# Magnetic Moments of Hidden-Bottom Pentaquark States

Halil Mutuk<EMAIL_ADDRESS>Department of Physics, Faculty of Sciences,
Ondokuz Mayis University, 55200 Samsun, Türkiye

###### Abstract

We study systematically magnetic moments of hiddden-bottom pentaquark states
with quantum numbers $J^{P}=\frac{1}{2}^{\pm}$, $J^{P}=\frac{3}{2}^{\pm}$, and
$J^{P}=\frac{5}{2}^{\pm}$ with molecular, diquark-diquark-antiquark, and
diquark-triquark models. The numerical results show that magnetic moments are
different within the same model according to same quantum numbers and spin-
orbit couplings. The results are also different when different models are
taken into account with the same angular momentum. The magnetic moments encode
valuable information about inner structures. We believe that our results may
be helpful for experimental studies.

##  I Introduction

According to quark model, conventional mesons are quark-antiquark states and
baryons are three-quark states. Since its introduction in the 60s, this model
explained successfully the spectrum of mesons and baryons. This paradigm has
changed with the beginning of the millennium. The discoveries of the
multiquark states have opened a new era in the particle physics. These
unconventional objects are called exotic hadrons and do not fit the quark
model picture due to the some extra ordinary features. The quantum
chromodynamics (QCD) which is the theory of strong interactions, allows those
unconventional hadrons such as multiquark states $(qq\bar{q}\bar{q}$,
$qqqq\bar{q}$, $qqq\bar{q}\bar{q}\bar{q}$, $qqqqqq,\cdots)$, glueballs
$(gg,ggg,\cdots)$, and quark-gluon hybrids $(q\bar{q}g)$.

The first exotic state $\chi_{c1}(3872)$ which is a four-quark state is
observed in 2003 by Belle Collaboration [1]. Since then, many exotic hadrons
(mostly four-quark states) have been observed in the various experiments.
Among these exotic hadrons, pentaquark family has a few members. In 2015, LHCb
Collaboration reported two hidden-charm pentaquark states $P_{c}(4380)$ and
$P_{c}(4450)$ after investigating $J/\psi p$ 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)

Four years later after this observation, LHCb Collaboration reported three new
pentaquark states namely $P_{c}(4312)$, $P_{c}(4440)$ and $P_{c}(4457)$ using
updated data [3]. The previously reported $P_{c}(4450)$ 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)

The continuous search on pentaquark states resulted a further discovery. A new
pentaquark state with strange quark content namely $P_{cs}(4459)^{0}$ is
reported by the LHCb collaboration via analyzing of $J\psi\Lambda$ invariant
mass distribution in $\Xi_{b}^{-}\rightarrow J/\psi K^{-}\Lambda$ decays [4].
The mass and width for this state is

$m_{P_{cs}(4459)^{0}}=4458.8\pm
2.9^{+4.7}_{-1.1}~{}\mathrm{MeV},\quad\Gamma_{P_{cs}(4459)^{0}}=17.3\pm
6.5^{+8.0}_{-5.7}~{}\mathrm{MeV},$ (6)

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

$m_{P_{cs}(4338)}=4338.2\pm 0.7\pm
0.4~{}\mathrm{MeV},\quad\Gamma_{P_{cs}(4338)}=7.0\pm 1.2\pm
1.3~{}\mathrm{MeV},$ (7)

respectively, [5].

The observation of the above pentaquark states triggered theoretical and
phenomenological studies in order to identify their various properties.
Although there are numerous studies regarding spectroscopic and decay
parameters of these states, the inner structures and quantum numbers are still
unclear. Consecutive studies on these properties provide valuable information
about understanding the nonperturbative nature of QCD. In addition to this,
such theoretical and phenomenological studies may help experimental studies.

The aforementioned pentaquark states are investigated by using different
models which have different substructures such as diquark-diquark-antiquark
states [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], diquark-
triquark states [21, 6, 19], meson-baryon molecular states [22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] and topological soliton model
[37].

Doubly-heavy tetraquarks $(QQ\bar{q}\bar{q})$ should be stable when the ratio
$m_{QQ}/m_{\bar{q}\bar{q}}$ is sufficiently large [38, 39]. Before and after
the observation of $T_{cc}^{+}$ [40, 41] which is a tetraquark state of
$cc\bar{u}\bar{d}$ quark content, lattice QCD studies found some evidences
about doubly-heavy tetraquarks with $b$ quark content [42, 43, 44, 45, 46].
Same logic can be applied to hidden-charm pentaquark states by replacing $c$
quarks with $b$ quarks. Hidden-charm pentaquark states were studied in the
literature with various methods. Here we collect some studies that are devoted
to hidden-bottom pentaquarks. In addition to $P_{c}(4380)^{+}$ state, Ref.
[47] contains mass predictions of hidden-bottom pentaquark states with a mass
around $11.08-11.11$ GeV and $J^{P}=3/2^{-}$ quantum number is favored.
Furthermore, in the same work it was mentioned that there may exist some
loosely-bound molecular pentaquark states with heavy quark contents of
$c\bar{b}$, $b\bar{c}$ and $b\bar{b}$. Such pentaquark masses were calculated
as $M_{\bar{c}c}=4678~{}\text{MeV}$, $M_{\bar{b}c}=7789~{}\text{MeV}$ and
$M_{\bar{b}b}=11198~{}\text{MeV}$ in Ref. [48] by using a variant of D4-D8
brane model [49]. Masses and residues of spin 3/2 and 5/2 hidden-bottom
pentaquark states with both negative and positive parities are calculated via
QCD sum rule formalism [50]. The obtained mass values are
$M=10.93^{+0.82}_{-0.85}~{}\text{GeV}$ for $J^{P}=\frac{3}{2}^{+}$,
$M=10.96^{+0.84}_{-0.88}~{}\text{GeV}$ for $J^{P}=\frac{3}{2}^{-}$,
$M=11.94^{+0.84}_{-0.82}~{}\text{GeV}$ for $J^{P}=\frac{5}{2}^{+}$, and
$M=10.98^{+0.82}_{-0.82}~{}\text{GeV}$ for $J^{P}=\frac{5}{2}^{-}$. Ref. [51]
prompted chiral quark model to the hidden-bottom pentaquark states with spin-
parity quantum numbers $J^{P}=\frac{1}{2}^{\pm}$, $J^{P}=\frac{3}{2}^{\pm}$
and $J^{P}=\frac{5}{2}^{\pm}$ and in the $\frac{1}{2}$ and $\frac{3}{2}$
isospin sectors. They found no bound state candidates for positive parity
hidden-bottom pentaquark states. In Ref. [52], possible hidden-bottom
pentaquark states from coupled-channel
$\Sigma^{(*)}_{b}B^{(*)}-\Lambda_{b}B^{(*)}$ interaction are investigated by
using quasipotential Bethe-Salpeter equation approach. They found seven
molecular states can be produced from the interactions. Five-quark systems
composed of $qqs\bar{Q}Q$ configuration with ($q=u$ or $d$, $Q=b$ or $c$) are
studied in the chiral quark model [53]. They observed that color-octet
structure yielded more bounding energy than color-singlet structure. In Ref.
[54], production of non-strange hidden-bottom pentaquark states near threshold
$\Upsilon(1S)$ meson photoproduction is studied. Hidden-heavy pentaquark and
$P_{cs}$ states are studied in the framework of MIT bag model [55]. Apart from
$P_{cs}$ states, hadron properties and decay channels of $P_{bss}$ and
$P_{bsss}$ are obtained. Masses, magnetic moments and partial widths of
hidden-bottom pentaquarks are studied by using extension of the Gursey-
Radicati mass formula, effective mass and screened charge schemes [56].

In the light of experimental and theoretical developments, it is natural to
expect bottom analogues of the observed hidden-charm penta quark states.
Indeed, historically the observation of hadrons with $c$ quarks is followed by
similar structures with $b$ quarks content. In this path, LHCb Collaboration
sparked a light for pentaquark states containing a single $b$ quark which
decay weakly via the $b\to c\overline{c}s$ transition in the $J/\psi
K^{+}\pi^{-}p$, $J/\psi K^{-}\pi^{-}p$, $J/\psi K^{-}\pi^{+}p$, $J/\psi\phi p$
final states [57]. They reported about similar explorations for hidden-bottom
pentaquark states should be expected in the near future.

The literature review reflects that the majority of studies has concentrated
on spectroscopic (such as mass) and decay (such as pole residue) parameters of
hidden-bottom pentaquark states. Such properties may be insufficient to
elucidate the inner structures of these states. Therefore further studies that
cover electromagnetic properties, radiative and weak decays are required.
Unveiling the nature of the unconventional hadronic states is a major concern
of high energy physics. Physical quantities associated with the
electromagnetic properties are valuable parameters. In this respect, magnetic
moments may provide valuable insights about the internal structure of both
conventional and unconventional hadrons. They present information about charge
distributions and shape of the hadrons. Magnetic moment is also an excellent
probe for quark-gluon dynamics inside the hadron since it is the leading-order
response of a bound state to a weak external magnetic field [58].

In this present work, we systematically study magnetic moments of hidden-
bottom pentaquarks using diquark-diquark-antiquark scheme, diquark-tiruqark
scheme and meson-baryon molecule scheme. In Section II, we present the models
that are used in this study. Section III is related to wave functions of
different models. Sections IV, V, and V are devoted to magnetic moment results
of molecular, diquark-diquark-antiquark, and diquark-triquark respectively.
Last two sections are reserved for discussion and conclusion.

##  II Models

In this section, we categorize three different models that are used to study
magnetic moments of hidden-bottom pentaquark states. Pentaquark states can be
decomposed into two or three clusters in quark level and color configurations.
Furthermore, these clusters should combine in a way to become color neutral.

