# Spectrum of the molecular pentaquarks

Bo Wang  0000-0003-0985-2958 College of Physics Science & Technology, Hebei
University, Baoding 071002, China  Hebei Key Laboratory of High-precision
Computation and Application of Quantum Field Theory, Baoding, 071002, China
Hebei Research Center of the Basic Discipline for Computational Physics,
Baoding, 071002, China  Kan Chen  0000-0002-1435-6564<EMAIL_ADDRESS>School of Physics, Northwest University, Xi’an 710127, China  Lu Meng
0000-0001-9791-7138<EMAIL_ADDRESS>Institut für Theoretische Physik II, Ruhr-
Universität Bochum, D-44780 Bochum, Germany  Shi-Lin Zhu  0000-0002-4055-6906
<EMAIL_ADDRESS>School of Physics and Center of High Energy Physics, Peking
University, Beijing 100871, China

###### Abstract

We investigate the mass spectrum of the molecular pentaquarks composed of a
baryon and a meson. We establish the underlying relations among the near-
threshold interactions of the molecular tetraquark and pentaquark systems. We
find the existence of the molecule candidates in the
$\Sigma_{c}\bar{D}^{(\ast)}$, $DD^{\ast}$, and $D\bar{D}^{\ast}$ systems
indicates a substantial presence of the hadronic molecules in the heavy baryon
plus heavy meson systems (heavy refers to the hadrons with the $c$ and/or $s$
quarks). We make an exhaustive prediction of the possible bound/virtual
molecular states in the systems: $\Sigma_{c}^{(\ast)}\bar{D}^{(\ast)}$,
$\Sigma_{c}^{(\ast)}D^{(\ast)}$, $\Xi_{c}^{(\prime,\ast)}\bar{D}^{(\ast)}$,
$\Xi_{c}^{(\prime,\ast)}D^{(\ast)}$, $\Sigma_{c}^{(\ast)}K^{\ast}$,
$\Sigma_{c}^{(\ast)}\bar{K}^{\ast}$, $\Xi_{c}^{(\prime,\ast)}K^{\ast}$,
$\Xi_{c}^{(\prime,\ast)}\bar{K}^{\ast}$, $\Xi_{cc}^{(\ast)}\bar{D}^{(\ast)}$,
$\Xi_{cc}^{(\ast)}D^{(\ast)}$, $\Xi_{cc}^{(\ast)}K^{\ast}$,
$\Xi_{cc}^{(\ast)}\bar{K}^{\ast}$, $\Sigma^{(\ast)}\bar{D}^{(\ast)}$,
$\Sigma^{(\ast)}D^{(\ast)}$, $\Xi^{(\ast)}\bar{D}^{(\ast)}$,
$\Xi^{(\ast)}D^{(\ast)}$, $\Sigma^{(\ast)}K^{\ast}$,
$\Sigma^{(\ast)}\bar{K}^{\ast}$, $\Xi^{(\ast)}K^{\ast}$,
$\Xi^{(\ast)}\bar{K}^{\ast}$. Hunting for the predicted states in experiments
will significantly deepen our understanding of the formation mechanism of the
hadronic molecules, and shed light on the manifestation of flavor symmetry in
the low-energy residual strong interactions.

##  I Introduction

An abundance of the exotic near-threshold states has been observed in
experiments [1], starting with the ground-breaking discovery of the $X(3872)$
in 2003 [2]. The unconventional nature of these states, characterized by their
unique tetraquark and pentaquark compositions, defies the conventional meson
and baryon configurations. Unraveling these novel structures has been a
vibrant research field of hadron physics over the past two decades. Various
theoretical frameworks have been proposed to elucidate these experimental
findings, offering different perspectives on the nature of these states. These
include the compact multiquark models, kinematic effects, and the existence of
the molecular states bound by the residual strong interactions [3, 4, 5, 6, 7,
8, 9, 10, 11, 12].

The LHCb Collaboration made significant breakthroughs in the study of the
hidden-charm pentaquarks, starting with the observation of the
$P_{\psi}^{N}(4380)$ and $P_{\psi}^{N}(4450)$ in the $J/\psi p$ invariant mass
spectrum of the $\Lambda_{b}^{0}\to J/\psi pK^{-}$ process in 2015 [13]. Four
years later, with the updated data, the LHCb identified a new state,
$P_{\psi}^{N}(4312)$, and revealed that the previously reported
$P_{\psi}^{N}(4450)$ actually consists of two distinct states,
$P_{\psi}^{N}(4440)$ and $P_{\psi}^{N}(4457)$ [14]. The $J^{P}$ quantum
numbers of $P_{\psi}^{N}(4312)$, $P_{\psi}^{N}(4440)$, and
$P_{\psi}^{N}(4457)$ remain undetermined. These states are often interpreted
as the bound states of $\Sigma_{c}\bar{D}$ and $\Sigma_{c}\bar{D}^{\ast}$,
respectively, given that their masses lie below the corresponding thresholds
by a few to tens of MeVs [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34]. The discovery of these $P_{\psi}^{N}$ states has
also stimulated predictions for the hidden-charm pentaquarks with strangeness
[35, 36, 37, 38, 39, 40].

More recently, the LHCb Collaboration reported evidence of a new state,
$P_{\psi s}^{\Lambda}(4459)$, near the $\Xi_{c}\bar{D}^{\ast}$ threshold in
the $J/\psi\Lambda$ invariant mass spectrum from the $\Xi_{b}^{-}\to
J/\psi\Lambda K^{-}$ process [41]. The measured mass of $P_{\psi
s}^{\Lambda}(4459)$ is in excellent agreement with our predictions for the
$\Xi_{c}\bar{D}^{\ast}$ molecular states in Ref. [40]. The
$\Xi_{c}\bar{D}^{\ast}$ system is expected to have two states with spin-parity
$\frac{1}{2}^{-}$ and $\frac{3}{2}^{-}$, respectively. However, the current
data sample in Ref. [41] is insufficient to distinguish between these two
structures. Additionally, the LHCb recently observed a state, $P_{\psi
s}^{\Lambda}(4338)$, with high significance in the $J/\psi\Lambda$ invariant
mass distribution of the $B^{-}\to J/\psi\Lambda\bar{p}$ process [42]. The
mass of $P_{\psi s}^{\Lambda}(4338)$ is very close to the $\Xi_{c}\bar{D}$
threshold, and experimental analyses favor a spin-parity assignment of
$\frac{1}{2}^{-}$ [42]. The observation of $P_{\psi s}^{\Lambda}(4459)$ and
$P_{\psi s}^{\Lambda}(4338)$ has sparked intensive studies on their mass
spectra [43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56], strong
decays [57, 58, 59, 60], experimental lineshapes [61, 62, 63, 64], productions
[65, 66, 67], and magnetic moments [68, 69, 70, 71, 72].

The LHCb Collaboration has also made remarkable progress in the search for
tetraquark states in recent years. The discoveries of the $T_{cs0,1}(2900)$,
$T_{cc}(3875)$, and $T_{c\bar{s}0}^{a}(2900)$ have provided valuable insights
into the existence and properties of these exotic hadrons. The
$T_{cs0,1}(2900)$ tetraquark states were observed in the $D^{-}K^{+}$
invariant mass spectrum of the $B^{+}\to D^{+}D^{-}K^{+}$ decay. These states
have strangeness and fully open flavor, and their masses are close to the
$\bar{D}^{\ast}K^{\ast}$ threshold [73, 74]. The $T_{cc}(3875)$ is the first
observed double-charm tetraquark, discovered in the $D^{0}D^{0}\pi^{+}$ final
state [75, 76]. This state is a narrow resonance that lies below the
$D^{\ast+}D^{0}$ threshold by a few hundred keV. More recently, LHCb observed
the isovector tetraquark state $T_{c\bar{s}0}^{a}(2900)$ in the
$D_{s}^{+}\pi^{+}$ and $D_{s}^{+}\pi^{-}$ invariant mass distributions of the
$B$ decay processes $B^{+}\to D^{-}D_{s}^{+}\pi^{+}$ and
$B^{0}\to\bar{D}^{0}D_{s}^{+}\pi^{-}$ [77, 78]. This state was measured to
have a spin-parity of $0^{+}$. Similar to the $T_{cs0}(2900)$, the
$T_{c\bar{s}0}^{a}(2900)$ is located near the $D^{\ast}K^{\ast}$ threshold.
These observations of tetraquark states with different flavors and quantum
numbers have opened up new avenues for studying and understanding the nature
of exotic hadrons.

The near-threshold characteristics of these exotic states suggest that they
are highly plausible molecular candidates of the corresponding di-hadron
systems. Moreover, these states exhibit a significant isospin tropism, with a
majority of them being observed in the lowest isospin channels. The isospin
tropism indicates the important role of the one-boson exchange interactions.
We expect these molecule candidates do not exist as independent entities, but
rather with underlying connections. We have made significant advancements
based on our prior researches [79, 80, 81]. We have developed a systematic
framework that delineates the near-threshold S-wave interactions of the heavy-
light di-hadron systems [82]. In this framework, we assume that the strange
quark can be regarded as an inert source or a heavy quark due to its limited
involvement in the near-threshold residual strong interactions. Therefore, the
existence of these states is heavily reliant on the correlations of the light
quark ($u$ and $d$) pairs between the distinct heavy-light hadrons. This
framework categorizes the exotic states into two distinct groups, each
governed by the unique light quark pair correlations: the $qq$ type and the
$\bar{q}q$ type ($q=u,d$), respectively. Detailed information regarding these
interactions can be found in our recent work [82]. Within this comprehensive
framework, we have established connections among several observed states, such
as $P_{\psi}^{N}$, $T_{cc}$, and $T_{cs}$, as well as those among $X(3872)$,
$Z_{c}(3900)$, and $T_{c\bar{s}}$ [82]. For instance, the recently discovered
$T_{cs0}(2900)$ and $T_{c\bar{s}0}^{a}(2900)$ can be confidently identified as
the charm-strange counterparts of $T_{cc}(3875)$ and $Z_{c}(3900)$,
respectively [82].

In our recent paper [82], we have successfully constructed the complete mass
spectrum of the molecular tetraquarks, involving the $D^{(\ast)}D^{(\ast)}$,
$D^{(\ast)}\bar{D}^{(\ast)}$, $D^{(\ast)}K^{\ast}$, and
$\bar{D}^{(\ast)}K^{\ast}$ systems. Building upon this progress, we will now
extend our calculations to include the spectrum of the molecular pentaquarks,
which will encompass additional systems such as $\Sigma_{c}^{(\ast)}K^{\ast}$
and $\Sigma_{c}^{(\ast)}\bar{K}^{\ast}$, etc. (see Table 1 for a comprehensive
list of the systems under consideration).

This paper is structured as follows: Section II focuses on deriving the
effective potentials of the di-hadron systems through the utilization of
quark-level potentials. Additionally, we establish the correspondences between
the various systems under the framework of heavy quark symmetry. In Section
III, we present the calculated mass spectra of the pentaquarks, along with a
selection of the important strong decay channels. Finally, in Section IV, we
provide a concise summary of our results and conclude this work.

