# Prediction of $P_{cc}$ states in quark model

Ye Yan1<EMAIL_ADDRESS>Hongxia Huang1 hxhuang@njnu.edu.cn(Corresponding
author) Xinmei Zhu2 xmzhu@yzu.edu.cn(Corresponding author) Jialun Ping1
<EMAIL_ADDRESS>1School of Physics and Technology, Nanjing Normal
University, Nanjing 210097, People’s Republic of China  2Department of
Physics, Yangzhou University, Yangzhou 225009, People’s Republic of China

###### Abstract

Inspired by the observation of hidden-charm pentaquark $P_{c}$ and $P_{cs}$
states by the LHCb Collaboration, we explore the $qqc\bar{c}c$ ($q~{}=~{}u$ or
$d$) pentaquark systems in the quark delocalization color screening model. The
interaction between baryons and mesons and the influence of channel coupling
are studied in this work. Three compact $qqc\bar{c}c$ pentaquark states are
obtained, whose masses are 5259 MeV with $I(J^{P})$ = $0(1/2^{-})$, 5396 MeV
with $I(J^{P})$ = $1(1/2^{-})$, and 5465 MeV with $I(J^{P})$ = $1(3/2^{-})$.
Two molecular states are obtained, which are $I(J^{P})$ = $0(1/2^{-})$
$\Lambda_{c}J/\psi$ with 5367 MeV and $I(J^{P})$ = $0(5/2^{-})$
$\Xi_{cc}^{*}\bar{D}^{*}$ with 5690 MeV. These predicted states may provide
important information for future experimental search.

##  I Introduction

In the recent years, regarding the exotic hadron studies, one of the most
noteworthy experimental progresses is the observation of the hidden-charm
pentaquark states, which brings great interest in pentaquark investigations.
In 2015, the LHCb Collaboration reported two states $P_{c}(4380)$ and
$P_{c}(4450)$ in the $J/\psi p$ invariant mass spectrum of
$\Lambda^{0}_{b}\rightarrow J/\psi K^{-}p$ LHCbPc2015 . Subsequently, the LHCb
Collaboration updated their results in 2019, a new states $P_{c}(4312)$ was
proposed, and the $P_{c}(4450)$ was split to $P_{c}(4440)$ and $P_{c}(4457)$
LHCbPc2019 .

Before the LHCb collaboration’s observations, several theoretical works have
predicted the existence of the hidden-charm pentaquark states Wu:2010jy ;
Wang:2011rga ; Yang:2011wz ; Xiao:2013yca . After the experimental report, a
wide range of theoretical researches on these $P_{c}$ states were initiated
Chen:2015moa ; Wang:2015epa ; Wang:2015ava ; Azizi:2016dhy ; Azizi:2018bdv ;
Chen:2019bip ; Zhang:2019xtu ; Xu:2019zme ; Wang:2019got ; Azizi:2020ogm ;
Xu:2020flp ; Ozdem:2018qeh ; Lu:2015fva ; Lu:2016nnt ; Xiao:2019mvs ;
Wu:2019rog ; Wang:2019dsi ; Zhang:2020erj ; Xu:2020gjl ; Li:2023zag ;
Roca:2015dva ; Shimizu:2016rrd ; Meng:2019ilv ; He:2015cea ; He:2019ify ;
Oset:2016nvf ; Wang:2019ato ; Peng:2020xrf ; Wang:2015qlf ; Ali:2016dkf ;
Shimizu:2019ptd ; Du:2021fmf ; Guo:2015umn ; Liu:2015fea ; Guo:2016bkl ;
Santopinto:2016pkp ; Ortega:2016syt ; Park:2017jbn ; Zhu:2019iwm ;
Phumphan:2021tta ; Yang:2015bmv ; Dong:2020nwk ; Chen:2015loa ; Liu:2019zvb ;
Chen:2019asm ; Yalikun:2021bfm ; Deng:2016rus ; HuangPc1 ; HuangPc2 ;
Zhu:2023hyh ; Scoccola:2015nia ; Wu:2017weo ; Weng:2019ynv ; Zhu:2015bba ;
Wang:2016dzu ; Hiyama:2018ukv ; Burns:2019iih ; Yang:2022ezl ;
Meissner:2015mza ; Mutuk:2019snd ; Yamaguchi:2019seo ; Kubarovsky:2015aaa ;
Wang:2015jsa ; HillerBlin:2016odx ; Paryev:2018fyv ; Guo:2019fdo ;
Shen:2016tzq ; Burns:2015dwa . These states have been investigated using the
QCD (Quantum Chromodynamics) sum rules Chen:2015moa ; Wang:2015epa ;
Wang:2015ava ; Azizi:2016dhy ; Azizi:2018bdv ; Chen:2019bip ; Zhang:2019xtu ;
Xu:2019zme ; Wang:2019got ; Azizi:2020ogm ; Xu:2020flp , light-cone QCD sum
rules Ozdem:2018qeh , effective Lagrangian approach Lu:2015fva ; Lu:2016nnt ;
Xiao:2019mvs ; Wu:2019rog ; Wang:2019dsi ; Zhang:2020erj ; Xu:2020gjl ;
Li:2023zag , chiral Lagrangian approach Roca:2015dva ; Shimizu:2016rrd ,
chiral perturbation theory Meng:2019ilv , quasipotential Bethe–Saltpeter
approach He:2015cea ; He:2019ify , chiral unitary approach Oset:2016nvf ,
effective field theory Wang:2019ato ; Peng:2020xrf , heavy quark spin symmetry
Wang:2015qlf ; Ali:2016dkf ; Shimizu:2019ptd ; Du:2021fmf , triangle
singularity Guo:2015umn ; Liu:2015fea ; Guo:2016bkl , Gürsey-Radicati inspired
mass formula Santopinto:2016pkp , constituent quark model Ortega:2016syt ;
Park:2017jbn ; Zhu:2019iwm ; Phumphan:2021tta ; Yang:2015bmv ; Dong:2020nwk ;
HuangPc1 ; HuangPc2 ; Zhu:2023hyh , one-boson exchange model Chen:2015loa ;
Liu:2019zvb ; Chen:2019asm ; Yalikun:2021bfm , color flux-tube model
Deng:2016rus , topological soliton model Scoccola:2015nia , color-magnetic
interaction model Wu:2017weo ; Weng:2019ynv , molecular models Zhu:2015bba ;
Wang:2016dzu ; Hiyama:2018ukv ; Burns:2019iih ; Yang:2022ezl , and various
other models Meissner:2015mza ; Mutuk:2019snd ; Yamaguchi:2019seo . In
addition, the production mechanism of these states is also investigated in
Refs. Kubarovsky:2015aaa ; Wang:2015jsa ; HillerBlin:2016odx ; Paryev:2018fyv
. Given the near-threshold nature of the $P_{c}$ states, most of the
theoretical works interpret them as molecular states. For instance, in Ref.
Chen:2019asm , the authors performed a direct calculation using the one-boson
exchange model and explicitly demonstrated that the $P_{c}(4312)$,
$P_{c}(4440)$ and $P_{c}(4457)$ correspond to the loosely bound
$\Sigma_{c}\bar{D}$ with ($I$ = 1/2, $J^{P}=1/2^{-}$), $\Sigma_{c}\bar{D}^{*}$
with ($I$ = 1/2, $J^{P}=1/2^{-}$), and $\Sigma_{c}\bar{D}^{*}$ with ($I$ =
1/2, $J^{P}=3/2^{-}$ ), respectively. More detailed reviews on the $P_{c}$
states can also be found in Refs. ChenHX0 ; LiuYR0 ; YangG0 .

