# Emergence of molecular-type characteristic spectrum of hidden-charm
pentaquark with strangeness embodied in the $P_{\psi s}^{\Lambda}(4338)$ and
$P_{cs}(4459)$

Fu-Lai Wang1,2,3<EMAIL_ADDRESS>Xiang Liu1,2,3<EMAIL_ADDRESS>1School of Physical Science and Technology, Lanzhou University, Lanzhou
730000, China
2Research Center for Hadron and CSR Physics, Lanzhou University and Institute
of Modern Physics of CAS, Lanzhou 730000, China
3Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, and Frontiers Science Center for Rare Isotopes, Lanzhou
University, Lanzhou 730000, China

###### Abstract

Inspired by the newly observed $P_{\psi s}^{\Lambda}(4338)$ and the reported
$P_{cs}(4459)$, we indicate the existence of the molecular-type characteristic
mass spectrum for hidden-charm pentaquark with strangeness. It shows that the
$P_{cs}(4459)$ may contain two substructures corresponding to the
$\Xi_{c}\bar{D}^{*}$ molecular states with $J^{P}=1/2^{-}$ and $3/2^{-}$,
while there exists the corresponding $\Xi_{c}\bar{D}$ molecular state with
$J^{P}=1/2^{-}$. As the prediction, we present another characteristic mass
spectrum of the $\Xi_{c}^{\prime}\bar{D}^{(*)}$ molecular states. Experimental
confirmation of these characteristic mass spectra is a crucial step of
constructing hadronic molecular family.

##  I Introduction

Since the birth of the quark model GellMann:1964nj ; Zweig:1981pd , the
concept of exotic hadronic matter has been proposed, which has attracted
extensive attention from both experimentalists and theorists. Especially, with
the accumulation of experimental data, the observations of charmoniumlike
$XYZ$ states and $P_{c}$ states in the past around twenty years make this
issue becomes hot spot of hadron physics up till now Liu:2013waa ;
Hosaka:2016pey ; Chen:2016qju ; Richard:2016eis ; Lebed:2016hpi ;
Brambilla:2019esw ; Liu:2019zoy ; Chen:2022asf ; Olsen:2017bmm ; Guo:2017jvc ;
Meng:2022ozq . These studies are helpful to deepen our understanding of non-
perturbative quantum chromodynamics (QCD). Among different exotic hadronic
matters, the hadronic molecular state was popularly applied to explain these
novel phenomena relevant to new hadronic states since most of observed
hadronic states are close the thresholds of hadron pair. A typical example is
the LHCb’s discoveries of the $P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$
in the $J/\psi p$ invariant mass spectrum of the $\Lambda_{b}\to J/\psi pK$
Aaij:2019vzc  weak decay, which show a characteristic mass spectrum consistent
with that of hidden-charm molecular pentaquark, which was predicted in Refs.
Li:2014gra ; Karliner:2015ina ; Wu:2010jy ; Wang:2011rga ; Yang:2011wz ;
Wu:2012md ; Chen:2015loa . Finally, it provides strong evidence to support the
existence of the hidden-charm meson-baryon molecular states. Of course, there
also exist other possible interpretations to the $P_{c}(4312)$, $P_{c}(4440)$,
and $P_{c}(4457)$ Liu:2013waa ; Hosaka:2016pey ; Chen:2016qju ;
Richard:2016eis ; Lebed:2016hpi ; Brambilla:2019esw ; Liu:2019zoy ;
Chen:2022asf ; Olsen:2017bmm ; Guo:2017jvc ; Meng:2022ozq ; Burns:2022uiv .

Very recently, the LHCb Collaboration announced the observed of a
$J/\psi\Lambda$ resonance in the $B^{-}\to J/\psi\Lambda\bar{p}$ process
Collaboration:2022boa , which has resonance parameters

$\displaystyle M=4338.2\pm 0.7\pm 0.4\,\mathrm{MeV},~{}~{}\Gamma=7.0\pm 1.2\pm
1.3\,\mathrm{MeV},$

and significance larger than $10\sigma$. Thus, this $J/\psi\Lambda$ resonance
is refereed to be the $P_{\psi s}^{\Lambda}(4338)$, which is the candidate of
hidden-charm pentaquark with strangeness, as predicted by former theoretical
studies Chen:2016ryt ; Wu:2010vk ; Hofmann:2005sw ; Anisovich:2015zqa ;
Wang:2015wsa ; Feijoo:2015kts ; Lu:2016roh ; Xiao:2019gjd ; Shen:2020gpw ;
Chen:2015sxa ; Zhang:2020cdi ; Wang:2019nvm ; Chen:2020uif ; Peng:2020hql ;
Chen:2020opr ; Liu:2020hcv ; Dong:2021juy ; Chen:2022onm ; Chen:2020kco ;
Chen:2021cfl ; Wang:2022neq ; Chen:2021spf ; Du:2021bgb ; Hu:2021nvs ;
Xiao:2021rgp ; Zhu:2021lhd ; Weng:2019ynv . Before observing the $P_{\psi
s}^{\Lambda}(4338)$, LHCb once reported an evidence of the enhancement
structure ($P_{cs}(4459)$) in the $J/\psi\Lambda$ invariant mass spectrum of
the $\Xi_{b}^{-}\to J/\psi\Lambda K$ LHCb:2020jpq , which is the candidate of
hidden-charm pentaquark with strangeness Chen:2016ryt ; Wu:2010vk ;
Hofmann:2005sw ; Anisovich:2015zqa ; Wang:2015wsa ; Feijoo:2015kts ;
Lu:2016roh ; Xiao:2019gjd ; Shen:2020gpw ; Chen:2015sxa ; Zhang:2020cdi ;
Wang:2019nvm ; Chen:2020uif ; Peng:2020hql ; Chen:2020opr ; Liu:2020hcv ;
Dong:2021juy ; Chen:2022onm ; Chen:2020kco ; Chen:2021cfl ; Wang:2022neq ;
Chen:2021spf ; Du:2021bgb ; Hu:2021nvs ; Xiao:2021rgp ; Zhu:2021lhd ;
Weng:2019ynv .

Figure 1: Comparison of the mass spectrum of three $P_{c}$ states Aaij:2019vzc
and that of the $P_{\psi s}^{\Lambda}(4338)$ Collaboration:2022boa  and
$P_{cs}(4459)$ LHCb:2020jpq . Here, we also list the experimental data of the
$P_{cs}(4459)$ from LHCb LHCb:2020jpq .

In this work, we indicate that there exists a new characteristic mass spectrum
applied to identify the molecular-type hidden-charm pentaquark with
strangeness, which is inspired by the observed $P_{\psi s}^{\Lambda}(4338)$
and the reported $P_{cs}(4459)$. As shown in Fig. 1, the gap between the
$P_{c}(4312)$ mass and the average mass of the $P_{c}(4440)$ and $P_{c}(4457)$
is 137 MeV, which is similar to the mass gap of the $P_{cs}(4459)$ and the
$P_{\psi s}^{\Lambda}(4338)$. This is analogous to the similarity of the mass
gaps for the $\omega$ and $\phi$ meson families Wang:2012wa . We also notice
an interesting phenomenon, and there exists corresponding relations of these
$P_{c}$ and $P_{cs}$ states, i.e., the $P_{\psi s}^{\Lambda}(4338)$ should
correspond to the $P_{c}(4312)$, while the $P_{cs}(4459)$ structure
corresponds to the $P_{c}(4440)$ and $P_{c}(4457)$. This fact makes us
conjecture that the $P_{cs}(4459)$ enhancement structure should contain two
substructures. If checking the LHCb data of the $J/\psi\Lambda$ invariant mass
spectrum of the $\Xi_{b}^{-}\to J/\psi\Lambda K$ LHCb:2020jpq , we find that
there exists possible double peak structure slightly below the
$\Xi_{c}\bar{D}^{*}$ threshold as indicated in Fig. 1, which just correspond
to the $\Xi_{c}\bar{D}^{*}$ molecular states with $J^{P}=1/2^{-}$ and
$3/2^{-}$. Such scenario can be tested in future experiment like the LHCb. As
the partner of $P_{c}(4312)$, the $\Xi_{c}\bar{D}$ molecular state with
$J^{P}=1/2^{-}$ was studied in this work. We should indicate the fact that the
central value of the mass of the newly reported $P_{\psi s}^{\Lambda}(4338)$
is above the $\Xi_{c}\bar{D}$ threshold, which is the difficulty to directly
assign the newly observed $P_{\psi s}^{\Lambda}(4338)$ as the $\Xi_{c}\bar{D}$
molecular state with $J^{P}=1/2^{-}$. How to solving this puzzling phenomenon
is also an interesting topic. In the final section, we will address this
point. Inspired by the established molecular-type characteristic mass spectrum
of three $P_{c}$ states in 2019 Aaij:2019vzc  and the molecular-type
characteristic mass spectrum of the $P_{\psi s}^{\Lambda}(4338)$ and the
$P_{cs}(4459)$ enhancement structure found in this work, we further give a new
characteristic mass spectrum of the $\Xi_{c}^{\prime}\bar{D}^{(*)}$ molecular
states, which will be accessible at experiment.

This paper is organized as the follows. After introduction, the calculation of
the interactions of these discussed $\Xi_{c}^{(\prime)}\bar{D}^{(*)}$ systems
will be given by adopting the OBE model in Sec. II. With this preparation, we
discuss the characteristic mass spectra of the
$\Xi_{c}^{(\prime)}\bar{D}^{(*)}$ molecular pentaquarks in Sec. III. Finally,
we will give the discussion and conclusion in Sec. IV.