The color interaction plays important role when determining the configurations
of clusters and consequently pentaquark states. In quark potential model,
color factor enters through the Coulombic part. The color factor $f$ in the
quark-antiquark color interaction is

$\kappa=-\frac{1}{4}\sum_{i<j,a=1}^{8}\lambda_{i}^{a}\lambda_{j}^{a},$ (8)

where $\lambda_{a}^{i}$ is the color generator of the ith quark. Then the
corresponding Coulombic potential is

$V_{q\bar{q}}(r)\simeq\kappa\frac{\alpha_{s}}{r}.$ (9)

The quark-quark color factor is given by

$\tilde{\kappa}=\frac{1}{4}\sum_{i<j,a=1}^{8}\lambda_{i}^{a}\lambda_{j}^{a},$
(10)

and the Coulombic potential is

$V_{qq}(r)\simeq\tilde{\kappa}\frac{\alpha_{s}}{r}.$ (11)

The SU(3) color symmetry of QCD implies that, when we combine a quark and an
antiquark in the fundamental color representation, we obtain
$|q\bar{q}\rangle:3\otimes\bar{3}=1\oplus 8$. This representation gives the
color factor for the color singlet as $\kappa=-4/3$ of the quark-antiquark
system. When we combine two quarks in the fundamental color representation, it
reduces to $|qq\rangle:3\otimes 3=\bar{3}\oplus 6$, a color antitriplet
$\bar{3}$ and a color sextet $6$. The antitriplet state has a color factor
$\kappa=-2/3$ which is attractive whereas the sextet state has a color factor
$\kappa=+1/3$ which is repulsive. Since an attractive potential is responsible
for binding quarks together, color factors are important to determine the
configurations for hadronic states. Table 1 presents the color factor values.

State | Quark configuration | Color configuration | Color factor
---|---|---|---
Meson | $|q\bar{q}\rangle$ | $3_{c}\otimes\bar{3}_{c}=1_{c}\oplus 8_{c}$ | $1_{c}\oplus 8_{c}$
$-\frac{4}{3}\quad\frac{1}{6}$
Diquark | $|qq\rangle$ | $3_{c}\otimes 3_{c}=6_{c}\oplus\bar{3}_{c}$ | $6_{c}\oplus\bar{3}_{c}$
$\frac{1}{3}\quad-\frac{2}{3}$
Baryon | $|qqq\rangle$ | $3_{c}\otimes 3_{c}\otimes 3_{c}=1_{c}\oplus 8_{1c}\oplus 8_{2c}\oplus 10_{c}$ | $1_{c}\oplus 8_{1c}\oplus 8_{2c}\oplus 10_{c}$
$-2$ $-\frac{1}{2}$ $-\frac{1}{2}$ 1
Triquark | $|qq\bar{q}\rangle$ | $3_{c}\otimes 3_{c}\otimes\bar{3}_{c}=3_{1c}\oplus 3_{2c}\oplus\bar{6}_{c}\oplus 15_{c}$ | $3_{1c}\oplus 3_{2c}\oplus\bar{6}_{c}\oplus 15_{c}$
$-\frac{4}{3}$ $-\frac{4}{3}$ $-\frac{1}{3}$ $2$
Table 1:  Color factors of hadronic configurations. Color factors are given
beneath the corresponding color structures.

Since color confinement restricts hadrons to be color singlets, one can
classify pentaquarks into three categories:

  1. 1.

Molecular model

Molecular states of pentaquarks are assumed to be composed of mesons and
baryons. Each cluster in the molecular model tends to be color singlets.
Looking color factors of mesons, it can be seen that color factor of $1_{c}$
is negative and while it is positive for $8_{c}$. In other words
$\kappa_{1_{c}}<\kappa_{8_{c}}$ so that a singlet state is more likely to form
than octet state. Looking color factors of baryons, color factors of
$(1_{c},8_{1c},8_{2c})$ color representations are negative whilst $10_{c}$
color representation has positive color factor. As mentioned before, negative
values of color factor are attractive and positive values are repulsive. The
color factors are ordered as
$\kappa_{1_{c}}<\kappa_{8_{1c}}=\kappa_{8_{1c}}<\kappa_{10_{c}}$. This implies
that it is easier to form a singlet state. As a result, there are two
configurations for molecular model: $(b\bar{b})(q_{1}q_{2}q_{3})$ and
$(\bar{b}q_{1})(bq_{2}q_{3})$, where $q$ denotes light quarks, $u,d,s$.

  2. 2.

Diquark-diquark-antiquark model

The diquark cluster has color factor of $\kappa=-\frac{2}{3}$ for
$\bar{3}_{c}$ color representation and $\kappa=\frac{1}{3}$ for $6_{c}$ color
representation. The diquark prefers to form $\bar{3}_{c}$, therefore each
diquark cluster recasts as $\bar{3}_{c}\otimes\bar{3}_{c}$ to form $3_{c}$
color representation. The antiquark has also $\bar{3}_{c}$ to form color
neutral state. This consideration brings us to have
$\bar{3}_{c}({\text{Diquark}})\otimes\bar{3}_{c}(\text{Diquark})\otimes\bar{3}_{c}(\text{Antiquark})$
configuration. Hence, the pentaquark configuration
$(bq_{1})(q_{2}q_{3})(\bar{b})$ is the nontrivial one according to diquark-
diquark-antiquark model.

  3. 3.

Diquark-triquark model

As in the diquark-diquark-antiquark model, diquark prefers $\bar{3}_{c}$ color
representation. Triquark has two quarks and one antiquark which distinguish it
from a conventional baryon. Looking color factors of triquark, it can be seen
that
$\kappa_{3_{1_{c}}}=\kappa_{3_{2c}}<\kappa_{\bar{6}_{c}}<\kappa_{\bar{15}_{c}}$.
It is more likely to bind in $3_{c}$ color representation. As a result we have
$3_{c}({\text{Triquark})}\otimes\bar{3}_{c}({\text{Diquark})}$ to recast in a
color neutral state. Thus, the pentaquark configurations related to diquark-
triquark model are $(bq_{1})(\bar{b}q_{2}q_{3})$ and
$(\bar{b}bq_{1})(q_{2}q_{3})$.

Some pentaquark configurations may be omitted due to some kinematic and
dynamical reasons. The clusters $(b\bar{b})$ and $(\bar{b}q_{1})$ tend to the
color singlet states. $(b\bar{b})$ is the botomonium and $\bar{b}q_{1}$ is a
heavy-light meson. There is no strong attraction between a bottomonium and a
light baryon $(q_{1}q_{2}q_{3})$. As a result, it is hard to form a loosely
bound molecular state composed of $(b\bar{b})$ and $(q_{1}q_{2}q_{3})$.
Furthermore, as pointed out in Ref. [3], the separation of charm $c$ and anti-
charm $\bar{c}$ into distinct confinement volumes provides a natural
suppression mechanism for the pentaquark widths. Since the separation of
bottom $b$ and anti-bottom $\bar{b}$ in the botomonium is less then the
charmonium, such suppresion may occur in the hidden-bottom pentaquark states.
Therefore, we do not consider $(b\bar{b})(q_{1}q_{2}q_{3})$ configuration in
molecular model and $(\bar{b}bq_{1})(q_{2}q_{3})$ configuration in diquark-
triquark model.

The aforementioned models presented good description for the experimentally
observed hidden-charm pentaquark states in terms of mass and decay properties.
However, mass and decay width alone cannot distinguish inner structures of
these pentaquark states. Magnetic moment is an important dynamical observable
for a hadron which reflects indications about charge distributions, shape and
magnetization. Although hidden-bottom pentaquark states are not experimentally
observed, any study regarding magnetic moment may present underlying reasons
for distinguishing models and identify inner structure.

##  III Wave Function of Hidden-Bottom Pentaquark States

There are four degrees of freedom in the quark level: color, spin, flavor and
space. Accordingly one can write the wavefunction of a hadronic state as

$\Psi_{\text{Wavefunction}}=\psi_{\text{flavor}}\psi_{\text{spin}}\psi_{\text{color}}\psi_{\text{space}}.$
(12)

The overall wave function should be antisymmetric due to the Fermi statistics.
We construct hidden-bottom pentaquark states in the $SU(3)_{f}$ frame. The
$q_{2}q_{3}$ forms $\bar{3}_{f}$ and $6_{f}$ flavor representation with the
total spin $S=0$ and $S=1$, respectively. If $q_{2}q_{3}$ forms in $6_{f}$,
then $q_{1}$ combines with $q_{2}q_{3}$ to form $6_{f}\otimes
3_{f}=10_{f}\oplus 8_{1f}$. If $q_{2}q_{3}$ forms in $\bar{3}_{f}$, then
$q_{1}$ combines with $q_{2}q_{3}$ to form $\bar{3}_{f}\otimes
3_{f}=8_{2f}\oplus 1_{f}$. After inserting $b$ quark and $\bar{b}$ antiquark
together with the Clebsch-Gordan coefficients, we obtain the flavor wave
function in the molecular model with the $(\bar{b}q_{1})(bq_{2}q_{3})$
configuration. The same method can be applied to diquark-diquark-antiquark
$(bq_{1})(q_{2}q_{3})(\bar{b})$ and diquark-triquark
$(bq_{1})(\bar{b}q_{2}q_{3})$ configurations. We list the wave functions in
Table 2.

