# Unveiling the Structure of Hidden-Bottom Strange Pentaquarks via Magnetic
Moments

Halil Mutuk<EMAIL_ADDRESS>Department of Physics, Faculty of Sciences,
Ondokuz Mayis University, 55200 Samsun, Türkiye  Xian-Wei Kang
<EMAIL_ADDRESS>Key Laboratory of Beam Technology of the Ministry of
Education, College of Nuclear Science and Technology, Beijing Normal
University, Beijing 100875, China  Institute of Radiation Technology, Beijing
Academy of Science and Technology, Beijing 100875, China

###### Abstract

Motivated by the discovery of hidden-charm strange pentaquarks, we conduct a
systematic study of the magnetic moments of the hidden-bottom strange
pentaquarks in molecular picture. We calculate magnetic moments of hidden-
bottom strange pentaquarks with strangeness-1 and 2. Magnetic moment gives
valuable information about the inner structure and shape of the hadron. The
obtained results may be helpful to determine the inner structure of these
hypothetical states.

Pentaquarks, Hidden-bottom strange pentaquarks, Magnetic moments, Hadronic
molecule

##  I Motivation

Although there were theoretical predictions about the existence of multiquark
states apart from conventional hadrons $(q\bar{q},qqq)$, the discovery of
$\chi_{c1}(3872)$ by the Belle Collaboration in 2003 [1] opened a new era in
the hadron physics. This particle was the first observed four-quark state. A
large number of non-conventional hadrons have been observed by the
experimental facilities in the past two decades. These non-conventional
hadronic states are called exotic hadrons, since their physical features
cannot be fit to the quark model.

The paradigm of multiquark states turned into a more interesting path when the
pentaquark states are observed. In 2015, the LHCb Collaboration announced the
observation of two peaks in the $J/\psi p$ invariant mass spectrum of
$\Lambda_{b}^{0}\to J/\psi pK^{-}$ process with masses and widths [2]:

$\displaystyle m_{P_{c}(4380)^{+}}$ $\displaystyle=$ $\displaystyle 4380\pm
8\pm 29~{}\mathrm{MeV},\quad\Gamma_{P_{c}(4380)^{+}}=205\pm 18\pm
86~{}\mathrm{MeV},$ (1) $\displaystyle m_{P_{c}(4450)^{+}}$ $\displaystyle=$
$\displaystyle 4449.8\pm 1.7\pm
2.5~{}\mathrm{MeV},\quad\Gamma_{P_{c}(4450)^{+}}=39\pm 5\pm
19~{}\mathrm{MeV}.$ (2)

Using an updated data, in 2019 LHCb Collaboration reported three new
pentaquark states named as $P_{c}(4312)$, $P_{c}(4440)$ and $P_{c}(4457)$ [3].
The previously reported $P_{c}(4450)$ state was resolved into two narrow
states as $P_{c}(4440)$ and $P_{c}(4457)$. The masses and widhts of these new
pentaquark states are as follows

$\displaystyle m_{P_{c}(4312)^{+}}$ $\displaystyle=$ $\displaystyle 4311.9\pm
0.7^{+6.8}_{-0.6}~{}\mathrm{MeV},\quad\Gamma_{P_{c}(4312)^{+}}=9.8\pm
2.7^{+3.7}_{-4.5}~{}\mathrm{MeV},$ (3) $\displaystyle m_{P_{c}(4440)^{+}}$
$\displaystyle=$ $\displaystyle 4440.3\pm
1.3^{+4.1}_{-4.7}~{}\mathrm{MeV},\quad\Gamma_{P_{c}(4440)^{+}}=20.6\pm
4.9^{+8.7}_{-10.1}~{}\mathrm{MeV},$ (4) $\displaystyle m_{P_{c}(4457)^{+}}$
$\displaystyle=$ $\displaystyle 4457.3\pm
0.6^{+4.1}_{-1.7}~{}\mathrm{MeV},\quad\Gamma_{P_{c}(4457)^{+}}=6.4\pm
2.0^{+5.7}_{-1.9}~{}\mathrm{MeV}.$ (5)

According to exotic hadron naming convention [4], these three states are
denoted as

