# Investigating $\Xi$ resonances from pentaquark perspective

Ye Yan1<EMAIL_ADDRESS>Qi Huang1<EMAIL_ADDRESS>Xinmei Zhu2
xmzhu@yzu.edu.cn(Corresponding author) Hongxia Huang1
hxhuang@njnu.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

We have investigated the $qss\bar{q}q$ ($q=u$ or $d$) system to find possible
pentaquark explanations for the $\Xi$ resonances. The bound state calculation
is carried out within the framework of the quark delocalization color
screening model. The scattering processes are also studied to examine the
possible resonance states. The current results indicate that the $\Xi(1950)$
can be interpreted as $\Lambda\bar{K}^{*}$ state with $J^{P}=1/2^{-}$. Three
states are identified that match the $\Xi(2250)$, which are
$\Sigma^{*}\bar{K}^{*}$ state with $J^{P}=3/2^{-}$, $\Sigma^{*}\bar{K}^{*}$
state with $J^{P}=5/2^{-}$, and $\Xi^{*}\rho$ state with $J^{P}=5/2^{-}$. This
may explain the conflicting experimental values for the width of the
$\Xi(2250)$. A new $\Xi$ resonance is predicted, whose mass and width are
2066–2079 MeV and 186–189 MeV, respectively. These results contribute to
understanding the nature of the $\Xi$ resonances and to the future search for
new $\Xi$ resonances. Moreover, it is meaningful to further investigate the
$\Xi$ resonances from an unquenched picture on the basis of pentaquark
investigation.

##  I Introduction

The study of strange baryons forms a bridge between light-flavor baryons and
heavy-flavor baryons, playing a critical role in our comprehension of the
baryon spectrum. Through the efforts of experimental collaborations and the
accumulation of experimental data, the information about strange baryons has
been continually unveiled SIDDHARTA:2011dsy ; CLAS:2013rjt ; CLAS:2014tbc ;
BESIII:2015dvj ; Belle:2018mqs ; Belle:2019zco ; Belle:2018lws ;
BESIII:2019cuv ; Belle:2021gtf ; LHCb:2020jpq ; BGOOD:2021sog ; ALICE:2022yyh
; J-PARCE31:2022plu ; Amsterdam-CERN-Nijmegen-Oxford:1977bvi . However,
compared with $\Lambda$ and $\Sigma$ baryons in the $S=-1$ sector, the
information regarding $\Xi$ baryons in the $S=-2$ sector is even more obscure.
So far, only the ground octet and decuplet states with four-star ratings,
$\Xi(1320)$ with $J^{P}=1/2^{+}$ and $\Xi(1530)$ with $J^{P}=3/2^{+}$, as well
as the three-star-rated $\Xi(1820)$ with $J^{P}=3/2^{-}$, have determined
spin-parity quantum numbers Workman:2022ynf . Our understanding of the
remaining eight $\Xi$ resonances is still insufficient.

Understanding these $\Xi$ resonances and explaining their properties and
structures has always been a focal point of theoretical work, helping us gain
a better comprehension of strong interactions and QCD (quantum
chromodynamics). The most direct approach is to understand these states from
the perspective of traditional three-quark configuration. The $\Xi$ spectrum
has been investigated as three-quark configuration in the framework of various
quark models Chao:1980em ; Capstick:1986ter ; Glozman:1995fu ; Glozman:1997ag
; Valcarce:2005rr ; Bijker:2000gq ; Arifi:2022ntc ; Pervin:2007wa ;
Xiao:2013xi ; Menapara:2021dzi , Skyrme model Oh:2007cr , lattice QCD
Engel:2013ig , QCD sum rules Jido:1996zw ; Lee:2002jb ; Aliev:2018hre , and
large-$N_{c}$ analysis Semay:2007cv ; Schat:2001xr ; Goity:2003ab ;
Matagne:2004pm ; Matagne:2006zf . According to the numerical results, the
interpretation of $\Xi(1318)$, $\Xi(1530)$, $\Xi(1820)$, and $\Xi(1950)$ as
three-quark excited states is consistent with the $\Xi$ baryon spectrum
obtained by most of the aforementioned approaches. In addition, interpreting
$\Xi(1620)$ as a $J^{P}=1/2^{-}$ state is supported in Ref. Oh:2007cr  using
the Skyrme model and Ref. Jido:1996zw  using the QCD sum rules. In Ref.
Arifi:2022ntc , the spin-parity quantum numbers of $J^{P}=1/2^{-}$ is also
favored for $\Xi(1620)$ by studying its property of strong decay. As for
$\Xi(1690)$, in the Skyrme model Oh:2007cr , it is predicted to have
$J^{P}=1/2^{-}$. The same $J^{P}=1/2^{-}$ prediction for $\Xi(1690)$ can be
found in other theoretical works, by using different quark models
Arifi:2022ntc ; Pervin:2007wa ; Xiao:2013xi  and the QCD sum rules
Aliev:2018hre . A $J^{P}=1/2^{+}$ $\Xi$ state derived from the nonrelativistic
quark model also matches well with $\Xi(1690)$ Chao:1980em . In Ref.
Xiao:2013xi , $\Xi(1950)$ might correspond to several different $\Xi$
resonances, and $\Xi(2030)$ seems to have $J^{P}=3/2^{+}$, although this
conflicts with the spin quantum number provided by moment analysis Amsterdam-
CERN-Nijmegen-Oxford:1977bvi . In Ref. PavonValderrama:2011gp  analysis, based
on the experimental measurements, it is proposed that $\Xi(1950)$ might
correspond to three $\Xi$ resonances, which are $J^{P}=1/2^{-}$ decuplet,
$J^{P}=5/2^{+}$ octet and $J^{P}=5/2^{-}$ octet.

In recent years, many states have been observed by experimental collaborations
that are difficult to explain using traditional hadronic states. In
particular, the $P_{c}$ and $P_{cs}$ states have been widely regarded as
promising candidates for pentaquarks in researches LHCb:2020jpq ; LHCbPc2015 ;
LHCbPc2019 ; LHCb:2022ogu ; Chen:2016qju ; Liu:2019zoy ; Yang:2020atz .
Additionally, the $\Lambda(1405)$ has also attracted extensive attention
CLAS:2013rjt ; CLAS:2014tbc ; BGOOD:2021sog ; ALICE:2022yyh ;
J-PARCE31:2022plu . While the pole position and the structure of the
$\Lambda(1405)$ are still in controversy, many work support its two poles
nature and the possible molecular explanation Jido:2003cb ; Hyodo:2011ur ;
Meissner:2020khl ; Hyodo:2020czb ; Qin:2020gxr ; Mai:2020ltx ;
BaryonScatteringBaSc:2023zvt . The progress inspires us to explore $\Xi$
resonances from a broader perspective, suggesting possibilities beyond the
traditional three-quark configuration, including pentaquark configuration,
dynamical effect, or unquenched picture.

Some theoretical studies have delved into the interpretation of $\Xi$
resonances beyond the conventional three-quark configuration. In the framework
of the one-boson-exchange model, the $\Xi(1620)$ can be explained as a
$\bar{K}\Lambda$ molecular state, and a $\bar{K}\Sigma$ molecular state is
also predicted Chen:2019uvv . Using the same model, it is suggested that
$\Xi(2030)$ can be assigned as a $P$-wave
$\bar{K}\Sigma/\rho\Xi/\bar{K}\Lambda/\phi\Xi/\omega\Xi$ molecular state with
$J^{P}=5/2^{+}$ Hei:2023eqz . In Ref. Sarti:2023wlg , femtoscopic data is used
to constrain the parameters of a low-energy effective QCD Lagrangian and the
primary composition of the $\Xi(1620)$ is suggested to be $\eta\Xi$.
Additionally, the decays and the production of the $\Xi(1620)$ are studied in
the framework of the effective Lagrangian approach. The radiative decays of
the $\Xi(1620)$ is studied by assuming that it is a
$\Lambda\bar{K}-\Sigma\bar{K}$ molecular state in Ref. Huang:2020taj . In Ref.
Guo:2023gvx , the production of the $\Xi(1620)$ in the $K^{-}p$ scattering
process is investigated, where the $\Xi(1620)$ is considered as a
$\bar{K}\Lambda$ molecular state. The production of the $\Xi(1690)$ from the
$K^{-}p\rightarrow K^{+}K^{-}\Lambda$ reaction is also studied within the
effective Lagrangian approach in Ref. Ahn:2018hbc  Based on the Bethe-Salpeter
equation approach Garcia-Recio:2003ejq , it is suggested that the quantum
numbers $J^{P}=1/2^{-}$ be assigned to the $\Xi(1620)$ and the $\Xi(1690)$.
Using a similar approach Wang:2019krq , the $\Xi(1620)$ can be explained as
$\Lambda\bar{K}$ and $\Sigma\bar{K}$ bound states with $J^{P}=1/2^{-}$,
respectively. In Ref. Feijoo:2023wua , the $\Xi(1620)$ and the $\Xi(1690)$ are
studied as molecular states from $S=-2$ meson-baryon interaction using an
extended Unitarized Chiral Perturbation Theory. In Ref. Shevchenko:2015oea ,
$\Xi(1950)$ is explained as a $\bar{K}\bar{K}N$ state using Faddeev-type AGS
equations.

