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# The $1D$-wave bottom-strange baryons and possible interpretation of
$\Xi_{b}(6327)^{0}$ and $\Xi_{b}(6333)^{0}$

Wen-Jia Wang1, Yu-Hui Zhou1, Li-Ye Xiao1 111E-mail<EMAIL_ADDRESS>Xian-
Hui Zhong2,3 222E-mail<EMAIL_ADDRESS>1)Institute of Theoretical
Physics, University of Science and Technology Beijing, Beijing 100083, China
2) Department of Physics, Hunan Normal University, and Key Laboratory of Low-
Dimensional Quantum Structures and Quantum Control of Ministry of Education,
Changsha 410081, China  3) Synergetic Innovation Center for Quantum Effects
and Applications (SICQEA), Hunan Normal University, Changsha 410081, China

###### Abstract

Inspired by the LHCb’s newest observation of two new excited $\Xi_{b}^{0}$
states, we systematically study the strong decays of the low-lying $\lambda$\-
and $\rho$-modes $1D$-wave $\Xi_{b}$ and $\Xi^{{}^{\prime}}_{b}$ baryons using
the chiral quark model within the $j$-$j$ coupling scheme. Based on the
measured masses and strong decay properties of $\Xi_{b}(6327)^{0}$ and
$\Xi_{b}(6333)^{0}$, we explain the two states as the $\lambda$-mode $1D$
$\Xi_{b}$ states with $J^{P}=3/2^{+}$ and $J^{P}=5/2^{+}$, respectively.
Moreover, under this assignment, another dominant decay channel of
$\Xi_{b}(6327)^{0}$ is $\Xi^{\prime}_{b}\pi$ and that of $\Xi_{b}(6333)^{0}$
is $\Xi_{b}^{*}\pi$. Hence, the decay modes $\Xi^{\prime}_{b}\pi$ and
$\Xi_{b}^{*}\pi$ may be another ideal channels as well to decode the inner
structure of $\Xi_{b}(6327)^{0}$ and $\Xi_{b}(6333)^{0}$, respectively. For
other unseen $1D$ $\Xi_{b}$ and $\Xi^{\prime}_{b}$ states, our results
indicate: (i) $\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\rho}$ and
$\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\rho}$ are most likely to be narrow
states with a width of $\Gamma\simeq(12-30)$ MeV, and dominantly decay into
$\Sigma_{b}K$ and $\Sigma^{*}_{b}K$, respectively; (ii) The $1D$
$\Xi^{\prime}_{b}$ baryons are not broad states, and the widths vary in the
range of $\Gamma\simeq(14-46)$ MeV. These states have a good potential to be
observed in their dominant decay processes.

##  I Introduction

Establishing and improving hadron spectroscopy always is a key subject in
hadron physics. Completing this subject can help us to understand the hadron
structure, and then improve our understanding of the dynamics of Quantum
Chromodynamics (QCD). As an indispensable part of hadrons, single heavy
baryons play an important role since heavy quark symmetry is a good
approximation and can provide some qualitative properties, especially in the
baryons containing the bottom ($b$) quark. However, differing from the charmed
baryons, searching for the bottom states is quite difficult for experiment
since higher energy and higher luminance of the beams are required to produce
them. Fortunately, experimenters have made important progress in searching for
the bottom baryons in recent years [1], which provides us good opportunities
to establish an abundant spectrum by decoding the inner structures of these
newly observed bottom baryons.

In 2012, two narrow $1P$ $\Lambda_{b}^{0}$ baryons, denoted as
$\Lambda_{b}(5912)^{0}$ and $\Lambda_{b}(5920)^{0}$, were firstly observed by
the LHCb Collaboration [2] and confirmed by the CDF Collaboration [3] the
following year. Later, two bottom baryons, i.e., $\Xi_{b}(6227)^{-}$ [4] and
$\Sigma_{b}(6097)^{\pm}$ [5], were found by the LHCb Collaboration. Recently,
two $1D$ $\Lambda_{b}^{0}$ candidates, $\Lambda_{b}(6146)^{0}$ and
$\Lambda_{b}(6152)^{0}$, were discovered by the LHCb Collaboration in the
$\Lambda_{b}^{0}\pi^{+}\pi^{-}$ spectrum [6]. This may be the first time that
the low-lying $D$-wave singly bottom baryons are observed in experiment. In
addition, four extremely narrow excited $\Omega_{b}$ states,
$\Omega_{b}(6316)^{-}$, $\Omega_{b}(6330)^{-}$, $\Omega_{b}(6340)^{-}$ and
$\Omega_{b}(6350)^{-}$, were announced by the LHCb Collaboration in the
$\Xi_{b}^{0}K^{-}$ mass spectrum[7] in 2020. These observed single bottom
baryons have stimulated a wide discussion [19, 18, 21, 20, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 22, 23, 24, 25, 26].

Very recently, the LHCb Collaboration again reported their new discovery of
two new excited $\Xi_{b}^{0}$ states in the $\Lambda_{b}^{0}K^{-}\pi^{+}$ mass
spectrum using a data sample of $pp$ collisions [27]. The measured masses and
decay widths of these two states are

$\displaystyle m(\Xi_{b}(6327)^{0})=6327.28^{+0.23}_{-0.21}\pm 0.08\pm
0.24~{}\text{MeV},$ (1) $\displaystyle
m(\Xi_{b}(6333)^{0})=6332.69^{+0.17}_{-0.18}\pm 0.03\pm 0.22~{}\text{MeV},$
(2) $\displaystyle\Gamma(\Xi_{b}(6327)^{0})<2.20(2.56)~{}\text{MeV},$ (3)
$\displaystyle\Gamma(\Xi_{b}(6333)^{0})<1.55(1.85)~{}\text{MeV},$ (4)

where the natural widths $\Gamma$ correspond to $90\%(95\%)$ confidence level
upper limits. It is also found that the $\Xi_{b}(6327)^{0}$ resonance observed
in the $\Lambda_{b}^{0}K^{-}\pi^{+}$ final states is dominantly contributed by
the intermediate channel $\Sigma_{b}^{+}K^{-}$, while the $\Xi_{b}(6333)^{0}$
resonance is significantly contributed by the intermediate channel
$\Sigma_{b}^{*+}K^{-}$. To decode their inner structure and analysis their
dynamic mechanism in theory is necessary. Before the LHCb’s measurement [27],
there exist many theoretical predictions of the mass spectra of the $\Xi_{b}$
and $\Xi_{b}^{\prime}$ baryons with various models in the literature [23, 11,
28, 24, 25, 10, 26, 29, 30, 31, 32, 33, 34], etc. We collect the some
theoretical predictions of the spectrum for the $1D$ $\Xi_{b}$ and
$\Xi_{b}^{\prime}$ baryons in Table 1. From the table, the two new exited
$\Xi_{b}^{0}$ states observed by the LHCb Collaboration [27] are in the
predicted mass region of the $\lambda$-mode $1D$ $\Xi_{b}$ resonances with
spin-parity $J^{P}=3/2^{+}$ and $J^{P}=5/2^{+}$ [16, 29, 30, 31, 32, 33].
Except mass spectrum, decay property is one of the important bases for
determining hadron’s properties. However, there are only a few discussions of
the strong decays of the $1D$ bottom baryons [16, 17, 22]. It should be
pointed out that the predicted strong decay properties of the $\lambda$-mode
$1D$ $\Xi_{b}$ resonances with spin-parity $J^{P}=3/2^{+}$ and $J^{P}=5/2^{+}$
in Refs. [16, 17] are in good agreement with the properties of the two new
exited $\Xi_{b}^{0}$ states observed by the LHCb Collaboration [27]. Recently,
Bijker et al. [35] assigned the newly observed $\Xi_{b}(6327)^{0}$ and
$\Xi_{b}(6333)^{0}$ states as the $\lambda$-mode $1D$ $\Xi_{b}$ resonances
with spin-parity $J^{P}=3/2^{+}$ and $J^{P}=5/2^{+}$ as well both within the
elementary emission model and the quark-pair creation model. The same
theoretical results were obtained in Ref. [10] with the method of QCD sum
rules.