$P_{c}(4312)\to P_{\psi}^{N}(4312),~{}P_{c}(4440)\to
P_{\psi}^{N}(4440),~{}P_{c}(4457)\to P_{\psi}^{N}(4457).$ (6)

In 2021, LHCb reported evidence of the $P_{cs}(4459)$ ($P_{\psi
s}^{\Lambda}(4459)$ in the new naming convention) with charm quark content by
analyzing of $J/\psi\Lambda$ invariant mass distribution in
$\Xi_{b}^{-}\rightarrow J/\psi K^{-}\Lambda$ decays [5] with the mass and
width

$m_{P_{\psi s}^{\Lambda}(4459)}=4458.8\pm
2.9^{+4.7}_{-1.1}~{}\mathrm{MeV},\quad\Gamma_{P_{\psi
s}^{\Lambda}(4459)}=17.3\pm 6.5^{+8.0}_{-5.7}~{}\mathrm{MeV},$ (7)

respectively. Very recently, $P_{\psi s}^{\Lambda}(4338)$ $(P_{cs}(4338))$
state is observed in the $B^{-}\rightarrow J/\psi\Lambda\bar{p}$ process with
mass and width

$m_{P_{\psi s}^{\Lambda}(4338)}=4338.2\pm 0.7\pm
0.4~{}\mathrm{MeV},\quad\Gamma_{P_{\psi s}^{\Lambda}(4338)}=7.0\pm 1.2\pm
1.3~{}\mathrm{MeV},$ (8)

respectively, [6]. The spin-parities of $P_{\psi s}^{\Lambda}(4459)$ and
$P_{\psi s}^{\Lambda}(4338)$ states are preferably $\frac{3}{2}^{-}$ and
$\frac{1}{2}^{-}$.

After the observation of hidden-charm strange pentaquark states, various
studies with different models are devoted to understand the spectroscopic
parameters such as by using effective field theories [7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18], line shapes [19, 16], QCD sum rules [20, 21, 22, 23, 24],
phenomenological quark models [25, 26, 27, 28, 29, 30, 31, 32, 19, 33, 34],
and machine learning tools [35]. According to these studies, $P_{\psi
s}^{\Lambda}(4338)$ state can be identified as a $\Xi_{c}\bar{D}^{(*)}$
molecular structure whereas the $P_{\psi s}^{\Lambda}(4459)$ seems better to
be a $\Xi^{(^{\prime},*)}_{c}\bar{D}^{(*)}$ hadronic molecule. Apart from
these models, different approaches arouse to elucidate inner structure of
these states such as mixed configurations [36, 12], compact structure analysis
[37, 38] and triangle singularities [39]. The production and decay properties
of these pentaquark states are also studied in Refs. [40, 41, 42, 43, 44, 45,
46].

The aforementioned studies present information about the underlying structures
of the $P_{\psi s}^{\Lambda}(4338)$ and $P_{\psi s}^{\Lambda}(4459)$ states.
However, some models may not distinguish the color and spin configurations in
the same multiplet. In order to shed light on the inner structure of these
states, apart from radiative decays and weak decays, electromagnetic
properties are needed to be obtained. Studying electromagnetic properties of
the related hadron paves the way for this task. They encode valuable
information about charge distribution and response to magnetic field.
Electromagnetic properties of $P_{\psi s}^{\Lambda}(4338)$ and $P_{\psi
s}^{\Lambda}(4459)$ states are studied in Refs. [47, 48, 49, 50, 51, 52, 33,
53, 54, 55, 56]. The magnetic moments of different exotic states are
successfully obtained in Refs. [57, 58, 59, 60, 61]. For a good introduction
of molecular interpretation about hadronic molecules, see Ref. [62].

Historically, the observation of hadrons containing charm valence quarks is
followed by the identification of similar hadrons containing a bottom quark
content. Accordingly, it is natural to expect possible observation of the
bottom analogues of the observed hidden-charm strange pentaquark states.
Spectroscopic and electromagnetic properties together with strong and weak
decays maintain valuable information for future experimental and lattice QCD
studies. Furthermore, theoretical and phenomenological studies may be helpful
to get insights into the underlying structure and dynamics of these particles.
This motivation led us to study electromagnetic properties of bottom analogues
of hidden-charm strange pentaquark states.