##  II The OBE effective potentials of the $\Xi_{c}^{(\prime)}\bar{D}^{(*)}$
systems

For getting the information of the interactions of the
$\Xi_{c}^{(\prime)}\bar{D}^{(*)}$ systems, the one-boson-exchange (OBE) model
is adopted. In the OBE model, the interaction between two hadrons is a direct
consequence of the exchange of the allowed light mesons Chen:2016qju ;
Liu:2019zoy , which is one of the effective ways to investigate the
interactions between hadrons. Such a formalism is a straightforward extension
of the traditional meson exchange model involved in the nuclear force
Yukawa:1935xg . Up to now, the OBE model has successfully applied the
exploration of a series of hadronic molecular states including the $P_{c}$
states and the $T_{cc}$ state Chen:2016qju ; Liu:2019zoy . We should indicate
that it is not the only way to get the interactions between hadrons, since
there also exist other approaches like the chiral effective field theory
Meng:2022ozq . For the chiral effective field theory, the contact term is
usually introduced, which is different from the treatment of the OBE model on
this point. In fact, this difference reflects the different treatment to the
short distant interactions between hadrons under two approaches. Despite all
this, the OBE model and the chiral effective field theory usually can reach
the same conclusion for the same hadronic molecular states.

In general, there exists three typical steps for deducing the effective
potentials of these discussed $\Xi_{c}^{(\prime)}\bar{D}_{s}^{(*)}$ systems
within the OBE model. As the first step, we should write out the scattering
amplitudes $\mathcal{M}(h_{1}h_{2}\to h_{3}h_{4})$ of the scattering processes
$h_{1}h_{2}\to h_{3}h_{4}$ by considering the effective Lagrangian approach.

According to the heavy quark spin symmetry, the hadrons containing single
heavy quark with total spin $J_{\pm}=j_{l}\pm 1/2$ (except $j_{l}=0$) come
into doublets, which can be written as the super-fields when constructing the
compact effective Lagrangians. For the pseudoscalar anti-charmed meson
$\bar{D}$ with $I(J^{P})=1/2(0^{-})$ and vector anti-charmed meson
$\bar{D}^{*}$ with $I(J^{P})=1/2(1^{-})$, they are degenerated in the heavy
quark spin symmetry, which can be written as the super-field
$H^{(\overline{Q})}_{a}=(\bar{D}^{*(\overline{Q})\mu}_{a}\gamma_{\mu}-\bar{D}^{(\overline{Q})}_{a}\gamma_{5})\frac{(1-{v}\\!\\!\\!/)}{2}$
Ding:2008gr . In the heavy quark limit, the charm baryons can be classified in
terms of the flavor symmetry of the diquark, the spin of the charm baryon in
the $6_{F}$ flavor representation is either 1/2 or 3/2, while the spin of the
charm baryon in the $\bar{3}_{F}$ flavor representation is only $1/2$. Here,
we need to mention that $\Xi_{c}$ with $J^{P}=1/2^{+}$ denotes the $S$-wave
charm baryon in $\bar{3}_{F}$ flavor representation, while $\Xi_{c}^{\prime}$
with $J^{P}=1/2^{+}$ is the $S$-wave charmed baryon in $6_{F}$ flavor
representation. The involved effective Lagrangians for depicting the heavy
hadrons $\mathcal{B}_{c}/\bar{D}^{(*)}$ coupling with the light scalar,
pesudoscalar, and vector mesons read as Wise:1992hn ; Casalbuoni:1992gi ;
Casalbuoni:1996pg ; Yan:1992gz ; Bando:1987br ; Harada:2003jx ; Ding:2008gr ;
Chen:2017xat

$\displaystyle\mathcal{L}_{\mathcal{B}_{\bar{3}}\mathcal{B}_{\bar{3}}\sigma}$
$\displaystyle=$ $\displaystyle
l_{B}\langle\bar{\mathcal{B}}_{\bar{3}}\sigma\mathcal{B}_{\bar{3}}\rangle,$
(1)
$\displaystyle\mathcal{L}_{\mathcal{B}_{\bar{3}}\mathcal{B}_{\bar{3}}\mathbb{V}}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\beta_{B}g_{V}\langle\bar{\mathcal{B}}_{\bar{3}}v\cdot\mathbb{V}\mathcal{B}_{\bar{3}}\rangle,$
(2) $\displaystyle\mathcal{L}_{\mathcal{B}_{6}\mathcal{B}_{6}\sigma}$
$\displaystyle=$ $\displaystyle-
l_{S}\langle\bar{\mathcal{B}}_{6}\sigma\mathcal{B}_{6}\rangle,$ (3)
$\displaystyle\mathcal{L}_{\mathcal{B}_{6}\mathcal{B}_{6}\mathbb{P}}$
$\displaystyle=$ $\displaystyle
i\frac{g_{1}}{2f_{\pi}}\varepsilon^{\mu\nu\lambda\kappa}v_{\kappa}\langle\bar{\mathcal{B}}_{6}\gamma_{\mu}\gamma_{\lambda}\partial_{\nu}\mathbb{P}\mathcal{B}_{6}\rangle,$
(4) $\displaystyle\mathcal{L}_{\mathcal{B}_{6}\mathcal{B}_{6}\mathbb{V}}$
$\displaystyle=$
$\displaystyle-\frac{\beta_{S}g_{V}}{\sqrt{2}}\langle\bar{\mathcal{B}}_{6}v\cdot\mathbb{V}\mathcal{B}_{6}\rangle$
(5)
$\displaystyle-i\frac{\lambda_{S}g_{V}}{3\sqrt{2}}\langle\bar{\mathcal{B}}_{6}\gamma_{\mu}\gamma_{\nu}\left(\partial^{\mu}\mathbb{V}^{\nu}-\partial^{\nu}\mathbb{V}^{\mu}\right)\mathcal{B}_{6}\rangle,$
$\displaystyle\mathcal{L}_{\mathcal{B}_{\bar{3}}\mathcal{B}_{6}\mathbb{P}}$
$\displaystyle=$
$\displaystyle-\sqrt{\frac{1}{3}}\frac{g_{4}}{f_{\pi}}\langle\bar{\mathcal{B}}_{6}\gamma^{5}\left(\gamma^{\mu}+v^{\mu}\right)\partial_{\mu}\mathbb{P}\mathcal{B}_{\bar{3}}\rangle+h.c.,$
(6)
$\displaystyle\mathcal{L}_{\mathcal{B}_{\bar{3}}\mathcal{B}_{6}\mathbb{V}}$
$\displaystyle=$
$\displaystyle-\frac{\lambda_{I}g_{V}}{\sqrt{6}}\varepsilon^{\mu\nu\lambda\kappa}v_{\mu}\langle\bar{\mathcal{B}}_{6}\gamma^{5}\gamma_{\nu}\left(\partial_{\lambda}\mathbb{V}_{\kappa}-\partial_{\kappa}\mathbb{V}_{\lambda}\right)\mathcal{B}_{\bar{3}}\rangle$
(7) $\displaystyle+h.c.,$
$\displaystyle\mathcal{L}_{{\bar{D}}{\bar{D}}\sigma}$ $\displaystyle=$
$\displaystyle-2g_{\sigma}{\bar{D}}_{a}{\bar{D}}_{a}^{{\dagger}}\sigma,$ (8)
$\displaystyle\mathcal{L}_{{\bar{D}}^{*}{\bar{D}}^{*}\sigma}$ $\displaystyle=$
$\displaystyle
2g_{\sigma}{\bar{D}}_{a\mu}^{*}{\bar{D}}_{a}^{*\mu{\dagger}}\sigma,$ (9)
$\displaystyle\mathcal{L}_{{\bar{D}}^{*}{\bar{D}}^{*}\mathbb{P}}$
$\displaystyle=$
$\displaystyle\frac{2ig}{f_{\pi}}v^{\alpha}\varepsilon_{\alpha\mu\nu\lambda}{\bar{D}}_{a}^{*\mu{\dagger}}{\bar{D}}_{b}^{*\lambda}\partial^{\nu}{\mathbb{P}}_{ab},$
(10) $\displaystyle\mathcal{L}_{{\bar{D}}{\bar{D}}\mathbb{V}}$
$\displaystyle=$ $\displaystyle\sqrt{2}\beta
g_{V}{\bar{D}}_{a}{\bar{D}}_{b}^{{\dagger}}v\cdot\mathbb{V}_{ab},$ (11)
$\displaystyle\mathcal{L}_{{\bar{D}}^{*}{\bar{D}}^{*}\mathbb{V}}$
$\displaystyle=$ $\displaystyle-\sqrt{2}\beta
g_{V}{\bar{D}}_{a\mu}^{*}{\bar{D}}_{b}^{*\mu{\dagger}}v\cdot\mathbb{V}_{ab}$
(12) $\displaystyle-2\sqrt{2}i\lambda
g_{V}{\bar{D}}_{a}^{*\mu{\dagger}}{\bar{D}}_{b}^{*\nu}\left(\partial_{\mu}\mathbb{V}_{\nu}-\partial_{\nu}\mathbb{V}_{\mu}\right)_{ab}.$

Here, $v_{\mu}=(1,\bm{0})$ is the four velocity under the non-relativistic
approximation, and $\sigma$ stands for the scalar singlet meson. Additionally,
the matrices $\mathcal{B}_{\bar{3}}$, $\mathcal{B}_{6}$, ${\mathbb{P}}$, and
$\mathbb{V}_{\mu}$ have the standard forms as listed below

$\displaystyle\mathcal{B}_{\bar{3}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&\Lambda_{c}^{+}&\Xi_{c}^{+}\\\
-\Lambda_{c}^{+}&0&\Xi_{c}^{0}\\\
-\Xi_{c}^{+}&-\Xi_{c}^{0}&0\end{array}\right),$ (16)
$\displaystyle\mathcal{B}_{6}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\Sigma_{c}^{++}&\frac{\Sigma_{c}^{+}}{\sqrt{2}}&\frac{\Xi_{c}^{(^{\prime})+}}{\sqrt{2}}\\\
\frac{\Sigma_{c}^{+}}{\sqrt{2}}&\Sigma_{c}^{0}&\frac{\Xi_{c}^{(^{\prime})0}}{\sqrt{2}}\\\
\frac{\Xi_{c}^{(^{\prime})+}}{\sqrt{2}}&\frac{\Xi_{c}^{(^{\prime})0}}{\sqrt{2}}&\Omega_{c}^{0}\end{array}\right),$
(20) $\displaystyle{\mathbb{P}}$ $\displaystyle=$
$\displaystyle{\left(\begin{array}[]{cc}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&\pi^{+}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}\end{array}\right)},$
(23) $\displaystyle{\mathbb{V}}_{\mu}$ $\displaystyle=$
$\displaystyle{\left(\begin{array}[]{cc}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&\rho^{+}\\\
\rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}\end{array}\right)}_{\mu},$
(26)

respectively.