In 2020, the LHCb Collaboration reported a 3$\sigma$ hidden-charm strange
pentaquark structure $P_{cs}(4459)$ in the $\Xi^{-}_{b}\rightarrow
J/\psi\Lambda K^{-}$ decay LHCb:2020jpq . In 2022, the LHCb Collaboration
reported their results about the $B^{-}\rightarrow J/\psi\Lambda\bar{p}$
decay, which indicates the existence of a new hidden-charm strange pentaquark
state $P_{cs}(4338)$ LHCb:2022ogu . Given that the mass of the $P_{cs}(4459)$
and $P_{cs}(4338)$ is close to the threshold of $\Xi_{c}\bar{D}^{*}$ and
$\Xi_{c}\bar{D}$, respectively, it has led to various theoretical studies on
the $P_{cs}$ statesChen:2020uif ; Wang:2020eep ; Azizi:2021utt ; Wang:2021itn
; Wang:2022neq ; Azizi:2023iym ; Ozdem:2021ugy ; Ozdem:2022kei ; Ozdem:2023htj
; Peng:2020hql ; Liu:2020hcv ; Yan:2021nio ; Yan:2022wuz ; Cheng:2021gca ;
Yang:2021pio ; Wu:2021caw ; Clymton:2021thh ; Zhu:2021lhd ; Zhu:2022wpi ;
Burns:2022uha ; Feijoo:2022rxf ; Xiao:2021rgp ; Chen:2020kco ; Chen:2021tip ;
Wang:2022mxy ; Chen:2022onm ; Li:2023aui ; Hu:2021nvs ; Deng:2022vkv ;
Wang:2022tib ; Ortega:2022uyu ; Gao:2021hmv ; Meng:2022wgl ; Giachino:2022pws
; Paryev:2022zdx ; Chen:2022wkh ; Paryev:2023icm ; Nakamura:2022gtu ;
Liu:2020ajv . These states have been investigated in the framework of the QCD
sum rules Chen:2020uif ; Wang:2020eep ; Azizi:2021utt ; Wang:2021itn ;
Wang:2022neq ; Azizi:2023iym , light-cone QCD sum rules Ozdem:2021ugy ;
Ozdem:2022kei ; Ozdem:2023htj , effective field theory Peng:2020hql ;
Liu:2020hcv ; Yan:2021nio ; Yan:2022wuz , effective Lagrangian approach
Cheng:2021gca ; Yang:2021pio ; Wu:2021caw ; Clymton:2021thh , quasipotential
Bethe-Salpeter equation approach Zhu:2021lhd ; Zhu:2022wpi , triangle
singularity Burns:2022uha , heavy quark spin symmetry Feijoo:2022rxf , coupled
channel unitary approach Xiao:2021rgp , constituent quark model Wang:2022tib ;
Ortega:2022uyu ; Gao:2021hmv ; Hu:2021nvs , one-boson exchange model
Chen:2020kco ; Chen:2021tip ; Wang:2022mxy ; Chen:2022onm , the color-magnetic
interaction model Li:2023aui , color flux-tube model Deng:2022vkv , zero-range
model and the Flatté model Meng:2022wgl , and other models Giachino:2022pws ;
Paryev:2022zdx ; Chen:2022wkh ; Paryev:2023icm ; Nakamura:2022gtu . One of the
focuses of these theoretical works is to determine the structure of the
$P_{cs}$ states. For the $P_{cs}(4459)$, conclusion of molecular configuration
is supported in Refs. Chen:2020uif ; Wang:2021itn ; Peng:2020hql ; Yan:2022wuz
; Cheng:2021gca ; Yang:2021pio ; Wu:2021caw ; Xiao:2021rgp ; Zhu:2021lhd ;
Feijoo:2022rxf ; Chen:2020kco ; Chen:2021tip ; Wang:2022mxy ; Hu:2021nvs ;
Ortega:2022uyu , while there are also theoretical works that interpret the
$P_{cs}(4459)$ as a compact pentaquark Wang:2020eep ; Azizi:2021utt ;
Li:2023aui ; Deng:2022vkv . For the $P_{cs}(4338)$, molecular configuration is
preferred in Refs. Wang:2022neq ; Azizi:2023iym ; Yan:2022wuz ; Zhu:2022wpi ;
Feijoo:2022rxf ; Wang:2022mxy ; Giachino:2022pws ; Ortega:2022uyu . However,
it can be interpreted as $udsc\bar{c}$ compact pentaquark state according to
Ref. Li:2023aui . Determining the structure of different states in the same
theoretical framework is a meaningful subject. According to the QCD sum rules
results of Refs. Azizi:2021utt ; Azizi:2023iym , the compact pentaquark nature
of diquark-diquark-antiquark form with $J^{P}=1/2^{-}$ is favored for the
$P_{cs}(4459)$ state, and the $\Xi_{c}\bar{D}$ molecular nature with
$J^{P}=1/2^{-}$ is favored for the $P_{cs}(4338)$ state.