The paper is organized as follows: in Section II, we introduce wave functions
in molecular picture. We briefly introduce magnetic moment concept in Section
III and calculate magnetic moments in Section IV. Last section contains a
brief discussion and concluding remarks.

##  II Wave Functions

There are different pictures while studying hidden-heavy pentaquark states:
molecular, diquark-diquark-antiquark, and diquark-triquark pictures. Among
these pictures, molecular models are interesting since as outlined in Ref.
[63], a clear-cut method for identifying a multiquark state (including
hadronic molecule configurations) is to observe resonances decaying into a
heavy quarkonium plus a light-quark baryon or meson with nonzero isospin.
Studying electromagnetic properties of pentaquark states in molecular models
presents useful clues about the internal structure.

Wave function of a hadron can be written in terms of four degrees of freedom
as

$\Psi_{total}=\psi_{flavor}\psi_{spin}\psi_{color}\psi_{spatial},$ (9)

where the total wave function should be antisymmetric due to the Fermi
statistics. We focus on $S$\- wave hidden-bottom pentaquark states with
strangeness $\mathcal{S}=-1$. In the ground state, color wave function
$\psi_{color}$ is antisymmetric whereas space wave function $\psi_{spatial}$
is symmetric. Therefore the symmetry requirement of spin-flavor is adequate
for studying magnetic moments.

A pentaquark state in molecular picture is made up of a baryon and a meson.
The possible quark arrangements are $(b\bar{b})(q_{1}q_{2}q_{3})$ and
$(\bar{b}q_{3})(bq_{1}q_{2})$. In the former case, it is hard to form a
loosely bound molecular state between bottomonium and a light baryon.
Moreover, as mentioned in Ref. [3], narrow $P_{c}^{+}$ states could arise by
binding a narrow baryon and a narrow meson such that separation of $c$ and
$\bar{c}$ into distinct confinement volumes maintains a natural suppresion
mechanism for the widths of $P_{c}^{+}$ states. Since the separation of $b$
and $\bar{b}$ is smaller then the separation of $c$ and $\bar{c}$ in
charmonium, the same suppression may take place in the hidden-bottom
pentaquark states. As a result, we take into account
$(\bar{b}q_{3})(bq_{1}q_{2})$ configuration and do not consider
$(b\bar{b})(q_{1}q_{2}q_{3})$ configuration in the molecular picture.

According to aforesaid discussion, baryon inside the hidden-bottom strange
pentaquark should be a singly-bottom baryon and meson inside the baryon should
be bottom-light meson. The two light quarks inside the singly-bottom baryon
can be either symmetric or antisymmetric. In the symmetric case, $3_{f}$
flavor representation of bottom-light meson couples with the $6_{f}$
representation of the singly-bottom baryon to form $6_{f}\otimes
3_{f}=10_{f}\oplus 8_{1f}$. In the antisymmetric case, the singly-bottom
baryon has $\bar{3}_{f}$ flavor representation which couples to $3_{f}$ flavor
representation of bottom-light meson to form $\bar{3}_{f}\otimes
3_{f}=1_{f}\oplus 8_{2f}$. The so far observed pentaquark states are likely to
be a member of octet states. For example, in the molecular scheme, the masses
of $P_{c}(4380)$ and $P_{c}(4450)$ states with isospin $I=\frac{1}{2}$ are
very close the mass threshold of $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ and belong to
the $8_{1f}$ representation since they were detected in the $J/\psi p$ channel
[64]. Therefore it is useful to investigate spectroscopic and electromagnetic
properties of the hidden-heavy strange pentaquark states in molecular model
[65].

In Table 1, we respectively list the flavor wave functions of hidden-bottom
strange molecular pentaquarks under $SU(3)$ symmetry.