As an effective way, the OBE model was widely used to investigate the
interactions between hadrons. In calculation, the involved coupling constants
are as input when discussing the problem of the hadronic molecular state.
Usually, we prefer to fix the coupling constant if there exists the
corresponding experimental data. For some coupling constants, we can estimate
them by phenomenological models, where the relevant experimental information
is absent. These coupling constants adopted in this work can be extracted from
the experimental data or by the theoretical model, and the corresponding signs
between these coupling constants can be fixed by the quark model Riska:2000gd
. Thus, we take $g_{\sigma}=0.76$, $l_{B}=-3.65$, $l_{S}=6.20$, $g=0.59$,
$g_{4}=1.06$, $g_{1}=0.94$, $f_{\pi}=132~{}\rm{MeV}$, $\beta g_{V}=-5.25$,
$\beta_{B}g_{V}=-6.00$, $\beta_{S}g_{V}=12.00$, $\lambda
g_{V}=-3.27~{}\rm{GeV}^{-1}$, $\lambda_{I}g_{V}=-6.80~{}\rm{GeV}^{-1}$, and
$\lambda_{S}g_{V}=19.20~{}\rm{GeV}^{-1}$ in the following numerical analysis
Chen:2017xat ; Chen:2019asm ; Chen:2020kco . Here, $g_{\sigma}$ can be deduced
by the relation $g_{\sigma}=\widetilde{g}/2\sqrt{6}$ with $\widetilde{g}=3.73$
due to the spontaneously broken chiral symmetry Bardeen:2003kt . The pionic
coupling constant $g$ is determined by reproducing the experimental width of
the $D^{*+}\to D^{0}\pi^{+}$ process Falk:1992cx ; Isola:2003fh . According to
the vector meson dominance mechanism, $\beta$ and $g_{V}$ are fixed
Isola:2003fh ; Cleven:2016qbn . $\lambda$ can be obtained by comparing the
form factor obtained from the lattice QCD with this calculated via the light
cone sum rule Isola:2003fh . $g_{1}$ is related to another coupling constant
$g_{4}$ by the relation $g_{1}=\sqrt{8}g_{4}/3$ Liu:2011xc , and $g_{4}$ can
be extracted by the decay width of the $\Sigma_{c}^{*}\to\Lambda\pi$ process.
$l_{S}$ is extracted with the chiral multiplet assumption Liu:2011xc . For the
coupling constants $l_{B}$, $\beta_{B}$, $\beta_{S}$, $\lambda_{I}$, and
$\lambda_{S}$, we can calculate them by the quark model Riska:2000gd , which
is a popular approach to determine the coupling constants. Here, for the
discussed vertex, we can write out its amplitude at hadronic and quark levels.
By connecting these two amplitudes, finally the coupling constant can be
obtained. Here, we need to indicate that these coupling constants adopted in
this work were applied to reproduce the masses of the observed three $P_{c}$
states under the $S$-wave isoscalar $\Sigma_{c}\bar{D}^{(*)}$ molecular
assignment Chen:2019asm .

With the above preparation, the obtained effective potentials in the momentum
space can be related to the corresponding scattering amplitudes via the Breit
approximation Breitapproximation . Finally, the effective potentials in the
coordinate space can be obtained by the Fourier transformation. Since the
discussed hadrons are not pointlike particles, we should introduce the
monopole type form factor
$\mathcal{F}(q^{2},m_{E}^{2})=(\Lambda^{2}-m_{E}^{2})/(\Lambda^{2}-q^{2})$
Tornqvist:1993ng ; Tornqvist:1993vu  in each interaction vertex, which can
compensate the effect from the off-shell exchanged mesons and depict the
structure effect of interaction vertex.

In order to obtain the concrete effective potentials for these discussed
hidden-charm molecular pentaquark systems with strangeness, we need to
construct their wave functions. For these discussed
$\Xi_{c}^{(\prime)}\bar{D}^{(*)}$ systems, their spin-orbital wave functions
can be expressed as

$\displaystyle|\Xi_{c}^{(\prime)}\bar{D}({}^{2S+1}L_{J})\rangle$
$\displaystyle=$
$\displaystyle\sum_{m,m_{L}}C^{J,M}_{\frac{1}{2}m,Lm_{L}}\chi_{\frac{1}{2}m}|Y_{L,m_{L}}\rangle,$
(27) $\displaystyle|\Xi_{c}^{(\prime)}\bar{D}^{*}({}^{2S+1}L_{J})\rangle$
$\displaystyle=$
$\displaystyle\sum_{m,m^{\prime},m_{S},m_{L}}C^{S,m_{S}}_{\frac{1}{2}m,1m^{\prime}}C^{J,M}_{Sm_{S},Lm_{L}}\chi_{\frac{1}{2}m}$
(28) $\displaystyle\times\epsilon_{m^{\prime}}^{\mu}|Y_{L,m_{L}}\rangle.$

In the above expressions, the constant $C^{e,f}_{ab,cd}$ denotes the Clebsch-
Gordan coefficient, and $|Y_{L,m_{L}}\rangle$ is the spherical harmonics
function. The polarization vector $\epsilon_{m}^{\mu}\,(m=0,\,\pm 1)$ with
spin-1 field is written as $\epsilon_{\pm}^{\mu}=\left(0,\,\pm
1,\,i,\,0\right)/\sqrt{2}$ and $\epsilon_{0}^{\mu}=\left(0,0,0,-1\right)$ in
the static limit, and the $\chi_{\frac{1}{2}m}$ stands for the spin wave
function for the charmed baryons $\Xi_{c}^{(\prime)}$. And then, we summarize
the flavor wave functions $|I,I_{3}\rangle$ for these discussed
$\Xi_{c}^{(\prime)}\bar{D}^{(*)}$ systems in Table 1.

Table 1: Flavor wave functions for these discussed $\Xi_{c}^{(\prime)}\bar{D}^{(*)}$ systems. Here, $I$ and $I_{3}$ are their isospins and the third components, respectively.  $|I,I_{3}\rangle$ | Flavor wave functions
---|---
$|1,1\rangle$ | $\left|\Xi_{c}^{(\prime)+}\bar{D}^{(*)0}\right\rangle$
$|1,0\rangle$ | $\sqrt{\frac{1}{2}}\left|\Xi_{c}^{(\prime)+}{D}^{(*)-}\right\rangle+\sqrt{\frac{1}{2}}\left|\Xi_{c}^{(\prime)0}\bar{D}^{(*)0}\right\rangle$
$|1,-1\rangle$ | $\left|\Xi_{c}^{(\prime)0}{D}^{(*)-}\right\rangle$
$|0,0\rangle$ | $\sqrt{\frac{1}{2}}\left|\Xi_{c}^{(\prime)+}{D}^{(*)-}\right\rangle-\sqrt{\frac{1}{2}}\left|\Xi_{c}^{(\prime)0}\bar{D}^{(*)0}\right\rangle$

With the standard procedures of the OBE model, the expressions of the
effective potentials in the coordinate space for these investigated isoscalar
$\Xi_{c}^{(\prime)}\bar{D}_{s}^{(*)}$ systems are given by

$\displaystyle\mathcal{V}^{\Xi_{c}\bar{D}}$ $\displaystyle=$ $\displaystyle
2l_{B}g_{\sigma}Y_{\sigma}-\frac{3\beta\beta_{B}g_{V}^{2}}{4}Y_{\rho}+\frac{\beta\beta_{B}g_{V}^{2}}{4}Y_{\omega},$
$\displaystyle\mathcal{V}^{\Xi_{c}^{\prime}\bar{D}}$ $\displaystyle=$
$\displaystyle-
l_{S}g_{\sigma}Y_{\sigma}+\frac{3\beta\beta_{S}g_{V}^{2}}{8}Y_{\rho}-\frac{\beta\beta_{S}g_{V}^{2}}{8}Y_{\omega},$
$\displaystyle\mathcal{V}^{\Xi_{c}\bar{D}^{*}}$ $\displaystyle=$
$\displaystyle
2l_{B}g_{\sigma}\mathcal{A}_{1}Y_{\sigma}-\frac{3\beta\beta_{B}g_{V}^{2}}{4}\mathcal{A}_{1}Y_{\rho}+\frac{\beta\beta_{B}g_{V}^{2}}{4}\mathcal{A}_{1}Y_{\omega},$
$\displaystyle\mathcal{V}^{\Xi_{c}^{\prime}\bar{D}^{*}}$ $\displaystyle=$
$\displaystyle-
l_{S}g_{\sigma}\mathcal{A}_{1}Y_{\sigma}-\frac{g_{1}g}{4f_{\pi}^{2}}\left[\mathcal{A}_{2}\mathcal{O}_{r}+\mathcal{A}_{3}\mathcal{P}_{r}\right]Y_{\pi}$
$\displaystyle-$
$\displaystyle\frac{g_{1}g}{36f_{\pi}^{2}}\left[\mathcal{A}_{2}\mathcal{O}_{r}+\mathcal{A}_{3}\mathcal{P}_{r}\right]Y_{\eta}$
$\displaystyle+$
$\displaystyle\frac{3\beta\beta_{S}g_{V}^{2}}{8}\mathcal{A}_{1}Y_{\rho}+\frac{3\lambda\lambda_{S}g_{V}^{2}}{18}\left[2\mathcal{A}_{2}\mathcal{O}_{r}-\mathcal{A}_{3}\mathcal{P}_{r}\right]Y_{\rho}$
$\displaystyle-$
$\displaystyle\frac{\beta\beta_{S}g_{V}^{2}}{8}\mathcal{A}_{1}Y_{\omega}-\frac{\lambda\lambda_{S}g_{V}^{2}}{18}\left[2\mathcal{A}_{2}\mathcal{O}_{r}-\mathcal{A}_{3}\mathcal{P}_{r}\right]Y_{\omega}.$