##  II Near-threshold effective potentials of the di-hadron systems

As we have elucidated in the Ref. [82], the near-threshold interactions of two
heavy-light hadrons are dominantly undertaken by the light $qq$ and $\bar{q}q$
($q=u,d$) correlations. The non-relativistic effective potentials for the
light $qq$ and $\bar{q}q$ can be formulated in the following forms [82]

$\displaystyle V_{qq}$ $\displaystyle=$
$\displaystyle\left(\bm{\tau}_{1}\cdot\bm{\tau}_{2}+\frac{1}{2}\tau_{0,1}\cdot\tau_{0,2}\right)(c_{s}+c_{a}\bm{\sigma}_{1}\cdot\bm{\sigma}_{2}),$
(1) $\displaystyle V_{\bar{q}q}$ $\displaystyle=$
$\displaystyle\left(-\bm{\tau}_{1}^{\ast}\cdot\bm{\tau}_{2}+\frac{1}{2}\tau_{0,1}\cdot\tau_{0,2}\right)(\tilde{c}_{s}+\tilde{c}_{a}\bm{\sigma}_{1}\cdot\bm{\sigma}_{2}),$
(2)

where $\tau_{0}$ denotes the $2\times 2$ identity matrix, $\bm{\tau}$ and
$\bm{\sigma}$ represent the Pauli matrices in the isospin and spin spaces,
respectively. The $c_{s}$ ($\tilde{c}_{s}$) and $c_{a}$ ($\tilde{c}_{a}$) are
the low-energy constants (LECs), and their values (ranges) are determined with
the well-established states in experiments [82],

$\displaystyle c_{s}=146.4\pm 10.8\textrm{ GeV}^{-2},$ $\displaystyle
c_{a}=-7.3\pm 10.5\textrm{ GeV}^{-2};$ (3) $\displaystyle 184.3\textrm{
GeV}^{-2}<\tilde{c}_{a}+\tilde{c}_{s}<187.5\textrm{ GeV}^{-2},$ $\displaystyle
78.1\textrm{ GeV}^{-2}<\tilde{c}_{a}-\tilde{c}_{s}<180.3\textrm{ GeV}^{-2}.$
(4)

The quark-level potentials $V_{qq}$ ($V_{\bar{q}q}$) can be translated into
the hadron-level potentials by sandwiching them between the initial and final
di-hadron states. For example, in the case of a system composed of the hadrons
$\mathscr{H}_{1}$ and $\mathscr{H}_{2}$ (with only $qq$ couplings), the
hadron-level potentials can be expressed as

$\displaystyle V_{\mathscr{H}_{1}\mathscr{H}_{2}}^{I,J}$ $\displaystyle=$
$\displaystyle\left\langle[\mathscr{H}_{1}\mathscr{H}_{2}]_{J}^{I}\left|V_{qq}\right|[\mathscr{H}_{1}\mathscr{H}_{2}]_{J}^{I}\right\rangle,$
(5)

where the $[\mathscr{H}_{1}\mathscr{H}_{2}]_{J}^{I}$ represents the spin-
flavor wave function of the $\mathscr{H}_{1}\mathscr{H}_{2}$ di-hadron systems
with the specified total angular momentum $J$ and total isospin $I$ [53, 54,
56].

The molecular pentaquark systems considered in this study are listed in Table
1. We simultaneously analyze two types of the di-hadron systems, where their
interactions are governed by the $qq$ and $\bar{q}q$ couplings respectively.
Our previous study has illustrated that the residual strong interactions at
the near-threshold energy scale are too weak to excite the strange quark(s)
within the hadrons [82]. Therefore, it is reasonable to treat the strange
quark as having a similar role to the charm quark. Based on this
approximation, we establish the correspondences for the di-hadron systems
presented in Table 1, as illustrated in Fig. 1.

Table 1: The considered pentaquark systems and the corresponding quark contents. The interactions of the Type-I and Type-II systems are assumed to be governed by the $V_{qq}$ and $V_{\bar{q}q}$, respectively. Type-I: $V_{qq}$ | Type-II: $V_{\bar{q}q}$
---|---
$\Sigma_{c}^{(\ast)}\bar{D}^{(\ast)}$ | $[cqq][\bar{c}q]$ | $\Sigma_{c}^{(\ast)}D^{(\ast)}$ | $[cqq][c\bar{q}]$
$\Xi_{c}^{(\prime,\ast)}\bar{D}^{(\ast)}$ | $[csq][\bar{c}q]$ | $\Xi_{c}^{(\prime,\ast)}D^{(\ast)}$ | $[csq][c\bar{q}]$
$\Sigma_{c}^{(\ast)}K^{\ast}$ | $[cqq][\bar{s}q]$ | $\Sigma_{c}^{(\ast)}\bar{K}^{\ast}$ | $[cqq][s\bar{q}]$
$\Xi_{c}^{(\prime,\ast)}K^{\ast}$ | $[csq][\bar{s}q]$ | $\Xi_{c}^{(\prime,\ast)}\bar{K}^{\ast}$ | $[csq][s\bar{q}]$
$\Xi_{cc}^{(\ast)}\bar{D}^{(\ast)}$ | $[ccq][\bar{c}q]$ | $\Xi_{cc}^{(\ast)}D^{(\ast)}$ | $[ccq][c\bar{q}]$
$\Xi_{cc}^{(\ast)}K^{\ast}$ | $[ccq][\bar{s}q]$ | $\Xi_{cc}^{(\ast)}\bar{K}^{\ast}$ | $[ccq][s\bar{q}]$
$\Sigma^{(\ast)}\bar{D}^{(\ast)}$ | $[sqq][\bar{c}q]$ | $\Sigma^{(\ast)}D^{(\ast)}$ | $[sqq][c\bar{q}]$
$\Xi^{(\ast)}\bar{D}^{(\ast)}$ | $[ssq][\bar{c}q]$ | $\Xi^{(\ast)}D^{(\ast)}$ | $[ssq][c\bar{q}]$
$\Sigma^{(\ast)}K^{\ast}$ | $[sqq][\bar{s}q]$ | $\Sigma^{(\ast)}\bar{K}^{\ast}$ | $[sqq][s\bar{q}]$
$\Xi^{(\ast)}K^{\ast}$ | $[ssq][\bar{s}q]$ | $\Xi^{(\ast)}\bar{K}^{\ast}$ | $[ssq][s\bar{q}]$

Figure 1: The correspondences among the systems in Table 1 under the heavy
($c$ and $s$) quark symmetry.

The effective potentials for the $\Sigma_{c}^{(\ast)}\bar{D}^{(\ast)}$ and
$\Xi_{c}^{(\prime,\ast)}\bar{D}^{(\ast)}$ systems are presented in Table 2. To
obtain the effective potentials for the corresponding
$\Sigma_{c}^{(\ast)}D^{(\ast)}$ and $\Xi_{c}^{(\prime,\ast)}D^{(\ast)}$
systems in Type-II, the following replacements should be made:

$\displaystyle c_{s}\to\tilde{c}_{s},\qquad c_{a}\to\tilde{c}_{a}.$ (6)

For the other systems in Type-I(II), the effective potentials can be directly
obtained using the correspondences depicted in Fig. 1. For example, for the
Type-I systems,
$V_{\Sigma_{c}\bar{D}^{\ast}}^{I,J}=V_{\Sigma_{c}K^{\ast}}^{I,J}=V_{\Sigma\bar{D}^{\ast}}^{I,J}=V_{\Sigma
K^{\ast}}^{I,J}$, and
$V_{\Xi_{c}^{\prime}\bar{D}^{\ast}}^{I,J}=V_{\Xi_{c}^{\prime}K^{\ast}}^{I,J}=V_{\Xi_{cc}\bar{D}^{\ast}}^{I,J}=V_{\Xi_{cc}K^{\ast}}^{I,J}=V_{\Xi\bar{D}^{\ast}}^{I,J}=V_{\Xi
K^{\ast}}^{I,J}$. Similar relations hold for the Type-II systems. The matrix
elements of the isospin-isospin and spin-spin coupling operators for the di-
hadrons with a definite isospin $I$ and angular momentum $J$ are listed in
Table 3, which allows for the projection of the effective potentials onto a
specific $I(J^{P})$ state.

Table 2: The effective potentials of the Type-I systems, where $\mathbf{I}_{1}$ ($\mathbf{S}_{1}$) and $\mathbf{I}_{2}$ ($\mathbf{S}_{2}$) denote the isospin (spin) of the heavy baryons and heavy mesons, respectively. Systems | Effective potentials | $|\mathbf{I}_{1}\otimes\mathbf{I}_{2}\rangle,|\mathbf{S}_{1}\otimes\mathbf{S}_{2}\rangle$
---|---|---
$\Sigma_{c}\bar{D}$ | $\left(1+4\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)c_{s}$ | $|1\otimes\frac{1}{2}\rangle,|\frac{1}{2}\otimes 0\rangle$
$\Sigma_{c}\bar{D}^{\ast}$ | $\left(1+4\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)\left(c_{s}+\frac{4}{3}c_{a}\mathbf{S}_{1}\cdot\mathbf{S}_{2}\right)$ | $|1\otimes\frac{1}{2}\rangle,|\frac{1}{2}\otimes 1\rangle$
$\Sigma_{c}^{\ast}\bar{D}$ | $\left(1+4\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)c_{s}$ | $|1\otimes\frac{1}{2}\rangle,|\frac{3}{2}\otimes 0\rangle$
$\Sigma_{c}^{\ast}\bar{D}^{\ast}$ | $\left(1+4\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)\left(c_{s}+\frac{2}{3}c_{a}\mathbf{S}_{1}\cdot\mathbf{S}_{2}\right)$ | $|1\otimes\frac{1}{2}\rangle,|\frac{3}{2}\otimes 1\rangle$
$\Xi_{c}\bar{D}$ | $\left(\frac{1}{2}+4\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)c_{s}$ | $|\frac{1}{2}\otimes\frac{1}{2}\rangle,|\frac{1}{2}\otimes 0\rangle$
$\Xi_{c}\bar{D}^{\ast}$ | $\left(\frac{1}{2}+4\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)\left(c_{s}+2c_{a}\mathbf{S}_{1}\cdot\mathbf{S}_{2}\right)$ | $|\frac{1}{2}\otimes\frac{1}{2}\rangle,|\frac{1}{2}\otimes 1\rangle$
$\Xi_{c}^{\prime}\bar{D}$ | $\left(\frac{1}{2}+4\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)c_{s}$ | $|\frac{1}{2}\otimes\frac{1}{2}\rangle,|\frac{1}{2}\otimes 0\rangle$
$\Xi_{c}^{\prime}\bar{D}^{\ast}$ | $\left(\frac{1}{2}+4\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)\left(c_{s}-\frac{2}{3}c_{a}\mathbf{S}_{1}\cdot\mathbf{S}_{2}\right)$ | $|\frac{1}{2}\otimes\frac{1}{2}\rangle,|\frac{1}{2}\otimes 1\rangle$
$\Xi_{c}^{\ast}\bar{D}$ | $\left(\frac{1}{2}+4\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)c_{s}$ | $|\frac{1}{2}\otimes\frac{1}{2}\rangle,|\frac{3}{2}\otimes 0\rangle$
$\Xi_{c}^{\ast}\bar{D}^{\ast}$ | $\left(\frac{1}{2}+4\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)\left(c_{s}+\frac{2}{3}c_{a}\mathbf{S}_{1}\cdot\mathbf{S}_{2}\right)$ | $|\frac{1}{2}\otimes\frac{1}{2}\rangle,|\frac{3}{2}\otimes 1\rangle$
Table 3: Matrix elements of the isospin-isospin and spin-spin coupling operators in Table 2. $|\mathbf{I}_{1}\otimes\mathbf{I}_{2}\rangle$ |  $|\frac{1}{2}\otimes\frac{1}{2}\rangle$ |  $|\frac{1}{2}\otimes 1\rangle$ |
---|---|---|---
$I$ | $0$ | $1$ | $\frac{1}{2}$ | $\frac{3}{2}$ |
$\langle\mathbf{I}_{1}\cdot\mathbf{I}_{2}\rangle$ | $-\frac{3}{4}$ | $\frac{1}{4}$ | $-1$ | $\frac{1}{2}$ |
$|\mathbf{S}_{1}\otimes\mathbf{S}_{2}\rangle$ |  $|\frac{1}{2}\otimes 1\rangle$ | $|\frac{3}{2}\otimes 1\rangle$
$J$ | $\frac{1}{2}$ | $\frac{3}{2}$ | $\frac{1}{2}$ | $\frac{3}{2}$ | $\frac{5}{2}$
$\langle\mathbf{S}_{1}\cdot\mathbf{S}_{2}\rangle$ | $-1$ | $\frac{1}{2}$ | $-\frac{5}{2}$ | $-1$ | $\frac{3}{2}$

In this study, we exclude the systems that involve an isospin singlet
$\Lambda$ or $\Lambda_{c}$ baryon. Previous researches have shown that the
interactions in the systems such as $\Lambda_{c}\bar{D}^{(\ast)}$ are feeble
[83, 84, 40] due to the absence of the isospin couplings. In our calculations,
we have $V_{\Lambda_{c}\bar{D}^{(\ast)}}=c_{s}>0$, which means that no bound
states can be formed in these systems.

