# Exploring the mass spectrum and electromagnetic property of the
$\Xi_{cc}K^{(*)}$ and $\Xi_{cc}\bar{K}^{(*)}$ molecules

Li-Cheng Sheng1 Jin-Yu Huo1 Rui Chen1111Corresponding author
<EMAIL_ADDRESS>Fu-Lai Wang2,3,4,6 Xiang Liu2,3,4,5,6<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
2School of Physical Science and Technology, Lanzhou University, Lanzhou
730000, China
3Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, Lanzhou University, Lanzhou 730000, China
4Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou University,
Lanzhou 730000, China
5MoE Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou
730000, China
6Research Center for Hadron and CSR Physics, Lanzhou University and Institute
of Modern Physics of CAS, Lanzhou 730000, China

###### Abstract

Using the one-boson-exchange model, we investigate the interactions between
the doubly charmed baryon $\Xi_{cc}(3621)$ and the $S-$wave (anti-)kaon
accounting for the $S-D$ wave mixing and coupled-channel effects. We find the
coupled $\Xi_{cc}K/\Xi_{cc}K^{*}$ state with $I(J^{P})=0(1/2^{-})$, the
$\Xi_{cc}K^{*}$ state with $0(1/2^{-})$, the $\Xi_{cc}\bar{K}$ state with
$0(1/2^{-})$, and the $\Xi_{cc}\bar{K}^{*}$ states with $0(1/2^{-},3/2^{-})$
can be recommended as good doubly charmed molecular candidates with
strangeness $|S|=1$. We further examine their M1 radiative decay behaviors and
magnetic moments within the constituent quark model framework. This
information can enhance our understanding of their inner structures, including
the distribution of electric charge and the orientation of the constituent
quarks’ spins.

##  I Introduction

In recent decades, experimental discoveries of new hadron matter have
significantly advanced the study of hadronic spectroscopy, making it a
prominent research focus in particle physics (see reviews Chen:2016qju ;
Liu:2019zoy ; Chen:2016spr ; Guo:2017jvc ; Chen:2022asf ; Liu:2013waa ;
Hosaka:2016pey  for more details). Delving deeply into hadronic spectroscopy
cannot only reveals the inner structures and formation mechanisms of new
hadronic states but also provide insights into the complex non-perturbative
dynamics of quantum chromodynamics (QCD).

Recently, the LHCb Collaboration observed two open-charm exotic structures,
$T_{c\bar{s}}^{a0(++)}(2900)$ in the $D_{s}^{+}\pi^{-}(\pi^{+})$ final states
LHCb:2022sfr . Their quantum number configurations are $I(J^{P})=1(0^{+})$.
The masses and decay widths are $M_{T_{c\bar{s}}^{a0}(2900)}=2892\pm 14\pm 15$
MeV, $\Gamma_{T_{c\bar{s}}^{a0}(2900)}=119\pm 26\pm 12$ MeV,
$M_{T_{c\bar{s}}^{a++}(2900)}=2921\pm 17\pm 19$ MeV, and
$\Gamma_{T_{c\bar{s}}^{a++}(2900)}=137\pm 32\pm 14$ MeV. Due to their near-
threshold behaviors and quantum numbers, the $T_{c\bar{s}}^{a0(++)}(2900)$ are
proposed as isovector $D^{\ast}K^{\ast}$ molecules with $J^{P}=0^{+}$. The
hadronic molecular explanations remind us of the famous charm-strange states,
the $D_{s0}(2317)$ and $D_{s1}(2460)$. These states were observed by the BaBar
and CLEO experiments BaBar:2003oey ; CLEO:2003ggt  in the $D_{s}^{+}\pi^{0}$
and $D_{s}^{*+}\pi^{0}$ decay modes, respectively. These states were further
confirmed by the Belle and BaBar experiments Belle:2003kup ; BaBar:2006eep ;
BaBar:2003cdx . Theoretically, since they are located below the $DK$ and
$D^{*}K$ mass thresholds, $D_{s0}(2317)$ and $D_{s1}(2460)$ are often
interpreted as $S$-wave isoscalar $DK$ and $D^{*}K$ molecular states,
respectively Barnes:2003dj ; Chen:2004dy ; Guo:2006fu ; Guo:2006rp ;
Navarra:2015iea ; Faessler:2007us ; Xie:2010zza ; Feng:2012zze ; Yang:2021tvc
.

Additionally, it is worth mentioning that in 2020, the LHCb Collaboration
first observed two other structures in the $D^{-}K^{+}$ invariant mass
distribution, namely, the $X_{0}(2900)$ and $X_{1}(2900)$, by analyzing the
decay amplitude for the $B^{+}\to D^{+}D^{-}K^{+}$ process LHCb:2020pxc . The
corresponding masses and decay widths are $M_{X_{0}(2900)}=2866\pm 7\pm 2$
MeV, $\Gamma_{X_{0}(2900)}=57\pm 12\pm 4$ MeV, $M_{X_{1}(2900)}=2904\pm 5\pm
1$ MeV, and $\Gamma_{X_{1}(2900)}=110\pm 11\pm 4$ MeV, respectively. Their
spin-parities are $J^{P}=0^{+}$ and $1^{-}$, respectively. The $X_{0}(2900)$
and $X_{1}(2900)$ are also good candidates for open-charm tetraquark states,
as the valence quark components in the final states are $[\bar{c}suu]$.
Because the $X_{0,1}(2900)$ are very close to the mass thresholds for the
$\bar{D}^{*}K^{*}$ and $\bar{D}_{1}K$ systems, they are often regarded as
isoscalar meson-meson hadron molecules Molina:2020hde ; Kong:2021ohg ;
Wang:2021lwy ; Chen:2020aos ; Agaev:2020nrc ; He:2020btl ; Qi:2021iyv ;
Chen:2021tad .

In the bottom sector, there is an interesting discussion on the existence of
open-bottom tetraquarks. Experimentally, in 2016, the DØ Collaboration
reported a narrow structure, $X(5568)$, in the $B_{s}^{0}\pi^{\pm}$ invariant
mass spectrum D0:2016mwd . However, it was not confirmed by the ATLAS, CDF,
CMS, and LHCb collaborations ATLAS:2018udc ; CDF:2017dwr ; CMS:2017hfy ;
LHCb:2016dxl . Theoretically, the hadronic explanations for the $X(5568)$ have
been examined in several studies Xiao:2016mho ; Agaev:2016urs ; Burns:2016gvy
; Albaladejo:2016eps ; Chen:2016ypj ; Lu:2016kxm ; Sun:2016tmz . In
particular, in our previous work Chen:2016ypj , we found that the $X(5568)$
cannot be assigned as an isovector $B\bar{K}$ molecular state, however, there
can exist possible isoscalar $B^{(*)}\bar{K}^{(*)}$ molecular states. In
addition, the $B_{sJ}(6158)$ was explained as a $\bar{B}K^{*}$ molecular state
with $1^{+}$ Kong:2021ohg .

$\displaystyle\begin{array}[]{ccccc}\text{Single
charm}&\rightleftharpoons&\text{Double charm}&\rightleftharpoons&\text{Single
bottom}\\\ \hline\cr D_{s0}(2317)\sim
DK&&P_{cc\bar{s}}\sim\Xi_{cc}K^{(*)}&&X(5568)\sim\bar{B}{K}\\\
D_{s1}(2460)\sim D^{*}K&&&&B_{sJ}(6158)\sim\bar{B}K^{*}\\\
T_{c\bar{s}}(2900)\sim D^{*}K^{*}&&&&\\\
X(2900)\sim\bar{D}^{*}K^{*}&&P_{cc{s}}\sim\Xi_{cc}\bar{K}^{(*)}&&\\\
\hline\cr\\\ \end{array}$

As shown above, it is very interesting to explore the $\Xi_{cc}K^{(*)}$ and
$\Xi_{cc}\bar{K}^{(*)}$ interactions to search for possible doubly charmed
molecular candidates with strangeness $|S|=1$. In fact, if we ignore the
coupling between the heavy quarks and the light quarks, the interactions
between the double charmed baryon and kaon are very similar to the
interactions between the charmed meson and kaon. Due to the heavier reduced
masses, there is a potential for the existence of the $\Xi_{cc}K^{(*)}$ and
$\Xi_{cc}\bar{K}^{(*)}$ bound states.