Here, $\mathcal{O}_{r}=\frac{1}{r^{2}}\frac{\partial}{\partial
r}r^{2}\frac{\partial}{\partial r}$ and
$\mathcal{P}_{r}=r\frac{\partial}{\partial
r}\frac{1}{r}\frac{\partial}{\partial r}$. The function $Y_{i}$ is defined as

$\displaystyle Y_{i}\equiv\dfrac{e^{-m_{i}r}-e^{-\Lambda_{i}r}}{4\pi
r}-\dfrac{\Lambda_{i}^{2}-m_{i}^{2}}{8\pi\Lambda_{i}}e^{-\Lambda_{i}r}.$ (29)

Additionally, we also define several operators, which include
$\mathcal{A}_{1}=\chi^{\dagger}_{3}\left({\bm{\epsilon}^{\dagger}_{4}}\cdot{\bm{\epsilon}_{2}}\right)\chi_{1}$,
$\mathcal{A}_{2}=\chi^{\dagger}_{3}\left[{\bm{\sigma}}\cdot\left(i{\bm{\epsilon}_{2}}\times{\bm{\epsilon}^{\dagger}_{4}}\right)\right]\chi_{1}$,
and
$\mathcal{A}_{3}=\chi^{\dagger}_{3}T({\bm{\sigma}},i{\bm{\epsilon}_{2}}\times{\bm{\epsilon}^{\dagger}_{4}},\hat{\bm{r}})\chi_{1}$.
Here, the tensor force operator is expressed as
$T({\bm{x}},{\bm{y}},\hat{\bm{r}})=3\left(\hat{\bm{r}}\cdot{\bm{x}}\right)\left(\hat{\bm{r}}\cdot{\bm{y}}\right)-{\bm{x}}\cdot{\bm{y}}$
with $\hat{\bm{r}}={\bm{r}}/|{\bm{r}}|$. In the effective potentials, the
corresponding matrix elements $\langle f|\mathcal{A}_{k}|i\rangle$ can be
obtained by these operators $\mathcal{A}_{k}$ sandwiched between the relevant
spin-orbit wave functions of the initial and final states. For example, the
matrices element
$\langle\Xi_{c}^{(\prime)}\bar{D}^{*}|\mathcal{A}_{1}|\Xi_{c}^{(\prime)}\bar{D}^{*}\rangle$
with $J=1/2$ can be expressed as
${\left(\begin{array}[]{cc}\mathcal{A}_{1}^{11}&\mathcal{A}_{1}^{12}\\\
\mathcal{A}_{1}^{21}&\mathcal{A}_{1}^{22}\\\ \end{array}\right)}$ with

$\displaystyle\mathcal{A}_{1}^{11}$ $\displaystyle=$
$\displaystyle\langle\Xi_{c}^{(\prime)}\bar{D}^{*}({}^{2}\mathbb{S}_{\frac{1}{2}})|\mathcal{A}_{1}|\Xi_{c}^{(\prime)}\bar{D}^{*}({}^{2}\mathbb{S}_{\frac{1}{2}})\rangle,$
$\displaystyle\mathcal{A}_{1}^{12}$ $\displaystyle=$
$\displaystyle\langle\Xi_{c}^{(\prime)}\bar{D}^{*}({}^{4}\mathbb{D}_{\frac{1}{2}})|\mathcal{A}_{1}|\Xi_{c}^{(\prime)}\bar{D}^{*}({}^{2}\mathbb{S}_{\frac{1}{2}})\rangle,$
$\displaystyle\mathcal{A}_{1}^{21}$ $\displaystyle=$
$\displaystyle\langle\Xi_{c}^{(\prime)}\bar{D}^{*}({}^{2}\mathbb{S}_{\frac{1}{2}})|\mathcal{A}_{1}|\Xi_{c}^{(\prime)}\bar{D}^{*}({}^{4}\mathbb{D}_{\frac{1}{2}})\rangle,$
$\displaystyle\mathcal{A}_{1}^{22}$ $\displaystyle=$
$\displaystyle\langle\Xi_{c}^{(\prime)}\bar{D}^{*}({}^{4}\mathbb{D}_{\frac{1}{2}})|\mathcal{A}_{1}|\Xi_{c}^{(\prime)}\bar{D}^{*}({}^{4}\mathbb{D}_{\frac{1}{2}})\rangle.$

In Table 2, we collect the obtained relevant operator matrix elements $\langle
f|\mathcal{A}_{k}|i\rangle$, which are used in our calculation.

Table 2: The relevant operator matrix elements $\langle f|\mathcal{A}_{k}|i\rangle$, which are obtained by sandwiching these operators between the relevant spin-orbit wave functions. Spin | $J=1/2$ | $J=3/2$
---|---|---
$\langle\Xi_{c}^{(\prime)}\bar{D}^{*}|\mathcal{A}_{1}|\Xi_{c}^{(\prime)}\bar{D}^{*}\rangle$ | $\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right)$ | $\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&1\end{array}\right)$
$\langle\Xi_{c}^{\prime}\bar{D}^{*}|\mathcal{A}_{2}|\Xi_{c}^{\prime}\bar{D}^{*}\rangle$ | $\left(\begin{array}[]{cc}-2&0\\\ 0&1\end{array}\right)$ | $\left(\begin{array}[]{ccc}1&0&0\\\ 0&-2&0\\\ 0&0&1\end{array}\right)$
$\langle\Xi_{c}^{\prime}\bar{D}^{*}|\mathcal{A}_{3}|\Xi_{c}^{\prime}\bar{D}^{*}\rangle$ | $\left(\begin{array}[]{cc}0&-\sqrt{2}\\\ -\sqrt{2}&-2\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&1&2\\\ 1&0&-1\\\ 2&-1&0\end{array}\right)$

When discussing the bound state properties of the $S$-wave isoscalar
$\Xi_{c}\bar{D}^{*}$ states with $J^{P}=1/2^{-}$ and $3/2^{-}$ after including
the coupled channels $\Xi_{c}\bar{D}^{*}$ and $\Xi_{c}^{\prime}\bar{D}^{*}$,
we need the effective potentials in the coordinate space for the
$\Xi_{c}\bar{D}^{*}\to\Xi_{c}^{\prime}\bar{D}^{*}$ process, i.e,

$\displaystyle\mathcal{V}^{\Xi_{c}\bar{D}^{*}\to\Xi_{c}^{\prime}\bar{D}^{*}}$
$\displaystyle=$
$\displaystyle-\frac{gg_{4}}{2\sqrt{6}f_{\pi}^{2}}\left[\mathcal{A}_{2}\mathcal{O}_{r}+\mathcal{A}_{3}\mathcal{P}_{r}\right]Y_{\pi
0}$ $\displaystyle+$
$\displaystyle\frac{gg_{4}}{6\sqrt{6}f_{\pi}^{2}}\left[\mathcal{A}_{2}\mathcal{O}_{r}+\mathcal{A}_{3}\mathcal{P}_{r}\right]Y_{\eta
0}$ $\displaystyle-$
$\displaystyle\frac{\lambda\lambda_{I}g_{V}^{2}}{\sqrt{6}}\left[2\mathcal{A}_{2}\mathcal{O}_{r}-\mathcal{A}_{3}\mathcal{P}_{r}\right]Y_{\rho
0}$ $\displaystyle+$
$\displaystyle\frac{\lambda\lambda_{I}g_{V}^{2}}{3\sqrt{6}}\left[2\mathcal{A}_{2}\mathcal{O}_{r}-\mathcal{A}_{3}\mathcal{P}_{r}\right]Y_{\omega
0}.$

Here, $m_{0}=\sqrt{m^{2}-q_{0}^{2}}$ and
$\Lambda_{0}=\sqrt{\Lambda^{2}-q_{0}^{2}}$ with $q_{0}=0.06\,{\rm{GeV}}$.

##  III The characteristic mass spectra of the
$\Xi_{c}^{(\prime)}\bar{D}^{(*)}$ molecular pentaquarks

Based on the obtained effective potentials in the coordinate space for these
discussed hidden-charm pentaquarks with strangeness, we can obtain the bound
state solutions by solving the coupled channel Schr$\rm{\ddot{o}}$dinger
equation, where the obtained bound state solutions mainly include the binding
energy $E$, the root-mean-square radius $r_{\rm RMS}$, and the probabilities
for different components $P_{i}$, which may provide us the critical
information to judge whether these discussed hidden-charm molecular pentaquark
states with strangeness exist or not. In the coupled channel analysis, the
probabilities for different components $P_{i}$ are useful information to
reflect the properties of the hadronic molecular states, except for the
binding energy $E$ and the root-mean-square radius $r_{\rm RMS}$. If the
coupled system is not dominant by the lowest mass threshold channel among the
selected coupled channels, where usually the obtained root-mean-square radius
$r_{{\rm RMS}}$ is too small, this coupled system is not recommended to be a
hadronic molecular state (see Ref. Chen:2017xat  for more details).