Table 1: The flavor wave functions of hidden-bottom strange molecular pentaquark states. $I$ represents isospin and $I_{3}$ represents its third component. Here, $\\{q_{1}q_{2}\\}=\frac{1}{\sqrt{2}}(q_{1}q_{2}+q_{2}q_{1})$ and $[q_{1}q_{2}]=\frac{1}{\sqrt{2}}(q_{1}q_{2}-q_{2}q_{1})$. State | Flavor | $(I,I_{3})$ | Wave function
---|---|---|---
$P_{bs}^{1}$ | $8_{1f}$ | $(1,1)$ | $\sqrt{\frac{1}{3}}({\bar{b}}u)(b\\{us\\})-\sqrt{\frac{2}{3}}({\bar{b}}s)(b\\{uu\\})$
| $8_{2f}$ |  | $({\bar{b}}u)(b[us])$
$P_{bs}^{2}$ | $8_{1f}$ | $(1,0)$ | $\sqrt{\frac{1}{6}}[({\bar{b}}d)(b\\{us\\})+({\bar{b}}u)(b\\{ds\\})]-\sqrt{\frac{2}{3}}({\bar{b}}s)(b\\{ud\\})$
| $8_{2f}$ |  | $\frac{1}{\sqrt{2}}\\{({\bar{b}}d)(b[us])+({\bar{b}}u)(b[ds])\\}$
$P_{bs}^{3}$ | $8_{1f}$ | $(1,-1)$ | $\sqrt{\frac{1}{3}}({\bar{b}}d)(b\\{ds\\})-\sqrt{\frac{2}{3}}({\bar{b}}s)(b\\{dd\\})$
| $8_{2f}$ |  | $({\bar{b}}d)(b[ds])$
$P_{bs}^{4}$ | $8_{1f}$ | $(0,0)$ | $\sqrt{\frac{1}{2}}[({\bar{b}}u)(b\\{ds\\})-({\bar{b}}d)(b\\{us\\})]$
| $8_{2f}$ |  | $\frac{1}{\sqrt{6}}\\{({\bar{b}}d)(b[us])-({\bar{b}}u)(b[ds])-2(\bar{b}s)(b[ud])\\}$
$P_{bss}^{1}$ | $8_{1f}$ | $(\frac{1}{2},\frac{1}{2})$ | $\sqrt{\frac{1}{3}}({\bar{b}}s)(b\\{us\\})-\sqrt{\frac{2}{3}}({\bar{b}}u)(b\\{ss\\})$
| $8_{2f}$ |  | $({\bar{b}}s)(b[us])$
$P_{bss}^{2}$ | $8_{1f}$ | $(\frac{1}{2},-\frac{1}{2})$ | $\sqrt{\frac{1}{3}}({\bar{b}}s)(b\\{ds\\})-\sqrt{\frac{2}{3}}({\bar{b}}d)(b\\{ss\\})$
| $8_{2f}$ |  | $({\bar{b}}s)(b[ds])$

One can understand these flavor representations as follows: $8_{1f}$ flavor
representations are combinations or mixings of two states whereas $8_{2f}$
flavor representations are pure states except $I_{3}=0$.

##  III Magnetic Moments

Since quarks are Dirac fermions, at the quark level the operators of the
magnetic moments are

$\hat{\mu}=\sum_{i}\frac{Q_{i}}{2m_{i}}\hat{\sigma}_{i},$ (10)

where $Q_{i}$, $m_{i}$, and $\hat{\sigma}_{i}$ denote charge, mass, and
Pauli’s spin matrix of the $i$th quark, respectively. The total magnetic
moment formula for the $S$-wave molecular pentaquarks can be written as

$\displaystyle\hat{\mu}=\hat{\mu}_{B}+\hat{\mu}_{M},$ (11)

where the subscripts $B$ and $M$ represent the baryon and meson, respectively.
The baryon and meson magnetic moments are