Table 1: The classifications and masses(MeV) of the $1D$-wave $\Xi_{b}$ and $\Xi^{{}^{\prime}}_{b}$ states within the $j$-$j$ coupling scheme. $\Xi_{b}$ and $\Xi_{b}^{\prime}$ stand for the bottom baryons belonging to flavor antitriplet $\mathbf{\bar{3}}_{F}$ and flavor sextet $\mathbf{6}_{F}$, respectively. State | Quantum number |  | Mass |
---|---|---|---|---
$\Xi^{(^{\prime})}_{b}|J^{P},j\rangle_{\lambda(\rho)}$ | $l_{\lambda}$ | $l_{\rho}$ | $L$ | $s_{\rho}$ | $j$ | $J^{P}$ |  | RQM [29] | QM [30] | CQC [31] | CQM [32] | hCQM [33] | QPM [16] |
$\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda}$ | 2 | 0 | 2 | 0 | 2 | $\frac{3}{2}^{+}$ |  | 6366 | 6311 | 6373 | 6359 | 6386 | 6327 |
$\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}$ | 2 | 0 | 2 | 0 | 2 | $\frac{5}{2}^{+}$ |  | 6373 | 6300 |  | 6365 | 6369 | 6330 |
$\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\rho}$ | 0 | 2 | 2 | 0 | 2 | $\frac{3}{2}^{+}$ |  |  |  |  |  |  |  |
$\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\rho}$ | 0 | 2 | 2 | 0 | 2 | $\frac{5}{2}^{+}$ |  |  |  |  |  |  |  |
$\Xi^{\prime}_{b}|J^{P}=\frac{1}{2}^{+},1\rangle_{\lambda}$ | 2 | 0 | 2 | 1 | 1 | $\frac{1}{2}^{+}$ |  | 6447 |  |  | 6420 |  | 6486 |
$\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},1\rangle_{\lambda}$ | 2 | 0 | 2 | 1 | 1 | $\frac{3}{2}^{+}$ |  | 6459 |  |  | 6410 |  | 6488 |
$\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda}$ | 2 | 0 | 2 | 1 | 2 | $\frac{3}{2}^{+}$ |  | 6431 |  |  | 6412 |  | 6456 |
$\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}$ | 2 | 0 | 2 | 1 | 2 | $\frac{5}{2}^{+}$ |  | 6432 | 6402 |  | 6403 |  | 6457 |
$\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},3\rangle_{\lambda}$ | 2 | 0 | 2 | 1 | 3 | $\frac{5}{2}^{+}$ |  | 6420 |  |  | 6377 |  | 6407 |
$\Xi^{\prime}_{b}|J^{P}=\frac{7}{2}^{+},3\rangle_{\lambda}$ | 2 | 0 | 2 | 1 | 3 | $\frac{7}{2}^{+}$ |  | 6414 | 6405 |  | 6390 |  | 6408 |
$\Xi^{\prime}_{b}|J^{P}=\frac{1}{2}^{+},1\rangle_{\rho}$ | 0 | 2 | 2 | 1 | 1 | $\frac{1}{2}^{+}$ |  |  |  |  |  |  |  |
$\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},1\rangle_{\rho}$ | 0 | 2 | 2 | 1 | 1 | $\frac{3}{2}^{+}$ |  |  |  |  |  |  |  |
$\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\rho}$ | 0 | 2 | 2 | 1 | 2 | $\frac{3}{2}^{+}$ |  |  |  |  |  |  |  |
$\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\rho}$ | 0 | 2 | 2 | 1 | 2 | $\frac{5}{2}^{+}$ |  |  |  |  |  |  |  |
$\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},3\rangle_{\rho}$ | 0 | 2 | 2 | 1 | 3 | $\frac{5}{2}^{+}$ |  |  |  |  |  |  |  |
$\Xi^{\prime}_{b}|J^{P}=\frac{7}{2}^{+},3\rangle_{\rho}$ | 0 | 2 | 2 | 1 | 3 | $\frac{7}{2}^{+}$ |  |  |  |  |  |  |  |

In the present work, we will further investigate the probable assignments of
the new $\Xi_{b}(6327)^{0}$ and $\Xi_{b}(6333)^{0}$ states. Furthermore,
considering the powerful detecting ability of LHCb, etc., more and more $1D$
$\Xi_{b}$ and $\Xi^{\prime}_{b}$ baryons are excepted to be observed in the
near future. Thus, it is necessary for us to carry out a systematical study of
the strong decay properties of the $1D$ $\Xi_{b}$ and $\Xi^{\prime}_{b}$
baryons including both the $\rho$\- and $\lambda$-mode excitations. By
analyzing the decay properties of the $1D$ $\Xi_{b}$ and $\Xi^{\prime}_{b}$
baryons, we will suggest ideal decay channels to establish missing states in
the follow-up experiments. It should be mentioned that proper consideration of
the heavy quark symmetry is necessary for the bottom baryons, wherein the
states can more favor the $j$-$j$ coupling scheme [36, 19, 30]. Hence, we
study the strong decays of the $\rho$\- and $\lambda$-mode $1D$ $\Xi_{b}$ and
$\Xi^{\prime}_{b}$ baryons within the $j$-$j$ scheme. With this coupling
scheme, the classifications of the $1D$ $\Xi_{b}$ and $\Xi^{\prime}_{b}$
states investigated in this work are listed in Table 1.

The paper is organized as follows. In Sec. II, we give a brief review of our
theoretical method and the relationship between the $j$-$j$ coupling scheme
and the $L$-$S$ coupling scheme. In Sec. III, we investigate the strong decay
properties of the $\rho$\- and $\lambda$-mode $1D$ $\Xi_{b}$ and
$\Xi^{\prime}_{b}$ baryons within the $j$-$j$ scheme, where we attempt to
decode the properties of the newly observed $\Xi_{b}(6327)^{0}$ and
$\Xi_{b}(6333)^{0}$ states. Finally, we present a short summarization in Sec.
IV.

##  II Chiral quark model

In this work, we systematically investigate the strong decay properties of
both the $\rho$\- and $\lambda$-modes $1D$ $\Xi_{b}$ and $\Xi^{\prime}_{b}$
baryons within the chiral quark model [37, 38, 39]. In the chiral quark model,
the effective low energy quark-pseudoscalar-meson coupling in the SU(3) flavor
basis at tree level is given by [37]

$\displaystyle
H_{m}=\sum_{j}\frac{1}{f_{m}}\bar{\psi}_{j}\gamma^{j}_{\mu}\gamma^{j}_{5}\psi_{j}\vec{\tau}\cdot\partial^{\mu}\phi_{m},$
(5)

where $f_{m}$ represents the pseudoscalar meson decay constant and $\psi_{j}$
denotes the $j$-th quark field in a baryon. $\phi_{m}$ stands for the
pseudoscalar meson octet and reads

$\displaystyle\phi_{m}=\begin{pmatrix}\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&\pi^{+}&K^{+}\cr\pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&K^{0}\cr
K^{-}&\bar{K}^{0}&-\sqrt{\frac{2}{3}\eta}\end{pmatrix}.$ (6)

Considering the harmonic oscillator spatial wave function of baryons in this
work being nonrelativistic form, the quark-pseudoscalar-meson coupling is
adopt the nonrelativistic form as well and is described by [40, 41, 42]

$\displaystyle
H^{nr}_{m}=\sum_{j}\Big{\\{}\frac{\omega_{m}}{E_{f}+M_{f}}\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{P}_{f}+\frac{\omega_{m}}{E_{i}+M_{i}}\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{P}_{i}$
(7)
$\displaystyle-\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{q}+\frac{\omega_{m}}{2\mu_{q}}\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{p}^{\prime}_{j}\Big{\\}}I_{j}\phi_{m}.$

Here, $\omega_{m}$ and q correspond to the energy and three-vector momentum of
the meson, respectively; $E_{i(f)}$, $M_{i(f)}$ and $\textbf{P}_{i(f)}$
represent the energy, mass and three-vector momentum of the initial (final)
baryon; The $\mbox{\boldmath$\sigma$\unboldmath}_{j}$ and $\mu_{q}$ denote the
Pauli spin vector and the reduced mass of the $j$-th quark in the initial and
final baryons, respectively.
$\textbf{p}^{\prime}_{j}=\textbf{p}_{j}-(m_{j}/M)\textbf{P}_{c.m.}$ is the
internal momentum of the $j$-th quark in the baryon rest frame and $I_{j}$ is
the isospin operator associated with the pseudoscalar meson;
$\varphi_{m}=e^{(-)i\textbf{q}\cdot\textbf{r}_{j}}$ for absorbing (emitting) a
meson.