Of course, when judging whether the loosely bound state is the possible
hadronic molecular candidate, we also need to specify two points: (1) As a
free parameter, the cutoff value of the form factor cannot be determined
exactly due to the lack of relevant experiment data. In realistic calculation,
one should find bound state solution by changing the cutoff value. According
to the experience of studying the deuteron by the OBE model, the cutoff value
should not be far away from 1 GeV Machleidt:1987hj ; Epelbaum:2008ga ;
Esposito:2014rxa ; Chen:2016qju . Furthermore, we can reproduce the masses of
the observed three $P_{c}$ states Chen:2019asm  and $T_{cc}$ state Li:2021zbw
under the hadronic molecular picture when taking the cutoff values around 1
GeV. In the present work, we still take the cutoff range around 1 GeV to
discuss the bound state properties of the $S$-wave isoscalar
$\Xi_{c}^{(\prime)}\bar{D}^{(*)}$ systems; (2) The reasonable binding energy
for the possible hadronic molecular candidate should be at most tens of MeV,
and the corresponding typical size should be larger than the size of all the
included component hadrons Chen:2017xat . These criteria may provide useful
hints to identify the hidden-charm molecular pentaquark candidates with
strangeness.

In our calculation, the masses of these involved hadrons are
$m_{\sigma}=600.00$ MeV, $m_{\pi}=137.27$ MeV, $m_{\eta}=547.86$ MeV,
$m_{\rho}=775.26$ , $m_{\omega}=782.66$ MeV, $m_{D}=1867.25$ MeV,
$m_{D^{*}}=2008.56$ MeV, $m_{\Xi_{c}}=2469.08$ MeV, and
$m_{\Xi_{c}^{\prime}}=2578.45$ MeV, which are taken from the Particle Data
Group ParticleDataGroup:2022pth .

###  III.1 The characteristic mass spectrum of the $\Xi_{c}\bar{D}^{(*)}$
molecular pentaquarks

For the $S$-wave isoscalar $\Xi_{c}\bar{D}^{(*)}$ systems, we list the
obtained bound state solutions by considering the $S$-$D$ wave mixing effect
in Table. 3. For the $S$-wave isoscalar $\Xi_{c}\bar{D}$ state with
$J^{P}=1/2^{-}$, we obtain the bound state solution by setting the cutoff
parameter $\Lambda$ around 1.41 GeV. For the $S$-wave isoscalar
$\Xi_{c}\bar{D}^{*}$ states with $J^{P}=1/2^{-}$ and $3/2^{-}$, there exist
the bound state solutions when the cutoff parameter is taken to be around 1.39
GeV. Here, the probabilities for the $D$-wave components are zero for these
two states, since the contribution of the tensor forces from the $S$-$D$ wave
mixing effect disappears for the case of the isoscalar $\Xi_{c}\bar{D}^{*}$
interactions. In addition, there exists mass degeneration for the $S$-wave
isoscalar $\Xi_{c}\bar{D}^{*}$ states with $J^{P}=1/2^{-}$ and $3/2^{-}$ when
adopting same cutoff value in the $S$-$D$ wave mixing analysis.

Table 3: Bound state properties for the $S$-wave isoscalar
$\Xi_{c}\bar{D}^{(*)}$ systems by considering the $S$-$D$ wave mixing effect.
Here, the cutoff $\Lambda$, binding energy $E$, and root-mean-square radius
$r_{{\rm RMS}}$ are in units of $\rm{GeV}$, $\rm{MeV}$, and $\rm{fm}$,
respectively. $\Xi_{c}\bar{D}(J^{P}=1/2^{-})$
---
$\Lambda$ | $E$ | $r_{\rm RMS}$ |
1.41 | $-0.35$ | 4.73 |
1.61 | $-4.82$ | 1.64 |
1.79 | $-12.49$ | 1.10 |
$\Xi_{c}\bar{D}^{*}(J^{P}=1/2^{-})$
$\Lambda$ | $E$ | $r_{\rm RMS}$ | P(${}^{2}\mathbb{S}_{\frac{1}{2}}/{}^{4}\mathbb{D}_{\frac{1}{2}}$)
1.39 | $-0.34$ | 4.70 |  100.00/$o(0)$
1.57 | $-4.71$ | 1.63 |  100.00/$o(0)$
1.74 | $-12.21$ | 1.10 |  100.00/$o(0)$
$\Xi_{c}\bar{D}^{*}(J^{P}=3/2^{-})$
$\Lambda$ | $E$ | $r_{\rm RMS}$ | P(${}^{4}\mathbb{S}_{\frac{3}{2}}/{}^{2}\mathbb{D}_{\frac{3}{2}}/{}^{4}\mathbb{D}_{\frac{3}{2}}$)
1.39 | $-0.34$ | 4.70 |  100.00/$o(0)$/$o(0)$
1.57 | $-4.71$ | 1.63 |  100.00/$o(0)$/$o(0)$
1.74 | $-12.21$ | 1.10 |  100.00/$o(0)$/$o(0)$

In the above discussions, we indicate that these discussed $S$-wave isoscalar
$\Xi_{c}\bar{D}^{(*)}$ systems can be viewed as the hidden-charm molecular
pentaquark candidates with single strangeness within the OBE model by
considering the $S$-$D$ wave mixing effect. However, there exist mass
degeneration for the $S$-wave isoscalar $\Xi_{c}\bar{D}^{*}$ states with
$J^{P}=1/2^{-}$ and $3/2^{-}$ when adopting same cutoff value in the $S$-$D$
wave mixing analysis. Since the observation of the charmoniumlike state
$X(3872)$, various corrections Chen:2016qju  including the coupled channel
effect Li:2012bt ; Li:2012cs ; Li:2012ss  have been introduced to discuss the
fine properties of the hadron-hadron interactions in the OBE model, which may
decorate the bound state properties of hadronic molecular states Li:2012bt ;
Li:2012cs ; Li:2012ss ; Chen:2015add ; Chen:2017jjn ; Chen:2017xat ;
Chen:2018pzd ; Wang:2019nwt ; Chen:2019asm ; Wang:2019aoc ; Du:2019pij ;
Yamaguchi:2019seo ; Wang:2020dya ; Wang:2020bjt ; Chen:2020kco ; Wang:2021ajy
; Chen:2021kad ; Wang:2021hql ; Wang:2021yld ; Li:2021zbw ; Yang:2021sue ;
Du:2021fmf ; Chen:2022onm ; Burns:2022uiv ; Roca:2006sz ; Oset:2010tof ;
Liu:2011xc ; Xie:2012np ; Xiao:2013yca ; Xiao:2013jla ; Ozpineci:2013zas ;
Liang:2014eba ; Roca:2015dva ; Moir:2016srx ; Kang:2016zmv ;
Albaladejo:2016eps ; Shimizu:2016rrd ; Shimizu:2017xrg ; Yamaguchi:2016ote ;
Lu:2016kxm ; Shimizu:2018ran ; Pavao:2018xub ; Sekihara:2018tsb ;
Valderrama:2018bmv ; Dias:2018qhp ; He:2018plt ; He:2017lhy ; Shen:2017ayv ;
Montana:2017kjw ; Meng:2019ilv ; Bruns:2019fwi ; Yang:2018amd ; Yu:2019yfr ;
Burns:2019iih ; Xiao:2019gjd ; Xiao:2019aya ; Yu:2018yxl ; Wang:2019niy ;
Ortega:2018cnm ; Zhu:2020vto ; Yang:2019rgw ; Chen:2020yvq ; Chen:2021vhg ;
Ikeno:2020mra ; Zhu:2021lhd ; Yalikun:2021bfm ; Peng:2020hql ; Hao:2022vwt ;
Ikeno:2021mcb ; Kamiya:2022thy ; Dai:2022ulk ; Zhu:2022fyb ; Wang:2022oof ;
Yalikun:2021dpk ; Albaladejo:2021vln ; Wang:2021lwy .

For the $\Xi_{c}\bar{D}^{(*)}$ molecular systems, the authors of Refs.
Chen:2020kco ; Chen:2022onm  once presented a coupled channel analysis within
the OBE model. Here, for the $S$-wave isoscalar $\Xi_{c}\bar{D}^{*}$ state
with $J^{P}=3/2^{-}$, the coupled channel effect is helpful to form this
molecular state Chen:2020kco , since the cutoff value becomes smaller after
including the coupled channel effect compared with the situation without
considering the coupled channel effect. However, when discussing the isoscalar
$\Xi_{c}\bar{D}^{*}$ state with $J^{P}=1/2^{-}$ with a coupled channel
analysis, a puzzling phenomenon appears Chen:2020kco , where a higher
$\Xi_{c}^{\prime}\bar{D}^{*}$ channel among these selected coupled channels
becomes dominant. Obviously, a jump occurs which is not reasonable. The reason
resulting in this puzzling phenomenon is that the author of Ref. Chen:2020kco
took the same cutoff value for these considered channels in their coupled
channel analysis. Obviously, this treatment should be improved to ensure that
the coupled channel effect only plays the role to decorate the discussed pure
state, which was not realised in these early studies Chen:2022onm ;
Chen:2021kad ; Wang:2021hql ; Chen:2020kco ; Wang:2019aoc ; Chen:2017xat ;
Wang:2021yld ; Wang:2020dya .

An approaches to overcome such problem is proposed in this work. In practice,
the cutoff values for the involved coupled channels can be different. Along
this line, we discuss the bound state properties of the isoscalar
$\Xi_{c}\bar{D}^{*}$ molecular states with $J^{P}=1/2^{-}$ and $3/2^{-}$,
where we adopt different cutoff values for the $\Xi_{c}\bar{D}^{*}$ and
$\Xi_{c}^{\prime}\bar{D}^{*}$ channels. In Table 4, we present the bound state
solutions of the $S$-wave isoscalar $\Xi_{c}\bar{D}^{*}$ states with
$J^{P}=1/2^{-}$ and $3/2^{-}$ by the coupled channel analysis. The above fact
shows that the coupled channel effect indeed may decorate the bound state
properties of the $S$-wave isoscalar $\Xi_{c}\bar{D}^{*}$ state with
$J^{P}=1/2^{-}$, and ensure that the jump phenomenon mentioned above does not
happen. Thus, the conclusion of the existence of the $S$-wave isoscalar
$\Xi_{c}\bar{D}^{*}$ molecular state with $J^{P}=1/2^{-}$ does not change when
considering the coupled channel effect. We should indicate that the former
studies in Refs. Zhu:2021lhd ; Wang:2019nvm ; Chen:2021cfl  also support the
existence of the $S$-wave isoscalar $\Xi_{c}\bar{D}^{*}$ molecular states with
$J^{P}=1/2^{-}$ and $3/2^{-}$. In this work, we can find the bound state
solutions for the $S$-wave isoscalar $\Xi_{c}\bar{D}^{*}$ states with
$J^{P}=1/2^{-}$ and $3/2^{-}$, where the $\Xi_{c}\bar{D}^{*}$ is the dominant
channel. Different from the case of the single channel analysis, there exists
the mass difference for the $S$-wave isoscalar $\Xi_{c}\bar{D}^{*}$ states
with $J^{P}=1/2^{-}$ and $3/2^{-}$ when considering the coupled channel
effect.