In this work, we adopt the one-boson-exchange (OBE) model to derive the
effective potentials for the $\Xi_{cc}K^{(*)}$ and $\Xi_{cc}\bar{K}^{(*)}$
systems, including the $\pi$, $\sigma$, $\eta$, $\rho$, and $\omega$ exchanges
interactions. In our calculations, we also consider the $S-D$ wave mixing
effects and the coupled-channel effects. Using these OBE effective potentials,
we solve the coupled channel Schrödinger equations to search for possible
loosely bound state solutions.

Our study can not only predict the existence of possible $\Xi_{cc}K^{(*)}$ and
$\Xi_{cc}\bar{K}^{(*)}$ molecular candidates, but also deepen our
understanding of the interactions between the doubly charmed baryon and the
(anti-)strange meson. In addition, this study will test the molecular-state
picture for the open-charm exotic structures with strangeness to some extent.

Afterward, we use the obtained bound state solutions to explore the M1
radiative decay behaviors and the magnetic moments of the predicted molecular
candidates. And we employ the constituent quark model, which is often adopted
to study the electromagnetic properties of the baryons Schlumpf:1993rm ;
Cheng:1997kr ; Majethiya:2009vx ; Sharma:2010vv ; Ghalenovi:2014swa ;
Girdhar:2015gsa ; Thakkar:2016sog ; Dahiya:2018ahb ; Mohan:2022sxm  and
multiquarks Liu:2003ab ; Huang:2004tn ; Zhu:2004xa ; Wang:2016dzu ; Li:2021ryu
; Wang:2022tib ; Zhou:2022gra ; Gao:2021hmv ; Wang:2022nqs ; Wang:2023aob ;
An:2022qpt ; Wu:2022gie ; Guo:2023fih ; Wang:2023bek ; Wang:2023ael ;
Li:2024wxr ; Li:2024jlq ; Lai:2024jfe . These efforts can provide valuable
clues to understand the inner structures of the open heavy flavor multiquarks.

This paper is organized as follows. In Sec. II, we introduce the theoretical
frameworks of the OBE effective potentials, the M1 radiative decay behaviors,
and the magnetic moments for the $\Xi_{cc}K^{(*)}$ and $\Xi_{cc}\bar{K}^{(*)}$
systems. In Sec. III, we present the corresponding numerical results. The
paper ends with a summary in Sec. IV.

##  II Theoretical frameworks

###  II.1 OBE effective potentials

In this subsection, we study the $\Xi_{cc}K^{(*)}$ and $\Xi_{cc}\bar{K}^{(*)}$
interactions using the OBE model. First, we introduce the relevant
Lagrangians. For the interactions between the strange mesons and the light
mesons $(\sigma,\pi,\eta,\rho,\omega)$, we construct the effective Lagrangians
based on $SU(3)$ symmetry Lin:1999ad ; Nagahiro:2008mn ; Chen:2019uvv , i.e.,

$\displaystyle\mathcal{L}_{PP\sigma}$ $\displaystyle=$ $\displaystyle
g_{\sigma}m_{P}\langle PP\sigma\rangle,$ (2)
$\displaystyle\mathcal{L}_{VV\sigma}$ $\displaystyle=$ $\displaystyle
g_{\sigma}m_{V}\langle VV\sigma\rangle,$ (3) $\displaystyle\mathcal{L}_{PPV}$
$\displaystyle=$
$\displaystyle\frac{ig}{\sqrt{2}}\langle\partial^{\mu}{P}(PV_{\mu}-V_{\mu}P)\rangle,$
(4) $\displaystyle\mathcal{L}_{VVP}$ $\displaystyle=$
$\displaystyle\frac{g_{VVP}}{\sqrt{2}}\varepsilon^{\mu\nu\alpha\beta}\langle\partial_{\mu}{V_{\nu}}\partial_{\alpha}{V_{\beta}}P\rangle,$
(5) $\displaystyle\mathcal{L}_{VVV}$ $\displaystyle=$
$\displaystyle\frac{ig}{2\sqrt{2}}\langle\partial^{\mu}{V^{\nu}}(V_{\mu}V_{\nu}-V_{\nu}V_{\mu})\rangle.$
(6)

Here, $P$ and $V$ are the $SU(3)$ matrices containing the octets of vector and
pseudoscalar mesons, respectively, which are given as follows:

$\displaystyle P$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&K^{0}\\\
K^{-}&\bar{K}^{0}&-\sqrt{\frac{2}{3}}\eta\end{array}\right),$ $\displaystyle
V$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&\rho^{+}&K^{*+}\\\
\rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\end{array}\right).$

Once expanding the Eqs. (2)$-$(6), we can obtain

$\displaystyle\mathcal{L}_{K^{(*)}K^{(*)}\sigma}$ $\displaystyle=$
$\displaystyle
g^{\prime}_{\sigma}m_{K}\bar{K}K\sigma-g^{\prime}_{\sigma}m_{K^{*}}\bar{K^{*}}\cdot{K^{*}}\sigma,$
(9) $\displaystyle\mathcal{L}_{PKK^{*}}$ $\displaystyle=$
$\displaystyle\frac{ig}{4}\left[\left(\bar{K}^{*\mu}K-\bar{K}K^{*\mu}\right)\left(\bm{\tau}\cdot{\partial_{\mu}{\bm{\pi}}}+\frac{\partial_{\mu}{\eta}}{\sqrt{3}}\right)\right]$
(10)
$\displaystyle+\frac{ig}{4}\left[\left(\partial_{\mu}{\bar{K}}K^{*\mu}-\bar{K}^{*\mu}\partial_{\mu}{K}\right)\left(\bm{\tau}\cdot{\bm{\pi}}+\frac{\eta}{\sqrt{3}}\right)\right],$
$\displaystyle\mathcal{L}_{VKK}$ $\displaystyle=$
$\displaystyle\frac{ig}{4}\left[\bar{K}\partial_{\mu}{K}-\partial_{\mu}{\bar{K}K}\right]\left(\bm{\tau}\cdot{\bm{\rho}^{\mu}}+\omega^{\mu}\right),$
(11) $\displaystyle\mathcal{L}_{VK^{*}K^{*}}$ $\displaystyle=$
$\displaystyle\frac{ig}{4}\left[\left(\bar{K}^{*}_{\mu}\partial^{\mu}{K}^{*\nu}-\partial^{\mu}{\bar{K}^{*\nu}}K^{*}_{\mu}\right)\left(\bm{\tau}\cdot{\bm{\rho}_{\nu}}+\omega_{\nu}\right)\right]$
(12)
$\displaystyle+\frac{ig}{4}\left[\left(\partial^{\mu}{\bar{K}}^{*\nu}K^{*}_{\nu}-\bar{K}^{*}_{\nu}\partial^{\mu}{\bar{K}^{*\nu}}\right)\left(\bm{\tau}\cdot{\bm{\rho}_{\mu}}+\omega_{\mu}\right)\right]$
$\displaystyle+\frac{ig}{4}\left[\left(\bar{K}^{*}_{\nu}K^{*}_{\mu}-\bar{K}^{*}_{\mu}K^{*}_{\nu}\right)\left(\bm{\tau}\cdot{\partial^{\mu}{\bm{\rho}}^{\nu}+\partial^{\mu}{\omega}^{\nu}}\right)\right],$
$\displaystyle\mathcal{L}_{PK^{*}K^{*}}$ $\displaystyle=$ $\displaystyle
g_{VVP}\varepsilon_{\mu\nu\alpha\beta}\partial^{\mu}\bar{K}^{*\nu}\partial^{\alpha}{K}^{*\beta}\left(\bm{\tau}\cdot{\bm{\pi}}+\frac{\eta}{\sqrt{3}}\right),$
(13) $\displaystyle\mathcal{L}_{VKK^{*}}$ $\displaystyle=$ $\displaystyle
g_{VVP}\varepsilon_{\mu\nu\alpha\beta}\left(\partial^{\mu}{\bar{K}^{*\nu}}K+\bar{K}\partial^{\mu}{K}^{*\nu}\right)$
(14)
$\displaystyle\left(\bm{\tau}\cdot{\partial^{\alpha}{\bm{\rho}^{\beta}}}+\partial^{\alpha}{\omega^{\beta}}\right).$

Drawing from the experience from the nucleon-nucleon interactions, we
construct the effective Lagrangians between the doubly charmed baryons and the
light mesons as follows Meng:2017fwb ; Meng:2017udf :