$\displaystyle\mu$ $\displaystyle=$
$\displaystyle\langle\psi|\hat{\mu}_{B}+\hat{\mu}_{M}|\psi\rangle$ (12)
$\displaystyle=$ $\displaystyle\sum_{SS_{z},ll_{z}}\langle
SS_{z},ll_{z}|JJ_{z}\rangle^{2}\left\\{\sum_{S_{B},S_{M}}\ \langle\
S_{B}\widetilde{S}_{B},S_{M}\widetilde{S}_{M}|SS_{z}\
\rangle^{2}\Bigg{[}\widetilde{S}_{M}\bigg{(}\mu_{\bar{b}}+\mu_{q_{3}}\bigg{)}\right.$
$\displaystyle+$ $\displaystyle\left.\sum_{\widetilde{S}_{b}}\ \langle\
S_{b}\widetilde{S}_{b},S_{Di}\widetilde{S}_{B}-\widetilde{S}_{b}|S_{B}\widetilde{S}_{B}\rangle^{2}\bigg{(}g\mu_{b}\widetilde{S}_{b}+(\widetilde{S}_{B}-\widetilde{S}_{b})(\mu_{q_{1}}+\mu_{q_{2}})\bigg{)}\Bigg{]}\right\\},$

where $\psi$ represents the flavor wave function in molecular model listed in
in Table 1, $S_{M}$, $S_{B}$, $S_{Di}$, $S_{b}$ are the meson, baryon, the
diquark spin inside the baryon, and bottom quark, respectively.
$\widetilde{S}$ is the third spin component.

We take $8_{2f}$ of $P_{bs}^{4}$ hidden-bottom strange molecular state for an
example. Let us take $J^{P}=\frac{1}{2}(\frac{1}{2}^{+}\otimes 1^{-})$ where
$(J_{B}^{P_{b}}\otimes J_{M}^{P_{m}})$ correspond to the angular momentum and
parity of baryon and meson. The wave function

$|P_{bs}^{4}\rangle=\frac{1}{\sqrt{6}}\\{({\bar{b}}d)(b[us])-({\bar{b}}u)(b[ds])-2(\bar{b}s)(b[ud])\\},$
(13)

gives magnetic moment as

$\displaystyle\mu$ $\displaystyle=\langle
P_{bs}^{4}|\hat{\mu}_{B}+\hat{\mu}_{M}\ |P_{bs}^{4}\rangle$
$\displaystyle=\langle\frac{1}{2}\frac{1}{2},10|\frac{1}{2}\frac{1}{2}\rangle^{2}\Bigg{[}\langle\frac{1}{2}\frac{1}{2},00|\frac{1}{2}\frac{1}{2}\rangle^{2}\Bigg{(}\frac{1}{6}(\frac{1}{2}g\mu_{b})+\frac{1}{6}(\frac{1}{2}g\mu_{b})+\frac{4}{6}(\frac{1}{2}g\mu_{b})\Bigg{)}\Bigg{]}+$
$\displaystyle\ \ \ \ \
\langle\frac{1}{2}-\frac{1}{2},11|\frac{1}{2}\frac{1}{2}\rangle^{2}\Bigg{[}\Bigg{(}\frac{1}{6}(\frac{1}{2}g\mu_{\bar{b}}+\frac{1}{2}g\mu_{d})+\frac{1}{6}(\frac{1}{2}g\mu_{\bar{b}}+\frac{1}{2}g\mu_{u})+\frac{4}{6}(\frac{1}{2}g\mu_{\bar{b}}+\frac{1}{2}g\mu_{s})\Bigg{)}+$
$\displaystyle\ \ \ \ \
\langle\frac{1}{2}-\frac{1}{2},00|\frac{1}{2}-\frac{1}{2}\rangle^{2}\Bigg{(}\frac{1}{6}(-\frac{1}{2}g\mu_{b})+\frac{1}{6}(-\frac{1}{2}g\mu_{b})+\frac{4}{6}(-\frac{1}{2}g\mu_{b})\Bigg{)}\Bigg{]}$
$\displaystyle=\frac{1}{9}(\mu_{u}+\mu_{d}+4\mu_{s}-9\mu_{b}).$ (14)

We use the following constituent quark masses for numerical analysis [66]

$\displaystyle m_{u}\ =0.338\ \mbox{GeV},\ m_{d}\ =\ 0.350\ \mbox{GeV},\
m_{s}\ =\ 0.500\ \mbox{MeV},\ \ m_{b}\ =\ 4.67\ \mbox{GeV}.$ (15)

##  IV Numerical Results and Discussion

Before calculating magnetic moments of hidden-bottom strange pentaquarks, it
will be beneficial to calculate magnetic moments of octet baryons which are
experimentally observed. In Table 2, we present our results with the ones
observed in experiments. Our results are in rough agreement with the
experimental results.

