

After calculating two different structure coupling, we continue to add the
third structure into the coupling. As one can see, after coupling all three
structures, the lowest energy of the whole system is 2 MeV lower than the
energy of the threshold $\Xi_{c}^{*}\bar{K}$. Since the $\Xi_{c}^{*}\bar{K}$
is the lowest threshold of the $ssc\bar{q}q$ system with $J^{P}=3/2^{-}$, a
stable bound state is obtained and its corrected mass is 3138 MeV. Besides,
the bound state conclusion can also be confirmed in the scattering process. In
Fig 4, as the incident energy approaches 0 MeV, the phase shift of the open
channel $\Xi_{c}^{*}\bar{K}$ tends to 180 degrees, which conforms to the
characteristics of a bound state. According to the further calculation, this
state is dominated by $\Xi_{c}^{*}\bar{K}$ and the RMS calculation is 1.8 fm.
The mass is close to the mass of $\Omega_{c}(3120)$, which is $3119.1\pm
0.3\pm 0.9\pm 0.3$ MeV. In addition, the bound state $\Xi_{c}^{*}\bar{K}$ can
still decay to some $D$-wave channels, such as $\Xi_{c}\bar{K}$, through the
tensor force coupling, which is the next step of our research in the future.
However, the decay width of this type of decay is usually very narrow,
according to our previous research Chen:2011zzb . This corresponds to the
decay width of $\Omega_{c}(3120)$, which is $0.60\pm 0.63$ MeV. In this case,
$\Omega_{c}(3120)$ could be interpreted as a $\Xi_{c}^{*}\bar{K}$ molecular
state with $J^{P}=3/2^{-}$ in present calculation.

Figure 4:  The phase shift of the open channel $\Xi_{c}^{*}\bar{K}$ with
$J^{P}=\frac{3}{2}^{-}$.

Furthermore, the scattering process is studied to examine whether
$\Omega_{c}\omega$ and $\Omega_{c}^{*}\omega$ can form resonance states. The
phase shifts of different $S-$wave open channels with channel coupling are
shown in Fig 4 and Fig 5. However, the phase shifts of all open channels do
not show a sharp increase around the energies of the quasi-bound state
$\Omega_{c}\omega$ or $\Omega_{c}^{*}\omega$. The result shows that the
$\Omega_{c}\omega$ and $\Omega_{c}^{*}\omega$ become scattering states rather
than resonance states after being coupled to other channels.

Figure 5:  The phase shifts of different open channels with
$J^{P}=\frac{3}{2}^{-}$.

###  III.3 $J^{P}=\frac{5}{2}^{-}$ sector

For the $ssc\bar{q}q$ system with $J^{P}=5/2^{-}$, there is three channels
$\Xi^{*}D^{*}$, $\Xi_{c}^{*}\bar{K}^{*}$ and $\Omega_{c}^{*}\omega$. The
energies obtained in the single-channel calculation are presented in Table 7.
The $\Omega_{c}^{*}\omega$ forms a bound state, which will be examined later
to see if it is a resonance state.

Table 7: The single-channel energies of the $ssc\bar{q}q$ pentaquark system with $J^{P}=\frac{5}{2}^{-}$ (unit: MeV). Structure |  $\chi^{f_{i}}$ |  $\chi^{\sigma_{j}}$ |  Channel |  $E_{th}^{Theo}$ |  $E_{sc}$ |  $E_{B}$ |  $E_{th}^{Exp}$ |  $E^{\prime}$
---|---|---|---|---|---|---|---|---
$qss-\bar{q}c$ | $i=2$ | $j=7$ | $\Xi^{*}D^{*}$ | 3441 | 3444 | ub | 3543 | 3546
$qsc-\bar{q}s$ | $i=2$ | $j=7$ | $\Xi^{*}_{c}\bar{K}^{*}$ | 3466 | 3474 | ub | 3537 | 3545
$ssc-\bar{q}q$ | $i=1$ | $j=7$ | $\Omega_{c}^{*}\omega$ | 3558 | 3555 | -3 | 3548 | 3545

As is shown in Table 8, since each structure has only one channel, the channel
coupling of one structure is not needed here. After coupling the
$qss-\bar{q}c$ and the $ssc-\bar{q}q$ structure, the energy obtained is still
above the threshold. Besides, after coupling all three channels, a bound state
is formed. The corrected mass of this state is 3527 MeV and the value of RMS
of this state is 1.7 fm. According to the RMS of this state, it tends to be
molecular structure and its main composition is $\Xi^{*}_{c}\bar{K}^{*}$.
Therefore, a $J^{P}=5/2^{-}$ $ssc\bar{q}q$ pentaquark state is predicted here,
whose mass is 3527 MeV. Although it can decay to some $D$-wave channels, such
as $\Xi D$ and $\Xi_{c}^{\prime}\bar{K}$, it is still possible to be a
resonance, which is worthy of experimental search and research.

Table 8: The coupled-channel energies of the $ssc\bar{q}q$ pentaquark system with $J^{P}=\frac{5}{2}^{-}$ (unit: MeV). Coupled-structure |  $E_{th}^{Theo}$ (Channel) |  $E_{cc}$ |  $E_{B}$ |  $E_{th}^{Exp}$ |  $E^{\prime}$
---|---|---|---|---|---
$qss-\bar{q}s,~{}ssc-\bar{q}q$ | 3466 ($\Xi^{*}_{c}\bar{K}^{*}$) | 3469 | ub | 3537 | 3540
$qss-\bar{q}c,~{}qsc-\bar{q}s,~{}ssc-\bar{q}q$ | 3441 ($\Xi^{*}D^{*}$) | 3430 | -11 | 3537 | 3527

The scattering process is studied to examine whether $\Omega_{c}^{*}\omega$
could be a resonance state. The phase shifts of the $S-$wave open channels
$\Xi^{*}D^{*}$ and $\Xi_{c}^{*}\bar{K}^{*}$ are shown in Fig 6. However, there
is no sharp increase structure of phase shift around the energy of the
$\Omega_{c}^{*}\omega$ single channel. This indicates that the
$\Omega_{c}^{*}\omega$ does not form a resonance state, but rather a
scattering state. In addition, when the incident energy approaches zero, the
behavior phase shift also confirms the existence of the bound state.