$\displaystyle\mathcal{L}_{\Xi_{cc}\Xi_{cc}\sigma}$ $\displaystyle=$
$\displaystyle g_{\sigma}\bar{\Xi}_{cc}\sigma\Xi_{cc},$ (15)
$\displaystyle\mathcal{L}_{\Xi_{cc}\Xi_{cc}P}$ $\displaystyle=$ $\displaystyle
g_{\pi}\bar{\Xi}_{cc}i\gamma_{5}P\Xi_{cc},$ (16)
$\displaystyle\mathcal{L}_{\Xi_{cc}\Xi_{cc}V}$ $\displaystyle=$ $\displaystyle
h_{v}\bar{\Xi}_{cc}\gamma_{\mu}V^{\mu}\Xi_{cc}+\frac{f_{v}}{2m_{\Xi_{cc}}}\bar{\Xi}_{cc}\sigma_{\mu\nu}\partial^{\mu}{V^{\nu}}\Xi_{cc}.$
(17)

We estimate the coupling constants in Eqs. (2)-(17) as follows. With the help
of the the quark model, we can obtain the relationship between the coupling
constants of double-charm baryons and those of nucleon-nucleon interaction
Machleidt:2000ge ; Machleidt:1987hj ; Cao:2010km ,
$g_{\sigma}=\frac{1}{3}g_{{\sigma}NN}$,
$g_{\pi}=-\frac{\sqrt{2}m_{\Xi_{cc}}}{5m_{N}}g_{{\pi}NN}$,
$h_{v}=\sqrt{2}g_{{\rho}NN}$, and
$h_{v}+f_{v}=-\frac{\sqrt{2}m_{\Xi_{cc}}}{5m_{N}}(g_{{\rho}NN}+f_{{\rho}NN})$.
$g^{\prime}_{\sigma}=-3.65$ Chen:2023qlx , $g=12.0$ Chen:2017xat ,
$g_{VVP}=3g^{2}/(32\sqrt{2}\pi^{2}f_{\pi})$, and $f_{\pi}=132$ MeV are
estimated by using the vector meson dominance model Kaymakcalan:1983qq . We
further fix the phase between all the coupling constants in the quark model.
These values are summarized in Table 1.

Table 1: Coupling constants adopted in our calculation. The coupling constants for the nucleon-nucleon interactions are taken from Refs. Machleidt:2000ge ; Machleidt:1987hj ; Cao:2010km . $\sigma$ | $\pi/\eta$ | $\rho/\omega$ |
---|---|---|---
$\frac{g^{2}_{{\sigma}NN}}{4\pi}=5.69$ | $\frac{g^{2}_{{\pi}NN}}{4\pi}=13.60$ | $\frac{g^{2}_{{\rho}NN}}{4\pi}=0.84$ | $\frac{f_{{\rho}NN}}{g_{{\rho}NN}}=6.10$
$g_{\sigma}=-2.82$ | $g_{\pi}=14.26$ | $h_{v}=4.65$ | $f_{v}=-28.11$

With the preparations of the effective Lagrangians, we next derive the OBE
effective potentials for the $\Xi_{cc}K^{(*)}$ systems, which relate to the
scattering amplitudes in the Breit approximation:

$\displaystyle\mathcal{V}_{E}^{h_{1}h_{2}\to h_{3}h_{4}}(\bm{q})$
$\displaystyle=$ $\displaystyle-\frac{\mathcal{M}(h_{1}h_{2}\to
h_{3}h_{4})}{\sqrt{\prod_{i}2M_{i}\prod_{f}2M_{f}}}.$ (18)

Here, $\mathcal{M}(h_{1}h_{2}\to h_{3}h_{4})$ denotes the scattering amplitude
for the $h_{1}h_{2}\to h_{3}h_{4}$ process by exchanging the light meson
$(\sigma,\pi,\eta,\rho,\omega)$ in $t$ channel. $M_{i}$ and $M_{f}$ are the
masses of the initial states ($h_{1}$, $h_{2}$) and final states ($h_{3}$,
$h_{4}$), respectively. By performing a Fourier transformation, we can obtain
the effective potential in the coordinate space $\mathcal{V}(\bm{r})$:

$\displaystyle\mathcal{V}_{E}^{h_{1}h_{2}\to
h_{3}h_{4}}(\bm{r})=\int\frac{d^{3}\bm{q}}{(2\pi)^{3}}e^{i\bm{q}\cdot\bm{r}}\mathcal{V}_{E}^{h_{1}h_{2}\to
h_{3}h_{4}}(\bm{q})\mathcal{F}^{2}(q^{2},m_{E}^{2}).$

Here, we introduce a monopole form factor
$\mathcal{F}(q^{2},m_{E}^{2})=(\Lambda^{2}-m_{E}^{2})/(\Lambda^{2}-q^{2})$ at
each interactive vertex, which can reflect the off-shell effect of the
exchanged boson. $\Lambda$, $m_{E}$, and $q$ are the cutoff, mass, and four-
momentum of the exchanged meson, respectively. In our OBE effective
potentials, the cutoff $\Lambda$ is the only free parameter, relating to the
typical hadronic scale or the intrinsic size of hadrons. Based on the
experience of the deuteron Tornqvist:1993ng ; Tornqvist:1993vu , a reasonable
cutoff value is around $\Lambda\sim 1.00$ GeV.

In order to obtain the total OBE effective potentials, we further construct
the wave functions for the investigated systems. For the $\Xi_{cc}K^{(*)}$
systems, the wave function $|\Psi\rangle$ can be constructed as the direct
product of the color wave function $|\phi_{c}\rangle$, the flavor wave
function $|I,I_{3}\rangle$, the spin-orbit wave function
$|{}^{2S+1}L_{J}\rangle$, and the radial wave function $|\psi(r)\rangle$:

$\displaystyle|\Psi\rangle$ $\displaystyle=$
$\displaystyle|\phi_{c}\rangle|I,I_{3}\rangle|{}^{2S+1}L_{J}\rangle|\psi(r)\rangle.$
(19)

Here, the color wave functions is 1, indicating that the color configuration
is a singlet. The isospin $I$ can be either 0 or 1 for the $\Xi_{cc}K^{(*)}$
systems. We can construct their flavor function as follows:

$\displaystyle|0,0\rangle$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}\left(\left|\Xi_{cc}^{++}K^{(*)0}\right\rangle-\left|\Xi_{cc}^{+}K^{(*)+}\right\rangle\right),$
$\displaystyle|1,1\rangle$ $\displaystyle=$
$\displaystyle\left|\Xi_{cc}^{++}K^{(*)+}\right\rangle,$
$\displaystyle|1,0\rangle$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}\left(\left|\Xi_{cc}^{++}K^{(*)0}\right\rangle+\left|\Xi_{cc}^{+}K^{(*)+}\right\rangle\right),$
$\displaystyle|1,-1\rangle$ $\displaystyle=$
$\displaystyle\left|\Xi_{cc}^{+}K^{(*)0}\right\rangle.$

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

$\displaystyle\Xi_{cc}K:$ $\displaystyle J^{P}=\frac{1}{2}^{-}$
$\displaystyle\left|{}^{2}S_{\frac{1}{2}}\right\rangle,$ (20)
$\displaystyle\Xi_{cc}K^{*}:$ $\displaystyle J^{P}=\frac{1}{2}^{-}$
$\displaystyle\left|{}^{{2}}S_{\frac{1}{2}}\right\rangle,\quad\left|{}^{4}D_{\frac{1}{2}}\right\rangle,$
(21) $\displaystyle J^{P}=\frac{3}{2}^{-}$
$\displaystyle\left|{}^{4}S_{\frac{3}{2}}\right\rangle,\quad\left|{}^{2}D_{\frac{3}{2}}\right\rangle,\quad\left|{}^{4}D_{\frac{3}{2}}\right\rangle.$
(22)

The general expressions can be expressed as follows:

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

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

Finally, the total OBE effective potentials for the $\Xi_{cc}K\to\Xi_{cc}K$,
$\Xi_{cc}K^{*}\to\Xi_{cc}K^{*}$, and $\Xi_{cc}K\to\Xi_{cc}K^{*}$ processes,
denoted as $V_{11}$, $V_{22}$, and $V_{12}$, respectively, can be written as