To search for the possible bound or virtual states, we solve the Lippmann-
Schwinger equations (LSEs) given by

$\displaystyle t=v+vGt,$ (7)

where $v$ denotes the $V_{\mathscr{H}_{1}\mathscr{H}_{2}}^{I,J}$ derived
above, and $G$ is the non-relativistic two-body propagator. The explicit form
of $G$ is given by

$\displaystyle G(E+i\epsilon)$ $\displaystyle=$
$\displaystyle\int_{0}^{\Lambda}\frac{k^{2}dk}{(2\pi)^{3}}\frac{2\mu}{p^{2}-k^{2}+i\epsilon}$
(8) $\displaystyle=$
$\displaystyle\frac{2\mu}{(2\pi)^{3}}\left[p\tanh^{-1}\left(\frac{p}{\Lambda}\right)-\Lambda-\frac{i\pi}{2}p\right],$

where a sharp cutoff $\Lambda$ is introduced to regularize the loop integral.
We will use $\Lambda=0.4$ GeV, as in Ref. [82]. $\mu$ represents the reduced
mass of the corresponding two-body system, and
$p=[2\mu(E-m_{\text{th}})]^{1/2}$, where $E$ is the center-of-mass energy and
$m_{\text{th}}$ is the threshold.

The integral equation (7) can be reduced to an algebraic equation for the
energy-independent effective potentials,

$\displaystyle t^{-1}$ $\displaystyle=$ $\displaystyle v^{-1}-G.$ (9)

The bound and virtual states exist as the zeros of the inverse T-matrix
$t^{-1}$ in the corresponding physical and unphysical Riemann sheets,
respectively. The physical and unphysical sheets are associated with the
following replacements for the $G$ function:

Physical $\displaystyle:$ $\displaystyle G(E+i\epsilon),$ Unphysical
$\displaystyle:$ $\displaystyle G(E+i\epsilon)+i\frac{\mu}{4\pi^{2}}p,$ (10)

where the $G(E+i\epsilon)$ is given by Eq. (8).

##  III Spectrum of the molecular pentaquarks

###  III.1 $\Sigma_{c}^{(\ast)}\bar{D}^{(\ast)}$,
$\Xi_{c}^{(\prime,\ast)}\bar{D}^{(\ast)}$ and $\Sigma_{c}^{(\ast)}D^{(\ast)}$,
$\Xi_{c}^{(\prime,\ast)}D^{(\ast)}$ systems

The mass spectra and strong decay patterns of the molecular pentaquarks in the
$\Sigma_{c}^{(\ast)}\bar{D}^{(\ast)}$,
$\Xi_{c}^{(\prime,\ast)}\bar{D}^{(\ast)}$, $\Sigma_{c}^{(\ast)}D^{(\ast)}$,
and $\Xi_{c}^{(\prime,\ast)}D^{(\ast)}$ systems are presented in Table 4.
These results are also summarized in the first two rows of Fig. 2, which
illustrate the mass spectra and relative positions of the relevant thresholds.
In the subsequent analysis, we will examine the outcomes of these systems in a
systematic manner.

The three $P_{\psi}^{N}$ states within the $\Sigma_{c}\bar{D}^{(\ast)}$
systems are considered as the inputs in our study [82]. We predict the
existence of four bound states in the isospin-$\frac{1}{2}$ channels of the
$\Sigma_{c}^{\ast}\bar{D}^{(\ast)}$ systems, which is consistent with the
findings of several previous investigations [16, 17, 19, 26, 27, 85]. Notably,
the bound state with a mass around $4380.3$ MeV in the
$\Sigma_{c}^{\ast}\bar{D}$ system shall correspond to the previously reported
$P_{\psi}^{N}(4380)$ [13]. However, no molecular states are found in the
isospin-$\frac{3}{2}$ channels. The discovery of the $P_{\psi}^{N}$ states has
sparked significant interest in the investigation of their mass spectra,
strong decays, production mechanisms, and experimental line-shapes, leading to
a plethora of studies in this field (see the recent review [12]).

In total, there are ten bound states identified in the isoscalar
$\Xi_{c}^{(\prime,\ast)}\bar{D}^{(\ast)}$ systems, which is consistent with
our previous predictions [40]. In Ref. [40], the mass of the $\Xi_{c}\bar{D}$
bound state was estimated to be approximately $4320$ MeV, about $15$ MeV lower
than the $\Xi_{c}\bar{D}$ threshold. However, the recent observation of the
$P_{\psi s}^{\Lambda}(4338)$ state indicates that it is very close to the
$\Xi_{c}\bar{D}$ threshold [42]. This discrepancy can be naturally resolved
within the framework of heavy quark symmetry. By considering this symmetry,
the $\Xi_{c}\bar{D}$ interaction is governed by a single pair $qq$ coupling,
similar to the $DD^{\ast}$ system, where the $T_{cc}$ state exists. From Table
2 and Ref. [82], the potentials for the isoscalar $\Xi_{c}\bar{D}$ and
$DD^{\ast}$ respectively read111The contribution from crossed diagrams in
$DD^{\ast}\to DD^{\ast}$ scattering will give rise to an effective spin
transition between $D$ and $D^{\ast}$, resulting in an additional $c_{a}$ term
in the $DD^{\ast}$ potentials.

$\displaystyle V_{\Xi_{c}\bar{D}}^{0,1/2}=-\frac{5}{2}c_{s},\qquad
V_{DD^{\ast}}^{0,1}=-\frac{5}{2}c_{s}+\frac{5}{2}c_{a}.$ (11)

Considering $|c_{s}|\gg|c_{a}|$ [e.g., see the Eq. (3)], it is evident that
$V_{\Xi_{c}\bar{D}}^{0,1/2}\approx V_{DD^{\ast}}^{0,1}$. Thus, a
straightforward connection can be established between the $T_{cc}(3875)$ and
$P_{\psi s}^{\Lambda}(4338)$ states, and their near-threshold characteristics
can be simultaneously explained within a unified framework. Additionally, it
is worth noting that the other bound states in the
$\Xi_{c}^{(\prime,\ast)}\bar{D}^{(\ast)}$ systems are characterized by weak
binding energies, typically a few MeVs. The $0(\frac{1}{2}^{-})$ or (and)
$0(\frac{3}{2}^{-})$ state(s) in the $\Xi_{c}\bar{D}^{\ast}$ system is
expected to correspond to the recently observed signal of the $P_{\psi
s}^{\Lambda}(4459)$ [41].

A series of studies have been conducted on the possible existence of double-
charm pentaquarks, following the discovery of the $P_{\psi}^{N}$ states [86,
87, 88, 89, 90, 81, 91, 92, 93, 94]. For instance, Chen et al. investigated
the spectrum of $\Sigma_{c}^{(\ast)}D^{(\ast)}$ molecules using the chiral
effective field theory and found a similar pattern to that of the
$P_{\psi}^{N}$ states, with seven bound states in the isospin-$\frac{1}{2}$
channels [81]. Similar results were also obtained in Ref. [91]. By employing
the one-boson exchange (OBE) model, several isospin-$\frac{1}{2}$ states in
the coupled systems of $\Sigma_{c}^{(\ast)}D^{(\ast)}$ were obtained in Refs.
[86, 92, 93]. Additionally, the chiral quark model was utilized to predict
several narrow resonances in $\Sigma_{c}^{(\ast)}D^{(\ast)}$ systems [90, 95].

Our calculations indicate that the spectrum of $\Sigma_{c}^{(\ast)}D^{(\ast)}$
differs significantly from that of $\Sigma_{c}^{(\ast)}\bar{D}^{(\ast)}$.
Notably, not all isospin-$\frac{1}{2}$ channels can form molecular states (see
Table 4 and Fig. 2). The most promising candidates are the
$\frac{1}{2}(\frac{3}{2}^{-})$ and $\frac{3}{2}(\frac{1}{2}^{-})$ states in
$\Sigma_{c}D^{\ast}$, as well as the $\frac{1}{2}(\frac{5}{2}^{-})$,
$\frac{3}{2}(\frac{1}{2}^{-})$, and $\frac{3}{2}(\frac{3}{2}^{-})$ states in
$\Sigma_{c}^{\ast}D^{\ast}$. The two isospin-$\frac{1}{2}$ states in
$\Sigma_{c}^{(\ast)}D$ systems may exist as virtual states or disappear when
$|\tilde{c}_{s}|$ is small.

We also examined the spectrum of double-charm pentaquarks composed of
$\Xi_{c}^{(\prime,\ast)}$ and $D^{(\ast)}$, which can be regarded as the
double-charm counterparts of the $P_{\psi s}^{\Lambda}$ states. Our analysis
reveals that the $0(\frac{3}{2}^{-})$ and $1(\frac{1}{2}^{-})$ states in
$\Xi_{c}D^{\ast}$, the $0(\frac{1}{2}^{-})$ state in
$\Xi_{c}^{\prime}D^{\ast}$, as well as the $0(\frac{5}{2}^{-})$,
$1(\frac{1}{2}^{-})$, and $1(\frac{3}{2}^{-})$ states in
$\Xi_{c}^{\ast}D^{\ast}$ are promising candidates of the molecular states.
However, the isoscalar states in $\Xi_{c}^{(\prime,\ast)}D$ systems may exist
as virtual states or not exist at all. Recently, Wang et al. also examined the
interactions of $\Xi_{c}^{(\prime,\ast)}D^{(\ast)}$ systems using the OBE
model and identified ten hadronic molecules in the isoscalar channels [96].
The spectrum obtained in Ref. [96] follows a similar pattern to that of the
$P_{\psi s}^{\Lambda}$ spectrum.

It is worth noting that the double-charm molecular pentaquarks in
$\Sigma_{c}^{(\ast)}D^{(\ast)}$ and $\Xi_{c}^{(\prime,\ast)}D^{(\ast)}$
systems can decay into the double-charm baryons
[$\Xi_{cc}^{(\ast)},\Omega_{cc}^{(\ast)}$] in combination with light hadrons
($\pi,\rho,\omega,\dots$). Therefore, these hadronic molecules have the
potential to serve as the sources of the production of the double-charm
baryons.