Given the existence of the $P_{c}$ $(qqqc\bar{c})$ and $P_{cs}$
$(qqsc\bar{c})$ states, one may wonder if there are other types of pentaquark
states that contain a pair of $c\bar{c}$. If the strange quark in the $P_{cs}$
state is replaced by the charm quark, it is interesting to explore if there
exists the $P_{cc}$ state $(qqcc\bar{c})$. Additionally, according to Ref.
Gershon:2022xnn , $P_{cc}$ state can also be named $P_{\psi c}^{\Lambda}$ and
$P_{\psi c}^{\Sigma}$ based on the isospin. Several theoretical studies have
been carried out concerning the $(qqqc\bar{c})$ system Hofmann:2005sw ;
Chen:2017jjn ; Wang:2018ihk ; Wang:2019aoc ; An:2019idk . In Ref. Chen:2017jjn
; Wang:2019aoc , the $\Xi_{cc}\bar{D}$, $\Xi_{cc}\bar{D}^{*}$,
$\Xi_{cc}\bar{D}_{1}$, and $\Xi_{cc}\bar{D}_{2}^{*}$ interactions are studied
from molecular picture by using the one-boson exchange model, and several
possible states were found, which are $\Xi_{cc}\bar{D}^{*}$ with
$I(J^{P})=0(1/2^{-})$, $\Xi_{cc}\bar{D}_{1}$ with
$I(J^{P})=0(1/2^{+},3/2^{+})$, and $\Xi_{cc}\bar{D}_{2}^{*}$ with
$I(J^{P})=0(3/2^{+},5/2^{+})$. In Ref. Wang:2018ihk , the scalar-diquark-
scalar-diquark-antiquark type current is constructed to interpolate the
$P_{cc\bar{c}ud}$ pentaquark states with $J^{P}=1/2^{\pm}$. In Ref. An:2019idk
, in the framework of the color-magnetic interaction model, the $qqcc\bar{c}$
$(q=u,d)$ with $I(J^{P})=0(5/2^{-})$ is determined to be a possible stable
state.

The quark delocalization color screening model (QDCSM) was developed with the
aim of explaining the similarities between nuclear and molecular forces
Wu:1996fm . This model gives a good description of the $NN$ 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 Huang:2023jec . Studies show that color screening
is an effective description of the hidden-color channel coupling ChenLZ ;
Huang:2011kf . When the LHCb collaboration reported the $P_{c}$ states for the
first time, the QDCSM is employed to study this system and seven states are
obtained HuangPc1 ; HuangPc2 . Three of them can be used to explain the
updated results of $P_{c}$ states reported by the LHCb collaboration in 2019.
Therefore, it is feasible and meaningful to extend this model to investigate
the $qqcc\bar{c}$ pentaquark system.

In this work, we systematically investigate the $qqc\bar{c}c$ pentaquark
systems in order to find the possible states. First, the effective potential
is studied to understand the interaction between baryon and meson. The five-
body system is calculated by means of the resonating group method to search
for bound states. The influence of channel coupling is discussed based on the
current results. In addition, we calculate the scattering phase shift to
examine the possible resonance states.

This paper is organized as follows. After introduction, the details of QDCSM
are presented in Sec. II. The effective potential, the bound-state
calculation, and the scattering process are presented in Sec. III, along with
the discussion and analysis of the results. Finally, the paper ends with a
summary in Sec. 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 multiquark 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 section, we
mainly introduce the salient features of this model. The general form of the
pentaquark Hamiltonian is given by

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

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

$\displaystyle V(\boldsymbol{r}_{ij})=$ $\displaystyle
V_{\mathrm{CON}}(\boldsymbol{r}_{ij})+V_{\mathrm{OGE}}(\boldsymbol{r}_{ij})+V_{\chi}(\boldsymbol{r}_{ij}).$
(2)

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

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

$\displaystyle V_{\mathrm{CON}}(\boldsymbol{r}_{ij})=$ $\displaystyle-
a_{c}\boldsymbol{\lambda}_{i}^{c}\cdot\boldsymbol{\lambda}_{j}^{c}\left[f(\boldsymbol{r}_{ij})+V_{0}\right],$
(3) $\displaystyle f(\boldsymbol{r}_{ij})=$
$\displaystyle\left\\{\begin{array}[]{l}\boldsymbol{r}_{ij}^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}i,j~{}\text{occur
in the same cluster }\\\
\frac{1-e^{-\mu_{q_{i}q_{j}}\boldsymbol{r}_{ij}^{2}}}{\mu_{q_{i}q_{j}}},~{}~{}~{}i,j~{}\text{occur
in different cluster }\end{array}\right.$ (6)

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 $\mu_{qq}=0.45$,
$\mu_{qs}=0.19$, and $\mu_{ss}=0.08~{}$fm-2, satisfying the
relation—$\mu_{qs}^{2}=\mu_{qq}\mu_{ss}$ ChenM . Additionally, 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 an adjustable parameter from 0.01 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 $\mu_{cc}=0.01$ and $\mu_{qc}=0.067~{}$fm-2, also
satisfying 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 the $S$-wave, so the spin-orbit and tensor
interactions are not included. The one-gluon exchange potential
($V_{\mathrm{OGE}}$), which includes Coulomb and chromomagnetic interactions,
is written as

$\displaystyle V_{\mathrm{OGE}}(\boldsymbol{r}_{ij})=$
$\displaystyle\frac{1}{4}\alpha_{s_{q_{i}q_{j}}}\boldsymbol{\lambda}_{i}^{c}\cdot\boldsymbol{\lambda}_{j}^{c}$
(7)
$\displaystyle\cdot\left[\frac{1}{r_{ij}}-\frac{\pi}{2}\delta\left(\mathbf{r}_{ij}\right)\left(\frac{1}{m_{i}^{2}}+\frac{1}{m_{j}^{2}}+\frac{4\boldsymbol{\sigma}_{i}\cdot\boldsymbol{\sigma}_{j}}{3m_{i}m_{j}}\right)\right],$

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 $\bar{D}$ and
$\bar{D}^{*}$ is determined by the chromomagnetic interaction in Eq. (II), so
the parameter $\alpha_{s_{\bar{c}q}}$ is determined by fitting the mass
difference between $\bar{D}$ and $\bar{D}^{*}$.

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

$\displaystyle V_{\chi}(\boldsymbol{r}_{ij})=$ $\displaystyle
V_{\pi}(\boldsymbol{r}_{ij})+V_{K}(\boldsymbol{r}_{ij})+V_{\eta}(\boldsymbol{r}_{ij}).$
(8)

Among them

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

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 Workman:2022ynf . The
chiral coupling constant $g_{ch}$, is determined from the $\pi NN$ coupling
constant through

$\displaystyle\frac{g_{ch}^{2}}{4\pi}$
$\displaystyle=\left(\frac{3}{5}\right)^{2}\frac{g_{\pi
NN}^{2}}{4\pi}\frac{m_{u,d}^{2}}{m_{N}^{2}}.$ (12)

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-state baryons and mesons. Table 2 shows the
masses of the baryons and mesons used in this work. Since it is very difficult
to fit well all ground-state hadrons with limited parameters, we give priority
to fitting lighter baryons and mesons when setting parameters. As a result,
the mass gaps between theoretical and experimental values of heavier baryons
and mesons are larger.