$\displaystyle V_{11}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}g_{\sigma}g_{\sigma}^{\prime}Y(\Lambda,m_{\sigma},r)$
(23)
$\displaystyle+\frac{\sqrt{2}}{8}gh_{V}\mathcal{Y}^{I}_{1}(\Lambda,m_{\rho},m_{\omega},r)$
$\displaystyle+\frac{\sqrt{2}f_{V}g}{32m^{2}_{\Xi_{cc}}}\nabla^{2}\mathcal{Y}^{I}_{1}(\Lambda,m_{\rho},m_{\omega},r),$
$\displaystyle V_{22}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}g_{\sigma}g_{\sigma}^{\prime}(\epsilon^{{\dagger}}_{4}\cdot\epsilon_{2})Y(\Lambda,m_{\sigma},r)$
$\displaystyle+\frac{\sqrt{2}g_{\pi}g_{VVP}}{24m_{\Xi_{cc}}}\mathcal{F}(r,i\bm{\sigma},\bm{\epsilon}_{2}\times\bm{\epsilon}_{4}^{\dagger})\mathcal{Y}^{I}_{2}(\Lambda,m_{\pi},m_{\eta},r)$
$\displaystyle+\frac{\sqrt{2}h_{V}g}{8}(\bm{\epsilon_{4}^{\dagger}}\cdot\bm{\epsilon_{2}})\mathcal{Y}^{I}_{1}(\Lambda,m_{\rho},m_{\omega},r)$
$\displaystyle+\frac{\sqrt{2}h_{V}g}{32m^{2}_{\Xi_{cc}}}(\bm{\epsilon_{4}^{\dagger}}\cdot\bm{\epsilon_{2}})\nabla^{2}\mathcal{Y}^{I}_{1}(\Lambda,m_{\rho},m_{\omega},r)$
$\displaystyle+\frac{\sqrt{2}}{64}\frac{(h_{V}+f_{V})g}{m_{\Xi_{cc}}m_{K^{*}}}\mathcal{F}^{\prime}(r,i\bm{\sigma},\bm{\epsilon}_{2}\times\bm{\epsilon}_{4}^{\dagger})\mathcal{Y}^{I}_{1}(\Lambda,m_{\rho},m_{\omega},r),$
$\displaystyle V_{12}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{2}}{24}\frac{h_{V}g_{VVP}}{24m_{\Xi_{cc}}}\sqrt{\frac{m_{K^{*}}}{m_{K}}}\mathcal{F}^{\prime}(r,\bm{\sigma},\bm{\epsilon}_{4}^{\dagger})\mathcal{Y}^{I}_{1}(\Lambda_{0},m_{\rho
0},m_{\omega 0},r)$
$\displaystyle-\frac{\sqrt{2}}{24}\frac{f_{V}g_{VVP}}{24m_{\Xi_{cc}}}\sqrt{\frac{m_{K^{*}}}{m_{K}}}\mathcal{F}^{\prime}(r,\bm{\sigma},\bm{\epsilon}_{4}^{\dagger})\mathcal{Y}^{I}_{1}(\Lambda_{0},m_{\rho
0},m_{\omega 0},r)$
$\displaystyle-\frac{\sqrt{2}}{64}\frac{g_{\pi}g}{m_{\Xi_{cc}}\sqrt{m_{K}m_{K^{*}}}}\mathcal{F}(r,\bm{\sigma},\bm{\epsilon}_{4}^{\dagger})\mathcal{Y}^{I}_{2}(\Lambda_{0},m_{\pi
0},m_{\eta 0},r).$

Here, the variables in the Eq. (LABEL:potential3) are
$\Lambda_{0}^{2}=\Lambda^{2}-q_{0}^{2}$, $m_{0}^{2}=m^{2}-q_{0}^{2}$, and
$q_{0}=\frac{m_{K^{*}}^{2}-m_{K}^{2}}{2(m_{K^{*}}+m_{\Xi_{cc}})}$. And we
define serval useful functions:

$\displaystyle\mathcal{F}(r,\bm{a},\bm{b})$ $\displaystyle=$
$\displaystyle\bm{a}\cdot\bm{b}\nabla^{2}+S(\hat{r},\bm{a},\bm{b})r\frac{\partial}{\partial
r}\frac{1}{r}\frac{\partial}{\partial r},$
$\displaystyle\mathcal{F}^{\prime}(r,\bm{a},\bm{b})$ $\displaystyle=$
$\displaystyle
2\bm{a}\cdot\bm{b}\nabla^{2}-S(\hat{r},\bm{a},\bm{b})r\frac{\partial}{\partial
r}\frac{1}{r}\frac{\partial}{\partial r},$ $\displaystyle
S(\hat{r},\bm{a},\bm{b})$ $\displaystyle=$ $\displaystyle
3(\hat{r}\cdot\bm{a})(\hat{r}\cdot\bm{b})-\bm{a}\cdot\bm{b},$
$\displaystyle\mathcal{Y}^{I}_{1}(\Lambda,m_{1},m_{2},r)$ $\displaystyle=$
$\displaystyle\mathcal{G}(I)Y(\Lambda,m_{1},r)+Y(\Lambda,m_{1},r),$
$\displaystyle\mathcal{Y}^{I}_{2}(\Lambda,m_{1},m_{2},r)$ $\displaystyle=$
$\displaystyle\mathcal{G}(I)Y(\Lambda,m_{2},r)+\frac{1}{3}Y(\Lambda,m_{2},r),$

with $Y(\Lambda,m,r)=\frac{e^{-mr}-e^{-\Lambda r}}{4\pi
r}-\frac{\Lambda^{2}-m^{2}}{8\pi\Lambda}e^{-\Lambda r}$, the isospin factors
$\mathcal{G}(0)=-3$ and $\mathcal{G}(1)=1$. The $\bm{a}\cdot\bm{b}$ and
$S(\hat{r},\bm{a},\bm{b})$ are the spin-spin interactions and tensor force
operators, respectively. In Table 2, we collect the matrices elements for the
spin-spin interactions and tensor force operators involved in the OBE
effective potentials as showed in Eqs. (II.1)-(LABEL:potential3), they can be
obtained by sandwiched the discussed spin-orbit wave functions in Eqs.
(20)-(22).

Table 2: Matrices elements for the spin-spin interactions and tensor force operators in the OBE effective potentials. | $J=\frac{1}{2}$ | $J=\frac{3}{2}$
---|---|---
$\langle\bm{\epsilon_{4}^{\dagger}}\cdot\bm{\epsilon_{2}}\rangle$ | $\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right)$ | $\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&1\end{array}\right)$
$\langle i\bm{\sigma}\bm{\cdot}\left(\bm{\epsilon_{2}}\times\bm{\epsilon_{4}^{{\dagger}}}\right)\rangle$ | $\left(\begin{array}[]{cc}-2&0\\\ 0&1\end{array}\right)$ | $\left(\begin{array}[]{ccc}1&0&0\\\ 0&-2&0\\\ 0&0&1\end{array}\right)$
$\langle S(\hat{r},\bm{\sigma,i\epsilon_{2}\times\epsilon_{4}^{\dagger}})\rangle$ | $\left(\begin{array}[]{cc}0&-\sqrt{2}\\\ -\sqrt{2}&-2\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&1&2\\\ 1&0&-1\\\ 2&-1&0\end{array}\right)$
$\langle\bm{\sigma}\bm{\cdot}\bm{\epsilon_{4}^{\dagger}}\rangle$ | $\left(\begin{array}[]{c}\sqrt{3}\\\ 0\end{array}\right)$ | …
$\langle S(\hat{r},\bm{\sigma},\bm{\epsilon_{4}^{\dagger}})\rangle$ | $\left(\begin{array}[]{c}0\\\ -\sqrt{6}\\\ \end{array}\right)$ | …

###  II.2 The M1 radiative decay behaviors and the magnetic moments

In this work, we further discuss the M1 radiative decay behaviors and the
magnetic moments of the possible $\Xi_{cc}K^{(*)}$ molecules. If the possible
$\Xi_{cc}K$ and $\Xi_{cc}K^{*}$ molecules can coexist, we can estimate the M1
radiative decay width for the $\Xi_{cc}K^{*}\to\Xi_{cc}K+\gamma$ process
Wang:2022nqs :

$\displaystyle\Gamma_{H\to H^{\prime}\gamma}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{EM}}{2J_{H}+1}\frac{k^{3}}{m_{p}^{2}}\frac{\sum_{J_{H^{\prime}z},J_{Hz}}\left(\begin{array}[]{ccc}J_{H^{\prime}}&1&J_{H}\\\
-J_{H^{\prime}z}&0&J_{Hz}\end{array}\right)^{2}}{\left(\begin{array}[]{ccc}J_{H^{\prime}}&1&J_{H}\\\
-J_{z}&0&J_{z}\end{array}\right)^{2}}\frac{|\mu_{H\to
H^{\prime}}|^{2}}{\mu_{N}^{2}}.$ (30)