Table 4: The mass spectra and selected decay channels of the molecular pentaquarks in $\Sigma_{c}^{(\ast)}\bar{D}^{(\ast)}$, $\Xi_{c}^{(\prime,\ast)}\bar{D}^{(\ast)}$ and $\Sigma_{c}^{(\ast)}D^{(\ast)}$, $\Xi_{c}^{(\prime,\ast)}D^{(\ast)}$ systems. The isospin average masses of the corresponding systems are given in the brackets in the second column, e.g., in the form $\Sigma_{c}\bar{D}~{}[4320.7]$. The subscripts $B/V$ denote the bound/virtual state solutions. The superscript $\sharp$ means that this state might be nonexistent in the range of parameters, while the $\dagger$ represents that this state can also be the near-threshold bound (virtual) state if it is labeled as the virtual (bound) state in this table. The (pole) masses are given in units of MeV. $I(J^{P})$ | Systems [Thr.] | $E_{B}/E_{V}$ | Decay channels | Systems | $E_{B}/E_{V}$ | Decay channels
---|---|---|---|---|---|---
$\frac{1}{2}\left(\frac{1}{2}^{-}\right)$ | $\Sigma_{c}\bar{D}~{}[4320.7]$ | $\left[4311.9\pm 0.7_{-0.6}^{+6.8}\right]_{B}$ | $J/\psi N,\eta_{c}N,\Lambda_{c}\bar{D}^{(\ast)}$ | $\Sigma_{c}D$ | $\left[\sim 4308.4\right]_{V}^{\sharp}$ | $\Lambda_{c}D^{(\ast)},\Xi_{cc}\pi$
$\frac{3}{2}\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$\frac{1}{2}\left(\frac{1}{2}^{-}\right)$ | $\Sigma_{c}\bar{D}^{\ast}~{}[4462.0]$ | $\left[4440.3\pm 1.3_{-4.7}^{+4.1}\right]_{B}$ | $J/\psi N,\eta_{c}N,\Lambda_{c}\bar{D}^{(\ast)}$ | $\Sigma_{c}D^{\ast}$ | $-$ | $-$
$\frac{1}{2}\left(\frac{3}{2}^{-}\right)$ | $\left[4457.3\pm 0.6_{-1.7}^{+4.1}\right]_{B}$ | $J/\psi N,\Lambda_{c}\bar{D}^{\ast}$ | $\left[4457.0,4459.8\right]_{B}$ | $\Lambda_{c}D^{\ast},\Xi_{cc}^{\ast}\pi,\Xi_{cc}\rho(\omega)$
$\frac{3}{2}\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $\left[4436.8,4460.0\right]_{B}$ | $\Sigma_{c}D,\Xi_{cc}\pi,\Xi_{cc}\rho$
$\frac{3}{2}\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$\frac{1}{2}\left(\frac{3}{2}^{-}\right)$ | $\Sigma_{c}^{\ast}\bar{D}~{}[4385.6]$ | $\left[4380.3_{-1.8}^{+1.7}\right]_{B}$ | $J/\psi N,\Lambda_{c}\bar{D}^{\ast}$ | $\Sigma_{c}^{\ast}D$ | $\left[\sim 4373.7\right]_{V}^{\sharp}$ | $\Lambda_{c}D^{\ast},\Xi_{cc}^{\ast}\pi,\Xi_{cc}\rho(\omega)$
$\frac{3}{2}\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$\frac{1}{2}\left(\frac{1}{2}^{-}\right)$ | $\Sigma_{c}^{\ast}\bar{D}^{\ast}~{}[4526.7]$ | $\left[4518.7_{-5.6}^{+4.7}\right]_{B}$ | $J/\psi N,\eta_{c}N,\Lambda_{c}\bar{D}^{(\ast)}$ | $\Sigma_{c}^{\ast}D^{\ast}$ | $-$ | $-$
$\frac{1}{2}\left(\frac{3}{2}^{-}\right)$ | $\left[4520.0_{-3.3}^{+2.9}\right]_{B}$ | $J/\psi N,\Lambda_{c}\bar{D}^{\ast}$ | $-$ | $-$
$\frac{1}{2}\left(\frac{5}{2}^{-}\right)$ | $\left[4522.1_{-3.7}^{+3.0}\right]_{B}$ | $J/\psi N,\eta_{c}N,\Lambda_{c}\bar{D}^{(\ast)}$ | $\left[4513.0,4513.5\right]_{B}$ | $\Lambda_{c}D^{(\ast)},\Xi_{cc}^{(\ast)}\pi,\Xi_{cc}^{(\ast)}\rho(\omega)$
$\frac{3}{2}\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $\left[4486.6,4517.3\right]_{B}$ | $\Sigma_{c}D,\Xi_{cc}\pi,\Xi_{cc}\rho$
$\frac{3}{2}\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $\left[4483.8,4526.3\right]_{V}^{\dagger}$ | $\Sigma_{c}D^{\ast},\Xi_{cc}^{\ast}\pi,\Xi_{cc}\rho$
$\frac{3}{2}\left(\frac{5}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$0\left(\frac{1}{2}^{-}\right)$ | $\Xi_{c}\bar{D}~{}[4336.3]$ | $\left[4334.8_{-1.1}^{+0.8}\right]_{B}$ | $J/\psi\Lambda,\eta_{c}\Lambda,\bar{D}_{s}\Lambda_{c}$ | $\Xi_{c}D$ | $\left[\sim 4313.5\right]_{V}^{\sharp}$ | $\Xi_{cc}\bar{K},\Omega_{cc}\pi^{+}\pi^{-}\pi^{0}$
$1\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$0\left(\frac{1}{2}^{-}\right)$ | $\Xi_{c}\bar{D}^{\ast}~{}[4477.6]$ | $\left[4473.9_{-4.5}^{+3.1}\right]_{B}$ | $J/\psi\Lambda,\eta_{c}\Lambda,\bar{D}_{s}^{(\ast)}\Lambda_{c}$ | $\Xi_{c}D^{\ast}$ | $-$ | $-$
$0\left(\frac{3}{2}^{-}\right)$ | $\left[4476.3_{-2.3}^{+1.3}\right]_{B}$ | $J/\psi\Lambda,\bar{D}_{s}^{\ast}\Lambda_{c}$ | $\left[4470.3,4470.7\right]_{B}$ | $\Xi_{cc}^{\ast}\bar{K},\Omega_{cc}\pi^{+}\pi^{-}\pi^{0}$
$1\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $\left[4465.3,4477.2\right]_{B}$ | $\Xi_{cc}\bar{K},\Omega_{cc}\pi$
$1\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$0\left(\frac{1}{2}^{-}\right)$ | $\Xi_{c}^{\prime}\bar{D}~{}[4445.6]$ | $\left[4443.7_{-1.1}^{+0.9}\right]_{B}$ | $J/\psi\Lambda,\eta_{c}\Lambda,\bar{D}_{s}^{(\ast)}\Lambda_{c}$ | $\Xi_{c}^{\prime}D$ | $\left[\sim 4424.4\right]_{V}^{\sharp}$ | $\Xi_{cc}\bar{K},\Omega_{cc}\pi^{+}\pi^{-}\pi^{0}$
$1\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$0\left(\frac{1}{2}^{-}\right)$ | $\Xi_{c}^{\prime}\bar{D}^{\ast}~{}[4586.8]$ | $\left[4585.0_{-2.0}^{+1.4}\right]_{B}$ | $J/\psi\Lambda,\eta_{c}\Lambda,\bar{D}_{s}^{(\ast)}\Lambda_{c}$ | $\Xi_{c}^{\prime}D^{\ast}$ | $\left[4584.8,4586.3\right]_{B}$ | $\Xi_{c}D,\Xi_{cc}\bar{K},\Omega_{cc}\omega$
$0\left(\frac{3}{2}^{-}\right)$ | $\left[4584.2_{-1.7}^{+1.4}\right]_{B}$ | $J/\psi\Lambda,\bar{D}_{s}^{\ast}\Lambda_{c}$ | $-$ | $-$
$1\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$1\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $\left[\sim 4526.5\right]_{V}^{\sharp}$ | $\Xi_{cc}^{\ast}\bar{K},\Omega_{cc}^{\ast}\pi$
$0\left(\frac{3}{2}^{-}\right)$ | $\Xi_{c}^{\ast}\bar{D}~{}[4512.4]$ | $\left[4510.5_{-1.2}^{+0.9}\right]_{B}$ | $J/\psi\Lambda,\bar{D}_{s}^{\ast}\Lambda_{c}$ | $\Xi_{c}^{\ast}D$ | $\left[\sim 4492.2\right]_{V}^{\sharp}$ | $\Xi_{cc}^{\ast}\bar{K},\Omega_{cc}\pi^{+}\pi^{-}\pi^{0}$
$1\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$0\left(\frac{1}{2}^{-}\right)$ | $\Xi_{c}^{\ast}\bar{D}^{\ast}~{}[4653.7]$ | $\left[4649.8_{-4.0}^{+3.0}\right]_{B}$ | $J/\psi\Lambda,\eta_{c}\Lambda,\bar{D}_{s}^{(\ast)}\Lambda_{c}$ | $\Xi_{c}^{\ast}D^{\ast}$ | $-$ | $-$
$0\left(\frac{3}{2}^{-}\right)$ | $\left[4650.7_{-2.3}^{+1.8}\right]_{B}$ | $J/\psi\Lambda,\bar{D}_{s}^{\ast}\Lambda_{c}$ | $-$ | $-$
$0\left(\frac{5}{2}^{-}\right)$ | $\left[4652.0_{-2.4}^{+1.5}\right]_{B}$ | $J/\psi\Lambda,\eta_{c}\Lambda,\bar{D}_{s}^{(\ast)}\Lambda_{c}$ | $\left[4645.7,4646.1\right]_{B}$ | $\Xi_{c}D,\Xi_{cc}^{(\ast)}\bar{K},\Omega_{cc}^{(\ast)}\omega$
$1\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $\left[4646.6,4653.7\right]_{B}^{\dagger}$ | $\Xi_{c}D,\Xi_{cc}\bar{K},\Omega_{cc}\pi$
$1\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $\left[4459.3,4648.1\right]_{V}$ | $\Xi_{cc}^{\ast}\bar{K},\Omega_{cc}^{\ast}\pi,\Omega_{cc}\rho$
$1\left(\frac{5}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$

Figure 2: Spectrum of the molecular pentaquarks with a charmed baryon. The
$x$\- and $y$-axes represent the $I(J^{P})$ quantum numbers and the masses (in
units of MeV), respectively. The corresponding di-hadron systems are marked in
each subfigure. We use the red, blue and black rectangles to denote the bound
states, virtual states and the possibly nonexistent states, respectively,
while the red (blue) rectangles with blue (red) edges means that the states
can also be the near-threshold virtual (bound) states. The dark gray
rectangles represent the experimental data. We stress that only the vertical
widthes of red rectangles qualitatively reflect the corresponding mass ranges.
The corresponding thresholds are denoted with the horizontal dashed lines. The
experimental data for $P_{\psi}^{N}$ and $P_{\psi s}^{\Lambda}$ states are
taken from Refs. [14, 41, 42], where the results with two-peak hypothesis for
$P_{\psi s}^{\Lambda}(4459)$ [41] are quoted.

###  III.2 $\Sigma_{c}^{(\ast)}K^{\ast}$, $\Xi_{c}^{(\prime,\ast)}K^{\ast}$
and $\Sigma_{c}^{(\ast)}\bar{K}^{\ast}$,
$\Xi_{c}^{(\prime,\ast)}\bar{K}^{\ast}$ systems

The recently observed $T_{cs0}(2900)$ and $T_{c\bar{s}0}^{a}(2900)$ states
have emerged as compelling candidates for the $\bar{D}^{\ast}K^{\ast}$ and
$D^{\ast}K^{\ast}$ molecules, respectively, as evidenced in Refs. [82, 97] and
the references therein. This leads to an intriguing exploration of potential
molecular states in systems such as $\Sigma_{c}^{(\ast)}K^{\ast}$,
$\Xi_{c}^{(\prime,\ast)}K^{\ast}$, $\Sigma_{c}^{(\ast)}\bar{K}^{\ast}$, and
$\Xi_{c}^{(\prime,\ast)}\bar{K}^{\ast}$. Very recently, Chen et al. obtained
two molecular states with $\frac{1}{2}(\frac{1}{2}^{-})$ and
$\frac{1}{2}(\frac{3}{2}^{-})$ in the coupled
$\Lambda_{c}K^{\ast}$$-$$\Sigma_{c}K^{\ast}$ system through the OBE model
[98]. Additionally, Liu et al. predicted two isospin-$\frac{1}{2}$ resonances
in the $\Sigma_{c}K^{\ast}$ system employing a chiral quark model [99].

In the context of heavy quark symmetry, these hadronic molecules are expected
to be counterparts to the aforementioned states discussed in Sec. III.1, as
depicted in Fig. 1. The mass spectra and selected decay channels for these
systems are presented in Table 5, while an overview of the spectrum can be
found in the third row of Fig. 2.

Within our framework, the effective potentials for
$\Sigma_{c}^{(\ast)}K^{\ast}$ systems are analogous to those for
$\Sigma_{c}^{(\ast)}\bar{D}^{\ast}$. Consequently, the mass spectra of these
systems exhibit striking similarities. Notably, the molecular states in
$\Sigma_{c}^{(\ast)}K^{\ast}$ survive in the isospin-$\frac{1}{2}$ channels,
just as their counterparts, and are likely to exist as near-threshold virtual
(bound) states due to their lower reduced masses. The
$\frac{1}{2}(\frac{1}{2}^{-})$ and $\frac{1}{2}(\frac{3}{2}^{-})$ states in
$\Sigma_{c}K^{\ast}$ can be identified as charm-strange partners to
$P_{\psi}^{N}(4440)$ and $P_{\psi}^{N}(4457)$, with their $D_{s}N$ and
$D_{s}^{\ast}N$ decay channels corresponding to the $\eta_{c}N$ and $J/\psi N$
decay channels of the $P_{\psi}^{N}$ states, respectively. It is reasonable to
expect that a similar broad peak, akin to those observed for $T_{cs0}(2900)$
and $T_{c\bar{s}0}^{a}(2900)$, may manifest in the $D_{s}^{(\ast)}N$ invariant
mass distributions near the $\Sigma_{c}^{(\ast)}K^{\ast}$ thresholds.