According to the non-relativistic operator of quark-pseudoscalar-meson
coupling, the partial decay amplitudes $M_{J_{iz},J_{fz}}$ of a light
pseudoscalar meson emission in a baryon strong decays can be worked out. Here,
$J_{iz}$ and $J_{fz}$ stand for the third components of the total angular
momenta of the initial and final baryons, respectively. Then, the strong decay
width can be calculated by

$\Gamma=\left(\frac{\delta}{f_{m}}\right)^{2}\frac{(E_{f}+M_{f})|q|}{4\pi
M_{i}}\frac{1}{2J_{i}+1}\sum_{J_{iz}J_{fz}}|M_{J_{iz},J_{fz}}|^{2},$ (8)

where $\delta$ is a global parameter accounting for the strength of the quark-
meson couplings, which has been determined by experimental data in works [38,
39]. Here, we fix its value the same as that in Refs. [38, 39], i.e.
$\delta=0.557$ MeV.

For the single bottom baryons, decoding their inner structure within the
$j$-$j$ coupling scheme is considered preferable to that within the $L$-$S$
coupling scheme (listed in Table 2). In the heavy quark symmetry limit [36],
the states within the the $j$-$j$ coupling scheme can be expressed as linear
combinations of the states within the $L$-$S$ coupling scheme via the
following relationship [30]:

$\begin{split}\big{|}[[(l_{\rho}l_{\lambda})_{L}s_{\rho}]_{j}s_{Q}]_{J}\big{\rangle}=(-1)^{L+s_{\rho}+1/2+J}\sqrt{2j+1}\sum_{S}\sqrt{2S+1}\\\
\left(\begin{array}[]{ccc}L&s_{\rho}&j\\\ s_{Q}&J&S\\\
\end{array}\right)\big{|}[(l_{\rho}l_{\lambda})_{L}(s_{\rho}s_{Q})_{S}]_{J^{P}}\big{\rangle}.\end{split}$
(9)

In the expression, $l_{\rho}$ and $l_{\lambda}$ correspond to the quantum
numbers of the orbital angular momenta for the $\rho$-mode and $\lambda$-mode
(see Fig. 1) oscillators, respectively. $L$
($=|l_{\rho}-l_{\lambda}|,\cdot\cdot\cdot,l_{\rho}+l_{\lambda}$) corresponds
to the quantum number of the total orbital angular momentum. $s_{\rho}$ is the
quantum numbers of the total spin of the two light quarks and $s_{Q}$ is the
spin of the heavy quark. $S$
($=|s_{\rho}-s_{Q}|,\cdot\cdot\cdot,s_{\rho}+s_{Q}$) denotes the quantum
number of the total spin angular momentum.

Figure 1: The $\rho$\- and $\lambda$-mode excitations in a single bottom baryon. Here $\rho$ and $\lambda$ are the Jacobi coordinates defined as $\rho=\frac{1}{\sqrt{2}}(\mathbf{r}_{1}-\mathbf{r}_{2})$ and $\lambda=\frac{1}{\sqrt{6}}(\mathbf{r}_{1}+\mathbf{r}_{2}-2\mathbf{r}_{3})$, respectively. $q_{1}$ and $q_{2}$ stand for the light quarks($u$, $s$), and $Q_{3}$ stands for the bottom quark($b$).  Table 2: The classifications of the $1D$-wave $\Xi_{b}$ and $\Xi_{b}^{\prime}$ states within the $L$-$S$ coupling scheme. $\Xi_{b}$ and $\Xi_{b}^{\prime}$ stand for the bottom baryons belonging to flavor antitriplet $\mathbf{\bar{3}}_{F}$ and flavor sextet $\mathbf{6}_{F}$, respectively. Notation | $l_{\lambda}$ | $l_{\rho}$ | $L$ | $s_{\rho}$ | $s_{Q}$ | $S$ | $J^{P}$ | $\rm Wave\ function$
---|---|---|---|---|---|---|---|---
${\ket{\Xi_{b}{{}^{2}}D_{\lambda\lambda}\frac{3}{2}^{+}}}$ | $0$ | $2$ | $2$ | $0$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{3}{2}^{+}$ | ${}^{2}\Psi^{\lambda\lambda}_{2L_{z}}\chi^{\rho}_{s_{z}}\phi_{\Xi_{b}}$
${\ket{\Xi_{b}{{}^{2}}D_{\lambda\lambda}\frac{5}{2}^{+}}}$ | $2$ | $0$ | $2$ | $0$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{5}{2}^{+}$ |
${\ket{\Xi_{b}{{}^{2}}D_{\rho\rho}\frac{3}{2}^{+}}}$ | $0$ | $2$ | $2$ | $0$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{3}{2}^{+}$ | ${}^{2}\Psi^{\rho\rho}_{2L_{z}}\chi^{\rho}_{s_{z}}\phi_{\Xi_{b}}$
${\ket{\Xi_{b}{{}^{2}}D_{\rho\rho}\frac{5}{2}^{+}}}$ | $0$ | $2$ | $2$ | $0$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{5}{2}^{+}$ |
${\ket{\Xi_{b}^{{}^{\prime}}{{}^{2}}D_{\lambda\lambda}\frac{3}{2}^{+}}}$ | $2$ | $0$ | $2$ | $1$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{3}{2}^{+}$ | ${}^{2}\Psi^{\lambda\lambda}_{2L_{z}}\chi^{\lambda}_{s_{z}}\phi_{\Xi^{{}^{\prime}}_{b}}$
${\ket{\Xi_{b}^{{}^{\prime}}{{}^{2}}D_{\lambda\lambda}\frac{5}{2}^{+}}}$ | $2$ | $0$ | $2$ | $1$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{5}{2}^{+}$ |
${\ket{\Xi_{b}^{{}^{\prime}}{{}^{4}}D_{\lambda\lambda}\frac{1}{2}^{+}}}$ | $2$ | $0$ | $2$ | $1$ | $\frac{1}{2}$ | $\frac{3}{2}$ | $\frac{1}{2}^{+}$ | ${}^{2}\Psi^{\lambda\lambda}_{2L_{z}}\chi^{S}_{s_{z}}\phi_{\Xi^{{}^{\prime}}_{b}}$
${\ket{\Xi_{b}^{{}^{\prime}}{{}^{4}}D_{\lambda\lambda}\frac{3}{2}^{+}}}$ | $2$ | $0$ | $2$ | $1$ | $\frac{1}{2}$ | $\frac{3}{2}$ | $\frac{3}{2}^{+}$ |
${\ket{\Xi_{b}^{{}^{\prime}}{{}^{4}}D_{\lambda\lambda}\frac{5}{2}^{+}}}$ | $2$ | $0$ | $2$ | $1$ | $\frac{1}{2}$ | $\frac{3}{2}$ | $\frac{5}{2}^{+}$ |
${\ket{\Xi_{b}^{{}^{\prime}}{{}^{4}}D_{\lambda\lambda}\frac{7}{2}^{+}}}$ | $2$ | $0$ | $2$ | $1$ | $\frac{1}{2}$ | $\frac{3}{2}$ | $\frac{7}{2}^{+}$ |
${\ket{\Xi_{b}^{{}^{\prime}}{{}^{2}}D_{\rho\rho}\frac{3}{2}^{+}}}$ | $0$ | $2$ | $2$ | $1$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{3}{2}^{+}$ | ${}^{2}\Psi^{\rho\rho}_{2L_{z}}\chi^{\lambda}_{s_{z}}\phi_{\Xi^{{}^{\prime}}_{b}}$
${\ket{\Xi^{{}^{\prime}}_{b}{{}^{2}}D_{\rho\rho}\frac{5}{2}^{+}}}$ | $0$ | $2$ | $2$ | $1$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{5}{2}^{+}$ |
${\ket{\Xi_{b}^{{}^{\prime}}{{}^{4}}D_{\rho\rho}\frac{1}{2}^{+}}}$ | $0$ | $2$ | $2$ | $1$ | $\frac{1}{2}$ | $\frac{3}{2}$ | $\frac{1}{2}^{+}$ | ${}^{2}\Psi^{\rho\rho}_{2L_{z}}\chi^{S}_{s_{z}}\phi_{\Xi^{{}^{\prime}}_{b}}$
${\ket{\Xi_{b}^{{}^{\prime}}{{}^{4}}D_{\rho\rho}\frac{3}{2}^{+}}}$ | $2$ | $0$ | $2$ | $1$ | $\frac{1}{2}$ | $\frac{3}{2}$ | $\frac{3}{2}^{+}$ |
${\ket{\Xi_{b}^{{}^{\prime}}{{}^{4}}D_{\rho\rho}\frac{5}{2}^{+}}}$ | $2$ | $0$ | $2$ | $1$ | $\frac{1}{2}$ | $\frac{3}{2}$ | $\frac{5}{2}^{+}$ |
${\ket{\Xi_{b}^{{}^{\prime}}{{}^{4}}D_{\rho\rho}\frac{7}{2}^{+}}}$ | $2$ | $0$ | $2$ | $1$ | $\frac{1}{2}$ | $\frac{3}{2}$ | $\frac{7}{2}^{+}$ |