In the above formula, $\alpha_{EM}\approx 1/137$ is the electromagnetic fine
structure constant, $k$ is the moment of the emitted photon with
$k=(m_{H}^{2}-m_{H^{\prime}}^{2})/2m_{H}$, $J_{A}$ and $J_{Az}$ with
$A=(H,H^{\prime})$ denote the total and the $z-$component spin of the involved
hadrons, respectively. $\mu_{N}=e/2m_{p}$ with $m_{p}=0.938$ GeV
ParticleDataGroup:2022pth .
$\left(\begin{array}[]{ccc}J_{H^{\prime}}&1&J_{H}\\\
-J_{H^{\prime}z}&0&J_{Hz}\end{array}\right)$ and
$\left(\begin{array}[]{ccc}J_{H^{\prime}}&1&J_{H}\\\
-J_{z}&0&J_{z}\end{array}\right)$ stand for the $3-j$ coefficients, and
$J_{z}$ is the minimun value of the total spin $J_{A}$ of the involved
hadrons. $\mu_{H\to H^{\prime}}$ is the corresponding transition magnetic
moment, which can be expressed as

$\displaystyle\mu_{H\to H^{\prime}}$ $\displaystyle=$
$\displaystyle\left\langle
J_{H^{\prime}},J_{z}\right|\sum_{j^{\prime}}\hat{\mu}_{zj^{\prime}}^{\text{spin}}e^{-i\bm{k}\cdot\bm{r}_{j^{\prime}}}+\hat{\mu}_{z}^{\text{orbital}}\left|J_{H},J_{z}\right\rangle.$
(32)

Here, $e^{-i\bm{k}\cdot\bm{r}_{j^{\prime}}}$ stands for the spatial wave
function of the emitted photon. In our calculations,
$e^{-i\bm{k}\cdot\bm{r}_{j^{\prime}}}$ is expanded as a serial of series
contening the spherical Bessel functions $j_{l}(x)$ and the spherical harmonic
functions $Y_{lm}(\Omega)$, i.e.,

$\displaystyle e^{-i\bm{k}\cdot\bm{r}_{j^{\prime}}}$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\sum_{m=-l}^{l}4\pi(-i)^{l}j_{l}(kr_{j^{\prime}})Y^{*}_{lm}(\Omega_{\bm{k}})Y_{lm}(\Omega_{{\bm{r}}_{j^{\prime}}}).$
(33)

In order to gain a more comprehensive understanding of the spatial wave
functions of the initial and final states, it is necessary to consider the
spatial wave functions of the molecular states, the doubly charmed baryon, and
the (anti-)kaon. In the case of the molecular states, we utilise the precise
spatial wave functions by solving the Schrödinger equation. In the case of the
doubly charmed baryon and the (anti-)kaon, we adopt the simple harmonic
oscillator wave function, given by the expression $\psi_{n,l,m}(\beta,{\bf
r})$, to describe their spatial wave functions. This can be written as

$\displaystyle\psi_{n,l,m}(\beta,{\bf r})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2n!}{\Gamma(n+l+\frac{3}{2})}}L_{n}^{l+\frac{1}{2}}(\beta^{2}r^{2})\beta^{l+\frac{3}{2}}$
(34)
$\displaystyle\times{\mathrm{e}}^{-\frac{\beta^{2}r^{2}}{2}}r^{l}Y_{lm}(\Omega).$

In this context, the radial, orbital, and magnetic quantum numbers of the
hadrons are designated as $n$, $l$, and $m$, respectively.
$L_{n}^{l+\frac{1}{2}}(x)$ is used to represent the associated Laguerre
polynomial. The oscillating parameters $\beta$ is taken to be $0.4~{}{\rm
GeV}$ in the realistic calculations Wang:2022nqs .

In the constituent quark model, the spin magnetic moment for all the involved
quarks $(\hat{\mu}_{z}^{\text{spin}})$ and the orbital magnetic moment linking
the molecular components $(\hat{\mu}_{z}^{\text{orbital}})$ can be expressed
as Wang:2024sbw ; Wang:2023ael ; Lai:2024jfe ; Wang:2022nqs :

$\displaystyle\hat{\mu}_{j^{\prime}z}^{\text{spin}}$ $\displaystyle=$
$\displaystyle\frac{e_{j^{\prime}}\hat{\sigma}_{zj^{\prime}}}{2m_{j^{\prime}}},$
(35) $\displaystyle\hat{\mu}_{z}^{\text{orbital}}$ $\displaystyle=$
$\displaystyle\left(\frac{m_{M}}{m_{B}+m_{M}}\frac{e_{B}}{2m_{B}}+\frac{m_{B}}{m_{B}+m_{M}}\frac{e_{M}}{2m_{M}}\right)\hat{L}_{z}.$
(36)

Here, $e_{j^{\prime},B,M}$ and $m_{j^{\prime},B,M}$ are the charges and the
masses of the $j^{\prime}-$th constituent quarks of the hadron, the
constituent baryon $B$ and meson $M$, respectively.
$\hat{\sigma}_{zj^{\prime}}$ and $\hat{L}_{z}$ are the $z$-component of the
Pauli spin operator of the $j^{\prime}-$th constituent quarks of the hadron
and the orbital angular momentum operator between the baryon and meson,
respectively.

In general, the hadronic magnetic moment $\mu_{H}$ can be estimated by the
expectation values of the $z-$component of the total magnetic moment operator
$\hat{\mu}_{z}$, i.e.,
$\hat{\mu}_{z}=\sum_{j^{\prime}}\hat{\mu}_{j^{\prime}z}^{\text{spin}}+\hat{\mu}_{z}^{\text{orbital}}$.
For the possible $\Xi_{cc}K^{(*)}$ molecules, their magnetic moments read as
$\mu_{H}=\langle\Psi_{H}|\hat{\mu}_{z}|\Psi_{H}\rangle$, where
$|\Psi_{H}\rangle$ is the wave function of the hadron $H$. For example, the
magnetic moment for the $S-$wave $\Xi_{cc}K^{*}$ state with
$I(J^{P})=0(1/2^{-})$ can be written as

$\displaystyle\mu_{\Xi_{cc}K^{*}}$ $\displaystyle=$
$\displaystyle\langle\Psi_{\Xi_{cc}K^{*}}|\hat{\mu}_{z}|\Psi_{\Xi_{cc}K^{*}}\rangle$
(37) $\displaystyle=$ $\displaystyle\langle
I^{\prime},I_{3}^{\prime}||I,I_{3}\rangle\langle\psi^{\prime}(r)||\psi(r)\rangle\langle{}^{2}S_{1/2}|\hat{\mu}_{z}|{}^{2}S_{1/2}\rangle.$

Once we construct the spin and flavor wave functions for the doubly charmed
baryon $\Xi_{cc}[ccq]$ and strange mesons $K^{(*)}[q\bar{s}]$, one can easily
write down the corresponding magnetic moments. As collected in Table 3, we
summary the spin and flavor wave functions and the magnetic moments for the
doubly charmed baryon $\Xi_{cc}$ and strange mesons $K^{(*)}$. The masses of
the constituent quarks can be taken as $m_{u}=0.336$ GeV, $m_{d}=0.336$ GeV,
$m_{s}=0.450$ GeV, and $m_{c}=1.680$ GeV Kumar:2005ei .

Table 3: The spin and flavor wave functions and the magnetic moments for the doubly charmed baryon $\Xi_{cc}$ and strange mesons $K^{(*)}$. Hadron | $\left|S,S_{z}\right\rangle$ | $\left|I,I_{3}\right\rangle$ | $\mu_{H}$
---|---|---|---
$\Xi_{cc}^{+}$ | $\left|1/2,1/2\right\rangle$ | $\left|1/2,-1/2\right\rangle$ | $\frac{4}{3}\mu_{c}-\frac{1}{3}\mu_{d}$
| $\left|1/2,-1/2\right\rangle$ |  | $-\frac{4}{3}\mu_{c}+\frac{1}{3}\mu_{d}$
$\Xi_{cc}^{++}$ | $\left|1/2,1/2\right\rangle$ | $\left|1/2,1/2\right\rangle$ | $\frac{4}{3}\mu_{c}-\frac{1}{3}\mu_{u}$
| $\left|1/2,-1/2\right\rangle$ |  | $-\frac{4}{3}\mu_{c}+\frac{1}{3}\mu_{u}$
$K^{*0}$ | $\left|1,1\right\rangle$ | $\left|1/2,-1/2\right\rangle$ | $\mu_{d}+\mu_{\bar{s}}$
| $\left|1,0\right\rangle$ |  | $0$
| $\left|1,-1\right\rangle$ |  | $-\mu_{d}-\mu_{\bar{s}}$
$K^{*+}$ | $\left|1,1\right\rangle$ | $\left|1/2,1/2\right\rangle$ | $\mu_{\bar{s}}+\mu_{u}$
| $\left|1,0\right\rangle$ |  | $0$
| $\left|1,-1\right\rangle$ |  | $-\mu_{\bar{s}}-\mu_{u}$
$K^{0}/K^{+}$ | $\left|0,0\right\rangle$ | $\left|1/2,-1/2(1/2)\right\rangle$ | $0$

##  III Numerical results

After deriving the total OBE effective potentials, we search for the bound
state solutions (binding energy $E$ and root-mean-square (RMS) radius
$r_{RMS}$) for the $\Xi_{cc}K^{(*)}$ and $\Xi_{cc}\bar{K}^{(*)}$ systems by
solving the coupled channel Schrödinger equations. For a hadronic molecular
state, reasonable bound state solutions have a binding energy $E$ ranging from
several to several tens of MeV and an RMS radius $r_{RMS}$ around 1.00 fm or
larger.