Model | Multiplet | $(I,I_{3})$ | Wave function
---|---|---|---
Molecular |  $10_{f}$ |  $(\frac{3}{2},-\frac{1}{2})$ |  $\sqrt{\frac{2}{3}}({\bar{b}}d)(b\\{ud\\})+\sqrt{\frac{1}{3}}({\bar{b}}u)(b\\{dd\\})$
|  $8_{1f}$ |  $(\frac{1}{2},-\frac{1}{2})$ |  $\sqrt{\frac{1}{3}}({\bar{b}}d)(b\\{ud\\})-\sqrt{\frac{2}{3}}({\bar{b}}u)(b\\{dd\\})$
|  $8_{2f}$ |  $(\frac{1}{2},-\frac{1}{2})$ |  $({\bar{b}}d)(b[ud])$
Diquark-diquark-antiquark |  $10_{f}$ |  $(\frac{3}{2},-\frac{1}{2})$ |  $\sqrt{\frac{2}{3}}({b}d)\\{ud\\}{\bar{b}}+\sqrt{\frac{1}{3}}({b}u)\\{dd\\}{\bar{b}}$
|  $8_{1f}$ |  $(\frac{1}{2},-\frac{1}{2})$ |  $\sqrt{\frac{1}{3}}(bd)\\{ud\\}{\bar{b}}-\sqrt{\frac{2}{3}}({b}u)\\{dd\\}{\bar{b}}$
|  $8_{2f}$ |  $(\frac{1}{2},-\frac{1}{2})$ |  $(bd)[ud]{\bar{b}}$
Diquark-triquark |  $10_{f}$ |  $(\frac{3}{2},-\frac{1}{2})$ |  $\sqrt{\frac{2}{3}}({b}d)(\bar{b}\\{ud\\})+\sqrt{\frac{1}{3}}({b}u)(\bar{b}\\{dd\\})$
|  $8_{1f}$ |  $(\frac{1}{2},-\frac{1}{2})$ |  $\sqrt{\frac{1}{3}}({b}d)(\bar{b}\\{ud\\})-\sqrt{\frac{2}{3}}({b}u)(\bar{b}\\{dd\\})$
|  $8_{2f}$ |  $(\frac{1}{2},-\frac{1}{2})$ |  $({b}d)(\bar{b}[ud])$
Table 2: The flavor wave functions of the hidden-bottom pentaquark states in
different models. Here,
$\\{q_{1}q_{2}\\}=\frac{1}{\sqrt{2}}(q_{1}q_{2}+q_{2}q_{1})$ and
$[q_{1}q_{2}]=\frac{1}{\sqrt{2}}(q_{1}q_{2}-q_{2}q_{1})$. $I$ and $I_{3}$ are
the isospin and its third component, respectively.

In this present paper, we obtain magnetic moments of hidden-bottom pentaquark
states with strangeness zero. In other words, the pentaquark states of this
work do not contain $s$ quark.

##  IV Magnetic Moments of Hidden-Bottom Pentaquark States in Molecular Model

The magnetic moment of the pentaquark state with the molecular configuration
$(\bar{b}q_{1})(bq_{2}q_{3})$ can be written as

$\hat{\mu}=\ \hat{\mu}_{B}+\hat{\mu}_{B}+\hat{\mu}_{l},$ (13)

where the subscripts $B$ and $M$ represent the baryon and meson, respectively,
and $l$ is the orbital excitation between the meson and baryon. We assume that
the orbital excitation occurs between the meson and baryon and do not consider
the orbital excitation in the bound state. The expressions for the magnetic
moments of meson, baryon and orbital excitation are given as

$\displaystyle\hat{\mu}_{B}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\mu_{i}g_{i}\hat{S}_{i},$ (14)
$\displaystyle\hat{\mu}_{M}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2}\mu_{i}g_{i}\hat{S}_{i},$ (15)
$\displaystyle\hat{\mu}_{l}=\mu_{l}\hat{l}$ $\displaystyle=$
$\displaystyle\frac{M_{M}\mu_{B}+M_{B}\mu_{M}}{M_{M}+M_{B}}\hat{l}.$ (16)

where $g_{i}$ is the Lande factor, $M_{M}$ and $M_{B}$ are the masses of meson
and baryon, respectively. The magnetic moment expression for
$(\bar{b}q_{1})(bq_{2}q_{3})$ molecular configuration is

$\displaystyle\mu$ $\displaystyle=$ $\displaystyle\langle\ \psi_{Pentaquark}\
|\ \hat{\mu}_{B}+\hat{\mu}_{M}+\hat{\mu}_{l}\ |\ \psi_{Pentaquark}\ \rangle$
(17) $\displaystyle=$ $\displaystyle\sum_{SS_{z},ll_{z}}\ \langle\
SS_{z},ll_{z}|JJ_{z}\
\rangle^{2}\left\\{\mu_{l}l_{z}+\sum_{S^{\prime}_{B},S^{\prime}_{M}}\ \langle\
S_{B}S^{\prime}_{B},S_{M}S^{\prime}_{M}|SS_{z}\
\rangle^{2}\Bigg{[}S^{\prime}_{M}\bigg{(}\mu_{\bar{b}}+\mu_{q_{1}}\bigg{)}\right.$
$\displaystyle+$ $\displaystyle\left.\sum_{S^{\prime}_{b}}\ \langle\
S_{b}S^{\prime}_{b},S_{D}S^{\prime}_{B}-S^{\prime}_{b}|S_{B}S^{\prime}_{B}\rangle^{2}\bigg{(}g\mu_{b}S^{\prime}_{b}+(S^{\prime}_{B}-S^{\prime}_{b})(\mu_{q_{2}}+\mu_{q_{3}})\bigg{)}\Bigg{]}\right\\}.$

Here $\psi$ represents the wave function listed in Table 2. $S_{M}$, $S_{B}$,
$S_{D}$ are the meson, baryon, and the diquark spin inside the baryon,
respectively, and $S^{\prime}$ is the third spin component.

We use the following constituent quark masses [59]:

$\displaystyle m_{u}\ =\ m_{d}\ =\ 336\ \mbox{MeV},\ m_{s}\ =\ 540\
\mbox{MeV},\ m_{b}\ =\ 4700\ \mbox{MeV},$

which lead to the magnetic moments of the octet baryons in rough agreement
with the experimental values listed in Table 3.

Baryon | Magnetic moment | This work | Experiment [60]
---|---|---|---
$p$ |  $\frac{4}{3}\mu_{u}-\frac{1}{3}\mu_{d}$ |  $2.792$ |  $2.793$
$n$ |  $\frac{4}{3}\mu_{d}-\frac{1}{3}\mu_{u}$ |  $-1.861$ |  $-1.913$
$\Lambda$ |  $\mu_{s}$ |  $-0.579$ |  $-0.613\pm 0.006$
$\Sigma^{+}$ |  $\frac{4}{3}\mu_{u}-\frac{1}{3}\mu_{s}$ |  $2.674$ |  $2.460\pm 0.006$
$\Sigma^{-}$ |  $\frac{4}{3}\mu_{d}-\frac{1}{3}\mu_{s}$ |  $-1.048$ |  $-1.160\pm 0.025$
$\Xi^{0}$ |  $\frac{4}{3}\mu_{s}-\frac{1}{3}\mu_{u}$ |  $-1.392$ |  $-1.250\pm 0.014$
$\Xi^{-}$ |  $\frac{4}{3}\mu_{s}-\frac{1}{3}\mu_{d}$ |  $-0.462$ |  $-0.651\pm 0.0025$
$\Omega^{-}$ |  $3\mu_{s}$ |  $-1.737$ |  $-2.020\pm 0.05$
Table 3: Magnetic moments of the octet baryons. The results are listed in unit
of nuclear magneton $\mu_{N}$.

Consider the $8_{1f}$ flavor representation of
$\sqrt{\frac{1}{3}}(b\\{ud\\})({\bar{b}}d)-\sqrt{\frac{2}{3}}(b\\{dd\\})({\bar{b}}u)$
as an example. Let’s take $J^{P}=\frac{1}{2}^{-}({\frac{1}{2}}^{+}\otimes
0^{-}\otimes 0^{+})$ where $J_{B}^{P_{B}}\otimes J_{M}^{P_{M}}\otimes
J_{l}^{P_{l}}$ correspond to the angular momentum and parity of baryon, meson
and orbital, respectively. We obtain the magnetic moment expression as

$\displaystyle\mu$
$\displaystyle=\left\langle\psi_{Pentaquark}\left|g_{M}\mu_{M}{\vec{S}_{M}}+g_{B}\mu_{B}{\vec{S}}_{B}+{\mu_{l}}\vec{l}\,\,\right|\psi_{Pentaquark}\right\rangle$
$\displaystyle=\langle
00,\frac{1}{2}\frac{1}{2}|\frac{1}{2}\frac{1}{2}\rangle^{2}\Big{[}\langle\frac{1}{2}\frac{1}{2},00|\frac{1}{2}\frac{1}{2}\rangle^{2}\big{(}\langle\frac{1}{2}\frac{1}{2},10|\frac{1}{2}\frac{1}{2}\rangle^{2}\mu_{b}$
$\displaystyle\
+\langle\frac{1}{2}-\frac{1}{2},11|\frac{1}{2}\frac{1}{2}\rangle^{2}(-\mu_{b}+\frac{1}{3}\mu_{u}+\frac{5}{3}\mu_{d})\big{)}\Big{]}$
$\displaystyle=\frac{1}{9}(2\mu_{u}+10\mu_{d}-3\mu_{b}).$ (18)

The magnetic moments of hidden-bottom pentaquark states in molecular model are
listed in Table 4 for $8_{1f}$, Table 5 for $8_{2f}$, and Table 6 for $10_{f}$
representations, respectively.