In the $\Xi_{c}^{(\prime,\ast)}K^{\ast}$ systems, we predict the presence of
near-threshold virtual states in the isoscalar channels. Serving as the
counterparts to the $\Xi_{c}^{(\prime,\ast)}\bar{D}^{\ast}$ states, the
spectrum of the $\Xi_{c}^{(\prime,\ast)}K^{\ast}$ systems mirrors that of the
$P_{\psi s}^{\Lambda}$ states outlined in Table 4 and Fig. 2. Specifically,
the $0(\frac{1}{2}^{-})$ and $0(\frac{3}{2}^{-})$ states in
$\Xi_{c}^{(\prime)}K^{\ast}$, along with the $0(\frac{1}{2}^{-})$,
$0(\frac{3}{2}^{-})$, and $0(\frac{5}{2}^{-})$ states in
$\Xi_{c}^{\ast}K^{\ast}$, emerge as the promising candidates of the molecular
states.

In the $\Sigma_{c}^{(\ast)}\bar{K}^{\ast}$ systems, five molecular states are
predicted: the $\frac{1}{2}(\frac{3}{2}^{-})$ and
$\frac{3}{2}(\frac{1}{2}^{-})$ states in $\Sigma_{c}\bar{K}^{\ast}$, as well
as the $\frac{1}{2}(\frac{5}{2}^{-})$, $\frac{3}{2}(\frac{1}{2}^{-})$, and
$\frac{3}{2}(\frac{3}{2}^{-})$ states in $\Sigma_{c}^{\ast}\bar{K}^{\ast}$.

Lastly, in the $\Xi_{c}^{(\prime,\ast)}\bar{K}^{\ast}$ systems, the
$0(\frac{3}{2}^{-})$ and $1(\frac{1}{2}^{-})$ states in
$\Xi_{c}\bar{K}^{\ast}$, the $0(\frac{1}{2}^{-})$ state in
$\Xi_{c}^{\prime}\bar{K}^{\ast}$, and the $0(\frac{5}{2}^{-})$ and
$1(\frac{1}{2}^{-})$ states in $\Xi_{c}^{\ast}\bar{K}^{\ast}$ are all
promising candidates of the molecular states. See also the coupled-channel
calculations within the OBE model in Ref. [100].

Table 5: The mass spectra and selected decay channels of the molecular pentaquarks in $\Sigma_{c}^{(\ast)}K^{\ast}$, $\Xi_{c}^{(\prime,\ast)}K^{\ast}$ and $\Sigma_{c}^{(\ast)}\bar{K}^{\ast}$, $\Xi_{c}^{(\prime,\ast)}\bar{K}^{\ast}$ systems. The notations are the same as those in Table 4. $I(J^{P})$ | Systems [th.] | $E_{B}/E_{V}$ | Decay channels | Systems | $E_{B}/E_{V}$ | Decay channels
---|---|---|---|---|---|---
$\frac{1}{2}\left(\frac{1}{2}^{-}\right)$ | $\Sigma_{c}K^{\ast}$ $[3345.5]$ | $\left[3345.4_{-1.8}^{+0.1}\right]_{V}^{\dagger}$ | $D_{s}^{(\ast)}N,\Lambda_{c}K$ | $\Sigma_{c}\bar{K}^{\ast}$ | $-$ | $-$
$\frac{1}{2}\left(\frac{3}{2}^{-}\right)$ | $\left[3344.8_{-2.5}^{+0.6}\right]_{V}^{\dagger}$ | $D_{s}^{\ast}N,\Lambda_{c}K^{\ast}$ | $\left[3342.4,3345.0\right]_{V}$ | $D^{\ast}\Lambda(\Sigma),\Xi_{c}\rho(\omega),\Xi_{c}^{\ast}\pi,\Lambda_{c}(\Sigma_{c})\bar{K}\pi$
$\frac{3}{2}\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $\left[3337.0,3345.5\right]_{B}^{\dagger}$ | $D^{(\ast)}\Sigma,\Xi_{c}\pi,\Xi_{c}\rho,\Sigma_{c}\bar{K}\pi$
$\frac{3}{2}\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$\frac{1}{2}\left(\frac{1}{2}^{-}\right)$ | $\Sigma_{c}^{\ast}K^{\ast}$ $[3410.1]$ | $\left[3409.5_{-1.2}^{+0.6}\right]_{V}^{\dagger}$ | $D_{s}^{(\ast)}N,\Lambda_{c}K$ | $\Sigma_{c}^{\ast}\bar{K}^{\ast}$ | $-$ | $-$
$\frac{1}{2}\left(\frac{3}{2}^{-}\right)$ | $\left[3409.2_{-1.0}^{+0.3}\right]_{V}^{\dagger}$ | $D_{s}^{\ast}N,\Lambda_{c}K^{\ast}$ | $-$ | $-$
$\frac{1}{2}\left(\frac{5}{2}^{-}\right)$ | $\left[3409.0_{-3.6}^{+0.7}\right]_{V}^{\dagger}$ | $D_{s}^{(\ast)}N,\Lambda_{c}K^{(\ast)}$ | $\left[3408.6,3408.8\right]_{B}$ | $D^{(\ast)}\Lambda(\Sigma),\Xi_{c}\rho(\omega),\Xi_{c}^{(\ast)}\pi,\Lambda_{c}(\Sigma_{c})\bar{K}$
$\frac{3}{2}\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $\left[3390.0,3409.9\right]_{B}$ | $D^{(\ast)}\Sigma,\Xi_{c}\pi,\Xi_{c}\rho,\Sigma_{c}\bar{K}$
$\frac{3}{2}\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $\left[3161.7,3406.2\right]_{V}$ | $D^{\ast}\Sigma,\Xi_{c}^{\ast}\pi,\Xi_{c}\rho,\Sigma_{c}\bar{K}\pi$
$\frac{3}{2}\left(\frac{5}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$0\left(\frac{1}{2}^{-}\right)$ | $\Xi_{c}K^{\ast}$ $[3361.1]$ | $\left[3359.7_{-6.6}^{+1.3}\right]_{V}$ | $D_{s}^{(\ast)}\Lambda,\Lambda_{c}\phi(\eta),\Xi_{c}K\pi$ | $\Xi_{c}\bar{K}^{\ast}$ | $-$ | $-$
$0\left(\frac{3}{2}^{-}\right)$ | $\left[3356.0_{-7.6}^{+3.7}\right]_{V}^{\dagger}$ | $D_{s}^{\ast}\Lambda,\Lambda_{c}\phi,\Xi_{c}K\pi$ | $\left[3361.0,3361.1\right]_{V}$ | $D^{\ast}\Xi,\Xi_{c}\bar{K}\pi,\Omega_{c}\pi^{+}\pi^{-}\pi^{0}$
$1\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $\left[3351.9,3361.1\right]_{V}^{\dagger}$ | $D^{(\ast)}\Xi,\Xi_{c}\bar{K}\pi,\Omega_{c}\pi\pi$
$1\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$0\left(\frac{1}{2}^{-}\right)$ | $\Xi_{c}^{\prime}K^{\ast}$ $[3470.2]$ | $\left[3466.1_{-5.3}^{+2.8}\right]_{V}$ | $D_{s}^{(\ast)}\Lambda,\Lambda_{c}\phi(\eta),\Xi_{c}K$ | $\Xi_{c}^{\prime}\bar{K}^{\ast}$ | $\left[3460.9,3466.4\right]_{V}$ | $D^{(\ast)}\Xi,\Xi_{c}\bar{K},\Omega_{c}\pi^{+}\pi^{-}\pi^{0}$
$0\left(\frac{3}{2}^{-}\right)$ | $\left[3467.5_{-3.1}^{+1.8}\right]_{V}$ | $D_{s}^{\ast}\Lambda,\Lambda_{c}\phi,\Xi_{c}K\pi$ | $-$ | $-$
$1\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$1\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $\left[\sim 3134.1\right]_{V}^{\sharp}$ | $D^{\ast}\Xi,\Xi_{c}\bar{K}\pi,\Omega_{c}\pi\pi$
$0\left(\frac{1}{2}^{-}\right)$ | $\Xi_{c}^{\ast}K^{\ast}$ $[3537.1]$ | $\left[3535.8_{-5.5}^{+1.3}\right]_{V}$ | $D_{s}^{(\ast)}\Lambda,\Lambda_{c}\phi(\eta),\Xi_{c}K$ | $\Xi_{c}^{\ast}\bar{K}^{\ast}$ | $-$ | $-$
$0\left(\frac{3}{2}^{-}\right)$ | $\left[3534.9_{-3.6}^{+1.7}\right]_{V}$ | $D_{s}^{\ast}\Lambda,\Lambda_{c}\phi,\Xi_{c}K\pi$ | $-$ | $-$
$0\left(\frac{5}{2}^{-}\right)$ | $\left[3532.7_{-7.1}^{+3.3}\right]_{V}$ | $D_{s}^{(\ast)}\Lambda,\Lambda_{c}\phi(\eta),\Xi_{c}K$ | $\left[3536.5,3536.8\right]_{B}^{\dagger}$ | $D^{(\ast)}\Xi,\Xi_{c}\bar{K},\Omega_{c}\omega$
$1\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $\left[3514.2,3537.1\right]_{V}$ | $D^{(\ast)}\Xi,\Xi_{c}\bar{K},\Omega_{c}\pi$
$1\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $\left[\sim 3473.8\right]_{V}^{\sharp}$ | $D^{\ast}\Xi,\Xi_{c}\bar{K}\pi,\Omega_{c}^{\ast}\pi,\Omega_{c}\rho$
$1\left(\frac{5}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$

###  III.3 $\Xi_{cc}^{(\ast)}\bar{D}^{(\ast)}$, $\Xi_{cc}^{(\ast)}K^{\ast}$
and $\Xi_{cc}^{(\ast)}D^{(\ast)}$, $\Xi_{cc}^{(\ast)}\bar{K}^{\ast}$ systems

The molecular pentaquark systems involving a double-charm baryon have
attracted significant interest due to the presence of the heavy diquark-
antiquark symmetry (HDAS) [12, 101, 102, 103]. According to HDAS, the double-
heavy baryons $\Xi_{cc}^{(\ast)}$ can be correlated with the anti-charmed
mesons $\bar{D}^{(\ast)}$. Consequently, the existence of hadronic molecules
in the $DD^{\ast}$ system—the $T_{cc}(3875)$, and the $D\bar{D}^{\ast}$
system—the $X(3872)$ and $Z_{c}(3900)$, implies the potential presence of
pentaquarks in the $\Xi_{cc}^{(\ast)}\bar{D}^{(\ast)}$ and
$\Xi_{cc}^{(\ast)}D^{(\ast)}$ systems, respectively. Previous studies have
predicted several molecular states in these systems. For example, Guo et al.
predicted a bound state with $0(\frac{5}{2}^{-})$ in the
$\Xi_{cc}^{\ast}D^{\ast}$ system by utilizing the $X(3872)$ as an input [104].
Additionally, Chen et al. derived bound states with $0(\frac{1}{2}^{-})$ in
the $\Xi_{cc}\bar{D}^{\ast}$ system, $0(\frac{1}{2}^{-})$ in the $\Xi_{cc}D$
system, and $0(\frac{3}{2}^{-})$ in the $\Xi_{cc}D^{\ast}$ system [105].
Recently, Asanuma et al. proposed a connection between $T_{cc}$ and the
molecules in $\Xi_{cc}^{(\ast)}D^{(\ast)}$, and they predicted a bound state
with $0(\frac{1}{2}^{-})$ [106]. The investigation of triple-heavy pentaquarks
has also been pursued using various approaches such as the QCD sum rule [107],
the chromomagnetic interaction model [108, 109], the OBE model [110, 111], and
the quark model [112].