Figure 6:  The phase shifts of different open channels with
$J^{P}=\frac{5}{2}^{-}$.

Considering that there have been a few theoretical works on the newly reported
$\Omega_{c}^{0}(3185)$ and $\Omega_{c}^{0}(3327)$ states, we make a brief
review here. For the $\Omega_{c}^{0}(3185)$ state, Refs. Yu:2023bxn ;
Karliner:2023okv ; Jakhad:2023mni  interpret it as a three-quark excited
state. The quantum number assignment $J^{P}~{}(nL)$ for the
$\Omega_{c}^{0}(3185)$ state could be: $J^{P}=3/2^{+}~{}(2S$) Yu:2023bxn ;
Jakhad:2023mni  and $J^{P}=1/2^{+}~{}(2S$) Karliner:2023okv . On the other
hand, the explanation of the $\Omega_{c}^{0}(3185)$ as a $\Xi D$ molecular
state with $J^{P}=1/2^{-}$ can be found in Refs. Yan:2023ttx ; Xin:2023gkf .

As for the $\Omega_{c}^{0}(3327)$ state, the three-quark excitation
explanation can be found in Refs. Luo:2023sra ; Yu:2023bxn ; Wang:2023wii ;
Karliner:2023okv ; Jakhad:2023mni . The quantum number assignment for the
$\Omega_{c}^{0}(3327)$ state could be: $J^{P}=5/2^{+}~{}(1D$) Luo:2023sra ,
$J^{P}=3/2^{+}~{}(1D$) Yu:2023bxn ; Jakhad:2023mni , $J^{P}=1/2^{+},3/2^{+}$
or $5/2^{+}$ ($D-$wave) Wang:2023wii  and $J^{P}=3/2^{+}~{}(2S$)
Karliner:2023okv . Refs. Feng:2023ixl ; Yan:2023ttx ; Xin:2023gkf  also
support the $\Omega_{c}^{0}(3327)$ state to be interpreted as a $\Xi D^{*}$
molecular state with $J^{P}=3/2^{-}$. The conclusions of the studies on the
two newly discovered $\Omega_{c}$ states from different theoretical groups are
summarized in Table 9. According to our calculation, the
$\Omega_{c}^{0}(3185)$ can be well interpreted as a $\Xi D$ molecular state
with $J^{P}=1/2^{-}$, whereas $\Omega_{c}(3327)$ is not found within the
multi-quark framework. Therefore, we propose to explain $\Omega_{c}(3327)$
from the perspective of the three-quark excitation, and at the same time, the
investigation of these states in an unquenched picture could be beneficial.

Table 9: The conclusions of the studies on the two newly discovered $\Omega_{c}$ states. | $\Omega_{c}^{0}(3185)$ | $\Omega_{c}^{0}(3327)$
---|---|---
Ref | three-quark |  molecular |  three-quark |  molecular
Ref. Luo:2023sra  |  |  | $\checkmark$ |
Ref. Yu:2023bxn  | $\checkmark$ |  | $\checkmark$ |
Ref. Wang:2023wii  |  |  | $\checkmark$ |
Ref. Feng:2023ixl  |  | $\checkmark$ |  | $\checkmark$
Ref. Karliner:2023okv  | $\checkmark$ |  | $\checkmark$ |
Ref. Yan:2023ttx  |  | $\checkmark$ |  | $\checkmark$
Ref. Jakhad:2023mni  | $\checkmark$ |  | $\checkmark$ |
Ref. Xin:2023gkf  |  | $\checkmark$ |  | $\checkmark$
This Work |  | $\checkmark$ |  |

In addition to exploring the pentaquark explanation of the two newly
discovered $\Omega_{c}$ states, the study of the $ssc\bar{q}q$ system has also
led to some other results. In this work, three states are obtained, including
one resonance state and two bound states. We have summarized the obtained
states in Table 10. Considering that the resonance energy of $\Xi D$ is
obtained in the scattering phase shift, the resonance energies obtained in
different open channels are not exactly the same. Therefore, the mass of $\Xi
D$ state has a range. One may notice that there is no value for the decay
widths of $\Xi_{c}^{*}\bar{K}$ and $\Xi^{*}_{c}\bar{K}^{*}$ in Table 10. Since
$\Xi_{c}^{*}\bar{K}$ and $\Xi^{*}_{c}\bar{K}^{*}$ cannot decay to $S$-wave
channels, the decay process of the two states to $D$-wave channels will be
studied in our future works. In addition, the decay width to the $D$-wave
channels is usually narrow according to our previous research Chen:2011zzb .

Table 10:  The states obtained in this work. $J^{P}$ |  Main Composition |  Corrected Mass |  decay width
---|---|---|---
$1/2^{-}$ | $\Xi D$ | 3174$\sim$3182 MeV | 42 MeV
$3/2^{-}$ | $\Xi_{c}^{*}\bar{K}$ | 3138 MeV |
$5/2^{-}$ | $\Xi^{*}_{c}\bar{K}^{*}$ | 3527 MeV |

##  IV Summary

In this work, we investigate the excited $\Omega_{c}$ states from the
pentaquark perspective. The $S$-wave pentaquark systems $ssc\bar{q}q$ with $I$
= 0, $J^{P}=1/2^{-}$, $3/2^{-}$ and $5/2^{-}$ are studied in the framework of
the QDCSM. The dynamic bound state calculation is carried out to search for
bound states in the $ssc\bar{q}q$ systems. Both the single-channel and the
coupled-channel calculations are performed to explore the effect of the multi-
channel coupling. Meanwhile, the study of the scattering process of the open
channels is carried out to confirm possible resonance states. We also
calculate the RMS of cluster spacing to further study the structure of the
obtained states.