$J^{P}$ | ${}^{2s+1}L_{j}$ |  $J_{B}^{P_{B}}\otimes J_{M}^{P_{M}}\otimes J_{l}^{P_{l}}$ |  Magnetic moment
---|---|---|---
$\frac{1}{2}^{-}$ |  $\quad{}^{2}S_{\frac{1}{2}}$ |  ${\frac{1}{2}}^{+}\otimes 0^{-}\otimes 0^{+}$ |  $-0.598$
|  |  ${\frac{1}{2}}^{+}\otimes 1^{-}\otimes 0^{+}$ |  $0.864$
|  |  ${\frac{3}{2}}^{+}\otimes 1^{-}\otimes 0^{+}$ |  $-0.886$
$\frac{3}{2}^{-}$ |  ${}^{4}S_{\frac{3}{2}}$ |  ${\frac{1}{2}}^{+}\otimes 1^{+}\otimes 0^{-}$ |  $0.399$
|  |  ${\frac{3}{2}}^{+}\otimes 0^{-}\otimes 0^{+}$ |  $-0.997$
|  |  ${\frac{3}{2}}^{+}\otimes 1^{+}\otimes 0^{-}$ |  $0.333$
$\frac{5}{2}^{-}$ |  ${}^{6}S_{\frac{5}{2}}$ |  ${\frac{3}{2}}^{+}\otimes 1^{-}\otimes 0^{+}$ |  $0.237$
$\frac{1}{2}^{+}$ |  ${}^{2}P_{\frac{1}{2}}$ |  ${\frac{1}{2}}^{+}\otimes 0^{-}\otimes 1^{-}$ |  $-0.756$
|  |  ${(\frac{1}{2}}^{+}\otimes 1^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $-1.243$
|  ${}^{4}P_{\frac{1}{2}}$ |  $({\frac{1}{2}}^{+}\otimes 1^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.699$
|  |  ${\frac{3}{2}}^{+}\otimes 0^{-}\otimes 1^{-}$ |  $-0.041$
|  ${}^{2}P_{\frac{1}{2}}$ |  ${(\frac{3}{2}}^{+}\otimes 1^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $1.152$
|  ${}^{4}P_{\frac{1}{2}}$ |  ${(\frac{3}{2}}^{+}\otimes 1^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.293$
$\frac{3}{2}^{+}$ |  ${}^{2}P_{\frac{3}{2}}$ |  ${\frac{1}{2}}^{+}\otimes 0^{-}\otimes 1^{-}$ |  $-2.031$
|  |  ${(\frac{1}{2}}^{+}\otimes 1^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $-0.568$
|  ${}^{4}P_{\frac{3}{2}}$ |  ${(\frac{1}{2}}^{+}\otimes 1^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.280$
|  |  ${\frac{3}{2}}^{+}\otimes 0^{-}\otimes 1^{-}$ |  $-1.304$
|  ${}^{2}P_{\frac{3}{2}}$ |  ${(\frac{3}{2}}^{+}\otimes 1^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.399$
|  ${}^{4}P_{\frac{3}{2}}$ |  ${(\frac{3}{2}}^{+}\otimes 1^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.270$
|  ${}^{6}P_{\frac{3}{2}}$ |  ${(\frac{3}{2}}^{+}\otimes 1^{-})_{\frac{5}{2}}\otimes 1^{-}$ |  $0.860$
$\frac{5}{2}^{+}$ |  ${}^{4}P_{\frac{5}{2}}$ |  ${\frac{1}{2}}^{+}\otimes 1^{-}\otimes 1^{-}$ |  $-1.034$
|  |  ${\frac{3}{2}}^{+}\otimes 0^{-}\otimes 1^{-}$ |  $-2.430$
|  |  ${(\frac{3}{2}}^{+}\otimes 1^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $-1.765$
|  ${}^{6}P_{\frac{5}{2}}$ |  ${(\frac{3}{2}}^{+}\otimes 1^{-})_{\frac{5}{2}}\otimes 1^{-}$ |  $-0.409$
Table 4: Magnetic moments of the $8_{1f}$ representation of the molecular hidden-bottom pentaquark state. $J_{B}^{P_{B}}$ corresponds to the angular momentum and parity of baryon, $J_{M}^{P_{M}}$ is for meson and $J_{l}^{P_{l}}$ is for orbital excitation between meson and baryon. The results are listed in unit of nuclear magneton $\mu_{N}$. $J^{P}$ |  ${}^{2s+1}L_{j}$ |  $J_{B}^{P_{B}}\otimes J_{M}^{P_{M}}\otimes J_{l}^{P_{l}}$ |  Magnetic moment
---|---|---|---
$\frac{1}{2}^{-}$ |  ${}^{2}S_{\frac{1}{2}}$ |  ${\frac{1}{2}}^{+}\otimes 0^{-}\otimes 0^{+}$ |  $-0.066$
|  |  ${\frac{1}{2}}^{+}\otimes 1^{-}\otimes 0^{+}$ |  $-0.532$
$\frac{3}{2}^{-}$ |  ${}^{4}S_{\frac{3}{2}}$ |  ${\frac{1}{2}}^{+}\otimes 1^{-}\otimes 0^{+}$ |  $-0.930$
$\frac{1}{2}^{+}$ |  ${}^{2}P_{\frac{1}{2}}$ |  ${\frac{1}{2}}^{+}\otimes 0^{-}\otimes 1^{-}$ |  $-0.218$
|  |  ${(\frac{1}{2}}^{+}\otimes 1^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $-0.055$
|  ${}^{4}P_{\frac{1}{2}}$ |  $({\frac{1}{2}}^{+}\otimes 1^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.397$
$\frac{3}{2}^{+}$ |  ${}^{2}P_{\frac{3}{2}}$ |  ${\frac{1}{2}}^{+}\otimes 0^{-}\otimes 1^{-}$ |  $-0.427$
|  |  ${(\frac{1}{2}}^{+}\otimes 1^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $-0.914$
|  ${}^{4}P_{\frac{3}{2}}$ |  ${(\frac{1}{2}}^{+}\otimes 1^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.826$
$\frac{5}{2}^{+}$ |  ${}^{4}P_{\frac{5}{2}}$ |  ${\frac{1}{2}}^{+}\otimes 1^{-}\otimes 1^{-}$ |  $-1.291$
Table 5: Magnetic moments of the $8_{2f}$ representation of the molecular hidden-bottom pentaquark state. $J_{B}^{P_{B}}$ corresponds to the angular momentum and parity of baryon, $J_{M}^{P_{M}}$ is for meson and $J_{l}^{P_{l}}$ is for orbital excitation between meson and baryon. The results are listed in unit of nuclear magneton $\mu_{N}$. $J^{P}$ |  ${}^{2s+1}L_{j}$ |  $J_{B}^{P_{B}}\otimes J_{M}^{P_{M}}\otimes J_{l}^{P_{l}}$ |  Magnetic moment
---|---|---|---
$\frac{1}{2}^{-}$ |  ${}^{2}S_{\frac{1}{2}}$ |  ${\frac{1}{2}}^{+}\otimes 0^{-}\otimes 0^{+}$ |  $0.022$
|  |  ${\frac{1}{2}}^{+}\otimes 1^{-}\otimes 0^{+}$ |  $0.037$
|  |  ${\frac{3}{2}}^{+}\otimes 1^{-}\otimes 0^{+}$ |  $-0.059$
$\frac{3}{2}^{-}$ |  ${}^{4}S_{\frac{3}{2}}$ |  ${\frac{1}{2}}^{+}\otimes 1^{+}\otimes 0^{-}$ |  $0.089$
|  |  ${\frac{3}{2}}^{+}\otimes 0^{-}\otimes 0^{+}$ |  $-0.066$
|  |  ${\frac{3}{2}}^{+}\otimes 1^{+}\otimes 0^{-}$ |  $-0.022$
$\frac{5}{2}^{-}$ |  ${}^{6}S_{\frac{5}{2}}$ |  ${\frac{3}{2}}^{+}\otimes 1^{-}\otimes 0^{+}$ |  $-0.127$
$\frac{1}{2}^{+}$ |  ${}^{2}P_{\frac{1}{2}}$ |  ${\frac{1}{2}}^{+}\otimes 0^{-}\otimes 1^{-}$ |  $-0.522$
|  |  ${(\frac{1}{2}}^{+}\otimes 1^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $-0.526$
|  ${}^{4}P_{\frac{1}{2}}$ |  $({\frac{1}{2}}^{+}\otimes 1^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.306$
|  |  ${\frac{3}{2}}^{+}\otimes 0^{-}\otimes 1^{-}$ |  $0.220$
|  ${}^{2}P_{\frac{1}{2}}$ |  ${(\frac{3}{2}}^{+}\otimes 1^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $-0.494$
|  ${}^{4}P_{\frac{1}{2}}$ |  ${(\frac{3}{2}}^{+}\otimes 1^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.245$
$\frac{3}{2}^{+}$ |  ${}^{2}P_{\frac{3}{2}}$ |  ${\frac{1}{2}}^{+}\otimes 0^{-}\otimes 1^{-}$ |  $-0.749$
|  |  ${(\frac{1}{2}}^{+}\otimes 1^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $-0.734$
|  ${}^{4}P_{\frac{3}{2}}$ |  ${(\frac{1}{2}}^{+}\otimes 1^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.243$
|  |  ${\frac{3}{2}}^{+}\otimes 0^{-}\otimes 1^{-}$ |  $-0.357$
|  ${}^{2}P_{\frac{3}{2}}$ |  ${(\frac{3}{2}}^{+}\otimes 1^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $-0.830$
|  ${}^{4}P_{\frac{3}{2}}$ |  ${(\frac{3}{2}}^{+}\otimes 1^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.325$
|  ${}^{6}P_{\frac{3}{2}}$ |  ${(\frac{3}{2}}^{+}\otimes 1^{-})_{\frac{5}{2}}\otimes 1^{-}$ |  $-0.285$
$\frac{5}{2}^{+}$ |  ${}^{4}P_{\frac{5}{2}}$ |  ${\frac{1}{2}}^{+}\otimes 1^{-}\otimes 1^{-}$ |  $-0.683$
|  |  ${\frac{3}{2}}^{+}\otimes 0^{-}\otimes 1^{-}$ |  $-0.838$
|  |  ${(\frac{3}{2}}^{+}\otimes 1^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.793$
|  ${}^{6}P_{\frac{5}{2}}$ |  ${(\frac{3}{2}}^{+}\otimes 1^{-})_{\frac{5}{2}}\otimes 1^{-}$ |  $-0.220$
Table 6: Magnetic moments of the $10_{f}$ representation of the molecular
hidden-bottom pentaquark state. $J_{B}^{P_{B}}$ corresponds to the angular
momentum and parity of baryon, $J_{M}^{P_{M}}$ is for meson and
$J_{l}^{P_{l}}$ is for orbital excitation between meson and baryon. The
results are listed in unit of nuclear magneton $\mu_{N}$.