In this study, we extend our analysis to include pentaquark systems with
double-heavy baryons beyond the triple-charm
$\Xi_{cc}^{(\ast)}\bar{D}^{(\ast)}$ and $\Xi_{cc}^{(\ast)}D^{(\ast)}$ systems.
We also consider the $\Xi_{cc}^{(\ast)}K^{\ast}$ and
$\Xi_{cc}^{(\ast)}\bar{K}^{\ast}$ systems. The numerical results for these
systems are presented in Table 6, and the corresponding spectrum is
illustrated in Fig. 3. It is important to note that the mass of the
$\Xi_{cc}^{++}$ has been measured to be approximately $3621.5$ MeV [1], while
the $\Xi_{cc}^{\ast}$ is unobserved yet. Within the framework of HDAS, the
mass splitting
$m_{\Xi_{cc}^{\ast}}-m_{\Xi_{cc}}=\frac{3}{4}(m_{D^{\ast}}-m_{D})\simeq 106$
MeV [101, 102]. Quark model calculations, such as in Ref. [113], predicted the
$m_{\Xi_{cc}^{\ast}}-m_{\Xi_{cc}}$ to be around $70$ MeV. In our calculations,
we consider $m_{\Xi_{cc}^{\ast}}-m_{\Xi_{cc}}$ values in the range of
$[70,100]$ MeV to cover the estimates from HDAS and quark models. From Table
6, it can be observed that the notation $\pm 15$ MeV is used to represent the
uncertainties associated with the mass of $\Xi_{cc}^{\ast}$.

In our framework, the connections between the $DD^{\ast}$ and
$\Xi_{cc}^{(\ast)}\bar{D}^{(\ast)}$ systems are presented in a more direct
manner through their interacting types, i.e., the $V_{qq}$ interactions. Thus,
the existence of the molecule candidate $T_{cc}$ in the $DD^{\ast}$ system
naturally leads to the existence of molecules in the
$\Xi_{cc}^{(\ast)}\bar{D}^{(\ast)}$ systems. The mass spectrum of the
$\Xi_{cc}^{(\ast)}\bar{D}^{(\ast)}$ systems exhibits similarities to that of
the $\Xi_{c}^{\prime,\ast}\bar{D}^{(\ast)}$ systems. For instance, all seven
bound states exist in the isoscalar channels, while there are no molecules in
the isovector channels. In addition to the $0(\frac{1}{2}^{-})$ states
reported in Refs. [105, 106], we also obtain $0(\frac{3}{2}^{-})$ and
$0(\frac{5}{2}^{-})$ states. The seven molecules with $0(\frac{1}{2}^{-})$ in
$\Xi_{cc}\bar{D}$, $0(\frac{1}{2}^{-})$ and $0(\frac{3}{2}^{-})$ in
$\Xi_{cc}\bar{D}^{\ast}$, $0(\frac{3}{2}^{-})$ in $\Xi_{cc}^{\ast}\bar{D}$,
and $0(\frac{1}{2}^{-})$, $0(\frac{3}{2}^{-})$, and $0(\frac{5}{2}^{-})$ in
$\Xi_{cc}^{\ast}\bar{D}^{\ast}$ are good candidates of the molecular states.
These states can be reconstructed, for example, from the $J/\psi\Lambda_{c}$
invariant mass distributions, and they can be regarded as analogues of the
$J/\psi\Lambda$ channel for the $P_{\psi s}^{\Lambda}$ states. The mass
spectrum in the $\Xi_{cc}^{(\ast)}K^{\ast}$ systems follows a similar pattern,
although the five isoscalar molecules become virtual states due to their low
reduced masses.

Regarding the $\Xi_{cc}^{(\ast)}D^{(\ast)}$ systems, we also find a
$0(\frac{5}{2}^{-})$ state in $\Xi_{cc}^{\ast}D^{\ast}$ with a binding energy
of approximately $10$ MeV222One can easily verify the potential of
$\Xi_{cc}^{\ast}D^{\ast}$ with $0(\frac{5}{2}^{-})$ is the same as that of
$D\bar{D}^{\ast}$ with $0^{+}(1^{++})$ [where the $X(3872)$ exists], and they
both equal to $-\frac{5}{2}\tilde{c}_{s}-\frac{5}{2}\tilde{c}_{a}$. See Table
2, Fig. 1 and Ref. [82].. The prediction in Ref. [104] using the HDAS is
consistent with our calculations. Additionally, we identify two more bound
states: one with $0(\frac{1}{2}^{-})$ in $\Xi_{cc}D^{\ast}$ and another with
$1(\frac{1}{2}^{-})$ in $\Xi_{cc}^{\ast}D^{\ast}$. We also find one virtual
state with $1(\frac{3}{2}^{-})$ in $\Xi_{cc}^{\ast}D^{\ast}$. The molecule
spectrum of the $\Xi_{cc}^{(\ast)}\bar{K}^{\ast}$ system is similar to that of
the $\Xi_{cc}^{(\ast)}D^{\ast}$ system. Further details can be found in Table
6 and Fig. 3.

Table 6: The mass spectra and selected decay channels of the molecular pentaquarks in $\Xi_{cc}^{(\ast)}\bar{D}^{(\ast)}$, $\Xi_{cc}^{(\ast)}K^{\ast}$ and $\Xi_{cc}^{(\ast)}D^{(\ast)}$, $\Xi_{cc}^{(\ast)}\bar{K}^{\ast}$ systems. The notations are the same as those in Table 4. $I(J^{P})$ | Systems [th.] | $E_{B}/E_{V}$ | Decay channels | Systems | $E_{B}/E_{V}$ | Decay channels
---|---|---|---|---|---|---
$0\left(\frac{1}{2}^{-}\right)$ | $\Xi_{cc}\bar{D}$ $[5488.8]$ | $\left[5485.3_{-1.4}^{+1.3}\right]_{B}$ | $J/\psi\Lambda_{c},\eta_{c}\Lambda_{c}$ | $\Xi_{cc}D$ | $\left[\sim 5476.9\right]_{V}^{\sharp}$ | $\Omega_{ccc}\pi^{+}\pi^{-}\pi^{0}$
$1\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$0\left(\frac{1}{2}^{-}\right)$ | $\Xi_{cc}\bar{D}^{\ast}$ $[5630.1]$ | $\left[5626.5_{-2.5}^{+2.0}\right]_{B}$ | $J/\psi\Lambda_{c},\eta_{c}\Lambda_{c}$ | $\Xi_{cc}D^{\ast}$ | $\left[5626.3,5628.5\right]_{B}$ | $\Xi_{cc}D\pi,\Omega_{ccc}\pi^{+}\pi^{-}\pi^{0}$
$0\left(\frac{3}{2}^{-}\right)$ | $\left[5625.5_{-2.1}^{+1.8}\right]_{B}$ | $J/\psi\Lambda_{c}$ | $-$ | $-$
$1\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$1\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $\left[\sim 5591.4\right]_{V}^{\sharp}$ | $\Xi_{cc}D\pi,\Omega_{ccc}\pi\pi$
$0\left(\frac{3}{2}^{-}\right)$ | $\Xi_{cc}^{\ast}\bar{D}$ $[5573.8\pm 15]$ | $\left[5570.2_{-1.5}^{+1.3}\pm 15\right]_{B}$ | $J/\psi\Lambda_{c}$ | $\Xi_{cc}^{\ast}D$ | $\left[\sim 5562.3\pm 15\right]_{V}^{\sharp}$ | $\Omega_{ccc}\pi^{+}\pi^{-}\pi^{0}$
$1\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$0\left(\frac{1}{2}^{-}\right)$ | $\Xi_{cc}^{\ast}\bar{D}^{\ast}$$[5715.1\pm 15]$ | $\left[5709.0_{-4.6}^{+3.8}\pm 15\right]_{B}$ | $J/\psi\Lambda_{c},\eta_{c}\Lambda_{c},\Xi_{cc}\bar{D}$ | $\Xi_{cc}^{\ast}D^{\ast}$ | $-$ | $-$
$0\left(\frac{3}{2}^{-}\right)$ | $\left[5710.0_{-2.7}^{+2.3}\pm 15\right]_{B}$ | $J/\psi\Lambda_{c},\Xi_{cc}\bar{D}^{\ast},\Xi_{cc}^{\ast}\bar{D}$ | $-$ | $-$
$0\left(\frac{5}{2}^{-}\right)$ | $\left[5711.7_{-3.0}^{+2.3}\pm 15\right]_{B}$ | $J/\psi\Lambda_{c},\eta_{c}\Lambda_{c},\Xi_{cc}\bar{D}$ | $\left[5704.6\pm 0.2\pm 15\right]_{B}$ | $\Xi_{cc}D,\Omega_{ccc}\omega$
$1\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $\left[5710.0\pm 5.0\pm 15\right]_{B}$ | $\Xi_{cc}D,\Omega_{ccc}\pi,\Omega_{ccc}\rho$
$1\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $\left[5649.7\pm 63.3\pm 15\right]_{V}$ | $\Xi_{cc}D\pi,\Omega_{ccc}\rho$
$1\left(\frac{5}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$0\left(\frac{1}{2}^{-}\right)$ | $\Xi_{cc}K^{\ast}$ $[4513.6]$ | $\left[4511.6_{-3.6}^{+1.6}\right]_{V}$ | $D_{s}^{(\ast)}\Lambda_{c},\Xi_{cc}K,\Xi_{cc}K\pi$ | $\Xi_{cc}\bar{K}^{\ast}$ | $\left[4508.0,4511.8\right]_{V}$ | $D^{(\ast)}\Xi_{c},\Xi_{cc}\bar{K}\pi,\Omega_{cc}\pi^{+}\pi^{-}\pi^{0}$
$0\left(\frac{3}{2}^{-}\right)$ | $\left[4512.5_{-2.0}^{+0.9}\right]_{V}$ | $D_{s}^{\ast}\Lambda_{c},\Xi_{cc}K\pi$ | $-$ | $-$
$1\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$1\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $\left[\sim 4242.6\right]_{V}^{\sharp}$ | $D^{\ast}\Xi_{c},\Xi_{cc}\bar{K}\pi,\Omega_{cc}^{\ast}\pi,\Omega_{cc}\pi\pi$
$0\left(\frac{1}{2}^{-}\right)$ | $\Xi_{cc}^{\ast}K^{\ast}$ $[4598.6\pm 15]$ | $\left[4598.2_{-3.5}^{+0.3}\pm 15\right]_{V}$ | $D_{s}^{(\ast)}\Lambda_{c},\Xi_{cc}K$ | $\Xi_{cc}^{\ast}\bar{K}^{\ast}$ | $-$ | $-$
$0\left(\frac{3}{2}^{-}\right)$ | $\left[4597.7_{-2.4}^{+0.8}\pm 15\right]_{V}$ | $D_{s}^{\ast}\Lambda_{c},\Xi_{cc}K\pi$ | $-$ | $-$
$0\left(\frac{5}{2}^{-}\right)$ | $\left[4596.4_{-5.0}^{+2.0}\pm 15\right]_{V}$ | $D_{s}^{(\ast)}\Lambda_{c},\Xi_{cc}K$ | $\left[4598.1\pm 0.1\pm 15\right]_{B}$ | $D^{(\ast)}\Xi_{c},\Xi_{cc}\bar{K},\Omega_{cc}\omega$
$1\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $\left[4590.6\pm 8.0\pm 15\right]_{V}^{\dagger}$ | $D^{(\ast)}\Xi_{c},\Xi_{cc}\bar{K},\Omega_{cc}\pi,\Omega_{cc}\rho$
$1\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $\left[\sim 4550.7\pm 15\right]_{V}^{\sharp}$ | $D^{\ast}\Xi_{c},\Xi_{cc}\bar{K}\pi,\Omega_{cc}^{\ast}\pi,\Omega_{cc}\rho$
$1\left(\frac{5}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$

Figure 3: Spectrum of the molecular pentaquarks with a double-charm baryon.
The horizontal bands with light gray represent the uncertainties of
$\Xi_{cc}^{\ast}$ mass. Other notations are the same as those in Fig. 2.

