# Possible open charm molecular pentaquarks from
$\Lambda_{c}K^{(*)}/\Sigma_{c}K^{(*)}$ interactions

Rui Chen1<EMAIL_ADDRESS>Qi Huang2111Corresponding author
<EMAIL_ADDRESS>1Key Laboratory of Low-Dimensional Quantum Structures and
Quantum Control of Ministry of Education, Department of Physics and Synergetic
Innovation Center for Quantum Effects and Applications, Hunan Normal
University, Changsha 410081, China
2Department of Physics, Nanjing Normal University, Nanjing 210023, China

###### Abstract

In this work, we adopt the one-boson-exchange model to study the
$Y_{c}K^{(*)}(Y_{c}=\Lambda_{c},\Sigma_{c})$ interactions. After considering
both of the $S-D$ wave mixing effects and the coupled channel effects, we can
predict several possible open-charm molecular pentaquarks, i.e., the single
$\Sigma_{c}K^{*}$ molecular states with $I(J^{P})=1/2(1/2^{-})$,
$1/2(3/2^{-})$ and $3/2(1/2^{-})$, the coupled
$\Lambda_{c}K^{*}/\Sigma_{c}K^{*}$ molecular states with $1/2(1/2^{-})$ and
$1/2(3/2^{-})$, and the coupled $\Sigma_{c}K/\Lambda_{c}K^{*}/\Sigma_{c}K^{*}$
molecular state with $1/2(1/2^{-})$. Meanwhile, we extend our study to the
$Y_{c}\bar{K}^{(*)}$ interactions, our results suggest the $\Sigma_{c}\bar{K}$
system with $I(J^{P})=1/2(1/2^{-})$, the $\Sigma_{c}\bar{K}^{*}$ systems with
$1/2(1/2^{-})$, $1/2(3/2^{-})$, and $3/2(3/2^{-})$, the coupled
$\Lambda_{c}\bar{K}^{*}/\Sigma_{c}\bar{K}^{*}$ system with $1/2(1/2^{-})$, and
the $\Sigma_{c}\bar{K}/\Lambda_{c}\bar{K}^{*}/\Sigma_{c}\bar{K}^{*}$ system
with $1/2(1/2^{-})$ can be the prime molecular candidates.

###### pacs:

12.39.Pn, 14.20.Pt, 13.75.Jz

##  I introduction

In the past decades, the observations of $X/Y/Z/P_{c}/T_{cc}$ structures have
stimulated theorist’s extensive interest in exploring the properties of exotic
states. Among the possible configurations, the hadronic molecular state, which
is composed of the color-singlet hadrons, plays an important role in
explaining the observed exotic structures. The main reason of introducing such
a configuration is that many observed $X/Y/Z/P_{c}/T_{cc}$ structures are near
some specific mass thresholds of the hadron pairs, which leads to answers
whether these observations can be explained under the framework of the
molecular state (one can see Refs. Chen:2016qju ; Liu:2019zoy ; Chen:2016spr ;
Guo:2017jvc ; Chen:2022asf  for a detailed review). Thus, carrying out the
study of the hadronic molecular state has became an active and important
research field in the hadron physics. It is not only helpful to reveal the
underlying structures of these near thresholds $X/Y/Z/P_{c}/T_{cc}$
structures, but also can improve our knowledge of the non-perturbative
behavior of the quantum chromodynamics (QCD).

Very recently, the LHCb collaboration continued to report their observations
of two open heavy flavor multiquark candidates, $T_{c\bar{s}}^{a0}(2900)$ and
$T_{c\bar{s}}^{a++}(2900)$, where the superscript $a$ means that their quantum
numbers are both $I(J^{P})=1(0^{+})$ LHCb:Tcc ; LHCb:Qian . For the
$T_{c\bar{s}}^{a0}(2900)$, the discovered channel is $D_{s}^{+}\pi^{-}$, the
mass and width are $2892\pm 14\pm 15$ MeV and $119\pm 26\pm 12$ MeV,
respectively, while for the $T_{c\bar{s}}^{a++}(2900)$, the discovered
channel, the mass, and the width are $D_{s}^{+}\pi^{+}$, $2921\pm 17\pm 19$
MeV and $137\pm 32\pm 14$ MeV, respectively. According to their channels, mass
positions and quantum numbers, it is easy to guess that the
$T_{c\bar{s}}^{a0}(2900)$ and $T_{c\bar{s}}^{a++}(2900)$ belong to the same
isovector triplet. Furthermore, the LHCb collaboration also determined their
averaged masses and decay widths, which are $2908\pm 11\pm 20$ MeV and $136\pm
23\pm 11$ MeV, respectively.

Due to the charged property of $T_{c\bar{s}}^{a0(++)}(2900)$, their minimal
valance quark components are naturally inferred to be $c\bar{s}q\bar{q}$
($q=u,~{}d$). Since they are very close to the $D^{*}K^{*}$ mass threshold, it
is natural conjecture whether the $T_{c\bar{s}}^{a0(++)}(2900)$ states can be
the isovector $D^{*}K^{*}$ molecules with $J^{P}=0^{+}$. In fact, in our
former work Chen:2016ypj , we can not only reproduce the $D_{s0}^{\ast}(2317)$
and $D_{s1}(2460)$ in the $S-$wave $DK$ and $D^{*}K$ molecular scenario, but
also find the one-boson-exchange (OBE) effective potentials are strong enough
to form loosely bound molecular states for the $D^{*}K^{*}$ systems with
$I(J^{P})=0(0^{+},1^{+},2^{+})$, and $1(0^{+})$. Therefore, the $D^{*}K^{*}$
hadronic molecular explanations for the $T_{c\bar{s}}^{a0(++)}(2900)$ states
cannot be excluded. In addition, there are other different theoretical
explanations to the $T_{c\bar{s}}^{a0(++)}(2900)$ states, like the compact
open-charm pentaquark Chen:2017rhl ; Guo:2021mja ; Cheng:2020nho  and the
$D^{*}\rho$ molecule Agaev:2022duz .

Besides the $T_{c\bar{s}}^{a0(++)}(2900)$, another two open-charm states
$X_{0}(2900)$ and $X_{1}(2900)$, which were observed by the LHCb collaboration
in the $D^{-}K^{+}$ final states of the $B^{+}\to D^{+}D^{-}K^{+}$ decay
process LHCb:2020bls ; LHCb:2020pxc , are also interesting. Their spin-
parities $J^{P}$ are $0^{+}$ and $1^{+}$, respectively. Because their mass
positions are very close to the $\bar{D}^{*}K^{*}$ and $\bar{D}_{1}K$ mass
thresholds, respectively, many theorists propose the $X_{0}(2900)$ and
$X_{1}(2900)$ states as the hadronic molecular states Molina:2020hde ;
Hu:2020mxp ; Liu:2020nil ; Kong:2021ohg ; Wang:2021lwy ; Xiao:2020ltm ;
Huang:2020ptc ; Chen:2020aos ; Agaev:2020nrc ; Qi:2021iyv ; Chen:2021tad . At
present, the inner structures for the $T_{c\bar{s}}^{a0(++)}(2900)$ and
$X_{0,1}(2900)$ are still on discussion (one can see Refs. Chen:2022asf ).

As is well known, the light diquark in the heavy baryons
$Y_{c}=(\Lambda_{c},\Sigma_{c})$ has the same color structure $\bar{3}_{c}$
with the light anti-quark in the heavy meson $Q\bar{q}$ An:2022vtg . If the
$T_{c\bar{s}}^{a0(++)}(2900)$ can be assigned as the loosely bound hadronic
molecular states composed by the charmed meson and kaon, it is natural to
conjecture whether there exist possible open charm molecular pentaquarks
counterpart of the $T_{c\bar{s}}^{a0(++)}(2900)$, which are near the
thresholds of the $\Lambda_{c}K^{(*)}$ and $\Sigma_{c}K^{(*)}$, respectively.
In this work, we search for such open charm molecular partners composed by
$\Lambda_{c}K^{(*)}$ and $\Sigma_{c}K^{(*)}$, which can not only enrich the
family of the exotic states, but also help us to understand the nature of the
newly $T_{c\bar{s}}^{a0(++)}(2900)$.

Apart from searching for possible $\Lambda_{c}K^{(*)}$ and $\Sigma_{c}K^{(*)}$
molecular states, in this work, we also study the interactions between the
$S-$wave charmed baryon $Y_{c}=(\Lambda_{c},\Sigma_{c})$ and the anti-strange
meson $\bar{K}^{(*)}$ by adopting the OBE model and considering both of the
S-D mixing effects and the coupled channel effects. After solving the coupled
channel Schrödinger equations, we can search for the possible charmed-strange
molecular pentaquarks counterpart of the $X_{0,1}(2900)$. Our study will not
only provide valuable information to experimental search for exotic open charm
hadronic molecular pentaquarks, but also give indirect test of molecular state
picture for the $T_{c\bar{s}}^{a0(++)}(2900)$ and $X_{0,1}(2900)$.

This paper is organized as follows. After this introduction, we introduce the
relevant effective Lagrangians and the OBE model in Sec. II. In Sec. III, we
present the OBE effective potentials and the corresponding numerical results.
The paper ends with a summary in Sec. IV.

##  II Lagrangians and OBE model

In this work, we deduce the OBE effective potentials for the $Y_{c}K^{(*)}$
systems by employing the effective Lagrangian approach at the hadronic level.
The relevant Lagrangians describing the interactions between the heavy baryons
and light mesons are constructed in terms of the heavy quark limit and chiral
symmetry Liu:2011xc , i.e.,

$\displaystyle\mathcal{L}_{\mathcal{B}_{\bar{3}}}$ $\displaystyle=$
$\displaystyle
l_{B}\langle\bar{\mathcal{B}}_{\bar{3}}\sigma\mathcal{B}_{\bar{3}}\rangle+i\beta_{B}\langle\bar{\mathcal{B}}_{\bar{3}}v^{\mu}(\mathcal{V}_{\mu}-\rho_{\mu})\mathcal{B}_{\bar{3}}\rangle,$
(1) $\displaystyle\mathcal{L}_{\mathcal{B}_{6}}$ $\displaystyle=$
$\displaystyle
l_{S}\langle\bar{\mathcal{S}}_{\mu}\sigma\mathcal{S}^{\mu}\rangle-\frac{3}{2}g_{1}\varepsilon^{\mu\nu\lambda\kappa}v_{\kappa}\langle\bar{\mathcal{S}}_{\mu}A_{\nu}\mathcal{S}_{\lambda}\rangle$
(2)
$\displaystyle+i\beta_{S}\langle\bar{\mathcal{S}}_{\mu}v_{\alpha}\left(\mathcal{V}_{ab}^{\alpha}-\rho_{ab}^{\alpha}\right)\mathcal{S}^{\mu}\rangle+\lambda_{S}\langle\bar{\mathcal{S}}_{\mu}F^{\mu\nu}(\rho)\mathcal{S}_{\nu}\rangle,~{}$
$\displaystyle\mathcal{L}_{\mathcal{B}_{\bar{3}}\mathcal{B}_{6}}$
$\displaystyle=$ $\displaystyle
ig_{4}\langle\bar{\mathcal{S}^{\mu}}A_{\mu}\mathcal{B}_{\bar{3}}\rangle+i\lambda_{I}\varepsilon^{\mu\nu\lambda\kappa}v_{\mu}\langle\bar{\mathcal{S}}_{\nu}F_{\lambda\kappa}\mathcal{B}_{\bar{3}}\rangle+h.c..$
(3)