##  V Magnetic Moments of Hidden-Bottom Pentaquark States in Diquark-Diquark-
Antiquark Model

The diquark-diquark-antiquark configuration is
$(bq_{1})(q_{2}q_{3})(\bar{b})$. In this model, $P$-wave excitation includes
$\rho$ mode excitation and $\lambda$ mode excitation. The $P$-wave orbital
excitation of the $\rho$ mode lies between the $(bq_{1})$ diquark and
$(q_{2}q_{3})$ diquark. The $P$-wave orbital excitation of the $\lambda$ mode
lies between the antibottom quark $(\bar{b})$ and the center of mass of system
of $(bq_{1})$ diquark and $(q_{2}q_{3})$ diquark.

The magnetic moment of the diquark-diquark-antiquark model can be written as

$\displaystyle\hat{\mu}=\
\hat{\mu}_{H}+\hat{\mu}_{L}+\hat{\mu}_{\bar{b}}+\hat{\mu}_{l},$ (19)

where the subscripts $H$ and $L$ represent a heavy diquark $(bq_{1})$ and
light diquark $(q_{2}q_{3})$, respectively, and $l$ is the orbital excitation.
In the diquark-diquark-antiquark model, the specific magnetic moments formula
of the pentaquark $(bq_{1})(q_{2}q_{3})\bar{b}$ is:

$\displaystyle\mu$ $\displaystyle=$ $\displaystyle\langle\ \psi_{Pentaquark}\
|\ \hat{\mu}_{H}+\hat{\mu}_{L}+\hat{\mu}_{\bar{b}}+\hat{\mu}_{l}\ |\
\psi_{Pentaquark}\ \rangle$ (20) $\displaystyle=$
$\displaystyle\sum_{S_{z},l_{z}}\ \langle\ SS_{z},ll_{z}|JJ_{z}\
\rangle^{2}\left\\{\mu_{l}l_{z}+\sum_{\widetilde{S}_{\bar{b}}}\ \langle\
S_{\bar{b}}\widetilde{S}_{\bar{b}},S_{HL}\widetilde{S}_{HL}|SS_{z}\
\rangle^{2}\Bigg{[}g\widetilde{S}_{\bar{b}}\mu_{\bar{b}}\right.$
$\displaystyle+$
$\displaystyle\left.\sum_{\widetilde{S}_{H},\widetilde{S}_{L}}\ \langle\
S_{H}\widetilde{S}_{H},S_{L}\widetilde{S}_{L}|S_{HL}\widetilde{S}_{HL}\rangle^{2}\bigg{(}\widetilde{S}_{H}(\mu_{b}+\mu_{q_{1}})+\widetilde{S}_{L}(\mu_{q_{2}}+\mu_{q_{3}})\bigg{)}\Bigg{]}\right\\}.$

where $S_{H}$ is the spin of the $(bq_{1})$ diquark, $S_{L}$ is the spin of
the $(q_{2}q_{3})$ diquark, $S_{HL}$ is the spin of the $(bq_{1})(q_{2}q_{3})$
system and $\widetilde{S}$ is the third spin component. We use the following
diquark masses from Ref. [61]:

$\displaystyle[q,q]$ $\displaystyle=$ $\displaystyle 710~{}\text{MeV},\ \ \ \
\\{q,q\\}=909~{}\text{MeV},$ $\displaystyle\ [b,q]$ $\displaystyle=$
$\displaystyle 5359~{}\text{MeV},\ \ \ \\{b,q\\}=5381~{}\text{MeV},$ (21)

where $[\cdots]$ denotes antisymmetric and $\\{\cdots\\}$ denotes symmetric
representation in flavour, respectively.

As mentioned before, there are orbital excitations in the $\rho$ mode and
$\lambda$ mode. The corresponding magnetic moments $\mu_{l}$ in $\rho$ mode
and $\lambda$ mode can be written as

$\displaystyle\mu_{l,\rho}$ $\displaystyle=$
$\displaystyle\frac{{m_{q_{2}q_{3}}}\mu_{bq_{1}}+{m_{bq_{1}}}\mu_{q_{2}q_{3}}}{m_{bq_{1}}+m_{q_{1}q_{2}}},$
(22) $\displaystyle\mu_{l,\lambda}$ $\displaystyle=$
$\displaystyle\frac{{m_{\bar{b}}}\mu_{(bq_{1})(q_{2}q_{3})}+{m_{(bq_{1})(q_{2}q_{3})}}\mu_{\bar{b}}}{m_{(bq_{1})(q_{2}q_{3})}+m_{\bar{b}}},$
(23)

where
${m_{(bq_{1})(q_{2}q_{3})}}=\frac{{m_{q_{2}q_{3}}}{m_{bq_{1}}}}{m_{bq_{1}}+m_{q_{2}q_{3}}}$,
$m$ and $\mu$ represent the mass and magnetic moment of the corresponding
clusters represented by their subscripts: $m_{bq_{1}}$ is the mass of $bq_{1}$
diquark and $m_{q_{2}q_{3}}$ is the mass of $q_{2}q_{3}$ diquark. Since the
$S$-wave systems are more accessible in the experiments, especially for $b$
quarks, we obtain magnetic moments of $S$-wave diquark-diquark-antiquark
states. The results are listed in Table 7. Excited $P$-wave magnetic moments
are given in Table 8 for $8_{1f}$ representation, Table 9 for $8_{2f}$
representation, and Table 10 for $10_{f}$ representation, respectively.

