# Molecular-type $QQss\bar{s}$ pentaquarks predicted by the extended chiral
unitary approach

Zhong-Yu Wang1,2<EMAIL_ADDRESS>Chu-Wen Xiao3,4,5<EMAIL_ADDRESS>Zhi-Feng Sun1,2,6,7<EMAIL_ADDRESS>Xiang Liu1,2,6,7<EMAIL_ADDRESS>1School of Physical Science and Technology, Lanzhou University, Lanzhou
730000, China
2Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, and Key Laboratory of Quantum Theory and Applications of
the Ministry of Education, Lanzhou University, Lanzhou, 730000, China
3Department of Physics, Guangxi Normal University, Guilin 541004, China
4Guangxi Key Laboratory of Nuclear Physics and Technology, Guangxi Normal
University, Guilin 541004, China
5School of Physics, Central South University, Changsha 410083, China
6MoE Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou
730000, China
7Research Center for Hadron and CSR Physics, Lanzhou University and Institute
of Modern Physics of CAS, Lanzhou 730000, China

###### Abstract

In this work, we investigate the double-heavy molecular pentaquark states with
the quark contents $ccss\bar{s}$, $bbss\bar{s}$, and $bcss\bar{s}$ by using
the coupled channel approach. The extended local hidden gauge Lagrangians are
used to obtain the meson-baryon interactions by exchanging the vector mesons.
We predict some candidates for the molecular states with the quantum numbers
$I(J^{P})=0(1/2^{-},3/2^{-},5/2^{-})$, whose binding energies are of the order
of $20-30$ MeV and whose widths are all less than $8$ MeV. These predicted
exotic double-heavy molecular pentaquark states may be accessible in future
experiments such as LHCb.

##  I Introduction

How to quantitatively describe the strong interaction at the low energy region
is a challenging problem in particle physics. Hadron spectroscopy can provide
some insights to deepen our understanding of the non-perturbative behavior of
the strong interaction. Usually, there are conventional mesons and baryons,
which have the quark components of $q\bar{q}$ and $qqq$, respectively. These
conventional hadrons form the bulk of the reported hadrons, as shown in the
Particle Data Group (PDG) ParticleDataGroup:2022pth . It is the first stage in
the construction of the particle zoo.

Since 2003, we have entered a new phase. More and more heavy flavor new
hadronic states have been found in experiments, which cannot be explained by
the quenched quark model. For example, there were famous low-mass puzzles for
these reported $D_{s0}^{*}(2317)$ BaBar:2003oey , $D_{s1}(2460)$ CLEO:2003ggt
, and $X(3872)$ Belle:2003nnu . In 2015, the LHCb Collaboration discovered the
pentaquark-like states $P_{c}(4380)^{+}$ and $P_{c}(4450)^{+}$ in the decay of
$\Lambda_{b}^{0}\rightarrow J/\psi K^{-}p$ LHCb:2015yax , which were
considered as competing candidates for the hadronic molecular states and
attracted intense interest in both theory and experiment. In 2017, the
observation of the doubly charmed baryon $\Xi_{cc}^{++}$ in the
$\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+}$ invariant mass spectrum by the LHCb
Collaboration LHCb:2017iph  captured the continuing enthusiasm for the heavy
states. Furthermore, with the discovery of several narrow states $\Omega_{c}$
LHCb:2017uwr ; Belle:2017ext ; LHCb:2021ptx ; LHCb:2023rtu  and $\Omega_{b}$
LHCb:2020tqd ; Matiunin:2020xbg , the LHCb Collaboration tried to search for
more doubly heavy baryons, such as $\Xi_{bc}$, $\Omega_{cc}$, and
$\Omega_{bc}$, etc LHCb:2021xba . More results can be found in these recent
review articles Liu:2013waa ; Hosaka:2016pey ; Chen:2016qju ; Richard:2016eis
; Lebed:2016hpi ; Olsen:2017bmm ; Guo:2017jvc ; Liu:2019zoy ;
Brambilla:2019esw ; Meng:2022ozq ; Chen:2022asf . These new hadronic states
not only provide a good opportunity to continue building the particle zoo, but
also make it possible to search for exotic states.

In fact, the study of heavy flavor pentaquarks has always been a focus point
in the search for exotic states, as reflected in the recent observations of
$P_{cs}(4338)^{0}$ and $P_{cs}(4459)^{0}$, which are from the $B^{-}\to
J/\psi\Lambda\bar{p}$ and $\Xi_{b}^{-}\to J/\psi\Lambda K^{-}$ processes,
respectively LHCb:2022ogu ; LHCb:2020jpq . The observation of the
characteristic spectrum of $P_{c}(4312)^{+}$, $P_{c}(4440)^{+}$, and
$P_{c}(4457)^{+}$ LHCb:2015yax ; LHCb:2019kea  supports the existence of the
molecular-type hidden-charm pentaquark Wu:2010jy ; Wang:2011rga ; Yang:2011wz
; Li:2014gra ; Chen:2015loa ; Karliner:2015ina , and other such states were
predicted in Refs. Chen:2015moa ; Ozdem:2018qeh ; Du:2019pij ; Azizi:2022qll .
The discovered $T_{cc}(3875)$ LHCb:2021vvq  is a good candidate for a double-
charm tetraquark Manohar:1992nd ; Ericson:1993wy ; Tornqvist:1993ng ;
Janc:2004qn ; Ding:2009vj ; Molina:2010tx ; Ohkoda:2012hv ; Li:2012ss ;
Xu:2017tsr ; Liu:2019stu ; Tang:2019nwv ; Ding:2020dio . These phenomena
mentioned above give us sufficient reason to believe that there should be
double-charm molecular pentaquarks Hofmann:2005sw ; Romanets:2012hm ;
Zhou:2018bkn ; Dong:2021bvy ; Wang:2022aga ; Wang:2023aob , which is the
starting point of the present work.