Here, $v=(1,\textbf{0})$ is the four velocity,
$\rho_{ba}^{\mu}=ig_{V}{V}_{ba}^{\mu}/\sqrt{2}$, and
$F^{\mu\nu}(\rho)=\partial^{\mu}\rho^{\nu}-\partial^{\nu}\rho^{\mu}+\left[\rho^{\mu},\rho^{\nu}\right]$.
$A_{\mu}$ and $\mathcal{V}_{\mu}$ stand for the axial current and vector
current, respectively. They can be written as

$\displaystyle A_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\xi^{{\dagger}}\partial_{\mu}\xi-\xi\partial_{\mu}\xi^{{\dagger}})=\frac{i}{f_{\pi}}\partial_{\mu}{P}+\ldots,$
$\displaystyle\mathcal{V}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\xi^{{\dagger}}\partial_{\mu}\xi+\xi\partial_{\mu}\xi^{{\dagger}})=\frac{i}{2f_{\pi}^{2}}\left[{P},\partial_{\mu}{P}\right]+\ldots,$

respectively. Here, $\xi=\text{exp}(i{P}/f_{\pi})$ and $f_{\pi}=132$ MeV.
$\mathcal{B}_{\bar{3}}$ and
$\mathcal{S}_{\mu}=-\sqrt{\frac{1}{3}}(\gamma_{\mu}+v_{\mu})\gamma^{5}\mathcal{B}_{6}+\mathcal{B}_{6\mu}^{*}$
denote the ground heavy baryons multiplets with their light quarks in the
$\bar{3}$ and $6$ flavor representation, respectively. The matrices
$\mathcal{B}_{\bar{3}}$, $\mathcal{B}_{6}$, ${P}$, and ${V}$ read as

$\displaystyle\left.\begin{array}[]{ll}\mathcal{B}_{\bar{3}}=\left(\begin{array}[]{ccc}0&\Lambda_{c}^{+}\\\
-\Lambda_{c}^{+}&0\end{array}\right),&\mathcal{B}_{6}=\left(\begin{array}[]{ccc}\Sigma_{c}^{++}&\frac{\Sigma_{c}^{+}}{\sqrt{2}}\\\
\frac{\Sigma_{c}^{+}}{\sqrt{2}}&\Sigma_{c}^{0}\end{array}\right),\\\
{P}=\left(\begin{array}[]{cc}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&\pi^{+}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}\end{array}\right),&{V}=\left(\begin{array}[]{cc}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&\rho^{+}\\\
\rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}\end{array}\right).\end{array}\right.$

The effective Lagrangians describing the interactions between the strange
mesons and light mesons are constructed in the $SU(3)$ symmetry Lin:1999ad ;
Nagahiro:2008mn , i.e.,

$\displaystyle\mathcal{L}_{PPV}$ $\displaystyle=$
$\displaystyle\frac{ig}{2\sqrt{2}}\langle\partial^{\mu}P\left(PV_{\mu}-V_{\mu}P\right\rangle,$
(7) $\displaystyle\mathcal{L}_{VVP}$ $\displaystyle=$
$\displaystyle\frac{g_{VVP}}{\sqrt{2}}\epsilon^{\mu\nu\alpha\beta}\left\langle\partial_{\mu}V_{\nu}\partial_{\alpha}V_{\beta}P\right\rangle,$
(8) $\displaystyle\mathcal{L}_{VVV}$ $\displaystyle=$
$\displaystyle\frac{ig}{2\sqrt{2}}\langle\partial^{\mu}V^{\nu}\left(V_{\mu}V_{\nu}-V_{\nu}V_{\mu}\right)\rangle.$
(9)

After expanding Eqs. (1)-(9), we can further obtain

$\displaystyle\mathcal{L}_{\sigma}$ $\displaystyle=$ $\displaystyle
l_{B}\langle\bar{\mathcal{B}}_{\bar{3}}\sigma\mathcal{B}_{\bar{3}}\rangle-
l_{S}\langle\bar{\mathcal{B}}_{6}\sigma\mathcal{B}_{6}\rangle,$ (10)
$\displaystyle\mathcal{L}_{{P}}$ $\displaystyle=$ $\displaystyle
i\frac{g_{1}}{2f_{\pi}}\varepsilon^{\mu\nu\lambda\kappa}v_{\kappa}\langle\bar{\mathcal{B}}_{6}\gamma_{\mu}\gamma_{\lambda}\partial_{\nu}{P}\mathcal{B}_{6}\rangle$
(11)
$\displaystyle-\sqrt{\frac{1}{3}}\frac{g_{4}}{f_{\pi}}\langle\bar{\mathcal{B}}_{6}\gamma^{5}\left(\gamma^{\mu}+v^{\mu}\right)\partial_{\mu}{P}\mathcal{B}_{\bar{3}}\rangle+h.c.,$
$\displaystyle\mathcal{L}_{{V}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\beta_{B}g_{V}\langle\bar{\mathcal{B}}_{\bar{3}}v\cdot{V}\mathcal{B}_{\bar{3}}\rangle-\frac{\beta_{S}g_{V}}{\sqrt{2}}\langle\bar{\mathcal{B}}_{6}v\cdot{V}\mathcal{B}_{6}\rangle$
(12)
$\displaystyle-\frac{\lambda_{I}g_{V}}{\sqrt{6}}\varepsilon^{\mu\nu\lambda\kappa}v_{\mu}\langle\bar{\mathcal{B}}_{6}\gamma^{5}\gamma_{\nu}\left(\partial_{\lambda}{V}_{\kappa}-\partial_{\kappa}{V}_{\lambda}\right)\mathcal{B}_{\bar{3}}\rangle+h.c.$
$\displaystyle-i\frac{\lambda
g_{V}}{3\sqrt{2}}\langle\bar{\mathcal{B}}_{6}\gamma_{\mu}\gamma_{\nu}\left(\partial^{\mu}{V}^{\nu}-\partial^{\nu}{V}^{\mu}\right)\mathcal{B}_{6}\rangle,$
$\displaystyle\mathcal{L}_{K^{(*)}K^{(*)}\sigma}$ $\displaystyle=$
$\displaystyle g_{\sigma}m_{K}\bar{K}K\sigma-
g_{\sigma}m_{K^{*}}\bar{K}^{*}\cdot K^{*}\sigma,$ (13)
$\displaystyle\mathcal{L}_{PKK^{*}}$ $\displaystyle=$
$\displaystyle\frac{ig}{4}\left[\left(\bar{K}^{*\mu}K-\bar{K}K^{*\mu}\right)\left(\bm{\tau}\cdot\partial_{\mu}\bm{\pi}+\frac{\partial_{\mu}{\eta}}{\sqrt{3}}\right)\right.$
(14)
$\displaystyle\left.+\left(\partial_{\mu}\bar{K}K^{*\mu}-\bar{K}^{*\mu}\partial_{\mu}K\right)\left(\bm{\tau}\cdot\bm{\pi}+\frac{\eta}{\sqrt{3}}\right)\right],$
$\displaystyle\mathcal{L}_{{V}KK}$ $\displaystyle=$
$\displaystyle\frac{ig}{4}\left[\bar{K}\partial_{\mu}K-\partial_{\mu}\bar{K}K\right]\left(\bm{\tau}\cdot\bm{\rho}^{\mu}+{\omega}^{\mu}\right),$
(15) $\displaystyle\mathcal{L}_{{V}K^{*}K^{*}}$ $\displaystyle=$
$\displaystyle\frac{ig}{4}\left[\left(\bar{K}_{\mu}^{*}\partial^{\mu}K^{*\nu}-\partial^{\mu}\bar{K}^{*\nu}K_{\mu}^{*}\right)\left(\bm{\tau}\cdot\bm{\rho}_{\nu}+\omega_{\nu}\right)\right.$
(16)
$\displaystyle\left.+\left(\partial^{\mu}\bar{K}^{*\nu}K_{\nu}^{*}-\bar{K}_{\nu}^{*}\partial^{\mu}K^{*\nu}\right)\left(\bm{\tau}\cdot\bm{\rho}_{\mu}+\omega_{\mu}\right)\right.$
$\displaystyle\left.+\left(\bar{K}_{\nu}^{*}K^{*}_{\mu}-\bar{K}_{\mu}^{*}K^{*}_{\nu}\right)\left(\bm{\tau}\cdot\partial^{\mu}\bm{\rho}^{\nu}+\partial^{\mu}\omega^{\nu}\right)\right],$
$\displaystyle\mathcal{L}_{PK^{*}K^{*}}$ $\displaystyle=$ $\displaystyle
g_{VVP}\varepsilon_{\mu\nu\alpha\beta}\partial^{\mu}\bar{K}^{*\nu}\partial^{\alpha}K^{*\beta}\left(\bm{\tau}\cdot\bm{\pi}+\frac{\eta}{\sqrt{3}}\right),$
(17) $\displaystyle\mathcal{L}_{VKK^{*}}$ $\displaystyle=$ $\displaystyle
g_{VVP}\varepsilon_{\mu\nu\alpha\beta}\left(\partial^{\mu}\bar{K}^{*\nu}K+\bar{K}\partial^{\mu}{K}^{*\nu}\right)$
(18)
$\displaystyle\left(\bm{\tau}\cdot\partial^{\alpha}\bm{\rho}^{\beta}+\partial^{\alpha}{\omega}^{\beta}\right).$

Coupling constants in the above Lagrangians are estimated with the quark model
Liu:2011xc ; Chen:2017xat , $l_{S}=-2l_{B}=7.3$,
$g_{1}=(\sqrt{8}/3)g_{4}=1.0$, $\beta_{S}g_{V}=-2\beta_{B}g_{V}=12.0$,
$\lambda_{S}g_{V}=-2\sqrt{2}\lambda_{I}g_{V}=19.2~{}\text{GeV}^{-1}$,
$g_{\sigma}=-3.65$, and $g=12.00$. $g_{VVP}=3g^{2}/(32\sqrt{2}\pi^{2}f_{\pi})$
Kaymakcalan:1983qq .