Table 2:  Magnetic moments of the octet baryons. The results are listed in unit of nuclear magneton $\mu_{N}$. State | Magnetic moment | This work | Experiment [67]
---|---|---|---
$p$ | $\frac{4}{3}\mu_{u}-\frac{1}{3}\mu_{d}$ | $2.764$ | $2.793$
$n$ | $\frac{4}{3}\mu_{d}-\frac{1}{3}\mu_{u}$ | $-1.808$ | $-1.913$
$\Lambda$ | $\mu_{s}$ | $-0.625$ | $-0.613\pm 0.006$
$\Sigma^{+}$ | $\frac{4}{3}\mu_{u}-\frac{1}{3}\mu_{s}$ | $2.675$ | $2.460\pm 0.006$
$\Sigma^{-}$ | $\frac{4}{3}\mu_{d}-\frac{1}{3}\mu_{s}$ | $-0.982$ | $-1.160\pm 0.025$
$\Xi^{0}$ | $\frac{4}{3}\mu_{s}-\frac{1}{3}\mu_{u}$ | $-1.450$ | $-1.250\pm 0.014$
$\Xi^{-}$ | $\frac{4}{3}\mu_{s}-\frac{1}{3}\mu_{d}$ | $-0.536$ | $-0.651\pm 0.0025$
$\Omega^{-}$ | $3\mu_{s}$ | $-1.876$ | $-2.020\pm 0.05$

Motivated with this picture, we calculate magnetic moments of hidden-bottom
strange pentaquarks. The results are presented in Table 3.

Table 3:  Magnetic moments of the molecular hidden-bottom strange pentaquark states. The results are listed in unit of nuclear magneton $\mu_{N}$. State | $J_{B}^{P_{b}}\otimes J_{M}^{P_{m}}$ | $I(J^{P})$ | Magnetic moment $(8_{1f})$ | Magnetic moment $(8_{2f})$
---|---|---|---|---
$P_{bs}^{1}$ | $\frac{1}{2}^{+}\otimes 0^{-}$ | $1(\frac{1}{2})^{-}$ | $1.939$ | $-0.067$
| $\frac{1}{2}^{+}\otimes 1^{-}$ | $1(\frac{1}{2})^{-}$ | $-0.468$ | $1.300$
|  | $1(\frac{3}{2})^{-}$ | $2.206$ | $1.850$
$P_{bs}^{2}$ | $\frac{1}{2}^{+}\otimes 0^{-}$ | $1(\frac{1}{2})^{-}$ | $0.415$ | $-0.067$
| $\frac{1}{2}^{+}\otimes 1^{-}$ | $1(\frac{1}{2})^{-}$ | $-0.265$ | $0.386$
|  | $1(\frac{3}{2})^{-}$ | $0.224$ | $0.478$
$P_{bs}^{3}$ | $\frac{1}{2}^{+}\otimes 0^{-}$ | $1(\frac{1}{2})^{-}$ | $-1.109$ | $-0.067$
| $\frac{1}{2}^{+}\otimes 1^{-}$ | $1(\frac{1}{2})^{-}$ | $-0.062$ | $-0.573$
|  | $1(\frac{3}{2})^{-}$ | $-1.757$ | $-0.893$
$P_{bs}^{4}$ | $\frac{1}{2}^{+}\otimes 0^{-}$ | $0(\frac{1}{2})^{-}$ | $-0.076$ | $-0.067$
| $\frac{1}{2}^{+}\otimes 1^{-}$ | $0(\frac{1}{2})^{-}$ | $0.389$ | $-0.105$
|  | $0(\frac{3}{2})^{-}$ | $0.470$ | $-0.257$
$P_{bss}^{1}$ | $\frac{1}{2}^{+}\otimes 0^{-}$ | $\frac{1}{2}(\frac{1}{2})^{-}$ | $-0.261$ | $-0.067$
| $\frac{1}{2}^{+}\otimes 1^{-}$ | $\frac{1}{2}(\frac{1}{2})^{-}$ | $0.815$ | $-0.350$
|  | $\frac{1}{2}(\frac{3}{2})^{-}$ | $0.741$ | $-0.625$
$P_{bss}^{2}$ | $\frac{1}{2}^{+}\otimes 0^{-}$ | $\frac{1}{2}(\frac{1}{2})^{-}$ | $-0.871$ | $-0.067$
| $\frac{1}{2}^{+}\otimes 1^{-}$ | $\frac{1}{2}(\frac{1}{2})^{-}$ | $-0.201$ | $-0.350$
|  | $\frac{1}{2}(\frac{3}{2})^{-}$ | $-1.608$ | $-0.625$