$J^{P}$ |  ${}^{2s+1}L_{j}$ |  $J_{H}^{P_{H}}\otimes J_{L}^{P_{L}}\otimes J_{\bar{b}}^{P_{\bar{b}}}\otimes J_{l}^{P_{l}}$ |  Magnetic moment
---|---|---|---
|  | $8_{1f}$ |
$\frac{1}{2}^{-}$ |  ${}^{2}S_{\frac{1}{2}}$ |  $0^{+}\otimes 1^{+}\otimes{\frac{1}{2}}^{-}\otimes 0^{+}$ |  $-1.928$
|  |  $(1^{+}\otimes 1^{+})_{0}\otimes{\frac{1}{2}}^{-}\otimes 0^{+}$ |  $0.066$
|  |  $(1^{+}\otimes 1^{+})_{1}\otimes{\frac{1}{2}}^{-}\otimes 0^{+}$ |  $-0.044$
$\frac{3}{2}^{-}$ |  ${}^{4}S_{\frac{3}{2}}$ |  $(0^{+}\otimes 1^{+})\otimes{\frac{1}{2}}^{-}\otimes 0^{+}$ |  $-0.864$
|  |  $(1^{+}\otimes 1^{+})_{1}\otimes{\frac{1}{2}}^{-}\otimes 0^{+}$ |  $0.033$
|  |  $(1^{+}\otimes 1^{+})_{2}\otimes{\frac{1}{2}}^{-}\otimes 0^{+}$ |  $-0.099$
$\frac{5}{2}^{-}$ |  ${}^{6}S_{\frac{5}{2}}$ |  $1^{+}\otimes 1^{+}\otimes{\frac{1}{2}}^{-}\otimes 0^{+}$ |  $2.792$
|  | $8_{2f}$ |
$\frac{1}{2}^{-}$ |  ${}^{2}S_{\frac{1}{2}}$ |  $0^{+}\otimes 0^{+}\otimes{\frac{1}{2}}^{-}\otimes 0^{+}$ |  $0.066$
|  |  $1^{+}\otimes 0^{+}\otimes{\frac{1}{2}}^{-}\otimes 0^{+}$ |  $-0.687$
$\frac{3}{2}^{-}$ |  ${}^{4}S_{\frac{3}{2}}$ |  $1^{+}\otimes 0^{+}\otimes{\frac{1}{2}}^{-}\otimes 0^{+}$ |  $-0.930$
|  | $10_{f}$ |
$\frac{1}{2}^{-}$ |  ${}^{2}S_{\frac{1}{2}}$ |  $0^{+}\otimes 1^{+}\otimes{\frac{1}{2}}^{-}\otimes 0^{+}$ |  $-0.066$
|  |  $(1^{+}\otimes 1^{+})_{0}\otimes{\frac{1}{2}}^{-}\otimes 0^{+}$ |  $0.066$
|  |  $(1^{+}\otimes 1^{+})_{0}\otimes{\frac{1}{2}}^{-}\otimes 0^{+}$ |  $-0.044$
$\frac{3}{2}^{-}$ |  ${}^{4}S_{\frac{3}{2}}$ |  $(0^{+}\otimes 1^{+})\otimes{\frac{1}{2}}^{-}\otimes 0^{+}$ |  $0.066$
|  |  $(1^{+}\otimes 1^{+})_{1}\otimes{\frac{1}{2}}^{-}\otimes 0^{+}$ |  $0.033$
|  |  $(1^{+}\otimes 1^{+})_{2}\otimes{\frac{1}{2}}^{-}\otimes 0^{+}$ |  $-0.099$
$\frac{5}{2}^{-}$ |  ${}^{6}S_{\frac{3}{2}}$ |  $1^{+}\otimes 1^{+}\otimes{\frac{1}{2}}^{-}\otimes 0^{+}$ |  $2.792$
Table 7: Magnetic moments of the $S$-wave diquark-diquark-antiquark hidden-bottom pentaquark states in $8_{1f}$, $8_{2f}$ and $10_{f}$ representations. $J_{H}^{P_{H}}$ corresponds to the angular momentum and parity of $(bq_{1})$, $J_{L}^{P_{L}}$ is for $(q_{2}q_{3})$, $J_{\bar{b}}^{P_{\bar{b}}}$ is for $\bar{b}$, and $J_{l}^{P_{l}}$ is for orbital. The results are listed in unit of nuclear magneton $\mu_{N}$. $J^{P}$ |  ${}^{2s+1}L_{j}$ |  $J_{H}^{P_{H}}\otimes J_{L}^{P_{L}}\otimes J_{\bar{b}}^{P_{\bar{b}}}\otimes J_{l}^{P_{l}}$ |  Magnetic moment $(\lambda)$ |  Magnetic moment $(\rho)$
---|---|---|---|---
$\frac{1}{2}^{+}$ |  ${{}^{2}P_{\frac{1}{2}}}$ |  $(0^{+}\otimes 1^{+}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.347$ |  $0.104$
|  |  $((1^{+}\otimes 1^{+})_{0}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.108$ |  $-0.043$
|  |  $((1^{+}\otimes 1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.145$ |  $-0.006$
|  ${{}^{4}P_{\frac{1}{2}}}$ |  $(0^{+}\otimes 1^{+}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.371$ |  $-0.331$
|  |  $((1^{+}\otimes 1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.130$ |  $-0.090$
|  |  $((1^{+}\otimes 1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.204$ |  $-0.189$
$\frac{3}{2}^{+}$ |  ${{}^{2}P_{\frac{3}{2}}}$ |  $(0^{+}\otimes 1^{+}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $-0.332$ |  $-0.452$
|  |  $((1^{+}\otimes 1^{+})_{0}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.139$ |  $0.018$
|  |  $((1^{+}\otimes 1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.080$ |  $-0.040$
|  ${{}^{4}P_{\frac{3}{2}}}$ |  $(0^{+}\otimes 1^{+}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.553$ |  $-0.644$
|  |  $((1^{+}\otimes 1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.102$ |  $0.011$
|  |  $((1^{+}\otimes 1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.04$ |  $-0.086$
|  ${{}^{6}P_{\frac{3}{2}}}$ |  $((1^{+}\otimes 1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{5}{2}}\otimes 1^{-}$ |  $-0.462$ |  $-0.327$
$\frac{5}{2}^{+}$ |  ${{}^{4}P_{\frac{5}{2}}}$ |  $(0^{+}\otimes 1^{+}\otimes{\frac{1}{2}}^{-})\otimes 1^{-}$ |  $-0.664$ |  $-0.890$
|  |  $((1^{+}\otimes 1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.227$ |  $0.001$
|  |  $((1^{+}\otimes 1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.157$ |  $0.036$
|  ${{}^{6}P_{\frac{5}{2}}}$ |  $((1^{+}\otimes 1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{5}{2}}\otimes 1^{-}$ |  $0.055$ |  $-0.008$
Table 8: Magnetic moments of the excited diquark-diquark-antiquark hidden-bottom pentaquark states in $8_{1f}$ representation. $J_{H}^{P_{H}}$ corresponds to the angular momentum and parity of $(bq_{1})$, $J_{L}^{P_{L}}$ is for $(q_{2}q_{3})$, $J_{\bar{b}}^{P_{\bar{b}}}$ is for $\bar{b}$, and $J_{l}^{P_{l}}$ is for orbital. The results are listed in unit of nuclear magneton $\mu_{N}$. $J^{P}$ |  ${}^{2s+1}L_{j}$ |  $J_{H}^{P_{H}}\otimes J_{L}^{P_{L}}\otimes J_{\bar{b}}^{P_{\bar{b}}}\otimes J_{l}^{P_{l}}$ |  Magnetic moment $(\lambda)$ |  Magnetic moment $(\rho)$
---|---|---|---|---
$\frac{1}{2}^{+}$ |  ${{}^{2}P_{\frac{1}{2}}}$ |  $0^{+}\otimes 0^{+}\otimes{\frac{1}{2}}^{-}\otimes 1^{-}$ |  $0.154$ |  $0.236$
|  |  $(1^{+}\otimes 0^{+}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.401$ |  $0.309$
|  ${{}^{4}P_{\frac{1}{2}}}$ |  $(1^{+}\otimes 0^{+}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $-1.437$ |  $-0.556$
$\frac{3}{2}^{+}$ |  ${{}^{2}P_{\frac{3}{2}}}$ |  $0^{+}\otimes 0^{+}\otimes{\frac{1}{2}}^{-}\otimes 1^{-}$ |  $0.331$ |  $0.186$
|  |  $(1^{+}\otimes 0^{+}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $-0.429$ |  $-0.566$
|  ${{}^{4}P_{\frac{3}{2}}}$ |  $(1^{+}\otimes 0^{+}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.579$ |  $-0.634$
$\frac{5}{2}^{+}$ |  ${{}^{4}P_{\frac{5}{2}}}$ |  $1^{+}\otimes 0^{+}\otimes{\frac{1}{2}}^{-}\otimes 1^{-}$ |  $-0.672$ |  $-0.810$
Table 9: Magnetic moments of the excited diquark-diquark-antiquark hidden-bottom pentaquark states in $8_{2f}$ representation. $J_{H}^{P_{H}}$ corresponds to the angular momentum and parity of $(bq_{1})$, $J_{L}^{P_{L}}$ is for $(q_{2}q_{3})$, $J_{\bar{b}}^{P_{\bar{b}}}$ is for $\bar{b}$, and $J_{l}^{P_{l}}$ is for orbital. The results are listed in unit of nuclear magneton $\mu_{N}$. $J^{P}$ |  ${}^{2s+1}L_{j}$ |  $J_{H}^{P_{H}}\otimes J_{L}^{P_{L}}\otimes J_{\bar{b}}^{P_{\bar{b}}}\otimes J_{l}^{P_{l}}$ |  Magnetic moment $(\lambda)$ |  Magnetic moment $(\rho)$
---|---|---|---|---
$\frac{1}{2}^{+}$ |  ${{}^{2}P_{\frac{1}{2}}}$ |  $(0^{+}\otimes 1^{+}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.141$ |  $0.030$
|  |  $((1^{+}\otimes 1^{+})_{0}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.108$ |  $-0.001$
|  |  $((1^{+}\otimes 1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.145$ |  $0.035$
|  ${{}^{4}P_{\frac{1}{2}}}$ |  $(0^{+}\otimes 1^{+}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.030$ |  $0.025$
|  |  $((1^{+}\otimes 1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.046$ |  $0.007$
|  |  $((1^{+}\otimes 1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.292$ |  $0.346$
$\frac{3}{2}^{+}$ |  ${{}^{2}P_{\frac{3}{2}}}$ |  $(0^{+}\otimes 1^{+}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.178$ |  $0.012$
|  |  $((1^{+}\otimes 1^{+})_{0}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.260$ |  $0.097$
|  |  $((1^{+}\otimes 1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.150$ |  $-0.013$
|  ${{}^{4}P_{\frac{3}{2}}}$ |  $(0^{+}\otimes 1^{+}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.129$ |  $0.062$
|  |  $((1^{+}\otimes 1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.102$ |  $0.176$
|  |  $((1^{+}\otimes 1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.004$ |  $-0.060$
|  ${{}^{6}P_{\frac{3}{2}}}$ |  $((1^{+}\otimes 1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{5}{2}}\otimes 1^{-}$ |  $-0.462$ |  $-0.365$
$\frac{5}{2}^{+}$ |  ${{}^{4}P_{\frac{5}{2}}}$ |  $(0^{+}\otimes 1^{+}\otimes{\frac{1}{2}}^{-})\otimes 1^{-}$ |  $0.267$ |  $0.100$
|  |  $((1^{+}\otimes 1^{+})_{1}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.227$ |  $0.064$
|  |  $((1^{+}\otimes 1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.294$ |  $0.131$
|  ${{}^{6}P_{\frac{5}{2}}}$ |  $((1^{+}\otimes 1^{+})_{2}\otimes{\frac{1}{2}}^{-})_{\frac{5}{2}}\otimes 1^{-}$ |  $0.055$ |  $0.006$
Table 10: Magnetic moments of the excited diquark-diquark-antiquark hidden-
bottom pentaquark states in $10_{f}$ representation. $J_{H}^{P_{H}}$
corresponds to the angular momentum and parity of $(bq_{1})$, $J_{L}^{P_{L}}$
is for $(q_{2}q_{3})$, $J_{\bar{b}}^{P_{\bar{b}}}$ is for $\bar{b}$, and
$J_{l}^{P_{l}}$ is for orbital. The results are listed in unit of nuclear
magneton $\mu_{N}$.