In this work, we choose a special double-heavy molecular pentaquark system.
This system contains most strange quarks. For quantitatively obtaining the
information of the mass spectrum of this focused pentaquark system, we adopt
an extension of the chiral unitary approach (ChUA) based on the local hidden
gauge approach. The ChUA was proposed early on to describe the states
$f_{0}(500)$, $f_{0}(980)$, $a_{0}(980)$, and $\Lambda(1405)$ in the molecular
configuration Kaiser:1995eg ; Oller:1997ti ; Oset:1997it ; Kaiser:1998fi ;
Oller:1998hw ; Oller:2000ma ; Oller:2000fj . Note that the scattering
amplitudes of the coupled channels were calculated by solving the Bethe-
Salpeter equation using the on-shell approximation in Refs. Oller:1997ti ;
Oset:1997it ; Oller:2000fj . This approach has been widely applied in the
meson-meson and meson-baryon interactions, with great success, for example in
the predictions of the $P_{c}$, $P_{cs}$ states Wu:2010jy ; Wu:2010vk ;
Xiao:2013yca ; Xiao:2019gjd  and the interpretation of the $T_{cc}$ state
Feijoo:2021ppq ; Dai:2023cyo . Using this approach, Ref. Debastiani:2017ewu
dynamically reproduced three $\Omega_{c}$ states observed in the
$\Xi_{c}^{+}K^{-}$ invariant mass spectrum by the LHCb Collaboration
LHCb:2017uwr . Moreover, taking the advantage of this method, it has been used
to predict the molecular states in the heavy sector, such as the resonances
$\Omega_{b}$ Liang:2017ejq ; Liang:2020dxr , $\Xi_{cc}$ Dias:2018qhp ,
$\Xi_{c}$ and $\Xi_{b}$ Yu:2018yxl , $\Xi_{bc}$ Yu:2019yfr , $\Xi_{bb}$ and
$\Omega_{bbb}$ Dias:2019klk , and so on. Note that the systems with the quark
components $ccsq\bar{q}$, $bbsq\bar{q}$, and $bcsq\bar{q}$ were studied in the
work of Wang:2022aga , where some states of $\Omega_{cc}$, $\Omega_{bb}$,
$\Omega_{bc}$ were predicted.

In order to provide more theoretical support to experiment, in the present
work we study the double-heavy molecular pentaquark systems with the quark
components $ccss\bar{s}$, $bbss\bar{s}$, and $bcss\bar{s}$, which are
dynamically generated from the $s$-wave interactions of meson and baryon using
an extension of the ChUA based on the local hidden gauge approach. For the
meson-baryon interactions, the dominant potentials are obtained by the vector
meson exchange mechanism from the extended local hidden gauge Lagrangians as
done in Ref. Debastiani:2017ewu . In fact, the diagonal terms of the matrix
for the interaction potentials exchange only the light vector mesons, and thus
in the interaction procedures the heavy quarks act as spectators, which
automatically respects the heavy quark symmetry. On the other hand, the non-
diagonal terms that exchange the heavy vector mesons do not fulfil the heavy
quark symmetry, but neither should they since they correspond to subleading
terms in the heavy quark counting, see further discussion in Ref.
Debastiani:2017ewu .

The rest of the paper is organized as follows. The framework for the
potentials of meson-baryon interactions derived from the local hidden gauge
Lagrangians is introduced in Sec. II. The numerical results of the scattering
amplitudes solved by the coupled channel Bethe-Salpeter equation are presented
in Sec. III. Finally, we conclude this work with a summary in Section IV.

##  II The molecular-type double-heavy pentaquarks with three strange quarks

In this section, we will briefly introduce the formalism. First, following
Ref. Wang:2022aga , we construct the coupled channels of the systems with the
quark components $ccss\bar{s}$, $bbss\bar{s}$, and $bcss\bar{s}$.

In the $ccss\bar{s}$ sector, We divide the coupled channels into four blocks,
$PB(1/2^{+})$, $PB(3/2^{+})$, $VB(1/2^{+})$, and $VB(3/2^{+})$, as shown in
Table 1, where $P$ stands for the pseudoscalar meson, $V$ for the vector
meson, $B(1/2^{+})$ for the $J^{P}=1/2^{+}$ ground state baryons, and
$B(3/2^{+})$ for the $J^{P}=3/2^{+}$ ground state baryons. Note that there is
no channel containing the $\eta$ meson as well as the quark components
$ccsq\bar{q}$ $(q=u,d)$, as these systems were studied in Ref. Wang:2022aga ,
and the thresholds are much lower than those of the present work. A similar
situation occurs in the $bbss\bar{s}$ and $bcss\bar{s}$ sectors.

Table 1: Coupled channels in different sectors. The symbol $P$ stands for the pseudoscalar meson, $V$ for the vector meson, $B(1/2^{+})$ for the $J^{P}=1/2^{+}$ ground state baryons, and $B(3/2^{+})$ for the $J^{P}=3/2^{+}$ ground state baryons. | $ccss\bar{s}$ | $bbss\bar{s}$
---|---|---
$PB(1/2^{+})$ | $\eta^{{}^{\prime}}\Omega_{cc}$ | $D_{s}\Omega_{c}$ | $\eta^{{}^{\prime}}\Omega_{bb}$ | $\bar{B}_{s}\Omega_{b}$
$PB(3/2^{+})$ | $\eta^{{}^{\prime}}\Omega_{cc}^{*}$ | $D_{s}\Omega_{c}^{*}$ | $\eta^{{}^{\prime}}\Omega_{bb}^{*}$ | $\bar{B}_{s}\Omega_{b}^{*}$
$VB(1/2^{+})$ | $\phi\Omega_{cc}$ | $D_{s}^{*}\Omega_{c}$ | $\phi\Omega_{bb}$ | $\bar{B}_{s}^{*}\Omega_{b}$
$VB(3/2^{+})$ | $\phi\Omega_{cc}^{*}$ | $D_{s}^{*}\Omega_{c}^{*}$ | $\phi\Omega_{bb}^{*}$ | $\bar{B}_{s}^{*}\Omega_{b}^{*}$
| $bcss\bar{s}$
$PB(1/2^{+})$ | $\eta^{{}^{\prime}}\Omega_{bc}$ | $\eta^{{}^{\prime}}\Omega_{bc}^{{}^{\prime}}$ | $D_{s}\Omega_{b}$ | $\bar{B}_{s}\Omega_{c}$
$PB(3/2^{+})$ | $\eta^{{}^{\prime}}\Omega_{bc}^{*}$ | $D_{s}\Omega_{b}^{*}$ | $\bar{B}_{s}\Omega_{c}^{*}$ |
$VB(1/2^{+})$ | $\phi\Omega_{bc}$ | $\phi\Omega_{bc}^{{}^{\prime}}$ | $D_{s}^{*}\Omega_{b}$ | $\bar{B}_{s}^{*}\Omega_{c}$
$VB(3/2^{+})$ | $\phi\Omega_{bc}^{*}$ | $D_{s}^{*}\Omega_{b}^{*}$ | $\bar{B}_{s}^{*}\Omega_{c}^{*}$ |

The evaluation of the potentials for the meson-baryon interactions is given by
the extended local hidden gauge symmetry approach Wang:2022aga . The
interaction mechanism in each block is constructed from the dominant process
of the vector meson exchange. The local hidden gauge Lagrangians Bando:1984ej
; Bando:1987br ; Meissner:1987ge  describing the vertices $VPP$, $VVV$, and
$VBB$ are given by