With these prepared effective Lagrangians, we can easily write down the
scattering amplitudes for the $B_{1}M_{2}\to B_{3}M_{4}$ processes in the
$t-$channel, where $B_{1}$ and $B_{3}$ stand for the initial and final
baryons, respectively, and $M_{2}$ and $M_{4}$ stand for the inial and final
mesons, respectively. The corresponding effective potentials can be related to
the scattering amplitudes by the Breit approximation,

$\displaystyle\mathcal{V}_{E}^{B_{1}M_{2}\to B_{3}M_{4}}(\bm{q})$
$\displaystyle=$ $\displaystyle-\frac{\mathcal{M}(B_{1}M_{2}\to
B_{3}M_{4})}{4\sqrt{m_{B_{1}}m_{M_{2}}m_{B_{3}}m_{M_{4}}}}.$ (19)

Here, $m_{i}$ is the mass of the interaction hadron.
$\mathcal{M}(B_{1}M_{2}\to B_{3}M_{4})$ denotes the scattering amplitude for
the $B_{1}M_{2}\to B_{3}M_{4}$ process by exchanging the light mesons
($\sigma$, $\pi$, $\eta$, $\rho$, and $\omega$). Next, we perform the Fourier
transformation to obtain the effective potentials in the coordinate space
$\mathcal{V}(\bm{r})$,

$\displaystyle\mathcal{V}_{E}(\bm{r})=\int\frac{d^{3}\bm{q}}{(2\pi)^{3}}e^{i\bm{q}\cdot\bm{r}}\mathcal{V}_{E}(\bm{q})\mathcal{F}^{2}(q^{2},m_{E}^{2}).$

In order to compensate the off-shell effect of the exchanged meson, we
introduce a monopole form factor
$\mathcal{F}(q^{2},m_{E}^{2})=(\Lambda^{2}-m_{E}^{2})/(\Lambda^{2}-q^{2})$ at
every interactive vertex, where $\Lambda$, $m_{E}$, and $q$ are the cutoff
parameter, the mass and four-momentum of the exchanged meson, respectively. In
our numerical calculations, we vary the cutoff value in the range of
$0.8\leq\Lambda\leq 5.0$ GeV. According to the deuteron experience
Tornqvist:1993ng ; Tornqvist:1993vu , the reasonable cutoff value is taken
around 1.00 GeV. In the following discussion, the loosely bound state with the
cutoff value around 1.00 GeV can be recommended as the prime hadronic
molecular candidate.

For the $\Lambda_{c}K^{(*)}$ systems, the flavor wave function
$|I,I_{3}\rangle$ can be expressed as
$|1/2,1/2\rangle=|\Lambda_{c}^{+}K^{(*)+}\rangle$ and
$|1/2,-1/2\rangle=|\Lambda_{c}^{+}K^{(*)0}\rangle$. For the
$\Sigma_{c}K^{(*)}$ systems, their isospin $I$ can be taken as $1/2$ or $3/2$.
The corresponding flavor wave functions $|I,I_{3}\rangle$ are

$\displaystyle\begin{array}[]{c}\left|\frac{1}{2},\frac{1}{2}\right\rangle=\sqrt{\frac{2}{3}}\left|\Sigma_{c}^{++}{K}^{(*)0}\right\rangle-\frac{1}{\sqrt{3}}\left|\Sigma_{c}^{+}{K}^{(*)+}\right\rangle,\\\
\left|\frac{1}{2},-\frac{1}{2}\right\rangle=\frac{1}{\sqrt{3}}\left|\Sigma_{c}^{+}{K}^{(*)0}\right\rangle-\sqrt{\frac{2}{3}}\left|\Sigma_{c}^{0}{K}^{(*)+}\right\rangle,\end{array}$
$\displaystyle\begin{array}[]{l}\left|\frac{3}{2},\frac{3}{2}\right\rangle=\left|\Sigma_{c}^{++}{K}^{(*)+}\right\rangle,\\\
\left|\frac{3}{2},\frac{1}{2}\right\rangle=\frac{1}{\sqrt{3}}\left|\Sigma_{c}^{++}{K}^{(*)0}\right\rangle+\sqrt{\frac{2}{3}}\left|\Sigma_{c}^{+}{K}^{(*)+}\right\rangle,\\\
\left|\frac{3}{2},-\frac{1}{2}\right\rangle=\sqrt{\frac{2}{3}}\left|\Sigma_{c}^{+}{K}^{(*)0}\right\rangle+\frac{1}{\sqrt{3}}\left|\Sigma_{c}^{0}{K}^{(*)+}\right\rangle,\\\
\left|\frac{3}{2},-\frac{3}{2}\right\rangle=\left|\Sigma_{c}^{0}{K}^{(*)0}\right\rangle,\end{array}$

respectively. When we consider the $S-D$ wave mixing effects, the spin-orbit
wave functions $|{}^{2S+1}L_{J}\rangle$ are

$\displaystyle\left.\begin{array}[]{llll}Y_{c}K[J^{P}=1/2^{-}]:&|{}^{2}S_{1/2}\rangle,\\\
Y_{c}K^{*}[J^{P}=1/2^{-}]:&|{}^{2}S_{1/2}\rangle,&|{}^{4}D_{1/2}\rangle,\\\
Y_{c}K^{*}[J^{P}=3/2^{-}]:&|{}^{4}S_{3/2}\rangle,&|{}^{2}D_{3/2}\rangle,&|{}^{4}D_{3/2}\rangle.\end{array}\right.$
(25)

The general expressions of the spin-orbit wave functions
$|{}^{2S+1}L_{J}\rangle$ for the $Y_{c}K^{(*)}$ systems read as

$\displaystyle Y_{c}K:\,\left|{}^{2S+1}L_{J}\right\rangle$ $\displaystyle=$
$\displaystyle\sum_{m_{S},m_{L}}C^{J,M}_{\frac{1}{2}m_{S},Lm_{L}}\chi_{\frac{1}{2}m}|Y_{L,m_{L}}\rangle,$
$\displaystyle Y_{c}{K}^{*}:\left|{}^{2S+1}L_{J}\right\rangle$
$\displaystyle=$
$\displaystyle\sum_{m,m^{\prime}}^{m_{S},m_{L}}C^{S,m_{S}}_{\frac{1}{2}m,1m^{\prime}}C^{J,M}_{Sm_{S},Lm_{L}}\chi_{\frac{1}{2}m}\epsilon^{m^{\prime}}|Y_{L,m_{L}}\rangle.$

Here, $C^{J,M}_{\frac{1}{2}m_{S},Lm_{L}}$,
$C^{S,m_{S}}_{\frac{1}{2}m,1m^{\prime}}$, and $C^{J,M}_{Sm_{S},Lm_{L}}$ are
the Clebsch-Gordan coefficients. $\chi_{\frac{1}{2}m}$ and $Y_{L,m_{L}}$ stand
for the spin wave function and the spherical harmonics function, respectively.
$\epsilon$ is the polarization vector for the vector meson with
$\epsilon_{\pm}^{m}=\mp\frac{1}{\sqrt{2}}\left(\epsilon_{x}^{m}{\pm}i\epsilon_{y}^{m}\right)$
and $\epsilon_{0}^{m}=\epsilon_{z}^{m}$, which satisfies $\epsilon_{\pm
1}=\frac{1}{\sqrt{2}}\left(0,\pm 1,i,0\right)$ and
$\epsilon_{0}=\left(0,0,0,-1\right)$.

##  III The OBE effective potentials and the numerical results

Following the above procedures, we can deduce the concrete OBE effective
potentials for the $Y_{c}K^{(*)}$ systems with different quantum
configurations. After that, we adopt the obtained OBE effective potentials to
solve the coupled channel Schrödinger equations. By doing this, we can search
for the bound state solutions. A system with the reasonable bound state
solutions can be recommended as the good hadronic molecular candidate, where
the binding energy is taken from several MeV to several tens MeV, and the
root-mean-square (RMS) radius is a few fm or larger.

###  III.1 The $\Lambda_{c}K^{(*)}$ systems

The total OBE effective potentials for the single $\Lambda_{c}K$ system can be
written as

$\displaystyle V_{\Lambda_{c}K\to\Lambda_{c}K}$ $\displaystyle=$
$\displaystyle
l_{B}g_{\sigma}\chi_{3}^{{\dagger}}\chi_{1}Y(\Lambda,m_{\sigma},r)$ (26)
$\displaystyle+\frac{\beta_{B}g_{V}g}{4}\chi_{3}^{{\dagger}}\chi_{1}Y(\Lambda,m_{\omega},r).$

Here, we define

$\displaystyle Y(\Lambda,m,{r})$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
r}(e^{-mr}-e^{-\Lambda r})-\frac{\Lambda^{2}-m^{2}}{8\pi\Lambda}e^{-\Lambda
r}.$ (27)

As shown in Eq. (26), there exist the $\sigma$ exchange and $\omega$ exchange
interactions, which contribute in the intermediate range and the short range,
respectively. The $\sigma$ exchange provides an attractive interaction,
whereas the $\omega$ exchange interaction is repulsive. Here, the $\rho$
exchange interaction is strongly suppressed as the isospin forbidden in the
$\Lambda_{c}-\Lambda_{c}-\rho$ coupling. Since the $KK\pi(\eta)$ coupling is
forbidden by the spin-parity conservation, the pseudoscalr meson $(\pi/\eta)$
exchanges interactions are strongly suppressed, either.

After solving the Schrödinger equation, we don’t find bound state solutions in
the cutoff region $0.8\leq\Lambda\leq 5.0$ GeV. Thus, the OBE effective
potentials for the $\Lambda_{c}K$ system is not strong enough to bind a bound
state.

For the single $S-$wave $\Lambda_{c}K^{*}$ systems with $J^{P}=1/2^{-}$ and
$3/2^{-}$, their OBE effective potentials are the same, i.e.,

$\displaystyle V_{\Lambda_{c}K^{*}\to\Lambda_{c}K^{*}}$ $\displaystyle=$
$\displaystyle
l_{B}g_{\sigma}(\bm{\epsilon}_{2}\cdot\bm{\epsilon}_{4}^{{\dagger}})\chi_{3}^{{\dagger}}\chi_{1}Y(\Lambda,m_{\sigma},r)$
(28)
$\displaystyle+\frac{\beta_{B}g_{V}g}{4}(\bm{\epsilon}_{2}\cdot\bm{\epsilon}_{4}^{{\dagger}})\chi_{3}^{{\dagger}}\chi_{1}Y(\Lambda,m_{\omega},r).$

When we consider the $S-D$ wave mixing effects, the operator
$\bm{\epsilon}_{2}\cdot\bm{\epsilon}_{4}^{{\dagger}}$ will be replaced by the
unit matrix
$\mathcal{I}=\langle{}^{2S^{\prime}+1}L^{\prime}_{J^{\prime}}|\bm{\epsilon}_{2}\cdot\bm{\epsilon}_{4}^{{\dagger}}|{}^{2S+1}L_{J}\rangle$
in the numerical calculations, which indicates the OBE effective potentials
are the exactly the same with those for the $\Lambda_{c}K$ system with
$1/2^{-}$. In the cutoff region $0.8\leq\Lambda\leq 5.0$ GeV, we cannot find
the bound state solutions, either.