###  III.1 The $\Xi_{cc}K^{(*)}$ systems

In this subsection, we first study the mass spectrum behavior of the
$\Xi_{cc}K^{(*)}$. We present our numerical results in six cases:

  * •

Case I, II, and III correspond to the numerical results after considering the
one-pion-exchange (OPE) interactions, the scalar together with vector meson
exchanges (OSVE) interactions, and the one-boson-exchange (OBE) interactions
in the single-channel analysis, respectively.

  * •

Case IV, V, and VI correspond to the numerical results after considering the
OPE, the OSVE, and the OBE interactions in the coupled-channel analysis,
respectively.

With these discussions, one can further explore the roles of the OPE, the OSVE
interactions, and the coupled-channel effects in forming the $\Xi_{cc}K^{(*)}$
bound states.

For the $\Xi_{cc}K$ systems, the vertex ${K}-{K}-\pi$ violates the spin-parity
conservation, so there don’t exist the OPE interactions as shown in Eq.
(II.1). Therefore, one cannot obtain the bound state solutions in case I. The
bound state solutions for the single $\Xi_{cc}K$ system in case II are the
same as those in case III.

When we vary the cutoff value from 0.80 GeV to 5.00 GeV, we find the single
$\Xi_{cc}K$ state with $I(J^{P})=0(1/2^{-})$ can be the only bound state. The
loosely bound state solutions appear with the cutoff value $\Lambda>3.60$ GeV,
which is obviously far away from the reasonable value $\Lambda\sim 1.00$ GeV.
Therefore, the OSVE interactions cannot provide strong enough interactions to
bind the single $\Xi_{cc}K$ systems.

When we take into account the coupled channel effects, there are three
channels, the $\Xi_{cc}K({}^{2}S_{1/2})$, $\Xi_{cc}K^{*}({}^{2}S_{1/2})$, and
$\Xi_{cc}K^{*}({}^{4}D_{1/2})$ channels. The OPE interactions exist in the
$\Xi_{cc}K\to\Xi_{cc}K^{*}$ and the $\Xi_{cc}K^{*}\to\Xi_{cc}K^{*}$ processes.
As shown in Table 4, we collect the bound state solutions for the coupled
$\Xi_{cc}K/\Xi_{cc}K^{*}$ systems with $I(J^{P})=0,1(1/2^{-})$ in cases IV, V,
and VI. Here, we find that for the coupled $\Xi_{cc}K/\Xi_{cc}K^{*}$ system
with $I(J^{P})=0(1/2^{-})$, the loosely bound state solutions in cases IV, V,
and VI emerge at the cutoff value around 2.50 GeV, 1.70 GeV, and 1.45 GeV,
respectively. The dominant channel is the $\Xi_{cc}K({}^{2}S_{1/2})$ channel.
Therefore, the solo OPE interactions cannot provide the strong enough
interactions to bind the coupled $\Xi_{cc}K/\Xi_{cc}K^{*}$ system with
$I(J^{P})=0(1/2^{-})$. Compared to the OPE interactions, the OSVE interactions
are much stronger attractive. When we consider the OBE interactions, the
coupled $\Xi_{cc}K/\Xi_{cc}K^{*}$ system with $I(J^{P})=0(1/2^{-})$ can be
recommended as a good molecular candidate, as the cutoff value is very close
to the reasonable range $\Lambda\sim 1.00$ GeV. In addition, the coupled
channel effects play an important role in bind this coupled channel bound
state.

For the coupled $\Xi_{cc}K/\Xi_{cc}K^{*}$ system with $I(J^{P})=1(1/2^{-})$,
we cannot find the bound state solutions when considering only the OPE
interactions. However, in OSVE case, the loosely bound state solutions appear
when the cutoff is taken around 2.60 GeV. If we further consider the OPE
interactions together with the OSVE interactions, this bound state binds
slightly deeper. Thus, the OSVE interactions contribute significantly to the
formation of this bound state, while the OPE interactions play a tiny but
positive role. In comparison with the results in the single channel case, the
cutoff value in the coupled channel case is much smaller, indicating that the
coupled channel effects also play a positive role here.

Table 4: The bound state properties (the binding energy $E$, the RMS radius $r_{RMS}$, the probabilities $p_{i}(\%)$ for all the discussed channels) for the coupled $\Xi_{cc}K/\Xi_{cc}K^{*}$ systems with $I(J^{P})=0(1/2^{-})$ and $1(1/2^{-})$ after adopting the OPE, OSVE, and OBE effective potentials. Here, $E$, $r_{RMS}$, and $\Lambda$ are in units of MeV, fm, and GeV, respectively. $I(J^{P})$ |  | Case I | Case II | Case III
---|---|---|---|---
$0(1/2^{-})$ | $\Lambda$ | $1.96$ | $2.08$ | $2.20$ | $1.46$ | $1.56$ | $1.66$ | $1.14$ | $1.21$ | $1.28$
| $E$ | $-0.29$ | $-4.50$ | $-14.04$ | $-0.52$ | $-5.70$ | $-16.99$ | $-0.30$ | $-4.83$ | $-14.80$
| $r_{RMS}$ | $5.11$ | $1.80$ | $1.06$ | $4.65$ | $1.74$ | $1.07$ | $5.19$ | $1.86$ | $1.14$
| $\Xi_{cc}K^{*}({}^{2}S_{1/2})$ | $99.72$ | $99.34$ | $99.14$ | $99.99$ | $99.96$ | $99.92$ | $99.89$ | $99.79$ | $99.78$
| $\Xi_{cc}K^{*}({}^{4}D_{1/2})$ | $0.28$ | $0.66$ | $0.86$ | $0.01$ | $0.04$ | $0.08$ | $0.11$ | $0.21$ | $0.22$
$0(3/2^{-})$ | $\Lambda$ | $4.90$ | $5.09$ | $5.28$ | $1.90$ | $2.30$ | $2.70$ | $2.30$ | $3.30$ | $4.30$
| $E$ | $-0.81$ | $-3.97$ | $-10.09$ | $-0.32$ | $-4.83$ | $-13.28$ | $-0.17$ | $-5.40$ | $-14.13$
| $r_{RMS}$ | $4.03$ | $2.03$ | $1.32$ | $5.23$ | $1.95$ | $1.27$ | $5.81$ | $1.90$ | $1.27$
| $\Xi_{cc}K^{*}({}^{4}S_{3/2})$ | $97.92$ | $95.96$ | $94.21$ | $99.94$ | $99.61$ | $99.06$ | $99.66$ | $99.04$ | $98.80$
| $\Xi_{cc}K^{*}({}^{2}D_{3/2})$ | $0.28$ | $0.54$ | $0.75$ | $0.01$ | $0.08$ | $0.21$ | $0.06$ | $0.16$ | $0.19$
| $\Xi_{cc}K^{*}({}^{4}D_{3/2})$ | $1.79$ | $3.51$ | $5.05$ | $0.05$ | $0.30$ | $0.72$ | $0.28$ | $0.80$ | $1.00$
$1(3/2^{-})$ | $\Lambda$ | … | … | … | $3.61$ | $3.63$ | $3.65$ | $3.25$ | $3.27$ | $3.29$
| $E$ | … | … | … | $-0.93$ | $-4.88$ | $-10.91$ | $-0.30$ | $-4.04$ | $-10.38$
| $r_{RMS}$ | … | … | … | $3.48$ | $1.53$ | $1.01$ | $4.86$ | $1.66$ | $1.00$
| $\Xi_{cc}K^{*}({}^{4}S_{3/2})$ | … | … | … | $99.08$ | $98.35$ | $97.92$ | $99.95$ | $99.89$ | $99.85$
| $\Xi_{cc}K^{*}({}^{2}D_{3/2})$ | … | … | … | $0.13$ | $0.23$ | $0.29$ | $0.01$ | $0.02$ | $0.02$
| $\Xi_{cc}K^{*}({}^{4}D_{3/2})$ | … | … | … | $0.78$ | $1.42$ | $1.79$ | $0.04$ | $0.10$ | $0.13$
$I(J^{P})$ |  | Case IV | Case V | Case VI
$0(1/2^{-})$ | $\Lambda$ | $2.49$ | $2.51$ | $2.53$ | $1.70$ | $1.73$ | $1.76$ | $1.43$ | $1.45$ | $1.47$
| $E$ | $-0.62$ | $-4.81$ | $-11.92$ | $-0.46$ | $-3.86$ | $-11.60$ | $-0.54$ | $-4.56$ | $-13.01$
| $r_{RMS}$ | $4.63$ | $1.93$ | $1.16$ | $5.24$ | $2.51$ | $1.47$ | $5.03$ | $2.26$ | $1.33$
| $\Xi_{cc}K({}^{2}S_{1/2})$ | $89.37$ | $75.29$ | $64.89$ | $98.33$ | $94.66$ | $88.99$ | $96.46$ | $89.46$ | $81.00$
| $\Xi_{cc}K^{*}({}^{2}S_{1/2})$ | $10.54$ | $24.56$ | $34.96$ | $1.62$ | $5.20$ | $10.79$ | $3.53$ | $10.53$ | $18.98$
| $\Xi_{cc}K^{*}({}^{4}D_{1/2})$ | $0.09$ | $0.14$ | $0.15$ | $0.06$ | $0.14$ | $0.22$ | $0.01$ | $0.02$ | $0.02$
$1(1/2^{-})$ | $\Lambda$ | … | … | … | $2.63$ | $2.65$ | $2.67$ | $2.58$ | $2.60$ | $2.62$
| $E$ | … | … | … | $-0.52$ | $-5.18$ | $-13.59$ | $-0.34$ | $-5.32$ | $-14.61$
| $r_{RMS}$ | … | … | … | $4.93$ | $1.99$ | $1.21$ | $5.27$ | $1.94$ | $1.15$
| $\Xi_{cc}K({}^{2}S_{1/2})$ | … | … | … | $96.83$ | $91.90$ | $88.02$ | $97.20$ | $91.98$ | $88.08$
| $\Xi_{cc}K^{*}({}^{2}S_{1/2})$ | … | … | … | $2.60$ | $6.72$ | $10.02$ | $2.60$ | $7.47$ | $11.12$
| $\Xi_{cc}K^{*}({}^{4}D_{1/2})$ | … | … | … | $0.57$ | $1.38$ | $1.96$ | $0.20$ | $0.56$ | $0.80$