By constructing the dynamical baryon-meson scattering processes in the
strangeness $S=-2$ sector, many results have been achieved in the framework of
the chiral unitary approach. According to the findings presented in Refs.
Ramos:2002xh ; Ramos:2003mu ; Nishibuchi:2023acl ; Miyahara:2016yyh ;
Li:2023olv ; Gamermann:2011mq , signals associated with the $\Xi(1620)$ are
dynamically generated, and the spin-parity quantum numbers of the $\Xi(1620)$
is identified to be $J^{P}=1/2^{-}$. Regarding the nature of this resonance,
in Refs. Ramos:2002xh ; Miyahara:2016yyh ; Li:2023olv , results support that
$\Xi(1620)$ couples strongly to the $\Xi\pi$ and the $\Lambda\bar{K}$
channels. In Ref. Gamermann:2011mq , $\Xi(1620)$ is considered to strongly
couple to the $\Xi\pi$ channel. It is demonstrated in Ref. Nishibuchi:2023acl
that the spectrum of the $\Xi(1620)$ is distorted by the effect of the nearby
$\Lambda\bar{K}$ threshold. In Ref. Sun:2021bgl , a $J^{P}=1/2^{-}$ resonance
state with a mass of about 1550 MeV and a decay width of 120–200 MeV is
dynamically generated, which is inconsistent with the $\Xi(2120)$. Meanwhile,
in the similar framework Miyahara:2016yyh ; Li:2023olv ; Liu:2023jwo ;
Sekihara:2015qqa ; Gamermann:2011mq ; Khemchandani:2016ftn  , a $\Xi$ state
with $J^{P}=1/2^{-}$ can be dynamically generated near the $\Sigma\bar{K}$
threshold, which can be identified with the $\Xi(1690)$ resonance. In Refs.
Gamermann:2011mq ; Sarkar:2004jh , $\Xi(1820)$ with $J^{P}=3/2^{-}$ can be
dynamically generated in the meson-baryon scattering process using the
coupled-channel unitary approach. A similar conclusion can also be found in
Ref. Kolomeitsev:2003kt , within the $\chi$-BS(3) approach. While in Ref.
Molina:2023uko , chiral unitary approach for baryon-meson interaction gives
rise to a narrow state and a wide state around the $\Xi(1820)$, indicating
that the $\Xi(1820)$ may actually be two states. For other $\Xi$ resonances,
$\Xi(1950)$ is identified with a spin-parity $J^{P}=1/2^{-}$ state
Gamermann:2011mq . A narrow $J^{P}=3/2^{-}$ state with mass $\sim$2050 MeV is
found, and related to the $\Xi(2120)$ in Ref. Khemchandani:2016ftn . In Ref.
Oset:2010tof , two states are generated that can be associated to the
$\Xi(1950)$ and the $\Xi(2120)$, using the formalism that produces doublets of
degenerate $J^{P}=1/2^{-},3/2^{-}$ states.

Studying hadron-hadron scattering processes is crucial for understanding
resonances. In addition to the above theoretical methods, the quark
delocalization color screening model (QDCSM) provides an alternative approach
for studying multiquark systems. It offers a robust description of the $NN$
and $YN$ interactions and the properties of the deuteron Ping:2000dx ;
Ping:1998si ; Wu:1998wu ; Pang:2001xx . Moreover, building on the foundation
of the Kohn-Hulthén-Kato (KHK) variational method Oka:1981ri , the model is
employed to compute hadron-hadron scattering phase shifts and investigate
exotic hadronic states Huang:2015uda ; Huang:2018wed ; Huang:2023jec . In Ref.
Huang:2018wed , hidden-charm pentaquark resonances are investigated in the
hadron-hadron scattering process, and three of the obtained states are
consistent with the $P_{c}$ states later reported by the LHCb collaboration in
2019.

Using this model, we have investigated the pentaquark interpretations of the
$\Lambda_{c}$ baryons Yan:2022nxp  and $\Omega_{c}$ baryons Yan:2023tvl ,
successfully obtaining masses and widths that agree with experimental values.
Therefore, we extend the application of this model to investigate the
$qss\bar{q}q$ system, aiming to explore potential molecular interpretations of
$\Xi$ resonances. In this work, we investigate the energy spectrum of the
pentaquark system. The scattering processes are studied to assess the presence
of resonance states and provide information about the characteristics of the
identified states.

This paper is organized as follows. In the next section, we provide a detailed
overview of QDCSM, scattering process, and the construction of wave function.
In Sec. III, numerical results and discussions are given, followed by a
summary in Sec. IV.

##  II THEORETICAL FORMALISM

###  II.1 Quark delocalization color screening model

Herein, the QDCSM is employed to investigate the properties of $qss\bar{q}q$
system. The QDCSM is an extension of the native quark cluster model
DeRujula:1975qlm ; Isgur:1978xj ; Isgur:1978wd ; Isgur:1979be  and has been
developed to address multiquark systems Ping:2000cb ; Huang:2013nba ;
Xue:2020vtq ; Yan:2023tvl . 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 .

In the present work, we mainly focus on the low-lying negative parity
$qss\bar{q}q$ 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.

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.

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. $b$ |  $m_{q}$ |  $m_{s}$ |  $V_{0_{qq}}$ |  $V_{0_{q\bar{q}}}$ |  $a_{c}$
---|---|---|---|---|---
(fm) | (MeV) | (MeV) | (fm-2) | (fm-2) |  (MeV fm-2)
0.518 | 313 | 573 |  $-$1.288 |  $-$0.201 | 58.03
$\alpha_{s_{qq}}$ | $\alpha_{s_{qs}}$ | $\alpha_{s_{ss}}$ | $\alpha_{s_{q\bar{q}}}$ | $\alpha_{s_{s\bar{q}}}$ |
0.565 | 0.524 | 0.451 | 1.793 | 1.783 |
Table 2: The masses of the baryons and mesons. Experimental values are taken from the Particle Data Group (PDG) Workman:2022ynf  (in MeV). Hadron |  $I(J^{P})$ |  $M^{\text{Exp}}$ |  $M^{\text{Theo}}$
---|---|---|---
$N$ | $1/2(1/2^{+})$ | 939 | 939
$\Delta$ | $3/2(3/2^{+})$ | 1232 | 1232
$\Lambda$ | $0(1/2^{+})$ | 1115 | 1123
$\Sigma$ | $1(1/2^{+})$ | 1189 | 1238
$\Sigma^{*}$ | $1(3/2^{+})$ | 1385 | 1360
$\Xi$ | $1/2(1/2^{+})$ | 1318 | 1374
$\Xi^{*}$ | $1/2(3/2^{+})$ | 1535 | 1496
$\eta$ | $0(0^{-})$ | 582 | 284
$\omega$ | $0(1^{-})$ | 782 | 842
$K$ | $1/2(1^{-})$ | 495 | 495
$K^{*}$ | $1/2(1^{-})$ | 892 | 892
$\pi$ | $1(0^{-})$ | 139 | 139
$\rho$ | $1(1^{-})$ | 770 | 890

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 .

###  II.2 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

$\psi_{5q}={\cal
A}\left[[\phi_{B}\phi_{M}]^{[\sigma]IS}\otimes\chi(\boldsymbol{R})\right]^{J},$
(14)

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:

$\int
H(\boldsymbol{\mathbf{R}},\boldsymbol{\mathbf{R^{\prime}}})\chi(\boldsymbol{\mathbf{R^{\prime}}})d\boldsymbol{\mathbf{R^{\prime}}}=E\int
N(\boldsymbol{\mathbf{R}},\boldsymbol{\mathbf{R^{\prime}}})\chi(\boldsymbol{\mathbf{R^{\prime}}})d\boldsymbol{\mathbf{R^{\prime}}},$
(15)

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

$\displaystyle\chi(\boldsymbol{R})=$
$\displaystyle\frac{1}{\sqrt{4\pi}}\left(\frac{6}{5\pi
b^{2}}\right)^{3/4}\sum_{i,L,M}C_{i,L}$
$\displaystyle\cdot\int\exp\left[-\frac{3}{5b^{2}}\left(\boldsymbol{R}-\boldsymbol{S}_{i}\right)^{2}\right]Y_{L,M}\left(\hat{\boldsymbol{S}}_{i}\right)d\Omega_{\boldsymbol{S}_{i}}$
(16)

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:

$\phi_{C}(\boldsymbol{R}_{C})=(\frac{5}{\pi
b^{2}})^{3/4}e^{-\frac{5\boldsymbol{R}^{2}_{C}}{2b^{2}}},$ (17)

the ansatz Eq. (14) can be rewritten as

$\displaystyle\psi_{5q}=$
$\displaystyle\mathcal{A}\sum_{i,L}C_{i,L}\int\frac{d\Omega_{\boldsymbol{S}_{i}}}{\sqrt{4\pi}}\prod_{\alpha=1}^{3}\phi_{\alpha}\left(\boldsymbol{S}_{i}\right)\prod_{\beta=4}^{5}\phi_{\beta}\left(-\boldsymbol{S}_{i}\right)$
$\displaystyle\cdot\left[\left[\chi_{I_{1}S_{1}}\left(B\right)\chi_{I_{2}S_{2}}\left(M\right)\right]^{IS}Y_{LM}\left(\hat{\boldsymbol{S}}_{i}\right)\right]^{J}$
$\displaystyle\cdot\left[\chi_{c}\left(B\right)\chi_{c}\left(M\right)\right]^{[\sigma]},$
(18)

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,

$\displaystyle\phi_{\alpha}\left(\boldsymbol{S}_{i}\right)$
$\displaystyle=\left(\frac{1}{\pi
b^{2}}\right)^{3/4}e^{-\frac{1}{2b^{2}}\left(r_{\alpha}-\frac{2}{5}\boldsymbol{S}_{i}\right)^{2}},$
$\displaystyle\phi_{\beta}\left(\boldsymbol{-S}_{i}\right)$
$\displaystyle=\left(\frac{1}{\pi
b^{2}}\right)^{3/4}e^{-\frac{1}{2b^{2}}\left(r_{\beta}+\frac{3}{5}\boldsymbol{S}_{i}\right)^{2}}.$
(19)

With the reformulated ansatz Eq. (18), the RGM Eq. (15) becomes an algebraic
eigenvalue equation:

$\sum_{j}C_{j}H_{i,j}=E\sum_{j}C_{j}N_{i,j},$ (20)