The numerical results show that a $\Xi D$ resonance state with $J^{P}=1/2^{-}$
and two bound states with $J^{P}=3/2^{-}$ and $5/2^{-}$ are obtained. The mass
and the decay width of the $\Xi D$ resonance state is 3174$\sim$3182 MeV and
42 MeV, respectively, which are close to the reported $\Omega_{c}^{0}(3185)$.
The RMS of the $\Xi D$ supports the molecular structure of this state. So the
recently reported $\Omega_{c}^{0}(3185)$ can be explained as the molecular
$\Xi D$ state with $J^{P}=1/2^{-}$. It would be very anticipated to see the
next experimental steps to determine the spin and parity of it. A bound
molecular state we obtained is $\Xi_{c}^{*}\bar{K}$ with $J^{P}=3/2^{-}$ and a
mass of 3138 MeV, which can be used to interpret the reported
$\Omega_{c}^{0}(3120)$. Besides, a new molecular state
$\Xi^{*}_{c}\bar{K}^{*}$ with $J^{P}=5/2^{-}$ and a mass of 3527 MeV is
predicted to exist, which is worth searching in the future. However, other
reported $\Omega_{c}$ states cannot be well described in the framework of
pentaquark systems in present work. The three-quark excitation, or the
unquenched picture may be a good explanation, which is worth further
exploration.

In addition, the present study shows that the channel coupling effect has to
be considered in describing the multi-quark system. Especially for the
possible resonance state, the coupling to the open channels will shift the
mass of the resonance state, or even destroy it. The $\Omega_{c}\omega$ and
$\Omega_{c}^{*}\omega$ bound states are obtained in the single-channel
calculation. However, the energies of $\Omega_{c}\omega$ and
$\Omega_{c}^{*}\omega$ are elevated by coupling to open channels, leading to
the disappearance of these two states. Based on this, we would like to
emphasize the importance of channel coupling effect in studying exotic hadron
states.

###### Acknowledgements.

This work is supported partly by the National Science Foundation of China
under Contract Nos. 11675080, 11775118, 11535005 and 11865019.

## Appendix A: Resonating group method for bound-state and scattering process

The resonating group method (RGM) RGM1 ; RGM  and generating coordinates
method GCM1 ; GCM2  are used to carry out a dynamical calculation. The main
feature of the RGM for two-cluster systems is that it assumes that 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

$\psi_{5q}={\cal
A}\left[[\phi_{B}\phi_{M}]^{[\sigma]IS}\otimes\chi(\boldsymbol{R})\right]^{J},$
(A1)

where the symbol ${\cal A}$ is the anti-symmetrization operator, and ${\cal
A}=1-P_{14}-P_{24}-P_{34}$. $[\sigma]=[222]$ 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
$\chi(\boldsymbol{\mathbf{R}})=\sum_{L}\chi_{L}(\boldsymbol{\mathbf{R}})$, one
obtains the RGM equation:

$\int
H(\boldsymbol{\mathbf{R}},\boldsymbol{\mathbf{R^{\prime}}})\chi(\boldsymbol{\mathbf{R^{\prime}}})d\boldsymbol{\mathbf{R^{\prime}}}=E\int
N(\boldsymbol{\mathbf{R}},\boldsymbol{\mathbf{R^{\prime}}})\chi(\boldsymbol{\mathbf{R^{\prime}}})d\boldsymbol{\mathbf{R^{\prime}}},$
(A2)

where $H(\boldsymbol{\mathbf{R}},\boldsymbol{\mathbf{R^{\prime}}})$ and
$N(\boldsymbol{\mathbf{R}},\boldsymbol{\mathbf{R^{\prime}}})$ 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
$\chi(\boldsymbol{\mathbf{R}})$ by using a set of gaussians with different
centers

$\displaystyle\chi(\boldsymbol{R})=$
$\displaystyle\frac{1}{\sqrt{4\pi}}\left(\frac{6}{5\pi
b^{2}}\right)^{3/4}\sum_{i,L,M}C_{i,L}$
$\displaystyle\cdot\int\exp\left[-\frac{3}{5b^{2}}\left(\boldsymbol{R}-\boldsymbol{S}_{i}\right)^{2}\right]Y_{L,M}\left(\hat{\boldsymbol{S}}_{i}\right)d\Omega_{\boldsymbol{S}_{i}}$
(A3)

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

$\phi_{C}(\boldsymbol{R}_{C})=(\frac{5}{\pi
b^{2}})^{3/4}e^{-\frac{5\boldsymbol{R}^{2}_{C}}{2b^{2}}},$ (A4)

the ansatz Eq. (A1) can be rewritten as

$\displaystyle\psi_{5q}=$
$\displaystyle\mathcal{A}\sum_{i,L}C_{i,L}\int\frac{d\Omega_{\boldsymbol{S}_{i}}}{\sqrt{4\pi}}\prod_{\alpha=1}^{3}\phi_{\alpha}\left(\boldsymbol{S}_{i}\right)\prod_{\beta=4}^{5}\phi_{\beta}\left(-\boldsymbol{S}_{i}\right)$
$\displaystyle\cdot\left[\left[\chi_{I_{1}S_{1}}\left(B\right)\chi_{I_{2}S_{2}}\left(M\right)\right]^{IS}Y_{LM}\left(\hat{\boldsymbol{S}}_{i}\right)\right]^{J}$
$\displaystyle\cdot\left[\chi_{c}\left(B\right)\chi_{c}\left(M\right)\right]^{[\sigma]},$
(A5)

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. $\phi_{\alpha}(\boldsymbol{S}_{i})$ and
$\phi_{\beta}(-\boldsymbol{S}_{i})$ are the single-particle orbital wave
functions with different reference centers:

$\displaystyle\phi_{\alpha}\left(\boldsymbol{S}_{i}\right)$
$\displaystyle=\left(\frac{1}{\pi
b^{2}}\right)^{3/4}e^{-\frac{1}{2b^{2}}\left(r_{\alpha}-\frac{2}{5}\boldsymbol{S}_{i}\right)^{2}},$
$\displaystyle\phi_{\beta}\left(\boldsymbol{-S}_{i}\right)$
$\displaystyle=\left(\frac{1}{\pi
b^{2}}\right)^{3/4}e^{-\frac{1}{2b^{2}}\left(r_{\beta}+\frac{3}{5}\boldsymbol{S}_{i}\right)^{2}}.$
(A6)

With the reformulated ansatz Eq. (A5), the RGM Eq. (A2) becomes an algebraic
eigenvalue equation:

$\sum_{j}C_{j}H_{i,j}=E\sum_{j}C_{j}N_{i,j},$ (A7)

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.