##  VI Magnetic Moments of Hidden-Bottom Pentaquark States in Diquark-Triquark
Model

The diquark-triquark configuration is $(bq_{1})(\bar{b}q_{2}q_{3})$. The total
magnetic moment in diquark-triquark model is

$\displaystyle\hat{\mu}=\ \hat{\mu}_{D}+\hat{\mu}_{T}+\hat{\mu}_{l},$ (24)

where the $D$ denotes diquark, $T$ denotes triquark, and $l$ is the orbital
excitation between the diquark and triquark. The magnetic moment formula can
be written as

$\displaystyle\mu$ $\displaystyle=$ $\displaystyle\langle\ \psi_{Pentaquark}\
|\ \hat{\mu}_{D}+\hat{\mu}_{T}+\hat{\mu}_{l}\ |\ \psi_{Pentaquark}\ \rangle$
(25) $\displaystyle=$ $\displaystyle\sum_{S_{z},l_{z}}\ \langle\
SS_{z},ll_{z}|JJ_{z}\
\rangle^{2}\left\\{\mu_{l}l_{z}+\sum_{\widetilde{S}_{\mathcal{D}},\widetilde{S}_{D}}\
\langle\
S_{\mathcal{D}}\widetilde{S}_{\mathcal{D}},S_{T}\widetilde{S}_{T}|SS_{z}\
\rangle^{2}\Bigg{[}\widetilde{S}_{D}\bigg{(}\mu_{b}+\mu_{q_{1}}\bigg{)}\right.$
$\displaystyle+$ $\displaystyle\left.\sum_{\widetilde{S}_{\bar{b}}}\ \langle\
S_{\bar{b}}\widetilde{S}_{\bar{b}},S_{m}\widetilde{S}_{T}-\widetilde{S}_{\bar{b}}|S_{T}\widetilde{S}_{T}\rangle^{2}\bigg{(}g\mu_{\bar{b}}\widetilde{S}_{\bar{b}}+(\widetilde{S}_{T}-\widetilde{S}_{\bar{b}})(\mu_{q_{2}}+\mu_{q_{3}})\bigg{)}\Bigg{]}\right\\},$

where $S_{D}$, $S_{T}$ and $S_{m}$ represent the diquark, triquark and the
light diquark spin inside the triquark, respectively. The orbital magnetic
moment in diquark-triquark model can be written as

$\displaystyle{\mu_{l}}$ $\displaystyle=$
$\displaystyle\frac{{m_{bq_{1}}}\mu_{\bar{b}q_{2}q_{3}}+{m_{\bar{b}q_{2}q_{2}}}\mu_{bq_{1}}}{m_{\bar{b}q_{2}q_{3}}+m_{bq_{1}}},$
(26)

where $m$ and $\mu$ are the mass and magnetic moment of the cluster
represented by their subscripts. The mass of the triquark is approximately
equal to the sum of the corresponding diquark mass and antiquark mass [19].
The results for the magnetic moments of diquark-triquark model are listed in
Table 11 for $8_{1f}$, Table 12 for $8_{2f}$, and Table 13 for $10_{f}$
representations, respectively.

$J^{P}$ |  ${}^{2s+1}L_{j}$ |  $J_{T}^{P_{T}}\otimes J_{D}^{P_{D}}\otimes J_{l}^{P_{l}}$ |  Magnetic moment
---|---|---|---
$\frac{1}{2}^{-}$ |  ${{}^{2}S_{\frac{1}{2}}}$ |  ${\frac{1}{2}}^{-}\otimes 0^{+}\otimes 0^{+}$ |  $-0.642$
|  |  ${\frac{1}{2}}^{-}\otimes 1^{+}\otimes 0^{+}$ |  $0.790$
|  |  ${\frac{3}{2}}^{-}\otimes 1^{+}\otimes 0^{+}$ |  $-0.768$
$\frac{3}{2}^{-}$ |  ${{}^{4}S_{\frac{3}{2}}}$ |  ${\frac{1}{2}}^{-}\otimes 1^{+}\otimes 0^{-}$ |  $0.221$
|  |  ${\frac{3}{2}}^{-}\otimes 0^{+}\otimes 0^{+}$ |  $-0.864$
|  |  ${\frac{3}{2}}^{-}\otimes 1^{+}\otimes 0^{-}$ |  $-0.288$
$\frac{5}{2}^{-}$ |  ${{}^{6}S_{\frac{5}{2}}}$ |  ${\frac{3}{2}}^{-}\otimes 1^{+}\otimes 0^{+}$ |  $-1.231$
$\frac{1}{2}^{+}$ |  ${{}^{2}P_{\frac{1}{2}}}$ |  ${\frac{1}{2}}^{-}\otimes 0^{+}\otimes 1^{-}$ |  $0.237$
|  |  $({\frac{1}{2}}^{-}\otimes 1^{+})_{\frac{1}{2}}\otimes 1^{-}$ |  $-0.257$
|  ${{}^{4}P_{\frac{1}{2}}}$ |  $({\frac{1}{2}}^{-}\otimes 1^{+})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.061$
|  |  ${\frac{3}{2}}^{-}\otimes 0^{+}\otimes 1^{-}$ |  $-0.491$
|  ${{}^{2}P_{\frac{1}{2}}}$ |  $({\frac{3}{2}}^{-}\otimes 1^{+})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.249$
|  ${{}^{4}P_{\frac{1}{2}}}$ |  $({\frac{3}{2}}^{-}\otimes 1^{+})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.157$
$\frac{3}{2}^{+}$ |  ${{}^{2}P_{\frac{3}{2}}}$ |  ${\frac{1}{2}}^{-}\otimes 0^{+}\otimes 1^{-}$ |  $-0.609$
|  |  $({\frac{1}{2}}^{-}\otimes 1^{+})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.779$
|  ${{}^{4}P_{\frac{3}{2}}}$ |  $({\frac{1}{2}}^{-}\otimes 1^{+})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.158$
|  |  ${\frac{3}{2}}^{-}\otimes 0^{+}\otimes 1^{-}$ |  $-0.620$
|  ${{}^{2}P_{\frac{3}{2}}}$ |  $({\frac{3}{2}}^{-}\otimes 1^{+})_{\frac{1}{2}}\otimes 1^{-}$ |  $-0.779$
|  ${{}^{4}P_{\frac{3}{2}}}$ |  $({\frac{3}{2}}^{-}\otimes 1^{+})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.215$
|  ${{}^{6}P_{\frac{3}{2}}}$ |  $({\frac{3}{2}}^{-}\otimes 1^{+})_{\frac{5}{2}}\otimes 1^{-}$ |  $0.006$
$\frac{5}{2}^{+}$ |  ${{}^{4}P_{\frac{5}{2}}}$ |  ${\frac{1}{2}}^{-}\otimes 1^{+}\otimes 1^{-}$ |  $0.211$
|  |  ${\frac{3}{2}}^{-}\otimes 0^{+}\otimes 1^{-}$ |  $-0.830$
|  |  $({\frac{3}{2}}^{-}\otimes 1^{+})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.298$
|  ${{}^{6}P_{\frac{5}{2}}}$ |  $({\frac{3}{2}}^{-}\otimes 1^{+})_{\frac{5}{2}}\otimes 1^{-}$ |  $-0.002$
Table 11: Magnetic moments of the diquark-triquark hidden-bottom pentaquark states in $8_{1f}$ representation. $J_{T}^{P_{T}}$ corresponds to the angular momentum and parity of triquark, $J_{D}^{P_{D}}$ is for diquark, $J_{\bar{b}}$ is for anti-bottom, and $J_{l}^{P_{l}}$ is for orbital. The results are listed in unit of nuclear magneton $\mu_{N}$. $J^{P}$ |  ${}^{2s+1}L_{j}$ |  $J_{T}^{P_{T}}\otimes J_{D}^{P_{D}}\otimes J_{l}^{P_{l}}$ |  Magnetic moment
---|---|---|---
$\frac{1}{2}^{-}$ |  ${{}^{2}S_{\frac{1}{2}}}$ |  ${\frac{1}{2}}^{-}\otimes 0^{+}\otimes 0^{+}$ |  $0.066$
|  |  ${\frac{1}{2}}^{-}\otimes 1^{+}\otimes 0^{+}$ |  $-0.366$
$\frac{3}{2}^{-}$ |  ${{}^{4}S_{\frac{3}{2}}}$ |  ${\frac{1}{2}}^{-}\otimes 1^{+}\otimes 0^{+}$ |  $-0.930$
$\frac{1}{2}^{+}$ |  ${{}^{2}P_{\frac{1}{2}}}$ |  ${\frac{1}{2}}^{-}\otimes 0^{+}\otimes 1^{-}$ |  $-0.017$
|  |  $({\frac{1}{2}}^{-}\otimes 1^{+})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.232$
|  ${{}^{4}P_{\frac{1}{2}}}$ |  $({\frac{1}{2}}^{-}\otimes 1^{+})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.519$
$\frac{3}{2}^{+}$ |  ${{}^{2}P_{\frac{3}{2}}}$ |  ${\frac{1}{2}}^{-}\otimes 0^{+}\otimes 1^{-}$ |  $0.074$
|  |  $({\frac{1}{2}}^{-}\otimes 1^{+})_{\frac{1}{2}}\otimes 1^{-}$ |  $-0.682$
|  ${{}^{4}P_{\frac{3}{2}}}$ |  $({\frac{1}{2}}^{-}\otimes 1^{+})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.226$
$\frac{5}{2}^{+}$ |  ${{}^{4}P_{\frac{5}{2}}}$ |  ${\frac{1}{2}}^{-}\otimes 1^{+}\otimes 1^{-}$ |  $-0.925$
Table 12: Magnetic moments of the diquark-triquark hidden-bottom pentaquark states in $8_{2f}$ representation. $J_{T}^{P_{T}}$ corresponds to the angular momentum and parity of triquark, $J_{D}^{P_{D}}$ is for diquark, $J_{\bar{b}}$ is for anti-bottom, and $J_{l}^{P_{l}}$ is for orbital. The results are listed in unit of nuclear magneton $\mu_{N}$. $J^{P}$ |  ${}^{2s+1}L_{j}$ |  $J_{T}^{P_{T}}\otimes J_{D}^{P_{D}}\otimes J_{l}^{P_{l}}$ |  Magnetic moment
---|---|---|---
$\frac{1}{2}^{-}$ |  ${}^{2}S_{\frac{1}{2}}$ |  ${\frac{1}{2}}^{-}\otimes 0^{+}\otimes 0^{+}$ |  $-0.022$
|  |  ${\frac{1}{2}}^{-}\otimes 1^{+}\otimes 0^{+}$ |  $-0.037$
|  |  ${\frac{3}{2}}^{-}\otimes 1^{+}\otimes 0^{+}$ |  $0.059$
$\frac{3}{2}^{-}$ |  ${}^{4}S_{\frac{3}{2}}$ |  ${\frac{1}{2}}^{-}\otimes 1^{+}\otimes 0^{-}$ |  $-0.089$
|  |  ${\frac{3}{2}}^{-}\otimes 0^{+}\otimes 0^{+}$ |  $0.066$
|  |  ${\frac{3}{2}}^{-}\otimes 1^{+}\otimes 0^{-}$ |  $0.022$
$\frac{5}{2}^{-}$ |  ${}^{6}S_{\frac{5}{2}}$ |  ${\frac{3}{2}}^{-}\otimes 1^{+}\otimes 0^{+}$ |  $0.127$
$\frac{1}{2}^{+}$ |  ${}^{2}P_{\frac{1}{2}}$ |  ${\frac{1}{2}}^{-}\otimes 0^{+}\otimes 1^{-}$ |  $0.467$
|  |  $({\frac{1}{2}}^{-}\otimes 1^{+})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.625$
|  ${}^{4}P_{\frac{1}{2}}$ |  $({\frac{1}{2}}^{-}\otimes 1^{+})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.286$
|  |  ${\frac{3}{2}}^{-}\otimes 0^{+}\otimes 1^{-}$ |  $-0.158$
|  ${}^{2}P_{\frac{1}{2}}$ |  $({\frac{3}{2}}^{-}\otimes 1^{+})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.567$
|  ${}^{4}P_{\frac{1}{2}}$ |  $({\frac{3}{2}}^{-}\otimes 1^{+})_{\frac{3}{2}}\otimes 1^{-}$ |  $-0.199$
$\frac{3}{2}^{+}$ |  ${}^{2}P_{\frac{3}{2}}$ |  ${\frac{1}{2}}^{-}\otimes 0^{+}\otimes 1^{-}$ |  $0.834$
|  |  $({\frac{1}{2}}^{-}\otimes 1^{+})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.684$
|  ${}^{4}P_{\frac{3}{2}}$ |  $({\frac{1}{2}}^{-}\otimes 1^{+})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.268$
|  |  ${\frac{3}{2}}^{-}\otimes 0^{+}\otimes 1^{-}$ |  $0.334$
|  ${}^{2}P_{\frac{3}{2}}$ |  $({\frac{3}{2}}^{-}\otimes 1^{+})_{\frac{1}{2}}\otimes 1^{-}$ |  $0.902$
|  ${}^{4}P_{\frac{3}{2}}$ |  $({\frac{3}{2}}^{-}\otimes 1^{+})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.396$
|  ${}^{6}P_{\frac{3}{2}}$ |  $({\frac{3}{2}}^{-}\otimes 1^{+})_{\frac{5}{2}}\otimes 1^{-}$ |  $0.246$
$\frac{5}{2}^{+}$ |  ${}^{4}P_{\frac{5}{2}}$ |  ${\frac{1}{2}}^{-}\otimes 1^{+}\otimes 1^{-}$ |  $0.692$
|  |  ${\frac{3}{2}}^{-}\otimes 0^{+}\otimes 1^{-}$ |  $0.823$
|  |  $({\frac{3}{2}}^{-}\otimes 1^{+})_{\frac{3}{2}}\otimes 1^{-}$ |  $0.751$
|  ${}^{6}P_{\frac{5}{2}}$ |  $({\frac{3}{2}}^{-}\otimes 1^{+})_{\frac{5}{2}}\otimes 1^{-}$ |  $0.233$
Table 13: Magnetic moments of the diquark-triquark hidden-bottom pentaquark
states in $10_{f}$ representation. $J_{T}^{P_{T}}$ corresponds to the angular
momentum and parity of triquark, $J_{D}^{P_{D}}$ is for diquark, $J_{\bar{b}}$
is for anti-bottom, and $J_{l}^{P_{l}}$ is for orbital. The results are listed
in unit of nuclear magneton $\mu_{N}$.