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

$\displaystyle\chi_{L}(\mathbf{R})$
$\displaystyle=\sum_{i}C_{i}\frac{\tilde{u}_{L}\left(\boldsymbol{R},\boldsymbol{S}_{i}\right)}{\boldsymbol{R}}Y_{L,M}(\hat{\boldsymbol{R}}),$
(21)

with

$\displaystyle\tilde{u}_{L}\left(\boldsymbol{R},\boldsymbol{S}_{i}\right)$
$\displaystyle=\left\\{\begin{array}[]{ll}\alpha_{i}u_{L}\left(\boldsymbol{R},\boldsymbol{S}_{i}\right),&\boldsymbol{R}\leq\boldsymbol{R}_{C}\\\
{\left[h_{L}^{-}(\boldsymbol{k},\boldsymbol{R})-s_{i}h_{L}^{+}(\boldsymbol{k},\boldsymbol{R})\right]R_{AB},}&\boldsymbol{R}\geq\boldsymbol{R}_{C}\end{array}\right.$
(24)

where

$\displaystyle u_{L}\left(\boldsymbol{R},\boldsymbol{S}_{i}\right)=$
$\displaystyle\sqrt{4\pi}\left(\frac{6}{5\pi
b^{2}}\right)^{3/4}\mathbf{R}e^{-\frac{3}{5b^{2}}\left(\boldsymbol{R}-\boldsymbol{S}_{i}\right)^{2}}$
$\displaystyle\cdot i^{L}j_{L}\left(-i\frac{6}{5b^{2}}S_{i}\right).$ (25)

$h^{\pm}_{L}$ are the $L$th spherical Hankel functions, $k$ is the momentum of
the relative motion with $k=\sqrt{2\mu E_{\text{{ie}}}}$, $\mu$ is the reduced
mass of two hadrons of the open channel, $E_{\text{ie}}$ is the incident
energy of the relevant open channels, which can be written as
$E_{\text{ie}}=E_{\text{{total}}}-E_{\text{{th}}}$, where $E_{\text{{total}}}$
denotes the total energy, and $E_{\text{{th}}}$ 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

$\displaystyle\sum_{j}\mathcal{L}_{ij}^{L}C_{j}$
$\displaystyle=\mathcal{M}_{i}^{L}(i=0,1,\ldots,n-1),$ (26)

with

$\displaystyle\mathcal{L}_{ij}^{L}$
$\displaystyle=\mathcal{K}_{ij}^{L}-\mathcal{K}_{i0}^{L}-\mathcal{K}_{0j}^{L}+\mathcal{K}_{00}^{L},$
$\displaystyle\mathcal{M}_{i}^{L}$
$\displaystyle=\mathcal{K}_{00}^{L}-\mathcal{K}_{i0}^{L},$ (27)

and

$\displaystyle\mathcal{K}_{ij}^{L}=$
$\displaystyle\left\langle\hat{\phi}_{A}\hat{\phi}_{B}\frac{\tilde{u}_{L}\left(\boldsymbol{R}^{\prime},\boldsymbol{S}_{i}\right)}{\boldsymbol{R}^{\prime}}Y_{L,M}\left(\boldsymbol{R}^{\prime}\right)|H-E|\right.$
$\displaystyle\left.\cdot\mathcal{A}\left[\hat{\phi}_{A}\hat{\phi}_{B}\frac{\tilde{u}_{L}\left(\boldsymbol{R},\boldsymbol{S}_{j}\right)}{\boldsymbol{R}}Y_{L,M}(\boldsymbol{R})\right]\right\rangle.$
(28)

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

$\displaystyle S_{L}$ $\displaystyle=e^{2i\delta_{L}}=\sum_{i}C_{i}s_{i}.$
(29)

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

$\displaystyle\sigma_{L}$
$\displaystyle=\frac{4\pi}{k^{2}}(2L+1)\sin^{2}\delta_{L}.$ (30)

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 $\pi/4$.

###  II.3 Wave function of $\boldsymbol{qss\bar{q}q}$ system

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. (35) and Eq. (67), respectively.

$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{\sigma}(3)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\ \hline\cr
3\\\
\cline{1-1}\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|}\hline\cr$\alpha$&$\alpha$\\\
\hline\cr$\beta$\\\
\cline{1-1}\cr\end{tabular}\,\big{>}=\frac{1}{\sqrt{6}}(2\alpha\alpha\beta-\alpha\beta\alpha-\beta\alpha\alpha),$
(35) $\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{\sigma\prime}(3)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&3\\\ \hline\cr
2\\\
\cline{1-1}\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|}\hline\cr$\alpha$&$\alpha$\\\
\hline\cr$\beta$\\\
\cline{1-1}\cr\end{tabular}\,\big{>}=\frac{1}{\sqrt{2}}(\alpha\beta\alpha-\beta\alpha\alpha),$
(40) $\displaystyle\chi_{\frac{1}{2},-\frac{1}{2}}^{\sigma}(3)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\ \hline\cr
3\\\
\cline{1-1}\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|}\hline\cr$\alpha$&$\beta$\\\
\hline\cr$\beta$\\\
\cline{1-1}\cr\end{tabular}\,\big{>}=\frac{1}{\sqrt{6}}(\alpha\beta\beta+\beta\alpha\beta-2\beta\beta\alpha),$
(45) $\displaystyle\chi_{\frac{1}{2},-\frac{1}{2}}^{\sigma\prime}(3)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&3\\\ \hline\cr
2\\\
\cline{1-1}\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|}\hline\cr$\alpha$&$\beta$\\\
\hline\cr$\beta$\\\
\cline{1-1}\cr\end{tabular}\,\big{>}=\frac{1}{\sqrt{2}}(\alpha\beta\beta-\beta\alpha\beta),$
(50) $\displaystyle\chi_{\frac{3}{2},\frac{3}{2}}^{\sigma}(3)$
$\displaystyle=\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&3\\\
\hline\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|c|}\hline\cr$\alpha$&$\alpha$&$\alpha$\\\
\hline\cr\end{tabular}\,\big{>}=\alpha\alpha\alpha,$ (53)
$\displaystyle\chi_{\frac{3}{2},\frac{1}{2}}^{\sigma}(3)$
$\displaystyle=\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&3\\\
\hline\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|c|}\hline\cr$\alpha$&$\alpha$&$\beta$\\\
\hline\cr\end{tabular}\,\big{>}=\frac{1}{\sqrt{3}}(\alpha\alpha\beta+\alpha\beta\alpha+\beta\alpha\alpha),$
(56) $\displaystyle\chi_{\frac{3}{2},-\frac{1}{2}}^{\sigma}(3)$
$\displaystyle=\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&3\\\
\hline\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|c|}\hline\cr$\alpha$&$\beta$&$\beta$\\\
\hline\cr\end{tabular}\,\big{>}=\frac{1}{\sqrt{3}}(\alpha\beta\beta+\beta\alpha\beta+\beta\beta\alpha),$
(59) $\displaystyle\chi_{\frac{3}{2},-\frac{3}{2}}^{\sigma}(3)$
$\displaystyle=\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&3\\\
\hline\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|c|}\hline\cr$\beta$&$\beta$&$\beta$\\\
\hline\cr\end{tabular}\,\big{>}=\beta\beta\beta.$ (62)
$\displaystyle\chi_{0,0}^{\sigma}(2)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|}\hline\cr 1\\\ \hline\cr 2\\\
\hline\cr\end{tabular}\,\,\begin{tabular}[]{|c|}\hline\cr$\alpha$\\\
\hline\cr$\beta$\\\
\hline\cr\end{tabular}\,\big{>}=\frac{1}{\sqrt{2}}(\alpha\beta-\beta\alpha),$
(67) $\displaystyle\chi_{1,1}^{\sigma}(2)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\
\hline\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|}\hline\cr$\alpha$&$\alpha$\\\
\hline\cr\end{tabular}\,\big{>}=\alpha\alpha,$ (70)
$\displaystyle\chi_{1,0}^{\sigma}(2)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\
\hline\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|}\hline\cr$\alpha$&$\beta$\\\
\hline\cr\end{tabular}\,\big{>}=\frac{1}{\sqrt{2}}(\alpha\beta+\beta\alpha),$
(73) $\displaystyle\chi_{1,-1}^{\sigma}(2)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\
\hline\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|}\hline\cr$\beta$&$\beta$\\\
\hline\cr\end{tabular}\,\big{>}=\beta\beta.$ (76)

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

$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{\sigma 1}(5)=$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{\sigma}(3)\chi_{0,0}^{\sigma}(2),$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{\sigma 2}(5)=$
$\displaystyle-\sqrt{\frac{2}{3}}\chi_{\frac{1}{2},-\frac{1}{2}}^{\sigma}(3)\chi_{1,1}^{\sigma}(2)+\sqrt{\frac{1}{3}}\chi_{\frac{1}{2},\frac{1}{2}}^{\sigma}(3)\chi_{1,0}^{\sigma}(2),$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{\sigma 3}(5)=$
$\displaystyle\sqrt{\frac{1}{6}}\chi_{\frac{3}{2},-\frac{1}{2}}^{\sigma}(3)\chi_{1,1}^{\sigma}(2)-\sqrt{\frac{1}{3}}\chi_{\frac{3}{2},\frac{1}{2}}^{\sigma}(3)\chi_{1,0}^{\sigma}(2)$
$\displaystyle+\sqrt{\frac{1}{2}}\chi_{\frac{3}{2},\frac{3}{2}}^{\sigma}(3)\chi_{1,-1}^{\sigma}(2),$
$\displaystyle\chi_{\frac{3}{2},\frac{3}{2}}^{\sigma 4}(5)=$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{\sigma}(3)\chi_{1,1}^{\sigma}(2),$
$\displaystyle\chi_{\frac{3}{2},\frac{3}{2}}^{\sigma 5}(5)=$
$\displaystyle\chi_{\frac{3}{2},\frac{3}{2}}^{\sigma}(3)\chi_{0,0}^{\sigma}(2),$
$\displaystyle\chi_{\frac{3}{2},\frac{3}{2}}^{\sigma 6}(5)=$
$\displaystyle\sqrt{\frac{3}{5}}\chi_{\frac{3}{2},\frac{3}{2}}^{\sigma}(3)\chi_{1,0}^{\sigma}(2)-\sqrt{\frac{2}{5}}\chi_{\frac{3}{2},\frac{1}{2}}^{\sigma}(3)\chi_{1,1}^{\sigma}(2),$
$\displaystyle\chi_{\frac{5}{2},\frac{5}{2}}^{\sigma 7}(5)=$
$\displaystyle\chi_{\frac{3}{2},\frac{3}{2}}^{\sigma}(3)\chi_{1,1}^{\sigma}(2).$
(77)