For a scattering problem, the relative wave function is expanded as

$\displaystyle\chi_{L}(\mathbf{R})$
$\displaystyle=\sum_{i}C_{i}\frac{\tilde{u}_{L}\left(\boldsymbol{R},\boldsymbol{S}_{i}\right)}{\boldsymbol{R}}Y_{L,M}(\hat{\boldsymbol{R}}),$
(A8)

with

$\displaystyle\tilde{u}_{L}\left(\boldsymbol{R},\boldsymbol{S}_{i}\right)$
$\displaystyle=\left\\{\begin{array}[]{ll}\alpha_{i}u_{L}\left(\boldsymbol{R},\boldsymbol{S}_{i}\right),&\boldsymbol{R}\leq\boldsymbol{R}_{C}\\\
{\left[h_{L}^{-}(\boldsymbol{k},\boldsymbol{R})-s_{i}h_{L}^{+}(\boldsymbol{k},\boldsymbol{R})\right]R_{AB},}&\boldsymbol{R}\geq\boldsymbol{R}_{C}\end{array}\right.$
(A11)

where

$\displaystyle u_{L}\left(\boldsymbol{R},\boldsymbol{S}_{i}\right)=$
$\displaystyle\sqrt{4\pi}\left(\frac{6}{5\pi
b^{2}}\right)^{3/4}\mathbf{R}e^{-\frac{3}{5b^{2}}\left(\boldsymbol{R}-\boldsymbol{S}_{i}\right)^{2}}$
$\displaystyle\cdot i^{L}j_{L}\left(-i\frac{6}{5b^{2}}S_{i}\right).$ (A12)

$h^{\pm}_{L}$ is the $L$-th spherical Hankel functions, $k$ is the momentum of
the relative motion with $k=\sqrt{2\mu E_{ie}}$, $\mu$ is the reduced mass of
two hadrons of the open channel, $E_{ie}$ is the incident energy of the
relevant open channels, which can be written as $E_{ie}=E_{total}-E_{th}$
where $E_{total}$ denotes the total energy and $E_{th}$ represents the
threshold of open channel. $R_{C}$ is a cutoff radius beyond which all the
strong interaction can be disregarded. Besides, $\alpha_{i}$ and $s_{i}$ are
complex parameters that are determined by the smoothness condition at
$R=R_{C}$ and $C_{i}$ satisfy $\sum_{i}C_{i}=1$. After performing the
variational procedure, a $L$-th partial-wave equationfor the scattering
problem can be deduced as

$\displaystyle\sum_{j}\mathcal{L}_{ij}^{L}C_{j}$
$\displaystyle=\mathcal{M}_{i}^{L}(i=0,1,\ldots,n-1),$ (A13)

with

$\displaystyle\mathcal{L}_{ij}^{L}$
$\displaystyle=\mathcal{K}_{ij}^{L}-\mathcal{K}_{i0}^{L}-\mathcal{K}_{0j}^{L}+\mathcal{K}_{00}^{L},$
$\displaystyle\mathcal{M}_{i}^{L}$
$\displaystyle=\mathcal{K}_{00}^{L}-\mathcal{K}_{i0}^{L},$ (A14)

and

$\displaystyle\mathcal{K}_{ij}^{L}=$
$\displaystyle\left\langle\hat{\phi}_{A}\hat{\phi}_{B}\frac{\tilde{u}_{L}\left(\boldsymbol{R}^{\prime},\boldsymbol{S}_{i}\right)}{\boldsymbol{R}^{\prime}}Y_{L,M}\left(\boldsymbol{R}^{\prime}\right)|H-E|\right.$
$\displaystyle\left.\cdot\mathcal{A}\left[\hat{\phi}_{A}\hat{\phi}_{B}\frac{\tilde{u}_{L}\left(\boldsymbol{R},\boldsymbol{S}_{j}\right)}{\boldsymbol{R}}Y_{L,M}(\boldsymbol{R})\right]\right\rangle.$
(A15)

By solving Eq. (A11), we can obtain the expansion coefficients $C_{i}$, then
the $S-$matrix element $S_{L}$ and the phase shifts $\delta_{L}$ are given by

$\displaystyle S_{L}$ $\displaystyle=e^{2i\delta_{L}}=\sum_{i}C_{i}s_{i}.$
(A16)

Resonances are unstable particles usually observed as bell-shaped structures
in scattering cross sections of their open channels. For a simple narrow
resonance, its fundamental properties correspond to the visible cross-section
features: mass $M$ is at the peak position, and decay width $\Gamma$ is the
half-width of the bell shape. The cross-section $\sigma_{L}$ and the
scattering phase shifts $\delta_{L}$ have relations:

$\displaystyle\sigma_{L}$
$\displaystyle=\frac{4\pi}{k^{2}}(2L+1)\sin^{2}\delta_{L}.$ (A17)

Therefore, resonances can also usually be observed in the scattering phase
shift, where the phase shift of the scattering channels rises through
$\frac{\pi}{2}$ at a resonance mass. We can obtain a resonance mass at the
position of the phase shift of $\frac{\pi}{2}$. The decay width is the mass
difference between the phase shift of $\frac{3\pi}{4}$ and $\frac{\pi}{4}$.