As can be seen in the table, the magnetic moments of the $8_{1f}$ and $8_{2f}$
flavor representations are different: no magnetic moment of $8_{1f}$ flavor
representation is equal to magnetic moment of $8_{2f}$ flavor representation.
We highlight our results as follows:

  * •

In the $P_{bs}^{1}$ states, the possible observation of magnetic moment may
serve to estimate the quantum numbers. According to the our results,
observation of a negative magnetic moment may support
$J^{P}=1(\frac{1}{2})^{-}$ assignment. Furthermore, a magnetic moment
measurement may also serve for determining the angular momentum of baryon and
meson in the molecular pentaquark state, $\frac{1}{2}^{+}\otimes 0^{-}$ or
$\frac{1}{2}^{+}\otimes 1^{-}$. The observation of the magnetic moment around
$-0.5~{}\mu_{N}$ may give a hint about the internal quantum numbers.
$J^{P}=\frac{3}{2}^{-}$ have the largest magnetic moments in both $8_{1f}$ and
$8_{2f}$ flavor representations.

  * •

In the $P_{bs}^{2}$ states, the sign of the magnetic moment for the quantum
number $J^{P}=\frac{1}{2}^{-}$ has opposite signs in $8_{1f}$ and $8_{2f}$
flavor representations. A possible observation of magnetic moment may identify
the angular momentum configuration. $J^{P}=\frac{3}{2}^{-}$ of $8_{2f}$ flavor
representation has the largest magnetic moment in magnitude.

  * •

All the magnetic moments of $P_{bs}^{3}$ states have negative sign. Therefore
measurement of magnetic moment may identify the quantum number and spin
configurations of these states. $J^{P}=\frac{3}{2}^{-}$ of $8_{1f}$ flavor
representation has the largest magnetic moment.

  * •

In the $P_{bs}^{4}$ states, the quantum numbers $J^{P}=\frac{1}{2}^{-}$ and
$J^{P}=\frac{3}{2}^{-}$ of $\frac{1}{2}^{+}\otimes 1^{-}$ spin configuration
in $8_{1f}$ flavor representation has positive magnetic moments whereas the
rest has negative magnetic moments. A possible observation of positive value
magnetic moment may identify spin configuration. $J^{P}=\frac{3}{2}^{-}$ of
$8_{1f}$ flavor representation has the largest magnetic moment in magnitude.

  * •

The quantum number $J^{P}=\frac{1}{2}^{-}$ of $\frac{1}{2}^{+}\otimes 0^{-}$
spin configuration has negative magnetic moments in $8_{1f}$ and $8_{2f}$
flavor representations of $P_{bss}^{1}$ states. $J^{P}=\frac{1}{2}^{-}$ of
$\frac{1}{2}^{+}\otimes 1^{-}$ spin configuration has positive magnetic moment
in $8_{1f}$ flavor representation whereas the magnetic moment is negative in
the $8_{2f}$ flavor representation. This is also valid for the quantum number
$J^{P}=\frac{3}{2}^{-}$. Unlike previous cases, $J^{P}=\frac{1}{2}^{-}$ of
$\frac{1}{2}^{+}\otimes 1^{-}$ in $8_{1f}$ flavor representation has the
largest magnetic moment in magnitude.