In this work, we further perform the coupled channel analysis on the
$\Lambda_{c}K^{*}/\Sigma_{c}K^{*}$ interactions, the corresponding OBE
effective potentials are

$\displaystyle V_{\Lambda_{c}K^{*}}^{\text{C}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}V_{\Lambda_{c}K^{*}\to\Lambda_{c}K^{*}}&V_{\Sigma_{c}K^{*}\to\Lambda_{c}K^{*}}\\\
V_{\Lambda_{c}K^{*}\to\Sigma_{c}K^{*}}&V_{\Sigma_{c}K^{*}\to\Sigma_{c}K^{*}}\end{array}\right),$
(31)

with

$\displaystyle V_{\Lambda_{c}K^{*}\to\Sigma_{c}K^{*}}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\frac{g_{4}g_{VVP}}{f_{\pi}}\mathcal{F}_{1}(r,\bm{\sigma},i\bm{\epsilon}_{2}\times\bm{\epsilon}_{4}^{{\dagger}})Y(\Lambda_{0},m_{\pi
0},r)$
$\displaystyle-\frac{1}{6\sqrt{2}}\frac{\lambda_{I}g_{V}g}{m_{K^{*}}}\mathcal{F}_{2}(r,\bm{\sigma},i\bm{\epsilon}_{2}\times\bm{\epsilon}_{4}^{{\dagger}})Y(\Lambda_{0},m_{\rho
0},r),$ $\displaystyle V_{\Sigma_{c}K^{*}\to\Sigma_{c}K^{*}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}l_{S}g_{\sigma}\chi_{3}^{{\dagger}}\chi_{1}\bm{\epsilon}_{2}\cdot\bm{\epsilon}_{3}^{{\dagger}}Y(\Lambda,m_{\sigma},r)$
(33)
$\displaystyle-\frac{g_{1}g_{VVP}}{6\sqrt{2}f_{\pi}}\mathcal{F}_{1}(r,\bm{\sigma},i\bm{\epsilon}_{2}\times\bm{\epsilon}_{4}^{{\dagger}})\mathcal{G}(I)Y(\Lambda,m_{\pi},r)$
$\displaystyle-\frac{g_{1}g_{VVP}}{18\sqrt{2}f_{\pi}}\mathcal{F}_{1}(r,\bm{\sigma},i\bm{\epsilon}_{2}\times\bm{\epsilon}_{4}^{{\dagger}})Y(\Lambda,m_{\eta},r)$
$\displaystyle+\frac{1}{8}\beta_{S}g_{V}g\chi_{3}^{{\dagger}}\chi_{1}\bm{\epsilon}_{2}\cdot\bm{\epsilon}_{3}^{{\dagger}}\mathcal{G}(I)Y(\Lambda,m_{\rho},r)$
$\displaystyle+\frac{\lambda_{S}g_{V}g}{8\sqrt{3}m_{\Sigma_{c}}}\chi_{3}^{{\dagger}}\chi_{1}\bm{\epsilon}_{2}\cdot\bm{\epsilon}_{3}^{{\dagger}}\mathcal{G}(I)\nabla^{2}Y(\Lambda,m_{\rho},r)$
$\displaystyle-\frac{\lambda_{S}g_{V}g}{24\sqrt{3}m_{K^{*}}}\mathcal{F}_{2}(r,\bm{\sigma},i\bm{\epsilon}_{2}\times\bm{\epsilon}_{4}^{{\dagger}})\mathcal{G}(I)Y(\Lambda,m_{\rho},r)$
$\displaystyle+\frac{1}{8}\beta_{S}g_{V}g\chi_{3}^{{\dagger}}\chi_{1}\bm{\epsilon}_{2}\cdot\bm{\epsilon}_{3}^{{\dagger}}Y(\Lambda,m_{\omega},r)$
$\displaystyle+\frac{\lambda_{S}g_{V}g}{8\sqrt{3}m_{\Sigma_{c}}}\chi_{3}^{{\dagger}}\chi_{1}\bm{\epsilon}_{2}\cdot\bm{\epsilon}_{3}^{{\dagger}}\nabla^{2}Y(\Lambda,m_{\omega},r)$
$\displaystyle-\frac{\lambda_{S}g_{V}g}{24\sqrt{3}m_{K^{*}}}\mathcal{F}_{2}(r,\bm{\sigma},i\bm{\epsilon}_{2}\times\bm{\epsilon}_{4}^{{\dagger}})Y(\Lambda,m_{\omega},r).$

Here, the value of the isospin factor $\mathcal{G}(I)$ is taken as
$\mathcal{G}(I=1/2)=-2$, $\mathcal{G}(I=3/2)=1$. The variables in Eq. (III.1)
are $\Lambda_{0}^{2}=\Lambda^{2}-q_{0}^{2}$, $m_{{\pi
0}}^{2}=m_{\pi}^{2}-q_{0}^{2}$, $m_{\rho 0}^{2}=m_{\rho}^{2}-q_{0}^{2}$ with
$q_{0}=\frac{M_{\Sigma_{c}}^{2}-M_{\Lambda_{c}}^{2}}{2(M_{\Sigma_{c}}+M_{K^{*}})}$.
And we define useful operators, i.e.,

$\displaystyle\mathcal{F}_{1}(r,\bm{a},\bm{b})$ $\displaystyle=$
$\displaystyle\chi_{3}^{{\dagger}}\left(\bm{a}\cdot\bm{b}\nabla^{2}+S(\hat{r},\bm{a},\bm{b})r\frac{\partial}{\partial
r}\frac{1}{r}\frac{\partial}{\partial r}\right)\chi_{1},$ (34)
$\displaystyle\mathcal{F}_{2}(r,\bm{a},\bm{b})$ $\displaystyle=$
$\displaystyle\chi_{3}^{{\dagger}}\left(2\bm{a}\cdot\bm{b}\nabla^{2}-S(\hat{r},\bm{a},\bm{b})r\frac{\partial}{\partial
r}\frac{1}{r}\frac{\partial}{\partial r}\right)\chi_{1}.$ (35)

Here, the $\bm{a}\cdot\bm{b}$ and $S(\hat{r},\bm{a},\bm{b})$ stand for the
spin-spin interactions and the tensor force operators, respectively. The
corresponding matrices elements can be obtained by sandwiched between the
spin-orbit wave functions as presented in the Eq. (25), i.e.,

$\displaystyle
i\bm{\sigma}\cdot(\bm{\epsilon}_{2}\times\bm{\epsilon}_{4}^{{\dagger}})$
$\displaystyle\mapsto$
$\displaystyle\left\\{\begin{array}[]{ll}\left(\begin{array}[]{cc}-2&0\\\
0&1\end{array}\right),&J^{P}=1/2^{-}\\\ \left(\begin{array}[]{ccc}1&0&0\\\
0&-2&0\\\ 0&0&1\end{array}\right),&J^{P}=3/2^{-}\end{array}\right.$ (43)
$\displaystyle
S(\hat{r},\bm{\sigma},i\bm{\epsilon}_{2}\times\bm{\epsilon}_{4}^{{\dagger}})$
$\displaystyle\mapsto$
$\displaystyle\left\\{\begin{array}[]{ll}\left(\begin{array}[]{cc}0&-\sqrt{2}\\\
-\sqrt{2}&-2\end{array}\right),&J^{P}=1/2^{-}\\\
\left(\begin{array}[]{ccc}0&1&2\\\ 1&0&-1\\\
2&-1&0\end{array}\right),&J^{P}=3/2^{-}\end{array}\right.$ (51)

Table 1: The bound state solutions (the binding energy $E$, the root-mean-square radius $r_{RMS}$, and the probabilities $P_{i}(\%)$ for all the discussed channels) for the coupled $\Lambda_{c}K^{*}/\Sigma_{c}K^{*}$ systems with $I(J^{P})=1/2(1/2^{-})$ and $1/2(3/2^{-})$. Here, $E$, $r_{RMS}$, and $\Lambda$ are in units of MeV, fm, and GeV, respectively. The dominant channels are labeled in a bold manner. $I(J^{P})$ | $\Lambda$ | $E$ | $r_{RMS}$ | $\Lambda_{c}K^{*}({}^{2}S_{1/2})$ | $\Lambda_{c}K^{*}({}^{4}D_{1/2})$ | $\Sigma_{c}K^{*}({}^{2}S_{1/2})$ | $\Sigma_{c}K^{*}({}^{4}D_{1/2})$ |  |
---|---|---|---|---|---|---|---|---|---
$1/2(1/2^{-})$ | 1.56 | $-0.14$ | 6.11 | 98.82 | $\sim 0$ | 1.14 | 0.04 |  |
| 1.58 | $-2.14$ | 2.62 | 97.11 | 0.01 | 2.82 | 0.06 |  |
| 1.60 | $-6.02$ | 1.56 | 95.12 | 0.02 | 4.79 | 0.07 |  |
| 1.62 | $-11.57$ | 1.12 | 93.18 | 0.03 | 6.72 | 0.07 |  |
$I(J^{P})$ | $\Lambda$ | $E$ | $r_{RMS}$ | $\Lambda_{c}K^{*}({}^{4}S_{3/2})$ | $\Lambda_{c}K^{*}({}^{2}D_{3/2})$ | $\Lambda_{c}K^{*}({}^{4}D_{3/2})$ | $\Sigma_{c}K^{*}({}^{4}S_{3/2})$ | $\Sigma_{c}K^{*}({}^{2}D_{3/2})$ | $\Sigma_{c}K^{*}({}^{4}D_{3/2})$
$1/2(3/2^{-})$ | 1.34 | $-0.07$ | 6.35 | 94.23 | 0.03 | 0.11 | 4.89 | 0.22 | 0.52
| 1.36 | $-3.06$ | 2.07 | 84.54 | 0.08 | 0.28 | 13.36 | 0.53 | 1.20
| 1.38 | $-8.93$ | 1.18 | 75.92 | 0.12 | 0.42 | 21.07 | 0.76 | 1.71
| 1.40 | $-16.80$ | 0.87 | 69.12 | 0.14 | 0.52 | 27.27 | 0.91 | 2.05

With these deduced effective potentials, we search for the bound state
solutions for the coupled $\Lambda_{c}K^{*}/\Sigma_{c}K^{*}$ systems in the
cutoff range $0.8\leq\Lambda\leq 5.0$ GeV. In Table 1, we collect the
corresponding numerical results, which include the cutoff dependence of the
binding energy $E$, the root-mean-square radius $r_{RMS}$, and the
probabilities $P_{i}(\%)$ for all the discussed channels.

For the coupled $\Lambda_{c}K^{*}/\Sigma_{c}K^{*}$ system with
$I(J^{P})=1/2(1/2^{-})$, there exist four channels, the
$\Lambda_{c}K^{*}({}^{2}S_{1/2},{}^{4}D_{1/2})$ channels and the
$\Sigma_{c}K^{*}({}^{2}S_{1/2},{}^{4}D_{1/2})$ channels after considering both
the $S-D$ wave mixing effects and the coupled channel effects. As presented in
Table 1, when the cutoff is taken as 1.56 GeV, the binding energy is $-0.14$
MeV, the RMS radius is 6.11 fm, and the probability for the
$\Lambda_{c}K^{*}({}^{2}S_{1/2})$ channel is 98.82%. As the cutoff $\Lambda$
increases to 1.62 GeV, the binding energy becomes $-11.57$ MeV, the RMS radius
becomes 1.12 fm, and the $\Lambda_{c}K^{*}({}^{2}S_{1/2})$ is still the
dominant channel with the probability around 93.18%. From the current
numerical results, in the cutoff range around 1.60 GeV, we can obtain the
weakly bound state with the reasonable loosely bound state solutions, and the
dominant channel is the $\Lambda_{c}K^{*}({}^{2}S_{1/2})$ with its probability
over 90%. Since the cutoff value is close to the empirical value $\Lambda\sim
1.00$ GeV for the deuteron Tornqvist:1993ng ; Tornqvist:1993vu , we conclude
that the coupled $\Lambda_{c}K^{*}/\Sigma_{c}K^{*}$ systems with
$I(J^{P})=1/2(1/2^{-})$ can be recommended as a good hadronic molecular
candidate.