## Appendix B: Constructing wave functions

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. (A16) and Eq. (A17), respectively

$\displaystyle\chi_{\frac{3}{2},\frac{3}{2}}^{\sigma}(3)$
$\displaystyle=\alpha\alpha\alpha,$
$\displaystyle\chi_{\frac{3}{2},\frac{1}{2}}^{\sigma}(3)$
$\displaystyle=\alpha\alpha\beta,$
$\displaystyle\chi_{\frac{3}{2},-\frac{1}{2}}^{\sigma}(3)$
$\displaystyle=\alpha\beta\beta,$
$\displaystyle\chi_{\frac{3}{2},-\frac{3}{2}}^{\sigma}(3)$
$\displaystyle=\beta\beta\beta,$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{\sigma 1}(3)$
$\displaystyle=\frac{1}{\sqrt{6}}(2\alpha\alpha\beta-\alpha\beta\alpha-\beta\alpha\alpha),$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{\sigma 2}(3)$
$\displaystyle=\frac{1}{\sqrt{2}}(\alpha\beta\alpha-\beta\alpha\alpha),$
$\displaystyle\chi_{\frac{1}{2},-\frac{1}{2}}^{\sigma 1}(3)$
$\displaystyle=\frac{1}{\sqrt{6}}(\alpha\beta\beta+\beta\alpha\beta-2\beta\beta\alpha),$
$\displaystyle\chi_{\frac{1}{2},-\frac{1}{2}}^{\sigma 2}(3)$
$\displaystyle=\frac{1}{\sqrt{2}}(\alpha\beta\beta-\beta\alpha\beta).$ (A18)
$\displaystyle\chi_{1,1}^{\sigma}(2)$ $\displaystyle=\alpha\alpha,$
$\displaystyle\chi_{1,0}^{\sigma}(2)$
$\displaystyle=\frac{1}{\sqrt{2}}(\alpha\beta+\beta\alpha),$
$\displaystyle\chi_{1,-1}^{\sigma}(2)$ $\displaystyle=\beta\beta,$
$\displaystyle\chi_{0,0}^{\sigma}(2)$
$\displaystyle=\frac{1}{\sqrt{2}}(\alpha\beta-\beta\alpha).$ (A19)

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 which can
distinguish the third component of the spin quantum number, so the wave
function of each spin quantum number can be written as follows

$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{\sigma 1}(5)=$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{\sigma}(3)\chi_{0,0}^{\sigma}(2),$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{\sigma 2}(5)=$
$\displaystyle-\sqrt{\frac{2}{3}}\chi_{\frac{1}{2},-\frac{1}{2}}^{\sigma}(3)\chi_{1,1}^{\sigma}(2)+\sqrt{\frac{1}{3}}\chi_{\frac{1}{2},\frac{1}{2}}^{\sigma}(3)\chi_{1,0}^{\sigma}(2),$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{\sigma 3}(5)=$
$\displaystyle\sqrt{\frac{1}{6}}\chi_{\frac{3}{2},-\frac{1}{2}}^{\sigma}(3)\chi_{1,1}^{\sigma}(2)-\sqrt{\frac{1}{3}}\chi_{\frac{3}{2},\frac{1}{2}}^{\sigma}(3)\chi_{1,0}^{\sigma}(2)$
$\displaystyle+\sqrt{\frac{1}{2}}\chi_{\frac{3}{2},\frac{3}{2}}^{\sigma}(3)\chi_{1,-1}^{\sigma}(2),$
$\displaystyle\chi_{\frac{3}{2},\frac{3}{2}}^{\sigma 4}(5)=$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{\sigma}(3)\chi_{1,1}^{\sigma}(2),$
$\displaystyle\chi_{\frac{3}{2},\frac{3}{2}}^{\sigma 5}(5)=$
$\displaystyle\chi_{\frac{3}{2},\frac{3}{2}}^{\sigma}(3)\chi_{0,0}^{\sigma}(2),$
$\displaystyle\chi_{\frac{3}{2},\frac{3}{2}}^{\sigma 6}(5)=$
$\displaystyle\sqrt{\frac{3}{5}}\chi_{\frac{3}{2},\frac{3}{2}}^{\sigma}(3)\chi_{1,0}^{\sigma}(2)-\sqrt{\frac{2}{5}}\chi_{\frac{3}{2},\frac{1}{2}}^{\sigma}(3)\chi_{1,1}^{\sigma}(2),$
$\displaystyle\chi_{\frac{5}{2},\frac{5}{2}}^{\sigma 7}(5)=$
$\displaystyle\chi_{\frac{3}{2},\frac{3}{2}}^{\sigma}(3)\chi_{1,1}^{\sigma}(2).$
(A20)

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

$\displaystyle\chi_{0,0}^{f1}(3)$ $\displaystyle=\frac{1}{\sqrt{6}}(2ssc-scs-
css),$ $\displaystyle\chi_{0,0}^{f2}(3)$
$\displaystyle=\frac{1}{\sqrt{2}}(scs-css),$ $\displaystyle\chi_{0,0}^{f3}(3)$
$\displaystyle=\frac{1}{\sqrt{3}}(ssc+scs+css),$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{f1}(3)$
$\displaystyle=\sqrt{\frac{1}{6}}(uss+sus-2ssu),$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{f2}(3)$
$\displaystyle=\sqrt{\frac{1}{2}}(uss-sus),$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{f3}(3)$
$\displaystyle=\sqrt{\frac{1}{3}}(uss+sus+ssu),$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{f4}(3)$
$\displaystyle=\sqrt{\frac{1}{12}}(2usc+2suc-csu-ucs-cus-scu),$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{f5}(3)$
$\displaystyle=\sqrt{\frac{1}{4}}(ucs+scu-csu-cus),$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{f6}(3)$
$\displaystyle=\sqrt{\frac{1}{4}}(ucs+cus-csu-scu),$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{f7}(3)$
$\displaystyle=\sqrt{\frac{1}{12}}(2usc-2suc+csu+ucs-cus-scu),$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{f8}(3)$
$\displaystyle=\sqrt{\frac{1}{6}}(usc+suc+csu+ucs+cus+scu),$
$\displaystyle\chi_{\frac{1}{2},-\frac{1}{2}}^{f1}(3)$
$\displaystyle=\sqrt{\frac{1}{6}}(dss+sds-2ssd),$
$\displaystyle\chi_{\frac{1}{2},-\frac{1}{2}}^{f2}(3)$
$\displaystyle=\sqrt{\frac{1}{2}}(dss-sds),$
$\displaystyle\chi_{\frac{1}{2},-\frac{1}{2}}^{f3}(3)$
$\displaystyle=\sqrt{\frac{1}{3}}(dss+sds+ssd),$
$\displaystyle\chi_{\frac{1}{2},-\frac{1}{2}}^{f4}(3)$
$\displaystyle=\sqrt{\frac{1}{12}}(2dsc+2sdc-csd-dcs-cds-scd),$
$\displaystyle\chi_{\frac{1}{2},-\frac{1}{2}}^{f5}(3)$
$\displaystyle=\sqrt{\frac{1}{4}}(dcs+scd-csd-cds),$
$\displaystyle\chi_{\frac{1}{2},-\frac{1}{2}}^{f6}(3)$
$\displaystyle=\sqrt{\frac{1}{4}}(dcs+cds-csd-scd),$
$\displaystyle\chi_{\frac{1}{2},-\frac{1}{2}}^{f7}(3)$
$\displaystyle=\sqrt{\frac{1}{12}}(2dsc-2sdc+csd+dcs-cds-scd),$
$\displaystyle\chi_{\frac{1}{2},-\frac{1}{2}}^{f8}(3)$
$\displaystyle=\sqrt{\frac{1}{6}}(dsc+sdc+csd+dcs+cds+scd).$