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

$\displaystyle\psi_{\alpha}(\boldsymbol{S_{i}},\epsilon)$ $\displaystyle=$
$\displaystyle\left(\phi_{\alpha}(\boldsymbol{S_{i}})+\epsilon\phi_{\alpha}(-\boldsymbol{S_{i}})\right)/N(\epsilon),$
$\displaystyle\psi_{\beta}(-\boldsymbol{S_{i}},\epsilon)$ $\displaystyle=$
$\displaystyle\left(\phi_{\beta}(-\boldsymbol{S_{i}})+\epsilon\phi_{\beta}(\boldsymbol{S_{i}})\right)/N(\epsilon),$
$\displaystyle N(S_{i},\epsilon)$ $\displaystyle=$
$\displaystyle\sqrt{1+\epsilon^{2}+2\epsilon e^{-S_{i}^{2}/4b^{2}}}.$ (13)

It is worth noting that the mixing parameter $\epsilon$ is not an adjusted one
but determined variationally by the dynamics of the multiquark system itself.
In this way, the multiquark system chooses its favorable configuration in the
interacting process. This mechanism has been used to explain the crossover
transition between the 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 are presented in Appendix, and the way of constructing wave functions
can be seen in Ref. Xia:2021tof .

Table 1: Model parameters used in this work: $m_{\pi}=0.7$, $m_{K}=2.51$, $m_{\eta}=2.77$, $\Lambda_{\pi}=4.2$, $\Lambda_{K}=5.2$, $\Lambda_{\eta}=5.2$ 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 | 0.207 | 58.03
$\alpha_{s_{\bar{c}q}}$ | $\alpha_{s_{\bar{c}c}}$ | $\alpha_{s_{qq}}$ | $\alpha_{s_{qc}}$ | $\alpha_{s_{cc}}$ |
1.970 | 2.357 | 0.524 | 0.467 | 0.213 |
Table 2: The masses (in MeV) of the baryons and mesons. Experimental values are taken from the Particle Data Group Workman:2022ynf . Hadron |  $I(J^{P})$ |  $M^{\text{Exp}}$ |  $M^{\text{Theo}}$
---|---|---|---
$\bar{D}$ | $1/2(0^{-})$ | 1869 | 1869
$\bar{D}^{*}$ | $1/2(1^{-})$ | 2007 | 2007
$\eta_{c}$ | $0(0^{-})$ | 2984 | 2984
$J/\psi$ | $0(1^{-})$ | 3097 | 3013
$N$ | $1/2(1/2^{+})$ | 939 | 939
$\Delta$ | $3/2(3/2^{+})$ | 1232 | 1232
$\Lambda_{c}$ | $0(1/2^{+})$ | 2286 | 2286
$\Sigma_{c}$ | $1(1/2^{+})$ | 2455 | 2465
$\Sigma^{*}_{c}$ | $1(3/2^{+})$ | 2490 | 2518
$\Xi_{cc}$ | $1/2(1/2^{+})$ | 3621 | 3766
$\Xi_{cc}^{*}$ | $1/2(3/2^{+})$ |  | 3791

##  III The results and discussions

In the present calculation, we systematically investigate the $S$-wave
$qqc\bar{c}c$ ($q=u~{}\text{or}~{}d)$ pentaquark systems in the framework of
the QDCSM. The quantum numbers $I$ = 0 and 1, $J^{P}=1/2^{-},3/2^{-}$, and
$5/2^{-}$ are considered. First, we study the effective potential of each
channel, which is presented in the Fig. 1 and Fig. 2. Moreover, to find out if
there exists any bound state, we carry out a dynamic bound-state calculation
of both single-channel and channel coupling. The root mean square (rms) of
cluster spacing of the obtained state is calculated to determine the spatial
configuration. Additionally, the scattering process is also studied to search
for the resonance state, and the detail of this process is introduced in
Appendix. In addition, we further investigate the influence of different
interaction terms in the effective potential and their contribution to the
binding energy, to explore the nature of the obtained states.

###  III.1 Effective potential

Since an attractive potential is necessary for forming a bound state, we first
calculate the effective potential. It is defined as
$V(S_{i})=E(S_{i})-E(\infty)$, where $S_{i}$ stands for the distance between
two clusters. $E(S_{i})$ and $E(\infty)$ are the energies of the system at the
generator coordinate $S_{i}$ and at a sufficient large distance, respectively.
$E(S_{i})$ is obtained as:

$\displaystyle E\left(S_{i}\right)$
$\displaystyle=\frac{\left\langle\Psi_{5q}\left(S_{i}\right)|H|\Psi_{5q}\left(S_{i}\right)\right\rangle}{\left\langle\Psi_{5q}\left(S_{i}\right)\mid\Psi_{5q}\left(S_{i}\right)\right\rangle},$
(14)

where $\Psi_{5q}(S_{i})$ represents the wave function of a certain channel,
$\left\langle\Psi_{5q}\left(S_{i}\right)|H|\Psi_{5q}\left(S_{i}\right)\right\rangle$
and
$\left\langle\Psi_{5q}\left(S_{i}\right)\mid\Psi_{5q}\left(S_{i}\right)\right\rangle$
are the diagonal matrix element of the Hamiltonian and the overlap of the
system. In order to investigate the interaction between baryons and mesons,
the effective potential of each channel with different quantum numbers is
presented in Fig. 1 and 2. In addition, after channel coupling, the lowest
energies are chosen to draw the effective potential of coupled-channel.

For the $I({J^{P}})=0(1/2^{-})$ system, in Fig. 1, there are five physical
channels: the $\Lambda_{c}\eta_{c}$, $\Lambda_{c}J/\psi$, $\Xi_{cc}\bar{D}$,
$\Xi_{cc}\bar{D}^{*}$, and $\Xi_{cc}^{*}\bar{D}^{*}$. One can see that the
effective potentials of the $\Lambda_{c}\eta_{c}$ and
$\Xi_{cc}^{*}\bar{D}^{*}$ channels are purely repulsive. Therefore, the
$\Lambda_{c}\eta_{c}$ and $\Xi_{cc}^{*}\bar{D}^{*}$ cannot form bound state in
the single channel calculation due to the lack of attraction. Additionally,
the effective potentials of the $\Xi_{cc}\bar{D}$ and $\Xi_{cc}\bar{D}^{*}$
show weakly attraction at about 1.2 fm, but have relatively large repulsion at
close range. Quantum mechanics tells us that the distance between two clusters
is probabilistic from close range to large range, so both interactions at
close range and medium range play a role in the formation of state. In this
case, the $\Xi_{cc}\bar{D}$ and $\Xi_{cc}\bar{D}^{*}$ are difficult to form
bound state. As for the $\Lambda_{c}J/\psi$, the effective potential at medium
range is attractive and there is no repulsion at close range. It is likely for
the $\Lambda_{c}J/\psi$ to form a bound state and a dynamic calculation is
carried out in the next subsection. After channel coupling, there is a strong
attraction, indicating that the coupled-channel is likely to be bound.