##  VII Discussion

We systematically study magnetic moments of hidden-bottom pentaquark states
with various quantum numbers and spin-couplings.

In the molecular model, one of the spin configurations of
$J^{P}=\frac{1}{2}^{-}$ is $\frac{1}{2}^{+}\otimes 0^{-}\otimes 0^{+}$. The
magnetic moments for this configuration are $\mu=-0.598~{}\mu_{N}$ for
$8_{1f}$, $\mu=-0.066~{}\mu_{N}$ for $8_{2f}$, and $\mu=0.022~{}\mu_{N}$ for
$10_{f}$ representations. The $\frac{1}{2}^{+}\otimes 0^{-}\otimes 1^{-}$
configuration of $J^{P}=\frac{1}{2}^{+}$ quantum number has magnetic moment
$\mu=0.699~{}\mu_{N}$ for $8_{1f}$, $\mu=-0.218~{}\mu_{N}$ for $8_{2f}$, and
$\mu=-0.522~{}\mu_{N}$ for $10_{f}$ representations. A measurement of magnetic
moment may present a clue for the representations and therefore quark
configurations both in these quantum numbers.

In the diquark-diquark-antiquark model, the $S$-wave magnetic moment results
have interesting patterns. Some magnetic moments value are same in $8_{1f}$,
$8_{2f}$ and $10_{f}$ representations. For example $1^{+}\otimes
1^{+}\otimes{\frac{1}{2}}^{-}\otimes 0^{+}$ configuration has same magnetic
moment value $\mu=0.066~{}\mu_{N}$ in $8_{1f}$ and $10_{f}$ representations.
In addition to this, $(1^{+}\otimes 1^{+})_{1}\otimes{\frac{1}{2}}^{-}\otimes
0^{+}$ configuration with $J^{P}=\frac{3}{2}^{-}$ has magnetic moment
$\mu=0.033~{}\mu_{N}$ in $8_{1f}$ and $10_{f}$ representations which cannot
distinguish inner structure. In the orbitally excited states, both the $\rho$
mode and $\lambda$ mode excitations lead difference in results.

In the diquark-triquark model, for example $\frac{1}{2}^{-}\otimes
0^{+}\otimes 0^{+}$ of $J^{P}=\frac{1}{2}^{-}$ has magnetic moment
$\mu=-0.642~{}\mu_{N}$ in $8_{1f}$, $\mu=0.066~{}\mu_{N}$ in $8_{2f}$ and
$\mu=-0.022~{}\mu_{N}$ in $10_{f}$ representations, respectively. A possible
measurement of magnetic moment may clarify the quark configuration for this
state.

As can be seen in the related tables, the magnetic moment of the hidden-bottom
pentaquark states with the same quantum number are different in three models.
Even within the same model, $J^{P}$ quantum numbers may correspond several
states with different configurations. As a result, magnetic moments are also
different.

It would be interesting to see how three models yield magnetic moments in the
same spin quantum number and configuration. For this purpose, we list
numerical results of $8_{1f}$ representations of the hidden-bottom pentaquark
states in Table 14. As can be seen in table, the results of the molecular
model and diquark-triquark model agree at an acceptable level whereas the
results of the diquark-diquark-antiquark model are significantly different.
The reason for this could be that the configuration of diquark-triquark model
is similar to molecular model except the color representation.

$J^{P}$ | ${}^{2s+1}L_{j}$ | Molecular | Diquark-diquark-antiquark | Diquark-Triquark
---|---|---|---|---
$\frac{1}{2}^{-}$ |  ${}^{2}S_{\frac{1}{2}}$ |  $-0.598$ |  $-1.928$ |  $-0.642$
|  |  $0.864$ |  $0.066$ |  $0.790$
|  |  $-0.886$ |  $-0.044$ |  $-0.768$
$\frac{3}{2}^{-}$ |  ${}^{4}S_{\frac{3}{2}}$ |  $0.399$ |  $-0.864$ |  $0.221$
|  |  $-0.997$ |  $0.033$ |  $-0.864$
|  |  $0.333$ |  $-0.099$ |  $-0.288$
$\frac{5}{2}^{-}$ |  ${}^{6}S_{\frac{5}{2}}$ |  $0.237$ |  $2.792$ |  $-1.231$
Table 14: Magnetic moments of the $8_{1f}$ hidden-bottom pentaquark states in
three different models. The results are listed in unit of nuclear magneton
$\mu_{N}$.

##  VIII Epilogue

In recent years, substantial experimental progress has been made in the study
of multiquark hadrons. The observation of hidden-charm pentaquark states is a
breakthrough in the era of multiquark states.

In this present work, we systematically study magnetic moments of hidden-
bottom pentaquark states in three models: molecular, diquark-diquark-antiquark
and diquark-triquark. All three models have different quark and and color
configurations. Therefore the interaction in the bound or resonance state is
quite different.

Our results show that magnetic moments of hidden-bottom pentaquark states with
different configurations are different. The results clearly show that the same
spin configuration has different magnetic moments in same models. The magnetic
moment is a guide to explore inner structures of the hadrons. A possible
measurement of magnetic moment may display the inner structure of the
corresponding state. Once hidden-bottom pentaquark states are observed
experimentally, measurement of magnetic moment can help distinguish their
inner structures. We believe that our calculations will be useful for
experimental studies.

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