###  III.4 $\Sigma^{(\ast)}\bar{D}^{(\ast)}$, $\Xi^{(\ast)}\bar{D}^{(\ast)}$
and $\Sigma^{(\ast)}D^{(\ast)}$, $\Xi^{(\ast)}D^{(\ast)}$ systems

In recent years, there have been significant advancements in the study of
pentaquark states with different quark configurations, particularly in the
field of hadronic molecules. One prominent example is the observation of
pentaquarks with the configurations $\bar{c}cqqq$ and $\bar{c}csqq$, known as
the $P_{\psi}^{N}$ and $P_{\psi s}^{\Lambda}$, respectively, which have been
confirmed through experiments conducted by LHCb. Prior to the discovery of
hidden-charm pentaquarks, Gignoux et al. [114] and Lipkin [115] had already
investigated anti-charm strange pentaquarks with the configuration
$\bar{c}sqqq$. It was denoted as the $P_{\bar{c}s}$ in Ref. [115] and it
corresponds to a bound state of the $\bar{D}_{s}N$ system, despite that the
residual strong interaction between $\bar{D}_{s}$ and $N$ should be too weak
to form a bound state. Later, Hofmann et al. predicted the existence of
isospin-$\frac{1}{2}$ bound states in the coupled systems of
$\bar{D}_{s}N$-$\bar{D}\Lambda$-$\bar{D}\Sigma$, as well as an isoscalar state
in the coupled $\bar{D}_{s}\Lambda$-$\bar{D}\Xi$ systems [116]. More recently,
Yalikun et al. obtained three bound states with isospin-$\frac{1}{2}$ in the
$\Sigma\bar{D}^{(\ast)}$ systems through the OBE model [117], and they
proposed to search for these states in the process
$B_{s}^{0}\to\bar{n}D_{s}^{-}p$. Additionally, Yan et al. obtained a similar
spectrum for the $\Sigma\bar{D}^{(\ast)}$ systems based on the light-meson
exchange saturation model [118].

Moreover, pentaquark systems with the quark configuration $cssq\bar{q}$, such
as the $\Xi D^{(\ast)}$, have been extensively studied in recent years to
understand some of the excited $\Omega_{c}$ states observed by the LHCb [119].
Various theoretical approaches, including the chiral unitary approach [120,
121, 122, 123], OBE related models [124, 118, 110], QCD sum rule [125], and
chiral quark model [126], have been employed to investigate the properties of
these systems.

In Table 7 and Fig. 4, we present the mass spectra and selected decay channels
for the $\Sigma^{(\ast)}\bar{D}^{(\ast)}$, $\Xi^{(\ast)}\bar{D}^{(\ast)}$,
$\Sigma^{(\ast)}D^{(\ast)}$, and $\Xi^{(\ast)}D^{(\ast)}$ systems. Our
calculation reveals the existence of several bound states and virtual states
in these systems. In the $\Sigma^{(\ast)}\bar{D}^{(\ast)}$ systems, we
identified seven molecules, all of which are present in the
isospin-$\frac{1}{2}$ channels. Among them, the three states in
$\Sigma\bar{D}^{(\ast)}$ can be considered near-threshold bound (or virtual)
states, while the molecules in $\Sigma^{\ast}\bar{D}^{(\ast)}$ correspond to
shallow bound states. Similarly, we derived seven hadronic molecules in the
isoscalar $\Xi^{(\ast)}\bar{D}^{(\ast)}$ systems, with most of them being
virtual states. In the $\Sigma^{(\ast)}D^{(\ast)}$ and
$\Xi^{(\ast)}D^{(\ast)}$ systems, we identified several states, such as the
$\frac{1}{2}(\frac{3}{2}^{-})$ and $\frac{3}{2}(\frac{1}{2}^{-})$ states in
$\Sigma D^{\ast}$, the $\frac{1}{2}(\frac{5}{2}^{-})$,
$\frac{3}{2}(\frac{1}{2}^{-})$, and $\frac{3}{2}(\frac{3}{2}^{-})$ states in
$\Sigma^{\ast}D^{\ast}$, and the $0(\frac{1}{2}^{-})$ state in $\Xi D^{\ast}$,
as well as the $0(\frac{5}{2}^{-})$, $1(\frac{1}{2}^{-})$, and
$1(\frac{3}{2}^{-})$ states in $\Xi^{\ast}D^{\ast}$, which are strong
candidates of the molecular states.

Our calculations indicate that there is no bound state in the $\Xi D$ system,
which disfavors the molecular interpretation of the excited $\Omega_{c}$ [119]
states discovered by LHCb. Nevertheless, it is worth mentioning that the LHCb
data also hint the presence of a broad structure around $3188$ MeV, close to
the $\Xi D$ threshold, which cannot be explained by a simple superposition of
other states [119]. In Ref. [124], this structure was considered to be a $\Xi
D$ bound state. In our work, there possibly exists a virtual state with
$0(\frac{1}{2}^{-})$ in $\Xi D$ (see Table 7), and it is known that the
virtual state pole will give rise to a peak exactly at the $\Xi D$ threshold
in the inelastic channels, e.g., the $\Xi_{c}\bar{K}$. This provides a
plausible explanation for this hint from the LHCb data [119].

Table 7: The mass spectra and selected decay channels of the molecular pentaquarks in $\Sigma^{(\ast)}\bar{D}^{(\ast)}$, $\Xi^{(\ast)}\bar{D}^{(\ast)}$ and $\Sigma^{(\ast)}D^{(\ast)}$, $\Xi^{(\ast)}D^{(\ast)}$ systems. The notations are the same as those in Table 4. $I(J^{P})$ | Systems [th.] | $E_{B}/E_{V}$ | Decay channels | Systems | $E_{B}/E_{V}$ | Decay channels
---|---|---|---|---|---|---
$\frac{1}{2}\left(\frac{1}{2}^{-}\right)$ | $\Sigma\bar{D}$ $[3060.4]$ | $\left[3060.0_{-0.4}^{+0.3}\right]_{B}^{\dagger}$ | $N\bar{D}_{s}^{(\ast)},\Lambda\bar{D}$ | $\Sigma D$ | $\left[\sim 3002.0\right]_{V}^{\sharp}$ | $\Lambda_{c}\bar{K},\Xi_{c}\pi$
$\frac{3}{2}\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$\frac{1}{2}\left(\frac{1}{2}^{-}\right)$ | $\Sigma\bar{D}^{\ast}$ $[3201.7]$ | $\left[3201.1_{-2.4}^{+0.6}\right]_{B}^{\dagger}$ | $N\bar{D}_{s}^{(\ast)},\Lambda\bar{D}^{(\ast)}$ | $\Sigma D^{\ast}$ | $-$ | $-$
$\frac{1}{2}\left(\frac{3}{2}^{-}\right)$ | $\left[3201.6_{-0.8}^{+0.1}\right]_{B}^{\dagger}$ | $N\bar{D}_{s}^{\ast},\Lambda\bar{D}^{\ast}$ | $\left[3201.2,3201.7\right]_{V}^{\dagger}$ | $\Lambda_{c}\bar{K}\pi,\Xi_{c}^{\ast}\pi,\Sigma D\pi$
$\frac{3}{2}\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $\left[3188.9,3201.7\right]_{B}^{\dagger}$ | $\Sigma_{c}\bar{K},\Xi_{c}\pi,\Sigma D\pi$
$\frac{3}{2}\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$\frac{1}{2}\left(\frac{3}{2}^{-}\right)$ | $\Sigma^{\ast}\bar{D}$ $[3251.8]$ | $\left[3251.3_{-0.9}^{+0.5}\right]_{B}$ | $N\bar{D}_{s}^{\ast},\Lambda\bar{D}^{\ast}$ | $\Sigma^{\ast}D$ | $\left[\sim 3209.8\right]_{V}^{\sharp}$ | $\Lambda_{c}\bar{K}\pi,\Xi_{c}^{\ast}\pi,\Lambda D^{\ast}$
$\frac{3}{2}\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$\frac{1}{2}\left(\frac{1}{2}^{-}\right)$ | $\Sigma^{\ast}\bar{D}^{\ast}$ $[3393.1]$ | $\left[3391.1_{-3.8}^{+2.0}\right]_{B}$ | $N\bar{D}_{s}^{(\ast)},\Lambda\bar{D}^{(\ast)}$ | $\Sigma^{\ast}D^{\ast}$ | $-$ | $-$
$\frac{1}{2}\left(\frac{3}{2}^{-}\right)$ | $\left[3391.9_{-2.0}^{+1.1}\right]_{B}$ | $N\bar{D}_{s}^{\ast},\Lambda\bar{D}^{\ast}$ | $-$ | $-$
$\frac{1}{2}\left(\frac{5}{2}^{-}\right)$ | $\left[3392.8_{-1.8}^{+0.4}\right]_{B}$ | $N\bar{D}_{s}^{(\ast)},\Lambda\bar{D}^{(\ast)}$ | $\left[3387.3,3387.7\right]_{B}$ | $\Lambda_{c}\bar{K},\Xi_{c}\pi,\Sigma D^{(\ast)},\Lambda D^{(\ast)}$
$\frac{3}{2}\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $\left[3364.3,3390.3\right]_{B}$ | $\Sigma_{c}\bar{K},\Xi_{c}\pi,\Sigma D$
$\frac{3}{2}\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $\left[3269.9,3393.0\right]_{V}$ | $\Sigma_{c}^{\ast}\bar{K},\Xi_{c}^{\ast}\pi,\Sigma D^{\ast}$
$\frac{3}{2}\left(\frac{5}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$0\left(\frac{1}{2}^{-}\right)$ | $\Xi\bar{D}$ $[3185.5]$ | $\left[3185.2_{-0.9}^{+0.3}\right]_{V}$ | $\Lambda\bar{D}_{s}$ | $\Xi D$ | $\left[\sim 3110.0\right]_{V}$ | $\Xi_{c}\bar{K},\Omega_{c}\pi^{+}\pi^{-}\pi^{0}$
$1\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$0\left(\frac{1}{2}^{-}\right)$ | $\Xi\bar{D}^{\ast}$ $[3326.8]$ | $[3326.5_{-1.9}^{+0.3}]_{V}$ | $\Lambda\bar{D}_{s}^{(\ast)},\Xi\bar{D}\pi$ | $\Xi D^{\ast}$ | $\left[3324.6,3326.5\right]_{V}$ | $\Xi_{c}\bar{K},\Xi D,\Omega_{c}\pi^{+}\pi^{-}\pi^{0}$
$0\left(\frac{3}{2}^{-}\right)$ | $\left[3326.7_{-0.8}^{+0.1}\right]_{V}$ | $\Lambda\bar{D}_{s}^{\ast},\Xi\bar{D}\pi$ | $-$ | $-$
$1\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$1\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $\left[\sim 3132.8\right]_{V}$ | $\Xi_{c}^{\ast}\bar{K},\Omega_{c}^{\ast}\pi$
$0\left(\frac{3}{2}^{-}\right)$ | $\Xi^{\ast}\bar{D}$ $[3400.7]$ | $\left[3400.0_{-0.3}^{+0.6}\right]_{V}$ | $\Lambda\bar{D}_{s}^{\ast},\Xi\bar{D}\pi$ | $\Xi^{\ast}D$ | $\left[\sim 3345.0\right]_{V}$ | $\Xi_{c}^{\ast}\bar{K},\Omega_{c}\pi^{+}\pi^{-}\pi^{0}$
$1\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$0\left(\frac{1}{2}^{-}\right)$ | $\Xi^{\ast}\bar{D}^{\ast}$ $[3542.0]$ | $\left[3541.5_{-2.2}^{+0.4}\right]_{B}^{\dagger}$ | $\Lambda\bar{D}_{s}^{(\ast)},\Xi\bar{D}$ | $\Xi^{\ast}D^{\ast}$ | $-$ | $-$
$0\left(\frac{3}{2}^{-}\right)$ | $\left[3541.8_{-0.9}^{+0.1}\right]_{B}^{\dagger}$ | $\Lambda\bar{D}_{s}^{\ast},\Xi\bar{D}\pi$ | $-$ | $-$
$0\left(\frac{5}{2}^{-}\right)$ | $\left[3541.9_{-1.4}^{+0.0}\right]_{V}^{\dagger}$ | $\Lambda\bar{D}_{s}^{(\ast)},\Xi\bar{D}$ | $\left[3539.3,3539.6\right]_{B}$ | $\Xi_{c}\bar{K},\Xi D,\Omega_{c}\omega$
$1\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $\left[3536.7,3541.9\right]_{V}^{\dagger}$ | $\Xi_{c}\bar{K},\Xi D,\Omega_{c}\pi$
$1\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $\left[3093.2,3519.9\right]_{V}$ | $\Xi_{c}^{\ast}\bar{K},\Omega_{c}^{\ast}\pi$
$1\left(\frac{5}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$

Figure 4: Spectrum of the molecular pentaquarks with a hyperon
$\Sigma^{(\ast)}$ or $\Xi^{(\ast)}$. The corresponding di-hadron systems are
marked in each subfigure. The notations are the same as those in Fig. 2.