In the calculation, the standard quark model parameters are adopted. Namely,
we set $m_{u}=m_{d}=330$ MeV, $m_{s}=450$ MeV and $m_{b}=5000$ MeV for the
constituent quark masses. The decay constants for $\pi$ and $K$ mesons are
taken as $f_{\pi}=132$ MeV and $f_{K}=160$ MeV, respectively. The masses of
the initial baryons($1D$ $\Xi_{b}$ and $\Xi^{\prime}_{b}$ baryons) are
estimated based on the various theoretical predictions listed in Table 1. The
masses of the final $S$-wave ground mesons and baryons used in the
calculations are adopted from the Particle Data Group [1], and those of the
final $P$-wave single-bottom baryons are taken the predictions in Ref. [29].
The spatial wave function ${}^{N}\Psi_{LL_{z}}$ of the baryons is taken the
form of non-relativistic harmonic oscillator spatial-wave function. The
harmonic oscillator parameter $\alpha_{\rho}$ for $us/ds$ system is taken as
$\alpha_{\rho}=420$ MeV [17, 18], and another harmonic oscillator parameter
$\alpha_{\lambda}$ is estimated by [38, 17, 18]

$\displaystyle\alpha^{2}_{\lambda}=\sqrt{\frac{3m_{Q}}{2m_{q}+m_{Q}}}\alpha^{2}_{\rho}.$
(10)

We notice that in the simplified case the ratio between oscillator frequencies
$\omega_{\lambda}$ and $\omega_{\rho}$ reads

$\displaystyle\frac{\omega_{\lambda}}{\omega_{\rho}}=\sqrt{\frac{1}{3}+\frac{2m_{q}}{3m_{q}}}<1.$
(11)

This expression indicates the excitation energy of the $\lambda$-mode is
smaller than that of the $\rho$-mode. Thus, the $\rho$-excitation modes are
heavier than the $\lambda$-excitation modes for the $1D$ $\Xi_{b}$ and
$\Xi^{\prime}_{b}$ baryons. The realistic potential is much more complicated,
while the general feature should be similar.

##  III Calculations and Results

We conduct a systematic investigation of strong decays of $\rho$\- and
$\lambda$-modes $1D$ $\Xi_{b}$ and $\Xi^{{}^{\prime}}_{b}$ within the $j$-$j$
coupling scheme in the framework of the chiral quark model, emphatically
explaining the two newly discovered states by LHCb Collaboration [27] and
giving predictions of other missing $1D$-wave states. Our theoretical results
are presented as follows.

###  III.1  $\Xi_{b}$ states

The flavor wave functions of the $\Xi_{b}$ baryons belonging to flavor
antitriplet $\mathbf{\bar{3}}_{F}$, $\phi_{\Xi_{b}}$, are antisymmetric, thus,
their spin-spatial wave functions must be antisymmetric as well. Hence,
according to the symmetry, there are two $\lambda$-mode $1D$ $\Xi_{b}$
baryons, $\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda}$ and
$\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}$, and two $\rho$-mode $1D$
$\Xi_{b}$ baryons, $\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\rho}$ and
$\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\rho}$. The predicted masses of the
two $\lambda$-mode $1D$ $\Xi_{b}$ baryons are about 6.3-6.4 GeV (see Table 1),
which are consistent with the measured masses of the newly observed
$\Xi_{b}(6327)^{0}$ and $\Xi_{b}(6333)^{0}$ states at LHCb [27]. As the
possible assignments, it is crucial to investigate the decay behaviors of the
two $\lambda$-mode $1D$ $\Xi_{b}$ baryons. For completeness, we present the
prediction of the strong decays of the other two $\rho$-mode $1D$ $\Xi_{b}$
baryons, and hope to provide some valuable reference for the future experiment
exploring.

For the $\lambda(\rho)$-mode $1D$ $\Xi_{b}$ baryons, there is one-to-one
correspondence between the $j$-$j$ coupling scheme and the $L$-$S$ coupling
scheme, namely,
$|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda(\rho)}=|{{}^{2}}D_{\lambda\lambda(\rho\rho)}\frac{3}{2}^{+}\rangle$
and
$|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda(\rho)}=|{{}^{2}}D_{\lambda\lambda(\rho\rho)}\frac{5}{2}^{+}\rangle$.

####  III.1.1 $\lambda$-mode excitations

Considering the uncertainties of the predicted masses of
$\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda}$ and
$\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}$, we plot the decay width as
a function of the mass in the range of $M=(6300-6400)$ MeV in Fig. 2. From the
figure, it is found that the two $\lambda$-mode $1D$ $\Xi_{b}$ baryons both
are narrow states with a width of several MeV. It’s important to note that the
partial decay widths of
$\Gamma[\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda}\rightarrow\Sigma_{b}K]$
and
$\Gamma[\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}\rightarrow\Sigma^{*}_{b}K]$
increase rapidly with the masses increasing(see Fig. 2). Similar results were
also obtained in the previous works [16, 17].

The $J^{P}=3/2^{+}$ state $\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda}$
mainly decays into the $\Sigma_{b}K$, $\Xi^{\prime}_{b}\pi$ and
$\Xi^{*}_{b}\pi$ channels. The
$\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda}$ is a good assignment of the
newly observed $\Xi_{b}(6327)^{0}$ in the $\Lambda_{b}^{0}K^{-}\pi^{+}$ final
state at LHCb [27], since it is dominantly contributed by the intermediate
channel $\Sigma_{b}^{+}K^{-}$. With the measured mass $M=6327$ MeV of
$\Xi_{b}(6327)^{0}$, our predicted decay properties of
$\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda}$ have been shown in Table 3.
It is seen that the total decay width

$\displaystyle\Gamma_{\text{Total}}\simeq 2.56\ \ \mathrm{MeV},$ (12)

is consistent with the observations. The branching fraction for the
$\Sigma_{b}K$ channel is

$\displaystyle\frac{\Gamma[\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda}\rightarrow\Sigma_{b}K]}{\Gamma_{\text{Total}}}\sim
23\%.$ (13)

In addition, we get that

$\displaystyle\frac{\Gamma[\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda}\rightarrow\Xi^{\prime}_{b}\pi]}{\Gamma_{\text{Total}}}\sim
51\%,$ (14)
$\displaystyle\frac{\Gamma[\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda}\rightarrow\Xi^{{}^{\prime}*}_{b}\pi]}{\Gamma_{\text{Total}}}\sim
26\%.$ (15)

If the newly observed $\Xi_{b}(6327)^{0}$ state corresponds to
$\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda}$ indeed, besides the
$\Sigma_{b}K$ channel, the $\Xi^{\prime}_{b}\pi$ and
$\Xi^{{}^{\prime}*}_{b}\pi$ may be another two interesting channels for the
observation of $\Xi_{b}(6327)$ in future experiments. The $\Xi_{b}(6327)$
resonance should be observed in the $\Xi_{b}\pi\pi$ final state as well.