Table 4: Bound state properties for the $S$-wave isoscalar $\Xi_{c}\bar{D}^{*}$ system by performing the coupled channel analysis. Here, the cutoff $\Lambda$, binding energy $E$, and root-mean-square radius $r_{{\rm RMS}}$ are in units of $\rm{GeV}$, $\rm{MeV}$, and $\rm{fm}$, respectively. Additionally, $\Lambda$ and $\Lambda^{\prime}$ denote the cutoff parameters of the $\Xi_{c}\bar{D}^{*}$ and $\Xi_{c}^{\prime}\bar{D}^{*}$ channels, respectively. $\Lambda$ | $\Lambda^{\prime}$ | $E$ | $r_{\rm RMS}$ | P($\Xi_{c}\bar{D}^{*}/\Xi_{c}^{\prime}\bar{D}^{*}$)
---|---|---|---|---
$\Xi_{c}\bar{D}^{*}(J^{P}=1/2^{-})$
1.12 | 0.92 | $-0.30$ | 4.74 |  97.75/2.25
1.16 | 0.96 | $-4.33$ | 1.58 |  89.46/10.54
1.20 | 1.00 | $-14.67$ | 0.89 |  77.76/22.24
$\Xi_{c}\bar{D}^{*}(J^{P}=3/2^{-})$
1.31 | 1.11 | $-0.29$ | 4.87 |  99.73/0.27
1.43 | 1.23 | $-4.52$ | 1.64 |  98.54/1.46
1.56 | 1.36 | $-15.01$ | 0.98 |  96.48/3.52

We also need to mention that the $P_{cs}(4459)$ existing in the
$J/\psi\Lambda$ invariant mass spectrum may be described by two peak
structures with the masses of 4454.9 MeV and 4467.9 MeV LHCb:2020jpq .
According to Table 4, the masses of the $S$-wave isoscalar
$\Xi_{c}\bar{D}^{*}$ state with $J^{P}=1/2^{-}$ may be around 4454.9 MeV, we
can reproduce this mass with the cutoff parameters of the $\Xi_{c}\bar{D}^{*}$
and $\Xi_{c}^{\prime}\bar{D}^{*}$ channels respectively around 1.22 GeV and
1.02 GeV, which are close to the reasonable range around 1.00 GeV
Tornqvist:1993ng ; Tornqvist:1993vu ; Wang:2019nwt ; Chen:2017jjn . Here, its
binding energy and root-mean-square radius are $-22.70$ MeV and 0.73 fm, and
the probabilities of the $\Xi_{c}\bar{D}^{*}$ and
$\Xi_{c}^{\prime}\bar{D}^{*}$ components are around 72% and 28%, respectively.

The corresponding thresholds of the $\Xi_{c}\bar{D}$ and $\Xi_{c}\bar{D}^{*}$
channels are 4336.33 MeV and 4477.64 MeV, so the masses of the $S$-wave
isoscalar $\Xi_{c}\bar{D}^{*}$ states are larger than that of the $S$-wave
isoscalar $\Xi_{c}\bar{D}$ state. Based on the analysis mentioned above, it is
clear that the masses of these discussed hidden-charm molecular pentaquarks
with strangeness satisfy the relation
$m[\Xi_{c}\bar{D}]<m[\Xi_{c}\bar{D}^{*}]$. Generally, the above analysis
presents a characteristic mass spectrum of hidden-charm pentaquark with
strangeness. Additionally, our obtained $S$-wave isoscalar $\Xi_{c}\bar{D}$
state with $J^{P}=1/2^{-}$ has the mass lower than that of the observed
$P_{\psi s}^{\Lambda}(4338)$ Collaboration:2022boa . However, the $P_{\psi
s}^{\Lambda}(4338)$ was reported in the $J/\psi\Lambda$ invariant mass
spectrum via the $B^{-}\to J/\psi\Lambda\bar{p}$ process with the width about
7 MeV. Thus, there is still some possibility that the origin of the $P_{\psi
s}^{\Lambda}(4338)$ can be due to the $S$-wave isoscalar $\Xi_{c}\bar{D}$
molecular state with $J^{P}=1/2^{-}$, which should be checked in future
experiment. Especially, confirming it via the $\Xi_{b}^{-}\to J/\psi\Lambda K$
process is strongly suggested here.

In Ref. Guo:2019kdc , the authors once estimated the probabilities of finding
the $\Sigma_{c}\bar{D}^{(*)}$ and $J/\psi p$ components inside the $P_{c}$
states within the effective range expansion and the resonance compositeness
relations, and concluded that the weight of the $\Sigma_{c}\bar{D}^{(*)}$
component around 99% is much larger than that of the $J/\psi p$ component.
Thus, most of the previous theoretical studies usually ignored the
contribution of the $J/\psi p$ channel when explaining the observed $P_{c}$
states as the $S$-wave isoscalar $\Sigma_{c}\bar{D}^{(*)}$ molecular states.
Here, we would like to mention that there exists similar spectroscopy behavior
for the $S$-wave isoscalar $\Sigma_{c}\bar{D}^{(*)}$ and
$\Xi_{c}\bar{D}^{(*)}$ molecular states, which reflects that the weight of the
$\Xi_{c}\bar{D}^{(*)}$ component should be much larger than that of the
$J/\psi\Lambda$ component for these discussed hidden-charm molecular
pentaquarks with single strangeness. In the present work, we do not consider
the contribution of the $J/\psi\Lambda$ channel when discussing the bound
state properties of the $S$-wave isoscalar $\Xi_{c}^{(\prime)}\bar{D}^{(*)}$
states.

###  III.2 The characteristic mass spectrum of the
$\Xi_{c}^{\prime}\bar{D}^{(*)}$ molecular pentaquarks

Following the procedure discussed above, we present the binding energy, root-
mean-square radius, and probabilities for different components for the
$S$-wave isoscalar $\Xi_{c}^{\prime}\bar{D}^{(*)}$ systems in Fig. 2. For the
$S$-wave isoscalar $\Xi_{c}^{\prime}\bar{D}$ state with $J^{P}=1/2^{-}$, the
bound state solution can be found when the cutoff parameter is fixed to be
larger than 1.45 GeV. For the $S$-wave isoscalar $\Xi_{c}^{\prime}\bar{D}^{*}$
system, the $\pi$, $\sigma$, $\eta$, $\rho$, and $\omega$ exchanges contribute
to the total effective potentials. For the $S$-wave isoscalar
$\Xi_{c}^{\prime}\bar{D}^{*}$ state with $J^{P}=1/2^{-}$, there exists the
bound state solutions with the cutoff parameter around 0.92 GeV, while we can
obtain loosely bound state solutions for the $S$-wave isoscalar
$\Xi_{c}^{\prime}\bar{D}^{*}$ state with $J^{P}=3/2^{-}$ when we tune the
cutoff parameter to be around 1.63 GeV. Thus, we can conclude these discussed
states are the isoscalar hidden-charm molecular pentaquark candidates with
strangeness, which is consistent with the conclusions in Refs. Zhu:2021lhd ;
Wang:2019nvm ; Chen:2021cfl ; Chen:2022onm . Additionally, the largest mass of
these isoscalar hidden-charm molecular pentaquark candidates with strangeness
is the $S$-wave isoscalar $\Xi_{c}^{\prime}\bar{D}^{*}$ state with
$J^{P}=3/2^{-}$, followed by the the $S$-wave isoscalar
$\Xi_{c}^{\prime}\bar{D}^{*}$ state with $J^{P}=1/2^{-}$ and the $S$-wave
isoscalar $\Xi_{c}^{\prime}\bar{D}$ state with $J^{P}=1/2^{-}$. Here, we need
to point out that the masses of the $S$-wave isoscalar
$\Xi_{c}^{\prime}\bar{D}^{*}$ state with $J^{P}=3/2^{-}$ and $1/2^{-}$ are
4582.3 MeV and 4568.7 MeV based on the chiral effective field theory
Wang:2019nvm , and the masses of the $S$-wave isoscalar
$\Xi_{c}^{\prime}\bar{D}^{*}$ state with $J^{P}=3/2^{-}$ and $1/2^{-}$ are
4582.1 MeV and 4564.9 MeV with a quark level interaction Chen:2021cfl , which
are comparable with our results. For the $\Xi_{c}^{\prime}\bar{D}^{(*)}$
pentaquark systems, there also exists a characteristic mass spectrum as
predicted above (see Fig. 2).

Figure 2: Bound state properties for the $S$-wave isoscalar
$\Xi_{c}^{\prime}\bar{D}^{(*)}$ systems. Here, the cutoff $\Lambda$, binding
energy $E$, and root-mean-square radius $r_{{\rm RMS}}$ are in units of
$\rm{GeV}$, $\rm{MeV}$, and $\rm{fm}$, respectively.