For the $I({J^{P}})=0(3/2^{-})$ system, the $\Xi_{cc}\bar{D}^{*}$,
$\Xi_{cc}^{*}\bar{D}$, and $\Xi_{cc}^{*}\bar{D}^{*}$ show very weak
attraction, which is not enough to form a bound state. The effective
potentials of the $\Lambda_{c}\eta_{c}$ and coupled-channel are all purely
repulsive. Besides, one might wonder why the effective potential of the
coupled-channel is not the lowest one. This is because effective potential is
obtained as $V(S_{i})=E(S_{i})-E(\infty)$, where $E(\infty)$ of different
channel is different. Although the $E(S_{i})$ of the coupled-channel the lower
than that of every single channel, the $E(\infty)$ of the coupled-channel is
also the lowest one. Therefore, sometimes the effective potential of the
coupled-channel is not the lowest one. Moreover, for the
$I({J^{P}})=0(3/2^{-})$ system, the main composition of the coupled-channel is
the $\Lambda_{c}J/\psi$. Compared with the $\Lambda_{c}J/\psi$, the repulsion
of the coupled-channel is not as strong, but it still cannot form a bound
state.

Since our current calculation is based on the $S$-wave system ($L=0$), there
is only $\Xi_{cc}^{*}\bar{D}^{*}$ can be coupled to a system with
$I({J^{P}})=0(5/2^{-})$. And there is no need for channel coupling. One can
find that the effective potential is attractive at medium range, indicating
that it is possible to form a bound state.

Figure 1:  The effective potential of $qqc\bar{c}c$ system with $I=0$.

For the $I({J^{P}})=1(1/2^{-})$ system, there are six channels, which are the
$\Sigma_{c}\eta_{c}$, $\Sigma_{c}J/\psi$, $\Sigma_{c}^{*}J/\psi$,
$\Xi_{cc}\bar{D}$, $\Xi_{cc}\bar{D}^{*}$, and $\Xi_{cc}^{*}\bar{D}^{*}$. Only
the $\Sigma_{c}^{*}J/\psi$ shows a weak attraction, and the repulsion at close
range is also very weak, making it still possible to be a bound state. After
channel coupling, the attraction of coupled-channel is deepened and there is
almost no repulsive interaction. It is likely for the $qqc\bar{c}c$ with
$I({J^{P}})=1(1/2^{-})$ to form bound state.

For the $I({J^{P}})=1(3/2^{-})$ system, the effective potential of all single-
channels are repulsive. However, the coupled-channel show an obvious
attraction, which may result in a bound state. As for the
$I({J^{P}})=1(5/2^{-})$ system, there are two channels the
$\Sigma_{c}^{*}J/\psi$ and $\Xi_{cc}^{*}\bar{D}^{*}$. The two channels
together with the coupled-channel are all purely repulsive. Therefore, the the
$qqc\bar{c}c$ with $I({J^{P}})=1(5/2^{-})$ cannot form any bound state.

Figure 2:  The effective potential of $qqc\bar{c}c$ system with $I=1$.

###  III.2 Bound-state calculation

In order to confirm whether any state with attractive interaction can form a
bound state, a dynamic calculation is carried out in this part. The numerical
results of pentaquark systems with different quantum numbers are listed in
Table 3. The first column headed with $I(J^{P})$ is the quantum number of each
system. The second column headed with ”Channel”, gives the physical channels
involved in the present work. The third column headed with $E_{\mathrm{th}}$
refers to the theoretical value of non-interacting baryon-meson threshold. The
forth column headed with $E_{\mathrm{sc}}$ shows the energy of each single
channel. The values of binding energy $E_{\mathrm{B}}$=
$E_{\mathrm{sc}}-E_{\mathrm{th}}^{\mathrm{Theo}}$ are listed in the fifth
column only if $E_{\mathrm{B}}<0$ MeV. Otherwise there will be ”Ub”, which
represents that the system is unbound. The experimental threshold
$E_{\mathrm{th}}^{\mathrm{Exp}}$, which is the sum of the experimental masses
of the corresponding baryon and meson, and the corrected mass
$E^{\prime}=E_{\mathrm{th}}^{\mathrm{Exp}}+E_{\mathrm{B}}$ are given in the
sixth and seventh columns. After channel coupling, the lowest energy of
coupled-channel $E_{\mathrm{cc}}$ and the corresponding binding energy
$E_{\mathrm{B}}$ is listed in the eighth column. Finally, the corrected mass
of coupled-channel $E^{\prime}_{\mathrm{cc}}$ is given in the last column. The
definitions of $E_{\mathrm{cc}}$ and $E_{\mathrm{B}}$ in channel coupling
calculation are similar to that in single-channel calculation. When we deal
with the mass correction of the coupled-channel, a modified formula Hu:2022pae
; Yan:2022nxp  is
used—$E^{\prime}_{\mathrm{cc}}=E_{\mathrm{cc}}+\sum_{n}p_{n}\left[E_{\mathrm{th}}^{\mathrm{Exp}}(n)-E_{\mathrm{th}}^{\mathrm{Theo}}(n)\right]$,
where $p_{n}$ is the proportion of $n$th physical channel. In addition, the
experimental value of $\Xi_{cc}^{*}$ has not been measured so far. Here we use
the calculated value of $\Xi_{cc}^{*}$ in Ref. Ortiz-Pacheco:2023kjn  for mass
correction.

It is worth noting that, only the lowest energy of each channel is presented
in the table, because whether the system can form bound state depends on
whether the lowest energy is below the threshold. First of all, from Table 3,
an intuitive analysis can be based on the results of the single-channel
calculation. For the $I({J^{P}})=0(1/2^{-})$ system, only the
$\Lambda_{c}J/\psi$ forms bound state in the single-channel calculation, and
other four channels are all scattering states. The binding energy of the
$\Lambda_{c}J/\psi$ is about $-$15 MeV. However, the $\Lambda_{c}J/\psi$ can
decay to $\Lambda_{c}\eta_{c}$ through strong interaction. In order to
determine whether the $\Lambda_{c}J/\psi$ can be a resonance state, the
scattering phase shift of this process is studied in the next part. Although
the $\Xi_{cc}\bar{D}$ and $\Xi_{cc}\bar{D}^{*}$ have attractive effective
potentials, it turns out that the weak attractions are not enough to form any
bound state. According to the effective potential of coupled-channel, the
interaction between baryon and meson is attractive. The numerical results also
show that the channel coupling forms a bound state which is 49 MeV below the
lowest threshold of the $I({J^{P}})=0(1/2^{-})$ system. The main composition
of this state is the $\Lambda_{c}\eta_{c}$ and $\Lambda_{c}J/\psi$ according
to the numerical calculation. Since in the current work we only consider
$S$-wave systems, this bound state cannot to other channels. Moreover, we
carry out the mass correction and the rms calculation of this bound state. The
corrected mass of the obtained state is 5259 MeV and the rms is 1.0 fm. On the
basis of its relatively small rms, it is a compact pentaquark state.