$\displaystyle\mathcal{L}_{VPP}=-ig\left\langle\left[P,\partial_{\mu}P\right]V^{\mu}\right\rangle,$
(1)
$\displaystyle\mathcal{L}_{VVV}=ig\left\langle\left(V^{\mu}\partial_{\nu}V_{\mu}-\partial_{\nu}V^{\mu}V_{\mu}\right)V^{\nu}\right\rangle,$
(2)
$\displaystyle\mathcal{L}_{VBB}=g\left(\left\langle\bar{B}\gamma_{\mu}\left[V^{\mu},B\right]\right\rangle+\left\langle\bar{B}\gamma_{\mu}B\right\rangle\left\langle
V^{\mu}\right\rangle\right),$ (3)

where $g$ is the coupling constant, which is taken as $M_{V}/(2f_{\pi})$ with
$M_{V}$ the mass of the exchanged light vector meson and $f_{\pi}=93$ MeV the
pion decay constant. The symbol $\left\langle\right\rangle$ denotes the trace
of flavor space. Note that Eq. (3) only holds for SU(3), light baryons, which
is not used in our case, see the discussions later. For the pseudoscalar $P$
and vector $V^{\mu}$ meson fields, they are easily extended from the SU(3) to
the SU(5) case following the work of Hofmann:2005sw ; Mizutani:2006vq ,

$\displaystyle
P=\left(\begin{array}[]{ccccc}\frac{\eta}{\sqrt{3}}+\frac{\eta^{\prime}}{\sqrt{6}}+\frac{\pi^{0}}{\sqrt{2}}&\pi^{+}&K^{+}&\bar{D}^{0}&B^{+}\\\
\pi^{-}&\frac{\eta}{\sqrt{3}}+\frac{\eta^{\prime}}{\sqrt{6}}-\frac{\pi^{0}}{\sqrt{2}}&K^{0}&D^{-}&B^{0}\\\
K^{-}&\bar{K}^{0}&-\frac{\eta}{\sqrt{3}}+\sqrt{\frac{2}{3}}\eta^{\prime}&D_{s}^{-}&B_{s}^{0}\\\
D^{0}&D^{+}&D_{s}^{+}&\eta_{c}&B_{c}^{+}\\\
B^{-}&\bar{B}^{0}&\bar{B}_{s}^{0}&B_{c}^{-}&\eta_{b}\end{array}\right),$ (4)
$\displaystyle
V^{\mu}=\left(\begin{array}[]{ccccc}\frac{\omega+\rho^{0}}{\sqrt{2}}&\rho^{+}&K^{*+}&\bar{D}^{*0}&B^{*+}\\\
\rho^{-}&\frac{\omega-\rho^{0}}{\sqrt{2}}&K^{*0}&D^{*-}&B^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi&D_{s}^{*-}&B_{s}^{*0}\\\
D^{*0}&D^{*+}&D_{s}^{*+}&J/\psi&B_{c}^{*+}\\\
B^{*-}&\bar{B}^{*0}&\bar{B}_{s}^{*0}&B_{c}^{*-}&\Upsilon\end{array}\right)^{\mu},$
(5)

but one is not making use of SU(5) symmetry, only of the $q\bar{q}$ character
of the mesons, as shown in Ref. Sakai:2017avl .

We can easily obtain the $VPP$ and $VVV$ vertices in the charm and bottom
sectors from Eqs. (1)-(2) and (4)-(5). However, it cannot directly extend the
SU(3) $VBB$ vertex to the charm and bottom sectors. According to Ref.
Debastiani:2017ewu , the $VBB$ vertex can be constructed through the flavor
and spin wave functions of mesons and baryons. In Table 2, we list the flavor
and spin wave functions of baryons used in this work, following the convention
of Ref. Close:1979 . Thus, for the $J^{P}=1/2^{+}$ ground state baryons in the
case of $S_{z}=+1/2$ we have

Table 2: Wave functions for baryons with $I(J^{P})=0(\frac{1}{2}^{+})$ and $0(\frac{3}{2}^{+})$, where $\chi_{MS}$ stands for mixed symmetric, $\chi_{MA}$ stands for mixed antisymmetric, and $\chi_{S}$ stands for fully symmetric. States | $I(J^{P})$ | Flavor | Spin
---|---|---|---
$\Omega_{cc}^{+}$ | $0(\frac{1}{2}^{+})$ | $ccs$ | $\chi_{MS}(12)$
$\Omega_{c}^{0}$ | $0(\frac{1}{2}^{+})$ | $css$ | $\chi_{MS}(23)$
$\Omega_{bb}^{-}$ | $0(\frac{1}{2}^{+})$ | $bbs$ | $\chi_{MS}(12)$
$\Omega_{b}^{-}$ | $0(\frac{1}{2}^{+})$ | $bss$ | $\chi_{MS}(23)$
$\Omega_{bc}^{0}$ | $0(\frac{1}{2}^{+})$ | $\frac{1}{\sqrt{2}}b(cs-sc)$ | $\chi_{MA}(23)$
$\Omega_{bc}^{{}^{\prime}0}$ | $0(\frac{1}{2}^{+})$ | $\frac{1}{\sqrt{2}}b(cs+sc)$ | $\chi_{MS}(23)$
$\Omega_{cc}^{*+}$ | $0(\frac{3}{2}^{+})$ | $ccs$ | $\chi_{S}$
$\Omega_{c}^{*0}$ | $0(\frac{3}{2}^{+})$ | $css$ | $\chi_{S}$
$\Omega_{bb}^{*-}$ | $0(\frac{3}{2}^{+})$ | $bbs$ | $\chi_{S}$
$\Omega_{b}^{*-}$ | $0(\frac{3}{2}^{+})$ | $bss$ | $\chi_{S}$
$\Omega_{bc}^{*0}$ | $0(\frac{3}{2}^{+})$ | $\frac{1}{\sqrt{2}}b(cs+sc)$ | $\chi_{S}$

$\displaystyle\chi_{MS}(12)=\frac{1}{\sqrt{6}}(\uparrow\downarrow\uparrow+\downarrow\uparrow\uparrow-2\uparrow\uparrow\downarrow),$
(6)
$\displaystyle\chi_{MS}(23)=\frac{1}{\sqrt{6}}(\uparrow\downarrow\uparrow+\uparrow\uparrow\downarrow-2\downarrow\uparrow\uparrow),$
(7)
$\displaystyle\chi_{MA}(23)=\frac{1}{\sqrt{2}}(\uparrow\uparrow\downarrow-\uparrow\downarrow\uparrow).$
(8)

Note that with our spin independent interaction, for the spin overlap between
some baryons, we need the following formulae

$\displaystyle\left\langle\chi_{MS}(12)|\chi_{MS}(23)\right\rangle=-\frac{1}{2},$
(9)
$\displaystyle\left\langle\chi_{MS}(12)|\chi_{MA}(23)\right\rangle=-\frac{\sqrt{3}}{2}.$
(10)

Furthermore, for the $J^{P}=3/2^{+}$ ground baryon states in the special case
$S_{z}=+3/2$, we obtain

$\displaystyle\chi_{S}=\uparrow\uparrow\uparrow.$ (11)

Figure 1: Diagram of the light vector meson $\phi$ exchange in the
$D_{s}\Omega_{c}\rightarrow D_{s}\Omega_{c}$ transition.