Here, both the light and heavy quarks are considered as identical particles
with the SU(4) extension. Then, the flavor wave functions of $\bar{q}q$
clusters are

$\displaystyle\chi_{1,1}^{f}(2)$ $\displaystyle=\bar{d}u,$
$\displaystyle\chi_{1,0}^{f}(2)$
$\displaystyle=\sqrt{\frac{1}{2}}(\bar{d}d-\bar{u}u),$
$\displaystyle\chi_{1,-1}^{f}(2)$ $\displaystyle=-\bar{u}d,$
$\displaystyle\chi_{0,0}^{f}(2)$
$\displaystyle=\sqrt{\frac{1}{2}}(\bar{d}d+\bar{u}u),$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{f}(2)$ $\displaystyle=\bar{d}s,$
$\displaystyle\chi_{\frac{1}{2},-\frac{1}{2}}^{f}(2)$
$\displaystyle=-\bar{u}s,$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{f}(2)$ $\displaystyle=\bar{d}c,$
$\displaystyle\chi_{\frac{1}{2},-\frac{1}{2}}^{f}(2)$
$\displaystyle=-\bar{u}c.$ (A22)

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

$\displaystyle\chi_{0,0}^{f1}(5)$
$\displaystyle=\sqrt{\frac{1}{2}}\chi_{\frac{1}{2},\frac{1}{2}}^{f}(3)\chi_{\frac{1}{2},-\frac{1}{2}}^{f}(2)-\sqrt{\frac{1}{2}}\chi_{\frac{1}{2},-\frac{1}{2}}^{f}(3)\chi_{\frac{1}{2},\frac{1}{2}}^{f}(2),$
$\displaystyle\chi_{0,0}^{f2}(5)$
$\displaystyle=\chi_{0,0}(3)\chi_{0,0}^{f}(2).$ (A23)

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

$\displaystyle\chi^{c}=$ $\displaystyle\frac{1}{\sqrt{6}}(rgb-rbg+gbr-grb+brg-
bgr)$ $\displaystyle\cdot\frac{1}{\sqrt{3}}(\bar{r}r+\bar{g}g+\bar{b}b).$
(A24)

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.