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

$\displaystyle\chi_{0,0}^{f}(3)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\ \hline\cr
3\\\
\cline{1-1}\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|}\hline\cr$u$&$s$\\\
\hline\cr$d$\\\
\cline{1-1}\cr\end{tabular}\,\big{>}=\sqrt{\frac{1}{4}}(usd+sud-sdu-dsu),$
(82) $\displaystyle\chi_{0,0}^{f\prime}(3)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&3\\\ \hline\cr
2\\\
\cline{1-1}\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|}\hline\cr$u$&$s$\\\
\hline\cr$d$\\\ \cline{1-1}\cr\end{tabular}\,\big{>}$ (87)
$\displaystyle=\sqrt{\frac{1}{12}}(2uds-2dus+sdu+usd-sud-dsu),$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{f}(3)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\ \hline\cr
3\\\
\cline{1-1}\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|}\hline\cr$u$&$s$\\\
\hline\cr$s$\\\
\cline{1-1}\cr\end{tabular}\,\big{>}=\sqrt{\frac{1}{6}}(uss+sus-2ssu),$ (92)
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{f\prime}(3)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&3\\\ \hline\cr
2\\\
\cline{1-1}\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|}\hline\cr$u$&$s$\\\
\hline\cr$s$\\\ \cline{1-1}\cr\end{tabular}\,\big{>}=\sqrt{\frac{1}{2}}(uss-
sus),$ (97) $\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{f\prime\prime}(3)$
$\displaystyle=\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&3\\\
\hline\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|c|}\hline\cr$u$&$s$&$s$\\\
\hline\cr\end{tabular}\,\big{>}=\sqrt{\frac{1}{3}}(uss+sus+ssu),$ (100)
$\displaystyle\chi_{\frac{1}{2},-\frac{1}{2}}^{f}(3)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\ \hline\cr
3\\\
\cline{1-1}\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|}\hline\cr$d$&$s$\\\
\hline\cr$s$\\\
\cline{1-1}\cr\end{tabular}\,\big{>}=\sqrt{\frac{1}{6}}(dss+sds-2ssd),$ (105)
$\displaystyle\chi_{\frac{1}{2},-\frac{1}{2}}^{f\prime}(3)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&3\\\ \hline\cr
2\\\
\cline{1-1}\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|}\hline\cr$d$&$s$\\\
\hline\cr$s$\\\ \cline{1-1}\cr\end{tabular}\,\big{>}=\sqrt{\frac{1}{2}}(dss-
sds),$ (110) $\displaystyle\chi_{\frac{1}{2},-\frac{1}{2}}^{f\prime\prime}(3)$
$\displaystyle=\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&3\\\
\hline\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|c|}\hline\cr$d$&$s$&$s$\\\
\hline\cr\end{tabular}\,\big{>}=\sqrt{\frac{1}{3}}(dss+sds+ssd),$ (113)
$\displaystyle\chi_{1,1}^{f}(3)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\ \hline\cr
3\\\
\cline{1-1}\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|}\hline\cr$u$&$u$\\\
\hline\cr$s$\\\ \cline{1-1}\cr\end{tabular}\,\big{>}=\sqrt{\frac{1}{6}}(2uus-
usu-suu),$ (118) $\displaystyle\chi_{1,1}^{f\prime}(3)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&3\\\ \hline\cr
2\\\
\cline{1-1}\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|}\hline\cr$u$&$u$\\\
\hline\cr$s$\\\ \cline{1-1}\cr\end{tabular}\,\big{>}=\sqrt{\frac{1}{2}}(usu-
suu),$ (123) $\displaystyle\chi_{1,1}^{f\prime\prime}(3)$
$\displaystyle=\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&3\\\
\hline\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|c|}\hline\cr$u$&$u$&$s$\\\
\hline\cr\end{tabular}\,\big{>}=\sqrt{\frac{1}{3}}(uus+usu+suu),$ (126)
$\displaystyle\chi_{1,0}^{f}(3)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\ \hline\cr
3\\\
\cline{1-1}\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|}\hline\cr$u$&$d$\\\
\hline\cr$s$\\\ \cline{1-1}\cr\end{tabular}\,\big{>}$ (131)
$\displaystyle=\sqrt{\frac{1}{12}}(2uds+2dus-sdu-usd-sud-dsu),$
$\displaystyle\chi_{1,0}^{f\prime}(3)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&3\\\ \hline\cr
2\\\
\cline{1-1}\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|}\hline\cr$u$&$d$\\\
\hline\cr$s$\\\
\cline{1-1}\cr\end{tabular}\,\big{>}=\sqrt{\frac{1}{4}}(usd+dsu-sdu-sud),$
(136) $\displaystyle\chi_{1,0}^{f\prime\prime}(3)$
$\displaystyle=\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&3\\\
\hline\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|c|}\hline\cr$u$&$d$&$s$\\\
\hline\cr\end{tabular}\,\big{>}$ (139)
$\displaystyle=\sqrt{\frac{1}{6}}(uds+dus+sdu+usd+sud+dsu),$
$\displaystyle\chi_{1,-1}^{f}(3)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\ \hline\cr
3\\\
\cline{1-1}\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|}\hline\cr$d$&$d$\\\
\hline\cr$s$\\\ \cline{1-1}\cr\end{tabular}\,\big{>}=\sqrt{\frac{1}{6}}(2dds-
dsd-sdd),$ (144) $\displaystyle\chi_{1,-1}^{f\prime}(3)$
$\displaystyle=\,\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&3\\\ \hline\cr
2\\\
\cline{1-1}\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|}\hline\cr$d$&$d$\\\
\hline\cr$s$\\\ \cline{1-1}\cr\end{tabular}\,\big{>}=\sqrt{\frac{1}{2}}(dsd-
sdd),$ (149) $\displaystyle\chi_{1,1}^{f\prime\prime}(3)$
$\displaystyle=\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&3\\\
\hline\cr\end{tabular}\,\,\begin{tabular}[]{|c|c|c|}\hline\cr$d$&$d$&$s$\\\
\hline\cr\end{tabular}\,\big{>}=\sqrt{\frac{1}{3}}(dds+dsd+sdd).$ (152)

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

$\displaystyle\chi_{0,0}^{f}(2)$
$\displaystyle=\sqrt{\frac{1}{2}}(\bar{d}d+\bar{u}u),$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{f}(2)$ $\displaystyle=\bar{d}s,$
$\displaystyle\chi_{\frac{1}{2},-\frac{1}{2}}^{f}(2)$
$\displaystyle=-\bar{u}s,$ $\displaystyle\chi_{1,1}^{f}(2)$
$\displaystyle=\bar{d}u,$ $\displaystyle\chi_{1,0}^{f}(2)$
$\displaystyle=\sqrt{\frac{1}{2}}(\bar{d}d-\bar{u}u),$
$\displaystyle\chi_{1,-1}^{f}(2)$ $\displaystyle=-\bar{u}d.$ (153)

As for the flavor degree of freedom, the isospin $I$ of pentaquark systems we
investigated in this work is $I=1/2$. The flavor wave functions of pentaquark
systems can be expressed as

$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{f1}(5)$
$\displaystyle=\chi_{0,0}^{f}(3)\chi_{\frac{1}{2},\frac{1}{2}}^{f}(2),$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{f2}(5)$
$\displaystyle=\chi_{\frac{1}{2},\frac{1}{2}}^{f}(3)\chi_{0,0}^{f}(2),$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{f3}(5)$
$\displaystyle=-\sqrt{\frac{2}{3}}\chi_{\frac{1}{2},-\frac{1}{2}}^{f}(3)\chi_{1,1}^{f}(2)+\sqrt{\frac{1}{3}}\chi_{\frac{1}{2},\frac{1}{2}}^{f}(3)\chi_{1,0}^{f}(2),$
$\displaystyle\chi_{\frac{1}{2},\frac{1}{2}}^{f4}(5)$
$\displaystyle=\sqrt{\frac{2}{3}}\chi_{1,1}^{f}(3)\chi_{\frac{1}{2},-\frac{1}{2}}^{f}(2)-\sqrt{\frac{1}{3}}\chi_{1,0}^{f}(3)\chi_{\frac{1}{2},\frac{1}{2}}^{f}(2).$
(154)

Studies show that color screening is an effective description of the hidden-
color channel coupling ChenLZ ; Huang:2011kf . Therefore, we only consider
color-singlet channel (two clusters are color-singlet), which can be obtained
by $1\otimes 1$:

$\displaystyle\chi^{c}=$ $\displaystyle\frac{1}{\sqrt{6}}(rgb-rbg+gbr-grb+brg-
bgr)$ $\displaystyle\cdot\frac{1}{\sqrt{3}}(\bar{r}r+\bar{g}g+\bar{b}b).$
(155)

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.

##  III The results and discussions

In the present calculation, we systematically investigate the $S$-wave
$qss\bar{q}q$ ($q=u~{}\text{or}~{}d)$ pentaquark systems in the framework of
the QDCSM. The quantum numbers $I$ = 1/2, $J^{P}=1/2^{-},3/2^{-}$, and
$5/2^{-}$ are considered. The effective potential of each channel is studied
and presented in the Fig. 1, Fig. 4, and Fig. 6 as the first step. In order 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 numerical results
are listed in Table 3, Table 5, and Table 7. Moreover, to verify whether the
quasibound channel form resonance state or scattering state after channel
coupling, the scattering process is also studied, which can be seen in Fig. 2,
Fig. 3, Fig. 5, and Fig. 7. The summary of the obtained states is presented in
Table 8.