Figure 2: Partial and total strong decay widths of the $\lambda$-mode $1D$ $\Xi_{b}$ states as a function of the masses. The solid curves stand for the total widths. Some decay channels are not shown in the figure for their small partial decay widths.  Table 3: The partial decay widths(MeV) of $\Xi_{b}(6327)^{0}$ and $\Xi_{b}(6333)^{0}$ assigned as $\lambda$-mode $1D$ $\Xi_{b}$ baryons $\Xi_{b}\Ket{J^{P}=\frac{3}{2}^{+},2}_{\lambda}$ and $\Xi_{b}\Ket{J^{P}=\frac{5}{2}^{+},2}_{\lambda}$, respectively. Decay width | $\underline{~{}~{}~{}~{}~{}~{}\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda}~{}~{}~{}~{}~{}~{}}$ | $\underline{~{}~{}~{}~{}~{}~{}\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}~{}~{}~{}~{}~{}~{}}$
---|---|---
$\Xi_{b}(6327)^{0}$ | $\Xi_{b}(6333)^{0}$
$\Gamma[\Sigma_{b}K]$ | 0.59 | 0.00
$\Gamma[\Sigma^{*}_{b}K]$ | - | 0.11
$\Gamma[\Xi^{{}^{\prime}}_{b}\pi]$ | 1.30 | 0.41
$\Gamma[\Xi^{{}^{\prime}*}_{b}\pi]$ | 0.67 | 1.64
$\Gamma_{\text{Total}}$ | 2.56 | 2.16
Expt. |  $<$2.20(2.56) |  $<$1.55(1.85)

For the $J^{P}=5/2^{+}$ state
$\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}$ (see Fig. 2), the mainly
decay channels are the $\Xi^{\prime}_{b}\pi$, $\Sigma^{*}_{b}K$ and
$\Xi^{*}_{b}\pi$ channels. Combining the natures of the newly observed state
$\Xi_{b}(6333)^{0}$, we obtain that this new state may be an assignment of
$\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}$. Fixing the mass of
$\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}$ at $M=6333$ MeV, we collect
its decay properties in Table 3 as well. It is found that the total decay
width

$\displaystyle\Gamma_{\text{Total}}\simeq 2.16\ \ \mathrm{MeV},$ (16)

is close to the upper limit of the observed one. The branching fractions for
the main decay channels are predicted to be

$\displaystyle\frac{\Gamma[\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}\rightarrow\Sigma^{*}_{b}K]}{\Gamma_{\text{Total}}}\sim
5\%,$ (17)
$\displaystyle\frac{\Gamma[\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}\rightarrow\Xi^{\prime}_{b}\pi]}{\Gamma_{\text{Total}}}\sim
19\%,$ (18)
$\displaystyle\frac{\Gamma[\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}\rightarrow\Xi^{{}^{\prime}*}_{b}\pi]}{\Gamma_{\text{Total}}}\sim
76\%.$ (19)

Thus, basing on our calculations, these strong decay processes may be measured
due to their significant branching fractions. To confirm the $\Xi_{b}(6333)$
resonance, the $\Xi_{b}\pi\pi$ final state is worth observing in experiments.

####  III.1.2 $\rho$-mode excitations

There are some qualitative discussions for the masses of $\rho$-mode $1D$
$\Xi_{b}$ states [34], and pointed out that the masses of $\rho$-mode
excitations were about 100 MeV heavier than those of $\lambda$-mode
excitations. Hence, the masses of the two $\rho$-mode $1D$ $\Xi_{b}$ states,
$\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\rho}$ and
$\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\rho}$, may vary in the range of
$M=(6400-6500)$ MeV, and we are most likely to exclude the two $\rho$-mode
states as assignments of $\Xi_{b}(6327)^{0}$ and $\Xi_{b}(6333)^{0}$ based on
the masses. We calculate the decay properties of the two $\rho$-mode $1D$
$\Xi_{b}$ states as a function of the mass within the possible range allowed,
as shown in Fig. 3. The total decay widths of the two $\rho$-mode $1D$
$\Xi_{b}$ states are about $\Gamma\simeq(12-30)$ MeV within the mass range
what we considered.

Figure 3: Partial and total strong decay widths of the $\rho$-mode $1D$
$\Xi_{b}$ states as a function of the masses. The solid curves stand for the
total widths. Some decay channels are not shown in the figure for their small
partial decay widths.

Meanwhile, we notice that $\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\rho}$
decays mainly through the $\Sigma_{b}K$ channel, and the predicted branching
ratio is

$\displaystyle\frac{\Gamma[\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\rho}\rightarrow\Sigma_{b}K]}{\Gamma_{\text{Total}}}\sim
51\%.$ (20)

For the $\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\rho}$ state, the main decay
channel is $\Sigma^{*}_{b}K$ and the corresponding branching fraction is

$\displaystyle\frac{\Gamma[\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\rho}\rightarrow\Sigma^{*}_{b}K]}{\Gamma_{\text{Total}}}\sim(53-58)\%.$
(21)

If we don’t care about masses of
$\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\rho}$ and
$\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\rho}$, the two $\rho$-mode $1D$
$\Xi_{b}$ states are good candidates of the newly observed states
$\Xi_{b}(6327)^{0}$ and $\Xi_{b}(6333)^{0}$, respectively, where the
$\Sigma_{b}K$ process dominates the decay of
$\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\rho}$, while the $\Sigma^{*}_{b}K$
channel dominates the decay of
$\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\rho}$. To further clarify the inner
structures of the two newly observed states and verify our predictions, more
experimental observations are needed.

###  III.2  $\Xi^{{}^{\prime}}_{b}$ states

The flavor wave functions of the $\Xi^{\prime}_{b}$ baryons belonging to
sextet $\mathbf{6}_{F}$, $\phi_{\Xi^{\prime}_{b}}$, are symmetric, thus, their
spin-spatial wave functions must be symmetric as well. Based on the symmetry,
there are six $\lambda$-mode $1D$ $\Xi^{\prime}_{b}$ baryons and six
$\rho$-mode $1D$ $\Xi^{\prime}_{b}$ baryons, as listed in Table 1. According
to the theoretical predictions by various quark model, the masses of
$\lambda$-mode $1D$ $\Xi^{{}^{\prime}}_{b}$ baryons vary in the region of
$M=(6380-6480)$ MeV. Considering the mass of the $\rho$-mode excitation being
$\sim 100$ MeV heavier than the $\lambda$-mode excitation, the $\rho$-mode
$1D$ $\Xi^{{}^{\prime}}_{b}$ baryons may be in the range of $M=(6480-6580)$
MeV. From the point of view of mass, the possibility of the newly observed
$\Xi^{0}_{b}(6327)$ and $\Xi^{0}_{b}(6333)$ states as the
$\Xi^{{}^{\prime}}_{b}$ state may be excluded.

From relationship given in Eq. (9), the $1D$ $\Xi^{\prime}_{b}$ baryon states
in the $j$-$j$ coupling scheme can be expressed with the linear combination of
the configurations in the $L$-$S$ coupling scheme:

$\Ket{J^{P}=\frac{1}{2}^{+},1}_{\lambda(\rho)}=\Ket{{{}^{4}}D_{\lambda\lambda(\rho\rho)}\frac{1}{2}^{+}},$
(22)
$\Ket{J^{P}=\frac{3}{2}^{+},1}_{\lambda(\rho)}=\sqrt{\frac{1}{2}}\Ket{{{}^{4}}D_{\lambda\lambda(\rho\rho)}\frac{3}{2}^{+}}-\sqrt{\frac{1}{2}}\Ket{{{}^{2}}D_{\lambda\lambda(\rho\rho)}\frac{3}{2}^{+}},$
(23)
$\Ket{J^{P}=\frac{3}{2}^{+},2}_{\lambda(\rho)}=\sqrt{\frac{1}{2}}\Ket{{{}^{4}}D_{\lambda\lambda(\rho\rho)}\frac{3}{2}^{+}}+\sqrt{\frac{1}{2}}\Ket{{{}^{2}}D_{\lambda\lambda(\rho\rho)}\frac{3}{2}^{+}},$
(24)
$\Ket{J^{P}=\frac{5}{2}^{+},2}_{\lambda(\rho)}=\frac{\sqrt{7}}{3}\Ket{{{}^{4}}D_{\lambda\lambda(\rho\rho)}\frac{5}{2}^{+}}-\frac{\sqrt{2}}{3}\Ket{{{}^{2}}D_{\lambda\lambda(\rho\rho)}\frac{5}{2}^{+}},$
(25)
$\Ket{J^{P}=\frac{5}{2}^{+},3}_{\lambda(\rho)}=\frac{\sqrt{2}}{3}\Ket{{{}^{4}}D_{\lambda\lambda(\rho\rho)}\frac{5}{2}^{+}}+\frac{\sqrt{7}}{3}\Ket{{{}^{2}}D_{\lambda\lambda(\rho\rho)}\frac{5}{2}^{+}},$
(26)
$\Ket{J^{P}=\frac{7}{2}^{+},3}_{\lambda(\rho)}=\Ket{{{}^{4}}D_{\lambda\lambda(\rho\rho)}\frac{7}{2}^{+}}.$
(27)

In the following, we present our theoretical predictions of the $1D$
$\Xi^{\prime}_{b}$ baryons within the $j$-$j$ coupling scheme.