For the coupled $\Lambda_{c}K^{*}/\Sigma_{c}K^{*}$ system with
$I(J^{P})=1/2(3/2^{-})$, there include the
$\Lambda_{c}K^{*}({}^{4}S_{3/2},{}^{2}D_{3/2},{}^{4}D_{3/2})$ channels and the
$\Sigma_{c}K^{*}({}^{4}S_{3/2},{}^{2}D_{3/2},{}^{4}D_{3/2})$ channels when we
consider both the coupled channel effects and the $S-D$ wave mixing effects.
As shown in Table 1, we can obtain loosely bound state solutions at the cutoff
larger than 1.34 GeV, where the binding energy is from several to ten MeV, and
the RMS radius is larger than 1.00 fm, the dominant channel is the
$\Lambda_{c}K^{*}({}^{4}S_{3/2})$ channel. As the increasing of the cutoff
value, the $\Sigma_{c}K^{*}({}^{4}S_{3/2})$ channel becomes more and more
important, when the cutoff is 1.40 GeV, the probability of the $S-$wave
$\Sigma_{c}K^{*}$ component turns into 27.27%. If we still adopt the
experience of the deuteron Tornqvist:1993ng ; Tornqvist:1993vu , the coupled
$\Lambda_{c}K^{*}/\Sigma_{c}K^{*}$ system with $1/2(3/2^{-})$ can be a good
hadronic molecular candidate, it is mainly composed by the
$\Lambda_{c}K^{*}({}^{4}S_{3/2})$ channel, followed by the
$\Sigma_{c}K^{*}({}^{4}S_{3/2})$ channel.

In addition, we find that the coupled channel effects play an important role
in forming these $\Lambda_{c}K^{*}/\Sigma_{c}K^{*}$ bound states with
$1/2(1/2^{-},3/2^{-})$, since there don’t exist bound state solutions in the
single $\Lambda_{c}K^{*}$ systems.

###  III.2 The $\Sigma_{c}K^{(*)}$ systems

The OBE effective potentials for the single $\Sigma_{c}K$ is

$\displaystyle V_{\Sigma_{c}K\to\Sigma_{c}K}$ $\displaystyle=$
$\displaystyle\frac{1}{2}l_{S}g_{\sigma}\chi_{3}^{{\dagger}}\chi_{1}Y(\Lambda,m_{\sigma},r)$
(52)
$\displaystyle+\frac{\mathcal{G}(I)}{8}\beta_{S}g_{V}g\chi_{3}^{{\dagger}}\chi_{1}Y(\Lambda,m_{\rho},r)$
$\displaystyle-\frac{\mathcal{G}(I)}{24m_{\Sigma_{c}}}\lambda_{S}g_{V}g\chi_{3}^{{\dagger}}\chi_{1}\nabla^{2}Y(\Lambda,m_{\rho},r)$
$\displaystyle+\frac{1}{8}\beta_{S}g_{V}g\chi_{3}^{{\dagger}}\chi_{1}Y(\Lambda,m_{\omega},r)$
$\displaystyle-\frac{1}{24m_{\Sigma_{c}}}\lambda_{S}g_{V}g\chi_{3}^{{\dagger}}\chi_{1}\nabla^{2}Y(\Lambda,m_{\omega},r).$

Here, there exists the extra $\rho$ exchange interaction in comparison to the
$\Lambda_{c}K$ system, and it provides the attractive and repulsive forces for
the $\Sigma_{c}K$ system with $I=1/2$ and $3/2$, respectively. Therefore, it
is possible to find the bound state solutions for the $\Sigma_{c}K$ system
with $I=1/2$ as the stronger attractive OBE effective potentials. After
solving the coupled channel Schrödinger equation, our results show that there
exist no bound state solutions for the iso-quartet $\Sigma_{c}K$ system. For
the iso-doublet $\Sigma_{c}K$ system, as presented in Table 2, we can obtain
the reasonable loosely bound state solutions when the cutoff $\Lambda$ is
larger than 2.00 GeV.

Table 2: The bound state solutions (the binding energy $E$, the root-mean-square radius $r_{RMS}$, and the probabilities $P_{i}(\%)$ for all the discussed channels) for the single $\Sigma_{c}K$ and the coupled $\Sigma_{c}K/\Lambda_{c}K^{*}/\Sigma_{c}K^{*}$ systems with $I(J^{P})=1/2(1/2^{-})$ and $3/2(1/2^{-})$. Here, $E$, $r_{RMS}$, and $\Lambda$ are in units of MeV, fm, and GeV, respectively. The dominant channels are labeled in a bold manner. | Single channel | Coupled channel
---|---|---
$I(J^{P})$ | $\Lambda$ | $E$ | $r_{RMS}$ | $\Lambda$ | $E$ | $r_{RMS}$ | $\Sigma_{c}K({}^{2}S_{1/2})$ | $\Lambda_{c}K^{*}({}^{2}S_{1/2})$ | $\Lambda_{c}K^{*}({}^{4}D_{1/2})$ | $\Sigma_{c}K^{*}({}^{2}S_{1/2})$ | $\Sigma_{c}K^{*}({}^{4}D_{1/2})$
$1/2(1/2^{-})$ | 2.00 | $-0.94$ | 4.78 | 0.90 | $-0.36$ | 6.14 | 98.85 | 0.61 | 0.47 | 0.01 | 0.06
| 2.20 | $-4.80$ | 2.44 | 0.95 | $-3.28$ | 3.04 | 97.61 | 1.34 | 0.92 | 0.02 | 0.12
| 2.40 | $-10.96$ | 1.68 | 1.00 | $-9.27$ | 1.91 | 96.11 | 2.25 | 1.42 | 0.04 | 0.18
| 2.60 | $-18.92$ | 1.31 | 1.05 | $-18.44$ | 1.42 | 94.60 | 3.18 | 1.92 | 0.06 | 0.24
$3/2(1/2^{-})$ | … | … | … | 1.28 | $-2.58$ | 2.85 | 92.77 | … | … | 7.11 | 0.12
| … | … | … | 1.29 | $-12.82$ | 1.23 | 85.78 | … | … | 13.99 | 0.22
| … | … | … | 1.30 | $-28.46$ | 0.80 | 80.03 | … | … | 19.67 | 0.30
| … | … | … | 1.31 | $-48.10$ | 0.61 | 75.36 | … | … | 24.29 | 0.35

When we further perform the coupled
$\Sigma_{c}K/\Lambda_{c}K^{*}/\Sigma_{c}K^{*}$ analysis. There can allow the
$\pi-$exchange interactions for both of the $\Lambda_{c}K^{*}\to\Sigma_{c}K$
and $\Sigma_{c}K^{*}\to\Sigma_{c}K$ process, which plays very important role
in binding the deuteron. The corresponding OBE effective potentials can be
expressed as

$\displaystyle V_{\Sigma_{c}K}^{\text{C}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}V_{\Sigma_{c}K\to\Sigma_{c}K}&V_{\Lambda_{c}K^{*}\to\Sigma_{c}K}&V_{\Sigma_{c}K^{*}\to\Sigma_{c}K}\\\
V_{\Sigma_{c}K\to\Lambda_{c}K^{*}}&V_{\Lambda_{c}K^{*}\to\Lambda_{c}K^{*}}&V_{\Sigma_{c}K^{*}\to\Lambda_{c}K^{*}}\\\
V_{\Sigma_{c}K\to\Sigma_{c}K^{*}}&V_{\Lambda_{c}K^{*}\to\Sigma_{c}K^{*}}&V_{\Sigma_{c}K^{*}\to\Sigma_{c}K^{*}}\end{array}\right),$
(56)

with

$\displaystyle V_{\Lambda_{c}K^{*}\to\Sigma_{c}K}$ $\displaystyle=$
$\displaystyle-\frac{1}{6}\frac{g_{4}g}{f_{\pi}\sqrt{m_{K}m_{K^{*}}}}\mathcal{F}_{1}(r,\bm{\sigma},\bm{\epsilon}_{2})U(\Lambda_{1},m_{\pi
1},r)$
$\displaystyle-\frac{\lambda_{I}g_{V}g_{VVP}}{3\sqrt{2}}\sqrt{\frac{m_{K^{*}}}{m_{K}}}\mathcal{F}_{2}(r,\bm{\sigma},\bm{\epsilon}_{2})Y(\Lambda_{1},m_{\rho
1},r),$ $\displaystyle V_{\Sigma_{c}K^{*}\to\Sigma_{c}K}$ $\displaystyle=$
$\displaystyle\frac{g_{1}g\mathcal{F}_{1}(r,\bm{\sigma},\bm{\epsilon}_{2})}{24\sqrt{2}f_{\pi}\sqrt{m_{K}m_{K^{*}}}}\mathcal{G}(I)Y(\Lambda_{2},m_{\pi
2},r)$
$\displaystyle+\frac{g_{1}g}{72\sqrt{2}f_{\pi}\sqrt{m_{K}m_{K^{*}}}}\mathcal{F}_{1}(r,\bm{\sigma},\bm{\epsilon}_{2})Y(\Lambda_{2},m_{\eta
2},r)$
$\displaystyle+\frac{\lambda_{S}g_{V}g_{VVP}}{6\sqrt{3}}\sqrt{\frac{m_{K^{*}}}{m_{K}}}\mathcal{F}_{2}(r,\bm{\sigma},\bm{\epsilon}_{2})\mathcal{G}(I)Y(\Lambda_{2},m_{\rho
2},r)$
$\displaystyle+\frac{\lambda_{S}g_{V}g_{VVP}}{6\sqrt{3}}\sqrt{\frac{m_{K^{*}}}{m_{K}}}\mathcal{F}_{2}(r,\bm{\sigma},\bm{\epsilon}_{2})Y(\Lambda_{2},m_{\omega
2},r).$

Here, we define an useful function in Eq. (III.2), i.e.,

$\displaystyle U(\Lambda,m,{r})$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
r}\left(\cos(mr)-e^{-\Lambda
r}\right)-\frac{\Lambda^{2}+m^{2}}{8\pi\Lambda}e^{-\Lambda r}.$

The variables in the above effective potentials (III.2)$-$(III.2) are defined
as
$q_{1}=\frac{M_{\Lambda_{c}}^{2}+M_{K}^{2}-M_{\Sigma_{c}}^{2}-M_{K^{*}}^{2}}{2(M_{\Sigma_{c}}+M_{K})}$,
$\Lambda_{1}^{2}=\Lambda^{2}-q_{1}^{2}$, $m_{\pi
1}^{2}=q_{1}^{2}-m_{\pi}^{2}$, $m_{\rho 1}^{2}=m_{\rho}^{2}-q_{1}^{2}$,
$q_{2}=\frac{M_{K^{*}}^{2}-M_{K}^{2}}{2(M_{\Sigma_{c}}+M_{K})}$,
$\Lambda_{2}^{2}=\Lambda^{2}-q_{2}^{2}$, $m_{\pi
2}^{2}=m_{\pi}^{2}-q_{2}^{2}$, $m_{\eta 2}^{2}=m_{\eta}^{2}-q_{2}^{2}$,
$m_{\rho 2}^{2}=m_{\rho}^{2}-q_{2}^{2}$, $m_{\omega
2}^{2}=m_{\omega}^{2}-q_{2}^{2}$. After considering the $S-D$ wave mixing
effects, the matrix elements for the spin-spin interaction and tensor force
operators read as
$\bm{\sigma}\cdot\bm{\epsilon_{2}}\mapsto(\begin{array}[]{cc}\sqrt{3}&0\end{array})$
and
$S(\hat{r},\bm{\sigma},\bm{\epsilon_{2}})\mapsto(\begin{array}[]{cc}0&-\sqrt{6}\end{array})$,
respectively.