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

The formation of states requires an attractive interaction, hence we first
calculate the effective potential between baryons and mesons. The effective
potential is quantified as $V(S_{i})=E(S_{i})-E(\infty)$, where $S_{i}$
represents the distance between baryons and mesons. Here, $E(S_{i})$ denotes
the energy of the system at the generator coordinate $S_{i}$, and $E(\infty)$
corresponds to the energy when the clusters are sufficiently far apart.
$E(S_{i})$ is determined through the following expression:

$\displaystyle E\left(S_{i}\right)$
$\displaystyle=\frac{\left\langle\Psi_{5q}\left(S_{i}\right)|H|\Psi_{5q}\left(S_{i}\right)\right\rangle}{\left\langle\Psi_{5q}\left(S_{i}\right)\mid\Psi_{5q}\left(S_{i}\right)\right\rangle},$
(156)

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,
respectively. The effective potentials of the $qss\bar{q}q$ system with
$J^{P}=1/2^{-}$ are presented in Fig. 1. There are eleven physical channels,
which are $\Lambda\bar{K}$, $\Lambda\bar{K}^{*}$, $\Sigma\bar{K}$,
$\Sigma\bar{K}^{*}$, $\Sigma^{*}\bar{K}^{*}$, $\Xi\eta$, $\Xi\pi$,
$\Xi\omega$, $\Xi\rho$, $\Xi^{*}\omega$, and $\Xi^{*}\rho$. Among them, the
interaction of $\Xi\pi$ and $\Xi\omega$ exhibit very weak attraction at medium
range. Although $\Sigma^{*}\bar{K}^{*}$ exhibits attractive interaction, the
repulsive interaction at close range is very stong. In this case, it is
difficult for $\Sigma^{*}\bar{K}^{*}$ to form a bound state. As for
$\Lambda\bar{K}^{*}$, $\Xi\rho$, $\Xi^{*}\omega$, and $\Xi^{*}\rho$, it is
likely for these channels to form single-channel bound states. However,
$\Lambda\bar{K}$, $\Sigma\bar{K}$, $\Lambda\bar{K}^{*}$ and $\Xi\eta$ channels
are likely to be scattering states considering their purely repulsive
interactions.

Figure 1:  The effective potentials of the $qss\bar{q}q$ system with
$J^{P}=\frac{1}{2}^{-}$.

To verify whether channels with attractive interactions form bound states, a
dynamic calculation was conducted, and numerical results are detailed in Table
3. The first column labeled ”Channel” identifies the relevant physical
channels. The second and third columns, $\chi^{f_{i}}$ and
$\chi^{\sigma_{j}}$, respectively represent the flavor and spin wave functions
of the corresponding physical channels, which are presented in Section II C.
The fourth and fifth columns, labeled $E_{\mathrm{th}}^{\mathrm{Exp}}$ and
$E_{\mathrm{th}}^{\mathrm{Theo}}$ respectively, represent the theoretical
threshold and experimental threshold of the system. The sixth column labeled
$E$ displays the energy of each single channel. The binding energies
$E_{\mathrm{B}}=E^{\mathrm{Theo}}-E_{\mathrm{th}}^{\mathrm{Theo}}$ are
recorded in the seventh column only when $E_{\mathrm{B}}<0$ MeV; otherwise,
”Ub” indicates that the system is unbound. After channel coupling, the lowest
energy of coupled-channel and the lowest threshold are listed in the last row.

Table 3:  The single-channel and the coupled-channel energies of the $qss\bar{q}q$ pentaquark systems with $J^{P}=\frac{1}{2}^{-}$ (in MeV). Channel |  $\chi^{f_{i}}$ |  $\chi^{\sigma_{j}}$ |  $E_{\text{th}}^{\text{Exp}}$ |  $E_{\mathrm{th}}^{\text{Theo}}$ |  $E^{\text{Theo}}$ |  $E_{\text{B}}$
---|---|---|---|---|---|---
$\Lambda\bar{K}$ | $i=1$ | $j=1$ | 1610 | 1613 | 1617 | Ub
$\Lambda\bar{K}^{*}$ | $i=1$ | $j=2$ | 2007 | 2010 | 2009 |  $-$1
$\Sigma\bar{K}$ | $i=4$ | $j=1$ | 1684 | 1727 | 1731 | Ub
$\Sigma\bar{K}^{*}$ | $i=4$ | $j=2$ | 2081 | 2124 | 2128 | Ub
$\Sigma^{*}\bar{K}^{*}$ | $i=4$ | $j=3$ | 2277 | 2247 | 2250 | Ub
$\Xi\eta$ | $i=2$ | $j=1$ | 1900 | 1679 | 1683 | Ub
$\Xi\pi$ | $i=3$ | $j=1$ | 1457 | 1534 | 1537 | Ub
$\Xi\omega$ | $i=2$ | $j=2$ | 2100 | 2238 | 2242 | Ub
$\Xi\rho$ | $i=3$ | $j=2$ | 2088 | 2286 | 2266 |  $-$20
$\Xi^{*}\omega$ | $i=2$ | $j=3$ | 2318 | 2360 | 2352 |  $-$8
$\Xi^{*}\rho$ | $i=3$ | $j=3$ | 2306 | 2408 | 2394 |  $-$14
Coupling |  |  | 1457 | 1534 | 1536 | Ub

It is worth noting that only the lowest energy of each channel is presented in
the table. This is because the formation of a bound state depends on whether
the lowest energy falls below the threshold. As is listed in Table 3, four
channels form quasibound states in the single-channel calculation, which are
$\Lambda\bar{K}^{*}$, $\Xi\rho$, $\Xi^{*}\omega$, and $\Xi^{*}\rho$.
$\Lambda\bar{K}^{*}$ form a loosely bound state with a binding energy of only
$-1$ MeV. The binding energies of $\Xi\rho$, $\Xi^{*}\omega$, and
$\Xi^{*}\rho$ are $-20$ MeV, $-8$ MeV, and $-14$ MeV, respectively. According
to the effective potential, the attractions of $\Sigma^{*}\bar{K}^{*}$,
$\Xi\pi$, and $\Xi\omega$ are very weak, and the single-channel calculation
results for these four channels are unbound. The attractive interaction of
$\Xi\rho$, $\Xi^{*}\omega$, and $\Xi^{*}\rho$ is stronger than that of
$\Lambda\bar{K}^{*}$, hence the binding energies of $\Xi\rho$,
$\Xi^{*}\omega$, and $\Xi^{*}\rho$ are deeper. As for channels with purely
repulsive interaction, $\Lambda\bar{K}$, $\Sigma\bar{K}^{*}$,
$\Sigma^{*}\bar{K}^{*}$ and $\Xi\eta$, they are all unbound. The numerical
results of the single-channel calculation are consistent with the analysis of
interaction.

While four quasibound states have been identified, none of them corresponds to
the system’s lowest threshold. In this scenario, due to interactions with
other channels after channel coupling, their energies might be elevated above
their respective thresholds, resulting in scattering states, or they may
remain below their respective thresholds, forming resonance states. This
process will be further investigated in subsequent scattering processes.
Furthermore, if the energy of the coupled channel is lower than the lowest
threshold $\Xi\pi$, a stable bound state is genuinely achieved in the current
system. After conducting the channel coupling calculation, the lowest energy
of coupled-channel is 1536 MeV, which exceeds the lowest threshold $\Xi\pi$ of
1534 MeV. Therefore, the $qss\bar{q}q$ system with $J^{P}=1/2^{-}$ does not
form a bound state.

To examine whether the four quasibound channels form scattering states or
resonance states, the scattering process is studied. The details of scattering
process can be found in section II B. In Fig. 2, the scattering phase shifts
of the open channel $\Xi\pi$ with varying degrees of channel coupling are
shown. We separately calculate the phase shifts of the open channel $\Xi\pi$
in different coupling scenarios: (1) without coupling (solid line with
circular symbols), (2) coupling the $\Xi\pi$ with a quasibound channel
$\Lambda\bar{K}^{*}$ (dashed line with square symbols), (3) coupling the
$\Xi\pi$ with a quasibound channel $\Xi\rho$ (dotted line with triangular
symbols), (4) coupling the $\Xi\pi$ with four quasibound channels (dash-dotted
line with diamond symbols).

Figure 2:  The phase shifts of the open channel $\Xi\pi$ with
$J^{P}=\frac{1}{2}^{-}$.

As one can see, the phase shift of the open channel $\Xi\pi$ without coupling
remains stable as the incident energy increases. When the open channel
$\Xi\pi$ is separately coupled with the channels $\Lambda\bar{K}^{*}$ and
$\Xi\rho$, the phase shifts exhibit sharp increases at incident energies of
472 MeV and 733 MeV, respectively. This indicates that the
$\Lambda\bar{K}^{*}$ and $\Xi\rho$ are resonance states, where the resonance
energy corresponds to the incident energy with a phase shift of $\pi/2$ plus
the threshold of the open channel $\Xi\pi$. The width of decay to $\Xi\pi$
corresponds to the difference in incident energies with phase shifts of
$\pi/4$ and $3\pi/4$. After coupling the $\Xi\pi$ with the four quasibound
channels, the phase shift exhibits sharp increase at an incident energy of 472
MeV and increases again at an incident energy of 740 MeV. It indicates that
the $\Lambda\bar{K}^{*}$ and $\Xi\rho$ form resonance states, which is
consistent with the previous coupling result. The obtained resonance energies
are slightly different from that in the coupling of one quasibound channel and
the open channel. This is caused by the coupling process not being exactly the
same, but this does not affect the conclusion of the resonance states.