## References

  * (1) B. Knapp, W. Y. Lee, P. Leung, S. D. Smith, A. Wijangco, J. Knauer, D. Yount, J. Bronstein, R. Coleman and G. Gladding, et al. Phys. Rev. Lett. 37, 882 (1976).
  * (2) H. Albrecht et al. [ARGUS], Phys. Lett. B 317, 227 (1993).
  * (3) K. W. Edwards et al. [CLEO], Phys. Rev. Lett. 74, 3331 (1995).
  * (4) H. Albrecht et al. [ARGUS], Phys. Lett. B 402, 207 (1997).
  * (5) P. L. Frabetti et al. [E687], Phys. Rev. Lett. 72, 961 (1994).
  * (6) M. Artuso et al. [CLEO], Phys. Rev. Lett. 86, 4479 (2001).
  * (7) R. Aaij et al. [LHCb], JHEP 05, 030 (2017).
  * (8) B. Aubert et al. [BaBar], Phys. Rev. Lett. 98, 012001 (2007).
  * (9) J. Yelton et al. [Belle], Phys. Rev. D 104, 052003 (2021).
  * (10) S. H. Lee et al. [Belle], Phys. Rev. D 89, 091102 (2014).
  * (11) V. V. Ammosov, I. L. Vasilev, A. A. Ivanilov, P. V. Ivanov, V. I. Konyushko, V. M. Korablev, V. A. Korotkov, V. V. Makeev, A. G. Myagkov and A. Y. Polyarush, et al. JETP Lett. 58, 247 (1993).
  * (12) G. Brandenburg et al. [CLEO], Phys. Rev. Lett. 78, 2304 (1997).
  * (13) R. Mizuk et al. [Belle], Phys. Rev. Lett. 94, 122002 (2005).
  * (14) Y. B. Li et al. [Belle], Phys. Rev. Lett. 130, 031901 (2023).
  * (15) R. Aaij et al. [LHCb], Phys. Rev. D 102, 071101 (2020).
  * (16) S. Acharya et al. [ALICE], Phys. Rev. Lett. 127, 272001 (2021).
  * (17) J. Yelton et al. [Belle], Phys. Rev. D 94, 052011 (2016).
  * (18) C. P. Jessop et al. [CLEO], Phys. Rev. Lett. 82, 492 (1999).
  * (19) Y. Kato et al. [Belle], Phys. Rev. D 89, 052003 (2014).
  * (20) J. Yelton et al. [Belle], Phys. Rev. D 102, 071103 (2020).
  * (21) R. Aaij et al. [LHCb], Phys. Rev. Lett. 124, 222001 (2020).
  * (22) T. J. Moon et al. [Belle], Phys. Rev. D 103, L111101 (2021).
  * (23) Y. Kato et al. [Belle], Phys. Rev. D 94, 032002 (2016).
  * (24) Y. Li et al. [Belle], Phys. Rev. D 104, 052005 (2021).
  * (25) B. Aubert et al. [BaBar], Phys. Rev. Lett. 97, 232001 (2006).
  * (26) R. Aaij et al. [LHCb], Phys. Rev. D 104, L091102 (2021).
  * (27) J. Yelton et al. [Belle], Phys. Rev. D 97, 051102 (2018).
  * (28) R. Aaij et al. [LHCb], Phys. Rev. Lett. 118, 182001 (2017).
  * (29) H. X. Chen, W. Chen, X. Liu and S. L. Zhu, Phys. Rept. 639, 1 (2016).
  * (30) E. S. Swanson, Phys. Rept. 429, 243 (2006).
  * (31) M. B. Voloshin, Prog. Part. Nucl. Phys. 61, 455 (2008).
  * (32) A. Esposito, A. Pilloni and A. D. Polosa, Phys. Rept. 668, 1 (2017).
  * (33) R. Chen, X. Liu and S. L. Zhu, Nucl. Phys. A 954, 406 (2016).
  * (34) H. X. Huang, C. R. Deng, X. J. Liu, Y. Tan and J. L. Ping, Symmetry 15, 1298 (2023).
  * (35) R. F. Lebed, R. E. Mitchell and E. S. Swanson, Prog. Part. Nucl. Phys. 93, 143 (2017).
  * (36) F. K. Guo, C. Hanhart, U. G. Meißner, Q. Wang, Q. Zhao and B. S. Zou, Rev. Mod. Phys. 90, no.1, 015004 (2018).
  * (37) M. Padmanath and N. Mathur, Phys. Rev. Lett. 119, 042001 (2017).
  * (38) H. Bahtiyar, K. U. Can, G. Erkol, P. Gubler, M. Oka and T. T. Takahashi, Phys. Rev. D 102, n054513 (2020).
  * (39) S. S. Agaev, K. Azizi and H. Sundu, EPL 118, 61001 (2017).
  * (40) Z. G. Wang, Eur. Phys. J. C 77, 325 (2017).
  * (41) T. M. Aliev, S. Bilmis and M. Savci, Mod. Phys. Lett. A 35, 1950344 (2020).
  * (42) S. S. Agaev, K. Azizi and H. Sundu, Eur. Phys. J. C 77, 395 (2017).
  * (43) Z. G. Wang, X. N. Wei and Z. H. Yan, Eur. Phys. J. C 77, 832 (2017).
  * (44) H. X. Chen, Q. Mao, W. Chen, A. Hosaka, X. Liu and S. L. Zhu, Phys. Rev. D 95, 094008 (2017).
  * (45) T. M. Aliev, S. Bilmis and M. Savci, Adv. High Energy Phys. 2018, 3637824 (2018).
  * (46) H. Y. Cheng and C. W. Chiang, Phys. Rev. D 95, 094018 (2017).
  * (47) J. Oudichhya, K. Gandhi and A. K. Rai, Phys. Rev. D 103, 114030 (2021).
  * (48) J. Oudichhya and A. K. Rai, Eur. Phys. J. A 59, 123 (2023).
  * (49) G. Yang and J. Ping, Phys. Rev. D 97, 034023 (2018).
  * (50) K. L. Wang, L. Y. Xiao, X. H. Zhong and Q. Zhao, Rev. D 95, 116010 (2017).
  * (51) K. L. Wang, Y. X. Yao, X. H. Zhong and Q. Zhao, Phys. Rev. D 96, 116016 (2017).
  * (52) Y. X. Yao, K. L. Wang and X. H. Zhong, Phys. Rev. D 98, 076015 (2018).
  * (53) W. Wang and R. L. Zhu, Phys. Rev. D 96, 014024 (2017).
  * (54) A. Ali, L. Maiani, A. V. Borisov, I. Ahmed, M. Jamil Aslam, A. Y. Parkhomenko, A. D. Polosa and A. Rehman, Eur. Phys. J. C 78, (2018).
  * (55) B. Chen and X. Liu, Phys. Rev. D 96, 094015 (2017).
  * (56) Z. Zhao, D. D. Ye and A. Zhang, Phys. Rev. D 95, 114024 (2017).
  * (57) H. Garcia-Tecocoatzi, A. Giachino, J. Li, A. Ramirez-Morales and E. Santopinto, Phys. Rev. D 107, 034031 (2023).
  * (58) H. C. Kim, M. V. Polyakov and M. Praszałowicz, Phys. Rev. D 96, 014009 (2017).
  * (59) H. C. Kim, M. V. Polyakov, M. Praszalowicz and G. S. Yang, Phys. Rev. D 96, 094021 (2017).