  * •

In the $P_{bss}^{2}$ states, all the magnetic moments have negative sign.
Therefore measurement of magnetic moment may identify the quantum number and
spin configurations of these states. $J^{P}=\frac{3}{2}^{-}$ of $8_{1f}$
flavor representation has the largest magnetic moment.

  * •

One of the interesting patterns is that all the $J^{P}=\frac{1}{2}^{-}$ of
$\frac{1}{2}^{+}\otimes 0^{-}$ spin configurations in the $8_{2f}$ flavor
representation has the same magnetic moment, $\mu=-0.067\mu_{N}$. The reason
for this could be that magnetic moments of these states are only determined by
$b$ quark.

  * •

The other interesting pattern is that the magnetic moments of $8_{2f}$ flavor
representations of $P_{bss}^{1}$ and $P_{bss}^{2}$ states are same for each
corresponding quantum numbers and spin configurations. Apart from
$J^{P}=\frac{1}{2}^{-}$ of $\frac{1}{2}^{+}\otimes 0^{-}$ spin configuration
in $8_{2f}$ of all states, magnetic moments could be determined by $s$ and $b$
quarks in the $\frac{1}{2}^{+}\otimes 1^{-}$ spin configurations.

Detailed analysis showed that the contribution of light quarks to the magnetic
moment cancel each other in some configurations. The hadron contains not only
valence quarks but also sea quarks, whose effects are embodied in the
constituent quark masses. Contributions of sea quarks to the magnetic moment
are studied in unquenched quark model [68, 69, 70]. These results showed that
the inclusion of the effects of sea quarks preserves the results of
constituent quark model for the magnetic moments of octet baryons.

##  V Final Remarks

The unabated pursuit for the discovery of new exotic hadron states may provide
some challenges for the inner structures. Although more than 20 years passed
over the discovery of $\chi_{c1}(3872)$, there is still no consensus about the
inner structure of this state.

The mass of a hadron is an important quantity since one can compare it to
corresponding thresholds. This may give an evidence for the stability of that
state. The magnetic moment of a hadron is an important quantity as its mass.
It contains valuable information about the inner structures. Motivated by the
observation of hidden-charm strange pentaquarks, in this work we investigate
the magnetic moments of hidden-bottom strange pentaquark states in molecular
picture.

Our results reflect some interesting features. For example, in all hidden-
bottom strange pentaquark states, $J^{P}=\frac{3}{2}^{-}$ of
$\frac{1}{2}^{+}\otimes 1^{-}$ has the largest magnetic moments, except in the
$P_{bss}^{1}$ states where the $J^{P}=\frac{1}{2}^{-}$ of
$\frac{1}{2}^{+}\otimes 1^{-}$ has the largest moment. The observation of
hidden-bottom strange pentaquarks with strangeness-2 may have some other
interesting properties. Our calculations may help to determine the
corresponding quantum numbers and spin configurations of hidden-bottom strange
pentaquark states.

In addition to magnetic moments, decay modes, decay channels and branching
ratios are also important for determining the inner structure. The magnetic
moments calculated here could be compared with other studies. We encourage
three-point QCD sum rule, light-cone QCD sum rule or lattice QCD studies for
comparison, since using different models for the same task may help for
understanding and elucidating the problem in hand. Lattice QCD studies consume
highly large times and therefore need advanced computational resources such as
supercomputing centres. The magnetic moments can also be calculated by the QCD
sum rules formalism where the bare quark masses are chosen as inputs for quark
masses. In constituent quark models, the quark masses are different than bare
quark masses. Quark masses are not accesible to direct experimental
measurements. From a physical perspective, the calculated quantities (magnetic
moments in this task) are identical, thus indicating that the outcomes should
be comparable. However, given that the two methodologies employ distinct
approximations and utilize distinct parameter sets, it is probable that
disparate outcomes will be observed. Only experimental outcomes may serve to
elucidate which model offers more consistent and superior outcomes, thereby
enabling the differentiation between these models.

We hope that our results may be helpful for theoretical and experimental
studies on the hidden-bottom strange pentaquark states. We also hope that this
research would contribute our understanding of the molecular states in hadron
physics.

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