However, after the incident energy exceeds 750 MeV, the phase shift remains
stable. No sharp increase in phase shift is shown at the incident energies
corresponding to the $\Xi^{*}\omega$ and $\Xi^{*}\rho$. We examined a series
of eigenvalues obtained after channel coupling, where the lowest energy states
predominantly composed of $\Lambda\bar{K}^{*}$ or $\Xi\rho$ remain below their
respective thresholds, while those predominantly composed of $\Xi^{*}\omega$
or $\Xi^{*}\rho$ are above their corresponding thresholds. Therefore, in the
current system, there exist two resonance states, $\Lambda\bar{K}^{*}$ and
$\Xi\rho$, while $\Xi^{*}\omega$ and $\Xi^{*}\rho$ are scattering states.

On the basis of the phase shift of $\Xi\pi$, the $\Lambda\bar{K}^{*}$
resonance state’s theoretical resonance mass, and decay width to $\Xi\pi$ are
2006 MeV and 0.1 MeV, respectively. The correction of the resonance mass is
based on the following equation:

$\displaystyle
M^{\text{Corr}}=M^{\text{Theo}}+\sum_{n}p_{n}\left[E_{\text{th}}^{\text{Exp}}(n)-E_{\text{th}}^{\text{Theo}}(n)\right],$
(157)

where $M^{\text{Corr}}$ and $M^{\mathrm{Theo}}$ are the corrected and
theoretical mass, $p_{n}$ is the proportion of the $n$th physical channel, and
$E_{\text{th}}^{\text{Exp}}(n)$ and $E_{\text{th}}^{\text{Theo}}(n)$ are the
experimental and theoretical thresholds of the $n$th physical channel.
Moreover, the composition of this resonancece state is overwhelmingly
predominated by the $\Lambda\bar{K}^{*}$. In this scenario, the influence of
other physical channels is exceedingly feeble. Hence, based on the phase shift
of the $\Xi\pi$, the corrected mass of the $\Lambda\bar{K}^{*}$ resonance
state is 2003 MeV. In the same way, the $\Xi\rho$ resonance state’s
theoretical mass, corrected mass, and decay width to the $\Xi\pi$ are 2278
MeV, 2079 MeV, and 0.2 MeV, respectively.

Further, the scattering phase shifts of other possible open channels are
calculated with the coupling of the four quasibound channels and the open
channel. The results are shown in Fig. 3. The $\Lambda\bar{K}^{*}$ and
$\Xi\rho$ resonance states can be simultaneously observed through the
scattering phase shifts of the $\Lambda\bar{K}$, $\Sigma\bar{K}$, and
$\Xi\eta$. Since the threshold of $\Sigma\bar{K}^{*}$ being higher than the
resonance energy of the $\Sigma\bar{K}$, only the $\Xi\rho$ is observed in the
scattering phase shift of $\Lambda\bar{K}^{*}$. The thresholds of
$\Sigma^{*}\bar{K}^{*}$ and $\Xi\omega$ are higher than the energies of the
two resonance states, thus no resonance state is observed.

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

By analyzing the scattering phase shifts, the theoretical resonance masses and
the decay widths to different open channels are obtained. The numerical
results, including the corrected masses, are presented in Table 4. On this
basis, the total decay widths $\Gamma_{\text{total}}$ can be estimated by
$\Gamma_{\text{total}}=\Gamma_{\text{el}}+\Gamma_{\text{inel}}$, where
$\Gamma_{\text{el}}$ is the elastic widths to open channels in Table 4, and
$\Gamma_{\text{inel}}$ is the inelastic width caused by the decay of the
hadron components in resonance states. Subsequently, we will introduce how to
estimate the inelastic width of the obtained states Ping:2008tp .

Table 4:  The masses and elastic widths of resonance states with the difference scattering processes (in MeV). $M^{\text{Theo}}_{\text{res}}$ and $M^{\text{Corr}}_{\text{res}}$ stand for the theoretical and corrected masses, respectively. $\Gamma_{\text{el}}$ is the elastic width of the resonance state decaying to an open channel. $J^{P}=1/2^{-}$ |  |  $\Lambda\bar{K}$ |  $\Sigma\bar{K}$ |  $\Sigma\bar{K}^{*}$ |  $\Xi\eta$ |  $\Xi\pi$
---|---|---|---|---|---|---
$\Lambda\bar{K}^{*}$ | $M^{\text{Theo}}$ | 1994 | 2007 | … | 2007 | 2006
| $M^{\text{Corr}}$ | 1991 | 2004 | … | 2004 | 2003
| $\Gamma_{\text{el}}$ | 8.2 | 3.7 | … | 0.3 | 0.1
$\Xi\rho$ | $M^{\text{Theo}}$ | 2265 | 2273 | 2270 | 2274 | 2278
| $M^{\text{Corr}}$ | 2066 | 2074 | 2071 | 2075 | 2079
| $\Gamma_{\text{el}}$ | 5.5 | 10.8 | 23.5 | 2.5 | 0.2

Taking the $\Lambda\bar{K}^{*}$ state for example, the inelastic width
$\Gamma_{\text{inel}}(\Lambda\bar{K}^{*})$ depends on the decaying
$\bar{K}^{*}$. The inelastic width can be related approximately to the
$\bar{K}^{*}$ width $\Gamma_{f\bar{K}^{*}}$ = 51 MeV in free space by only
accounting for the reduction in phase space available to decaying bound
$\bar{K}^{*}$, whose mass have been reduced to roughly

$\displaystyle M_{b\bar{K}^{*}}\approx
M_{f\bar{K}^{*}}-\frac{m_{\bar{K}^{*}}}{m_{\Lambda}+m_{\bar{K}^{*}}}E_{B},$
(158)

where $E_{B}$ is the binding energy of the bound state. Then,

$\displaystyle\Gamma_{b\bar{K}^{*}}\left(M_{b\bar{K}^{*}}\right)\approx\Gamma_{f\bar{K}^{*}}\frac{k_{b}^{2\ell}\rho\left(M_{b\bar{K}^{*}}\right)}{k_{f}^{2\ell}\rho\left(M_{f\bar{K}^{*}}\right)},$
(159)

where $k$ is the pion momentum in the rest frame of the decaying
$\bar{K}^{*}$, $\ell=1$ is the pion angular momentum, and

$\displaystyle\rho(M)=\pi\frac{kE_{\pi}E_{\bar{K}}}{M},$ (160)

is the two-body decay phase space at mass $M$ when each decay product has c.m.
energy $E_{i}$, $i=\pi,\bar{K}$. According to the above formulae, the
inelastic width $\Gamma_{\text{inel}}(\Lambda\bar{K}^{*})$ can be obtained as
$\Gamma_{\text{inel}}(\Lambda\bar{K}^{*})=\Gamma_{b\bar{K}^{*}}$, which falls
in the range of 47–49 MeV. Due to the resonance energies obtained in different
open channel scattering processes not being exactly the same, the binding
energy in Eq. (158) has a range, leading to a range of the inelastic width as
well. Furthermore, the total decay width
$\Gamma_{\text{total}}(\Lambda\bar{K}^{*})=\Gamma_{\text{el}}(\Lambda\bar{K}^{*})+\Gamma_{\text{inel}}(\Lambda\bar{K}^{*})$
is obtained, which is 59–61 MeV.

As listed in Table 4, the corrected mass of the $\Lambda\bar{K}^{*}$ is
1991–2004 MeV. This state can be associated with the $\Xi(1950)$. In fact, the
experimental information regarding $\Xi(1950)$ is still not sufficiently
clear. According to the PDG’s description of the mass of the $\Xi(1950)$
Workman:2022ynf , they list experiment data reported between 1875 and 2000 MeV
and suggest that there may be more than one $\Xi$ near this mass. In addition,
the decay width of the $\Lambda\bar{K}^{*}$ resonance is also well consistent
with the experimental data of the $\Xi(1950)$. Therefore, our calculations
suggest that the $\Xi(1950)$ can be interpreted a $J^{P}=1/2^{-}$
$\Lambda\bar{K}^{*}$ state. Among other theoretical works, in Ref
Gamermann:2011mq , $S=-2$ meson-baryon scattering in the $S$-wave is studied
within a coupled channel unitary approach, explaining the $\Xi(1950)$ as a
dynamically generated state with spin-parity $J^{P}=1/2^{-}$. The spin-parity
assignment of $J^{P}=1/2^{-}$ for the $\Xi(1950)$ is also suggested in Ref.
Oset:2010tof .

As for the $\Xi\rho$ resonance state, the corrected mass is 2066–2079 MeV.
Based on the inelastic width $\Gamma_{\text{inel}}$ caused by the decaying
$\rho$, the total decay width of the $\Xi\rho$ is 186–189 MeV. Although its
mass near the $\Xi(2030)$ and the $\Xi(2120)$, its decay width is much wider
than the $\Xi(2030)$ and the $\Xi(2120)$. Furthermore, its spin is also in
conflict with the experimental finding of $\Xi(2030)$, which suggest that
$J\geq 5/2$ Amsterdam-CERN-Nijmegen-Oxford:1977bvi . Therefore, the $\Xi\rho$
resonance state obtained in this work cannot serve as a candidate for the
$\Xi(2030)$ or the $\Xi(2120)$.

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

Similar to the procedure in the previous section, the effective potential of
the $qss\bar{q}q$ system with $J^{P}=3/2^{-}$ is studied. As is shown in Fig.
4, there are ten channels in this sector. The attractive interactions of
$\Xi\rho$, $\Xi^{*}\omega$, and $\Xi^{*}\rho$ channels are relatively strong,
while $\Sigma\bar{K}^{*}$, $\Sigma^{*}\bar{K}^{*}$, $\Xi\omega$, and
$\Xi^{*}\pi$ channels exhibit comparatively weak attractive interactions. The
interactions of $\Lambda\bar{K}^{*}$, $\Sigma^{*}\bar{K}$, and $\Xi^{*}\eta$
channels with $J^{P}=3/2^{-}$ are purely repulsive, indicating that they
cannot form any bound state.