For the $S$-wave loosely bound state, its bound state properties have close
relation to the effective potentials, where the $S$-wave interaction has
dominant contribution. For the $S$-wave $\Xi_{c}^{\prime}\bar{D}^{*}$ system,
the operator $\mathcal{A}_{2}$ has connection to the interaction strength,
i.e.,
$\langle\Xi_{c}^{\prime}\bar{D}^{*}|\mathcal{A}_{2}|\Xi_{c}^{\prime}\bar{D}^{*}\rangle=-2$
and $1$ with $J=1/2$ and $3/2$, respectively. Thus, we may conclude that the
cutoff value of the $S$-wave isoscalar $\Xi_{c}^{\prime}\bar{D}^{*}$ state
with $J^{P}=1/2^{-}$ should be smaller than that of the $S$-wave isoscalar
$\Xi_{c}^{\prime}\bar{D}^{*}$ state with $J^{P}=3/2^{-}$ if getting the same
binding energy. For the $S$-wave $\Xi_{c}\bar{D}^{*}$ system, there only
exists the operator $\mathcal{A}_{1}$ with
$\langle\Xi_{c}\bar{D}^{*}|\mathcal{A}_{1}|\Xi_{c}\bar{D}^{*}\rangle=1$ for
the cases of $J=1/2$ and $3/2$. Therefore, the cutoff for the $S$-wave
isoscalar $\Xi_{c}\bar{D}^{*}$ states with $J^{P}=1/2^{-}$ and $3/2^{-}$ has
the same value for getting the same binding energy when performing the $S$-$D$
wave mixing analysis.

At present, there does not exist the experimental information involved in the
coupling constant $g_{\sigma}$. Thus, we have to estimate the $g_{\sigma}$
value by the phenomenological model. Usually, there are two determined values
for coupling constant $g_{\sigma}$, which are $0.76$ and $2.82$, by the
spontaneously broken chiral symmetry Bardeen:2003kt  and the quark model
Riska:2000gd . Both of them were adopted in realistic study. As shown in Table
5, the loosely bound state solutions can be obtained with the cutoff values
closed to 1 GeV for the $S$-wave isoscalar $\Xi_{c}^{\prime}\bar{D}^{*}$ state
with $J^{P}=1/2^{-}$ by taking either $g_{\sigma}=0.76$ or $g_{\sigma}=2.82$.
The uncertainty resulted from the $g_{\sigma}$ value can be compensated by
smally changing $\Lambda$ value. Thus, the conclusion of existing the $S$-wave
isoscalar $\Xi_{c}^{\prime}\bar{D}^{*}$ molecular state with $J^{P}=1/2^{-}$
is still hold when considering the uncertainties of the coupling constant
$g_{\sigma}$.

Table 5: Bound state properties for the $S$-wave isoscalar
$\Xi_{c}^{\prime}\bar{D}^{*}$ state with $J^{P}=1/2^{-}$ when considering the
uncertainties of the coupling constant $g_{\sigma}$. Here, the cutoff
$\Lambda$, binding energy $E$, and root-mean-square radius $r_{{\rm RMS}}$ are
in units of $\rm{GeV}$, $\rm{MeV}$, and $\rm{fm}$, respectively.
$g_{\sigma}=0.76$
---
$\Lambda$ | $E$ | $r_{\rm RMS}$ | P(${}^{2}\mathbb{S}_{\frac{1}{2}}/{}^{4}\mathbb{D}_{\frac{1}{2}}$)
0.92 | $-0.50$ | 4.02 |  99.63/0.37
0.97 | $-4.18$ | 1.64 |  99.45/0.55
1.02 | $-11.90$ | 1.06 |  99.45/0.55
$g_{\sigma}=2.82$
$\Lambda$ | $E$ | $r_{\rm RMS}$ | P(${}^{2}\mathbb{S}_{\frac{1}{2}}/{}^{4}\mathbb{D}_{\frac{1}{2}}$)
0.83 | $-0.33$ | 4.62 |  99.71/0.29
0.88 | $-3.94$ | 1.74 |  99.53/0.47
0.93 | $-11.95$ | 1.11 |  99.51/0.49

We should indicate that the $P_{cs}(4459)$ and $P_{\psi s}^{\Lambda}(4338)$
are also close to the corresponding thresholds of baryon-meson channels like
the combination of the $\Lambda_{c}/\Sigma_{c}^{(*)}$ baryon and the
$D_{s}^{(*)}$ meson. Although the isovector
$\Sigma_{c}^{(*)}\bar{D}_{s}^{(*)}$ channel cannot couple with the discussed
isoscalar systems, the $\Lambda_{c}\bar{D}_{s}^{(*)}$ channel can interact
with the discussed systems, which is not addressed in the present coupled
channel analysis.

By above discussion, we may conclude that there exists the characteristic mass
spectra of the $S$-wave isoscalar $\Xi_{c}^{(\prime)}\bar{D}^{(*)}$-type
hidden-charm molecular pentaquarks with single strangeness. In Fig. 3, we
compare the masses between the observed hidden-charm pentaquarks with single
strangeness Collaboration:2022boa ; LHCb:2020jpq  and our predicted $S$-wave
isoscalar $\Xi_{c}^{(\prime)}\bar{D}^{(*)}$-type hidden-charm molecular
pentaquarks with single strangeness.

Figure 3: The comparison of the masses between the observed hidden-charm
pentaquarks with single strangeness Collaboration:2022boa ; LHCb:2020jpq  and
our predicted $S$-wave isoscalar $\Xi_{c}^{(\prime)}\bar{D}^{(*)}$-type
hidden-charm molecular pentaquarks with single strangeness. Here, the purple
dash-dotted lines denote the thresholds of these discussed hidden-charm
molecular pentaquarks with single strangeness, and the blue and red thick
solid lines represent the masses of the observed hidden-charm pentaquarks with
strangeness and our predicted $S$-wave isoscalar
$\Xi_{c}^{(\prime)}\bar{D}^{(*)}$-type hidden-charm molecular pentaquarks with
single strangeness, respectively.

In the following, we discuss the two-body hidden-charm decay behaviors of our
predicted $S$-wave isoscalar $\Xi_{c}^{(\prime)}\bar{D}^{(*)}$-type hidden-
charm molecular pentaquarks with single strangeness, which is inspired by the
observation from LHCb of the $P_{c}$ and $P_{cs}$ states Aaij:2015tga ;
Aaij:2019vzc ; Collaboration:2022boa ; LHCb:2020jpq  via the two-body hidden-
charm decay channels. In the past decades, the heavy quark symmetry was often
used to discuss the properties of the hadrons which contain the heavy quarks.
Thus, we may discuss the two-body hidden-charm decay behaviors of these
predicted hidden-charm molecular pentaquarks with single strangeness within
the heavy quark symmetry.

Figure 4: Diagram for the calculation of the 9-$j$ coefficients. Here, $j_{1}$
and $j_{2}$ denote the total angular momentum quantum numbers for the charmed
baryons and anti-charmed mesons, respectively. $s_{1}$ and $s_{2}$ stand for
the spin quantum numbers of the heavy quarks. $l_{1}$ and $l_{2}$ are the spin
quantum numbers of the light quarks. And then, $l_{1}=0$ and $1$ for the
$\Xi_{c}$ and $\Xi_{c}^{\prime}$ baryons, respectively.

As shown in Fig. 4, we expand the spin wave functions of the heavy hadron
systems $|\ell_{1}s_{1}j_{1},\ell_{2}s_{2}j_{2},JM\big{\rangle}$ in terms of
the heavy quark basis $|\ell_{1}\ell_{2}L,s_{1}s_{2}S,JM\big{\rangle}$, where
the general relation can be expressed as

$\displaystyle\left|\ell_{1}s_{1}j_{1},\ell_{2}s_{2}j_{2},JM\right\rangle$
$\displaystyle=$
$\displaystyle\sum_{S,L}\hat{S}\hat{L}\hat{j_{1}}\hat{j_{2}}\left\\{\begin{array}[]{ccc}\ell_{1}&\ell_{2}&L\\\
s_{1}&s_{2}&S\\\ j_{1}&j_{2}&J\end{array}\right\\}$ (34)
$\displaystyle\times|\ell_{1}\ell_{2}L,s_{1}s_{2}S,JM\big{\rangle}$

with $\hat{A}=\sqrt{2A+1}$. Thus, we obtain

$\displaystyle|\Xi_{c}\bar{D}(J=\frac{1}{2})\big{\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{2}|S_{c\bar{c}}=0,\,L_{sqq}=\frac{1}{2},\,J=\frac{1}{2}\big{\rangle}$
(35) $\displaystyle+$
$\displaystyle\frac{\sqrt{3}}{2}|S_{c\bar{c}}=1,\,L_{sqq}=\frac{1}{2},\,J=\frac{1}{2}\big{\rangle},$
$\displaystyle|\Xi_{c}\bar{D}^{*}(J=\frac{1}{2})\big{\rangle}$
$\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{2}|S_{c\bar{c}}=0,\,L_{sqq}=\frac{1}{2},\,J=\frac{1}{2}\big{\rangle}$
(36) $\displaystyle-$
$\displaystyle\frac{1}{2}|S_{c\bar{c}}=1,\,L_{sqq}=\frac{1}{2},\,J=\frac{1}{2}\big{\rangle},$
$\displaystyle|\Xi_{c}^{\prime}\bar{D}(J=\frac{1}{2})\big{\rangle}$
$\displaystyle=$
$\displaystyle\frac{1}{2}|S_{c\bar{c}}=0,\,L_{sqq}=\frac{1}{2},\,J=\frac{1}{2}\big{\rangle}$
(37) $\displaystyle-$
$\displaystyle\frac{1}{2\sqrt{3}}|S_{c\bar{c}}=1,\,L_{sqq}=\frac{1}{2},\,J=\frac{1}{2}\big{\rangle}$
$\displaystyle+$
$\displaystyle\sqrt{\frac{2}{3}}|S_{c\bar{c}}=1,\,L_{sqq}=\frac{3}{2},\,J=\frac{1}{2}\big{\rangle},$
$\displaystyle|\Xi_{c}^{\prime}\bar{D}^{*}(J=\frac{1}{2})\big{\rangle}$
$\displaystyle=$
$\displaystyle-\frac{1}{2\sqrt{3}}|S_{c\bar{c}}=0,\,L_{sqq}=\frac{1}{2},\,J=\frac{1}{2}\big{\rangle}$
(38) $\displaystyle+$
$\displaystyle\frac{5}{6}|S_{c\bar{c}}=1,\,L_{sqq}=\frac{1}{2},\,J=\frac{1}{2}\big{\rangle}$
$\displaystyle+$
$\displaystyle\frac{\sqrt{2}}{3}|S_{c\bar{c}}=1,\,L_{sqq}=\frac{3}{2},\,J=\frac{1}{2}\big{\rangle}.$