Table 3:  The energies of single-channels and coupled-channels (in MeV). $I(J^{P})$ |  Channel |  $E_{\mathrm{th}}^{\mathrm{Theo}}$ |  $E_{\mathrm{sc}}$ |  $E_{\mathrm{B}}$ |  $E_{\mathrm{th}}^{\mathrm{Exp}}$ |  $E_{\mathrm{sc}}^{\prime}$ |  $E_{\mathrm{cc}}/E_{\mathrm{B}}$ |  $E_{\mathrm{cc}}^{\prime}$
---|---|---|---|---|---|---|---|---
$0(1/2^{-})$ | $\Lambda_{c}\eta_{c}$ | 5232 | 5234 | Ub | 5270 | 5272 | 5183/$-$49 | 5259
| $\Lambda_{c}J/\psi$ | 5261 | 5246 |  $-$15 | 5383 | 5368 |  |
| $\Xi_{cc}\bar{D}$ | 5644 | 5646 | Ub | 5490 | 5492 |  |
| $\Xi_{cc}\bar{D}^{*}$ | 5782 | 5783 | Ub | 5628 | 5629 |  |
| $\Xi_{cc}^{*}\bar{D}^{*}$ | 5807 | 5809 | Ub | 5693 | 5695 |  |
$0(3/2^{-})$ | $\Lambda_{c}J/\psi$ | 5261 | 5363 | Ub | 5383 | 5385 | 5863/Ub | 5385
| $\Xi_{cc}\bar{D}^{*}$ | 5782 | 5784 | Ub | 5628 | 5630 |  |
| $\Xi_{cc}^{*}\bar{D}$ | 5669 | 5670 | Ub | 5555 | 5556 |  |
| $\Xi_{cc}^{*}\bar{D}^{*}$ | 5807 | 5809 | Ub | 5693 | 5695 |  |
$0(5/2^{-})$ | $\Xi_{cc}^{*}\bar{D}^{*}$ | 5807 | 5804 |  $-$3 | 5693 | 5690 | 5804/$-$3 | 5690
$1(1/2^{-})$ | $\Sigma_{c}\eta_{c}$ | 5411 | 5413 | Ub | 5439 | 5441 | 5341/$-$70 | 5396
| $\Sigma_{c}J/\psi$ | 5439 | 5442 | Ub | 5552 | 5555 |  |
| $\Sigma_{c}^{*}J/\psi$ | 5464 | 5463 |  $-$2 | 5617 | 5616 |  |
| $\Xi_{cc}\bar{D}$ | 5644 | 5647 | Ub | 5490 | 5493 |  |
| $\Xi_{cc}\bar{D}^{*}$ | 5782 | 5785 | Ub | 5628 | 5631 |  |
| $\Xi_{cc}^{*}\bar{D}^{*}$ | 5807 | 5809 | Ub | 5693 | 5695 |  |
$1(3/2^{-})$ | $\Sigma_{c}J/\psi$ | 5439 | 5442 | Ub | 5552 | 5555 | 5357/$-$78 | 5465
| $\Sigma_{c}^{*}\eta_{c}$ | 5435 | 5437 | Ub | 5504 | 5506 |  |
| $\Sigma_{c}^{*}J/\psi$ | 5464 | 5466 | Ub | 5617 | 5619 |  |
| $\Xi_{cc}\bar{D}^{*}$ | 5782 | 5785 | Ub | 5628 | 5631 |  |
| $\Xi_{cc}^{*}\bar{D}$ | 5669 | 5671 | Ub | 5555 | 5557 |  |
| $\Xi_{cc}^{*}\bar{D}^{*}$ | 5807 | 5809 | Ub | 5693 | 5695 |  |
$1(5/2^{-})$ | $\Sigma_{c}^{*}J/\psi$ | 5464 | 5466 | Ub | 5617 | 5619 | 5466/Ub | 5619
| $\Xi_{cc}^{*}\bar{D}^{*}$ | 5807 | 5809 | Ub | 5693 | 5695 |  |

For the $I({J^{P}})=0(3/2^{-})$ system, the energy of each single-channel is
above the corresponding threshold. Therefore, the four single-channels are all
scattering states and cannot be resonance states. After channel-coupling, the
lowest energy of coupled-channel is still higher than the threshold of the
lowest channel $\Lambda_{c}J/\psi$. Based on this, there is no stable state in
the $I({J^{P}})=0(3/2^{-})$ system, which is consistent with the behavior of
the corresponding effective potential.

For the $I({J^{P}})=0(5/2^{-})$ system, a bound state
$\Xi_{cc}^{*}\bar{D}^{*}$ with binding energy of $-$3 MeV is obtained. The
bound state $\Xi_{cc}^{*}\bar{D}^{*}$ can still decay to some $D$-wave
channels, such as the $\Lambda_{c}J/\psi$, through the tensor force coupling.
However, we focus on the $S$-wave systems in this work, thus the tensor force
coupling is not yet considered. The $D$-wave decay will be the next step of
our research in the future. According to our previous research Chen:2011zzb ,
the decay width of $D$-wave decay is usually very narrow. The corrected mass
and the rms of this $\Xi_{cc}^{*}\bar{D}^{*}$ state is 5690 MeV and 1.9 fm,
indicating that it is a molecular state. Herein, an $I({J^{P}})=0(5/2^{-})$
molecular state $\Xi_{cc}^{*}\bar{D}^{*}$ with energy of 5690 MeV is
predicted.

For the $I({J^{P}})=1(1/2^{-})$ system, the $\Sigma_{c}^{*}J/\psi$ forms a
bound state with binding energy of $-$2 MeV in the single-channel calculation.
The other five channels turn out to be scattering states. After channel
coupling, the energy of the coupled-channel is 70 MeV lower than the lowest
threshold, indicating the formation of a bound state. Additionally, this state
is dominated by the $\Sigma_{c}\eta_{c}$. According to the further
calculation, the corrected mass of this state is 5396 MeV and the rms is 0.9
fm. Therefore, there is a possible compact pentaquark $qqc\bar{c}c$ with
$I({J^{P}})=1(1/2^{-})$, whose mass is 5396 MeV.

For the $I({J^{P}})=1(3/2^{-})$ system, consistent with the behavior of
effective potential, none of the single-channels is bound. However, the
effective potential shows that there is a strong attraction after channel
coupling and the numerical result also confirms this. The coupled-channel
forms a deeply bound state, whose binding energy is $-78$ MeV. This state is
mainly composed of the $\Sigma_{c}J/\psi$ and $\Sigma_{c}^{*}\eta_{c}$ and its
corrected mass is 5468 MeV. As for the spatial configuration, it is determined
to be a compact pentaquark state, based on its rms of 0.9 fm.