Figure 4:  The effective potentials of the $qss\bar{q}q$ system with
$J^{P}=\frac{3}{2}^{-}$.

The single-channel energies of the $qss\bar{q}q$ system with $J^{P}=3/2^{-}$
and the lowest energy of coupled-channels are listed in Table 5. According to
the numerical results, there are four single-channel bound states, which are
$\Sigma^{*}\bar{K}^{*}$, $\Xi\rho$, $\Xi^{*}\omega$, and $\Xi^{*}\rho$. The
binding energy of $\Sigma^{*}\bar{K}^{*}$ is relatively small, at just $-4$
MeV. This is also consistent with the weakly attractive interaction displayed
by $\Sigma^{*}\bar{K}^{*}$ in the calculation of the effective potential. The
binding energies of $\Xi\rho$, $\Xi^{*}\omega$, and $\Xi^{*}\rho$ are $-15$
MeV, $-9$ MeV, and $-27$ MeV, respectively. We will further discuss whether
the four quasibound channels form resonance states or scattering states after
channel coupling. Before that, one can see that the lowest energy of the
coupled-channel is 1660 MeV, still higher than the $J^{P}=3/2^{-}$ system’s
lowest threshold $\Xi^{*}\pi$ of 1657 MeV. Therefore, no bound state is formed
in the $S$-wave $qss\bar{q}q$ system with $J^{P}=3/2^{-}$.

Table 5:  The single-channel and the coupled-channel energies of the $qss\bar{q}q$ pentaquark systems with $J^{P}=\frac{3}{2}^{-}$ (in MeV). Channel |  $\chi^{f_{i}}$ |  $\chi^{\sigma_{j}}$ |  $E_{\text{th}}^{\text{Exp}}$ |  $E_{\mathrm{th}}^{\text{Theo}}$ |  $E^{\text{Theo}}$ |  $E_{\text{B}}$
---|---|---|---|---|---|---
$\Lambda\bar{K}^{*}$ | $i=1$ | $j=4$ | 2007 | 2010 | 2014 | Ub
$\Sigma\bar{K}^{*}$ | $i=4$ | $j=4$ | 2081 | 2124 | 2128 | Ub
$\Sigma^{*}\bar{K}$ | $i=4$ | $j=5$ | 1880 | 1850 | 1855 | Ub
$\Sigma^{*}\bar{K}^{*}$ | $i=4$ | $j=6$ | 2277 | 2247 | 2243 |  $-$4
$\Xi\omega$ | $i=2$ | $j=4$ | 2100 | 2238 | 2241 | Ub
$\Xi\rho$ | $i=3$ | $j=4$ | 2088 | 2286 | 2271 |  $-$15
$\Xi^{*}\eta$ | $i=2$ | $j=5$ | 2118 | 1801 | 1804 | Ub
$\Xi^{*}\pi$ | $i=3$ | $j=5$ | 1675 | 1657 | 1661 | Ub
$\Xi^{*}\omega$ | $i=2$ | $j=6$ | 2318 | 2360 | 2351 |  $-$9
$\Xi^{*}\rho$ | $i=3$ | $j=6$ | 2306 | 2408 | 2381 |  $-$27
Coupling |  |  | 1675 | 1657 | 1660 | Ub

Four quasibound states are listed in Table 5, which are
$\Sigma^{*}\bar{K}^{*}$, $\Xi\rho$, $\Xi^{*}\omega$, and $\Xi^{*}\rho$. Fig. 5
shows the scattering phase shifts of different open channels. A sharp increase
of phase shift is observed in the scattering process of each open channel.
Based on the resonance energies, these resonance signals correspond to the
same resonance state, $\Sigma^{*}\bar{K}^{*}$. The other three channels,
$\Xi\rho$, $\Xi^{*}\omega$, and $\Xi^{*}\rho$, which form bound states in
single-channel calculations, are ultimately identified as scattering states.
The widths decaying into various open channels, along with the theoretical and
corrected resonance masses, are listed in the Table 6.

Figure 5:  The phase shifts of different open channels with $J^{P}=\frac{3}{2}^{-}$. Table 6:  The mass and elastic width of the resonance state with the difference scattering processes (in MeV). $J^{P}=3/2^{-}$ |  |  $\Lambda\bar{K}^{*}$ |  $\Sigma\bar{K}^{*}$ |  $\Sigma^{*}\bar{K}$ |  $\Xi\omega$ |  $\Xi^{*}\eta$ |  $\Xi^{*}\pi$
---|---|---|---|---|---|---|---
$\Sigma^{*}\bar{K}^{*}$ | $M^{\text{Theo}}$ | 2230 | 2234 | 2245 | 2239 | 2227 | 2229
| $M^{\text{Corr}}$ | 2260 | 2264 | 2275 | 2269 | 2257 | 2259
| $\Gamma_{\text{el}}$ | 4.0 | 11.0 | 5.5 | 2.1 | 8.7 | 8.5

Considering the inelastic width $\Gamma_{\text{inel}}$ caused by the decaying
$\Sigma^{*}$ and $\bar{K}^{*}$, the total decay width of the
$\Sigma^{*}\bar{K}^{*}$ is given by
$\Gamma_{\text{total}}=\Gamma_{\text{el}}+\Gamma_{\text{inel}}$, where
$\Gamma_{\text{el}}$ can be found in Table 6 and
$\Gamma_{\text{inel}}=\Gamma_{b\Sigma^{*}}+\Gamma_{b\bar{K}^{*}}$. On this
basis, the corrected mass and total width of the $\Sigma^{*}\bar{K}^{*}$
resonance state is obtained as 2257–2269 MeV and 116–122 MeV. This resonance
mass is consistent with the $\Xi(2250)$. According to the PDG Workman:2022ynf
, there are four set of experimental values on the $\Xi(2250)$ at present:
Ref. Bristol-Geneva-Heidelberg-Lausanne-QueenMaryColl-Rutherford:1986kqv : M =
2189 $\pm$ 7 MeV, $\Gamma$ = 46 MeV,
Ref. Jenkins:1983pm : M = 2214 $\pm$ 5 MeV, $\Gamma$ = ?
Ref. Goldwasser:1970fk : M = 2295 $\pm$ 15 MeV, $\Gamma<$ 30 MeV,
Ref. Aachen-Berlin-CERN-London-Vienna:1969bau : M = 2244 $\pm$ 52 MeV,
$\Gamma$ = 130 $\pm$ 80 MeV.
Given the conflicting experimental values for decay width, there may be more
than one $\Xi(2250)$. The mass and width of the $\Sigma^{*}\bar{K}^{*}$ with
$J^{P}=3/2^{-}$ ($M$ = 2257–2269 MeV, $\Gamma$ = 116–122 MeV) is consistent
with the experimental values in Ref. Aachen-Berlin-CERN-London-Vienna:1969bau
($M=2244\pm 52$ MeV, $\Gamma=130\pm 80$ MeV). All in all, this $J^{P}=3/2^{-}$
$\Sigma^{*}\bar{K}^{*}$ resonance state can serve as a candidate for the two-
star $\Xi(2250)$. In other theoretical work, using the chiral unitary
approach, the $\Xi(2250)$ is interpreted as a dynamically generated state with
a spin-parity assignment of $J^{P}=3/2^{-}$ Gamermann:2011mq .

One may notice that despite $\Xi\rho$, $\Xi^{*}\omega$, and $\Xi^{*}\rho$
forming bound states in the single-channel calculation, they ultimately
transition into scattering states. Similar situations also occur in the
$J^{P}=1/2^{-}$ sector, where single-channel bound states with higher energies
fail to form resonance states. By solving the coupled-channel Schrödinger
equation, we can obtain a series of eigenenergies. Taking the
$\Sigma^{*}\bar{K}^{*}$ with $J^{P}=3/2^{-}$ as an example, in coupled-channel
calculation, $\Sigma^{*}\bar{K}^{*}$ is influenced not only by interactions
from lower energies but also by interactions from higher energies. Overall,
the energy of the $\Sigma^{*}\bar{K}^{*}$ may remain stable or even be
suppressed. However, for states with higher energies, there are more
eigenenergies below them than above them, making it easier for them to be
elevated above their thresholds and become scattering states.

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

In the $J^{P}=5/2^{-}$ sector, there are only three channels needed to be
considered: $\Sigma^{*}\bar{K}^{*}$, $\Xi^{*}\omega$, and $\Xi^{*}\rho$. As
shown in Fig. 6, the baryon-meson interaction of $\Sigma^{*}\bar{K}^{*}$
channel is purely repulsive. The effective potentials of $\Xi^{*}\omega$
channel and $\Xi^{*}\rho$ channel exhibit attractions, with the attraction in
the $\Xi^{*}\omega$ channel being very weak and the attraction in the
$\Xi^{*}\rho$ channel being relatively strong.

Figure 6:  The effective potentials of the $qss\bar{q}q$ system with
$J^{P}=\frac{5}{2}^{-}$.

Then, the results of the single-channel and coupled-channel calculations are
listed in the Table 7. $\Sigma^{*}\bar{K}^{*}$ and $\Xi^{*}\omega$ channels
are unbound in the single-channel calculations, while $\Xi^{*}\rho$ channel
forms a quasibound state with a relatively deep binding energy of $-55$ MeV.
It is noteworthy that after the channel coupling, the lowest energy of the
$qss\bar{q}q$ system with $J^{P}=5/2^{-}$ is 2229 MeV, which is below the
lowest threshold $\Sigma^{*}\bar{K}^{*}$. Therefore, a bound state is obtained
here. According to numerical result, the predominant composition of this bound
state is $\Sigma^{*}\bar{K}^{*}$, with a minor presence of $\Xi^{*}\omega$,
while the contribution of $\Xi^{*}\rho$ is negligible. On this basis, the
corrected mass of this $qss\bar{q}q$ bound state with $J^{P}=5/2^{-}$ is 2233
MeV.