In Table 2, we collect the bound state solutions (the binding energy $E$, the
root-mean-square radius $r_{RMS}$, and the probabilities $P_{i}(\%)$ for all
the discussed channels) for the coupled
$\Sigma_{c}K/\Lambda_{c}K^{*}/\Sigma_{c}K^{*}$ systems with
$I(J^{P})=0,1(1/2^{-})$.

For the $\Sigma_{c}K/\Lambda_{c}K^{*}/\Sigma_{c}K^{*}$ system with
$I(J^{P})=1/2(1/2^{-})$, there exist the $\Sigma_{c}K({}^{2}S_{1/2})$ channel,
the $\Lambda_{c}K^{*}({}^{2}S_{1/2},{}^{4}D_{1/2})$ channels, and the
$\Sigma_{c}K^{*}({}^{2}S_{1/2},{}^{4}D_{1/2})$ channels. The reasonable
loosely bound state solutions emerge at the cutoff $\Lambda=0.90$ GeV, where
the binding energy is $-0.36$ MeV, the RMS radius is 4.78 fm, and the dominant
channel is the $\Sigma_{c}K({}^{2}S_{1/2})$ with the probability $P=98.85$%.
When the cutoff increases to 1.05 GeV, this bound state binds deeper, the
binding energy is $-18.44$ MeV, the RMS radius decreases to 1.42 fm, and the
$\Sigma_{c}K({}^{2}S_{1/2})$ channel is still the dominant channel with its
probability around 95%. For the remaining channels, their probabilities are
very tiny. Compared to the bound state properties in the single channel case,
the cutoff is very close to the reasonable value $\Lambda\sim$1.00 GeV.
Therefore, the coupled $\Sigma_{c}K/\Lambda_{c}K^{*}/\Sigma_{c}K^{*}$ system
with $I(J^{P})=1/2(1/2^{-})$ can be the prime molecular candidate, and the
coupled channel effects play an important role for the formation of this bound
state.

For the $\Sigma_{c}K/\Sigma_{c}K^{*}$ system with $I(J^{P})=3/2(1/2^{-})$,
there include the $\Sigma_{c}K({}^{2}S_{1/2})$ channel and the
$\Sigma_{c}K^{*}({}^{2}S_{1/2},{}^{4}D_{1/2})$ channels. We find a weakly
bound state at the cutoff $\Lambda=$1.28 GeV, the binding energy is $E=-2.58$
MeV, the RMS radius is $r_{RMS}=$2.85 fm, and the dominant channel is the
$\Sigma_{c}K({}^{2}S_{1/2})$ with the probability $P=92.77$%. With the
increasing of the cutoff value, the $\Sigma_{c}K^{*}$ channel becomes more and
more important. As the cutoff increases to 1.31 GeV, the binding energy turns
into $-48.10$ MeV, the RMS radius decreases to 0.61 fm, and the probability
for the $\Sigma_{c}K^{*}({}^{2}S_{1/2})$ is 24.29%. However, the binding
energy depends very sensitively with the cutoff. Thus, we cannot draw a
definite conclusion that the $\Sigma_{c}K/\Sigma_{c}K^{*}$ system with
$I(J^{P})=3/2(1/2^{-})$ as a good hadronic molecular candidate.

Table 3: The $\Lambda$ dependence of the obtained bound-state solutions (the binding energy $E$ and the root-mean-square radius $r_{RMS}$) for the single $\Sigma_{c}K^{*}$ systems. Here, $E$, $r_{RMS}$, and $\Lambda$ are in units of MeV, fm, and GeV, respectively. $I(J^{P})$ | $\Lambda$ | $E$ | $r_{RMS}$ | $I(J^{P})$ | $\Lambda$ | $E$ | $r_{RMS}$
---|---|---|---|---|---|---|---
$1/2(1/2^{-})$ | 1.70 | $-0.50$ | 5.32 | $1/2(3/2^{-})$ | 0.88 | $-0.25$ | 6.06
| 2.00 | $-3.32$ | 2.64 |  | 0.98 | $-3.32$ | 2.58
| 2.30 | $-8.31$ | 1.81 |  | 1.08 | $-10.72$ | 1.59
| 2.60 | $-15.30$ | 1.42 |  | 1.18 | $-23.27$ | 1.15
$3/2(1/2^{-})$ | 1.28 | $-0.11$ | 6.24 | $3/2(3/2^{-})$ | … | … | …
| 1.31 | $-2.42$ | 2.50 |  | … | … | …
| 1.34 | $-7.78$ | 1.43 |  | … | … | …
| 1.37 | $-16.71$ | 1.00 |  | … | … | …

For the $\Sigma_{c}K^{*}$ systems, the isospin and spin-parity configurations
$I(J^{P})$ include $1/2(1/2^{-})$, $1/2(3/2^{-})$, $3/2(1/2^{-})$, and
$3/2(3/2^{-})$ after considering the $S-D$ wave mixing effects. The relevant
OBE effective potentials are presented in Eq. (33). Our results indicate that
there exist the reasonable loosely bound state solutions for the
$\Sigma_{c}K^{*}$ states with $I(J^{P})=1/2(1/2^{-})$, $1/2(3/2^{-})$, and
$3/2(1/2^{-})$ in the cutoff range $0.80\leq\Lambda\leq 5.00$ GeV. As shown in
Table 3, for the $\Sigma_{c}K^{*}$ systems with $1/2(3/2^{-})$ and
$3/2(1/2^{-})$, the binding energy around several to several tens MeV and the
RMS radius around several fm appear at the cutoff around 1.00 GeV, which is
comparable to the value in the deuteron Tornqvist:1993ng ; Tornqvist:1993vu .
Therefore, these two states can be suggested as the good hadronic molecular
candidates. For the $\Sigma_{c}K^{*}$ system with $1/2(1/2^{-})$, the loosely
bound state solutions appear as the cutoff is larger than 1.70 GeV, which is
slightly far away from the empirical value for the deuteron Tornqvist:1993ng ;
Tornqvist:1993vu , in this work, we cannot exclude that the $\Sigma_{c}K^{*}$
system with $1/2(1/2^{-})$ as a suitable molecular candidate.

In summary, our results can predict several possible open charm molecular
pentaquarks, the coupled $\Lambda_{c}K^{*}/\Sigma_{c}K^{*}$ molecular states
with $I(J^{P})=1/2(1/2^{-},3/2^{-})$, the coupled
$\Sigma_{c}K/\Lambda_{c}K^{*}/\Sigma_{c}K^{*}$ molecular states with
$I(J^{P})=1/2(1/2^{-})$, and the single $\Sigma_{c}K^{*}$ states with
$I(J^{P})=1/2(1/2^{-},3/2^{-})$ and $3/2(1/2^{-})$. And the coupled channel
effects do play the very important role in generating these coupled channel
molecular candidates.

The study of the strong decay behaviors is very helpful to the search of these
predicted open flavor molecular pentaquarks. According to the conservation of
the quantum numbers and the limit of the phase space, we collect the important
strong decay decay channels as follows, i.e.,

$\displaystyle\Sigma_{c}K/\Lambda_{c}K^{*}/\Sigma_{c}K^{*}[1/2(1/2^{-})]$
$\displaystyle\to$ $\displaystyle\left\\{D_{s}N,\Lambda_{c}K\right\\},$
$\displaystyle\Lambda_{c}K^{*}/\Sigma_{c}K^{*}[1/2(1/2^{-})]$
$\displaystyle\to$
$\displaystyle\left\\{D_{s}^{(*)}N,\Lambda_{c}K,\Sigma_{c}K\right\\},$
$\displaystyle\Lambda_{c}K^{*}/\Sigma_{c}K^{*}[1/2(3/2^{-})]$
$\displaystyle\to$ $\displaystyle\left\\{D_{s}^{*}N\right\\},$
$\displaystyle\Sigma_{c}K^{*}[1/2(1/2^{-})]$ $\displaystyle\to$
$\displaystyle\left\\{D_{s}^{(*)}N,\Lambda_{c}K^{(*)},\Sigma_{c}K\right\\},$
$\displaystyle\Sigma_{c}K^{*}[1/2(3/2^{-})]$ $\displaystyle\to$
$\displaystyle\left\\{D_{s}^{*}N,\Lambda_{c}K^{*},\Sigma_{c}^{*}K\right\\},$
$\displaystyle\Sigma_{c}K^{*}[3/2(1/2^{-})]$ $\displaystyle\to$
$\displaystyle\left\\{D_{s}^{*}\Delta,\Sigma_{c}K\right\\}.$

###  III.3 The predictions of the possible $Y_{c}\bar{K}^{(*)}$ molecular
states

In this work, we further extend our study to the $\Lambda_{c}\bar{K}^{(*)}$
and $\Sigma_{c}\bar{K}^{(*)}$ systems, the corresponding OBE effective
potentials can be related to those for the $\Lambda_{c}{K}^{(*)}$ and
$\Sigma_{c}{K}^{(*)}$ systems by the $G-$parity rule Klempt:2002ap , i.e.,

$\displaystyle V_{B_{1}\bar{M}_{2}\to B_{3}\bar{M}_{4}}$ $\displaystyle=$
$\displaystyle(-1)^{G_{E}}V_{B_{1}M_{2}\to B_{3}M_{4}},$ (59)

where ${G_{E}}$ stands for the $G-$parity for the exchanged meson in the
$B_{1}M_{2}\to B_{3}M_{4}$ process, notations $\bar{M}_{i}$ and ${M}_{i}$
correspond to the anti-mesons and mesons, respectively. Therefore, the
effective potentials from the $\omega$ and $\pi$ exchanges are in completely
contrast between the $Y_{c}K^{(*)}$ and the $Y_{c}\bar{K}^{(*)}$ systems.