  * (60) Y. Liu and I. Zahed, Phys. Rev. D 96, 056027 (2017).
  * (61) J. Sonnenschein and D. Weissman, Nucl. Phys. B 927, 368 (2018).
  * (62) E. Santopinto, A. Giachino, J. Ferretti, H. García-Tecocoatzi, M. A. Bedolla, R. Bijker and E. Ortiz-Pacheco, Eur. Phys. J. C 79, 1012 (2019).
  * (63) C. K. Chua, Phys. Rev. D 100, 034025 (2019).
  * (64) D. Jia, J. H. Pan and C. Q. Pang, Eur. Phys. J. C 81, 434 (2021).
  * (65) S. Q. Luo and X. Liu, Phys. Rev. D 107, 074041 (2023).
  * (66) P. Jakhad, J. Oudichhya, K. Gandhi and A. K. Rai, Phys. Rev. D 108, 014011 (2023).
  * (67) M. Karliner and J. L. Rosner, Phys. Rev. D 95, 114012 (2017).
  * (68) E. Ortiz-Pacheco, R. Bijker, A. Giachino and E. Santopinto, J. Phys. Conf. Ser. 1610, 012011 (2020).
  * (69) Z. G. Wang and J. X. Zhang, Eur. Phys. J. C 78, 503 (2018).
  * (70) H. J. Wang, Z. Y. Di and Z. G. Wang, Commun. Theor. Phys. 73, 035201 (2021).
  * (71) H. Huang, J. Ping and F. Wang, Phys. Rev. D 97, 034027 (2018).
  * (72) C. S. An and H. Chen, Phys. Rev. D 96, 034012 (2017).
  * (73) V. V. Anisovich, M. A. Matveev, J. Nyiri and A. N. Semenova, Mod. Phys. Lett. A 32, 1750154 (2017).
  * (74) J. Nieves, R. Pavao and L. Tolos, Eur. Phys. J. C 78, 114 (2018).
  * (75) M. Z. Liu, T. W. Wu, J. J. Xie, M. Pavon Valderrama and L. S. Geng, Phys. Rev. D 98, 014014 (2018).
  * (76) G. Montaña, À. Ramos and A. Feijoo, Springer Proc. Phys. 238, 729 (2020).
  * (77) G. Montaña, A. Feijoo and À. Ramos, Eur. Phys. J. A 54, 64 (2018).
  * (78) A. Ramos, A. Feijoo, Q. Llorens and G. Montaña, Few Body Syst. 61, 34 (2020).
  * (79) M. Praszałowicz and M. Kucab, Phys. Rev. D 107, 034011 (2023).
  * (80) V. R. Debastiani, J. M. Dias, W. H. Liang and E. Oset, Phys. Rev. D 97, 094035 (2018).
  * (81) V. R. Debastiani, J. M. Dias, W. H. Liang and E. Oset, Phys. Rev. D 98, 094022 (2018).
  * (82) C. Wang, L. L. Liu, X. W. Kang, X. H. Guo and R. W. Wang, Eur. Phys. J. C 78, 407 (2018).
  * (83) Y. Huang, C. j. Xiao, Q. F. Lü, R. Wang, J. He and L. Geng, Phys. Rev. D 97, 094013 (2018).
  * (84) J. T. Zhu, S. Y. Kong, L. Q. Song and J. He, Phys. Rev. D 105, 094036 (2022).
  * (85) R. Aaij et al. [LHCb].
  * (86) S. Q. Luo and X. Liu, Phys. Rev. D 107, 074041 (2023).
  * (87) G. L. Yu, Y. Meng, Z. Y. Li, Z. G. Wang and L. Jie, [arXiv:2302.11758 [hep-ph]].
  * (88) Z. G. Wang, F. Lu and Y. Liu, [arXiv:2303.13976 [hep-ph]].
  * (89) J. Feng, C. Cheng, F. Yang and Y. Huang, [arXiv:2303.17770 [hep-ph]].
  * (90) M. Karliner and J. L. Rosner, Phys. Rev. D 108, 014006 (2023).
  * (91) M. J. Yan, F. Z. Peng and M. Pavon Valderrama, [arXiv:2304.14855 [hep-ph]].
  * (92) P. Jakhad, J. Oudichhya, K. Gandhi and A. K. Rai, Phys. Rev. D 108, 014011 (2023).
  * (93) Q. Xin, X. S. Yang and Z. G. Wang, [arXiv:2307.08926 [hep-ph]].
  * (94) G. H. Wu, L. J. Teng, J. L. Ping, F. Wang and J. T. Goldman, Phys. Rev. C 53, 1161 (1996).
  * (95) J. L. Ping, F. Wang and J. T. Goldman, Nucl. Phys. A 657, 95 (1999).
  * (96) G. h. Wu, J. L. Ping, L. j. Teng, F. Wang and J. T. Goldman, Nucl. Phys. A 673, 279 (2000).
  * (97) H. R. Pang, J. L. Ping, F. Wang and J. T. Goldman, Phys. Rev. C 65, 014003 (2002).
  * (98) J. L. Ping, F. Wang and J. T. Goldman, Phys. Rev. C 65, 044003 (2002).
  * (99) L. Z. Chen, H. R. Pang, H. X. Huang, J. L. Ping and F. Wang, Phys. Rev. C 76, 014001 (2007).
  * (100) H. Huang, P. Xu, J. Ping and F. Wang, Phys. Rev. C 84, 064001 (2011).
  * (101) Y. Yan, X. Hu, Y. Wu, H. Huang, J. Ping and Y. Yang, Eur. Phys. J. C 83, 524 (2023).
  * (102) A. De Rujula, H. Georgi and S. L. Glashow, Phys. Rev. D 12, 147 (1975).
  * (103) N. Isgur and G. Karl, Phys. Rev. D 18, 4187 (1978).
  * (104) N. Isgur and G. Karl, Phys. Rev. D 19, 2653 (1979).
  * (105) N. Isgur and G. Karl, Phys. Rev. D 20, 1191 (1979).
  * (106) J. l. Ping, F. Wang and J. T. Goldman, Nucl. Phys. A 688, 871 (2001).
  * (107) J. L. Ping, H. X. Huang, H. R. Pang, F. Wang and C. W. Wong, Phys. Rev. C 79, 024001 (2009).
  * (108) M. Chen, H. X. Huang, J. L. Ping and F. Wang, Phys. Rev. C 83, 015202 (2011).
  * (109) H. X. Huang, C. R. Deng, J. L. Ping and F. Wang, Eur. Phys. J. C 76, 624 (2016).
  * (110) H. X. Huang and J. L. Ping, Phys. Rev. D 99, 014010 (2019).
  * (111) P. A. Zyla et al. [Particle Data Group], PTEP 2020, 083C01 (2020).
  * (112) M. M. Xu, M. Yu and L. S. Liu, Phys. Rev. Lett. 100, 092301 (2008).
  * (113) M. Chen, H. Huang, J. Ping and F. Wang, Phys. Rev. C 83, 015202 (2011).
  * (114) J. A. Wheeler, Phys. Rev. 52, 1083 (1937).
  * (115) M. Kamimura, Prog. Theor. Phys. Suppl. 62, 236 (1977).
  * (116) D. L. Hill and J. A. Wheeler, Phys. Rev. 89, 1102 (1953).
  * (117) J. J. Griffin and J. A. Wheeler, Phys. Rev. 108, 311 (1957).