Table 7:  The single-channel and the coupled-channel energies of the $qss\bar{q}q$ pentaquark systems with $J^{P}=\frac{5}{2}^{-}$ (in MeV). Channel |  $\chi^{f_{i}}$ |  $\chi^{\sigma_{j}}$ |  $E_{\text{th}}^{\text{Exp}}$ |  $E_{\mathrm{th}}^{\text{Theo}}$ |  $E^{\text{Theo}}$ |  $E_{\text{B}}$
---|---|---|---|---|---|---
$\Sigma^{*}\bar{K}^{*}$ | $i=4$ | $j=7$ | 2277 | 2247 | 2252 | Ub
$\Xi^{*}\omega$ | $i=2$ | $j=7$ | 2318 | 2360 | 2364 | Ub
$\Xi^{*}\rho$ | $i=3$ | $j=7$ | 2306 | 2408 | 2353 |  $-$55
Coupling |  |  | 2277 | 2247 | 2229 |  $-$18

Since a bound state is obtained, without considering $D$-wave coupling, the
total decay width equals the inelastic width. Moreover, given that it contains
two nonnegligible compositions, it should be
$\Gamma_{\text{inel}}=\sum_{n}p_{n}\Gamma_{\text{inel}}(n)$, where $p_{n}$ and
$\Gamma_{\text{inel}}(n)$ is the proportion and the inelastic width of the
$n$th physical channel. For the composition $\Sigma^{*}\bar{K}^{*}$, the
inelastic width $\Gamma_{\text{inel}}(\Sigma^{*}\bar{K}^{*})$ depends on the
decaying $\Sigma^{*}$ and $\bar{K}^{*}$. Hence, the inelastic width
$\Gamma_{\text{inel}}(\Sigma^{*}\bar{K}^{*})$ can be obtained as
$\Gamma_{\text{inel}}(\Sigma^{*}\bar{K}^{*})=\Gamma_{b\Sigma^{*}}+\Gamma_{b\bar{K}^{*}}$.
As for the other composition $\Xi^{*}\omega$, the situation is different from
the previous calculation. The dominating decay mode of $\omega$ is a three-
body deacy, but Eq. (160) is used in a two-body decay process. However, due to
the narrow decay width of $\omega$, which is only 8.7 MeV, in the calculation
of Eq. (159), the difference between the width of decaying bound $\omega$ and
the width of $\omega$ in free space is very small. We can approximately
consider the width of $\omega$ in free space as the width of decaying bound
$\omega$ $\Gamma_{b\omega}$. Therefore,
$\Gamma_{\text{inel}}(\Xi\omega)=\Gamma_{b\Xi}+\Gamma_{b\omega}$. The total
decay width of the $\Sigma^{*}\bar{K}^{*}$ bound state is 53 MeV. As mentioned
the last paragraph, the corrected mass of this bound state is 2233 MeV, which
is consistent with the $\Xi(2250)$. At the same time, the decay width of this
bound state is consistent with one of the experimental values, $46\pm 27$ MeV
Bristol-Geneva-Heidelberg-Lausanne-QueenMaryColl-Rutherford:1986kqv .

Moreover, to determine the property of the quasibound state $\Xi^{*}\rho$, the
scattering shift of the $\Sigma^{*}\bar{K}^{*}$ is shown in Fig. 7. The
scattering shift increase at the incident energy around 70 MeV indicates that
$\Xi^{*}\rho$ channel forms a resonance state. According to the calculations,
the corrected mass and the total width of the $\Xi^{*}\rho$ is 2240 MeV and
161 MeV, respectively. The mass and width of the obtained $J^{P}=5/2^{-}$
$\Xi^{*}\rho$ state is very consistent with the experimental value of the
$\Xi(2250)$ in Ref. Aachen-Berlin-CERN-London-Vienna:1969bau  ($M=2244\pm
52~{}\text{MeV},~{}\Gamma=130\pm 80$ MeV). Additionally, one may notice that
after mass correction, the energy of $\Xi^{*}\rho$ falls below the
experimental threshold of $\Sigma^{*}\bar{K}^{*}$. This is because in our
calculations, the theoretical mass difference between the
$\Sigma^{*}\bar{K}^{*}$ and $\Xi^{*}\rho$ is larger than the experimental mass
difference between the $\Sigma^{*}\bar{K}^{*}$ and $\Xi^{*}\rho$, causing the
$\Xi^{*}\rho$ with a relatively deep binding energy to still be above the
theoretical threshold of the $\Sigma^{*}\bar{K}^{*}$. But the correction mass
of the $\Xi^{*}\rho$ is below the experimental threshold of the
$\Sigma^{*}\bar{K}^{*}$.

Figure 7:  The phase shift of the open channel $\Sigma^{*}\bar{K}^{*}$ with
$J^{P}=\frac{5}{2}^{-}$.

Finally, the obtained states in this work are summarized in Table 8. According
to our results, the $\Lambda\bar{K}^{*}$ with $J^{P}=1/2^{-}$ can serve as a
candidate for the $\Xi(1950)$. Both the $\Sigma^{*}\bar{K}^{*}$ state with
$J^{P}=3/2^{-}$ and the $\Xi^{*}\rho$ state with $J^{P}=5/2^{-}$ are
consistent with the experimental values about the $\Xi(2250)$ given in Ref.
Aachen-Berlin-CERN-London-Vienna:1969bau . Moreover, the
$\Sigma^{*}\bar{K}^{*}$ state with $J^{P}=5/2^{-}$ can explain the relatively
narrow $\Xi(2250)$ reported in Ref. Bristol-Geneva-Heidelberg-Lausanne-
QueenMaryColl-Rutherford:1986kqv . On this basis, three states are obtained
that can explain the $\Xi(2250)$. This might also explain the conflicting
experimental values for the width of the $\Xi(2250)$. Additionally, we obtain
a $\Xi\rho$ state with $J^{P}=1/2^{-}$, which cannot serve as a candidate for
the reported $\Xi$ resonances at present. Therefore, we here predict a new
$\Xi$ with $J^{P}=1/2^{-}$, whose mass and width are 2066–2079 MeV and 186–189
MeV, respectively.

Table 8:  The states obtained in this work (in MeV). $J^{P}$ | Main composition | Mass |  $\Gamma_{\text{el}}$ |  $\Gamma_{\text{inel}}$ |  $\Gamma_{\text{total}}$
---|---|---|---|---|---
$1/2^{-}$ | $\Lambda\bar{K}^{*}$ | 1991–2004 | 12 | 47–49 | 59–61
$1/2^{-}$ | $\Xi\rho$ | 2066–2079 | 43 | 143–146 | 186–189
$3/2^{-}$ | $\Sigma^{*}\bar{K}^{*}$ | 2257–2269 | 40 | 76–82 | 116–122
$5/2^{-}$ | $\Sigma^{*}\bar{K}^{*}$ | 2233 | … | 53 | 53
$5/2^{-}$ | $\Xi^{*}\rho$ | 2240 | 12 | 149 | 161

##  IV Summary

In this work, we investigate the $qss\bar{q}q$ system in the framework of the
QDCSM. The $S$-wave pentaquark system with $I$ = 1/2, $J^{P}$ = $1/2^{-}$,
$3/2^{-}$, and $5/2^{-}$ are considered. The effective potnetial is studied to
describe the baryon-meson interactions. The single-channel bound-state
calculation and scattering process study are carried out to search and confirm
resonance states. Meanwhile, the coupled-channel dynamic bound-state
calculation is performed to find bound state. In addition, both the elastic
width and the inelastic width are estimated. Based on the current results, the
conclusion can be drawn as follows.

(1) In our calculations, the $\Xi(1950)$ can be interpreted as
$\Lambda\bar{K}^{*}$ state with $J^{P}=1/2^{-}$.

(2) We obtain three states that match the mass of the $\Xi(2250)$. Among them,
the $\Sigma^{*}\bar{K}^{*}$ state with $J^{P}=3/2^{-}$ and the $\Xi^{*}\rho$
state with $J^{P}=5/2^{-}$ are consistent with the broad width reported in
Ref. Aachen-Berlin-CERN-London-Vienna:1969bau , while the
$\Sigma^{*}\bar{K}^{*}$ state with $J^{P}=5/2^{-}$ matches the relatively
narrow width reported in Ref. Bristol-Geneva-Heidelberg-Lausanne-
QueenMaryColl-Rutherford:1986kqv . We propose the existence of three
$\Xi(2250)$ states, which might also explain the conflicting experimental
values for the width.

(3) A new $\Xi$ resonance with mass and width of 2066–2079 MeV and 186–189
MeV, respectively, is predicted to exist. This resonance is identified as the
$\Xi\rho$ state with $J^{P}=1/2^{-}$.

Compared with the $\Lambda$ and $\Sigma$ resonances, there is currently a lack
of theoretical researches and experimental data on the $\Xi$ resonances. We
aim for our systematic calculations to provide valuable insights into
understanding the properties of the $\Xi$ resonances and discovering new $\Xi$
resonances. It is also necessary to understand the $\Xi$ resonances from the
perspective of three-quark excited states. Additionally, considering the
mixing of three-quark and five-quark from an unquenched picture is an
important aspect.

###### Acknowledgements.

This work is supported partly by the National Natural Science Foundation of
China under Contracts Nos. 11675080, No. 11775118, No. 12305087, and No.
11535005. Y. Y. is supported by the Postgraduate Research and Practice
Innovation Program of Jiangsu Province under Grant No. KYCX23 1675 and the
Doctoral Dissertation Topic Funding Program under Grant No. YXXT23-027. Q. H.
is supported by the Start-up Funds of Nanjing Normal University under Grant
No. 184080H201B20.

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