All energies of single-channels and coupled-channel of $I({J^{P}})=1(5/2^{-})$
system are higher than the corresponding thresholds. In other word, no stable
state is found in this system. The complete repulsion in the effective
potential is also consistent with the numerical result.

Further, one can find that none of the single-channels with
$I({J^{P}})=1(3/2^{-})$ are bound, but the coupled-channel is bound. In order
to figure out what causes this to happen, we investigate the effective
potential of each interaction term. Here we take the $qqc\bar{c}c$ with
$I({J^{P}})=1(3/2^{-})$ as an example. The effective potential of each
interaction term of the single-channel $\Sigma_{c}^{*}\eta_{c}$, which is the
lowest energy threshold, and the coupled-channel is shown in Fig. 3. As we can
see, the effect of $\pi$ exchange potential term ($V_{\pi}$) and $\eta$
exchange potential term ($V_{\eta}$) is very weak both before and after
channel coupling. The kinetic energy term always provides a large repulsive
effect at the close range. However, the repulsive effect is weakened after
channel coupling. The confinement potential term ($V_{\mathrm{CON}}$) and one-
gluon exchange ($V_{\mathrm{OGE}}$) potential term are repulsive before
channel coupling, but become obviously attractive after coupling. Therefore,
the mechanism of binding state formation is that channel coupling can weaken
the repulsion of kinetic energy term, while confinement potential term and
one-gluon exchange potential term provide strong attraction under the effect
of channel coupling. A similar situation also occurs in the $qqc\bar{c}c$
systems with $I({J^{P}})=0(1/2^{-})$ and $I({J^{P}})=1(1/2^{-})$.
Additionally, the contribution of each interaction term to the binding energy
is calculated and listed in Table 4. For the $qqc\bar{c}c$ systems with
$I({J^{P}})=0(1/2^{-})$, $I({J^{P}})=1(1/2^{-})$, and $I({J^{P}})=1(3/2^{-})$,
the repulsive contribution of kinetic energy term and the attractive
contribution of $V_{\mathrm{CON}}$ and $V_{\mathrm{OGE}}$ are consistent with
the result of effective potential.

Figure 3:  The effective potential of each interaction term of the $qqc\bar{c}c$ system with $I({J^{P}})=1(3/2^{-})$, where (a) and (b) are the results of the single-channel and coupled-channel, respectively. Table 4:  The contribution of each interaction term to the binding energy after channel coupling (in MeV). $I(J^{P})$ | mass |  kinetic |  $V_{\text{CON}}$ |  $V_{\text{OGE}}$ |  $V_{\pi,\eta}$ |  total
---|---|---|---|---|---|---
$0(1/2^{-})$ | 0 | +64.1 |  $-$71.8 |  $-$41.4 | +0.0 |  $-$49.1
$1(1/2^{-})$ | 0 | +75.1 |  $-$126.4 |  $-$20.3 | +1.5 |  $-$70.0
$1(3/2^{-})$ | 0 | +65.5 |  $-$92.1 |  $-$52.8 | +1.0 |  $-$78.4

Another factor that plays an important role in channel coupling is the
delocalization effect of the model. This is one of the features of our model,
which allows quarks to run between baryon and meson. The delocalization
parameter $\epsilon$ is determined variationally by the dynamics of the system
itself. Thus, the pentaquark system chooses its favorable configuration in the
interacting process. According to this, the variational space of the system is
expanded and the strength of interaction between two clusters (the baryon
cluster and meson cluster) can be reflected by the delocalization parameter
$\epsilon$. Here we continue to take the $I({J^{P}})=1(3/2^{-})$ system as an
example, the delocalization parameter $\epsilon$ of each channel is shown in
Fig. 4 (a). First, at extremely close range ($S_{i}<0.1$ fm), the
delocalization parameter $\epsilon$ approaches 1 because the baryon and meson
can no longer be divided into two clusters when they are very close. However,
based on the value of $\epsilon$, the role of delocalization effect in the
$I({J^{P}})=1(3/2^{-})$ system is still evident at close range (0.1 $<S_{i}<$
0.6 fm). The increase of the interaction strength between baryon and meson
caused by the delocalization effect also leads to the enhancement of the
channel coupling effect. Therefore, the repulsion of effective potential of
single-channel becomes attractive after channel coupling at close range.

On the other hand, the trend of change in the delocalization parameter
$\epsilon$ can also indirectly reflect the possible structure of the system.
Due to the influence of delocalization, there is mixing between the baryon and
meson clusters of the $I({J^{P}})=1(3/2^{-})$ system. Although the composition
calculation shows that the main components of the $I({J^{P}})=1(3/2^{-})$
system are $\Sigma_{c}J/\psi$ and $\Sigma_{c}^{*}\eta_{c}$, the real
components are not pure $\Sigma_{c}J/\psi$ and $\Sigma_{c}^{*}\eta_{c}$ due to
the delocalization effect. The mixing between baryons and mesons makes it more
of a compact pentaquark state. This conclusion is consistent with the small
value of rms = 0.9 fm, which also shows the characteristic of a compact
structure. As a comparison, the delocalization effect in the
$I({J^{P}})=0(5/2^{-})$ system is much smaller, which can be seen in Fig. 4
(b). The delocalization parameter has been approaching 0 since 0.2 fm. In this
case, this system retains its molecular structure. The rms of this state is
1.9 fm, confirming that it is a molecular state.

Figure 4:  The variation of delocalization parameter $\epsilon$ of each
channel with (a) $I({J^{P}})=1(3/2^{-})$ and (b) $I({J^{P}})=0(5/2^{-})$.

###  III.3 Scattering process and resonance states

As mentioned in the previous section, some quasibound states are obtained in
the single-channel calculation, but the energy of each quasibound state is
above the threshold of some channels of the corresponding system, which means
that the quasibound states will decay to the lower channels and become
resonance states or scattering states. Therefore, the scattering phase shift
calculation is performed to find resonance states. First, the scattering phase
shift of the $\Lambda_{c}\eta_{c}$ with $I({J^{P}})=0(1/2^{-})$ is presented
in Fig. 5 to test whether the $\Lambda_{c}J/\psi$ forms a resonance state. A
resonance state can usually be determined by the phase shift increased by
180∘. The phase shift of open channel $\Lambda_{c}\eta_{c}$ shows 180∘
increase around the resonance mass, indicating that the $\Lambda_{c}J/\psi$
forms a resonance state. Additionally, the way of identifying the mass and
decay width of the resonance state through scattering phase shift can be seen
in Appendix. The resonance mass, corrected mass, decay width, and the value of
the rms of this resonance state are summarized as follows:

$\displaystyle M^{\mathrm{Theo}}$ $\displaystyle=5245~{}\mathrm{MeV},$
$\displaystyle M^{\prime}$ $\displaystyle=5367~{}\mathrm{MeV},$
$\displaystyle\Gamma$ $\displaystyle=1~{}\mathrm{MeV},$ rms
$\displaystyle=1.5~{}\mathrm{fm}.$

It is worth noting that, although the wave function of a resonance state is
nonintegrable, we can calculate the rms of the main component of the resonance
state, whose wave function is integrable. According to the numerical result,
the rms of the resonance state $\Lambda_{c}J/\psi$ is 1.5 fm, indicating that
it is likely to be a molecular state. Thus, a narrow resonance state
$\Lambda_{c}J/\psi$ with molecular configuration, whose corrected mass and
decay width are 5367 and 1 MeV, is confirmed.