We can obtain the vertex $VBB$ with the wave functions of the baryons above.
Take the $D_{s}\Omega_{c}\rightarrow D_{s}\Omega_{c}$ process as an example,
as shown in Fig. 1, where the interaction of the transition is contributed by
the exchange of the light vector meson $\phi$ in the $t$-channel. Thus, for
the lower vertex we have

$\displaystyle\left\langle
css\chi_{MS}(23)\left|gs\bar{s}\right|css\chi_{MS}(23)\right\rangle=g.$ (12)

One can see Refs. Sakai:2017avl ; Dias:2018qhp  for more details. More details
of the $P_{i}B_{i}\rightarrow P_{j}B_{j}$ and $V_{i}B_{i}\rightarrow
V_{j}B_{j}$ transitions obtained through two vertices $VPP$ ($VVV$) and $VBB$
are given in the Appendix of Ref. Debastiani:2017ewu , and we do not repeat
them here. Note that we are concerned with the interaction near the threshold,
so we have neglected the three-momentum of the external particles in the
transitions. Therefore, for the polarizations of the two external vectors, we
have
$\epsilon_{1\mu}\epsilon_{3}^{\dagger\mu}=-\vec{\epsilon}_{1}\cdot\vec{\epsilon}_{3}^{\,\dagger}$
with $\epsilon^{0}=0$.

Finally, the explicit form of the potential from the vector meson exchange
mechanism, as shown in Fig. 1, is given by

$\displaystyle v_{ij}=-C_{ij}\frac{1}{4f_{\pi}^{2}}(p_{i}^{0}+p_{j}^{0}),$
(13)

where $p_{i}^{0}$ ($p_{j}^{0}$) is the energy of initial (final) meson. The
coefficient matrix elements $C_{ij}=C_{ji}$, which are given in Tables 3-5.
Their relativistically corrected expression in the $s$-wave is obtained as
Oset:2001cn

$\displaystyle
v_{ij}(\sqrt{s})=-C_{ij}\frac{2\sqrt{s}-M_{i}-M_{j}}{4f_{\pi}^{2}}\left(\frac{M_{i}+E_{i}}{2M_{i}}\right)^{1/2}\left(\frac{M_{j}+E_{j}}{2M_{j}}\right)^{1/2},$
(14)

where $M_{i}(M_{j})$ and $E_{i}(E_{j})$ denote the mass and center-of-mass
energy of the initial (final) baryon, respectively.

Table 3: Coefficient matrix elements $C_{ij}$ in the $ccss\bar{s}$ sector. $C_{ij}(PB(1/2^{+}),PB(3/2^{+}))$ |  | $\eta^{{}^{\prime}}\Omega_{cc}$ | $D_{s}\Omega_{c}$ |  | $\eta^{{}^{\prime}}\Omega_{cc}^{*}$ | $D_{s}\Omega_{c}^{*}$
---|---|---|---|---|---|---
$\eta^{{}^{\prime}}\Omega_{cc}$ | $0$ | $\frac{\lambda}{\sqrt{6}}$ | $\eta^{{}^{\prime}}\Omega_{cc}^{*}$ | $0$ | $\frac{-\sqrt{2}\lambda}{\sqrt{3}}$
$D_{s}\Omega_{c}$ | $\frac{\lambda}{\sqrt{6}}$ | $1$ | $D_{s}\Omega_{c}^{*}$ | $\frac{-\sqrt{2}\lambda}{\sqrt{3}}$ | $1$
$C_{ij}(VB(1/2^{+}),VB(3/2^{+}))$ |  | $\phi\Omega_{cc}$ | $D_{s}^{*}\Omega_{c}$ |  | $\phi\Omega_{cc}^{*}$ | $D_{s}^{*}\Omega_{c}^{*}$
$\phi\Omega_{cc}$ | $0$ | $\frac{\lambda}{2}$ | $\phi\Omega_{cc}^{*}$ | $0$ | $-\lambda$
$D_{s}^{*}\Omega_{c}$ | $\frac{\lambda}{2}$ | $1$ | $D_{s}^{*}\Omega_{c}^{*}$ | $-\lambda$ | $1$
Table 4: Coefficient matrix elements $C_{ij}$ in the $bbss\bar{s}$ sector. $C_{ij}(PB(1/2^{+}),PB(3/2^{+}))$ |  | $\eta^{{}^{\prime}}\Omega_{bb}$ | $\bar{B}_{s}\Omega_{b}$ |  | $\eta^{{}^{\prime}}\Omega_{bb}^{*}$ | $\bar{B}_{s}\Omega_{b}^{*}$
---|---|---|---|---|---|---
$\eta^{{}^{\prime}}\Omega_{bb}$ | $0$ | $0$ | $\eta^{{}^{\prime}}\Omega_{bb}^{*}$ | $0$ | $0$
$\bar{B}_{s}\Omega_{b}$ | $0$ | $1$ | $\bar{B}_{s}\Omega_{b}^{*}$ | $0$ | $1$
$C_{ij}(VB(1/2^{+}),VB(3/2^{+}))$ |  | $\phi\Omega_{bb}$ | $\bar{B}_{s}^{*}\Omega_{b}$ |  | $\phi\Omega_{bb}^{*}$ | $\bar{B}_{s}^{*}\Omega_{b}^{*}$
$\phi\Omega_{bb}$ | $0$ | $0$ | $\phi\Omega_{bb}^{*}$ | $0$ | $0$
$\bar{B}_{s}^{*}\Omega_{b}$ | $0$ | $1$ | $\bar{B}_{s}^{*}\Omega_{b}^{*}$ | $0$ | $1$
Table 5: Coefficient matrix elements $C_{ij}$ in the $bcss\bar{s}$ sector. $C_{ij}(PB(1/2^{+}))$ |  | $\eta^{{}^{\prime}}\Omega_{bc}$ | $\eta^{{}^{\prime}}\Omega_{bc}^{{}^{\prime}}$ | $D_{s}\Omega_{b}$ | $\bar{B}_{s}\Omega_{c}$
---|---|---|---|---|---
$\eta^{{}^{\prime}}\Omega_{bc}$ | $0$ | $0$ | $0$ | $0$
$\eta^{{}^{\prime}}\Omega_{bc}^{{}^{\prime}}$ | $0$ | $0$ | $\frac{-2\lambda}{\sqrt{3}}$ | $0$
$D_{s}\Omega_{b}$ | $0$ | $\frac{-2\lambda}{\sqrt{3}}$ | $1$ | $0$
$\bar{B}_{s}\Omega_{c}$ | $0$ | $0$ | $0$ | $1$
$C_{ij}(PB(3/2^{+}))$ |  | $\eta^{{}^{\prime}}\Omega_{bc}^{*}$ | $D_{s}\Omega_{b}^{*}$ | $\bar{B}_{s}\Omega_{c}^{*}$ |
$\eta^{{}^{\prime}}\Omega_{bc}^{*}$ | $0$ | $\frac{-2\lambda}{\sqrt{3}}$ | $0$ |
$D_{s}\Omega_{b}^{*}$ | $\frac{-2\lambda}{\sqrt{3}}$ | $1$ | $0$ |
$\bar{B}_{s}\Omega_{c}^{*}$ | $0$ | $0$ | $1$ |
$C_{ij}(VB(1/2^{+}))$ |  | $\phi\Omega_{bc}$ | $\phi\Omega_{bc}^{{}^{\prime}}$ | $D_{s}^{*}\Omega_{b}$ | $\bar{B}_{s}^{*}\Omega_{c}$
$\phi\Omega_{bc}$ | $0$ | $0$ | $0$ | $0$
$\phi\Omega_{bc}^{{}^{\prime}}$ | $0$ | $0$ | $-\sqrt{2}\lambda$ | $0$
$D_{s}^{*}\Omega_{b}$ | $0$ | $-\sqrt{2}\lambda$ | $1$ | $0$
$\bar{B}_{s}^{*}\Omega_{c}$ | $0$ | $0$ | $0$ | $1$
$C_{ij}(VB(3/2^{+}))$ |  | $\phi\Omega_{bc}^{*}$ | $D_{s}^{*}\Omega_{b}^{*}$ | $\bar{B}_{s}^{*}\Omega_{c}^{*}$ |
$\phi\Omega_{bc}^{*}$ | $0$ | $-\sqrt{2}\lambda$ | $0$ |
$D_{s}^{*}\Omega_{b}^{*}$ | $-\sqrt{2}\lambda$ | $1$ | $0$ |
$\bar{B}_{s}^{*}\Omega_{c}^{*}$ | $0$ | $0$ | $1$ |