####  III.2.1 $\lambda$-mode excitations

Firstly, we fix the masses of the $\lambda$-mode $1D$ $\Xi^{{}^{\prime}}_{b}$
states at the predictions within the nonrelativistic quark-diquark picture in
Ref. [29], and collect the decay properties in Table 4.

Table 4: The decay properties of the $\lambda$\- and $\rho-$ modes $1D$ $\Xi^{{}^{\prime}}_{b}$ states. $\Gamma_{\text{Total}}$ stands for the total decay width. The unit of the mass and width is MeV. In this work the $1P$-wave states $|\Lambda^{2}_{b}P_{\lambda}\frac{1}{2}^{-}\rangle$ and $|\Lambda^{2}_{b}P_{\lambda}\frac{3}{2}^{-}\rangle$ are assigned to $\Lambda_{b}(5912)$ and $\Lambda_{b}(5920)$, respectively. The masses for the unestablished $1P$-wave $\Xi_{b}$ and $\Xi_{b}^{\prime}$ states are taken the predictions in Ref. [29].  Decay width | $\underline{~{}~{}\Xi^{\prime}_{b}|J^{P}=\frac{1}{2}^{+},1\rangle_{\lambda}~{}~{}}$ | $\underline{~{}~{}\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},1\rangle_{\lambda}~{}~{}}$ | $\underline{~{}~{}\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda}~{}~{}}$ | $\underline{~{}~{}\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda~{}~{}}}$ | $\underline{~{}~{}\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},3\rangle_{\lambda}~{}~{}}$ | $\underline{~{}~{}\Xi^{\prime}_{b}|J^{P}=\frac{7}{2}^{+},3\rangle_{\lambda}~{}~{}}$
---|---|---|---|---|---|---
$M=6447$ | $M=6459$ | $M=6431$ | $M=6432$ | $M=6420$ | $M=6414$
$\Gamma$[$\Xi_{b}$$\pi$] | 2.21 | 1.96 | - | - | 9.60 | 9.11
$\Gamma$[$\Xi^{{}^{\prime}}_{b}$$\pi$] | 1.30 | 0.32 | 2.89 | 1.27 | 1.23 | 0.64
$\Gamma$[$\Xi^{{}^{\prime}*}_{b}$$\pi$] | 0.64 | 1.62 | 2.28 | 4.06 | 2.09 | 0.64
$\Gamma$[$\Lambda_{b}$K] | 2.41 | 2.10 | - | - | 5.77 | 5.43
$\Gamma$[$\Sigma_{b}$K] | 4.08 | 1.07 | 8.37 | 0.47 | 0.38 | 0.17
$\Gamma$[$\Sigma^{*}_{b}$K] | 1.76 | 4.86 | 1.84 | 6.83 | 0.77 | 0.10
$\Gamma$[$|\Lambda^{2}_{b}P_{\lambda}\frac{1}{2}^{-}\rangle$K] | 3.14 | 3.03 | - | - | - | 0.05
$\Gamma$[$|\Lambda^{2}_{b}P_{\lambda}\frac{3}{2}^{-}\rangle$K] | 6.24 | 6.24 | - | - | - | -
$\Gamma$[$|\Xi^{2}_{b}P_{\lambda}\frac{1}{2}^{-}\rangle$$\pi$] | 0.85 | 0.76 | - | - | 0.43 | 1.59
$\Gamma$[$|\Xi^{2}_{b}P_{\lambda}\frac{3}{2}^{-}\rangle$$\pi$] | 1.83 | 1.67 | - | - | 0.70 | 2.83
$\Gamma$[$|\Xi^{{}^{\prime}2}_{b}P_{\lambda}\frac{1}{2}^{-}\rangle$$\pi$] | 0.87 | 0.08 | 0.60 | 0.01 | 0.01 | 0.06
$\Gamma$[$|\Xi^{{}^{\prime}2}_{b}P_{\lambda}\frac{3}{2}^{-}\rangle$$\pi$] | 0.61 | 0.16 | 1.19 | 0.02 | 0.01 | 0.11
$\Gamma$[$|\Xi^{{}^{\prime}4}_{b}P_{\lambda}\frac{1}{2}^{-}\rangle$$\pi$] | 0.16 | 0.04 | 0.07 | 0.56 | 0.01 | 0.04
$\Gamma$[$|\Xi^{{}^{\prime}4}_{b}P_{\lambda}\frac{3}{2}^{-}\rangle$$\pi$] | 0.03 | 0.01 | 0.02 | 0.12 | - | 0.01
$\Gamma$[$|\Xi^{{}^{\prime}4}_{b}P_{\lambda}\frac{5}{2}^{-}\rangle$$\pi$] | 1.17 | 0.08 | 0.13 | 1.03 | 0.01 | 0.07
$\Gamma_{\text{Total}}$ | 26.87 | 23.93 | 17.39 | 14.37 | 21.27 | 21.09
Decay width | $\underline{~{}~{}\Xi^{\prime}_{b}|J^{P}=\frac{1}{2}^{+},1\rangle_{\rho}~{}~{}}$ | $\underline{~{}~{}\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},1\rangle_{\rho}~{}~{}}$ | $\underline{~{}~{}\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\rho}~{}~{}}$ | $\underline{~{}~{}\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\rho}~{}~{}}$ | $\underline{~{}~{}\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},3\rangle_{\rho}~{}~{}}$ | $\underline{~{}~{}\Xi^{\prime}_{b}|J^{P}=\frac{7}{2}^{+},3\rangle_{\rho}~{}~{}}$
$M=6547$ | $M=6559$ | $M=6531$ | $M=6532$ | $M=6520$ | $M=6514$
$\Gamma$[$\Xi_{b}$$\pi$] | 11.00 | 10.56 | - | - | 10.52 | 10.12
$\Gamma$[$\Xi^{{}^{\prime}}_{b}$$\pi$] | 4.34 | 1.09 | 9.61 | 2.00 | 2.03 | 1.08
$\Gamma$[$\Xi^{{}^{\prime}*}_{b}$$\pi$] | 2.13 | 5.38 | 4.83 | 11.44 | 5.26 | 2.66
$\Gamma$[$\Lambda_{b}$K] | 7.64 | 7.16 | - | - | 6.73 | 6.47
$\Gamma$[$\Sigma_{b}$K] | 11.06 | 2.81 | 24.15 | 1.79 | 1.70 | 0.87
$\Gamma$[$\Sigma^{*}_{b}$K] | 4.42 | 13.75 | 7.15 | 21.12 | 5.31 | 1.96
$\Gamma$[$|\Lambda^{2}_{b}P_{\lambda}\frac{1}{2}^{-}\rangle$K] | 0.49 | 0.58 | - | - | 0.03 | 0.02
$\Gamma$[$|\Lambda^{2}_{b}P_{\lambda}\frac{3}{2}^{-}\rangle$K] | 0.87 | 1.94 | 0.08 | - | 0.04 | 0.03
$\Gamma$[$|\Xi^{2}_{b}P_{\lambda}\frac{1}{2}^{-}\rangle$$\pi$] | 0.47 | 0.53 | - | - | 0.06 | 0.05
$\Gamma$[$|\Xi^{2}_{b}P_{\lambda}\frac{3}{2}^{-}\rangle$$\pi$] | 0.83 | 1.87 | 0.10 | - | 0.10 | 0.09
$\Gamma$[$|\Xi^{{}^{\prime}2}_{b}P_{\lambda}\frac{1}{2}^{-}\rangle$$\pi$] | 0.03 | 0.01 | 0.04 | - | - | -
$\Gamma$[$|\Xi^{{}^{\prime}2}_{b}P_{\lambda}\frac{3}{2}^{-}\rangle$$\pi$] | 0.05 | 0.02 | 0.08 | - | - | -
$\Gamma$[$|\Xi^{{}^{\prime}4}_{b}P_{\lambda}\frac{1}{2}^{-}\rangle$$\pi$] | 0.01 | - | 0.01 | 0.05 | - | -
$\Gamma$[$|\Xi^{{}^{\prime}4}_{b}P_{\lambda}\frac{3}{2}^{-}\rangle$$\pi$] | - | - | - | 0.01 | - | -
$\Gamma$[$|\Xi^{{}^{\prime}4}_{b}P_{\lambda}\frac{5}{2}^{-}\rangle$$\pi$] | 0.03 | 0.01 | 0.02 | 0.09 | - | -
$\Gamma_{\text{Total}}$ | 43.46 | 45.71 | 46.07 | 36.5 | 31.78 | 23.35
Figure 4: Partial and total strong decay widths of the $\lambda$-mode $1D$
$\Xi^{\prime}_{b}$ states as a function of the masses. Some decay channels are
not shown in the figure for their small partial decay widths.