In Fig. 6, we study the scattering phase shifts of the open channel
$\Sigma_{c}\eta_{c}$ and $\Sigma_{c}J/\psi$ to examine whether the
$\Sigma_{c}^{*}J/\psi$ can form a resonance state. However, the phase shifts
of open channels do not show a sharp increase around the energies of the
quasibound state $\Sigma_{c}^{*}J/\psi$. The result shows that the
$\Sigma_{c}^{*}J/\psi$ becomes scattering state rather than resonance state.
We further examine the eigenvalues obtained after channel coupling and find
that the lowest energy with the main component of the $\Sigma_{c}^{*}J/\psi$
is above its threshold. This indicates that the energy of the
$\Sigma_{c}^{*}J/\psi$ is elevated by the $\Sigma_{c}\eta_{c}$ and
$\Sigma_{c}J/\psi$ in channel coupling process, thus the
$\Sigma_{c}^{*}J/\psi$ becomes scattering state. This can be understood by the
fact that the binding energy of the $\Sigma_{c}^{*}J/\psi$ in the single-
channel calculation is very small ($\leq$ 2 MeV). So it can be easily pushed
above the corresponding threshold after the channel coupling and becomes a
scattering state. In addition, one can notice that in both Fig. 5 and 6, as
the incident energy approaches 0 MeV, the phase shifts of the open channels
$\Lambda_{c}\eta_{c}$ and $\Sigma_{c}\eta_{c}$ tend to 180∘, which conforms to
the characteristics of bound states.

Figure 5:  The phase shift of the open channel $\Lambda_{c}\eta_{c}$ with
$I(J^{P})=0(\frac{1}{2}^{-})$ channel coupling. Figure 6:  The phase shifts of
the open channel (a) $\Sigma_{c}\eta_{c}$ and (b) $\Sigma_{c}J/\psi$ with
$I(J^{P})=1(\frac{1}{2}^{-})$ channel coupling.

The obtained states in this work are summarized in Table 5. Additionally,
there have been some theoretical studies on the $qqc\bar{c}c$ pentaquark
states Chen:2017jjn ; Wang:2019aoc ; Wang:2018ihk ; An:2019idk . In Ref.
Wang:2019aoc ; Chen:2017jjn , several possible molecular states are obtained,
which are the $\Xi_{cc}\bar{D}^{*}$ with $I(J^{P})=0(1/2^{-})$,
$\Xi_{cc}\bar{D}_{1}$ with $I(J^{P})=0(1/2^{+},3/2^{+})$, and
$\Xi_{cc}\bar{D}_{2}^{*}$ with $I(J^{P})=0(3/2^{+},5/2^{+})$. However, the
composition of these possible states is not the same as that studied in our
present work. In Ref. Wang:2018ihk , the masses of the $P_{cc\bar{c}ud}$
pentaquark states with $J^{P}=1/2^{+}$ and $J^{P}=1/2^{-}$ are calculated,
which are about 5610 and 5720 MeV. In addition, the structure of the two
states is the scalar-diquark-scalar-diquark-antiquark type. In Ref. An:2019idk
, a stable $qqc\bar{c}c$ pentaquark state with $I(J^{P})=0(5/2^{-})$ is
obtained, whose mass is 5681 MeV. Our work also leads to a similar conclusion
that the $qqc\bar{c}c$ with $I(J^{P})=0(5/2^{-})$ can form a pentaquark state.
Moreover, according to our calculations, it has a molecular configuration.

Table 5:  The states obtained in this work. $I(J^{P})$ |  Mass (in MeV) |  Configuration
---|---|---
$0(1/2^{-})$ | 5259 | compact pentaquark
$0(1/2^{-})$ | 5367 | molecular state
$0(5/2^{-})$ | 5690 | molecular state
$1(1/2^{-})$ | 5396 | compact pentaquark
$1(3/2^{-})$ | 5465 | compact pentaquark

##  IV Summary

In this work, we investigate the hidden-charm $qqc\bar{c}c$ systems in the
framework of the QDCSM. The $S$-wave pentaquark systems with $I$ = 0 and 1,
$J^{P}$ = $1/2^{-}$, $3/2^{-}$, and $5/2^{-}$ are considered. The effective
potnetial is studied to describe the interaction between baryons and mesons.
Both the single-channel and the coupled-channel dynamic bound-state
calculation is carried out to search for possible states. 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 determine the structure of the obtained states.

According to our numerical results, we obtain three compact $qqc\bar{c}c$
pentaquark states. The masses and the quantum numbers of them are 5259 MeV
with $I(J^{P})$ = $0(1/2^{-})$, 5396 MeV with $I(J^{P})$ = $1(1/2^{-})$, and
5465 MeV with $I(J^{P})$ = $1(3/2^{-})$. In the decay channel
$\Lambda_{c}\eta_{c}$, we find an $I(J^{P})=0(1/2^{-})$ molecular state
$\Lambda_{c}J/\psi$ and its mass and decay width are 5367 and 1 MeV,
respectively. Another obtained molecular state is $I(J^{P})$ = $0(5/2^{-})$
$\Xi_{cc}^{*}\bar{D}^{*}$, whose mass is 5690 MeV. All these states are worthy
of further experimental exploration. In addition, the present study shows that
the influence of channel coupling is necessary in describing the multiquark
system. Several obtained states in this work result from the influence of
channel coupling and these states tend to have a compact configuration. Based
on this, we would like to emphasize the importance of channel coupling in
studying exotic hadron states.

###### Acknowledgements.

This work is supported partly by the National Science Foundation of China
under Contract Nos. and Postgraduate Research and Practice Innovation Program
of Jiangsu Province under Grant no.KYCX23 1675\.

## Appendix: 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 antisymmetrization 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 eigenproblem, 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)

$h^{\pm}_{L}$ are 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 the open channel. $R_{C}$ is a cutoff radius beyond which all the
strong interaction can be disregarded. Additionally, $\alpha_{i}$ 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 equation for 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 $\pi/2$
at a resonance mass. We can obtain a resonance mass at the position of the
phase shift of $\pi/2$. The decay width is the mass difference between the
phase shift of $3\pi/4$ and $\pi/4$.

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