In Tables 3-5, we keep the contributions of the light vector meson and
$D_{s}^{*}$ meson exchange, ignoring the heavier ones $J/\psi$, $B_{s}^{*}$,
etc. Compared to the light vector mesons, the contribution of exchanging
$D_{s}^{*}$ meson has a suppression factor of the order of
$m_{V}^{2}/m_{D_{s}^{*}}^{2}$, i.e., $\lambda$, which is approximated as
$0.25$ in the numerical calculations, see more details in Ref.
Debastiani:2017ewu .

With the potentials of the $s$-wave, one can obtain the scattering amplitudes
by solving the Bethe-Salpeter equation with the on-shell approximation
Oset:1997it , given by

$\displaystyle T=[1-vG]^{-1}v.$ (15)

The elements of the diagonal matrix $G$ in Eq. (15) are the loop functions of
intermediate mesons and baryons, whose the expression is given by

$\displaystyle
G_{l}=i\int\frac{d^{4}q}{(2\pi)^{4}}\frac{2M_{l}}{(P-q)^{2}-M_{l}^{2}+i\epsilon}\frac{1}{q^{2}-m_{l}^{2}+i\epsilon},$
(16)

where $P=k_{1}+p_{1}$ is the total momentum of the meson and baryon coupled
system. The loop function of Eq. (16) is logarithmically divergent, and
therefore it should be regularized, i.e., using the regularization of the
three-momentum cutoff method Oset:1997it , one obtains

$\displaystyle
G_{l}(s)=\int_{0}^{q_{max}}\frac{q^{2}dq}{2\pi^{2}}\frac{1}{2\omega_{l}(q)}\frac{M_{l}}{E_{l}(q)}\frac{1}{p^{0}+k^{0}-\omega_{l}(q)-E_{l}(q)+i\epsilon},$
(17)

where we define $p^{0}+k^{0}=\sqrt{s}$,
$\omega_{l}(q)=\sqrt{q^{2}+m_{l}^{2}}$, and $E_{l}(q)=\sqrt{q^{2}+M_{l}^{2}}$,
with $m_{l}$ and $M_{l}$ the masses of meson and baryon of the $l$ channel,
respectively, and $q_{max}$ the free parameter. Furthermore, the divergence
can also be treated with the dimensional regularization scheme Oller:2000fj ;
Jido:2003cb , having

$\displaystyle G_{l}(s)=$
$\displaystyle\frac{2M_{l}}{16\pi^{2}}\left\\{a(\mu)+\ln\frac{M_{l}^{2}}{\mu^{2}}+\frac{m_{l}^{2}-M_{l}^{2}+s}{2s}\ln\frac{m_{l}^{2}}{M_{l}^{2}}\right.$
(18)
$\displaystyle+\frac{q_{cml}(s)}{\sqrt{s}}\left[\ln\left(s-\left(M_{l}^{2}-m_{l}^{2}\right)+2q_{cml}(s)\sqrt{s}\right)\right.$
$\displaystyle+\ln\left(s+\left(M_{l}^{2}-m_{l}^{2}\right)+2q_{cml}(s)\sqrt{s}\right)$
$\displaystyle-\ln\left(-s-\left(M_{l}^{2}-m_{l}^{2}\right)+2q_{cml}(s)\sqrt{s}\right)$
$\displaystyle\left.\left.-\ln\left(-s+\left(M_{l}^{2}-m_{l}^{2}\right)+2q_{cml}(s)\sqrt{s}\right)\right]\right\\}.$

Here, $\mu$ is the regularization scale, $a(\mu)$ the subtraction constant,
and $q_{cml}(s)$ the three momentum of the particle in the center-of-mass
frame, given by

$\displaystyle
q_{cml}(s)=\frac{\lambda^{1/2}\left(s,M_{l}^{2},m_{l}^{2}\right)}{2\sqrt{s}}$
(19)

with the usual Källén triangle function
$\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2(ab+ac+bc)$. In the present work, we will
use the dimensional regularization scheme. The parameter of $\mu$ is chosen as
$\mu=q_{max}=650$ MeV, which is given in Ref. Debastiani:2017ewu  by
reproducing the three $\Omega_{c}$ states found in Ref. LHCb:2017uwr . And
$a(\mu)$ is determined by matching the values of $G_{l}(s)$ function from
these two methods at the threshold, as in Ref. Oset:2001cn .

In the ChUA, to search for the pole, one must extrapolate the $G_{l}(s)$
function into the second Riemann sheet for $\text{Re}(\sqrt{s})$ being greater
than the threshold of the $l$ channel ($m_{th}$), given by

$\displaystyle G_{l}^{(II)}(s)$ $\displaystyle=G_{l}(s)-2i\text{Im}G_{l}(s)$
(20) $\displaystyle=G_{l}(s)+\frac{i}{2\pi}\frac{M_{l}q_{cml}(s)}{\sqrt{s}}.$

Thus, $G_{l}^{(II)}(s)$ is taken when $\text{Re}(\sqrt{s})\geq m_{th}$, where
the corresponding channel is open for the decay channel, see the results
later. Note that the poles on the real energy axis and below the threshold on
the first Riemann sheet are bound states. In the coupled channels approach,
the quasi-bound states appear as poles off the real energy axis on the
unphysical Riemann sheet.