###  III.5 $\Sigma^{(\ast)}K^{\ast}$, $\Xi^{(\ast)}K^{\ast}$ and
$\Sigma^{(\ast)}\bar{K}^{\ast}$, $\Xi^{(\ast)}\bar{K}^{\ast}$ systems

Whether the existence of hidden-charm pentaquarks implies the existence of
hidden-strange pentaquarks is an intriguing question. In the early 2000s, Gao
et al. suggested that the QCD van der Waals force, mediated by multi-gluon
exchanges, is strong enough to form a bound state in the $\phi N$ system
[127]. Huang et al. extended this idea within the chiral SU(3) quark model and
demonstrated that the $\sigma$-exchange interaction in $\phi N$ combined with
the channel coupling of $\Lambda K^{\ast}$ can produce a quasi-bound state
[128]. Liu et al. using constituent quark models [129], and He et al. via the
effective Lagrangian approach [130], obtained similar results for the $\phi N$
bound state. Lattice QCD calculations also supported the existence of this
bound state [131], although it is important to note that the residual strong
interaction of $\phi N$ should be very similar to the $J/\psi N$. Yet, the
interaction in the $J/\psi N$ system is too weak to form a bound state due to
the Okubo-Zweig-Iizuka (OZI) rule, as shown in a recent lattice QCD
calculation [132].

Furthermore, Oset et al. obtained several dynamically generated resonances
from vector octet and baryon octet interactions [133], such as the
isospin-$\frac{1}{2}$ $\Sigma K^{\ast}$ and the isospin-$0$ $\Xi K^{\ast}$.
Huang et al. predicted several isospin-$\frac{1}{2}$ resonances in the
$\Sigma^{(\ast)}K^{(\ast)}$ systems [134], taking into account their couplings
to open channels. On the other hand, Yang et al. argued that there are no
hidden-strange pentaquarks in the $\Sigma^{(\ast)}K^{(\ast)}$ systems [135].
Lebed proposed a search for hidden-strange pentaquarks in the $\phi p$
invariant mass distributions through the singly Cabibbo-suppressed decay
$\Lambda_{c}^{+}\to\phi p\pi^{0}$, as well as open-strange ($s\bar{s}uds$)
pentaquarks in the decay $\Xi_{c}^{+}\to\phi\Lambda\pi^{+}$ [136]. However,
Xie et al. pointed out the difficulty of identifying hidden-strange
pentaquarks in the $\Lambda_{c}^{+}\to\phi p\pi^{0}$ decay due to triangle
singularities and small phase space [137]. The Belle Collaboration also
investigated the contribution of intermediate hidden-strange pentaquarks in
the decay $\Lambda_{c}^{+}\to\phi p\pi^{0}$ but found no evidence with the
currently available statistics [138].

In the following analyses, we will explore the possibility of molecular
pentaquarks in the $\Sigma^{(\ast)}K^{\ast}$, $\Xi^{(\ast)}K^{\ast}$,
$\Sigma^{(\ast)}\bar{K}^{\ast}$, and $\Xi^{(\ast)}\bar{K}^{\ast}$ systems, as
the correspondences shown in Fig. 1. The results are presented in Table 8 and
summarized in the last row of Fig. 4. We identified five virtual states in the
isospin-$\frac{1}{2}$ $\Sigma^{(\ast)}K^{\ast}$ system and another five
virtual states in the isoscalar $\Xi^{(\ast)}K^{\ast}$ system. These virtual
states can be considered as partners of the $P_{\psi}^{N}$ and $P_{\psi
s}^{\Lambda}$ states, respectively. Additionally, we found several states,
such as the $\frac{1}{2}(\frac{3}{2}^{-})$ and $\frac{3}{2}(\frac{1}{2}^{-})$
states in $\Sigma\bar{K}^{\ast}$, the $\frac{1}{2}(\frac{5}{2}^{-})$,
$\frac{3}{2}(\frac{1}{2}^{-})$, and $\frac{3}{2}(\frac{3}{2}^{-})$ states in
$\Sigma^{\ast}\bar{K}^{\ast}$, the $0(\frac{1}{2}^{-})$ state in
$\Xi\bar{K}^{\ast}$, and the $0(\frac{5}{2}^{-})$, $1(\frac{1}{2}^{-})$ states
in $\Xi^{\ast}\bar{K}^{\ast}$, which are strong candidates of the molecular
states.

Table 8: The mass spectra and selected decay channels of the molecular pentaquarks in $\Sigma^{(\ast)}K^{(\ast)}$, $\Xi^{(\ast)}K^{(\ast)}$ and $\Sigma^{(\ast)}\bar{K}^{(\ast)}$, $\Xi^{(\ast)}\bar{K}^{(\ast)}$ systems. The notations are the same as those in Table 4. $I(J^{P})$ | Systems [th.] | $E_{B}/E_{V}$ | Decay channels | Systems | $E_{B}/E_{V}$ | Decay channels
---|---|---|---|---|---|---
$\frac{1}{2}\left(\frac{1}{2}^{-}\right)$ | $\Sigma K^{\ast}$ $[2085.2]$ | $\left[2080.6_{-8.8}^{+3.7}\right]_{V}$ | $\phi N,\Lambda K,\Sigma K$ | $\Sigma\bar{K}^{\ast}$ | $-$ | $-$
$\frac{1}{2}\left(\frac{3}{2}^{-}\right)$ | $\left[2076.2_{-8.8}^{+5.1}\right]_{V}$ | $\phi N,\Lambda K^{\ast},\Sigma^{\ast}K$ | $\left[2067.6,2076.8\right]_{V}$ | $\Xi^{\ast}\pi,\Xi\pi\pi,\Xi\pi^{+}\pi^{-}\pi^{0}$
$\frac{3}{2}\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $\left[2083.2,2085.1\right]_{B}^{\dagger}$ | $\Xi\pi,\Sigma\bar{K}$
$\frac{3}{2}\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$\frac{1}{2}\left(\frac{1}{2}^{-}\right)$ | $\Sigma^{\ast}K^{\ast}$ $[2276.6]$ | $\left[2274.5_{-7.4}^{+2.1}\right]_{V}$ | $\phi N,\Lambda K,\Sigma K$ | $\Sigma^{\ast}\bar{K}^{\ast}$ | $-$ | $-$
$\frac{1}{2}\left(\frac{3}{2}^{-}\right)$ | $\left[2273.2_{-5.0}^{+2.5}\right]_{V}$ | $\phi N,\Lambda K^{\ast},\Sigma^{\ast}K$ | $-$ | $-$
$\frac{1}{2}\left(\frac{5}{2}^{-}\right)$ | $\left[2270.2_{-9.4}^{+4.6}\right]_{V}$ | $\phi N,\Lambda K^{(\ast)},\Sigma^{(\ast)}K$ | $\left[2276.4,2276.6\right]_{V}$ | $\Xi^{(\ast)}\pi,\Xi\rho(\omega),\Sigma\bar{K}$
$\frac{3}{2}\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $\left[2264.4,2276.6\right]_{B}^{\dagger}$ | $\Xi\pi,\Xi\rho,\Sigma\bar{K}$
$\frac{3}{2}\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $\left[1821.5,2261.7\right]_{V}$ | $\Xi^{\ast}\pi,\Xi\rho$
$\frac{3}{2}\left(\frac{5}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$0\left(\frac{1}{2}^{-}\right)$ | $\Xi K^{\ast}$ $[2210.3]$ | $\left[2192.8_{-12.2}^{+7.7}\right]_{V}$ | $\phi\Lambda,\Xi K$ | $\Xi\bar{K}^{\ast}$ | $\left[2180.8,2193.6\right]_{V}$ | $\Omega\pi^{+}\pi^{-}\pi^{0},\Xi\bar{K}$
$0\left(\frac{3}{2}^{-}\right)$ | $\left[2196.4_{-7.8}^{+5.4}\right]_{V}$ | $\phi\Lambda,\Xi K\pi$ | $-$ | $-$
$1\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$
$1\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $\left[\sim 1533.6\right]_{V}^{\sharp}$ | $\Omega\pi$
$0\left(\frac{1}{2}^{-}\right)$ | $\Xi^{\ast}K^{\ast}$ $[2425.4]$ | $\left[2418.7_{-11.6}^{+5.1}\right]_{V}$ | $\phi\Lambda,\Xi K$ | $\Xi^{\ast}\bar{K}^{\ast}$ | $-$ | $-$
$0\left(\frac{3}{2}^{-}\right)$ | $\left[2416.5_{-7.6}^{+4.6}\right]_{V}$ | $\phi\Lambda,\Xi K\pi$ | $-$ | $-$
$0\left(\frac{5}{2}^{-}\right)$ | $\left[2411.7_{-13.4}^{+7.5}\right]_{V}$ | $\phi\Lambda,\Xi K,\Xi K\pi$ | $\left[2423.5,2423.8\right]_{V}$ | $\Omega\pi^{+}\pi^{-}\pi^{0},\Xi\bar{K}$
$1\left(\frac{1}{2}^{-}\right)$ | $-$ | $-$ | $\left[2378.2,2423.2\right]_{V}$ | $\Omega\pi\pi,\Xi\bar{K}$
$1\left(\frac{3}{2}^{-}\right)$ | $-$ | $-$ | $\left[\sim 2310.6\right]_{V}^{\sharp}$ | $\Omega\pi$
$1\left(\frac{5}{2}^{-}\right)$ | $-$ | $-$ | $-$ | $-$

##  IV Summary

In summary, we extend the concept of heavy quark to cover the strange quark
for the near-threshold residual strong interactions. Hence we use the heavy to
denote the hadrons with strange and/or charm quarks. Our study focuses on a
systematic examination of the mass spectrum of molecular pentaquarks, composed
of a heavy baryon and a heavy meson. We consider a range of heavy baryons,
including charmed baryons $\Sigma_{c}^{(\ast)}$, $\Xi_{c}^{(\prime,\ast)}$,
double-charm baryons $\Xi_{cc}^{(\ast)}$, and hyperons $\Sigma^{(\ast)}$ and
$\Xi^{(\ast)}$. Meanwhile, the considered heavy mesons include the charmed
mesons $D^{(\ast)}$ and $\bar{D}^{(\ast)}$, as well as the strange vector
mesons, namely $K^{\ast}$ and $\bar{K}^{\ast}$. By combining these components,
we generate various pentaquark systems, which can be classified into two
groups based on their interaction types, as outlined in Table 1.

Experimental observations strongly suggest that the formation of the hadronic
molecules crucially depends on the correlations between light quark pairs
within separate hadrons. In our previous work [82], we have quantitatively
analyzed these interactions and demonstrated that the strange quarks inside
the hadrons can be approximated as heavy quarks due to their limited
involvement in near-threshold interactions. Consequently, we establish the
underlying connections between different systems under the framework of heavy
quark symmetry, as depicted in Fig. 1.

Our analysis reveals that the very near-threshold behavior of $P_{\psi
s}^{\Lambda}(4338)$ shares a common cause with the similarly near-threshold
$T_{cc}(3875)$. Furthermore, we find that the masses of the molecular
candidates in the $\Xi_{c}\bar{D}^{\ast}$ system should be closer to the
threshold than the currently available experimental data, specifically the
$P_{\psi s}^{\Lambda}(4459)$. Through our investigations, we have identified
numerous hadronic molecules within the twenty systems listed in Table 1. These
molecular states can be considered as counterparts to the $P_{\psi}^{N}$,
$T_{cc}$, $X(3872)$, and $Z_{c}(3900)$ states, expanding the family of
hadronic molecules.

The study of pentaquark systems with different quark configurations,
particularly in the realm of hadronic molecules, has provided valuable
insights into the nature of exotic hadrons and has helped to deepen our
understanding of the experimental observations in this field. The experimental
search for the predicted states in various decay channels continues to be of
great importance in advancing our understanding of the underlying physics.

## Acknowledgement

This work is supported by the National Natural Science Foundation of China
under Grants Nos. 12105072, 12305090, 11975033 and 12070131001. B. Wang was
also supported by the Youth Funds of Hebei Province (No. A2021201027) and the
Start-up Funds for Young Talents of Hebei University (No. 521100221021). This
project is also funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation, Project ID 196253076-TRR 110).

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