For the $\Xi_{cc}K^{*}$ systems, the discussed quantum number configurations
include $I(J^{P})=0(1/2^{-})$, $1(1/2^{-})$, $0(3/2^{-})$, and $1(3/2^{-})$.
By varying the cutoff value in the range of $0.80\leq\Lambda\leq 5.00$ GeV, we
cannot obtain the bound state solutions for the $\Xi_{cc}K^{*}$ system with
$1(1/2^{-})$. For the remaining three $\Xi_{cc}K^{*}$ systems, we present
their bound state solutions in Table 4. If taking the reasonable cutoff around
1.00 GeV, the $\Xi_{cc}K^{*}$ state with $0(1/2^{-})$ can be suggested as a
good molecular candidate, with the dominant channel being the
$\Xi_{cc}K^{*}({}^{2}S_{1/2})$. The OSVE interactions provide stronger
interactions than the OPE interactions, as the cutoff in case II is much
closer to the reasonable cutoff value.

For the isoscalar $\Xi_{cc}K^{*}$ system with $3/2^{-}$, the interactions
arising from the OPE, OSVE, and OBE processes are stronger than those in the
isovector systems. However, the cutoff in the isoscalar system is still a
little far away from the value $\Lambda\sim 1.00$ GeV. Therefore, the
likelihood of the isoscalar $\Xi_{cc}K^{*}$ systems with $3/2^{-}$ being a
hadronic molecular candidate is rather limited. Moreover, it is evident that
the OSVE interactions yield stronger attractive forces compared to the OPE
interactions, particularly when considering the cutoffs in cases I and II.

In summary, we can predict two promising candidates for hadronic molecules:
the coupled $\Xi_{cc}K/\Xi_{cc}K^{*}$ state with $I(J^{P})=0(1/2^{-})$ and the
$\Xi_{cc}K^{*}$ state with $0(1/2^{-})$. The formation of these two bound
states is notably influenced by the OSVE interactions, while the coupled
channel effects significantly contribute to generate the coupled
$\Xi_{cc}K/\Xi_{cc}K^{*}$ state with $I(J^{P})=0(1/2^{-})$.

After that, we study the radiative decay width for the single
$\Xi_{cc}K^{*}[0(1/2^{-})]$ molecule decaying into the coupled
$\Xi_{cc}K/\Xi_{cc}K^{*}[0(1/2^{-})]$ molecule and the photon. As is well
known, the radiative decay widths aren’t closely related to the radial wave
functions but the spin-orbit wave functions. As shown in the Table 4, the
probabilities of the $D-$wave channels are very tiny, which cannot
significantly influence the transition magnetic moments and the magnetic
moments Wang:2022nqs . In the following, we adopt their wave functions in the
$S-$wave channels to calculate the corresponding electromagnetic properties.
In Figure 1, we present the radiative decay width of the
$\Xi_{cc}K^{*}[0(1/2^{-})]\to\Xi_{cc}K/\Xi_{cc}K^{*}[0(1/2^{-})]+\gamma$
process, and we vary the binding energies for the single
$\Xi_{cc}K^{*}[0(1/2^{-})]$ molecule and the coupled
$\Xi_{cc}K/\Xi_{cc}K^{*}[0(1/2^{-})]$ molecule simultaneously. In the binding
energy region of $0$ to $-12$ MeV, the obtained radiative decay width of the
$\Xi_{cc}K^{*}[0(1/2^{-})]\to\Xi_{cc}K/\Xi_{cc}K^{*}[0(1/2^{-})]+\gamma$
process is around a few keV.

Figure 1: The decay width of the
$\Xi_{cc}K^{*}[0(1/2^{-})]\to\Xi_{cc}K/\Xi_{cc}K^{*}[0(1/2^{-})]+\gamma$
process.

In addition, we further investigate the magnetic moments for the coupled
$\Xi_{cc}K/\Xi_{cc}K^{*}$ molecule with $I(J^{P})=0(1/2^{-})$ and the
$\Xi_{cc}K^{*}$ molecule with $0(1/2^{-})$. As shown in Figure 2, when we vary
the binding energy from $0$ to $-12$ MeV, the magnetic moments for the coupled
$\Xi_{cc}K/\Xi_{cc}K^{*}$ molecule with $I(J^{P})=0(1/2^{-})$ and the single
$\Xi_{cc}K^{*}$ molecule with $0(1/2^{-})$ are around 0.34 $\mu_{N}$ and 0.66
$\mu_{N}$, respectively. Our results indicate that the magnetic moments are
insensitive with the corresponding binding energies.

Figure 2: The magnetic moments for the coupled $\Xi_{cc}K/\Xi_{cc}K^{*}$
molecule with $I(J^{P})=0(1/2^{-})$ and the $\Xi_{cc}K^{*}$ molecule with
$0(1/2^{-})$.

###  III.2 The $\Xi_{cc}\bar{K}^{(*)}$ systems

In this subsection, we extend our study to the $\Xi_{cc}\bar{K}^{(*)}$
interactions. The OBE effective potentials for the $\Xi_{cc}\bar{K}^{(*)}$
systems can be related to the $\Xi_{cc}K^{(*)}$ interactions by using the
$G-$parity rule Chen:2021kad , i.e.,
$V_{E}^{\Xi_{cc}\bar{K}^{(*)}\to\Xi_{cc}\bar{K}^{(*)}}=(-1)^{G_{E}}V_{E}^{\Xi_{cc}K^{(*)}\to\Xi_{cc}K^{(*)}}$,
where $G_{E}$ is the $G-$parity for the exchanged mesons. Following the same
procedures, we can first search for possible $\Xi_{cc}\bar{K}^{(*)}$ molecular
candidates, and then explore their electromagnetic properties.