In addition, the coupling constants $g_{i}$ can be calculated from the Laurent
expansion of the amplitudes near the pole $\sqrt{s_{p}}$ for different
channels Yamagata-Sekihara:2010kpd , written as

$T_{ij}=\frac{g_{i}g_{j}}{\sqrt{s}-\sqrt{s_{p}}}.$ (21)

##  III Numerical results

Table 6 shows the thresholds of different channels. The meson and baryon
masses used in the calculations are taken from Refs. Zhou:2018bkn ;
ParticleDataGroup:2022pth ; Yu:2022ymb . In Tables 7-9, we list the pole
positions and their coupling constants for each channel. In brackets after the
values of the pole position, the $+$ sign indicates that the corresponding
channel is closed, and the $-$ sign indicates that the corresponding channel
is open. The largest coupling constant is highlighted in bold. As mentioned in
the previous section, we take the free parameter $\mu=650$ MeV for all
systems.

Table 6: Thresholds (in MeV) of the relevant channels. $ccss\bar{s}$
---
Channels | $\eta^{{}^{\prime}}\Omega_{cc}$ | $D_{s}\Omega_{c}$ | $\eta^{{}^{\prime}}\Omega_{cc}^{*}$ | $D_{s}\Omega_{c}^{*}$ | $\phi\Omega_{cc}$ | $D_{s}^{*}\Omega_{c}$ | $\phi\Omega_{cc}^{*}$ | $D_{s}^{*}\Omega_{c}^{*}$
Thresholds | $4672.78$ | $4663.55$ | $4729.78$ | $4734.25$ | $4734.46$ | $4807.40$ | $4791.46$ | $4878.10$
$bbss\bar{s}$
Channels | $\eta^{{}^{\prime}}\Omega_{bb}$ | $\bar{B}_{s}\Omega_{b}$ | $\eta^{{}^{\prime}}\Omega_{bb}^{*}$ | $\bar{B}_{s}\Omega_{b}^{*}$ | $\phi\Omega_{bb}$ | $\bar{B}_{s}^{*}\Omega_{b}$ | $\phi\Omega_{bb}^{*}$ | $\bar{B}_{s}^{*}\Omega_{b}^{*}$
Thresholds | $11187.78$ | $11412.12$ | $11215.78$ | $11435.92$ | $11249.46$ | $11460.60$ | $11277.46$ | $11484.40$
$bcss\bar{s}$
Channels | $\eta^{{}^{\prime}}\Omega_{bc}$ | $\eta^{{}^{\prime}}\Omega_{bc}^{{}^{\prime}}$ | $D_{s}\Omega_{b}$ | $\bar{B}_{s}\Omega_{c}$ | $\eta^{{}^{\prime}}\Omega_{bc}^{*}$ | $D_{s}\Omega_{b}^{*}$ | $\bar{B}_{s}\Omega_{c}^{*}$ |
Thresholds | $7968.78$ | $8004.78$ | $8013.55$ | $8062.12$ | $8023.78$ | $8037.35$ | $8132.82$ |
Channels | $\phi\Omega_{bc}$ | $\phi\Omega_{bc}^{{}^{\prime}}$ | $D_{s}^{*}\Omega_{b}$ | $\bar{B}_{s}^{*}\Omega_{c}$ | $\phi\Omega_{bc}^{*}$ | $D_{s}^{*}\Omega_{b}^{*}$ | $\bar{B}_{s}^{*}\Omega_{c}^{*}$ |
Thresholds | $8030.46$ | $8066.46$ | $8157.40$ | $8110.60$ | $8085.46$ | $8181.20$ | $8181.30$ |

Figure 2: Modulus square of the amplitudes in the $ccss\bar{s}$ sector, where the ordinate is a logarithmic coordinate system. Table 7: Poles (in MeV) and their couplings for each channel in the $ccss\bar{s}$ sector. The sign $+$ indicates the corresponding channel is closed, and the sign $-$ indicates that the corresponding channel is open, in the same order as the channels in Table 1. The largest coupling constant is highlighted in bold. $I(J^{P})$ | Poles position | Couplings
---|---|---
$0(\frac{1}{2}^{-})$ |  $4640.95$ $(++)$ | $|g_{\eta^{{}^{\prime}}\Omega_{cc}}|=0.16$ | $|g_{D_{s}\Omega_{c}}|=\bf{2.18}$
$0(\frac{3}{2}^{-})$ |  $4710.21$ $(++)$ | $|g_{\eta^{{}^{\prime}}\Omega_{cc}^{*}}|=0.32$ | $|g_{D_{s}\Omega_{c}^{*}}|=\bf{2.17}$
$0(\frac{1}{2}^{-},\frac{3}{2}^{-})$ |  $4783.23-0.74i$ $(-+)$ | $|g_{\phi\Omega_{cc}}|=0.21$ | $|g_{D_{s}^{*}\Omega_{c}}|=\bf{2.24}$
$0(\frac{1}{2}^{-},\frac{3}{2}^{-},\frac{5}{2}^{-})$ |  $4850.69-3.01i$ $(-+)$ | $|g_{\phi\Omega_{cc}^{*}}|=0.39$ | $|g_{D_{s}^{*}\Omega_{c}^{*}}|=\bf{2.30}$

First, in Fig. 2, we have plotted the modulus square of the amplitudes
$|T|^{2}$ of the diagonal matrix elements in the $ccss\bar{s}$ system, where
four peak structures are clearly visible. Correspondingly, we find four poles
on the complex Riemann sheet, which are listed in Table 7. The first two poles
$4640.95$ MeV and $4710.21$ MeV are located on the first Riemann sheet without
imaginary parts, indicating that the widths of these two states are zero and
that these two states cannot decay into their coupled channel. They are mainly
coupled to the $D_{s}\Omega_{c}$ channel with quantum number
$I(J^{P})=0(1/2^{-})$ and the $D_{s}\Omega_{c}^{*}$ channel with
$I(J^{P})=0(3/2^{-})$, respectively, where the first state could mostly
qualify as a $D_{s}\Omega_{c}$ molecule and the second one as a
$D_{s}\Omega_{c}^{*}$ molecule, and both of them with a binding of about
$23-24$ MeV. The latter two poles are around $24-27$ MeV below the thresholds
of the most relevant channels, which are found on the second Riemann sheet and
the bound states of the $D_{s}^{*}\Omega_{c}$ and $D_{s}^{*}\Omega_{c}^{*}$
channels, respectively, with the widths of twice the imaginary part of the
pole. These two resonances can decay separately into the open channels
$\phi\Omega_{cc}$ and $\phi\Omega_{cc}^{*}$.