Table 4: The $\Lambda$ dependence of the obtained bound-state solutions (the binding energy $E$ and the root-mean-square radius $r_{RMS}$) for the single $Y_{c}\bar{K}^{(*)}$ systems. Here, $E$, $r_{RMS}$, and $\Lambda$ are in units of MeV, fm, and GeV, respectively.  Systems | $\Lambda$ | $E$ | $r_{RMS}$ | Systems | $\Lambda$ | $E$ | $r_{RMS}$ | Systems | $\Lambda$ | $E$ | $r_{RMS}$
---|---|---|---|---|---|---|---|---|---|---|---
$\Lambda_{c}\bar{K}[1/2(1/2^{-})]$ | … | … | … | $\Sigma_{c}\bar{K}[1/2(1/2^{-})]$ | 1.30 | $-0.54$ | 5.55 | $\Sigma_{c}\bar{K}^{*}[3/2(1/2^{-})]$ | 4.40 | $-0.10$ | 6.58
$\Lambda_{c}\bar{K}^{*}[1/2(1/2^{-})]$ | … | … | … |  | 1.35 | $-3.21$ | 2.97 |  | 4.60 | $-0.40$ | 5.43
$\Lambda_{c}\bar{K}^{*}[1/2(3/2^{-})]$ | … | … | … |  | 1.40 | $-7.92$ | 1.99 |  | 4.80 | $-0.84$ | 4.34
$\Sigma_{c}\bar{K}[3/2(1/2^{-})]$ | … | … | … |  | 1.45 | $-14.56$ | 1.52 |  | 5.00 | $-1.47$ | 3.47
$\Sigma_{c}\bar{K}^{*}[1/2(1/2^{-})]$ | 0.85 | $-0.37$ | 5.42 | $\Sigma_{c}\bar{K}^{*}[1/2(3/2^{-})]$ | 0.96 | $-0.40$ | 5.64 | $\Sigma_{c}\bar{K}^{*}[3/2(3/2^{-})]$ | 1.50 | $-0.75$ | 4.49
| 0.90 | $-3.49$ | 2.35 |  | 1.02 | $-3.26$ | 2.66 |  | 1.70 | $-4.46$ | 2.11
| 0.95 | $-9.56$ | 1.54 |  | 1.08 | $-9.90$ | 1.69 |  | 1.90 | $-11.09$ | 1.42
| 1.00 | $-18.11$ | 1.20 |  | 1.14 | $-21.21$ | 1.25 |  | 2.10 | $-20.62$ | 1.09

In the following, we also perform the single channel analysis and the coupled
channel analysis on the $Y_{c}\bar{K}^{(*)}$ systems. We summary the
corresponding numerical results in Table 4 and Table 5, respectively.

As shown in Table 4, we collect the bound state properties for the single
$Y_{c}\bar{K}^{(*)}$ systems. In the cutoff region $0.80\leq\Lambda\leq 5.00$
GeV, we can obtain five loosely bound states, the $\Sigma_{c}\bar{K}$ bound
state with $I(J^{P})=1/2(1/2^{-})$, the $\Sigma_{c}\bar{K}^{*}$ states with
$I(J^{P})=1/2(1/2^{-})$, $1/2(3/2^{-})$, $3/2(1/2^{-})$ and $3/2(3/2^{-})$.
Among these five bound states, we cannot recommend the $\Sigma_{c}\bar{K}^{*}$
state with $3/2(1/2^{-})$ as a good hadronic molecular candidate, as the
cutoff value is too far away from the empirical value $\Lambda\sim 1.00$ GeV.
For the remaining four bound states, we conclude that they can be prime
hadronic molecular candidates when we take the same cutoff criterion in the
deuteron.

Table 5: The bound state solutions (the binding energy $E$, the root-mean-square radius $r_{RMS}$, and the probabilities $P_{i}(\%)$ for all the discussed channels) for the coupled $Y_{c}\bar{K}^{(*)}$ systems. Here, $E$, $r_{RMS}$, and $\Lambda$ are in units of MeV, fm, and GeV, respectively. The dominant channels are labeled in a bold manner. $I(J^{P})$ | $\Lambda$ | $E$ | $r_{RMS}$ | $\Lambda_{c}\bar{K}^{*}({}^{2}S_{1/2})$ | $\Lambda_{c}\bar{K}^{*}({}^{4}D_{1/2})$ | $\Sigma_{c}\bar{K}^{*}({}^{2}S_{1/2})$ | $\Sigma_{c}\bar{K}^{*}({}^{4}D_{1/2})$ |  |
---|---|---|---|---|---|---|---|---|---
$1/2(1/2^{-})$ | 1.38 | $-0.90$ | 2.98 | 81.97 | 0.18 | 16.99 | 0.86 |  |
| 1.40 | $-4.77$ | 1.21 | 66.23 | 0.34 | 31.86 | 1.57 |  |
| 1.42 | $-9.42$ | 0.88 | 56.07 | 0.45 | 41.46 | 2.02 |  |
| 1.44 | $-14.44$ | 0.75 | 49.04 | 0.53 | 48.10 | 2.33 |  |
$I(J^{P})$ | $\Lambda$ | $E$ | $r_{RMS}$ | $\Lambda_{c}\bar{K}^{*}({}^{4}S_{3/2})$ | $\Lambda_{c}\bar{K}^{*}({}^{2}D_{3/2})$ | $\Lambda_{c}\bar{K}^{*}({}^{4}D_{3/2})$ | $\Sigma_{c}\bar{K}^{*}({}^{4}S_{3/2})$ | $\Sigma_{c}\bar{K}^{*}({}^{2}D_{3/2})$ | $\Sigma_{c}\bar{K}^{*}({}^{4}D_{3/2})$
$1/2(3/2^{-})$ | 1.38 | $-2.71$ | 1.51 | 40.22 | 0.27 | 2.16 | 48.94 | 0.46 | 7.96
| 1.39 | $-8.95$ | 0.86 | 28.77 | 0.32 | 2.53 | 58.55 | 0.53 | 9.29
| 1.40 | $-15.88$ | 0.72 | 23.23 | 0.34 | 2.69 | 63.27 | 0.56 | 9.90
| 1.41 | $-23.20$ | 0.66 | 19.83 | 0.36 | 2.77 | 66.19 | 0.58 | 10.27
$I(J^{P})$ | $\Lambda$ | $E$ | $r_{RMS}$ | $\Sigma_{c}\bar{K}({}^{2}S_{1/2})$ | $\Lambda_{c}\bar{K}^{*}({}^{2}S_{1/2})$ | $\Lambda_{c}\bar{K}^{*}({}^{4}D_{1/2})$ | $\Sigma_{c}\bar{K}^{*}({}^{2}S_{1/2})$ | $\Sigma_{c}\bar{K}^{*}({}^{4}D_{1/2})$ |
$1/2(1/2^{-})$ | 0.87 | $-0.34$ | 6.18 | 98.57 | 0.90 | 0.40 | 0.04 | 0.09 |
| 0.91 | $-3.71$ | 2.85 | 97.15 | 1.90 | 0.68 | 0.07 | 0.19 |
| 0.95 | $-11.61$ | 1.70 | 95.52 | 3.15 | 0.92 | 0.09 | 0.32 |
| 0.99 | $-25.39$ | 1.21 | 93.90 | 4.47 | 1.09 | 0.08 | 0.46 |
$I(J^{P})$ | $\Lambda$ | $E$ | $r_{RMS}$ | $\Sigma_{c}\bar{K}({}^{2}S_{1/2})$ | $\Lambda_{c}\bar{K}^{*}({}^{2}S_{1/2})$ | $\Lambda_{c}\bar{K}^{*}({}^{4}D_{1/2})$ | $\Sigma_{c}\bar{K}^{*}({}^{2}S_{1/2})$ | $\Sigma_{c}\bar{K}^{*}({}^{4}D_{1/2})$ |
$3/2(1/2^{-})$ | 3.80 | $-0.54$ | 5.47 | 99.43 | … | … | 0.28 | 0.29 |
| 4.00 | $-2.67$ | 3.12 | 99.00 | … | … | 0.50 | 0.50 |
| 4.20 | $-6.91$ | 2.00 | 98.42 | … | … | 0.82 | 0.76 |
| 4.40 | $-14.02$ | 1.43 | 97.68 | … | … | 1.23 | 1.09 |

When we consider the coupled channel analysis, we find four weakly bound
states by varying the cutoff from 0.80 GeV to 5.00 GeV as shown in Table 5.
For the $\Lambda_{c}\bar{K}^{*}/\Sigma_{c}\bar{K}^{*}$ coupled bound state
with $I(J^{P})=1/2(1/2^{-})$ and the
$\Sigma_{c}\bar{K}/\Lambda_{c}\bar{K}^{*}/\Sigma_{c}\bar{K}^{*}$ coupled bound
state with $I(J^{P})=1/2(1/2^{-})$, we can obtain the reasonable loosely bound
state properties at the cutoff taken around 1.00 GeV, the dominant channels
are the $\Lambda_{c}\bar{K}^{*}({}^{2}S_{1/2})$ and
$\Sigma_{c}\bar{K}({}^{2}S_{1/2})$ channels, respectively. Thus, these two
coupled bound states can be prime hadronic molecular candidates, which are
mainly composed by the $\Lambda_{c}\bar{K}^{*}$ and $\Sigma_{c}\bar{K}$
states, respectively.

Compared to the bound states solutions for the single $\Lambda_{c}\bar{K}^{*}$
and $\Sigma_{c}\bar{K}$ systems with $1/2(1/2^{-})$, we also find that the
coupled channel effects play an important role in generating the
$\Lambda_{c}\bar{K}^{*}$ state with $1/2(1/2^{-})$. However, it contributes
very little for the $\Sigma_{c}\bar{K}$ state with $1/2(1/2^{-})$. Thus, the
$\Sigma_{c}\bar{K}/\Lambda_{c}\bar{K}^{*}/\Sigma_{c}\bar{K}^{*}$ coupled bound
state with $I(J^{P})=1/2(1/2^{-})$ predicted here is not a new bound state but
has a close relation with the single $\Sigma_{c}\bar{K}$ molecule with
$1/2(1/2^{-})$.

For the $\Lambda_{c}\bar{K}^{*}/\Sigma_{c}\bar{K}^{*}$ coupled system with
$I(J^{P})=1/2(3/2^{-})$, its dominant channel is the
$\Sigma_{c}\bar{K}^{*}({}^{4}S_{3/2})$. As shown in Table 5, its size is much
smaller than those coupled channel bound states mainly made up by the lowest
system. As the dominant channel is the $\Sigma_{c}\bar{K}^{*}({}^{4}S_{3/2})$,
this bound state has a close relation to the $\Sigma_{c}\bar{K}^{*}$ molecule
with $1/2(3/2^{-})$.