When we adopt the OBE effective potentials and vary the cutoff around 1.00
GeV, we can obtain four loosely bound states, the single $\Xi_{cc}\bar{K}$
bound state with $0(1/2^{-})$, the coupled
$\Xi_{cc}\bar{K}/\Xi_{cc}\bar{K}^{*}$ bound state with $0(1/2^{-})$, and the
single $\Xi_{cc}\bar{K}^{*}$ bound states with $0(1/2^{-})$ and $0(3/2^{-})$.
In Figure 3, we present the binding energies dependence of the cutoff values.
It can be observed that the coupled channel effects play a positive role in
generating the coupled $\Xi_{cc}\bar{K}/\Xi_{cc}\bar{K}^{*}$ bound state with
$0(1/2^{-})$. The OBE interactions for the $\Xi_{cc}K^{*}$ bound state with
$0(1/2^{-})$ are weaker attractive compared to the $\Xi_{cc}K^{*}$
interactions with $0(3/2^{-})$ due to the larger cutoff values. The OBE
interactions for the isovector $\Xi_{cc}\bar{K}^{(*)}$ systems cannot be
strong enough to form loosely bound states with the reasonable cutoff values.

Figure 3: The binding energies dependence of the cutoff for the single
$\Xi_{cc}\bar{K}$ bound state with $0(1/2^{-})$, the coupled
$\Xi_{cc}\bar{K}/\Xi_{cc}\bar{K}^{*}$ bound state with $0(1/2^{-})$, and the
single $\Xi_{cc}\bar{K}^{*}$ bound states with $0(1/2^{-})$ and $0(3/2^{-})$.

Based on the reasonable cutoff values, the coupled
$\Xi_{cc}\bar{K}/\Xi_{cc}\bar{K}^{*}$ bound state with $0(1/2^{-})$, and the
single $\Xi_{cc}\bar{K}^{*}$ bound states with $0(1/2^{-})$ and $0(3/2^{-})$
can be recommended as the prime molecular candidates.

In the following, we analyze the radiative decay widths and the magnetic
moments of the coupled $\Xi_{cc}\bar{K}/\Xi_{cc}\bar{K}^{*}$ bound state with
$0(1/2^{-})$, and the single $\Xi_{cc}\bar{K}^{*}$ bound states with
$0(1/2^{-})$ and $0(3/2^{-})$. As shown in Figure 4, we find the radiative
decay widths of the
$\Xi_{cc}\bar{K}^{*}[0(1/2^{-})]\to\Xi_{cc}\bar{K}/\Xi_{cc}\bar{K}^{*}[0(1/2^{-})]+\gamma$
and
$\Xi_{cc}\bar{K}^{*}[0(3/2^{-})]\to\Xi_{cc}\bar{K}/\Xi_{cc}\bar{K}^{*}[0(1/2^{-})]+\gamma$
processes are around several keV.

Figure 4: The decay widths of the
$\Xi_{cc}\bar{K}^{*}[0(1/2^{-})]\to\Xi_{cc}\bar{K}/\Xi_{cc}\bar{K}^{*}[0(1/2^{-})]+\gamma$
and
$\Xi_{cc}\bar{K}^{*}[0(3/2^{-})]\to\Xi_{cc}\bar{K}/\Xi_{cc}\bar{K}^{*}[0(1/2^{-})]+\gamma$
processes.

In Figure 5, we present the binding energies dependence of the magnetic
moments for the coupled $\Xi_{cc}\bar{K}/\Xi_{cc}\bar{K}^{*}$ bound state with
$0(1/2^{-})$, and the single $\Xi_{cc}\bar{K}^{*}$ bound states with
$0(1/2^{-})$ and $0(3/2^{-})$. Here, we find the magnetic moments for the
$\Xi_{cc}\bar{K}^{*}$ molecules with $0(1/2^{-})$ and $0(3/2^{-})$ are around
$-0.87$ $\mu_{N}$ and $-0.81$ $\mu_{N}$, respectively. In addition, the
magnetic moment for the coupled $\Xi_{cc}\bar{K}/\Xi_{cc}\bar{K}^{*}$ molecule
with $0(1/2^{-})$ is around 0.34 $\mu_{N}$.

Figure 5: The magnetic moments for the coupled
$\Xi_{cc}\bar{K}/\Xi_{cc}\bar{K}^{*}$ bound state with $0(1/2^{-})$, and the
single $\Xi_{cc}\bar{K}^{*}$ bound states with $0(1/2^{-})$ and $0(3/2^{-})$.

##  IV Summary

The study of hadron spectroscopy is a fundamental issue in non-perturbative
strong interactions. Recent experimental observations, such as the doubly
charmed structures $\Xi_{cc}^{++}(3621)$ LHCb:2017iph  and $T_{cc}$
LHCb:2021vvq ; LHCb:2021auc , have sparked significant interest in exploring
other possible doubly charmed states. Theoretical investigations have proposed
various molecular states and examined their formation mechanisms and
properties Chen:2021vhg ; Wang:2023ael ; Feijoo:2021ppq ; Wang:2021yld ;
Chen:2019asm ; Wang:2023eng ; Yan:2023iie ; Wang:2023aob . In this work, we
focus on possible doubly charmed hadronic molecular states with strangeness
$|S|=1$ primarily composed of the $S-$wave $\Xi_{cc}K^{(*)}$ and
$\Xi_{cc}\bar{K}^{(*)}$ channels. We adopt the OBE model to study the relevant
interactions. When deducing the OBE effective potentials, we consider both the
$S-D$ wave mixing effects and the coupled channel effects.

Finally, we can predict five primary hadronic molecular candidates: the
coupled $\Xi_{cc}K/\Xi_{cc}K^{*}$ molecule with $I(J^{P})=0(1/2^{-})$, the
$\Xi_{cc}K^{*}$ molecule with $0(1/2^{-})$, the $\Xi_{cc}\bar{K}$ molecule
with $0(1/2^{-})$, and the $\Xi_{cc}\bar{K}^{*}$ molecules with
$0(1/2^{-},3/2^{-})$. The coupled channel effects play an essential role in
generating the coupled $\Xi_{cc}K/\Xi_{cc}K^{*}$ molecule with $0(1/2^{-})$.

Afterwards, we employ the constituent quark model to investigate the
electromagnetic properties of these predicted molecular candidates, including
their M1 radiative decay behaviors and magnetic moments between these
molecular candidates. Our results indicate that the magnetic moments of
molecular states are almost independent of their binding energies. The
magnetic moments for the coupled $\Xi_{cc}K/\Xi_{cc}K^{*}$ molecule with
$0(1/2^{-})$ and the coupled $\Xi_{cc}\bar{K}/\Xi_{cc}\bar{K}^{*}$ molecule
with $0(1/2^{-})$ are around 0.34 $\mu_{N}$, stemming from the doubly charmed
baryon. For the $\Xi_{cc}K^{*}$ molecule with $0(1/2^{-})$, the
$\Xi_{cc}\bar{K}^{*}$ molecules with $0(1/2^{-})$ and $0(3/2^{-})$, the
magnetic moments are around 0.66 $\mu_{N}$, $-0.87$ $\mu_{N}$, and $-0.81$
$\mu_{N}$, respectively. The decay widths for the
$\Xi_{cc}K^{*}[0(1/2^{-})]\to\Xi_{cc}K/\Xi_{cc}K^{*}[0(1/2^{-})]+\gamma$,
$\Xi_{cc}\bar{K}^{*}[0(1/2^{-},3/2^{-})]\to\Xi_{cc}\bar{K}/\Xi_{cc}\bar{K}^{*}[0(1/2^{-})]+\gamma$
processes are around a few keV.

Overall, our investigations can open up new avenues for exploring the
existence of possible doubly charmed molecular pentaquarks with strangeness
$|S|=1$. We look forward to experimental confirmation of these predictions.

## ACKNOWLEDGMENTS

R.C. is supported by the National Natural Science Foundation of China under
Grants No. 12305139 and the Xiaoxiang Scholars Programme of Hunan Normal
University. X.L. is supported by the National Natural Science Foundation of
China under Grant Nos. 12335001 and 12247101, National Key Research and
Development Program of China under Contract No. 2020YFA0406400, the 111
Project under Grant No. B20063, the fundamental Research Funds for the Central
Universities, and the project for top-notch innovative talents of Gansu
province. F.L.W. is supported by the China Postdoctoral Science Foundation
under Grant No. 2022M721440.

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