Note that Ref. Hofmann:2005sw  found a state located at $4571$ MeV with a
width of $90$ MeV and mainly coupled to the $D_{s}\Omega_{c}$ channel based on
the interactions from the chiral effective Lagrangians. As commented in Ref.
Marse-Valera:2022khy , the subtraction scales $\mu$ used in Ref.
Hofmann:2005sw  is unreasonably large to ensure that the heavy channel has a
negligible effect on the low-energy scattering of the light channel. In Ref.
Dong:2021bvy  four states with binding energies around $0.1$ MeV to $15$ MeV
were found with single channel interactions based on the interactions from the
heavy quark spin symmetry. The binding energies of the bound states obtained
by us are slightly higher than theirs. A main reason for this is that we use
the regularization scale $\mu=650$ MeV in the model, which is obtained by
fitting the experimental data in Ref. Debastiani:2017ewu . In addition, the
one-boson-exchange model was used to study the mass spectra of the
$D_{s}^{(*)}\Omega_{c}^{(*)}$-type molecular states in Ref. Wang:2023aob ,
where they obtained similar results using a slightly larger cutoff $\Lambda$
in the calculations.

Figure 3: Modulus square of amplitudes in the $bbss\bar{s}$ sector. Table 8: Poles (in MeV) and their couplings for every channel in the $bbss\bar{s}$ sector. $I(J^{P})$ | Poles position | Couplings
---|---|---
$0(\frac{1}{2}^{-})$ |  $11390.73$ $(++)$ | $|g_{\eta^{{}^{\prime}}\Omega_{bb}}|=0.00$ | $|g_{\bar{B}_{s}\Omega_{b}}|=\bf{1.77}$
$0(\frac{3}{2}^{-})$ |  $11414.56$ $(++)$ | $|g_{\eta^{{}^{\prime}}\Omega_{bb}^{*}}|=0.00$ | $|g_{\bar{B}_{s}\Omega_{b}^{*}}|=\bf{1.77}$
$0(\frac{1}{2}^{-},\frac{3}{2}^{-})$ |  $11439.27$ $(++)$ | $|g_{\phi\Omega_{bb}}|=0.00$ | $|g_{\bar{B}_{s}^{*}\Omega_{b}}|=\bf{1.77}$
$0(\frac{1}{2}^{-},\frac{3}{2}^{-},\frac{5}{2}^{-})$ |  $11463.09$ $(++)$ | $|g_{\phi\Omega_{bb}^{*}}|=0.00$ | $|g_{\bar{B}_{s}^{*}\Omega_{b}^{*}}|=\bf{1.77}$

Figure 4: Modulus square of amplitudes in the $bcss\bar{s}$ sector. Table 9: Poles (in MeV) and their couplings for every channel in the $bcss\bar{s}$ sector. $I(J^{P})$ | Poles position | Couplings
---|---|---
$0(\frac{1}{2}^{-})$ |  $7986.54$ $(++++)$ | $|g_{\eta^{{}^{\prime}}\Omega_{bc}}|=0.00$ | $|g_{\eta^{{}^{\prime}}\Omega_{bc}^{{}^{\prime}}}|=0.36$ | $|g_{D_{s}\Omega_{b}}|=\bf{1.81}$ | $|g_{\bar{B}_{s}\Omega_{c}}|=0.00$
$8036.70$ $(++++)$ | $|g_{\eta^{{}^{\prime}}\Omega_{bc}}|=0.00$ | $|g_{\eta^{{}^{\prime}}\Omega_{bc}^{{}^{\prime}}}|=0.00$ | $|g_{D_{s}\Omega_{b}}|=0.00$ | $|g_{\bar{B}_{s}\Omega_{c}}|=\bf{2.63}$
$0(\frac{3}{2}^{-})$ |  $8010.12$ $(+++)$ | $|g_{\eta^{{}^{\prime}}\Omega_{bc}^{*}}|=0.36$ | $|g_{D_{s}\Omega_{b}^{*}}|=\bf{1.80}$ | $|g_{\bar{B}_{s}\Omega_{c}^{*}}|=0.00$ |
$8107.45$ $(+++)$ | $|g_{\eta^{{}^{\prime}}\Omega_{bc}^{*}}|=0.00$ | $|g_{D_{s}\Omega_{b}^{*}}|=0.00$ | $|g_{\bar{B}_{s}\Omega_{c}^{*}}|=\bf{2.59}$ |
$0(\frac{1}{2}^{-},\frac{3}{2}^{-})$ |  $8126.30-3.82i$ $(--++)$ | $|g_{\phi\Omega_{bc}}|=0.00$ | $|g_{\phi\Omega_{bc}^{{}^{\prime}}}|=0.41$ | $|g_{D_{s}^{*}\Omega_{b}}|=\bf{1.91}$ | $|g_{\bar{B}_{s}^{*}\Omega_{c}}|=0.00$
$8085.20$ $(++++)$ | $|g_{\phi\Omega_{bc}}|=0.00$ | $|g_{\phi\Omega_{bc}^{{}^{\prime}}}|=0.00$ | $|g_{D_{s}^{*}\Omega_{b}}|=0.00$ | $|g_{\bar{B}_{s}^{*}\Omega_{c}}|=\bf{2.63}$
$0(\frac{1}{2}^{-},\frac{3}{2}^{-},\frac{5}{2}^{-})$ |  $8150.07-3.97i$ $(-++)$ | $|g_{\phi\Omega_{bc}^{*}}|=0.41$ | $|g_{D_{s}^{*}\Omega_{b}^{*}}|=\bf{1.91}$ | $|g_{\bar{B}_{s}^{*}\Omega_{c}^{*}}|=0.00$ |
$8155.95$ $(+++)$ | $|g_{\phi\Omega_{bc}^{*}}|=0.00$ | $|g_{D_{s}^{*}\Omega_{b}^{*}}|=0.00$ | $|g_{\bar{B}_{s}^{*}\Omega_{c}^{*}}|=\bf{2.60}$ |

Next, in the $bbss\bar{s}$ sector, we show our results in Fig. 3. From the
modulus square of the amplitudes we can see four distinct peaks around the
energy range $11380$ MeV to $11480$ MeV. The poles and their coupling
constants for each coupled channel are given in Table 8. The four states
obtained are the $\bar{B}_{s}\Omega_{b}$, $\bar{B}_{s}\Omega_{b}^{*}$,
$\bar{B}_{s}^{*}\Omega_{b}$, and $\bar{B}_{s}^{*}\Omega_{b}^{*}$ bound states,
respectively, which are all found on the first Riemann sheet. Due to the
absence of the exchange of light pseudoscalar and $B_{s}^{*}$ mesons, which
means that there are no coupled channels for each other, see the coefficients
in Table 4, the obtained poles are all without imaginary parts. The thresholds
of the $\bar{B}_{s}\Omega_{b}$, $\bar{B}_{s}\Omega_{b}^{*}$,
$\bar{B}_{s}^{*}\Omega_{b}$, and $\bar{B}_{s}^{*}\Omega_{b}^{*}$ channels are
$11412.12$ MeV, $11435.92$ MeV, $11460.60$ MeV, and $11484.40$ MeV,
respectively, see Table 6, and so we have these states all bound at about $21$
MeV.