For these predicted isoscalar $S$-wave $\Xi_{c}^{(\prime)}\bar{D}^{(*)}$-type
hidden-charm molecular pentaquarks with single strangeness, the allowed two-
body hidden-charm decay channels include the $J/\psi\Lambda$ and
$\eta_{c}(1S)\Lambda$. In the heavy quark symmetry, we can estimate

$\displaystyle\mathcal{R}_{\text{HQS}}^{1}$ $\displaystyle=$
$\displaystyle\frac{\Gamma[\Xi_{c}\bar{D}(1/2^{-})\to\eta_{c}(1S)\Lambda]}{\Gamma[\Xi_{c}\bar{D}(1/2^{-})\to
J/\psi\Lambda]}=\frac{1}{3},$ $\displaystyle\mathcal{R}_{\text{HQS}}^{2}$
$\displaystyle=$
$\displaystyle\frac{\Gamma[\Xi_{c}\bar{D}^{*}(1/2^{-})\to\eta_{c}(1S)\Lambda]}{\Gamma[\Xi_{c}\bar{D}^{*}(1/2^{-})\to
J/\psi\Lambda]}=3,$ $\displaystyle\mathcal{R}_{\text{HQS}}^{3}$
$\displaystyle=$
$\displaystyle\frac{\Gamma[\Xi_{c}^{\prime}\bar{D}(1/2^{-})\to\eta_{c}(1S)\Lambda]}{\Gamma[\Xi_{c}^{\prime}\bar{D}(1/2^{-})\to
J/\psi\Lambda]}=3,$ $\displaystyle\mathcal{R}_{\text{HQS}}^{4}$
$\displaystyle=$
$\displaystyle\frac{\Gamma[\Xi_{c}^{\prime}\bar{D}^{*}(1/2^{-})\to\eta_{c}(1S)\Lambda]}{\Gamma[\Xi_{c}^{\prime}\bar{D}^{*}(1/2^{-})\to
J/\psi\Lambda]}=\frac{3}{25}.$

For the $S$-wave $\Xi_{c}\bar{D}^{*}$ and $\Xi_{c}^{\prime}\bar{D}^{*}$
molecular states with $J^{P}={3}/{2}^{-}$, the $J/\psi\Lambda$ is the two-body
hidden-charm decay channel via the $S$-wave coupling, while the
$\eta_{c}(1S)\Lambda$ channel is suppressed since it is a $D$-wave
interaction. By this effort, we can find several significant two-body hidden-
charm decay channels for these predicted hidden-charm molecular pentaquarks
with single strangeness, which may provide crucial information of searching
for these predicted pentaquarks in future experiment. Besides the strong decay
behavior of these predicted pentaquarks, their radiative decay behavior has
aroused theorist’s interest as given in Ref. Wang:2022tib .

##  IV Discussion and conclusion

Very recently, the LHCb Collaboration announced the observation of the
$P_{\psi s}^{\Lambda}(4338)$ in the $J/\psi\Lambda$ invariant mass spectrum of
the $B^{-}\to J/\psi\Lambda\bar{p}$ process Collaboration:2022boa . The
$P_{\psi s}^{\Lambda}(4338)$ associated with the previously reported evidence
of the $P_{cs}(4459)$ by LHCb shows a molecular-type characteristic mass
spectrum of hidden-charm pentaquark with strangeness. Before that, a
molecular-type characteristic mass spectrum of hidden-charm pentaquark was
observed by discovering the $P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$ in
the $J/\psi p$ invariant mass spectrum of the $\Lambda_{b}\to J/\psi pK$
Aaij:2019vzc . In this work, we indicate the similarity of two molecular-type
characteristic mass spectra mentioned above.

Figure 5: The possible evidence of exiting substructures contained in the
$P_{cs}(4459)$ enhancement structure. Here, the data of $J/\psi\Lambda$
invariant mass spectrum are from the LHCb LHCb:2020jpq . The red solid line is
the $\Xi_{c}\bar{D}^{*}$ threshold. Near this threshold, there should exist
two $\Xi_{c}\bar{D}^{*}$ molecular states with $J^{P}=1/2^{-}$ and $3/2^{-}$.
The event cluster in the yellow ban shows possible double-peak evidence but it
is not obvious, which can be tested by more precise data.

For quantitatively depicting this characteristic mass spectrum of hidden-charm
pentaquark with strangeness, we apply the OBE model to obtain the interactions
of the $\Xi_{c}\bar{D}^{(*)}$ systems, where the effective potentials of the
focused systems can be deduced. By solving the Schrödinger equation, we find
out the bound state solutions of the $\Xi_{c}\bar{D}^{(*)}$ pentaquark
systems. The numerical result shows the existence of two $\Xi_{c}\bar{D}^{*}$
molecular pentaquarks with $J^{P}=1/2^{-}$ and $3/2^{-}$, which can be related
to the $P_{cs}(4459)$ structure. In fact, there exists a similar situation to
the $P_{c}(4450)$, which was observed firstly by LHCb in 2015 Aaij:2015tga .
When reanalyzing more precise data of the $\Lambda_{b}\to J/\psi pK$ decay in
2019, LHCb indicated that the $P_{c}(4450)$ structure contains two
substructures $P_{c}(4440)$ and $P_{c}(4457)$ Aaij:2019vzc . We may conjecture
that the double-peak phenomenon can be happened to the $P_{cs}(4459)$
structure111In former experimental analysis LHCb:2020jpq , LHCb tried to fit
the enhancement structure corresponding to the $P_{cs}(4459)$ by two
substructures LHCb:2020jpq . Due to the limit of precision of data, LHCb
cannot make firm conclusion of the existence of substructure phenomenon
LHCb:2020jpq . as illustrated in Fig. 5. Thus, we strongly suggest the
experimental colleagues to reanalyze the $J/\psi\Lambda$ invariant mass
spectrum of $\Xi_{b}^{-}\to J/\psi\Lambda K$ based on more precise data
collected with the running of high-luminosity LHC.

If checking the resonant parameter of the reported $P_{\psi
s}^{\Lambda}(4338)$ Collaboration:2022boa , we cannot ignore a fact that the
central value of the $P_{\psi s}^{\Lambda}(4338)$ mass is a little bit larger
than the threshold of the $\Xi_{c}\bar{D}$ channel. Thus, it seems unnatural
to assign the $P_{\psi s}^{\Lambda}(4338)$ as the $\Xi_{c}\bar{D}$ molecular
state with $J^{P}=1/2^{-}$ since the $\Xi_{c}\bar{D}$ molecular state has
negative binding energy. As shown in Fig. 1, the mass gap between the
$P_{cs}(4459)$ and the $P_{\psi s}^{\Lambda}(4338)$ is smaller than that
between the $P_{c}$ states. If considering the similarity of the mass gap
between the $P_{c}$ states and that between the $P_{cs}$ states. Taking the
mass gap $137$ MeV involved in the $P_{c}$ states and the mass of the
$P_{cs}(4459)$ as input, the $P_{cs}$ partner of the $P_{c}(4312)$ should be
located at $4322$ MeV, which is lower than the observed $P_{\psi
s}^{\Lambda}(4338)$ by LHCb via the $B^{-}\to J/\psi\Lambda\bar{p}$ process.
Thus, with more precise data, we also suggest to check such scenario in future
experiment like LHCb. Here, a high-priority task is to confirm the observation
of the $P_{\psi s}^{\Lambda}(4338)$ by the $\Xi_{b}^{-}\to J/\psi\Lambda K$
process.

As a prediction, in this work we also give another molecular-type
characteristic mass spectrum of hidden-charm $\Xi_{c}^{\prime}\bar{D}^{(*)}$
pentaquark systems. Our result shows that there exist a
$\Xi_{c}^{\prime}\bar{D}$ molecular pentaquark with $J^{P}=1/2^{-}$ and two
$\Xi_{c}^{\prime}\bar{D}^{*}$ molecular pentaquarks with $J^{P}=1/2^{-}$ and
$3/2^{-}$, which are near the thresholds of $\Xi_{c}^{\prime}\bar{D}$ and
$\Xi_{c}^{\prime}\bar{D}^{*}$, respectively. Thus, searching for this
predicted molecular-type characteristic mass spectrum of the
$\Xi_{c}^{\prime}\bar{D}^{(*)}$ pentaquark systems will be a new task for
further experimental exploration of pentaquarks.

Note added.–When preparing the present paper, we noticed a similar work from
Karliner and Rosner Karliner:2022erb  appeared in arXiv. They also indicated
the similarity of the mass gaps for these reported $P_{c}$ and $P_{cs}$
states. Different from Ref. Karliner:2022erb , in this work we addressed a
dynamical calculation within the OBE model to illustrate the importance of the
molecular-type characteristic mass spectra of hidden-charm
$\Xi_{c}^{(\prime)}\bar{D}^{(*)}$ pentaquark systems.

## Acknowledgement

This work is supported by the China National Funds for Distinguished Young
Scientists under Grant No. 11825503, National Key Research and Development
Program of China under Contract No. 2020YFA0406400, the 111 Project under
Grant No. B20063, and the National Natural Science Foundation of China under
Grant No. 12175091.

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