For the $\Sigma_{c}\bar{K}/\Sigma_{c}\bar{K}^{*}$ coupled system with
$I(J^{P})=3/2(1/2^{-})$, we can obtain the bound state solution as the cutoff
reaches up to 3.80 GeV. Obviously, the cutoff applied here is deviated from
the reasonable value 1.00 GeV. It cannot be a good molecular candidate.

All in all, our results can predict five $Y_{c}\bar{K}^{(*)}$ type hadronic
molecular candidates, the coupled
$\Lambda_{c}\bar{K}^{*}/\Sigma_{c}\bar{K}^{*}$ molecule with $1/2(1/2^{-})$,
the $\Sigma_{c}\bar{K}/\Lambda_{c}\bar{K}^{*}/\Sigma_{c}\bar{K}^{*}$ molecule
with $1/2(1/2^{-})$, the $\Sigma_{c}\bar{K}^{*}$ molecules with
$1/2(1/2^{-},3/2^{-})$, and $3/2(3/2^{-})$, where the coupled channel effects
play a vital role in binding the coupled
$\Lambda_{c}\bar{K}^{*}/\Sigma_{c}\bar{K}^{*}$ state with $1/2(1/2^{-})$.
Their important two-body strong decay channels are summarized as follows,
i.e.,

$\displaystyle\Sigma_{c}\bar{K}[1/2(1/2^{-})]$ $\displaystyle\to$
$\displaystyle\left\\{\Lambda_{c}\bar{K},\Xi_{c}^{(^{\prime})}\pi\right\\},$
$\displaystyle\Lambda_{c}\bar{K}^{*}/\Sigma_{c}\bar{K}^{*}[1/2(1/2^{-})]$
$\displaystyle\to$
$\displaystyle\left\\{\Lambda_{c}\bar{K},\Sigma_{c}\bar{K},D\Lambda,D\Sigma,\Xi_{c}^{(^{\prime})}\pi,\Xi_{c}^{(^{\prime})}\eta\right\\},$
$\displaystyle\Sigma_{c}\bar{K}^{*}[1/2(1/2^{-})]$ $\displaystyle\to$
$\displaystyle\left\\{\Lambda_{c}\bar{K}^{(*)},\Sigma_{c}\bar{K},D^{(*)}\Lambda,D^{(*)}\Sigma,\right.$
$\displaystyle\left.\Xi_{c}^{(^{\prime})}\pi,\Xi_{c}^{(^{\prime})}\eta,\Xi_{c}\rho,\Xi_{c}\omega\right\\},$
$\displaystyle\Sigma_{c}\bar{K}^{*}[1/2(3/2^{-})]$ $\displaystyle\to$
$\displaystyle\left\\{\Lambda_{c}\bar{K}^{*},\Sigma_{c}^{*}\bar{K},D^{*}\Lambda,D^{*}\Sigma,\right.$
$\displaystyle\left.\Xi_{c}\rho,\Xi_{c}\omega,\Xi_{c}^{*}\pi,\Xi_{c}^{*}\eta\right\\},$
$\displaystyle\Sigma_{c}\bar{K}^{*}[3/2(3/2^{-})]$ $\displaystyle\to$
$\displaystyle\left\\{\Sigma_{c}^{*}\bar{K},D^{*}\Sigma,\Xi_{c}\rho,\Xi_{c}^{*}\pi\right\\}.$

##  IV Summary

The study of the exotic states is an important and interesting issue in the
hadron physics. Searching for the hadronic molecular states can not only
enrich the family of the exotic states, but also help us to understand the
essential hadron-hadron interactions. Very recently, the LHCb collaboration
observed two open heavy flavor multiquarks $T_{c\bar{s}}^{a0(++)}$. Their near
threshold behavior inspires the isovector $D^{*}K^{*}$ molecular explanations
to them. In our former paper, we found the $D^{*}K^{*}$ state with
$I(J^{P})=1(0^{+})$ can be possible molecular candidate by adopting the OBE
effective potentials Chen:2016ypj .

Figure 1: A summary of the predicted $Y_{c}K^{(*)}$ and $Y_{c}\bar{K}^{*}$
molecular candidates. Here, the red and blue lines label the molecular
candidates predicted by the single channel analysis and the coupled channel
analysis. respectively.

In this work, we extend our study on the interactions between the $S-$wave
charmed baryon $Y_{c}=(\Lambda_{c},\Sigma_{c})$ and the strange meson
$K^{(*)}$ by using the OBE model, and we consider both of the $S-D$ wave
mixing effects and the coupled channel effects. As shown in Figure 1, our
results indicate the single $\Sigma_{c}K^{*}$ states with
$I(J^{P})=1/2(1/2^{-})$, $1/2(3/2^{-})$ and $3/2(1/2^{-})$ can be good open
charm molecular candidates. When we further consider the coupled channel
effects, we can predict another three prime open charm molecular candidates,
i.e., the coupled $\Lambda_{c}K^{*}/\Sigma_{c}K^{*}$ molecular states with
$1/2(1/2^{-})$ and $1/2(3/2^{-})$, and the coupled
$\Sigma_{c}K/\Lambda_{c}K^{*}/\Sigma_{c}K^{*}$ molecular state with
$1/2(1/2^{-})$, where the dominant channels correspond to the
$\Lambda_{c}K^{*}({}^{2}S_{1/2})$, $\Lambda_{c}K^{*}({}^{4}S_{3/2})$, and
$\Sigma_{c}K({}^{2}S_{1/2})$, respectively. And the coupled channel effects
play the essential role in binding these three coupled channel molecular
candidates.

As a byproduct, we further study the $Y_{c}\bar{K}^{(*)}$ interactions in the
same model. As shown in Figure 1, we can predict the existences of the
$Y_{c}\bar{K}^{(*)}$ type hadronic molecular states, i.e., the
$\Sigma_{c}\bar{K}$ molecule with $I(J^{P})=1/2(1/2^{-})$, the
$\Sigma_{c}\bar{K}^{*}$ molecules with $1/2(1/2^{-})$, $1/2(3/2^{-})$, and
$3/2(3/2^{-})$, the coupled $\Lambda_{c}\bar{K}^{*}/\Sigma_{c}\bar{K}^{*}$
molecule with $1/2(1/2^{-})$, and the
$\Sigma_{c}\bar{K}/\Lambda_{c}\bar{K}^{*}/\Sigma_{c}\bar{K}^{*}$ molecule with
$1/2(1/2^{-})$. We expect the experimentalists to search for these predicted
open charm molecular pentaquarks.

## ACKNOWLEDGMENTS

R. C. is supported by the Xiaoxiang Scholars Programme of Hunan Normal
University.

## References

  * (1) H. X. Chen, W. Chen, X. Liu and S. L. Zhu, Phys. Rept. 639, 1-121 (2016).
  * (2) Y. R. Liu, H. X. Chen, W. Chen, X. Liu and S. L. Zhu, Prog. Part. Nucl. Phys. 107, 237-320 (2019).
  * (3) H. X. Chen, W. Chen, X. Liu, Y. R. Liu and S. L. Zhu, Rept. Prog. Phys. 80, no.7, 076201 (2017).
  * (4) F. K. Guo, C. Hanhart, U. G. Meißner, Q. Wang, Q. Zhao and B. S. Zou, Rev. Mod. Phys. 90, no.1, 015004 (2018), [erratum: Rev. Mod. Phys. 94, no.2, 029901 (2022)].
  * (5) H. X. Chen, W. Chen, X. Liu, Y. R. Liu and S. L. Zhu, Rept. Prog. Phys. 86, no.2, 026201 (2023).
  * (6) C. Chen and E. S. Norella, https://indico.cern.ch/event/1176505/.
  * (7) W. B. Qian, https://indico.ihep.ac.cn/event/17185/.
  * (8) R. Chen and X. Liu, Phys. Rev. D 94, no.3, 034006 (2016).
  * (9) W. Chen, H. X. Chen, X. Liu, T. G. Steele and S. L. Zhu, Phys. Rev. D 95, no.11, 114005 (2017).
  * (10) T. Guo, J. Li, J. Zhao and L. He, Phys. Rev. D 105, no.5, 054018 (2022).
  * (11) J. B. Cheng, S. Y. Li, Y. R. Liu, Y. N. Liu, Z. G. Si and T. Yao, Phys. Rev. D 101, no.11, 114017 (2020).
  * (12) S. S. Agaev, K. Azizi and H. Sundu, J. Phys. G 50, no.5, 055002 (2023).
  * (13) R. Aaij et al. [LHCb], Phys. Rev. Lett. 125, 242001 (2020).
  * (14) R. Aaij et al. [LHCb], Phys. Rev. D 102, 112003 (2020).
  * (15) R. Molina and E. Oset, Phys. Lett. B 811, 135870 (2020).
  * (16) M. W. Hu, X. Y. Lao, P. Ling and Q. Wang, Chin. Phys. C 45, no.2, 021003 (2021).
  * (17) M. Z. Liu, J. J. Xie and L. S. Geng, Phys. Rev. D 102, no.9, 091502 (2020).
  * (18) S. Y. Kong, J. T. Zhu, D. Song and J. He, Phys. Rev. D 104, no.9, 094012 (2021).
  * (19) B. Wang and S. L. Zhu, Eur. Phys. J. C 82, no.5, 419 (2022).
  * (20) C. J. Xiao, D. Y. Chen, Y. B. Dong and G. W. Meng, Phys. Rev. D 103, no.3, 034004 (2021).
  * (21) Y. Huang, J. X. Lu, J. J. Xie and L. S. Geng, Eur. Phys. J. C 80, no.10, 973 (2020).
  * (22) H. X. Chen, W. Chen, R. R. Dong and N. Su, Chin. Phys. Lett. 37, no.10, 101201 (2020).
  * (23) S. S. Agaev, K. Azizi and H. Sundu, J. Phys. G 48, no.8, 085012 (2021).
  * (24) J. J. Qi, Z. Y. Wang, Z. F. Zhang and X. H. Guo, Eur. Phys. J. C 81, no.7, 639 (2021).
  * (25) H. Chen, H. R. Qi and H. Q. Zheng, Eur. Phys. J. C 81, no.9, 812 (2021).
  * (26) H. T. An, Z. W. Liu, F. S. Yu and X. Liu, Phys. Rev. D 106, no.11, L111501 (2022).
  * (27) Y. R. Liu and M. Oka, Phys. Rev. D 85, 014015 (2012).
  * (28) Z. W. Lin and C. M. Ko, Phys. Rev. C 62, 034903 (2000).
  * (29) H. Nagahiro, L. Roca and E. Oset, Eur. Phys. J. A 36, 73-84 (2008).
  * (30) R. Chen, A. Hosaka and X. Liu, Phys. Rev. D 97, no.3, 036016 (2018).
  * (31) O. Kaymakcalan, S. Rajeev and J. Schechter, Phys. Rev. D 30, 594 (1984)..
  * (32) N. A. Tornqvist, Z. Phys. C 61, 525-537 (1994).
  * (33) N. A. Tornqvist, Nuovo Cim. A 107, 2471-2476 (1994).
  * (34) E. Klempt, F. Bradamante, A. Martin and J. M. Richard, Phys. Rept. 368, 119-316 (2002).