Finally, there are eight bound states in the $bcss\bar{s}$ sector, as shown in
Fig. 4 and Table 9. In the case of $PB(1/2^{+})$, two poles couple most
strongly to the channels $D_{s}\Omega_{b}$ and $\bar{B}_{s}\Omega_{c}$, with
the binding energies $27$ MeV and $25$ MeV, respectively. In the case of
$PB(3/2^{+})$, the pole at $8010.12$ MeV would be mostly a
$D_{s}\Omega_{b}^{*}$ bound state with a binding energy of $27$ MeV, and the
pole at $8107.45$ MeV would be mostly a $\bar{B}_{s}\Omega_{c}^{*}$ state
bound at $25$ MeV. Note that these states have no width, although for the last
one, $8107.45$ MeV, there may be two lower channels open that are not coupled
to the $\bar{B}_{s}\Omega_{c}^{*}$ channel. The two states obtained in the
case of $VB(1/2^{+})$ sector degenerate in $J^{P}=1/2^{-}$ and
$J^{P}=3/2^{-}$. The one of $8126.30-3.82i$ MeV is a molecule of
$D_{s}^{*}\Omega_{b}$ and can decay into the open channel
$\phi\Omega_{bc}^{{}^{\prime}}$. The other one of $8085.20$ MeV is a molecule
of $\bar{B}_{s}^{*}\Omega_{c}$ with zero width, which is not coupled to the
other channels in this sector. Furthermore, the two states obtained in the
case of the $VB(3/2^{+})$ sector degenerate in $J^{P}=1/2^{-}$, $3/2^{-}$, and
$5/2^{-}$. The state $8150.07-3.97i$ MeV is located below the threshold of the
most relevant channel $D_{s}^{*}\Omega_{b}^{*}$ with the largest coupling
constant $|g_{D_{s}^{*}\Omega_{b}^{*}}|=1.91$, where its binding energy is
$31$ MeV, and the decay channel is $\phi\Omega_{bc}^{*}$. The last state in
Table 9 can be considered as a $\bar{B}_{s}^{*}\Omega_{c}^{*}$ bound state
with a binding energy of $25$ MeV and the coupling
$|g_{\bar{B}_{s}^{*}\Omega_{c}^{*}}|=2.60$, which has also no width since
there are no coupled channels to decay to.

Note that for the molecular states obtained above, the attractions of their
most relevant channels are provided by the exchange of the vector meson
$\phi$, while the interactions in nondiagonal terms are weaker due to the
exchange of the heavy vector meson $D_{s}^{*}$. It is very interesting that in
the absence of long-range interactions mediated by the exchange of $\pi$
meson, the bound states can only be formed by the exchange of the heavier
meson $\phi$. These types of doubly heavy pentaquarks with the most strange
quark contents $ccss\bar{s}$, $bbss\bar{s}$, and $bcss\bar{s}$ are excellent
systems to validate this mechanism. We look forward to the experimental search
for these states to confirm the model used in this paper. If these states
exist, it will deepen our understanding of the nonperturbative behavior of the
strong interaction.

##  IV Summary

In the present work, we make a study of $\Omega_{cc}$, $\Omega_{bb}$, and
$\Omega_{bc}$ molecular states dynamically generated from the meson-baryon
interaction in the charm and bottom sectors. With the extended local hidden
gauge approach, we obtain the dominant interactions through the vector meson
exchange mechanism, and then calculate the scattering amplitudes by solving
the Bethe-Salpeter equation in the on-shell approximation. By searching for
poles in the complex energy plane, the masses and widths of the molecular
states are determined. We find several candidates for hadronic molecular
states with the quark components $ccss\bar{s}$, $bbss\bar{s}$, and
$bcss\bar{s}$.

In the $ccss\bar{s}$ sector, we obtain four bound states mainly coupled to the
$D_{s}\Omega_{c}$, $D_{s}\Omega_{c}^{*}$, $D_{s}^{*}\Omega_{c}$, and
$D_{s}^{*}\Omega_{c}^{*}$ channels, respectively. In the $bbss\bar{s}$ sector,
we obtain four bound states mainly coupled to the $\bar{B}_{s}\Omega_{b}$,
$\bar{B}_{s}\Omega_{b}^{*}$, $\bar{B}_{s}^{*}\Omega_{b}$, and
$\bar{B}_{s}^{*}\Omega_{b}^{*}$ channels, respectively. In the $bcss\bar{s}$
sector, we obtain eight bound states mainly coupled to $D_{s}\Omega_{b}$,
$\bar{B}_{s}\Omega_{c}$, $D_{s}\Omega_{b}^{*}$, $\bar{B}_{s}\Omega_{c}^{*}$,
$D_{s}^{*}\Omega_{b}$, $\bar{B}_{s}^{*}\Omega_{c}$, $D_{s}^{*}\Omega_{b}^{*}$,
and $\bar{B}_{s}^{*}\Omega_{c}^{*}$ channels, respectively. In the cases of
the $PB(1/2^{+})$, $PB(3/2^{+})$, $VB(1/2^{+})$, and $VB(3/2^{+})$ sectors,
which carry the quantum numbers $I(J^{P})=0(1/2^{-})$, $0(3/2^{-})$,
$0(1/2^{-},3/2^{-})$, and $0(1/2^{-},3/2^{-},5/2^{-})$, respectively. All
these states have very small or zero widths, which are below the threshold of
the most dominant channel about $20-30$ MeV. The LHCb Collaboration is
carrying out the relevant measurement and analysis, and these predicted states
are likely to be detected in the near future.

## Acknowledgements

We would like to thank Fu-Lai Wang and Wen-Fei Wang for valuable discussions,
and acknowledge Profs. Eulogio Oset and Kazem Azizi for careful reading the
manuscript and useful comments. This work is supported by the China National
Funds for Distinguished Young Scientists under Grant No. 11825503, the
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 under Grant No. lzujbky-2022-sp02,
the project for top-notch innovative talents of Gansu province, and the
National Natural Science Foundation of China (NSFC) under Grants No. 12247101,
12335001, 11965016, 11705069 and 12047501 (Z.F.S.). This work is also partly
supported by the Natural Science Foundation of Changsha under Grants No.
kq2208257, the Natural Science Foundation of Hunan province under Grant No.
2023JJ30647, and the NSFC under Grant No. 12365019 (C.W.X.).

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