The $J^{P}=1/2^{+}$ state
$\Xi^{\prime}_{b}|J^{P}=\frac{1}{2}^{+},1\rangle_{\lambda}$ has a narrow width
of $\Gamma\simeq 27$ MeV and has relatively large decay rates into the
$\Lambda_{b}K$, $\Sigma_{b}K$, $\Xi_{b}\pi$, $\Lambda_{b}(5912)K$ and
$\Lambda_{b}(5920)K$ channels. The branching fractions for the $\Lambda_{b}K$,
$\Sigma_{b}K$, $\Xi_{b}\pi$ channels are predicted to be

$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{1}{2}^{+},1\rangle_{\lambda}\to\Lambda_{b}K]}{\Gamma_{\text{Total}}}\sim
9\%,$ (28)
$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{1}{2}^{+},1\rangle_{\lambda}\to\Sigma_{b}K]}{\Gamma_{\text{Total}}}\sim
15\%,$ (29)
$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{1}{2}^{+},1\rangle_{\lambda}\to\Xi_{b}\pi]}{\Gamma_{\text{Total}}}\sim
8\%.$ (30)

The $\Lambda_{b}K$, $\Sigma_{b}K$ and $\Xi_{b}\pi$ channels can be used to
search for the missing
$\Xi^{\prime}_{b}|J^{P}=\frac{1}{2}^{+},1\rangle_{\lambda}$ state.

For the $J^{P}=3/2^{+}$ state
$\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},1\rangle_{\lambda}$, its width is
predicted to be around $\Gamma\simeq 24$ MeV. This state has large decay rates
into $\Lambda_{b}(5920)K$, $\Sigma^{*}_{b}K$ and $\Lambda_{b}(5912)K$. The
branching fraction for the $\Sigma^{*}_{b}K$ channel can reach up to

$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},1\rangle_{\lambda}\to\Sigma^{*}_{b}K]}{\Gamma_{\text{Total}}}\sim
20\%.$ (31)

The $\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},1\rangle_{\lambda}$ may have a
large potential to be observed in the $\Lambda_{b}\pi K$ final state via the
decay chain
$\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},1\rangle_{\lambda}\to\Sigma^{*}_{b}K\to\Lambda_{b}\pi
K$ at LHCb.

The other $J^{P}=3/2^{+}$ state
$\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda}$ has a width of
$\Gamma\simeq 17$ MeV, and dominantly decay into $\Sigma_{b}K$ channel with a
branching fraction

$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda}\rightarrow\Sigma_{b}K]}{\Gamma_{\text{Total}}}\sim
48\%.$ (32)

This state may have a large potential to be observed in the $\Lambda_{b}\pi K$
final state as well via the decay chain
$\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},1\rangle_{\lambda}\to\Sigma_{b}K\to\Lambda_{b}\pi
K$ at LHCb.

For the $J^{P}=5/2^{+}$ state
$\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}$, the width is
predicted to be $\Gamma\simeq 14$ MeV. This state mainly decays into
$\Sigma^{*}_{b}K$ and $\Xi_{b}^{{}^{\prime}*}\pi$. Their branching fractions
are

$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}\rightarrow\Sigma^{*}_{b}K]}{\Gamma_{\text{Total}}}\sim
47\%,$ (33)
$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}\rightarrow\Xi_{b}^{{}^{\prime}*}\pi]}{\Gamma_{\text{Total}}}\sim
28\%.$ (34)

The $\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}$ may be
observed in the $\Lambda_{b}\pi K$ or/and $\Xi_{b}\pi\pi$ final states via the
decay chains
$\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}\to\Sigma_{b}^{*}K/\Xi_{b}^{{}^{\prime}*}\pi\to\Lambda_{b}\pi
K/\Xi_{b}\pi\pi$.

Both $\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},3\rangle_{\lambda}$ and
$\Xi^{\prime}_{b}|J^{P}=\frac{7}{2}^{+},3\rangle_{\lambda}$ have a similar
width of $\Gamma\simeq 21$ MeV, and dominantly decay into $\Xi_{b}\pi$ and
$\Lambda_{b}K$ channels. The branching fractions are predicted to be

$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+}(\frac{7}{2}^{+}),3\rangle_{\lambda}\to\Lambda_{b}K]}{\Gamma_{\text{Total}}}\sim
28\%(26\%),$ (35)
$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+}(\frac{7}{2}^{+}),3\rangle_{\lambda}\rightarrow\Xi_{b}\pi]}{\Gamma_{\text{Total}}}\sim
45\%(43\%).$ (36)

The $\Xi_{b}\pi$ and $\Lambda_{b}K$ may be good channels to search for the
$\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},3\rangle_{\lambda}$ and
$\Xi^{\prime}_{b}|J^{P}=\frac{7}{2}^{+},3\rangle_{\lambda}$ states.
Furthermore, it is found that the
$\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},3\rangle_{\lambda}$ state has sizeable
decay rates into the $\Xi_{b}^{{}^{\prime}*}\pi$ channel with a branching
fraction of

$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},3\rangle_{\lambda}\rightarrow\Xi^{{}^{\prime}*}_{b}\pi]}{\Gamma_{\text{Total}}}\sim
10\%.$ (37)

The $\Xi_{b}^{{}^{\prime}*}\pi$ channel which can be used to distinguish
$\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},3\rangle_{\lambda}$ from
$\Xi^{\prime}_{b}|J^{P}=\frac{7}{2}^{+},3\rangle_{\lambda}$ in future
experiments.

The predicted masses of the $\lambda-$ mode $1D$ $\Xi^{\prime}_{b}$ baryons
certainly have a large uncertainty, which may bring uncertainties to the
theoretical results. To investigate this effect, we plot the two-body strong
decay widths of the $\lambda$-mode $1D$ $\Xi^{\prime}_{b}$ baryons as a
function of the mass in Fig. 4. The sensitivities of the decay properties of
these states to their masses can be clearly seen from the figure. As a whole,
the $\lambda$-mode $1D$ $\Xi^{{}^{\prime}}_{b}$ states have a fairly narrow
width of $\Gamma\simeq(13-27)$ MeV. To looking for these missing states, the
$\Lambda_{b}K$, $\Xi_{b}\pi$, $\Lambda_{b}K\pi$ and $\Xi_{b}\pi\pi$ are worth
observing in future experiments.

####  III.2.2 $\rho$-mode excitations

Considering the masses of the $\rho$-mode excitations being about 100 MeV
heavier than the $\lambda$-mode excitations, thus, we fix the masses of the
six $\rho$-mode $1D$ $\Xi^{\prime}_{b}$ baryons in the range of
$M=(6480-6580)$ MeV, and listed the decay properties in Table 4 as well.

Figure 5: Partial and total strong decay widths of the $\rho$-mode $1D$
$\Xi^{\prime}_{b}$ states as a function of the masses. Some decay channels are
not shown in the figure for their small partial decay widths.

The three $\rho$-mode states
$\Xi^{\prime}_{b}|J^{P}=\frac{1}{2}^{+},1\rangle_{\rho}$,
$\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},1\rangle_{\rho}$ and
$\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\rho}$ have a comparable
width of $\Gamma\simeq 45$ MeV. While, we notice that the main decay channels
have some different among those three states. The
$\Xi^{\prime}_{b}|J^{P}=\frac{1}{2}^{+},1\rangle_{\rho}$ mainly decays into
$\Sigma_{b}K$, $\Lambda_{b}K$, and $\Xi_{b}\pi$ channels. Their branching
fractions are predicted to be

$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{1}{2}^{+},1\rangle_{\rho}\to\Sigma_{b}K]}{\Gamma_{\text{Total}}}\sim
25\%,$ (38)
$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{1}{2}^{+},1\rangle_{\rho}\to\Lambda_{b}K]}{\Gamma_{\text{Total}}}\sim
17\%,$ (39)
$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{1}{2}^{+},1\rangle_{\rho}\to\Xi_{b}\pi]}{\Gamma_{\text{Total}}}\sim
25\%.$ (40)

Meanwhile, the $\Xi^{\prime}_{b}|J^{P}=\frac{1}{2}^{+},1\rangle_{\rho}$ has
sizeable decay rates into $\Xi^{\prime}_{b}\pi$ and $\Sigma_{b}^{*}K$, their
branching fractions can reach up to $\sim 10\%$.

For the $\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},1\rangle_{\rho}$ state, the
dominant decay modes are $\Sigma^{*}_{b}K$ and $\Xi_{b}\pi$ with branching
fractions

$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},1\rangle_{\rho}\rightarrow\Sigma^{*}_{b}K]}{\Gamma_{\text{total}}}\sim
30\%,$ (41)
$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},1\rangle_{\rho}\rightarrow\Xi_{b}\pi]}{\Gamma_{\text{total}}}\sim
23\%.$ (42)

The decay rates of $\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},1\rangle_{\rho}$
into $\Xi_{b}^{{}^{\prime}*}\pi$ and $\Lambda_{b}K$ are fairly large as well.
Their branching fractions are predicted to be about $11\%$ and $15\%$,
respectively.

The decay of $\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\rho}$ is
governed by $\Sigma_{b}K$. The branching fraction is predicted to be

$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\rho}\rightarrow\Sigma_{b}K]}{\Gamma_{\text{total}}}\sim
52\%.$ (43)

The rates of $\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\rho}$ decaying
into $\Xi^{\prime}_{b}\pi$ and $\Sigma^{*}_{b}K$ are sizable and predicted to
be

$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\rho}\rightarrow\Xi^{\prime}_{b}\pi]}{\Gamma_{\text{total}}}\sim
21\%,$ (44)
$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\rho}\rightarrow\Sigma^{*}_{b}K]}{\Gamma_{\text{total}}}\sim
15\%.$ (45)

Compared with the three states above, the total decay widths of the two
$J^{P}=5/2^{+}$ states
$\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\rho}$ and
$\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},3\rangle_{\rho}$ are a little narrower
with a width of $\Gamma\simeq 35$ MeV. The
$\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\rho}$ state mainly decays
into $\Sigma^{*}_{b}K$ and $\Xi^{{}^{\prime}*}_{b}\pi$ with branching
fractions

$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\rho}\rightarrow\Sigma^{*}_{b}K]}{\Gamma_{\text{total}}}\sim
58\%,$ (46)
$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\rho}\rightarrow\Xi^{{}^{\prime}*}_{b}\pi]}{\Gamma_{\text{total}}}\sim
31\%.$ (47)

While, the strong decay of the state
$\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},3\rangle_{\rho}$ is dominated by the
$\Xi_{b}\pi$ channel, and the branching ratio is

$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},3\rangle_{\rho}\rightarrow\Xi_{b}\pi]}{\Gamma_{\text{total}}}\sim
33\%,$ (48)

which can be used to distinguish
$\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},3\rangle_{\rho}$ from
$\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\rho}$ in future
experiments. The $\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},3\rangle_{\rho}$ also
has relatively large decay rates into the $\Xi^{{}^{\prime}*}_{b}\pi$,
$\Lambda_{b}K$, and $\Sigma^{*}_{b}K$ channels with a comparable branching
fraction of $\sim 15\%$.

The $\Xi^{\prime}_{b}|J^{P}=\frac{7}{2}^{+},3\rangle_{\rho}$ may be the
narrowest one among the six $\rho$-mode $1D$ $\Xi^{\prime}_{b}$ baryons with a
width of $\Gamma\simeq 23$ MeV, and mainly decays into $\Xi_{b}\pi$ and
$\Lambda_{b}K$ channel. The predicted branching fractions are

$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{7}{2}^{+},3\rangle_{\rho}\to\Xi_{b}\pi]}{\Gamma_{\text{total}}}\sim
43\%.$ (49)
$\displaystyle\frac{\Gamma[\Xi^{\prime}_{b}|J^{P}=\frac{7}{2}^{+},3\rangle_{\rho}\to\Lambda_{b}K]}{\Gamma_{\text{total}}}\sim
28\%.$ (50)

Similarly, considering the uncertainties of the masses, we also plot the the
variations of the partial decay widths as a function of the mass, and show in
Fig. 5. As a whole the $\rho$-mode $1D$ $\Xi^{\prime}_{b}$ baryons may also
have good potentials to be observed in experiments due to their relatively
narrow widths. The ideal channels for observations are $\Xi_{b}\pi$,
$\Lambda_{b}K$, $\Sigma_{b}K$, $\Sigma^{*}_{b}K$ and
$\Xi^{{}^{\prime}*}_{b}\pi$.

##  IV Summary

Stimulated by the newly observed bottom baryon resonances $\Xi^{0}_{b}(6327)$
and $\Xi^{0}_{b}(6333)$ at LHCb, we carry out a systematic study on the two-
body strong decay behaviors of the $\rho$\- and $\lambda$-mode $1D$ $\Xi_{b}$
and $\Xi^{{}^{\prime}}_{b}$ baryons in the framework of chiral quark model
within the $j$-$j$ coupling scheme. For the newly observed states
$\Xi^{0}_{b}(6327)$ and $\Xi^{0}_{b}(6333)$, we give a possible theoretical
interpretation. Meanwhile, we give the predictions for the strong decay
properties of the missing $1D$ $\Xi_{b}$ and $\Xi^{{}^{\prime}}_{b}$ states,
and hope to provide helpful references in theory for the future experiment
exploring.

Our theoretical results indicate the newly observed $\Xi^{0}_{b}(6327)$ and
$\Xi^{0}_{b}(6333)$ may correspond to the $\mathbf{\bar{3}}_{F}$ assignments
$\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda}$ and
$\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}$, respectively, where the
$\Sigma_{b}K$ channel dominates the decay decay of
$\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda}$, while the partial decay
width of the $\Sigma^{*}_{b}K$ channel are sizable for
$\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}$. With this correspondence,
$\Xi^{\prime}_{b}\pi$ and $\Xi^{*}_{b}\pi$ may be another ideal channels for
investigating the nature of $\Xi^{0}_{b}(6327)$ and $\Xi^{0}_{b}(6333)$,
respectively, in future experiments.

The other two $\rho$-mode $\mathbf{\bar{3}}_{F}$ states
$\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\rho}$ and
$\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\rho}$ are most likely to be narrow
states with a total decay width of $\Gamma\simeq(12-30)$ MeV.
$\Xi_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\rho}$ mainly decays into
$\Sigma_{b}K$, while $\Xi_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\rho}$
dominantly decays into $\Sigma^{*}_{b}K$. Hence, those two states have a good
potential to be observed in their dominant decay process.

For the $1D$ $\Xi^{\prime}_{b}$ states belonging $\mathbf{6}_{F}$, the total
decay widths are not broad and vary in the range of $\Gamma\simeq(14-46)$ MeV.
Especially for $\Xi^{\prime}_{b}|J^{P}=\frac{3}{2}^{+},2\rangle_{\lambda}$ and
$\Xi^{\prime}_{b}|J^{P}=\frac{5}{2}^{+},2\rangle_{\lambda}$, their total decay
widths are about a dozen MeV.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China
under Grants No.12005013, No.11947048, No.12175065, No.U1832173 and
No.11775078.

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